Analysis, Design, and Optimization of
RF CMOS Polyphase Filters
Vom Fachbereich Ingenieurwissenschaften der
Universität Duisburg-Essen
zur Erlangung des akademischen Grades eines
Doktors der Ingenieurwissenschaften
genehmigte Dissertation
von
Zisan Zhang
aus
Guoyang, Anhui Province, China
Referent: Prof. Dr.-Ing. habil. Peter Jung
Korreferent: Prof. Dr.-Ing. Horst Fiedler
Tag der mündlichen Prüfung: 29. September 2005
To my wife, Hongjin and my son, Berkeley
Abstract
Polyphase filters (PPFs) are an efficient solution for high accuracy quadrature
generation in radio frequency (RF) CMOS design. Although there are some guidelines
for design of RF CMOS PPFs, they give too much freedom. With layout considerations,
optimization of RF CMOS PPFs cannot be reached by using analytical calculations
because of many constraints and tradeoffs in the design. Thus, in design of RF CMOS
PPFs, intensive trials and several design iterations are needed to reach given
specifications. In this situation, a computer-aided analysis and design optimization tool
for RF CMOS PPFs is of great help to the community of RF CMOS PPFs designers,
especially for industrial projects that are under time-to-market pressures. However, such
a dedicated tool has not been available until now, to the author’s best knowledge.
Focusing on computer-aided analysis and design optimization tools for RF CMOS
PPFs, this dissertation has the following scientific contributions:
• PPFANA, a dedicated computer-aided software tool for analysis of the
influences from the nonideal effects, namely, process tolerance, components
mismatch on the quadrature accuracy of the RF CMOS PPFs, was developed
by the author and presented in this dissertation.
• Physical design considerations for RF CMOS PPFs in system-on-a-chip (SoC)
solution were proposed by the author through the analysis on the influences
from parasitic effects in RF SoC.
• Silicon success of an RF CMOS PPF designed by the author for a low-IF
Bluetooth receiver using Fraunhofer-IMS 0.6 µm CMOS process was achieved.
• PPFOPTIMA, a computer-aided synthesis and optimization tool for RF CMOS
PPFs with dedicated worst-case simulation and Monte Carlo simulation
toolboxes for verification of the performance of RF CMOS PPFs, was
developed by the author and presented in this dissertation.
Zusammenfassung
Polyphasenfilter (PPF) sind eine leistungsfähige Lösung für die hochgenaue
Quadraturerzeugung im Hochfrequenz (HF)-CMOS-Entwurf. Obgleich es einige
Richtlinien für den Entwurf von HF-CMOS-PPF gibt, erlauben diese zu viele Freiheiten.
Eine Optimierung der Schaltung und der Layout kann mit Rücksicht auf viele technisch
und wirtschaftlich vorgegebene Randbedingungen und der daraus resultierenden
Kompromisse mit analytischen Methoden alleine nicht erreichen werden. Es werden
aufwändige Versuche und mehrere Entwurfsverbesserungen benötigt, um einen
zufriedenstellenden Entwurf zu erreichen. Deshalb ist ein computerbasiertes Werkzeug
zur Analyse und Optimierung des Entwurfs von HF-CMOS-PPF besonders für den
industriellen Anwendungsbereich wünschenswert. Nach bestem Wissen des Autors gab
es bislang ein solches Werkzeug nicht. Im Hinblick auf computerbasierte Werkzeuge
zur Analyse und Optimierung des Entwurfs von HF-CMOS-PPF bringt die vorliegende
Dissertation folgende wissenschaftliche Beiträge:
• PPFANA, ein spezielles Entwurfswerkzeug für die Simulation der Einflüsse
von den nichtidealen Effekten wie beispielsweise von Prozesstoleranzen, von
Fehlanpassungen der Komponenten und der Quadraturgenauigkeit, wurde vom
Autor entwickelt und vorgestellt.
• Richtlinien für den Entwurf von HF-CMOS-PPF auf „System-on-a-Chip“Lösungen werden vom Autor unter Berücksichtigung parasitärer Effekte
erarbeitet und vorgeschlagen.
• Die erfolgreiche Realisierung eines HF-CMOS-PPF für einen BluetoothEmpfänger mit niedriger Zwischenfrequenz wurde mit dem Fraunhofer-IMS
0.6 µm CMOS Prozeß nachgewiesen.
• PPFOPTIMA, ein Synthese- und Optimierungswerkzeug für HF-CMOS-PPF
auf Basis von Worst-Case- und Monte-Carlo-Simulationen wurde vom Autor
entwickelt und vorgestellt.
Acknowledgements
Time really flies; four years have passed. Looking back to the time I arrived at
Duisburg, the first colleague I met was Dr. Dieter Greifendorf, who picked my wife and
me up at the railway station of Duisburg in the middle of the night. In the last few years,
he has been the most important person who has given me countless helps in both my
academic works and daily life. My life here in Germany could not have been so easy if
there had not been a warm-hearted friend like him. Hereby I would like to give my
special thanks to him.
This dissertation would not have been completed; if there had not been continuous
supervisions and encouragements from my Ph.D supervisor, Professor Dr.-Ing. habil.
Peter Jung. His brilliance and solid academic foundations have given me invaluable
helps in the development of this dissertation.
Dr. Rainer Kokozinski, an expert in analog circuit design, has given me many
comments and suggestions in my research work. I cannot imagine how many difficulties
I should have had met, if there had not been his timely helps.
My group leaders Dr. Stephan Kolnsberg and Dr. Jörg Stammen have given me
many helps and supports. And I have enjoyed working under their supervision very
much.
I can never forget Dr. Thomas Kaiser, who accepted my application and gave me
the opportunity to working in Fraunhofer IMS. And he has given me countless support
in my academic development during the last four years.
My colleagues in the department of Wireless Chips and Systems (WCS),
Fraunhofer-Institute of Microelectronics Circuits and Systems have given me many
warm-hearted helps during my stay in Germany. Hereby I would like to give them my
sincere thanks.
Zisan Zhang
October, 2005
Duisburg, Germany
Table of Contents
List of Figures………………………………………………………..…………………………….V
List of Tables………………………………………………………….…………………………..IX
List of Abbreviations……………………………………………..…………………………….X
List of Symbols…………………………………………………………………………………XII
Chapter 1 Introduction……………………………………………...………………………1
1.1 Wireless communications with single chip transceivers………..……………………1
1.2 Analog conversion and complex mixing……………...……………...………………… 2
1.3 State-of-the-art of wireless transceiver architectures ………….…..…………………5
1.3.1 Introduction …….……………….……...………….…….………….……….…………5
1.3.2 Conventional heterodyne receiver architecture …………………………………6
1.3.3 Direct-conversion receiver architecture………………………………..………….8
1.3.4 Wideband IF double-conversion receiver architecture……………………….11
1.3.5 Low-IF receiver architecture………………………………………………...…….13
1.3.6 Selection of transceiver architecture in this work……………………….…….14
1.4 Analysis of RF quadrature generation techniques…………...…………………...….16
1.4.1 Introduction………………………………………………………………………...….16
1.4.2 Haven’s technique………………………………………………………………...….16
1.4.3 Frequency division……………………………………………………………….…..17
1.4.4 RC-CR network………………………………………………………………………18
1.4.5 RC polyphase filter…………………………………………………………………..19
1.5 Limitations of previous works …………………...……………………………………...23
1.6 Objectives and contributions of this dissertation …………………..………………..25
1.7 Outline of this dissertation …………………………………………...…………………..26
I
Chapter 2 Analysis of Nonidealities in RF CMOS Polyphase Filters
…………………….……………………………..………….29
2.1 Introduction………………………………….……………………………………………….29
2.2 Vector analysis of polyphase filters…………………………..………………………...31
2.3 PPFANA, a software tool for RF CMOS PPFs…………………..…………………..38
2.3.1 Overview………..……….…..…………..……..………..………………..………….38
2.3.2 Explicit transfer function of RF CMOS PPFs…………..……...…………….38
2.3.3 Development of PPFANA………………………………………………………..40
2.4 Analysis of quadrature accuracy in ideal PPFs……………………………….……....42
2.5 Analysis of quadrature accuracy influenced by process tolerances……………....46
2.6 Analysis of quadrature accuracy influenced by component mismatch….……….51
2.7 Analysis of influences from parasitic effects…….….………………………………...54
2.7.1 Simplified RF models of resistors and capacitors…………..……………….54
2.7.2 Substrate noise coupling………………………………………………………….58
2.8 Summary………………………….…………………………………………………………..62
Chapter 3 Design of an RF CMOS Polyphase Filter………..………………64
3.1 Overview…………………………………………….…………………………64
3.2 Schematic design………………...………………………………………………………….65
3.2.1
Derivation of specifications…………………………………..………………….65
3.2.2
Design guidelines………………………….………………………………………66
3.2.3
Tolerance design………………………………..………………………………….68
3.2.4
Analytical calculations……………………...…………….………………………69
3.2.5
Schematic simulation……………………………….……………………………..72
3.3 Physical layout design………………………………………...……………...……………73
3.3.1
Layout techniques……………………………….…………………………………73
3.3.2
On-chip noise isolation techniques………….…………………………...…….76
II
3.3.3
Layout and post-layout simulation……………………………………………..77
3.3.4
Chip photo…………………………………………………………………………...79
3.4 Measurement……………………...…………………………………………………………80
3.4.1 Introductory remarks……………………………………………………...…….…80
3.4.2 Modeling of test pads and bonding wires……………………………………..80
3.4.3 Design of the test PCB……………………………………………………………81
3.4.4 Measurement setup………………………………………………………………...83
3.4.5 Measurement results……………………………………………………………….84
3.5 Analysis of simulation and measurement results…………………...………………..88
3.6 Summary…………………………………...…………………………………………………88
Chapter 4 Optimization of RF CMOS Polyphase Filters…………………90
4.1 Introduction………………..………………………………………………………………...90
4.2 Design considerations and design flow…………………………….…………………..91
4.3 Problem formulation for RF CMOS PPFs optimization……………..…………….96
4.4 Analysis of possible optimization methods…………….…………..………………..100
4.4.1 Limitations of conventional optimization methods……………………..…100
4.4.2 Simulated annealing……………………………………………………………...102
4.4.3 Genetic algorithms……………………………………………………………….103
4.5 MATLAB implementation of genetic algorithms…………………………………105
4.5.1 Overview…………………………………………………...………………………105
4.5.2 Chromosome representation…...…………...………………………....105
4.5.3 Genetic operation……………………...……………………………....107
4.5.4 Initialization, termination and fitness functions.....…………………....111
4.6 Implementation of PPFOPTIMA………………………………..…………………….114
4.6.1 Introduction……………………………...……………………………………...…114
III
4.6.2 Construction of fitness functions…………………………………………...…116
4.6.3 Graphic user interface………………………………………………………...…116
4.6.4 Verification toolboxes……………………………………………...…...……….118
4.7 Experimental results…………………..…………………………………………….....…123
4.7.1 Overview…………………..………….………………………………………...….123
4.7.2 Experiments on RF CMOS PPFs for a Bluetooth receiver……………124
4.7.3 Automated synthesis and verification of RF CMOS PPFs
for different communication standards………………………………………135
4.8 Summary……………………………………..……………………………………………..146
Chapter 5 Summary and Outlook………………………………………………….149
5.1 Summary…..………………………………………..………………………………………..149
5.2 Outlook………………………………………….…………………………………………...151
References………………………………………………………………………………………...153
Appendix 1 Publications From the Author…………..…………………………..163
Appendix 2 Transfer Functions of Multi-Stage RF CMOS PPFs…….165
Appendix 3 Source Code of PPFANA ………………………………………...……167
Appendix 4 Source Code of PPFOPTIMA……………………………………… 222
IV
List of Figures
Figure 1.1 Mixing a real signal with a real sinusoid. ………………….….……………….…3
Figure 1.2 Mixing a real signal with a complex exponential. ……….….……………….…4
Figure 1.3 A practical implementation of the complex mixer. …………...…………….… 5
Figure 1.4 A generic heterodyne receiver……………………………………...…………….…6
Figure 1.5 A generic homodyne receiver…………………………………….…..………….… 8
Figure 1.6
Self-mixing of the LO and/or of an interfere in homodyne receiver frontend
……………………………………………….9
Figure 1.7 A wideband double-conversion receiver……………………….….…………….12
Figure 1.8 A generic low-IF receiver architecture………………………….…….………....14
Figure 1.9 Schematic of Haven’s technique. ………………………..……….………………17
Figure 1.10 A frequency divider as a quadrature generator. ……………………………...18
Figure 1.11 RC-CR networks as a quadrature generator. …………………………………19
Figure 1.12 Polyphase filter as a quadrature generator…………………………………… 20
Figure 2.1 One phase of a generalized N-phase PPFN………..……………………….…….32
Figure 2.2 One-stage RC sequence asymmetric PPFN……………………………...……....33
Figure 2.3
Quadrature signal generation using four-phase network.
(a) Circuit diagram. (b) Decomposition of the two-phase
input sequence into two four-phase symmetric sequences………………..…35
Figure 2.4 Graphic user interface of PPFANA for 4-stage RF CMOS PPF……….……41
Figure 2.5
Gain mismatch in ideal multi-stage RF CMOS PPFs tailored to
the Bluetooth frequency range 2.4GHz-2.48GHz...………………….……....43
Figure 2.6
Phase difference in ideal multi-stage RF PPFs tailored to
the Bluetooth frequency range 2.4GHz-2.48GHz..………...................…...…44
Figure 2.7
Gain mismatch influenced by process tolerance in a 2-stage
V
RF CMOS PPF (2.4GHz-2.48GHz) with regular design…………..…………48
Figure 2.8
Phase mismatch influenced by process tolerance in a 2-stage
RF CMOS PPF (2.4GHz-2.48GHz) with regular design….…………………48
Figure 2.9 Gain mismatch influenced by process tolerance in a 2-stage
RF CMOS PPF (2.4GHz-2.48GHz) with tolerance design. …………………50
Figure 2.10 Simplified RF model of a polysilicon resistor……….……….…………..……55
Figure 2.11 Simplified RF model of a capacitor……..……..…………………..…..………...56
Figure 2.12
Principle of substrate noise coupling
in system-on-a-chip……………………...……………………..……………..……58
Figure 2.13
Simplified model of substrate noise coupling model
in RF CMOS PPFs…………………………………………..……………………...59
Figure 3.1
Schematic of a 3-stage RF CMOS PPF
for a low-IF Bluetooth receiver…………………..………………………………..71
Figure 3.2 Simulation results of a 3-stage RF CMOS PPFs
obtained by using SpectreRF………………...…………………….…….....72
Figure 3.3 Simulation results of a 3-stage RF CMOS PPFs
obtained by using PPFANA……………………………………………………..…72
Figure 3.4 Layout of the four resistors in one-stage of RF CMOS PPF ……...………74
Figure 3.5
Layout of the four capacitors and wires
in one-stage of an RF CMOS PPF……….…..……………………….…………..75
Figure 3.6 Layout of a 3-stage RF CMOS PPF with test pads…………………....………78
Figure 3.7 Chip photo of an RF CMOS PPF
for a low-IF Bluetooth receiver……………………………………..…...………..79
VI
Figure 3.8 Model of test pads and bonding wires…………………………………….....…...81
Figure 3.9 Schematic of the printed circuit board (PCB)
for test of the RF CMOS PPF…………..……………………..……………....…...82
Figure 3.10 Photo of the fabricated test PCB…………………………….…….………...…...83
Figure 3.11 Diagram of the measurement setup for test of the RF CMOS PPF ….…....84
Figure 3.12 Time domain measurement results of the RF CMOS PPF ……….….….….85
Figure 3.13 Gain mismatch of the RF CMOS PPF measured in frequency domain…..86
Figure 3.14 Phase difference of the RF CMOS PPF
measured in frequency domain…………………………………………………….87
Figure 4.1
A design flow for specification-driven
automated synthesis of RF CMOS PPFs………..…………………………….…95
Figure 4.2
A three-dimensional plot of the amplitude mismatch
in a 2-stage RF CMOS PPF, expressed in the equation (4.20)…..…..……100
Figure 4.3 Mapping variables into chromosomes……………………..…...……………….106
Figure 4.4 Roulette wheel selection……………………………………….………………….109
Figure 4.5
Flowchart of the genetic algorithm implemented in PPFOPTIMA……....113
Figure 4.6
Framework of PPFOPTIMA…………………………………………….…….....115
Figure 4.7
Graphic user interface of PPFOPTIMA……..…………………………..……..117
Figure 4.8
Flowchart of Monte Carlo simulation option in PPFOPTIMA….….…….121
Figure 4.9
Flowchart of the novel worst-case simulation
methodology in PPFOPTIMA…………………………………………………..122
Figure 4.10 Synthesis result of a 2-stage RF CMOS PPF for a Bluetooth receiver….125
Figure 4.11 Worst-case simulation results of a 2-stage RF CMOS PPF for
a Bluetooth receiver…..………..………………………………………………….126
Figure 4.12 Monte Carlo simulation results of a 2-stage RF CMOS PPF
for a Bluetooth receiver……………………….……………………..…………...127
VII
Figure 4.13 Synthesis results of a 3-stage RF CMOS PPF for a Bluetooth receiver...29
Figure 4.14 Worst-case simulation results of a 3-stage RF CMOS PPF
for a Bluetooth receiver…….....…………………………...………...…………...130
Figure 4.15 Monte Carlo simulation results of a 3-stage RF CMOS PPF
for a Bluetooth receiver…..…..………………….……………….……………....131
Figure 4.16 Synthesis results of a 4-stage RF CMOS PPF for a Bluetooth receiver...132
Figure 4.17 Worst-case simulation results of a 4-stage RF CMOS PPF
for a Bluetooth receiver…………….…….………………………………..……..133
Figure 4.18 Monte Carlo simulation results of a 4-stage RF CMOS PPF
for a Bluetooth receiver…….…………………………..……………………......134
Figure 4.19 Graphic user interfaces of PPFOPTIMA
for automated synthesis of RF CMOS PPFs. ……………………………….136
Figure 4.20 Synthesis results of a 3-stage RF CMOS PPF for a GSM receiver….....137
Figure 4.21
Worst-case simulation results of
a 3-stage RF CMOS PPF for a GSM receiver…………………………...….138
Figure 4.22
Monte Carlo simulation results of
a 3-stage RF CMOS PPF for a GSM receiver..………………………….....139
Figure 4.23 Synthesis results of a 2-stage RF CMOS PPF for a DECT receiver….…140
Figure 4.24 Worst-case simulation results of
a 2-stage RF CMOS PPF for a DECT receiver………………………....…141
Figure 4.25 Monte Carlo simulation results of
a 2-stage RF CMOS PPF for a DECT receiver……………………………..42
Figure 4.26 Synthesis results of a 3-stage-RF CMOS PPF for a WCDMA receiver..144
Figure 4.27 Worst-case simulation results of
a 3-stage RF CMOS PPF for a WCDMA receiver……………………….145
Figure 4.28 Monte Carlo simulation results of
a 3-stage RF CMOS PPF for a WCDMA receiver………………………..146
VIII
List of Tables
Table 1.1 Advantages and disadvantages of different receiver architectures. …….……15
Table 2.1 Quadrature accuracy of ideal RF CMOS PPFs (2.4GHz-2.48GHz),
obtained by using PPFANA and SpectreRF……….…………………………….45
Table 2.2 Quadrature accuracy of RF CMOS PPFs (2.4GHz-2.48GHz)
with process tolerance, obtained by PPFANA and SpectreRF……….………50
Table 2.3 Quadrature accuracy influenced by component mismatch in
multi-stage RF CMOS PPFs (2.4GHz-2.48GHz). ……………..………………53
Table 3.1 Specification of RF CMOS PPF for a low-IF Bluetooth receiver……..…..…66
Table 3.2 Dimension of a 3-stage RF CMOS PPF
for a low-IF Bluetooth receiver………………………………….……………..….71
Table 3.3 Simulation results of a 3-stage RF CMOS PPFs obtained
by using PPFANA and SpectreRF……………………………..………………..…73
Table 3.4 Simulation results obtained by using Cadence
post-layout simulation and PPFANA……….…………………..……………..….78
Table 4.1 Main functions used in the optimization engine of PPFOPTIMA………….114
IX
List of Abbreviations
ABBREVIATION
FULL EXPRESSION
AM
Amplitude mismatch
BALUN
Balance unbalance converter
BiCMOS
Bipolar CMOS
C
Capacitor
CDMA
Code division multiple access
CEC
Compression and error correction
CM
Component mismatch
CMOS
Complementary metal oxide semiconductor
CPU
Central processing unit
DC
Direct current
DCR
Direct-down conversion receiver
DECT
Digitally enhanced cordless telephony
ESD
Electro static discharge
FM
Frequency modulation
GA
Genetic algorithms
GaAs
Gallium arsenide
GM
Gain mismatch
GSM
Global system for mobile communications
GUI
Graphical user interface
IC
Integrated circuit
IF
Intermediate frequency
IMS
Institut für Mikroelektronische Schaltungen
und Systeme
I/O
Input / output
I/Q
In phase/quadrature phase
IRF
Image reject filter
IRM
Image reject mixer
IRR
Image reject ratio
ISM
Industrial, scientific and medical
X
LNA
Low noise amplifier
LO
Local oscillator
LPF
Low pass filter
MC
Monte Carlo
MIM
Metal insulator metal
MLR
Maximum length of resistors
NF
Noise figure
PA
Power amplifier
PC
Personal computer
PCB
Printed circuit board
PD
Phase difference
PDF
Probability density function
PM
Phase mismatching
PPF
Polyphase filter
PPFANA
Polyphase filter analysis toolbox
PPFN
Polyphase filter network
PPFOPTIMA
Polyphase filter optimization toolbox
PS
Postlayout simulation
PT
Process tolerances
QAM
Quadrature amplitude modulation
QM
Quadrature modulation
QPSK
Quadrature phase shift keying
RC
Resistance capacitance
RF
Radio frequency
SA
Simulated annealing
SiGe
Silicon germanium
SoC
System on a chip
VCO
Voltage controlled oscillator
WLAN
Wireless local area network
WCDMA
Wideband code division multiple access
XI
List of Symbols
AI
Amplitude of the I path output signal in RF CMOS PPFs
AQ
Amplitude of the Q path output signal in RF CMOS PPFs
AR e
Fitting parameter for the standard deviation of σ R e
AR sh
Fitting parameter for the standard deviation of σ R sh
A∆w
Fitting parameter for the standard deviation of σ ∆w
CC
Parasitic capacitance between back plate of the capacitor
and the substrate
Ci
Capacitor of the ith stage in the RF CMOS PPFs
Ci , CM
Capacitor of the ith stage with component mismatch
between I path and Q path in the RF CMOS PPFs
Ci , PT
Capacitor of the ith stage with process tolerance
in the RF CMOS PPFs
CI , i
Capacitor of the ith stage in the I path of the RF CMOS PPFs
Cpara
Parasitic capacitance
CQ, i
Capacitor of the ith stage in the Q path of
the RF CMOS PPFs
CR
Parasitic capacitance between polysilicon resistor
and the substrate
Csum
Sum of parasitic capacitance between RF CMOS PPFs
and the substrate
CuaC
Parasitic capacitance between unit area capacitor
and the substrate
CuaCi
Parasitic capacitance between the capacitors in the ith stage
and the substrate
CuaRi
Parasitic capacitance between the resistors in the ith stage
and the substrate
XII
Cunit
Unit capacitance of the capacitor
d phase
Phase difference between I path and Q path in PPFs
f
Frequency of a signal
fa
Constant describing influences of area on mismatch
fp
Constant describing influences of periphery on mismatch
fd
Constant describing influences of distance on mismatch
Fi
Fitness of the individual i
H(jω)
Transfer function of polyphase filters
H I (jω)
Imaginary part of the transfer function of polyphase filters
H R (jω)
Real part of the transfer function of polyphase filters
Iin
Input current in RF CMOS PPFs
I k , in
The k th input current in RF CMOS PPFs
I k , out
The k th output current in RF CMOS PPFs
I out
Output current in RF CMOS PPFs
Lmax,Res
Maximum length of the resistor
LR
Length of the resistor
MC
Mismatch between the neighboring capacitors
M CMC
Component mismatch of the capacitors
M CMR
Component mismatch of the resistors
MR
Mismatch between the neighboring resistors
mgain,CM
Gain mismatch in RF CMOS PPFs
with component mismatch
mgain,ideal
Gain mismatch in the ideal RF CMOS PPFs
mgain, PT
Gain mismatch in RF CMOS PPFs
with process tolerances
mphase
Phase mismatch in RF CMOS PPFs
mphase,CM
Phase mismatch in RF CMOS PPFs
with component mismatch
XIII
mphase,ideal
Phase mismatch in the ideal RF CMOS PPFs
mphase,PT
Phase mismatch in RF CMOS PPFs
with process tolerances
N unit
Number of unit capacitor
Pj
Probability of selection of the j population
PR
Process tolerance of the resistors
Q
Quality factor
Re
End resistance of a resistor
Ri
Resistor of the ith stage in the RF CMOS PPFs
Ri , CM
Resistor of the ith stage with component mismatch
between I path and Q path in the RF CMOS PPFs
RI, i
Resistors of the ith stage in the I path of the RF CMOS PPFs
Ri , PT
Resistor of the ith stage with process tolerance
in the RF CMOS PPFs
RIR
Image reject ratio
RQ, i
Resistors of the ith stage in the Q path of the RF CMOS PPFs
Rsh
Sheet resistance of resistor
Rsub
Resistivity of the substrate
RUIR
Ultimately obtainable image reject ratio
S Ci
Area of the capacitor in the ith stage in RF CMOS PPFs
S Ri
Area of the resistor in the ith stage in RF CMOS PPFs
SR
Area of a resistor
Vin
Input voltage in RF CMOS PPFs
Vk , in
The k th input voltage in RF CMOS PPFs
Vk , out
The k th output voltage in RF CMOS PPFs
VI, in
Real part of the input signal in RF CMOS PPFs
VI, mid
Intermediate I path signal
XIV
VQ, mid
Intermediate Q path signal
VIout
I path output signal in RF CMOS PPFs
Vout
Output voltage in RF CMOS PPFs
VQ, in
Imaginary part of the input signal in RF CMOS PPFs
VQout
Q path output signal in RF CMOS PPFs
WR
Width of a resistor
wi
Weight of the ith item
Y
Admittances
ω
Angular frequency
ωIF
Angular frequency of the intermediate frequency signal
ωimage
Angular frequency of the image
ωLO
Angular frequency of the signal from the local oscillator
ωRF
Angular frequency of the radio frequency signal
ωsignal
Angular frequency of the signal
σa
Standard deviation of mismatch caused by area variation
σ component
Standard deviation of component mismatch
σd
Standard deviation of mismatch caused by distance
σp
Standard deviation of mismatch caused by periphery
σ Re
Standard deviation of the end resistance
σ R sh
Standard deviation of the sheet resistance
σ ∆w
Standard deviation of the width of a resistor
φ
Phase of a signal
θ
Relative phase difference in RF CMOS PPFs
∆A
Amplitude mismatch in volts
∆φ
Phase mismatch in degrees
XV
Chapter 1 Introduction
Chapter 1
Introduction
1.1 Wireless Communications With Single Chip Transceivers
The wireless communications market has grown substantially during the last
decades; there has always been pressure to reduce both the size and the cost of wireless
products. However, nowadays, in most designs, the analog part of receivers use multiple
packages and/or multiple integrated circuits (ICs) which are implemented in gallium
arsenide (GaAs) or silicon germanium (SiGe) bipolar processes, while baseband
components are fabricated in low-cost complementary metal oxide semiconductor
(CMOS) processes [1]. Since analog and digital components are designed for different
processes, it is cumbersome to integrate RF, analog and baseband in a single chip and
cost cannot be easily lowered.
Since a couple of decades ago, there have been tremendous endeavors in
developing CMOS wireless single chip transceivers, i.e., integrated transmitter and
receiver. Some pioneers, just name a few, Abidi [6, 8, 9] at University of California at
Los Angeles, Steyaert [23, 32] at Katholiek Universiteit Leuven, and Lee [2, 3] at
Stanford University, have contributed by both theoretical studies and practical system
implementations. Since the middle of the 1990s, more efforts have been put in
developing CMOS single chip transceivers with applications to various standards, for
example, global system for mobile communications (GSM), digitally enhanced cordless
telephony (DECT), Bluetooth, wireless local area network (WLAN), etc.
1
Chapter 1 Introduction
Even though there are a lot of GaAs, SiGe, bipolar circuits available, it is still
quite a challenge to implement similar circuits in CMOS processes. Oriented for digital
circuit design, a standard CMOS process has poor analog, and especially radio
frequency (RF) performance because of its process tolerances, parasitic effects, and low
quality (Q) factor passive components [2]. Thus, insights into transceiver architectures
which are integrateble and capable of providing good CMOS transceiver performances
become essential.
1.2 Analog Conversion and Complex Mixing
In RF receivers, frequency downconversion is an essential operation. Typically,
frequency downconversion of a passband signal is performed by multiplying, i.e.,
mixing, the signal with a sinusoid such as cos(2 πf LOt ) . In Figure 1.1 [24], case 1, f LO is
not equal to the center frequency of the desired signal. This setup is used in a
heterodyne receiver. Figure 1.1 case 2 shows the situation for f LO equals to the center
frequency of the desired signal. This principle is used in the homodyne receiver. In the
next section, we will discuss the architecture of heterodyne and homodyne receivers in
more details. The spectra of the inputs of the mixer and the corresponding spectrum of
the output of the mixer are shown in Figure 1.1 to help understanding the frequency
response.
For a real signal, the negative and positive Fourier components are complex
conjugates of each other. A multiplication in the time domain is equivalent to a
convolution in the frequency domain. Since the Fourier transform of a sinusoid contains
two equal-amplitude impulses symmetrically placed around zero on the frequency axis,
the spectrum of the mixer output signal is the superposition of the positive and negative
2
Chapter 1 Introduction
shifted versions of the spectrum of the input signal. From Figure 1.1, we can see that,
two frequency bands symmetric around the multiplying frequency are downconverted to
the same output band. The undesired input signal band, which will be superimposed on
the desired signal band after mixing, is called the ‘image band’. This is an important
issue in the design of a heterodyne receiver. It is necessary to suppress any signal in the
image band prior to the mixing operation. This is the task of the image reject (IR) filter
which usually precedes the mixer. In the special case where the multiplying frequency
is equal to the center frequency of the incoming signal, the image band is the same as
the signal band, and the image cannot be eliminated by filtering.
x(t )
Spectra of the signals
in the image band
y (t )
cos(2 πf LOt )
Case2
Case1
Amplitude
Amplitude
f
f
- f LO
- f LO
f LO
f
Spectra of the
desired signals
Figure 1.1 Mixing a real signal with a real sinusoid.
3
f LO
f
Chapter 1 Introduction
The image problem arises due to the fact that the frequency spectrum of a real
sinusoid contains impulses at both positive and negative frequencies. One way to avoid
this problem is to mix the signal with a complex exponential which can be expressed as:
e − j2π fLOt = cos(2π f LOt ) - jsin(2π f LO t ) .
(1.1)
Mixing a real signal with this negative frequency complex exponential gives a complex
signal whose spectrum is a shifted version of the real signal spectrum. Theoretically,
this eliminates the image problem associated with frequency shifting when mixing with
a real sinusoid. The realization of a multiplier for mixing a real signal with a complex
exponential requires two real multipliers as shown in Figure 1. 2.
yr (t )
cos(2 πf LOt )
xr (t )
0°
90°
cos(2 πf LOt )
sin ( 2πf LOt)
yi (t )
Figure 1.2 Mixing a real signal with a complex exponential [24].
Now, consider the mixing process shown in the above figure in a general way.
Mixing two complex signals x(t ) and z (t ) generates:
y (t ) = x(t ).z (t ) = ( xr (t ).zr (t ) - xi (t ).zi (t )) + j ( xr (t ).zi (t ) + xi (t ).zr (t )) ,
(1.2)
yr (t ) = xr (t ).zr (t ) - xi (t ).zi (t ) ,
(1.3)
yi (t ) = xr (t ).zi (t ) + xi (t ).zr (t ) .
(1.4)
4
Chapter 1 Introduction
A practical realization of this complex mixer using four real mixers and two
adders is shown in Figure 1.3. This realization is used in the wideband doubleconversion receiver discussed in the next section.
cos(2 πf LOt )
Adder
xr (t )
yr (t )
sin( 2 πf LOt )
xi (t )
yi (t )
Adder
cos( 2 πf LOt )
Figure 1.3 A practical implementation of the complex mixer [24].
1.3 State-of-the-Art of Wireless Transceiver Architectures
1.3.1 Introduction
In this section, we briefly review state-of-the-art receiver topologies and discuss
their advantages and disadvantages. For a given receiver architecture, there exists a
corresponding transmitter architecture with essentially the same fundamental building
blocks. Therefore, we focus on receiver structures.
5
Chapter 1 Introduction
1.3.2 Conventional Heterodyne Receiver Architecture
Most of today’s commercially available RF transceivers utilize some variant of
the conventional heterodyne architecture. In a heterodyne receiver, as shown in Figure
1.4, the RF front-end preselection filter serves to remove out-of band signal energy as
well as partially reject image band signals. After this prefiltering, the received signal is
amplified by a low-noise amplifier (LNA). The IR filter following the LNA further
attenuates the undesired signals at the image band frequencies. The desired signal at the
output of the IR filter is then downconverted from the carrier frequency to a fixed
intermediate frequency (IF) by mixing with the output of a local oscillator (LO).
Low-pass
Filter
Mixer
RF
ADC
Bandpass
Bandpass
LNA
Mixer
PGA
IR Filter
Filter
I
0°
90°
Preselection
Bandpass (RF)
Filter
D
S
P
LO 2
Q
LO1
ADC
Mixer
Low-pass
Filter
Figure 1.4 A generic heterodyne receiver.
In conventional heterodyne receivers, high performance, low phase noise voltagecontrolled oscillators (VCOs) employed as LOs are realized with discrete components
such as high quality (Q) factor inductors, varactors and diodes [3]. At the output of the
6
Chapter 1 Introduction
mixer, an IF filter, typically followed by an IF programmable-gain amplifier (PGA),
selects the desired channel and reduces the distortion and dynamic range requirements
of the subsequent receiver blocks [3]. The signal can be shifted to baseband and
demodulated or alternatively further downconverted to lower IFs, and then shifted to
baseband and demodulated. Since, at the carrier frequency the desired band and image
band are separated by twice the IF, it is desirable to choose a high IF to reduce the
requirements on the IR filter. In fact, if the IF is chosen high enough that the RF filter
can sufficiently attenuate the image band; it might be possible to directly connect the
output of the LNA to the mixer without including an IR filter [24]. On the other hand,
since channel selection in a heterodyne system is done at the IF, a low IF allows the
employment of higher quality channel-select filters. Therefore, the choice of the IF
depends on the trade-off between image rejection and channel selection.
Conventionally, all the filters used in the heterodyne system are high-Q discrete
component filters, such as surface acoustic wave (SAW) or ceramic filters. Compared to
other receiver architectures discussed below, the heterodyne receiver has superior
performance with respect to selectivity, a measure of a receiver’s ability to separate the
desired band around the carrier frequency from signals received at other frequencies,
and sensitivity, the minimal signal level at the receiver input for which there is
sufficient signal-to-noise ratio (SNR) at the receiver output. This is achieved with the
use of high-Q discrete components [3], which mainstream CMOS processes do not have.
Thus, the conventional heterodyne receiver is not well suited to be integrated onto a
CMOS single chip.
7
Chapter 1 Introduction
1.3.3 Direct-Conversion Receiver Architecture
Direct-conversion receiver (DCR), also known as homodyne receiver or zero-IF
receiver [84], is a natural approach to convert an RF signal directly down to baseband.
This architecture, shown in Figure 1.5, employs low-pass filtering in the baseband to
suppress nearby interferers and select the desired channel.
The homodyne architecture has several fundamental advantages over its
heterodyne counterpart [8]. The IF stages and the IR filters are removed, furthermore,
the absence of the bulky off-chip IR filters removes the requirement on the LNA to
drive a low impedance load. The functions of channel selection and subsequent
amplification at a nonzero IF are replaced by low-pass filtering and baseband
amplification, amenable to monolithic integration [8]. Despite this suitability for higher
levels of integration, a homodyne receiver has the following issues that either do not
exist or are not as serious in a heterodyne receiver.
Mixer
PGA
Low-pass
Filter
ADC
RF
LNA
I
0°
90°
Bandpass
RF Filter
D
S
P
LO
Q
ADC
Mixer
PGA
Low-pass
Filter
Figure 1.5 A generic homodyne receiver [8].
8
Chapter 1 Introduction
DC-Offsets — Perhaps the most serious problem is that of direct current (DC)
offsets in the baseband section of the homodyne receiver [4]. These extraneous offset
voltages can corrupt the desired signal and/or saturate the following stages. They arise
due to the self-mixing phenomenon of the local oscillator or the in-band interferer, aside
from the usual element mismatch in the signal path circuitry.
RF
Filter
LNA
Mixer
IF
LO Self - mixing
Figure 1.6 Self-mixing of the LO and/or of an interfere in homodyne receiver frontend.
To better understand the origin of these offsets, consider the received signal path
shown in Figure 1.6. First, the isolation between the LO port and the inputs of the mixer
and the LNA is not perfect, and a finite amount of feedthrough exists from the LO port
to the other mixer input and to the input of the LNA. Known as LO leakage, this effect
arises from capacitive and substrate coupling and, if the LO signal is provided
externally, through bond-wire coupling [21]. The leakage signal appearing at the inputs
of the LNA and the mixer is now mixed with the original LO signal, thus producing a
DC component at the output of the mixer [21]. This LO self-mixing can be quite severe,
and a time-varying DC-offset occurs when the LO signal leaks out through the antenna,
and is radiated and reflected from nearby objects back to the receiver [21]. A similar
9
Chapter 1 Introduction
effect occurs if a large interferer in the passband of the RF filter leaks from the LNA
output to the mixer LO port and gets multiplied by itself [24].
From the above discussion, we can see that direct-conversion receivers require
appropriate methods to remove undesired DC offsets. A simple approach is to use
alternate current (AC) coupling in the downconverted signal path [21]. However, the
spectra of all the spectrally efficient modulation schemes currently used exhibit
significant energy at and around DC, this kind of signals are corrupted by AC coupling
filters [4]. A better method is the use of baseband analog and/or digital signal
processing (DSP) techniques for offset estimation and cancellation [3][24]. However,
these techniques add complexity and do not solve the problems associated with 1/f noise
at low frequencies in CMOS implementations, another significant issue.
I/Q Mismatch — For most currently used modulation schemes, a homodyne
receiver must incorporate quadrature downconversion to avoid loss of information. This
requires shifting either the RF signal or the LO output by 90°, generating an in-phase (I)
path and a quadrature-phase (Q) path. Since phase-shifting the RF signal generally
entails severe noise, power, gain trade-offs [2] and is especially difficult for the
wideband signals used in high data-rate systems, it is often desirable to shift the LO
output. In either case, the errors in the nominally 90° phase shift and mismatches
between the amplitudes of the I and Q signal paths corrupt the downconverted signal
constellation, thereby increasing the bit error ratio (BER).
To gain more insight into the effect of I/Q mismatch, and show the versatility and
convenience of using complex formulation, consider the practical case where the
quadrature LO generates the complex signal:
xLO (t ) = cos(ωLO t ) - j(1 + ∆A) sin(ωLO t + ∆φ )
10
(1.5)
Chapter 1 Introduction
Here ∆A and ∆φ represent LO gain and phase errors. One can rewrite the quadrature
LO output as:
1
1
xLO (t ) = [1-(1 + ∆A)e j∆φ ]e jωLOt + [1+(1 + ∆A)e- j∆φ ]e- jωLOt
2
2
(1.6)
Ideally, the complex LO output should contain only the negative frequency. However,
from the above expression, it is apparent that, due to gain and phase errors, there is a
positive frequency component with a magnitude of |1 − (1 + ∆A)e j∆φ | / 2 . This component
causes interfering images and, if not compensated, can deteriorate the sensitivity of the
receiver.
Aside from the problems mentioned above, direct-conversion receivers are
sensitive to even order distortions. Also, since the downconverted spectrum is located
around zero frequency, the flicker (1/f) noise of the devices, a severe problem in CMOS
implementations [2], has a profound effect on the signal-to-noise ratio (SNR).
Furthermore, integrating the high-frequency low-phase-noise channel select frequency
synthesizer is difficult to achieve with low-Q VCOs available in integrated circuits [3].
1.3.4 Wideband IF Double-Conversion Receiver Architecture
Shown in Figure 1.7 is the wideband double-conversion receiver [23]. In this
receiver, after filtering and amplification, all potential RF channels are complex mixed
and downconverted to IF. A second complex mixing is done from IF to baseband, using
a tunable channel-select frequency-synthesizer. In this complex mixer, by properly
adding the outputs of the real multipliers in pairs, the image frequencies are canceled
while the desired channels add constructively. If the IF is chosen high enough,
additional image rejection may be obtained from the RF frontend filter [3].
11
Chapter 1 Introduction
Comparing the homodyne receiver and the wideband IF double-conversion
receiver, we can see that channel selection is performed in the baseband in both
architectures, allowing the possibility of a programmable integrated channel-select filter
for multi-standard receiver applications.
1st Mixing
Stage
2nd Mixing
Stage
Low-pass
Low-pass cos(2 πf LO2t )
Mixer Filter
PGA Filter
Adder
ADC
RF
I
LNA
0°
90°
Bandpass
Filter
D
S
P
sin(2 πf LO2t )
LO1
Q
ADC
Low-pass
Adder PGA Low-pass
Mixer Filter
cos(2 πf LO2t )
Filter
Figure 1.7 A wideband IF double-conversion receiver [23].
However, the wideband IF double-conversion receiver architecture has some
advantages over its homodyne counterpart [21]. Due to the fact that channel tuning is
performed using the IF LO, the RF LO can be implemented as a fixed-frequency
crystal-controlled oscillator [25]. Several techniques may be utilized to realize a lowphase-noise fixed LO with low-Q on-chip components [3]. Also, since tuning is
performed with the IF LO operating at a lower frequency, the phase-noise performance
12
Chapter 1 Introduction
of this oscillator can be significantly better than that of the tunable RF oscillator
employed in a homodyne receiver. Furthermore, since in the wideband IF system there
is no LO operating at the same frequency as the incoming RF carrier, the potential
problems associated with LO leakage and time-varying DC offsets are minimized [24].
Although in the wideband IF double-conversion receiver the second LO is at the same
frequency as the desired IF channel, the DC offset which results at the baseband from
self-mixing is relatively constant and may be cancelled using adaptive signal processing
methods [3].
The limitations of wideband IF double-conversion receivers are the following:
Since the first LO is fixed in frequency, all channels must pass through the IF stage. The
desired frequency channel is selected using the second LO. This has two problematic
implications: First, as a result of moving the channel selection to a lower frequency, the
IF synthesizer requires a VCO with the capability of tuning across a broader frequency
range than a small percentage of the nominal operating frequency. Second, by removing
the channel-select filter from the first mixing stage, strong adjacent channel interferers
are now a concern for the second mixer stage as well as the baseband blocks. This
implies a higher dynamic range requirement for these latter receiver stages [25]. Also,
I/Q phase and gain mismatch degrade the performance of the receiver [3].
1.3.5 Low-IF Receiver Architecture
The idea behind low-IF topologies is to combine the advantages of both
heterodyne and homodyne receivers [23]. Figure 1.8 shows a low-IF receiver
architecture [84], in which the IF is usually chosen at a low frequency, typically at a few
hundred KHz. Low-IF architectures are suitable for the integration since they provide
13
Chapter 1 Introduction
on-chip image rejection, and the channel selection can be done by using a low-Q filter
following the mixer. Unlike the zero-IF architecture, a low-IF receiver is not sensitive to
the parasitic DC offset, LO leakage, and even-order distortion [3]. Low-IF architectures
also provide flexibility for processing the signal in multiple ways. One of the drawbacks
of this architecture is its limited image rejection (~40 dB) due to the on-chip matching
between I and Q generators [84]. Without proper pre-filtering, the dynamic range and
resolution requirement on an analog-to-digital (A/D) converter can substantially
increase. Also, an increase in current consumption results when this topology is used for
wide-channel bandwidth application.
PGA
Mixer
RF
Phase Shifter
90°
LNA
I
0°
90°
Adder
LO
Mixer
D
S
P
Low-pass
Filter
Q
Band-pass
Filter
ADC
PGA
Figure 1.8 A generic low-IF receiver architecture [84].
1.3.6 Selection of Transceiver Architecture in This Work
In order to select a well suited transceiver architecture for this work on multistandard CMOS single-chip transceivers, based on the above analysis, the
aforementioned receiver architectures, namely, heterodyne, homodyne, wideband IF
14
Chapter 1 Introduction
double-conversion, and low-IF, were compared by using following Table 1.1. In this
table, eight important aspects in the receiver design are ranked as ‘+’ and ‘++’, which
－’ and ‘－－’, which stands for
stands for ‘advantage’ and ‘outstanding advantage’, ‘
‘disadvantage’ and ‘severe disadvantage’, or ‘NO’, which stands for ‘no need to
consider’. From Table 1.1, we can see that the low-IF receiver architecture has the most
advantages. Thus, the low-IF receiver architecture is chosen in this work.
Wideband Dual
Architecture
Heterodyne
Homodyne
Low-IF
Conversion
Items
Sensitivity
++
－
++
+
Image reject
++
NO
+
+
DC offset
NO
NO
NO
Flicker noise
NO
IQ mismatch
－
－－
－－
－－
－
－
－
－－
－－
++
++
++
++
++
++
++
－－
+
Multi-standard
compatibility
Integratebility
Simplicity
－－
－－
Table 1.1 Advantages and disadvantages of different receiver architectures.
15
Chapter 1 Introduction
1.4 Analysis of RF Quadrature Generation Techniques
1.4.1 Introduction
As discussed in section 1.2, quadrature conversion is dominant in modern
wireless transceiver design, because frequency modulation and phase modulation are
deployed in most digital communication systems. In the low-IF receiver architecture we
chose, quadrature signal generation is an important function block, because quadrature
accuracy, that is the accuracy of 90° phase-shifted signal, directly determines the image
reject ratio (IRR), an important specification determining the sensitivity of the receiver
[23].
Generally, there are four quadrature generation techniques; namely, Havens’
technique, Frequency division, RC-CR networks and RC polyphase filters (PPFs).
Although RC PPFs are cascaded RC-CR networks, there are significant differences
between them in CMOS implementation. Thus, we take RC PPFs as an independent
quadrature generation technique based on the ‘stagger-tuning’ techniques [55] used in
this circuitry.
1.4.2 Haven’s Technique
Figure 1.9 shows a schematic implementation of Haven’s technique [84]. This
schematic consists of a phase splitter (phase shifter), two first limiters, two adders, and
two further limiters. The phase splitter first splits the input signal Vin , creating 2 paths,
namely, VI, mid and VQ, mid , with approximately 90° phase difference, then adds and
subtracts them, producing the output signals VIout and VQout .
16
Chapter 1 Introduction
In Haven’s technique, if VI, mid and VQ, mid have equal amplitudes, the angle
between VIout and VQout is equal to 90° . The limiting stages will equalize the amplitudes
of VI, mid and VQ, mid by phase shift circuit. Moreover, the adders’ outputs will have
different amplitude if the phase difference between VI, mid and VQ, mid is not exactly 90° .
Another amplitude mismatch in Haven’s technique is the capacitive coupling between
the inputs of adders [23]. This may have significant effect in the phase imbalance at the
input, if output impedance of the limiter is not small.
Limiter
+
VI, mid
Phase
Splitter
Vin
Adder Limiter
VIout
+
≈90
o
VQ, mid
VQout
+
Limiter
Adder Limiter
Figure 1.9 Schematic of Haven’s technique [84].
1.4.3 Frequency Division
As shown in Figure 1.10, the idea of frequency division is to use a master-slave
flip-flop to divide a signal by a factor of 2, if Vin has an exact 50% duty cycle, then
there is an accurate 90° phase difference between VIout and VQout [23].
Generating quadrature signals using frequency division at radio frequency has the
following main disadvantages:
17
Chapter 1 Introduction
•
It consumes a substantial power to generate a reference signal whose frequency
is twice the operating frequency; for some CMOS processes with low cut-off
frequency, this is simply impossible. For example, in the Fraunhofer-Institut
für Mikroelektronische Schaltungen und Systeme (Fraunhofer-IMS) 0.6 µm
CMOS process, the cut-off frequency is only 4.5 GHz; it is impossible to use
frequency division for generating quadrature LO signal in the low-IF Bluetooth
receiver, because a 4.9 GHz reference signal is required.
•
There is a considerable phase mismatch when the input duty cycle deviates
from 50%, further more, mismatch in the signal path through the latches also
contribute to the phase mismatch.
Latch
VIout
VQout
Vin
Latch
Figure 1.10 A frequency divider as a quadrature generator [84].
1.4.4 RC-CR Networks
An RC-CR (R-resistor, C-capacitor) network is a quadrature generation technique
that shifts phases of the two signal paths by +
π
4
and −
π
4
respectively [23]. As shown
in Figure 1.11, this circuit consists of a low-pass filter and a high-pass RC filter in two
signal paths. In the RC-CR network, the phase differences between VIout and VQout is
18
Chapter 1 Introduction
90° for all frequencies, however, the amplitudes of the outputs are equal only at the
corner frequency of the RC filter, that is, ω equals to 1/( RC ) . As the absolute values of
the RC network vary with temperature and process, the corner frequency also varies; it
is thus impossible to get a robust quadrature signal in CMOS technology by using RCCR networks. Moreover, this technique is only suitable for narrow band operation,
because a good gain matching between two quadrature outputs can only be achieved
with a very narrow band centered on the corner frequency of this RC-CR network.
Low-pass
Filter
C
R
VIout
Vin
C
VQout
High-pass
Filter
R
Figure 1.11 RC-CR networks as a quadrature generator [84].
1.4.5 RC Polyphase Filter (PPF)
From the above discussions, we can see that traditional quadrature generation
techniques widely used in processes with high precision resistors and capacitors are not
well suited to CMOS implementation. Therefore, alternative circuitries, which are
19
Chapter 1 Introduction
robust against component mismatch and process variation, are needed. At this point, we
would like to point out that process variation is often also termed as process tolerances.
Originally, polyphase filters (PPFs) were developed by Gingell [45, 46] in 1971
for generating quadrature signals with even amplitude in audio. In 1994, integrated
PPFs were rediscovered as an efficient RF quadrature generation technique in CMOS
technology by Steyaert [55]. This has introduced PPFs into a new regime, which we
name as RF CMOS PPFs.
R1
Rm
Rn
VIout +
Vin +
C1
Cm
Cn
R1
Rm
Rn
VQout +
C1
Cm
Cn
R1
Rm
Rn
VIout −
Vin −
C1
Cm
Cn
R1
Rm
Rn
Cm
Cn
C1
stage 1
stage m
stage n
Figure 1.12 Polyphase filter as a quadrature generator [55].
20
VQout −
Chapter 1 Introduction
The topology of PPF is shown in Figure 1.12, and it can be viewed as ‘n’ stages
of RC networks in cascaded connection [55]. Each stage is composed of four identical
resistors and four identical capacitors. The resistors and capacitors vary from stage to
stage. In Figure 1.12, Vin + and Vin − are the differential inputs, VIout + and VIout − are
differential outputs of the I path, VQout + and VQout − are the differential outputs of the Q
path.
The principle of PPFs is a so-called ‘stagger tuning’ technique [55], that is, by
using two or more cascaded stages of RC-CR networks, the amplitude of the quadrature
signal generated can be smoothed, and the bandwidth can be extended. As a quadrature
generator, PPFs have the following advantages over other traditional quadrature
generation techniques [56]:
•
They are simple in implementation, because only resistors and capacitors are
used.
•
Since a stagger-tuning technique is used in PPFs, they can overcome the
problems brought by CMOS process variation without any other complicated
tuning circuitry.
•
They are well suited for wideband quadrature signal generation.
The above-mentioned features make PPFs well suited for the CMOS
implementation, and nowadays, PPFs are widely used as quadrature generators at radio
frequency (RF). In this work, we name them as RF CMOS PPFs for this specific
application.
Though PPFs are well suited to function as RF quadrature signal generator in a
CMOS process, they have the following disadvantages:
21
Chapter 1 Introduction
•
Being passive filters, RF CMOS PPFs have insertion loss of 3 dB/stage.
Buffers, i.e. amplifiers, are usually needed to compensate the signal attenuation.
For high accuracy quadrature signal generation, multiple cascaded stages are
needed in the PPFs, and the buffers are power hungry.
•
If PPFs are used in the signal path, thermal noise of the resistors deteriorates
the noise figure (NF) of the receiver.
The problems mentioned above can be overcome or reduced by the following
techniques:
•
In order to reduce power consumption of quadrature generation in wireless
transceiver design, direct-connection between voltage-controlled oscillator
(VCO) and RF CMOS PPFs can be used. Borremans [60] successfully
demonstrated that it is viable to connect RF CMOS PPF directly to VCO, thus
power-hungry buffers can be skipped.
•
If RF CMOS PPFs are used in the signal path, proper values of resistors can be
chosen to avoid deteriorating noise figure (NF) of the receiver. The values of
resistors in the first stage of a PPF should be determined by the output
impedance of the preceding component, for example, the LNA. And the values
of resistors in the last stage of the PPFs should be as close as possible to 50
Ohms. By using this configuration, influences of the resistors on NF of the
receivers can be controlled in an acceptable range [32].
From the above discussions, we find that RF CMOS PPFs have advantages over
other conventional quadrature generation techniques in simplicity, bandwidth and
robustness against CMOS process variation. Moreover, frequency division technique is
22
Chapter 1 Introduction
not applicable in our project, thus, PPF is chosen as the quadrature generator in this
dissertation.
1.5 Limitations of Previous Works
Originally developed for generating quadrature signals with even amplitude in
audio by Gingell [45, 46], PPFs have received a lot of attention in both theoretical
analysis and practical designs [45-54]. In his doctoral dissertation, Gingell [48] did
some simple analytical work for guiding the practical designs. As in discrete
components implementation, matching and absolute values of resistors and capacitors
are under control by properly selecting components, process variation is not an issue for
this application. And in low frequency applications of PPFs, parasitic effects are usually
not under consideration.
In 1994, integrated PPFs were rediscovered as an efficient RF quadrature
generation technique in CMOS technology by Steyaert [55]. Design of integrated RF
CMOS PPFs faces many challenges, which were not considered by the pioneers. In
1994 and 1995, Schmidt [53] and Tetsuo [54] analyzed the PPFs in a systematic way by
using vector analysis. Though vector analysis of PPFs provides insight into the
mathematic representation of PPFs, it is too abstract for the designer to get quantitative
results of the influences from process tolerance and components mismatch on the
quadrature accuracy of the RF CMOS PPFs.
The most comprehensive and important contribution so far in RF CMOS PPFs
design is from Behbahani [62]; he analyzed practical design aspects and presented some
23
Chapter 1 Introduction
design guidelines. For understanding the limitations of the design guidelines, let us
analyze the main factors that should be considered in the design of an RF CMOS PPF.
1. CMOS process variation can be as much as 20%. The bandwidth of RF CMOS
PPFs should be extended so that the band of interest can be covered even the
worst case of the process variation occurs.
2. Quadrature accuracy (QA) is determined by component matching between the I
path and the Q path. In order to get good component matching, the layout of
resistors and capacitors should have large areas.
3. Parasitic effects in the RF CMOS PPFs cause signal losses and noise coupling
problems. Specifically, parasitic capacitors cause noise coupling from both
substrate and neighboring components; parasitic resistors make signal loss
more severe than the theoretical value of 3 dB insertion losses in each stage of
passive RC filters. In order to reduce unwanted parasitic effects, the layout of
resistors and capacitors should be kept as small as possible.
4. Compared to active components, in CMOS RF transceivers, passive
components usually consume more chip area, which is a main factor of cost. It
is thus required to design chips as small as possible.
From the above discussions, we can see that the main specification of the RF
CMOS PPF, quadrature accuracy, is very layout dependent; and the requirements from
component matching is contradictory to that of minimization of noise coupling, signal
loss and chip area. Thus, a tradeoff, that is, an optimization, must be made in a specific
design. By simply following Behbahani’s guidelines [62], it takes intensive trials and
many design iterations to reach a good design. Furthermore, with layout considerations,
24
Chapter 1 Introduction
optimization of RF CMOS PPFs cannot be achieved by using analytical calculations,
because of many constraints and tradeoffs in the design.
In this situation, a computer-aided analysis and design optimization tool for RF
CMOS PPFs is of great help to the community of RF CMOS PPFs designers, especially
for industrial projects, which are under time-to-market pressures. However, until now,
there has not been such a dedicated tool available in the open literature, to the author’s
best knowledge.
1.6 Objectives and Contributions of This Dissertation
In order to solve the problems discussed in the previous section, the objective of
this dissertation is to contribute a computer-aided analysis and design optimization tool
for RF CMOS PPFs, which is in need for the community of RF CMOS PPFs designers.
Based on the whole design flow and the silicon success of an RF CMOS PPF for a
Bluetooth low-IF transceiver in the Fraunhofer-IMS 0.6 µm CMOS process, this
dissertation has the following important contributions:
1. PPFANA, a dedicated computer-aided software tool developed by the author
for analyzing the influences from the nonideal effects, namely, process
tolerances and component mismatch on the quadrature accuracy of the RF
CMOS PPFs, is presented.
2. Physical design considerations for RF CMOS PPFs in system-on-a-chip (SoC)
solution are proposed by the author through the analysis of the influences
from parasitic effects in SoC.
25
Chapter 1 Introduction
3. Silicon success of an RF CMOS PPF for a low-IF Bluetooth receiver using
the Fraunhofer-IMS 0.6 µm CMOS process is reported.
4. The methodology of optimization of RF CMOS PPFs with layout
considerations by using genetic algorithms (GAs) is illustrated.
5. PPFOPTIMA, a computer-aided synthesis and optimization tool for RF
CMOS PPFs developed by the author, is presented.
6. Dedicated worst-case simulation and Monte Carlo (MC) simulation toolboxes
for verification of the performance of RF CMOS PPFs are illustrated.
1.7 Outline of This Dissertation
Chapter 1 begins with a discussion on CMOS single chip wireless transceivers,
and then advantages and disadvantages of the state-of-the-art transceiver architectures
are analyzed. Based on the analysis, the low-IF receiver architecture is chosen for the
CMOS single-chip Bluetooth transceiver design. RF quadrature signal generation is an
important functional block in modern wireless transceiver design. Through the analysis
of quadrature generation techniques, PPF is chosen as the RF quadrature generator in
our low-IF Bluetooth receiver. Existing problems in analysis, design and optimization
of RF CMOS PPFs are analyzed, limitations of previous work in this topic are pointed
out, and then objectives and contributions of this dissertation are clarified.
In chapter 2, nonidealities in RF CMOS PPF design, namely, process tolerances
and component mismatch, are analyzed by using the transfer function of PPFs.
PPFANA, a computer-aided software tool for analysis of influences from process
tolerances and component mismatch on the quadrature accuracy (QA) of RF CMOS
26
Chapter 1 Introduction
PPFs developed by the author, is described. Based on the simplified RF models of
resistors and capacitors, parasitic effects and their influences on RF CMOS PPFs are
analyzed; techniques for minimizing unwanted noise coupling and signal losses are
proposed and evaluated.
Based on the analytical results from chapter 2, physical design considerations for
RF CMOS PPFs to be implemented in System-on-a-Chip (SoC) solutions are proposed
in chapter 3. The author designed an RF CMOS PPF for a low-IF Bluetooth receiver by
using the Fraunhofer-IMS 0.6 µm CMOS process. The designed chip has been
fabricated at the Fraunhofer-IMS in Duisburg. The author characterized the
performance of the fabricated RF CMOS PPF by measurements. The design, fabrication,
measurement procedure and measurement results are presented in chapter 3. Silicon
success of this design lays a solid foundation for this dissertation.
Quadrature accuracy (QA), the main specification in design of an RF CMOS PPF
is determined by component matching which requires large chip area; however, in order
to minimize unwanted noise coupling, signal loss and the cost, the chip area must be
minimized. Thus, a tradeoff among these design parameters, that is, an optimization,
must be made in the design. Chapter 4 is committed to the optimization of RF CMOS
PPFs. First, the optimization problem is formulated with design specifications and
constraints. Then, possible optimization algorithms like calculus-based optimization,
gradient descend, simulated annealing (SA) and genetic algorithms (GA) are analyzed
by the author. Based on the results of the analysis, GA is chosen as the optimization
engine because of its advantages in escaping from local optima. The author
implemented PPFOPTIMA, a GA based optimization tool for RF CMOS PPFs by using
MATLAB. In order to facilitate designers in design automation, this tool has been
27
Chapter 1 Introduction
further developed into a synthesis and verification environment for RF CMOS PPFs by
the author, and the detailed implementation of PPFOPTIMA is described in chapter 4.
Experiments on multistage RF CMOS PPFs for a given standard, and experiments on
automated synthesis of RF CMOS PPF for different standards, such as, global system
for mobile communications (GSM), digitally enhanced cordless telephony (DECT) and
wide-band code division multiple access (WCDMA) demonstrate that PPFOPTIMA is
effective and efficient in synthesis, optimization, and verification of RF CMOS PPFs.
In chapter 5, contributions and conclusions from this dissertation are summarized,
and directions for possible future work are pointed out.
28
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Chapter 2
Analysis of Nonidealities in RF CMOS PPFs
2.1 Introduction
Polyphase filters (PPFs) were invented by Gingell [45, 46] in 1971, for generation
of quadrature signals with even amplitude in audio. At that time, PPFs were
implemented by using discrete components. In his doctoral dissertation, Gingell [48] did
some simple analytical work for guiding the practical design. In 1994 and 1995,
Schmidt [53] and Tetsuo [54] analyzed the PPFN in a systematic way by using vector
analysis.
In 1994, integrated PPFs were rediscovered as an efficient RF quadrature
generation technique in CMOS technology by Stayaert [55], introducing PPFs into a
new regime, which we name as RF CMOS PPFs. The design of integrated RF CMOS
PPFs faces many challenges, which were not considered by the pioneers. In 2000, Galal
[58, 59] analyzed the sensitivity of the RF CMOS PPFs in RF integrated transceivers. In
2001, Behbahani [62] analyzed the application of RF CMOS PPFs in image rejection.
However, results from the open literature on the analysis of PPFs listed above
have the following limitations:
29
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
1. Gingell worked on audio frequency single sideband modulation [45-49]. At
such low operating frequencies, the influences from parasitic resistors and
capacitors are negligible. Thus, they were not considered in his publications.
2. In discrete components implementation of PPFs, components can be selected
with any given specifications. Thus, component mismatch was not analyzed in
[50-54].
3. References [55-61] provide useful information about the analysis and the
design of RF CMOS PPFs. However, until today, nonidealities, such as,
process tolerances, component mismatch and parasitic effects, which are
critical in RF CMOS PPF design and optimization, were not analyzed in a
systematic way.
4. Though vector analysis of PPFs [53, 54, 62] provides insight into the
mathematic representation of PPFs, it is too abstract for the designer to get
quantitative results about the influences from process tolerance and
components mismatch on the quadrature accuracy (QA) of RF CMOS PPFs. A
dedicated computer-aided software tool for solving this problem is required by
the design community of RF CMOS PPFs. However, this kind of computeraided tool has not been available to the author’s best knowledge.
In order to solve the shortcomings mentioned above, based on the explicit
transfer functions of PPFs, PPFANA, a dedicated computer-aided software tool for the
30
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
analysis and the simulation of influences from process tolerances and component
mismatch on the quadrature accuracy of the RF CMOS PPFs, has been developed by the
author, PPFANA will be presented in this chapter. By using simplified RF models of
resistors and capacitors, parasitic effects in RF CMOS PPFs are analyzed. Furthermore,
techniques for minimizing unwanted noise coupling and signal loss are proposed.
2.2 Vector Analysis of Polyphase Filters (PPFs)
A polyphase signal is a set of two or more vectors having the same frequency but
different phases [59]. If its vectors have the same magnitude and are equally spaced in
phase, it is said to be symmetric. Consequently, a symmetric four-phase signal consists
of two vectors of equal magnitudes having the same frequency and being separated in
phase by 90 degrees.
In general, polyphase filter networks (PPFNs) are networks having N input
terminals and N output terminals [54]. These networks are physically symmetric in a
way that the paths from each input to the corresponding output are symmetric. A PPFN
is driven by a polyphase signal consisting of a number of vectors that correspond to the
number of inputs of the PPFN. Although PPFNs are physically symmetric, they exhibit
asymmetric responses to inputs of opposite polarities [54]. This sequencediscrimination property is the reason behind the name ‘sequence asymmetric’ [54]. The
31
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
response of a four-phase PPFN driven by a symmetric four phases input signal will be
analyzed now. Since the four phases are symmetric, the chain matrix of a single phase
represents the chain matrix of the network. The response of the same network to
asymmetric inputs can be derived by resolving the input into a summation of symmetric
inputs. Applying the rules of linear superposition, the output of the PPFN is then
considered as the sum of the outputs cascaded by each symmetric input alone.
e jθ Vk ,in
Y2
I k ,in
Vk ,in
Y1
Y2
I k ,out
Vk ,out
e- jθ Vk ,out
Figure 2.1 One phase of a generalized N-phase PPFN.
Figure 2.1 shows the structure of one phase of a generalized N-phase PPFN,
where admittance Y1 is connected between the input and the corresponding output, and
admittance Y2 is skewed between the input and output of adjacent phases. Since the
input is symmetric and the PPFN is physically symmetric, we expect the output to be
also symmetric. The chain matrix of a single phase can be written as [58]:
Vk ,in 
1

=
jθ
 I k ,in  Y1 + e Y2
1  Vk ,out 
Y1 + Y2 ,



 2Y1Y2 (1 − cos θ ), Y1 + Y2   I k ,out 
32
(2.1)
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Where θ represents the relative phase difference between V1 and the neighboring inputs,
which in turn determines the polarity of the inputs. If θ <0, Vk , in will be leading e jθ Vk ,in
in phase, which causes the inputs to be positive. On the other hand, if θ > 0, Vk , in will
be lagging, thus causing the inputs to be negative.
Resistance–capacitance (RC) PPFNs are a special case of sequence asymmetric
PPFNs in which Y1 is equal to 1/ R and Y2 is equal to jωC . The structure of an RC
four-phase PPFN is shown in Figure 2.2. In the following figure, Vk ,in , Vk ,out , k=1,2,3,4,
are the four inputs and the four outputs of the RC PPFN, respectively.
R
V1,out
V1,in
C
R
V2,out
V2,in
C
R
V3,out
V3,in
C
R
V4,out
V4,in
C
Figure 2.2 One-stage of an RC sequence asymmetric PPFN.
33
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
In four-phase PPFNs such as shown in Figure 2.2, the value of θ is either π / 2
or − π / 2 according to the polarity of the inputs. For such PPFNs, the chain matrix for
positive inputs can be written as [59]:
Vk ,in 
1 + jω RC , R
 Vk ,out 
1

=
,

 I
+
2
j
ω
,1
j
ω
I
C
RC
+
1
ω
RC


,in
,out
k
k




(2.2)
while for negative inputs, we find [59]:
Vk ,in 
1 + jω RC , R  Vk ,out 
1

=
.


 I k ,in  1 − ω RC  2 jωC ,1 + jω RC   I k ,out 
(2.3)
The open-circuit voltage transfer function for the negative part of the inputs is
found for I k ,out equals to zero [59]:
H(ω ) =
Vk ,out
1- ω RC
=
,
Vk ,in 1 + jω RC
(2.4)
while for the positive part of the inputs, we yield [59]:
H( −ω ) =
Vk ,out 1 + ω RC
=
Vk ,in 1 + jω RC
(2.5)
Sequence-asymmetric PPFNs can be considered as a generalized structure of the
classical RC–CR network used for quadrature signal generation. Consider the M-stage
four-phase PPFN shown in Figure 2.3(a), having a positive part of the outputs transfer
function H(ω ) and a negative part of the outputs transfer function H( −ω ) , where
H(ω ) ≠ H( −ω ) . The PPFN is driven by only two phases representing a differential input
signal, while the other two inputs are connected as shown in Figure 2.3 (a). Figure 2.3(b)
34
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
shows the phasor representation of the input sequence which consists of two vectors
separated in phase by 180 degrees.
V1,in
V1,out
PPFN
V2,in
V
V3,in
V2,out
V3,out
H(ω )
V4,in
V4,out
(a)
V /2
V /4
V /4
1
1
1
=
4
2
jV / 4
3
3
−V / 2
−V / 4
Inputs
− jV / 4
+
2
4
jV / 4
− jV / 4
3
−V / 4
Negative part of the outputs
Positive part of the outputs
(b)
Figure 2.3 Quadrature signal generation using four-phase PPFNs
(a) Circuit diagram.
(b) Decomposition of the two-phase inputs into two parts of four-phase outputs.
To be able to understand how quadrature phases are generated from a single input,
the inputs can be represented as the sum of two parts of four-phase outputs of different
35
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
polarities. Using linear superposition, the PPFN output can be expressed as the sum of
the outputs, separately. It should be noted that the voltage transfer function for any
phase in the positive part is H(ω ) , while that for any phase in the negative part is
H( −ω ) . Thus, the output phases can be written as:
V
((H(ω ) + H(-ω )) ,
2
V
V2,out (ω ) = - j ((H(ω ) - H(-ω )) ,
2
V
V3,out (ω ) = - ((H(ω ) + H(-ω )) ,
2
V
V4,out (ω ) = j ((H(ω ) - H(-ω )) .
2
V1,out (ω ) =
(2.6)
(2.7)
(2.8)
(2.9)
If we consider the in-phase component to be V1,out (ω ) − V3,out (ω ) , and the
quadrature-phase component V2,out (ω ) - V4,out (ω ) , then we find:
VI (ω ) = V (H(ω ) + H(-ω )) ,
(2.10)
VQ (ω ) = - jV (H(ω ) - H(-ω )) ,
(2.11)
The ratio of the quadrature-phase to the in-phase outputs of an RC single stage
four-phase PPFN can be derived by substituting for H(ω ) and H( −ω ) . With (2.4) and
(2.5), we yield from (2.10) and (2.11),
VQ ( jω )
VI ( jω )
= jω RC
(2.12)
The disadvantage of the described PPFN is its limited frequency range within
which good gain matching can be achieved between quadrature outputs. The two
outputs, although in quadrature, have equal amplitudes only at a frequency equals to
1/ RC , which limits the use of this PPFN to narrow-band applications. Furthermore, the
36
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
gain and phase matching of the quadrature outputs is highly sensitive to the RC
mismatch.
To achieve broadband quadrature with better gain matching and reduced
sensitivity to components mismatch, a two-stage PPFN could be employed [65]. The
two outputs have equal amplitudes at two frequencies corresponding to the resonant
frequencies of each stage. The voltage transfer ratio can be derived by multiplying the
chain matrix of the two stages and evaluating the open-circuit voltage transfer function.
From (2.10) and (2.11), we yield [65]:
VQ ( jω )
VI ( jω )
=
jω ( R1C1 + R2C2 )
1 + ω 2 ( R1R1C2C2 )
(2.13)
This creates a wide-band quadrature generator defined by the resonant frequencies
1 / 2πR1C1 and 1 / 2πR2 C 2 . For any input signal with frequencies lying in this band, the
quadrature outputs are matched well in terms of gain and phase. The maximum gain
error occurs at frequency ωmax , corresponding to the geometric mean of the two
resonant frequencies. The maximum gain error at this frequency is given
VQ ( jω )
VI ( jω )
where ωmax is equal to
ωmax =
1
.
( R1 R2C2C2 )
37
R1C1 + R2C2
,
2 ( R1R2C2C2 )
(2.14)
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
2.3 PPFANA, a Software Tool for RF CMOS PPFs
2.3.1 Overview
As discussed in the first chapter, vector analysis is very helpful in understanding
the mathematical representation of PPFs; however, it is too abstract for the designer to
get quantitative calculation on the quadrature accuracy (QA) of the PPFs. Furthermore,
in design and optimization of RF CMOS PPFs, it is of great importance to analyze and
simulate QA influenced by process tolerances and component mismatch. Unfortunately,
there is not any efficient simulation tool openly available. Thus, PPFANA, a software
analysis tool for solving the problems mentioned above has been developed in this work.
The core of PPFANA is made up from explicit transfer functions, which will be derived
in the following subsection.
2.3.2 Explicit Transfer Function of RF CMOS PPFs
Explicit transfer functions of RF CMOS PPFs are obtained by multiplying chain
matrices of each cascaded stages, then separating the real and imaginary parts. The
relation between input and output can thus be expressed by values of resistors and
capacitors and the operating frequency, explicitly.
In CMOS wireless transceiver RF front-end design; 2-stage, 3-stage, and 4-stage
RF CMOS PPFs are most widely used [68]. RF CMOS PPFs with more than four
38
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
cascaded stages are rarely used because of their severe insertion loss. For passive filters,
there is a theoretical insertion loss of 3dB/stage. We therefore derive the explicit
transfer function of a 4-stge RF CMOS PPF. Those of 2-stage and 3-stage RF CMOS
PPFs can be derived by the same method used here. Details of the derivation are given
in appendix 2 of this dissertation.
The transfer function of 4-stage RF CMOS PPFs is set out from
R1 
 Vin 
1+jω R1C1
1
 Iin  = 1-ω R1C1  2jω C1 1+jω R1C1
 


R2 
1+jω R2C2
1

1-ω R2C2  2jω C2 1+jω R2C2 
R3 
1+jω R3C3
1

1-ω R3C3  2jω C3 1+jω R3C3 
(2.15)
R4   Vout 
1+jω R4C4
1
.

1-ω R4C4  2jω C4 1+jω R4C4   Iout 
The complex transfer function of the 4-stage RF CMOS PPF is given by
H(jω )=
N(jω )
,
D(jω )
N(jω )=(1-ω R1C1)(1-ω R2C2)(1-ω R3C3)(1-ω R4C4) ,
D(jω )=DR(ω )+jDI(ω ) ,
DR(ω )=1-ω 2 (R1R2C1C2+R1R3C1C3+R2R3C2C3
+R1R4C1C4+R2R4C2C4+R3R4C3C4
+2R1R3C1C4+2R2R3C2C4+2R1R2C2C4
+2R1R2C1C4+2R1R3C3C4+2R2R3C3C4
+2R1R3C2C3+2R1R2C2C3+2R1R4C2C4
+2R1R4C3C4+2R2R4C3C4+2R1R2C1C3
+4R1R3C2C4)+ω 4 (R1R2R3R4C1C2C3C4),
39
(2.16)
(2.17)
(2.18)
(2.19)
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
DI(ω )=ω (R1C1+R2C2+R3C3+R4C4+2R3C4+2R1C2+2R1C3
+2R2C3+2R2C4+2R1C4)-ω 3 (R1R2R3C1C2C3
+R1R2R4C1C2C4+R1R3R4C1C3C4+R2R3R4C2C3C4
+2R1R2R3C1C2C4+2R1R2R3C2C3C4+2R1R2R3C1C3C4
+2R1R3R4C2C3C4+2R1R2R4C2C3C4+2R1R2R4C1C3C4).
(2.20)
Equation (2.15) can also be expressed as:
H(jω )=H1(jω )+jH2(jω ) ,
H1(jω )=
(2.21)
1+aω 2 +bω 4
,
D(jω )
a=R1R2C1C2+R1R3C1C3+R2R3C2C3
+R1R4C1C4+R2R4C2C4+R3R4C3C4
b=R1R2R3R4C1C2C3C4 ,
H2(jω )=
cjω +djω 3
,
D(jω )
c=R1C1+R2C2+R3C3+R4C4 ,
d=R1R2R3C1C2C3+R1R2R4C1C2C4
.
+R1R3R4C1C3C4+R2R3R4C2C3C4
(2.22)
,
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
2.3.3 Development of PPFANA
With the explicit transfer functions we derived in subsection 2.3.2, the quadrature
accuracy of an RF CMOS PPF with given values of resistors, capacitors and operating
frequency can be calculated by using MATLAB. In order to make it easy to use,
graphical user interfaces (GUIs) have been developed for PPFANA. The GUI of
PPFANA for 4-stage RF CMOS PPFs is shown in Figure 2.4. The GUIs for the 2-stage
and 3-stage RF CMOS PPFs are similar to this one.
40
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Figure 2.4 Graphical user interface of PPFANA for 4-stage RF CMOS PPF.
In PPFANA, quadrature accuracy (QA), is measured by gain mismatch (GM) and
phase mismatch (PM), the smaller GM and PM are, the better QA is. From Figure 2.4,
we can see PPFANA has following features:
•
QA influenced by process tolerance (PT) and component mismatch (CM) can
be analyzed independently.
41
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
•
QA influenced by PT and CM can be analyzed at the same time.
•
Worst-case QA can be analyzed.
•
Default values for an RF CMOS PPF designed for a low-IF Bluetooth receiver
in Fraunhofer-IMS 0.6 µm CMOS technology are given as an example for new
users.
In the following sections, simulation results for QA influenced by PT and CM will
be presented. The simulation results will be compared with those obtained with the
commercial RF circuit simulator-Cadence SpectreRF to verify the effectiveness of
PPFANA.
2.4
Analysis of Quadrature Accuracy in Ideal PPFs
In this section, the quadrature accuracy (QA) of ideal PPFs is investigated by using
PPFANA. In the transfer function derived in the last section, H1 ( jω ) , H 2 ( jω ) are
complex signal of the I, Q path respectively. From the definition of complex numbers,
we can derive the module and the complex angle. Gain mismatch, mgain , and phase
mismatch, ∆φ , can be expressed by
mgain =| 20 log10 | H1 ( jω ) | −20log10 | H 2 ( jω ) || ,
(2.28)
mphase =|| ∠H1 ( jω ) - ∠H 2 ( jω ) | -90° | .
(2.29)
42
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Notice that in Cadence simulations, it is more convenient to get phase difference, d phase ,
than phase mismatch, ∆φ , between I path and Q path by using AC simulation. For the
purpose of comparison, we define a parameter, phase difference, d phase as
d phase =| ∠H1 ( jω ) - ∠H 2 ( jω ) | .
(2.30)
Please note that in all of the equations related to mgain , ∆φ , d phase used in this dissertation,
the angular frequency ω is used for the convenience of expression of complex. Whereas
in all of the figures related to mgain , mphase , d phase , frequency f is used for the purpose of
comparison with Cadence simulation results. It is well known that ω equals to 2πf .
Figure 2.5 Gain mismatch in ideal multi-stage RF CMOS PPFs tailored to
the Bluetooth frequency range 2.4 GH to 2.48 GHz.
43
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Three ideal multi-stage (2-stage, 3-stage and 4-stage) RF CMOS PPFs for the
frequency band of Bluetooth, i.e., 2.4 GHz to 2.48 GHz, were designed by the author
and they were simulated by using PPFANA. Simulation results for the gain mismatch
and phase difference in these ideal RF CMOS PPFs are shown in Figure 2.5 and Figure
2.6, respectively.
Figure 2.6 Phase difference in ideal multi-stage RF CMOS PPFs tailored to
the Bluetooth frequency range 2.4 GH to 2.48 GHz.
From above figures, we can see the following:
•
For ideal RF CMOS PPFs, at the desired frequency, gain mismatch is almost
zero, and phase difference is almost 90° .
•
With more cascaded stages, the useful bandwidth with less gain mismatch and
phase mismatch can be extended.
44
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
In order to compare simulation results, these ideal RF CMOS PPFs were
simulated with Cadence SpectreRF as well. Simulation results from Cadence SpectreRF
and from PPFANA are listed in the following Table 2.1.
Ideal RF
CMOS PPFs
∆φ /degrees
mgain /dB
PPFANA
SpectreRF
PPFANA
SpectreRF
2-stage
10−3
10−3
10−3
10−3
3-stage
< 10 −3
< 10 −3
< 10 −3
< 10 −3
4-stage
< 10 −3
< 10 −3
< 10 −3
< 10 −3
Table 2.1 Quadrature accuracy of ideal RF CMOS PPFs (2.4 GHz to 2.48 GHz),
obtained by using PPFANA and SpectreRF.
From these simulation results listed in the table above, we can see the following:
•
For ideal 2-stage RF CMOS PPFs, the gain mismatch is 10−3 dB and the phase
mismatch is 10−3 degrees.
•
For ideal RF CMOS PPFs with more than 2 cascaded stages, the gain mismatch
is less than 10−3 dB, and the phase mismatch is less than 10−3 degrees.
Considering computational errors, these values can be taken as zero. Thus, we
can say that ideal RF CMOS PPFs with more than two cascaded stages are
almost perfect in quadrature generation.
45
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
•
The simulation results of PPFANA are in accordance with those of Cadence
SpectreRF. This validates the effectiveness of PPFANA.
2.5
Analysis of Quadrature Accuracy Influenced by Process
Tolerance
In CMOS processes, the actual values of identically designed and fabricated
components can be different from wafer to wafer. The variation of the component
values in a given CMOS process is called process tolerances, which is also named
process variation. In this dissertation, these two terminologies are used without any
difference.
In high performance CMOS analog/RF circuit, tuning techniques are usually
needed to overcome process tolerances. Most tuning techniques use variable capacitors
or variable resistors and additional control circuits, making circuitries complicated. In
RF CMOS PPFs, a special stagger-tuning technique [55] is used. The idea of this
technique is to increase the bandwidth of the circuit, to guarantee a proper filter
operation despite process tolerances. By using this special and simple tuning technique,
conventional tuning techniques [55] are no longer needed in RF CMOS PPFs to
overcome the influence from process variations.
46
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Let us generalize the equations of GM and PM in multi-stage PPFs and derive
expressions for GM and PM influenced by process tolerances (PT). GM and PM in RF
CMOS PPFs with ideal components are expressed as mgain,ideal and ∆φ ,
mgain,ideal = G( R1 , C1 ,L Ri , Ci ,L Rn , Cn )
(2.31)
∆φideal = F( R1 , C1 ,L Ri , Ci ,L Rn , Cn )
(2.32)
Then we get GM and PM in RF CMOS PPFs with process tolerance mgain, PT and ∆φPT
according to
mgain,PT = G( RPT1 , CPT1 ,L RPTi , CPTi ,L RPTn , CPTn )
(2.33)
∆φPT = F( RPT1 , CPT1 ,L RPTi , CPTi ,L RPTn , CPTn )
(2.34)
where G , F are gain and phase expressions of RF CMOS PPFs’ transfer functions,
Ri , Ci , i = 1L n , are the nominal values of the resistors and capacitors, and RPTi , CPTi ,
i = 1L n , are the resistors and capacitors with process tolerance.
In this work, gain mismatch and phase mismatch influenced by process tolerances
in multi-stage RF CMOS PPFs have been simulated by using PPFANA. These RF
CMOS PPFs are designed for the frequency band of Bluetooth, ranging from 2.4 GHz
to 2.48 GHz, using the Fraunhofer-IMS 0.6 µm CMOS technology. In these simulations,
the resistors are polysilicon resistors, in which process tolerance is ± 16% , and the
capacitors are poly-N+ capacitors, in which process tolerance is ± 14% . Gain mismatch
and phase mismatch are the worst value in the frequency band of interest, i.e. 2.4 GHz
to 2.48 GHz.
47
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Figure 2.7 Gain mismatch influenced by process tolerance
in a 2-stage RF CMOS PPF (2.4 GHz to 2.48 GHz) with regular design.
Figure 2.8 Phase mismatch influenced by process tolerance
in a 2-stage RF CMOS PPF (2.4 GHz to 2.48 GHz) with regular design.
48
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Figure 2.7 and Figure 2.8 show the gain mismatch and the phase mismatch
influenced by process tolerance in a 2-stage RF CMOS PPFs with regular design, that is,
it is designed with lower corner frequency at 2.4 GHz and higher corner frequency at
2.48 GHz without any technique to overcome the influences from process tolerance.
From these figures, we can see the following:
•
Process tolerances have significant influence on the gain mismatch of the RF
CMOS PPF, whereas, their influences on the phase mismatch of the RF CMOS
PPF is not that obvious.
•
With regular design, RF CMOS PPF is not robust enough to overcome the
influence from process tolerance.
In order to get robust RF CMOS PPFs, the influence of process tolerances on the
quadrature accuracy should be overcome. As the ‘stagger tuning’ technique is used in
the polyphase filters, in practical design, we can extend the bandwidth to guarantee that
the signal frequency band can be covered by worst-case lower and upper corner
frequencies. This is the concept of tolerance design [102] in RF CMOS PPFs design.
The 2-stage RF CMOS PPF was redesigned with tolerance design, and the gain
mismatch influenced by process tolerances is shown in the following Figure 2.9. From
this figure, we can clearly see that, the gain mismatch is no longer sensitive to process
tolerance. Thus, tolerance design should be applied in design of RF CMOS PPFs.
49
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Figure 2.9 Gain mismatch influenced by process tolerance
in a 2-stage RF CMOS PPF (2.4 GHz to 2.48 GHz) with tolerance design.
RF CMOS
PPFs
∆φPT /degrees
mgain, PT /dB
PPFANA
SpectreRF
PPFANA
SpectreRF
2-stage
0.220
0.224
0.500
0.509
3-stage
0.015
0.017
0.050
0.053
4-stage
0.002
0.002
< 10 −3
< 10 −3
Table 2.2 Quadrature accuracy of RF CMOS PPFs (2.4 GHz to 2.48 GHz)
with process tolerance, obtained by PPFANA and SpectreRF
50
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
In order to compare the quadrature accuracy of multi-stage RF CMOS PPFs
influenced by process tolerance and to compare simulation results obtained by using
PPFANA and those by using Cadence SpectreRF, three RF CMOS PPFs with two, three
and four cascaded stages, respectively were designed for the frequency band of
Bluetooth with tolerance design. Simulation results for the gain mismatch and for the
phase mismatch of these RF CMOS PPFs are shown in Table 2.2, from which we can
see the following:
•
Quadrature accuracy of RF CMOS PPFs is influenced by process tolerances,
significantly.
•
By using more cascaded stages, the quadrature accuracy of the RF CMOS PPFs
can be improved.
•
The simulation results from PPFANA are in accordance with those from
Cadence SpectreRF.
2.6
Analysis
of
Quadrature
Accuracy
Influenced
by
Component Mismatch
In CMOS processes, two neighboring components that are ideally supposed to be
identical can have slight differences. This phenomenon is called component mismatch
(CM) [81]. Ideally, in RF CMOS PPFs, components in the I path and in the Q path
51
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
should be identical; however, in practice, this is impossible to achieve because of
component mismatch.
Now, let us generalize the transfer functions of multi-stage PPFs and analyze both
the GM and the PM influenced by component mismatch (CM) between I path and Q
path, we yield
mgain,ideal =| G I ( RI1 , CI1 ,L RIn , CIn ) - G Q ( RQ1 , CQ1 ,L RQn , CQn ) | ,
∆φideal =|| FI ( RI1 , CI1 ,L RIn , CIn ) - G Q ( RQ1 , CQ1 ,L RQn , CQn ) | -90o | .
(2.35)
(2.36)
Furthermore, we get
mgain,CM =| G I ( RICM1 , CICM1 ,L RICMn , CICMn ) - G Q ( RQCM1 , CQCM1 ,L RQCMn , CQCMn ) | ,
(2.37)
∆φCM =|| FI ( RICM1 , CICM1 ,L RICMn , CICMn ) - FQ ( RQCM1 , CQCM1 ,L RQCMn , CQCMn ) | -90o | .
(2.38)
In (2.37) and (2.38) RICMk equals (1 ± CM) RIk , RQCMk equals (1 m CM) RQk , CICMk is
(1 ± CM)CIk , and CQCMk is (1 m CM)CQk , k = 1L n , n is the number of cascaded stages
in a given RF CMOS PPF. In the equations above, mgain,ideal and ∆φideal are the gain
mismatch and the phase mismatch of RF CMOS PPFs with ideal resistors and
capacitors, mgain, CM and ∆φCM are the gain mismatch and the phase mismatch of RF
CMOS PPFs with component mismatch, FI , FQ , G I , G Q are I path and Q path gain and
phase expressions derived from multi-stage RF CMOS PPFs’ transfer functions,
RIk , CIk , RQk , CQk are nominal values of resistors and capacitors in I path and Q path,
52
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
RICMk , CICMk , RQCMk , CQCMk are values of resistors and capacitors in I path and Q path
with component mismatch.
In this work, quadrature accuracy influenced by component mismatch in multistage RF CMOS PPFs has been simulated by using PPFANA. These RF CMOS PPFs
were designed for the frequency band of Bluetooth, using Fraunhofer-IMS 0.6 µm
CMOS technology. In the simulations, the resistors are polysilicon resistors, and the
capacitors are poly-N+ capacitors. In order to compare the quadrature accuracy
influenced by different levels of component mismatch, 1% and 2% component
mismatch between neighboring resistors, and neighboring capacitors were simulated.
Gain and phase mismatches are the worst value in the band of interest, i.e., 2.4 GHz to
2.48 GHz. The simulation results are shown in Table 2.2.
RF CMOS
PPFs
resistors and capacitors with 1%
resistors and capacitors with 2%
component mismatch
component mismatch
mgain, CM /dB
∆φCM /degrees
mgain, CM /dB
∆φCM /degrees
PPFANA
PPFANA
PPFANA
PPFANA
2-stage
0.210
0.520
0.230
0.530
3-stage
0.025
0.030
0.060
0.090
4-stage
0.015
0.020
0.020
0.050
Table 2.3 Quadrature accuracy influenced by component mismatch in
multi-stage RF CMOS PPFs (2.4 GHz to 2.48 GHz).
53
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
From the simulation results in Table 2.3, we can see the following:
•
Component mismatch has significant influence on quadrature accuracy of RF
CMOS PPF.
•
Stronger component mismatch leads to worse quadrature accuracy, that is,
more gain mismatch and phase mismatch.
Though component mismatch cannot be totally eliminated in practice, layout
techniques, such as, inter-digitized [77] and common-centroid [77] layout structures, are
helpful to reduce the component mismatch to some extend. These techniques will be
discussed in the Chapter 3.
2.7 Analysis of Influences from Parasitic Effects
2.7.1 Simplified RF Models of Resistors and Capacitors
There are several types of resistors in CMOS processes; polysilicon resistors [81]
are the most favorable selection for RF applications for its area efficiency and
comparatively good robustness against parasitic influences. In compact models [75-80],
polysilicon resistors are usually modeled as distributed resistors and capacitors. As in
our application, the physical dimension of the resistors are much smaller than the
wavelength of the Bluetooth RF signal, for the purpose of analysis, we simplify the
distributed model into a simplified model as shown in Figure 2.8.
54
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
From Figure 2.10, we can see that polysilicon resistors with its parasitic
capacitance can form a low-pass filter. In RF CMOS PPFs, current flows through
resistors. If the corner frequency of the low pass filter formed is lower than that of the
signal frequency, the signal is then cut off. This phenomenon is unwanted and should be
avoided in RF CMOS PPF design.
R
Out
In
Cpara
Cpara
Rsub
Rsub
gnd
gnd
Figure 2.10 Simplified RF model of a polysilicon resistor [81].
Now, let us investigate how the problem mentioned above can be avoided in the
design of RF CMOS PPFs. We have [81]
f RCP =
1
1
1
=
=
,
2
L
2π RCpara 2π LWR C
2π L Rsh Cuap
sh uap
W
f op
L≤
≤f
RCP
,
1
,
2π f op Rsh Cuap
55
(2.39)
(2.40)
(2.41)
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Lmax,res =
1
,
2π f op Rsh Cuap
(2.42)
where f RCP is the corner frequency of the filter formed by the polysilicon resistor and its
parasitic capacitor to the substrate, f op is the operating signal frequency, Rsh is the sheet
resistance of resistor, L is the length of the resistor, Cuap is parasitic capacitance per
unit area to the substrate, Lmax, res is the maximum length of resistors can be used.
From the derivations above, we can see that for a certain CMOS process and a
given operating frequency of the RF CMOS PPF, before physical layout, the maximum
length of the resistor should be calculated to avoid inappropriate physical design.
Rp1
1
Cp1
Rp2
Cp1
2
Cp2
Cp2
Rsub
Rsub
gnd
gnd
Figure 2.11 Simplified RF model of a capacitor [81].
56
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
In different CMOS processes, there are different kinds of capacitors, e.g. polypoly capacitors [81], poly-well capacitors [81], MOS capacitors [81], and metalinsulator-metal (MIM) capacitors [81]. The first three kinds of capacitors are fairly
standard, whereas MIM capacitors are only available in advanced CMOS processes [81].
RF capacitors are modeled as distributed networks including parasitic resistors
and capacitors [81]. As in our application, the physical dimension of the capacitors are
much smaller than the wavelength of the Bluetooth RF signal, for the purpose of
analysis, we simplified the distributed model as the model shown in Figure 2.11. Now,
let us investigate the quality (Q) factors of the capacitor and make a comparison among
different capacitors mentioned above. The Q factor is given by [81]
1
,
RCω
(2.43)
R = Rsh N unit ,
(2.44)
Q(ω ) =
Insert (2.44) into (2.43), we get
Q(ω ) =
1
.
Rsh N unit Cω
(2.45)
In the above equations, R is the parasitic resistance of the capacitor, C is the nominal
capacitance, ω is the operating frequency, Rsh is the sheet resistance of the plates, N unit
is the number of unit squares.
From (2.45) we can see that for a given capacitance C and a given operating
frequency ω , there are two ways to improve quality factor of the capacitor:
•
Use capacitors with low sheet resistance plates.
57
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
•
Use a low number of unit squares, e.g. N unit equals 1.
As metal plates have less sheet resistance, the quality factor of metal-insulatormetal (MIM) capacitors is greater than that of poly-poly resistors, poly-well capacitors
and MOS capacitors [81]. Thus, if possible, the use of MIM capacitors is highly
recommended for design of RF CMOS PPFs. Furthermore, in the physical layout of
capacitors, square capacitors should be used to get a greater quality factor.
2.7.2 Substrate Noise Coupling
In CMOS single chip transceiver design, the whole transceiver system, including
RF, analog and digital circuits, is integrated into a chip [99]. This concept is call
system-on-a-chip (SoC) [99]. SoC design poses serious challenges different from
conventional multi-chip or multi-package design, because digital circuits are placed on
the same substrate with sensitive analog circuits [95].
Analog circuits
Noise
Coupling
Noise
Noise
Digital circuits
Noise
Coupling
Noise
Noise
Semiconductive
Substrate
Figure 2.12 Principle of substrate noise coupling in system-on-a-chip [99].
58
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
The most critical issue in SoC design is substrate noise coupling [99], which
occurs as a result of the conductive nature of silicon and the lack of perfect isolation
between devices and the substrate [100]. Below frequencies of 5 GHz, the capacitive
noise coupling is the main contributor to the substrate noise coupling [87, 88]. In this
dissertation, the operating frequency is 2.45 GHz, which is well below 5 GHz. Thus, we
consider only the capacitive noise coupling.
In SoC design, digital circuits are usually noisy. Noise generated by digital
circuits is injected into the substrate [99]. Due to the semi-conductive nature of the
silicon substrate, the noise is then sensed by the sensitive analog circuit via capacitive
coupling [99]. The principle of substrate noise coupling in SoC is shown in Figure 2.12
[99].
RF CMOS PPF
In
Out
Cpara
Cpara
Rsub
Rsub
Noise
Noise
gnd
gnd
Figure 2.13 Simplified model of substrate noise coupling in RF CMOS PPFs.
59
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
Shown in Figure 2.13 is the simplified model of substrate noise coupling in RF
CMOS PPFs. In this model, simplified RF models of resistors and capacitors together
with noisy resistive substrate model are used.
Now, let us analyze the noise coupled to the unshielded RF CMOS PPFs by the
substrate noise coupling. The parasitic capacitance between the polysilicon resistor and
the substrate ( CR ), and the parasitic capacitance between the capacitor and the substrate
( CC ) are
CR = LRWR CuaR = S R CuaR ,
(2.46)
CC = LCWCCuaC = SCCuaC .
(2.47)
Summing all capacitances, we get
∑
n
Csum = 4 ×
( SCi CuaCi + S Ri × CuaRi ) ,
(2.48)
i =1
where i = 1L n , n is the number of cascaded stages, CR is the parasitic capacitance
between the polysilicon resistors and the substrate, LR ,WR , SR are length, width and
area of the polysilicon resistors, respectively, CuaR is the unit area parasitic capacitance
of the resistors, CC is the parasitic capacitance between the capacitor back plate and the
substrate, LC ,WC , SC are length, width and area of the capacitors, respectively, and CuaC
is the unit area parasitic capacitance of the capacitors.
In order to minimize noise coupling from the substrate, the parasitic capacitance
between RF CMOS PPF and the substrate, i.e., Csum , should be minimized. In (2.48),
for a certain CMOS process, CuaC , CuaR are known constants. Thus, what the designer
60
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
can do is to minimize the parasitic capacitance by minimizing the chip area. However,
in practice, the chip area can only be reduced to a certain range, so noise coupling still
exists [99], and in practice, it is impossible to get sufficiently low substrate noise
coupling by only minimizing the chip area. Thus, noise isolation techniques, namely, NWell and guard ring should be applied [99].
For different types of substrates, these noise isolation techniques have different
effects [93]. Categorized by resistivity in the substrates, there exist a high-resistivity
substrate and a low-resistivity substrate in CMOS processes [93]. A high resistivity
substrate is composed of a uniformly doped layer with a resistivity coefficient of 20
Ωcm to 50 Ωcm [93]. A low resistivity substrate consists of a thick, high resistivity
epitaxial layer (10~15 Ωcm) and a low resistivity bulk (1 mΩcm) . Low resistivity
substrates have been widely adopted for desirable latch-up suppression properties [93].
It has been found that at low and medium frequencies typically lower than 5 GHz,
substrates show a resistive behavior [93]. At higher frequencies, the transport patterns
are too complex to be accurately modeled by resistive or resistive-capacitive meshes
[98]. In this dissertation, the operating frequency is well below 5 GHz, thus we use
resistive network to model the substrate. In high-resistivity substrates, distance and
guard rings are effective attenuation techniques to reduce signal interaction [93]. In low
resistivity substrates, the current tends to flow through low impedance paths located
deep in the chip’s lower layers; as a result, guard rings are generally ineffective in
61
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
blocking substrate currents [94]. In low-resistivity substrates the use of a very low
impedance backplate contact is often preferred [99]. From the discussions above, we
can see that the choice of the isolation techniques should be based on the property of a
given CMOS process and the specific application.
2.8 Summary
In this chapter, based on vector analysis, explicit transfer functions of multi-stage
PPFs were derived. Non-idealities in RF CMOS PPFs design, namely, process tolerance,
component mismatch, and parasitic effects, were analyzed. Important contributions and
conclusions in this chapter are:
•
PPFANA, a software tool developed by the author for the simulation of the
quadrature accuracy (QA) of RF CMOS PPFs, which are influenced by process
tolerance and component mismatch, is presented.
•
As is shown that the process tolerance has significant influences on quadrature
accuracy. In order to make an RF CMOS PPF robust against process tolerance,
tolerance design is used.
•
Component mismatch also has an influence on the quadrature accuracy (QA).
Solution for design: use layout technique to minimize component mismatch.
62
Chapter 2 Analysis of Nonidealities in RF CMOS PPFs
•
Parasitic effects in RF CMOS PPFs have been analyzed by using simplified RF
models of resistors and capacitors. Physical design considerations for
minimizing and isolating substrate noise coupling have been proposed.
63
Chapter 3 Design of an RF CMOS Polyphase Filter
Chapter 3
Design of an RF CMOS Polyphase Filter
3.1
Overview
In order to verify the effectiveness of PPFANA, an RF CMOS polyphase filter
(PPF) for a low-IF Bluetooth receiver using Fraunhofer-IMS 0.6 µm CMOS process has
been designed, fabricated and measured. Influences from process tolerances and
component mismatch on the quadrature accuracy of the RF CMOS PPF designed have
been analyzed by using PPFANA. Schematic and post-layout simulation results
obtained by using Cadence SpectreRF have been compared with those obtained by
using PPFANA.
Silicon success of this design lays a solid foundation for this dissertation in the
following two aspects:
•
Physical design considerations for RF CMOS PPFs in a system-on-a-chip (SoC)
solution have been proposed based on the design practice.
•
Design heuristics from this practical design lay the basis for optimization of RF
CMOS PPFs.
64
Chapter 3 Design of an RF CMOS Polyphase Filter
3.2
Schematic Design
3.2.1 Derivation of Specifications
The specifications of RF CMOS PPFs can be derived from the wireless
communication standards and the selected transceiver architecture. In our design of a
Bluetooth receiver, a low-IF receiver architecture has been chosen because it
incorporates advantages over heterodyne and homodyne receivers, providing high
integratebility.
The target specification related to the RF CMOS PPFs in the receiver is the
desired image reject ratio (IRR), which can be obtained from system level simulation. In
our low-IF Bluetooth receiver, the IRR is required to be greater than or equal to 36 dB
in the frequency band of Bluetooth, that is, between 2.4 GHz and 2.48 GHz. The
relationship between the IRR, the gain mismatch ∆A and the phase mismatch ∆φ can
be expressed as follows [84]:

∆A 2
∆φ 2 
RIR = -20 log10 
+

 16( A + A ) 2
4 
I
Q

(3.1)
In (3.1) RIR is the IRR, and AI and AQ are the voltage gain of the I path and Q path
output signal.
Setting out from (3.1), the main specifications of the RF CMOS PPF, i.e. the gain
mismatch and phase mismatch have been derived. In order to compare these target
specifications and the simulation results obtained by using Cadence SpectreRF and
PPFANA, the gain mismatch is expressed as:
65
Chapter 3 Design of an RF CMOS Polyphase Filter
mgain = -20log10 ∆A
(3.2)
Detailed specifications of this RF CMOS PPF are listed in the Table 3.1.
Frequency band / GHz
mgain / dB
∆φ / degrees
2.4-2.48
≤ 0.6
≤5
RF CMOS PPF
Table 3.1 Specifications of RF CMOS PPF for a low-IF Bluetooth receiver.
3.2.2 Design Guidelines
In 2001, Behbahani [62] proposed the following guidelines for the design of RF
CMOS PPFs:
[Step1] Calculate the number of cascaded stages required in the PPF from the
targeting image reject ratio (IRR) and the fractional bandwidth over which this
rejection is required.
[Step2] As an initial guess, place the two lowest and highest poles at the
boundaries of the rejection band. Space the remaining poles equally on the
logarithmic frequency axis. The actual pole locations are fine-tuned by
simulations.
[Step3] Specify the matching between the resistors and between the capacitors
based on the desired IRR. This determines the physical area of the filter.
66
Chapter 3 Design of an RF CMOS Polyphase Filter
[Step4]
Large resistors will lower the power consumption of the amplifier
driving the PPF input. Use the largest filter resistance, limited either by the
maximum noise at the output or by the cut-off frequency. In low-noise or low
frequency circuits, the first is the limiting factor, while in high-frequency PPFs
it is the second.
[Step5] To lower cascaded loss, taper down the resistance of the PPF stages
toward the input. The resulting impedance of the input stage of the PPF will
determine the drive requirements on the amplifier prior to the filter.
[Step6] Design the driving amplifier. If the cascade filter loss is still too large,
insert interstage amplifiers to preserve signal dynamic range within the PPF;
the collinear common-mode signal at each output shifts the zero crossings and
corrupts the final phase relationship among the outputs. Thus, the outputs must
be sensed using linear differential circuits that reject the common mode.
Though rather comprehensive, an important design consideration is missing in
these design guidelines, that is, process tolerance should be considered in selecting
lower and upper corner frequencies for the RF CMOS PPFs. The importance of
tolerance design in design of RF CMOS PPFs has already been analyzed in the previous
chapter 1; in this chapter we apply it to a practical design.
67
Chapter 3 Design of an RF CMOS Polyphase Filter
3.2.3 Tolerance Design
In the design of RF CMOS PPFs, to guarantee that the worst-case circuit
performance still satisfies the specifications, process tolerances should be considered by
using the concept of ‘tolerance design’ [102]. Specifically, by enlarging the bandwidth
of the PPF, the worst-case frequency range of the PPF can be guaranteed to be located
in a given operating frequency band. In the Fraunhofer-IMS 0.6 µm CMOS process,
sheet resistance of the polysilicon resistors used in this design varies from 36 Ω/square
to 50 Ω /square, while the nominal value is 43 Ω/square. The unit capacitance of the
poly_N+ capacitor in this process varies from 1.1 fF/µm 2 to 1.7 fF/µm 2 , whilst the
nominal value is 1.4 fF/µm 2 .
It is well known that the relationship between the corner frequency f corner of an
RC filter and the values of resistors and capacitors is given by
f corner =
1
.
2 πRC
(3.3)
From (3.3), we get the expression of RC time constant as follows:
RC =
1
,
2 πf corner
(3.4)
where R is the resistors value, C is the capacitor value, and f corner is the corner
frequency of this RC filter. For convenience of calculation of the RC time constant with
tolerance design, we define the RC time constant variation ratio
σ RC =
( R + ∆R)(C + ∆C )
× 100%
RC
68
(3.5)
Chapter 3 Design of an RF CMOS Polyphase Filter
In (3.5) ∆R and ∆C are the process variations of the resistors and capacitors,
respectively. Setting out from the Fraunhofer-IMS 0.6 µm CMOS process, we find σ RC
equal to 41.2%. In order to overcome the influence of σ RC , we need to decrease the
lower RC time constant, determined by higher corner frequency of the RC filter, and
increase the higher RC time constant, determined by lower corner frequency, by 41.2%
respectively. The lower corner frequency is set to 1.44 GHz, and the higher corner
frequency is set to 3.47 GHz with tolerance design of RF CMOS PPFs.
3.2.4 Analytical Calculations
In the schematic design, constraints in physical layout should also be considered
to avoid unreasonable circuit sizing. In the design of RF CMOS PPFs, the maximum
length of resistors, Lmax,res , allowed in a given CMOS process, and minimum
components area, required by a given standard deviation of the component mismatch,
need to be calculated.
•
Calculation of maximum resistor length. As discussed in section 2.7.1, at
radio frequency, parasitic effects have significant influences on the circuit
performance. By using the simplified model of RF resistors, the maximum
length of the resistor, Lmax,res , allowed in a given CMOS process has been
derived, and it is expressed in equation (2.42). Using this equation, at 2.45 GHz,
the maximum length Lmax,res of the polysilicon resistor in the Fraunhofer-IMS
0.6 µm CMOS process is equal to 132 µm.
69
Chapter 3 Design of an RF CMOS Polyphase Filter
•
Calculation of minimum component area. In the design of RF CMOS PPFs,
component matching is a critical factor to be considered. From Monte Carlo
simulations [73], it has been found that
RUIR > -20log10 (σ Component ) ,
(3.6)
where RUIR is the ultimately obtainable image reject ratio, σ component is the
variance of the component mismatch. It is known from the experimental study
[73] that the variance of adjacent on-chip resistors and capacitors is
proportional to the inverse of their surface area. Measurement results in
Fraunhofer-IMS 0.6 µm CMOS process show that for two neighboring
polysilicon resistors with width of 20 µm and length of 250 µm, the maximum
mismatch is 1%, for two neighboring poly-N+ capacitors with 300 µm 2 , the
maximum mismatch is 0.5%.
In our design, the target specification of the image reject ratio (IRR) is 36 dB.
Calculated by using equation (3.6), a component mismatch of 1% is needed to satisfy
this target specification. Based on the guidelines and the hand calculation results, the
range of length, width of the resistor and the value of the capacitor were calculated as a
preparation for the physical layout. The schematic of the RF CMOS PPF is shown in
Figure 3.1.
This design was based on intensive trials and it took several iterations between
schematic design and layout to obtain proper tradeoffs among design parameters, i.e.,
70
Chapter 3 Design of an RF CMOS Polyphase Filter
quadrature accuracy, signal loss, and chip area. The circuit dimensions are shown in
the following Table 3.2.
R0
R1
R2
I_LO+
C0
C1
C2
R0
R1
R2
LO+
Q_LO+
C0
C1
C2
R0
R1
R2
I_LO-
C0
C1
C2
R0
R1
R2
LO-
Q_LOC0
C1
Stage 1
Stage 2
C2
Stage 3
Figure 3.1 Schematic of a 3-stage RF CMOS PPF for a low-IF Bluetooth receiver.
Stage
Resistors
Value
Capacitors
Value
1
R1
217.6 Ω
C1
229.4 fF
2
R2
283.6 Ω
C2
229.4 fF
3
R3
347.6 Ω
C3
229.4 fF
Table 3.2 Dimensions of a 3-stage RF CMOS PPF for a low-IF Bluetooth receiver.
71
Chapter 3 Design of an RF CMOS Polyphase Filter
3.2.5 Schematic Simulation
Figure 3.2 Simulation results of a 3-stage RF CMOS PPF obtained by using SpectreRF
Figure 3.3 Simulation results of a 3-stage RF CMOS PPF obtained by using PPFANA.
72
Chapter 3 Design of an RF CMOS Polyphase Filter
mgain /dB
d phase /degrees
PPFANA
0.012
90.063
SpectreRF
0.010
90.061
Table 3.3 Simulation results of a 3-stage RF CMOS PPFs
obtained by using PPFANA and SpectreRF.
The simulation results obtained for the schematic by using Cadence SpectreRF
and PPFANA are shown in Figure 3.2 and Figure 3.3, respectively. Accurate results are
listed in Table 3.3. From figures and table above, we can see that the simulation results
match; the gain mismatch is approximately 0.01 dB and the phase difference is
approximately 90.06 degrees, measured between 2.4 GHz and 2.48 GHz, and ignoring
computational errors. As mentioned in the second chapter, the phase mismatch is phase
difference minus 90 degrees, thus phase mismatch in this RF CMOS PPF is
approximately 0.06 degrees. Compared with Table 3.1, we can see that the target
specifications are met in the schematic simulation.
3.3
Physical Layout Design
3.3.1 Layout Techniques
As discussed in the chapter 2, in the physical design, special attention must be
paid to the layout of the resistors and capacitors. The layout of the four resistors in one
73
Chapter 3 Design of an RF CMOS Polyphase Filter
stage of the RF CMOS PPF designed by the author is shown in Figure 3.4. To obtain
good matching, the following techniques are applied:
• Identical structures, including end contacts, are used for all the resistors.
• Resistors are placed very closely.
• All of the resistors have the same orientation.
• A common-centroid structure is used [77].
• Dummy resistors are used to minimize the boundary effects.
Dummy resistor
Resistor 1
Resistor 2
Resistor 3
Resistor 4
Dummy resistor
N-Well
Guard rings
Figure 3.4 Layout of the four resistors in one stage of an RF CMOS PPF.
The layout of the four capacitors in one stage of the RF CMOS PPF designed by
the author is shown in Figure 3.5. In order to obtain good matching, the following
techniques are applied:
74
Chapter 3 Design of an RF CMOS Polyphase Filter
• Identical structures, including terminals, are used for all capacitors.
• All of the capacitors to be matched are placed very closely.
• Capacitors to be matched are placed with the same geometrical orientation.
• A common-centroid structure is used [77].
• Dummy capacitors are placed around the matched array to minimize the
boundary effects.
• Input and output wires are placed as symmetrically as possible.
Guard ring
Capacitor 3
Capacitor 1
Wires
Capacitor 2
Capacitor 4
Note: Surrouding capacitors are dummy capacitors
Figure 3.5 Layout of the four capacitors and wires in one stage of an RF CMOS PPF.
Following the layout techniques mentioned above, component mismatch in RF
CMOS PPFs can be control well. The latest component mismatch model in [81] can be
75
Chapter 3 Design of an RF CMOS Polyphase Filter
used to estimate the standard deviation of the mismatch in the layout. Thus a layout of
this design is taken as the layout pattern used in the chapter 4.
3.3.2 On-Chip Noise Isolation Techniques
As discussed in chapter 2, in the design of RF CMOS PPFs, special attentions
must be paid to on-chip noise isolation techniques to reduce unwanted substrate noise
coupling. On-chip shielding should be used to protect the sensitive RF CMOS PPFs
from noise generated by other circuits. The CMOS process used in this design, the
Fraunhofer-IMS 0.6 µm CMOS process, is a P-substrate, single N-well process. In our
application, N-well and a guard ring are used.
N-Wells: N-Well is an epitaxial layer fabricated on the P-substrate to
differentiate it from the substrate. In an N-well process, i.e., a process with P-substrate
and N-well, the N-well is originally used for the fabrication of P-transistors. With
proper biasing, the N-well can be used to provide isolation between a component and
the substrate.
As shown in Figure 3.4, resistors in this design are protected by N-wells, the
region with grids. These resistors are built in the shallow P-region within the buried Nwell. A depletion layer exists on both sides of the N-well and this provides dielectric
isolation between the active devices and the bulk region. Since in our application the Nwell is connected to the low-impedance positive supply, the N-well performs a Farady
76
Chapter 3 Design of an RF CMOS Polyphase Filter
shield around the resistors. Thus, resistors are protected against noise coupling by Nwell.
However, the N-well is not well suited to use for the capacitors in this design,
because the bottom plate of the poly-N+ is already in the N+ region. Despite this fact,
poly-N+ capacitors are the best capacitor in the sense of linearity in the Fraunhofer-IMS
0.6 µm CMOS process.
Guard Rings: The principle of guard ring, also named as guarding band, is to
use a narrow ring of P+ or N+ with stable and noise-free connections forming a local
clean zone to reduce noise coupling through the substrate. In the Fraunhofer-IMS 0.6
µm CMOS process, guard rings can be connected to either ground or Vdd. A grounded
guard ring uses a low-resistance P+ area to connect to ground. A guard ring that
connects to Vdd uses an N-well and N+ region on the substrate. Positive N+
connections attract electrons, and the grounded P+ connections attract holes. Thus, they
can provide a barrier against substrate noise coupling. As shown in Figure 3.4 and
Figure 3.5, a P+ guard ring and a N+ guard ring are used in the layout of the resistors, a
P+ guard ring is used in the layout of capacitors, and these guard rings are placed as
close as possible to the resistors and capacitor.
3.3.3 Layout and Post-Layout Simulation
In order to test this chip independently, test pads are added to the layout of the RF
CMOS PPF; shielded ground pads (Abschirm_GND) are used to avoid cross-coupling
77
Chapter 3 Design of an RF CMOS Polyphase Filter
between different paths and wiring in the layout is kept as symmetrical as possible to
minimize additional mismatch. The final layout of the RF CMOS PPF with test pads is
shown in Figure 3.10.
Test pads
Stage 3 Stage 2 Stage 1
Decoupling capacitor
Figure 3.6 Layout of a 3-stage RF CMOS PPF with test pads.
mgain / dB
∆φ / degrees
0.071
0.26
0.27
1.8
Cadence postlayout simulation
PPFANA
Table 3.4 Simulation results obtained by using Cadence post-layout simulation and PPFANA.
78
Chapter 3 Design of an RF CMOS Polyphase Filter
After the layout, post-layout simulation (PS) of the RF CMOS PPF with
coefficient extracted networks was done by using Cadence SpectreRF. In this extraction
method, all kinds of parasitic resistances and capacitances, such as, parasitics between
neighboring components, wires, and parasitics to the substrate, are extracted. In this
way, we can provide realistic simulation results. Unfortunately, by this simulation
method, the influences from process tolerances and component mismatching cannot be
easily simulated. In this situation, PPFANA, the software the author developed in this
dissertation shows its advantages over Cadence post-layout simulation. The simulation
results by using PPFANA and Cadence post-layout simulation are shown in Table 3.4
for the purpose of comparison.
3.3.4 Chip Photo
`
Test pads
Stage 3
Stage 2
Stage 1
Decoupling capacitor
Figure 3.7 Chip photo of an RF CMOS PPF for a low-IF Bluetooth receiver.
79
Chapter 3 Design of an RF CMOS Polyphase Filter
A chip photo of the fabricated chip is shown in Figure 3.7. The squares in the top
of this photo are test pads; in the middle of the bottom part is the RF CMOS PPF
designed for a low-IF Bluetooth receiver.
3.4
Measurements
3.4.1 Introductory Remarks
As an RF CMOS PPF has three ports, namely one input port and two output ports,
and the wafer station in Fraunhofer-IMS can only measure components with two ports,
a printed circuit board (PCB) providing interfaces to the measurement equipments was
designed and manufactured. The fabricated chip was bonded to the test PCB to allow
measurements.
3.4.2 Modeling of Test Pads and Bonding Wires
As test pads and bonding wires have parasitic resistance, inductances and
capacitance, in order to simulate the influences from these parasitic components, a
model of test pads and bonding wires shown in Figure 3.8 was built by using the
Cadence package physical geometrical modeling tools. Both schematic simulation and
post-layout simulation of RF CMOS PPF with test pads and bonding wires show that
the parasitic components have no significant influences on the gain mismatch and the
80
Chapter 3 Design of an RF CMOS Polyphase Filter
phase mismatch of the RF CMOS PPF, because they are in a fully balanced
configuration and their layouts are symmetrical.
R_ESD
R_Bond
BeforePad
r=99
r = 200m
L_Bond
∩∩∩∩
AfterBond
l = 800p
f =2.45GHz
c = 183f
r = 174
C_Pad
R_Pad
gnd
Figure 3.8 Model of test pads and bonding wires.
3.4.3 Design of the Test PCB
The functionality of the test PCB is to provide an interface to the RF
measurement equipment. Specifically, the test PCB includes two types of conversions,
i.e., impedance conversion and differential to single-end conversion. A schematic of the
test PCB is shown in the Figure 3.9. From this schematic, we can see that the
differential input of the RF CMOS PPF is accomplished by using a balance-unbalance
converter (BALUN), and the differential outputs of I path and Q path are converted by
81
Chapter 3 Design of an RF CMOS Polyphase Filter
BALUNs to single-ended outputs respectively. In order to get maximum output power
gain, resistors and capacitors with optimized values are connected to the outputs as
loads.
gnd
r = 250 c = 240f
SMA
2:1
I_LO +
⊃
⊂
TOKO Balun
⊃
⊂
⊂
⊃
⊃⊂ 617DB-1023
in
I_LOgnd
gnd
SMA
1:2
in
TOKO Balun
617DB-1023
LO +
f_lo = 2.449GHz
⊃
⊂
⊃
⊂
i_DC
⊃
⊂
⊃⊂ = 0
r = 250 c = 240f
gnd r = 250 c = 240f
LO-
SMA
gnd
Q_LO +
2:1
⊃
⊂
TOKO Balun
⊃
⊂
⊃
⊃⊂
⊂ 617DB-1023
in
Q_LOgnd
gnd
r = 250 c = 240f
Figure 3.9 Schematic of the printed circuit board (PCB) for test of the RF CMOS PPF.
In Figure 3.10, the photo of the fabricated test PCB is shown. The metal
connectors are the SMA connectors. The black block in the center of the PCB is the
fabricated chip, which is bonded to the PCB.
82
Chapter 3 Design of an RF CMOS Polyphase Filter
Figure 3.10 Photo of the fabricated test PCB.
3.4.4 Measurement Setup
In the measurement setup by the author, an Agilent E8267C PSG vector signal
generator and a Lecroy wavermaster digital oscilloscope were used to measure the
performance of the fabricated chip. A diagram of the measurement setup is shown in
Figure 3.11.
The Agilent E8267C PSG vector signal generator generates a 2.45 GHz
continuous sinusoidal signal, which is input to the test PCB via RF cable. Then the I (inphase) path and Q (quadrature-phase) path were connected to the Lecroy wavemaster
digital oscilloscope via an RF cable. A feature of this Lecroy wavemaster digital
83
Chapter 3 Design of an RF CMOS Polyphase Filter
oscilloscope is its embedded real time fast Fourier transform (FFT). With this digital
oscilloscope, time domain and frequency domain measurements of the two port outputs
can be conducted in real time.
Agilent E8267 PSG
Vector signal generator
In
Out
P_in
RF CMOS PPFs
Test PCB
RF Cable
I_out
Q_out
RF Cable
RF Cable
P_out
P_in
P_out
In1
In2
Lecroy Wavemaster
Digital oscilloscope
Power Supply
Figure 3.11 Diagram of the measurement setup for test of the RF CMOS PPF.
3.4.5 Measurement Results
Two types of measurements, namely, time domain and frequency domain
measurements, on RF CMOS PPFs have been conducted in this work. Time domain
measurement results are shown in Figure 3.12. From the time domain signal, we can see
that the gain mismatch is approximately 0.17 dB, and the phase difference is
approximately 88.78 degrees, consequently, the phase mismatch is approximately 1.22
degrees.
84
Chapter 3 Design of an RF CMOS Polyphase Filter
Figure 3.12 Time domain measurement results of the RF CMOS PPF.
As time domain measurement is observed at only that time interval, it has
significant measurement inaccuracy in the RF domain.
85
Chapter 3 Design of an RF CMOS Polyphase Filter
In order to get more accurate measurement result, frequency domain
measurement was conducted.
Figure 3.13 Gain mismatch of the RF CMOS PPF measured in frequency domain.
86
Chapter 3 Design of an RF CMOS Polyphase Filter
The frequency domain measurement results of the gain mismatch and the phase
mismatch in the RF CMOS PPF are shown in Figure 3.13 and Figure 3.14 respectively.
The measurement results are 0.45 dB gain mismatch and 3.2 degrees phase mismatch.
Figure 3.14 Phase difference of the RF CMOS PPF measured in frequency domain.
87
Chapter 3 Design of an RF CMOS Polyphase Filter
3.5
Analysis of Simulation and Measurement Results
By comparing schematic simulation results obtained by using PPFANA and
Cadence SpectreRF, which are shown in Table 3.3, we can see that they are in good
accordance with each other; the effectiveness of PPFANA is thus verified. From the
schematic and post-layout simulation of RF CMOS PPFs with test pads, we have found
that test pads do not have significant influences on the quadrature accuracy of RF
CMOS PPFs because they have fully balanced configuration and symmetrical layout,
whereas bonding wires, external components, i.e., load resistors and capacitors, and
BALUN contribute certain mismatch.
By comparing the post-layout simulation results of RF CMOS PPFs obtained by
using PPFANA and Cadence SpectreRF and the measurement results, we can see that
the simulation results obtained by using PPFANA are closer to the measurement results
because the influences from process tolerances and component mismatch are simulated.
In this situation, PPFANA provides a useful design aid for RF CMOS PPF design.
3.6
Summary
This chapter concentrates on a practical design of an RF CMOS PPF for a low-IF
Bluetooth receiver. Detailed design considerations, analytical calculations and layout
techniques for minimizing component mismatch were analyzed.
88
Chapter 3 Design of an RF CMOS Polyphase Filter
The fabricated chip was bonded to the test PCB. Measurement results show that
PPFANA developed by the author provides more realistic simulation results than the
commercially available Cadence tools. Further contributions of this chapter are the
following:
1. Silicon success of an RF CMOS PPF for a low-IF Bluetooth receiver was
presented.
2. Design heuristics from design practice reported here lay a solid foundation for
the optimization of RF CMOS PPFs.
3.
The effectiveness of PPFANA was demonstrated.
89
Chapter 4 Optimization of RF CMOS Polyphase Filters
Chapter 4
Optimization of RF CMOS Polyphase Filters
4.1
Introduction
In the open literatures, there are some papers on analysis [55-60] and design[5568] of RF CMOS polyphase filters (PPFs). For instance, Behbahani [62] analyzed some
practical aspects in the design of RF CMOS PPFs and presented some design guidelines.
It is known that the main specification of the RF CMOS PPFs, quadrature accuracy, is
layout dependent. The requirements from component matching are contradictory to that
of minimization of noise coupling, signal loss and chip area [ZHA2]. Thus, a tradeoff
must be made in a specific design [ZHA2]. By simply following Behbahani’s guidelines
[62], it takes intensive trials and many design iterations to reach a good design [ZHA3].
Furthermore with layout considerations, optimization of RF CMOS PPFs cannot be
accomplished by using analytical calculations because of many constraints and tradeoffs
in the design [ZHA3]. In this situation, a computer-aided analysis and design
optimization tool for RF CMOS PPFs is of great help to the community of RF CMOS
PPFs designers, especially, for industrial projects, which are under time-to-market
pressures. Unfortunately, until now, there is not any such dedicated tool available
openly, to the author’s best knowledge. In this chapter, the author will present a novel
optimization tool for RF CMOS PPFs, based on MATLAB. The tool, which was
90
Chapter 4 Optimization of RF CMOS Polyphase Filters
developed by the author, is termed PPFOPTIMA and it takes layout constraints
explicitly into considerations [ZHA3]. A genetic algorithm (GA) is proposed as the
optimization engine of PPFOPTIMA for its capability in avoiding local optima. In order
to enable users to carry out an automated design, PPFOPTIMA provides automated
synthesis and verification incorporating dedicated worst-case and Monte Carlo
simulation options. Experimental results illustrated in this chapter demonstrate that
PPFOPTIMA is effective and efficient in the synthesis, the optimization and the
verification of RF CMOS PPFs.
4.2
Design Considerations and Design Flow
As analyzed in chapter 2, CMOS process tolerances, component mismatch and
parasitic effects should be considered in the design of RF CMOS PPFs in order to
accomplish a robust design. Design considerations and heuristics from our design
practice in chapter 3 will be discussed in this section.
CMOS process tolerances should be considered by using the concept of
‘tolerance design’ [102]. Specifically, by enlarging the bandwidth of the PPF, the
operating frequency can be covered by upper and lower corners frequency of the RF
CMOS PPFs, no matter how they shift with process variations [102].
91
Chapter 4 Optimization of RF CMOS Polyphase Filters
The most important specification for the design of an RF CMOS PPF is the
image reject ratio (IRR), which is related to the sensitivity of a receiver. From Monte
Carlo (MC) simulations, it has been found that [59]
RUIR > -20log10 (σ Component ) ,
(4.1)
where RUIR is the ultimately obtainable IRR, and σ component is the standard deviation of
the component mismatch. Typically, RUIR is in the range of 25 dB to 60 dB for RF
CMOS PPFs [ZHA3]. The standard deviation σ component typically between 3% and 0.5%
[81]. It is known from the experimental study [73] that the variance of adjacent on-chip
resistors and capacitors and their area are reciprocally proportional, that is, neighboring
components with larger area have better matching than those with small area [73]. By
using the latest resistor and capacitor mismatch models [81] of (4.2) and (4.3), the
minimum chip area required by a target specification of the IRR can be calculated
[ZHA3].
The standard deviation σ component of the component mismatch between two
neighboring resistors with identical layout can be calculated as follows [81]:
2
σ
2
∆R/R
=σ
2
R sh
2
2




L
L
L
2 
2 

 + σ Re 
 + σ ∆W 
 ,
 (W + ∆W ) 
 ( L ⋅ Rsh + Re ) 
 ( L ⋅ Rsh + Re ) 
(4.2)
with the standard deviation of the sheet resistance of the polysilicon resistors Rsh given
by
σ R sh =
AR sh
(WL)1 / 2
with the standard deviation of the resistor width
92
,
(4.3)
Chapter 4 Optimization of RF CMOS Polyphase Filters
σ ∆W =
A∆W
,
W 1/ 2
(4.4)
and with the standard deviation of the end resistance coefficient Re defined as
σ R = AR ,
e
e
(4.5)
In (4.2) to (4.5), W and L are the resistor width and the resistor length, and ∆W is the
resistor width offset, ARsh , A∆W , and ARe are fitting parameters obtained from massive
measurement results of σ Rsh , σ ∆W , σ Re , respectively.
For two neighboring capacitors with identical layout and capacitance C, the
variance σ ∆2c / C of the component mismatch between them can be calculated as follows
[81]:
2
σ ∆c/C
= σ p2 + σ a2 + σ d2 ,
(4.6)
where σ p denotes the standard deviation of the periphery, given by
σp =
fp
,
C 3/4
(4.7)
with the standard deviation caused by area variations defined as
σa =
fa
,
C 1/2
(4.8)
and with the standard deviation caused by distance fluctuations
σ d = f d .d ,
(4.9)
In (4.7) to (4.9), f p , f a , f d are constants describing the influences of periphery, area
and distance fluctuations.
In the component mismatch model of capacitors, the
periphery component models the effect of edge roughness, and it is most significant for
small capacitors, which have a relatively large amount of edge capacitance. The area
93
Chapter 4 Optimization of RF CMOS Polyphase Filters
component models the effect of short-range dielectric thickness variation, and it is most
significant for moderate size capacitors. The distance component models the effect of
global dielectric thickness variation across the wafer, and it becomes significant for
large capacitors or widely spaced capacitors [81].
Optimized for digital circuits, parasitic effects in CMOS processes are severe for
analog design. In the design of RF CMOS PPFs, there are two influences from parasitic
effects, namely [ZHA3]
• signal loss in addition to the theoretical 3 dB/stage caused by the lossy
substrate and
• capacitive noise coupling caused by parasitic capacitances.
For a given CMOS process, there are two ways to minimize the influences from
parasitic effects, one is to reduce the chip area as derived in Chapter 2 to decrease
capacitive noise coupling, the other is to use on-chip noise isolation techniques, namely
N-wells and guard rings, to protect noise sensitive analog circuits [ZHA2].
With considerations of parasitics, the maximum length of the resistor (MLR)
allowed in a CMOS process can be calculated by using the following equation [ZHA2]:
Lmax,res =
1
2 πf op Rsh Cp
(4.10)
where Lmax,res is the MLR, f op is the operating frequency of the RF CMOS PPF, Rsh is
the sheet resistance, and Cp is parasitic capacitance per unit area between resistor and
the substrate. Before accomplishing physical layout, the MLR must be calculated to
avoid an inappropriate physical design.
94
Chapter 4 Optimization of RF CMOS Polyphase Filters
Start
Specifications
RF CMOS PPF with 2 cascaded stages
Calculate minimum area of resistors and capacitors
Calculate values of resistors with
considerations of process tolerances
Increase the number of
cascaded stages by 1
Worst-case schematic simulation
No
Satisfy specifications?
Yes
Calculate maximum lengths of resistors
Tradeoff between lengths and widths of resistors
Physical layout
Worst-case post-layout simulation
No
Satisfy specifications?
Yes
End
Figure 4.1 A design flow for specification-driven automated synthesis of RF CMOS PPFs.
95
Chapter 4 Optimization of RF CMOS Polyphase Filters
Based on aforementioned design considerations and the design heuristics, a
design flow for specification-driven automated synthesis of RF CMOS PPFs, as shown
in Figure 4.1, is proposed in this chapter. From Figure 4.1, we can see that all the design
considerations and constraints we have just discussed are included in this design flow.
Setting out from a given target specification, we start with a 2-stage RF CMOS PPF,
going through schematic design. If in the schematic simulation, the target specification
cannot be met, we move on to an RF CMOS PPF with three cascaded stages, and
deduce the compliance with the target specification. If we fail again, an RF CMOS PPF
with four cascaded stage is taken. After completing the schematic design, we carry out
the physical layout design and check whether the target specification can be met in the
post-layout simulation. If we fail, we go back to the schematic design using an RF
CMOS PPF with more cascaded stages, until the target specifications are met in the
post-layout simulation. Then we finish this design.
4.3
Problem Formulation for RF CMOS PPFs Optimization
In design of an RF CMOS PPF, the design objectives are
• to satisfy the target specifications required by a given wireless communication
standard and transceiver architecture,
• to minimize the chip area,
• to maximize the immunity to noise coupling, and
96
Chapter 4 Optimization of RF CMOS Polyphase Filters
• to minimize additional signal losses.
Based on these design objectives, we can categorize the design considerations by
objective functions or constraints. Objective functions are the image reject ratio (IRR)
and the chip area. Here we use the relationship between the IRR RIR , the gain mismatch
mgain and the phase mismatch ∆φ as shown in the following equation (4.11) [84].
RIR = -20log10 (
2
mgain
16( AI + AQ )
+
∆φ 2
),
4
(4.11)
where the gain mismatch is defined by
mgain = AI − AQ ,
(4.12)
and where the phase mismatch is defined by
∆φ =|| φI − φQ | −90o | .
(4.13)
In (4.11) to (4.13), AI , AQ , φI , φQ are amplitudes and phases of the I path and the Q
path, respectively. All of these parameters can be calculated by using the transfer
functions of RF CMOS PPFs derived in chapter 3 of this dissertation.
In order to minimize the component mismatch, the layout pattern shown in Figure
3.4 and Figure 3.5 should be followed in the layout of resistors and capacitors. In this
layout pattern, the chip areas of an RF CMOS PPF with n cascaded stages is given by
[ZHA3]
n
S = ∑ (6 S Ri + 16 S Ci ) ,
i =1
(4.14)
where the chip area of resistors in the ith stage is
S Ri = LRi ⋅ WRi ,
97
(4.15)
Chapter 4 Optimization of RF CMOS Polyphase Filters
and where the chip area of the capacitor in the ith stage is
SCi =
Ci
Cunit ,
(4.16)
In (4.14) to (4.16), LRi , WRi are the length and width of resistors in the ith stage
respectively, Ci is the capacitor value in the ith stage, Cunit is the unit capacitance of the
capacitors, i=1,2,3,…n, n is the number of the cascaded stages in an RF CMOS PPF.
Though the chip areas defined in (4.14) is an objective to be minimized for the
reduction of substrate noise coupling, for a given target specification of the IRR and a
given CMOS process, there exist a minimum area of resistors and capacitors to avoid
bad component mismatch which cannot satisfy the specification of the IRR. Using the
equation (4.1), we find the corresponding standard deviation of resistors and capacitors
for a given target specification of IRR:
σ ∆R/R > 10- RIR / 20 ,
(4.17)
σ ∆C/C > 10- RIR /20 .
(4.18)
In (4.17) and (4.18), RIR is the IRR and σ ∆R/R , σ ∆C/C are standard deviations of
matched resistors and capacitors, respectively. These figures can be calculated by using
the latest component mismatch models proposed by Cheng [81] as shown in (4.2) and
(4.6). Another constraint in practice is that the length of resistors should be less than the
MLR calculated by using (4.10).
In order to choose a suitable optimization algorithm, let us investigate the
relationship between the IRR and component values. For the purpose of simplification,
we take the amplitude mismatch between the I path and the Q path in a 2-stage RF
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Chapter 4 Optimization of RF CMOS Polyphase Filters
CMOS PPF for a Bluetooth receiver as an example. In the design of this 2-stage RF
CMOS PPFs, if we determine the values of the resistors in each stage, then the
capacitors values are automatically determined by the relationship between corner
frequency and RC time constant shown in (3.4). In this situation the amplitude
mismatch ∆A can be expressed as:
1+
∆A =
ω2
ω1ω2
2

ω2   ω ω
R1 ω  
 1 −
+2

 + +
R2 ω2  
  ω1ω2   ω1 ω2

2
1/ 2
(4.19)
Where ω is the frequency of interest; ω1 and ω2 are the upper and the lower corner
frequency of this RF CMOS PPF, which are 2.4 GHz and 2.48 GHz respectively; R1
and R2 are resistors values in the first stage and the second stage, which are in the range
of 50 Ohms to 500 Ohms. Taking these into account, we can rewrite (4.19) as
1 + x2
z= 4
.
x − 2 x2 + 2x + 2 y
(4.20)
Where z is ∆A ; x is the normalized frequency of interest, which is in the range of [0.5,
5]; and y is the ratio between R1 and R2 , which is in the range of [0.5, 5]. The threedimensional plot of ∆A is shown in Figure 4.2. From this plot, we can clearly see that it
has several maxima. This means that, the function in equation (4.20) has multiple local
optima. For RF CMOS PPFs with more cascaded stages, the relationship between
quadrature accuracy, component values, and frequency of interest is even more complex.
In this situation, an in-depth analysis of possible optimization methods is necessary.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.2 A three-dimensional plot of the amplitude mismatch
in a 2-stage RF CMOS PPF, expressed in the equation (4.20).
4.4
Analysis of Possible Optimization Methodologies
4.4.1 Limitations of Conventional Optimization Methods
There are various optimization theories and techniques in engineering
mathematics [104]. Essentially, the targets of the optimization techniques are to search
for the maximum or the minimum values in a certain search space [104].
The easiest way to find the accurate maximum or minimum in a continuous
function is to use calculus-based search [104], in which the existence of the first order
and the second order derivatives are required [104]. However, this technique is not
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Chapter 4 Optimization of RF CMOS Polyphase Filters
suitable for solving the optimization problem in this dissertation for the following
reasons:
• Tendency to get trapped on local maxima. Even though a better solution may
exist, all moves from the local maxima seem to decrease the fitness of the
solution.
• Dependence on the existence of derivatives.
There is another optimization technique, gradient descent [104], which is also
referred to as hill-climbing [104]. This technique performs well on functions with only
one peak. However, for functions with multiple peaks, the first peak found is taken as
the maximum, no matter whether it is the highest peak or not [104]. Thus, hill-climbing
is not suitable for our optimization problem either.
For optimization problems with multiple peaks or multiple valleys, the direct way
is to use random search to find the optima. Unfortunately, for most of the real-life
optimization problems, this method is inefficient because of the vast amount of
computations and is therefore not selected by the author.
In order to save computational complexity, iterated climbing was proposed [104],
which is a combination of random search and gradient search. Once a peak has been
located, the hill-climbing is started again, but with another randomly chosen starting
point. However, since each random trial is performed in isolation, no overall idea of the
topology and photography of the search space is obtained. Furthermore, trials are
randomly allocated over the entire search space; many points in regions of low fitness
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Chapter 4 Optimization of RF CMOS Polyphase Filters
are evaluated as points in high fitness regions. Therefore, iterated climbing is not
considered as a proper candidate for the optimization of RF CMOS PPFs and is
therefore not selected by the author.
4.4.2 Simulated Annealing
Simulated annealing (SA) was invented by Kirkpatrick [109] in 1982 by
mimicking the process of cooling of a solid. Essentially, it is a modified version of hillclimbing. The search process starts from a random point in the search space. Then, a
random move is made. In the case of climbing reaches a higher point by this move, the
move is accepted. Otherwise, it is accepted only with probability p(t) (where t is time).
The probability function p(t), essentially relates to the temperature. A value of p(t) close
to one refers to a high temperature. Decreasing p(t) means lowering the temperature. At
time t equal to zero, p(0) is approximately one. As time t passes, p(t) is decreased,
approaching zero at the end of the SA operation. This procedure is an analogy with the
cooling of a solid. Therefore, initially, any moves are accepted, but as p(t) decreases, the
probability of accepting a negative move is lowered. This optimization algorithm has
been successfully applied in analog design automation [109-112], especially in IC
layout routing. However simulated annealing deals with only one candidate at a time.
Like random search, it does not build an overall picture of the search space, and no
information from previous moves is used to guide the selection of new moves. This
leads to a low efficiency in solving complicated optimization problems. Therefore, SA
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Chapter 4 Optimization of RF CMOS Polyphase Filters
does not seem promising for the optimization of RF CMOS PPFs and is hence not
selected by the author.
4.4.3 Genetic Algorithms
A. Introductory Remarks
Genetic algorithms (GAs) are adaptive methods based on the genetic processes of
biological organisms. For GA basics, please refer to the famous textbook written by
Goldberg [113]. Here we focus on the analysis of the suitability of GAs to our
optimization problems. The following four key schemes, which characterize GAs,
create the benefit of GAs over conventional optimization techniques and the SA we had
mentioned in the previous subsections.
B. Direct Manipulation of Coding [118]
GAs manipulate decisions or control variable representations at a string level to
exploit similarities among high-performance strings [118]. Other methods usually deal
with functions and their control variables directly. GAs deal with parameters of finite
length, which are encoded using a finite alphabet, rather than directly manipulating the
parameters. This means that the search is constrained neither by the continuity of the
function under investigation, nor by the existence of a derivative function. Moreover, by
exploring similarities in codings, GAs can deal effectively with a broader class of
functions than can many other procedures.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
C. Search from a Population, not a Single Point [118]
In GAs, the search sets out from a population of many individuals, rather than
starting from just one point. This parallelism means that the search will not become
trapped on a local maximum - especially if a measure of diversity-maintenance is
incorporated into the algorithm, even if one candidate may become trapped on a local
maximum, but the need to maintain diversity in the search population means that other
candidates will therefore avoid that particular area of the search space.
D. Search via Sampling, a Blind Search [118]
GAs remains general by exploiting information available in any search problem.
GAs process similarities in the coding with information and rank the structures
according to their survival capabilities in the current environment. By exploiting such
widely available information, GAs may be applied to virtually any problem.
E. Search Using Stochastic Operators, no Deterministic Rules [118]
The transition rules used by GAs are probabilistic, not deterministic. A
distinction, however, exists between the randomized operators of GAs and other
methods that are simple random walks. GAs use random choice to guide a highly
exploitative search.
From the analysis of GA, SA and conventional optimization techniques, we can
see that the GA is well suited for the solving of complicated optimization problems with
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Chapter 4 Optimization of RF CMOS Polyphase Filters
multiple objectives and, consequently, with multiple local optima. Since the
optimization of RF CMOS PPFs has multiple optimization objectives and multiple local
optima, conventional optimization algorithms are not recommendable. Therefore, the
GA was chosen for the optimization engine in PPFOPTIMA, the optimization tool for
RF CMOS PPFs developed by the author of this dissertation.
4.5
MATLAB Implementation of Genetic Algorithms
4.5.1 Overview
In this section, the MATLAB implementation of genetic algorithms is presented.
MATLAB is chosen as the programming platform for the following reasons:
• It provides many built-in auxiliary functions useful for function optimization.
• It is efficient for numerical computations.
4.5.2 Chromosome Representation
The process of GA begins with a set of potential solutions, termed chromosomes,
that are randomly generated or selected [113]. The entire set of those chromosomes
establishes a population [113]. For any GA, a chromosome represents an individual in
the population [113]. The representation scheme determines how the problem is
structured in the GA and how the genetic operators that are used [113]. Each
chromosome is made up of a sequence of genes, taken from a certain alphabet [113].
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Chapter 4 Optimization of RF CMOS Polyphase Filters
The sequence of genes makes the value of a particular chromosome. This value of a
chromosome is called genotypes [113]. The genotypes are uniquely mapped into the
decision variables, called phenotypic domain. The chromosomes evolve during several
iterations or generations.
The most commonly used representation in GAs is based on the binary alphabet
{0, 1}. In this case, each decision variable is encoded by a binary string. The binary
strings are concatenated to form a chromosome. For example, a problem with two
variables, X1 and X2, may be mapped onto the chromosome structure in the way shown
in Figure 4.3. X1 is encoded by 10 bits and X2 is represented by 15 bits. This encoding
may reflect the level of accuracy or range of the individual decision variables.
Examining the chromosome string in isolation yields no information about the problem
we are trying to solve. It is only with the decoding of the chromosome into its
phenotypic values that any meaning can be applied to the representation.
Chromosome comprising the values of two
concatenated decision variables X1 and X 2
1011010011 010111010100101
Decision variable
X1
Decision variable
X2
Figure 4.3 Mapping variables into chromosomes.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
4.5.3 Genetic Operation
A. Introductory Remarks
The genetic operation includes three basic operators, namely, reproduction,
crossover and mutation [113]. These three basic operators provide the basic search
mechanism of genetic algorithms (GAs) and they are used to create new solutions based
on existing solutions in the population [113].
B. Reproduction
Reproduction is the selection of individuals to produce successive generations
[113]. Reproduction plays an important role in a genetic algorithm. A probabilistic
selection is performed based upon the individual's fitness in such a way that the fitter
individuals have an increased chance of being selected. An individual in the population
can be selected more than once with all individuals in the population having a chance of
being selected to be reproduced into the next generation.
A common selection approach assigns a probability of selection, Pj , to each
individual, j, based on its fitness value [113]. In general, j is an integer, which can vary
between 1 and N [113]. A series of N random numbers is generated and compared
N
against the cumulative probability,
∑p
j
, of the population. The appropriate individual,
j =1
i, is selected and copied into the new population.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Various methods exist to assign probabilities to individuals, namely, roulette
wheel, linear ranking and geometric ranking. In the GA implemented in this work, the
roulette wheel mechanism was chosen to probabilistically select individuals based on
some measures of their performance.
The roulette wheel selection method, developed by Holland [113], was the first
selection method. The probability, Pj , for each individual is defined by
Pj =
Fj
(4.21)
N
∑F
j
j =1
Where Fj is the fitness of individual j, and N is the population size.
As shown in Figure 4.4, the circumference of the roulette wheel is the sum of all
six individual’s fitness values; the sizes of the sectors stand for the fitness values of the
individuals. Individual 5, which has the largest sector, is the fittest individual, whereas
individuals 6 and 4 are the least fit and, correspondingly, have the smallest sectors
within the roulette wheel. To select an individual, a uniformly distributed random
number is generated in the predefined interval, and the individual whose sector spans
the random number is selected. This process is repeated until the desired number of
individuals has been selected. The roulette wheel selection is not only used in genetic
algorithms in conjunction with maximization, when combined with e.g. windowing and
scaling, it is well suited for minimization and negativity. Thus, roulette wheel selection
method has been chosen by the author.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.4 Roulette wheel selections.
C. Crossover and Mutation
Crossover involves splitting two individuals and then combining one half of each
individual with the other pair [113]. Mutation involves altering one individual to
produce a single new solution [113]. If chromosomes are as binary types represented,
simple crossover and binary mutation can be adopted [113].
Let X and Y be two m-dimensional row vectors denoting individuals (parents)
from the population. Simple crossover generates a random sample r from a uniform
distribution from 1 to m and creates two new individuals according to equation (4.22)
and equation (4.23).
 xi, if i < r
x ′i = 
 yi, otherwise
(4.22)
 yi, if i < r
y′i = 
 xi, otherwise
(4.23)
For example, consider the two pairs of binary strings
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Chapter 4 Optimization of RF CMOS Polyphase Filters
P1 = 1 0 0 1 0 1 1 0,
(4.24)
P2 = 1 0 1 1 1 0 0 0.
(4.25)
If an integer position, i, is selected uniformly and randomly, and the genetic
information exchanged between the individuals about this point, then two new offspring
strings are produced. The two offspring below are produced when the crossover point i
= 5 is selected,
O1 = 1 0 0 1 0 0 0 0,
(4.26)
O2 = 1 0 1 1 1 1 1 0.
(4.27)
This crossover operation is not necessarily performed on all strings in the
population. Instead, it is applied with a probability Px when the pairs are chosen for
breeding.
Binary mutation flips each bit in every individual in the population with
probability Pm according to following equation (4. 28):
1 − xi, if U(0,1) < Pm
x ′i = 
 xi, otherwise
(4.28)
For example, mutating the fourth bit of O1 leads to the new string:
O1m = 1 0 0 0 0 0 0 0.
(4.29)
Mutation is generally considered to be a background operator, which ensures that,
the probability of searching a particular subspace of the problem space is never zero.
This has the effect of tending to inhibit the possibility of converging to a local optimum,
rather than the global optimum.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
4.5.4 Initialization, Termination and Fitness Functions
GAs simultaneously operate on a number of potential solutions, called a
population, consisting of some encoding of the parameter set [113]. The mode of
operation generates the required number of individuals using a random number
generator that uniformly distributes numbers in the desired range. However, since GAs
can iteratively improve existing solutions, i.e., solutions from other heuristics and/or
current practices, the initial population can be seeded with potentially good solutions,
with the remainder of the population being randomly generated solutions. One of the
variations is the extended random initialization procedure of Bramlette [113], whereby a
number of random initializations are tried for each individual and the one with the best
performance is chosen for the initial population. Typically, a population is composed of
between 30 and 100 individuals, which is about ten times the number of variables to be
optimized.
The GA moves from generation to generation, selecting parents and reproducing
offspring until a termination criterion is met. Because the GA is a stochastic search
method, it is difficult to formally specify convergence criteria. The most frequently used
stopping criterion is a specified maximum number of generations [118]. Another
termination strategy involves population convergence criteria. In general, GAs will
force much of the entire population to converge to a single solution. When the sum of
the deviations among individuals becomes smaller than some specified threshold, the
algorithm can be terminated. The algorithm can also be terminated due to a lack of
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Chapter 4 Optimization of RF CMOS Polyphase Filters
improvement in the best solution over a specified number of generations. Alternatively,
a target value for the evaluation measure can be established based on some arbitrarily
‘acceptable’ threshold. Several strategies can be used in conjunction with each other.
The objective function [118] is used to provide a measure of how individuals
have performed in the problem domain. In the case of a minimization problem, the
fittest individuals will have the lowest numerical value of the associated objective
function. This raw measure of fitness is usually only used as an intermediate stage in
determining the relative performance of individuals in a GA [118].
Another function, the fitness function, is normally used to transform the value of
the objective function f (x) into measure of relative fitness, thus:
F (x) = g (f (x))
(4.30)
where g (⋅) transforms the value of the objective function to a non-negative number and
F (⋅) is the resulting relative fitness. This mapping is always necessary when the
objective function is to be minimized as the lower objective function values correspond
to fitter individuals.
A linear transformation which offsets the objective function is often used prior to
fitness assignment, such that,
F (x) = a f (x) + b,
(4.31)
where a is a positive scaling factor. If we are minimizing, the offset b is used to ensure
that the resulting fitness values are non-negative.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Start
Create initial random population
End
Apply fitness measure to
individuals of the population
Design best
individual
for result
Create a new
population
Termination
criteria satisfied?
Yes
Select best
individuals
No
Select genetic operations
Select one
individual
based on fitness
Select two
individuals
based on fitness
Select one
individual
based on fitness
Perform
reproduction
Perform
crossover
Perform
mutation
Copy into new
population
Insert offsprings
into new
population
Insert mutation
into new
population
Figure 4.5 Flowchart of the genetic algorithm implemented in PPFOPTIMA.
The main functions used in the optimization engine of PPFOPTIMA are listed in
the Table 4.1. These GA elements have been integrated into an operational GA. The
flowchart of this GA is shown in Figure 4.5.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Main functions
Description
ga
Basic function
roulette
Selection function
simpleXover
Crossover function
binaryMutation
Mutation function.
initializega
Initialization function
maxGenTerm
Termination function
Table 4.1 Main functions used in the optimization engine of PPFOPTIMA.
4.6
Implementation of PPFOPTIMA
4.6.1 Introduction
In Figure 4.6, the framework of PPFOPTIMA is shown, from which we can see
that inputs needed from the user are specifications of the design, e.g. the operating
frequency, and process related information. Under these conditions, the GA used in the
optimization engine makes tradeoffs between resistors, capacitors and their
corresponding physical size. Values of the resistors and the capacitors are determined
by process tolerances and constraints from thermal noise. The physical sizes of the
resistors and the capacitors are determined by component mismatch properties and
constraints from noise coupling and insertion loss. The output of PPFOPTIMA
comprises the optimized circuit dimension for schematic and the physical layout, which
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Chapter 4 Optimization of RF CMOS Polyphase Filters
is convenient and efficient for designers under time-to-market pressures. Details of the
system implementation will be illustrated in the following subsections.
RF CMOS PPFs
design specifications
Quadrature
accuracy
Tolerance design
of resistors
and capacitors
Process
tolerance
CMOS
process files
Component
mismatch
Maximum area
of resistors
and capacitors
Design
constraints
Insertion
loss
Thermal
noise
Fewer stages,
Minimum area
Trade - off values of
resistors and capacitors
Constraints on
resistors values
Trade - off areas of
resistors and capacitors
Trade - off values and areas of resistors
and capacitors by using genetic algorithms
Circuit sizing for schmatic and layout
Figure 4.6 Framework of PPFOPTIMA.
115
Coupled
noise
Minimum
area
Chapter 4 Optimization of RF CMOS Polyphase Filters
4.6.2 Construction of Fitness Functions
Unlike using conventional optimization methods, the first step of the GA is to
establish a fitness function F, which is the criterion in selecting the suitable population
for the next generation. In PPFOPTIMA, the fitness function is a sum of IRR, chip area
and the component mismatch of the resistors and the capacitors with different weights:
F = w1S + w2 ( M R + M C ) + w3 RIR
(4.32)
The real numbers w1 , w2 and w3 are the weighting factors for chip area S, for the
component mismatch of resistors M R , and capacitors M C , and for the image reject ratio
(IRR) RIR , respectively. In our application, since S , M R and M C are in the order of
10−3 , w1 and w2 are chosen in the order of 104 to differentiate good population from
poor ones.
4.6.3 Graphical User Interface
In order to facilitate users input and visualize the outputs, a graphical user
interface (GUI) has been developed for PPFOPTIMA, which is depicted in Figure 4.7.
The blank area shown in Figure 4.7 will be filled by simulation results. The GUI has the
following features:
1. Explicit and regulated user input information: Operating frequency and
process information required are listed in the user input boxes. In order to
avoid possible errors caused by mis-operations, data types in input boxes are
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Chapter 4 Optimization of RF CMOS Polyphase Filters
regulated. For example, in the case of inputs that require numbers, strings will
not be accepted, and a popup window will notify the user about this.
Figure 4.7 Graphical user interface of PPFOPTIMA.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
2. Visualized and classified outputs: The optimization results are classified and
displayed in different categorized windows.
3. Programmable weights for the fitness function: Weighting factors for
component mismatch, chip area and image reject ratio are taken as inputs.
4. Preset values for tutorial purpose: In order to support novice users, input
parameters for RF CMOS PPF designed for a Bluetooth receiver in FraunhoferIMS 0.6 µm CMOS technology are provided in the GUI.
5. Integrated high-level verification toolboxes: In order to verify the optimization
results, high-level verification toolboxes, namely, Monte Carlo (MC)
simulation and worst-case simulation toolboxes, have been developed and
integrated into PPFOPTIMA.
4.6.4 Verification Toolboxes
A. Introductory Remarks
In order to verify the performance of the synthesized circuit, dedicated
verification toolboxes, namely, Monte Carlo (MC) simulation and worst-case simulation
were developed as already mentioned in section 4.6.3. In our application, the Monte
Carlo (MC) simulation provides an estimate of the discrete probability density of gain
mismatch and phase mismatch, which are related to the required IRR, the main
specification of RF CMOS PPFs. The worst-case simulation gives us the possible worst-
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Chapter 4 Optimization of RF CMOS Polyphase Filters
case values for both the gain mismatch and the phase mismatch versus frequency,
respectively.
B. Monte Carlo Simulation
In a real design, there are a lot of processes happening in probabilistic ways, and
it is hard to model or describe them by using ordinary or differential equations.
Therefore, a statistical representation is a suitable way to characterize them. In statistical
simulations, sequences of random numbers with a certain probability distribution
function (PDF) are used to model the stochastic process. Based on this modeling
technique, responses of systems to stochastic inputs can be simulated. Usually, many
statistical simulations runs are conducted and averaged to reach good accuracy of the
simulation results. This is the method of Monte Carlo (MC) simulation, which has
following primary components:
• Probability distribution functions (PDF) - the statistical description of the
physical, or mathematical, system.
• Random number generator - a source of random numbers to mimic stochastic
factors in the physical system.
• Sampling rule - a prescription for sampling from random numbers with the
specified PDF.
• Scoring (or tallying) - the accumulated outcomes to meet overall tallies or
scores for the quantities of interest.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
• Error estimation - an estimate of the statistical error (variance) as a function of
the number of trials and other quantities.
Process tolerances and component mismatch in integrated circuits are
consequences of stochastic processes within a certain range, and they are usually
available in CMOS process files derived by elaborate measurements. It is known that
both process tolerances and component mismatch have truncated Gaussian probability
distribution functions [102]. In our application, Monte Carlo (MC) simulation can be
applied to verify the statistical nature of the IRR with certain process tolerances and a
resultant component mismatch. In order to obtain comparatively stable and reliable
results, many simulation steps must be performed, and the results are taken as an
average over the number of Monte Carlo (MC) runs. A flowchart of the Monte Carlo
simulation toolbox implemented in PPFOPTIMA is shown in Figure 4.8.
The effect of the image reject ratio (IRR) used in this toolbox is represented by
the following equation:
p
RIR = F (mgain , mphase ) = [∑ F ( Ri ,1 , Ci ,1 ,L Ri ,n , Ci ,n )] / p ,
(4.33)
Ri , j = R j ∗ [1 ± PR ] ∗ [1 ± M R ] ,
(4.34)
Ci , j = C j ∗ [1 ± PC ] ∗ [1 ± M C ] ,
(4.35)
i =1
where PR , PC , M R and M C are vectors of random numbers normalized by process
tolerances and component mismatch of a given CMOS process. In (4.33) i = 1L p , p is
the number of Monte Carlo (MC) runs. In (4.34) and (4.35), j = 1L n , n is the number
of cascaded stages in an RF CMOS PPF.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Start
Generate random numbers
Normalize random numbers
according to probability distrinution
functions of process tolerances
and component mismatch
Take random values
of resitors and capacitors
to evaluate the IRR
No
Meet termination
criteria?
Yes
Computer output
by averaging the
statistical samples
Print out, and save output
End
Figure 4.8 Flowchart of the Monte Carlo simulation option in PPFOPTIMA
C
．Worst-case Simulation
In circuit design, it is required that the worst-case circuit performance satisfies a
given target specifications. For most circuits, worst-case circuit performance occurs as
the combination of worst-case process tolerance, temperature and component mismatch.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Start
Find worst image reject
ratio of RF CMOS PPFs
without process tolerances
Shift corner frequencies, to
make worst image reject ratio
fall into the operating band
Calculate component mismatch
according to the areas of
resistors and capacitors
Add mismatch into
neighboring components
Calculate, save and print
out simulation results
End
Figure 4.9 Flowchart of the novel worst-case simulation methodology in PPFOPTIMA.
To obtain worst-case simulation results with widely used commercial tools like,
e.g. Cadence SpectreRF, the user need to carry out the simulations with variety of
models. For example, slow-to-fast, fast-to-slow, fast-to-fast, slow-to-slow and slow-gain
models must be considered. The various simulation results must then be compared in
order to find the worst-case performance of the circuit. This procedure is inconvenient
and expensive with respect to computation and simulation time.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
RF CMOS PPFs with varying numbers of cascaded stages have varying worstcase boundaries. Through analysis, we have found that worst-case boundary of RF
CMOS PPFs does not necessarily to occur with the worst process variation. Instead, it
occurs in the case of a process variation where worst image reject ratio falls in the
operating frequency band. The verification toolboxes of PPFOPTIMA take this
observation into account, yielding a novel simulation methodology proposed by the
author of this dissertation. The flowchart of the novel worst-case simulation
methodology is shown in Figure 4.9.
4.7
Experimental Results
4.7.1 Overview
In order to verify effectiveness of PPFOPTIMA, two categories of experiments
have been conducted. The first category of experiments is to synthesize and verify
multi-stage RF CMOS PPFs for local oscillator (LO) quadrature generation in receivers
for a given communication standard. In these experiments, RF CMOS PPFs for LO
quadrature generation in a Bluetooth receiver are taken as examples. The second
category of experiments is the automated synthesis and verification of RF CMOS PPFs
in receivers for different communication standards, especially, for GSM, DECT, and
WCDMA receivers, by using PPFOPTIMA. The CMOS process used in all of these
experiments is Fraunhofer-IMS 0.6 µm CMOS process.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
4.7.2 Experiments on RF CMOS PPFs for a Bluetooth Receiver
A. Introductory Remarks
In this section, multi-stage RF CMOS PPFs following target specifications of a
low-IF Bluetooth receiver, which were synthesized and analyzed with PPFOPTIMA,
will be presented. These RF CMOS PPFs are designed for LO quadrature generation in
this low-IF Bluetooth receiver in which the LO frequency is 2.45 GHz. The
specification derived from the Bluetooth standard and the low-IF receiver architecture is
an image reject ratio (IRR) great than 30 dB, which can be translated into a gain
mismatch of less than 0.7 dB, and a phase mismatch of less than 5 degrees. Although
PPFOPTIMA is capable of synthesizing RF CMOS PPFs automatically, in these
experiments, RF CMOS PPFs have been synthesized and verified by using
PPFOPTIMA manually to verify the effectiveness of the tool.
The first step in using PPFOPTIMA is the dimensioning of the circuit by pressing
the button ‘Optimization by GA’ in the GUI. Synthesis results, which include values,
lengths, widths and layout areas of resistors and capacitors, are shown as ‘Optimization
Results’ in the GUI. After that worst-case simulations are used to check whether the
performance of the synthesized RF CMOS PPF satisfies the target specifications over
all of the process corners. Then, Monte Carlo simulations are used to check the
probability distribution of the gain mismatch and the phase difference, respectively.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
B. Experiments on a 2-Stage RF CMOS PPF for a Bluetooth Receiver
Shown in Figure 4.10 is the synthesis result of a 2-stage RF CMOS PPF designed
for LO quadrature generation in a low-IF Bluetooth receiver.
Figure 4.10 Synthesis result of a 2-stage RF CMOS PPF for a Bluetooth receiver.
The worst-case simulation results of the RF CMOS PPF synthesized in Figure
4.10 are shown in Figure 4.11. The gain mismatch, mgain , is 0.81 dB and the phase
125
Chapter 4 Optimization of RF CMOS Polyphase Filters
difference, d phase , is 92 degrees, i.e., the phase mismatch, mphase , is 2 degrees. As the
gain mismatch cannot meet the target specification, RF CMOS PPFs with more
cascaded stages must be tried in the following subsections.
Figure 4.11 Worst-case simulation results of a 2-stage RF CMOS PPF for a Bluetooth receiver.
Upper plot: gain mismatch mgain versus frequency f.
Lower plot: phase difference d phase versus frequency f.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
As worst-case simulation only provides a boundary of the circuit performance, in
order to know the statistical distribution of the circuit performance, Monte Carlo (MC)
simulation runs using 2000 samples of RF CMOS PPFs have been conducted.
Figure 4.12 Monte Carlo simulation results of a 2-stage RF CMOS PPF for a Bluetooth receiver.
Upper plot: histogram of the gain mismatch mgain .
Lower plot: histogram of the phase difference d phase .
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Chapter 4 Optimization of RF CMOS Polyphase Filters
The Monte Carlo simulation results are expressed as frequency of occurrence
histogram for different intervals of the gain mismatch and the phase difference, and they
are shown in Figure 4.12. In the MC simulations, after 200 MC runs, the simulation
results enter steady state. In 2000 samples of RF CMOS PPFs, 1880, that is, 94% of
them have a gain mismatch between 0.52 dB and 0.81 dB, and 1892, that is 94.6% of
them have a phase difference between 91.27 degrees and 91.97 degrees, that is, a phase
mismatch between 1.27 degrees and 1.97 degrees.
C. Experiments on a 3-Stage RF CMOS PPF for a Bluetooth Receiver
As the 2-stage RF CMOS PPF synthesized cannot satisfy the target specifications,
now let us check whether a 3-stage RF CMOS PPF can satisfy the specifications.
Synthesis results of a 3-stage RF CMOS PPF are shown in Figure 4.13. The
worst-case simulation results shown in Figure 4.14 reveal a gain mismatch of 0.083 dB,
and a phase difference of 90.15 degrees, i.e., a phase mismatch of 0.15 degrees. As the
target specifications of this design are met, a 3-stage RF CMOS PPF is chosen.
To verify the statistical distribution of gain mismatch and phase mismatch in this
design, MC simulation has been used. After 200 MC runs, the simulation results enter
steady state. The obtained results are shown in Figure 4.15. In 2000 samples of RF
CMOS PPFs, 90% of them have a gain mismatch of less than 0.047 dB and a phase
mismatch of less than 0.1 degrees.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.13 Synthesis results of a 3-stage RF CMOS PPF for a Bluetooth receiver.
The author would like to mention that in Monte Carlo simulations gain mismatch
and phase mismatch could be worse than that from worst-case simulation, because in
129
Chapter 4 Optimization of RF CMOS Polyphase Filters
the worst-case simulation, the worst image reject ratio (IRR), in which worst gain
mismatch and worst phase mismatch of the RF CMOS PPFs do not necessarily happen
at the same time.
Figure 4.14 Worst-case simulation results of a 3-stage RF CMOS PPF for a Bluetooth receiver.
Upper plot: gain mismatch mgain versus frequency f.
Lower plot: phase difference d phase versus frequency f.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.15 Monte Carlo simulation results of a 3-stage RF CMOS PPF for a Bluetooth receiver.
Upper plot: histogram of the gain mismatch mgain .
Lower plot: histogram of the phase difference d phase .
D. Experiments on a 4-stage RF CMOS PPF for a Bluetooth Receiver
In this section, a 4-stage RF CMOS PPF for a Bluetooth receiver was synthesized.
The synthesis results are shown as ‘Optimization Results’ in Figure 4.16.
131
Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.16 Synthesis results of a 4-stage RF CMOS PPF for a Bluetooth receiver.
132
Chapter 4 Optimization of RF CMOS Polyphase Filters
The worst-case simulation results of the RF CMOS PPF are shown in Figure 4.17,
in which the gain mismatch is 0.028 dB, and the phase difference is 90.031 degrees, i.e.,
the phase mismatch is equal to 0.031 degrees.
Figure 4.17 Worst-case simulation results of a 4-stage RF CMOS PPF for a Bluetooth receiver.
Upper plot: gain mismatch mgain versus frequency f.
Lower plot: phase difference d phase versus frequency f.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
Monte Carlo simulation results are shown in Figure 4.18, after 700 Monte Carlo
runs, the simulation result enter steady state. The MC simulation results show that in
2000 samples of RF CMOS PPFs, more than 90% of them have a gain mismatch less
than 0.005 dB and phase mismatch less than 0.01 degrees.
Figure 4.18 Monte Carlo simulation results of a 4-stage RF CMOS PPF for a Bluetooth receiver.
Upper plot: histogram of the gain mismatch mgain .
Lower plot: histogram of the phase difference d phase .
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Chapter 4 Optimization of RF CMOS Polyphase Filters
From the simulation results above, we can see that the performance of the 4-stage
RF CMOS PPF synthesized is more than sufficient for the target specifications in a lowIF Bluetooth receiver. However, for passive filters like RF CMOS PPFs, they have a
nominal insertion loss of 3 dB/stage. In order to compensate the insertion loss, buffer
amplifiers are usually necessary for RF CMOS PPFs with many cascaded stages, thus,
causing higher power consumption. In the design of CMOS wireless receivers, low
power consumption is required. Thus, the strategy for topology selection in
PPFOPTIMA is to use as few cascaded stages as possible to satisfy the target
specifications.
4.7.3 Automated Synthesis and Verification of RF CMOS PPFs for
Different Communication Standards
A. Introductory Remarks
In Figure 4.19 the GUI of PPFOPTIMA for automated synthesis of RF CMOS
PPFs is shown. The specification needed is image reject ratio (IRR) for CMOS wireless
receivers. With a given specification, a 2-stage RF CMOS PPF is chosen at first, and
optimized by PPFOPTIMA, and then the worst-case simulation is conducted to verify
whether the specifications can be met. If the specifications are met, a popup window
will notify the user that a 2-stage RF CMOS PPF is chosen and the optimized circuit
dimension is provided. If a 2-stage RF CMOS PPF cannot meet the specifications, the
135
Chapter 4 Optimization of RF CMOS Polyphase Filters
number of stages of the RF CMOS PPFs is increased by one until the specification can
be met. The final results are an RF CMOS PPF with a selected number of stages and
optimized circuit dimensions.
Figure 4.19 Graphic user interfaces of PPFOPTIMA
for automated synthesis of RF CMOS PPFs.
In order to verify the effectiveness of PPFOPTIMA in automated synthesis and
verification, RF CMOS PPFs for LO quadrature generation in receivers of different
communication standards, for example, GSM, DECT and WCDMA were synthesized
and verified. The results will be presented in the remainder of this section 4.7.3.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
B. Automated Synthesis and Verification of RF CMOS PPF for a GSM
Receiver
Figure 4.20 Synthesis results of a 3-stage-RF CMOS PPF for a GSM receiver.
137
Chapter 4 Optimization of RF CMOS Polyphase Filters
In this experiment, an RF CMOS PPF targeting an image reject ratio (IRR) of at
least 40 dB for a low-IF GSM receiver was synthesized and verified. A 3-stage RF
CMOS PPF was chosen by PPFOPTIMA, the synthesis results are shown in Figure 4.20.
Figure 4.21 Worst-case simulation results of a 3-stage RF CMOS PPF for a GSM receiver.
Upper plot: gain mismatch mgain versus frequency f.
Lower plot: phase difference d phase versus frequency f.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
The worst-case simulation results are shown in Figure 4.21, in which the IRR is
49.2 dB, the gain mismatch is 0.0512 dB and the phase difference is 90.1072 degrees,
i.e., the phase mismatch is 0.1072 degrees.
Figure 4.22 Monte Carlo simulation results of a 3-stage RF CMOS PPF for a GSM receiver.
Upper plot: histogram of the gain mismatch mgain .
Lower plot: histogram of the phase difference d phase .
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Chapter 4 Optimization of RF CMOS Polyphase Filters
In order to check the statistical distribution of gain mismatch and phase mismatch,
a Monte Carlo (MC) simulation in PPFOPTIMA was conducted. After 200 MC runs,
the results enter a steady state. The results of the MC simulation are shown in the Figure
4.22, from which we can see that in 2000 samples of RF CMOS PPFs, more than 90%
of them have a gain mismatch of less than 0.045 dB, and a phase mismatch of less than
0.1 degrees.
Figure 4.23 Synthesis results of a 2-stage-RF CMOS PPF for a DECT receiver.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
C. Automated Synthesis and Verification of RF CMOS PPFs for a
DECT Receiver
In this section, an RF CMOS PPF targeting 28 dB IRR for a low-IF DECT
receiver has been synthesized and verified. A 2-stage RF CMOS PPF was chosen by
PPFOPTIMA, the synthesis results are shown in Figure 4.23.
Figure 4.24 Worst-case simulation results of a 2-stage RF CMOS PPF for a DECT receiver.
Upper plot: gain mismatch mgain versus frequency f.
Lower plot: phase difference d phase versus frequency f.
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Chapter 4 Optimization of RF CMOS Polyphase Filters
The worst-case simulation results of the RF CMOS PPF synthesized according to
Figure 4.23 are shown in Figure 4.24. The worst-case IRR is 30.7 dB, the gain
mismatch is 0.612 dB, and the phase difference is 91.51 degrees, i.e., the phase
mismatch is 1.51 degrees.
Figure 4.25 Monte Carlo simulation results of a 2-stage RF CMOS PPF for a DECT receiver.
Upper plot: histogram of the gain mismatch mgain .
Lower plot: histogram of the phase difference d phase .
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Chapter 4 Optimization of RF CMOS Polyphase Filters
To check the statistical distribution of gain and the phase mismatch, Monte Carlo
simulation in PPFOPTIMA was applied. After 200 MC runs, the simulation results
enter a steady state. The MC simulation results are shown in Figure 4.25. In 2000
samples of RF CMOS PPFs, more than 90% of them have the gain mismatch between
0.45 dB to 0.61 dB, and the phase mismatch between 0.9 degrees and 1.5 degrees.
D. Automated Synthesis and Verification of RF CMOS PPFs for a
WCDMA Receiver
In this section, an RF CMOS PPF for LO quadrature generation in a low-IF
WCDMA receiver was synthesized and verified. The LO frequency is set to 1.90 GHz,
and the targeting specification of RF CMOS PPF is 36 dB IRR. A 3-stage RF CMOS
PPF was chosen by PPFOPTIMA, the synthesis results are shown in Figure 4.26.
The worst-case simulation results are shown in Figure 4.27, in which the worstcase IRR is 49.1 dB, the gain mismatch is 0.0541 dB, the phase difference is 90.1201
degrees, i.e., the phase mismatch is 0.1201 degrees.
To check the statistical distribution of the gain mismatch and the phase mismatch,
a Monte Carlo simulation in PPFOPTIMA has been conducted. The MC simulation
results shown in Figure 4.28 demonstrate that, in 2000 samples of RF CMOS PPFs,
more than 90% of them have the gain mismatch less than 0.04 dB, and the phase
mismatch less than 0.1 degrees.
143
Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.26 Synthesis results of a 3-stage-RF CMOS PPF for a WCDMA receiver.
144
Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.27 Worst-case simulation results of a 3-stage RF CMOS PPF for a WCDMA receiver.
Upper plot: gain mismatch mgain versus frequency f.
Lower plot: phase difference d phase versus frequency f.
145
Chapter 4 Optimization of RF CMOS Polyphase Filters
Figure 4.28 Monte Carlo simulation results of a 3-stage RF CMOS PPF for a WCDMA receiver.
Upper plot: histogram of the gain mismatch mgain .
Lower plot: histogram of the phase difference d phase .
4.8 Summary
In this chapter, the optimization problem for RF CMOS PPFs has been
formulated. Possible optimization algorithms like calculus-based optimization, gradient
146
Chapter 4 Optimization of RF CMOS Polyphase Filters
descend, simulated annealing (SA) and genetic algorithms (GA) have been analyzed.
GA has been selected for the optimization engine because of its advantages in escaping
from local optima.
PPFOPTIMA, a GA based optimization tool for RF CMOS PPFs has been
implemented by using MATLAB. In order to facilitate designers in design automation,
this tool has been developed further into a specification-driven automated synthesis and
verification environment incorporating dedicated worst-case and Monte Carlo
simulation.
In order to verify the effectiveness of PPFOPTIMA, two types of experiments
have been conducted. The first type of experiments consists of the synthesis and the
verification of multi-stage RF CMOS PPFs in a receiver for a given communication
standard. Here, a low-IF Bluetooth receiver has been taken as an example. The second
type of experiments consists of the synthesis and verification of the RF CMOS PPFs for
different communications standards, namely GSM, DECT, and WCDMA. The
experimental results demonstrate that PPFOPTIMA is effective in synthesis,
optimization and verification of RF CMOS PPFs.
Focusing on the optimization of RF CMOS PPFs, this chapter has the following
contributions:
1. The methodology of optimization of RF CMOS PPFs with layout
considerations by using genetic algorithm (GA) has been illustrated.
147
Chapter 4 Optimization of RF CMOS Polyphase Filters
2. PPFOPTIMA, a computer-aided synthesis and optimization tool for RF
CMOS PPFs, has been presented.
3. Dedicated worst-case simulation and Monte Carlo simulation toolboxes for
verification of the performance of RF CMOS PPF have been developed.
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Chapter 5 Summary and Outlook
Chapter 5
Summary and Outlook
5.1 Summary
RF CMOS polyphase filters (PPFs) is an efficient solution for high accuracy radio
frequency quadrature generation in CMOS technology. The main specification of the
RF CMOS PPF, namely the quadrature accuracy, is layout dependent; and the
requirements from component match are contradictory to that of minimization of noise
coupling, signal loss and chip area, thus a tradeoff must be made in a specific design.
By simply following guidelines proposed by Behbahani [62], it takes intensive trials and
many design iterations to reach a good design. Furthermore, with layout considerations,
the optimization of RF CMOS PPFs cannot be solved by using analytical calculations,
because of many constraints and tradeoffs in the design.
In this situation, a computer-aided analysis and design optimization tool for RF
CMOS PPFs is of great help to the community of RF CMOS PPFs designers, especially
for industrial projects, which are under time-to-market pressure. Unfortunately, until
now, there is no such dedicated tool available, to the author’s best knowledge.
Focused on analysis, design and optimization of RF CMOS PPFs, this dissertation
has following contributions:
1. PPFANA, a dedicated software tool for simulation of quadrature accuracy in
RF CMOS PPFs influenced by process tolerances and component mismatch,
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Chapter 5 Summary and Outlook
has been developed. It provides an efficient computer-aided analysis for RF
CMOS PPFs.
2. Based on the simplified RF CMOS models of resistors, and capacitors,
parasitic effects in RF CMOS PPFs have been analyzed. Techniques of
minimizing those unwanted influences from parasitics have been proposed, and
physical design considerations for RF CMOS polyphase filter in system-on-achip (SoC) solution have been analyzed.
3. Silicon success of an RF CMOS polyphase filter for a low-IF Bluetooth
receiver was achieved. Measurement results from this practical design further
verified the effectiveness of PPFANA. Design consideration and design
heuristics from this design practice lay a solid foundation for this dissertation.
4. Based on heuristics from the practical design of an RF CMOS PPF, a design
automation oriented design flow for RF CMOS PPFs has been proposed.
5. By analysis on existing optimization algorithms, the methodology of synthesis
and verification of RF CMOS PPFs with layout considerations by using genetic
algorithm (GA) has been proposed in this dissertation.
6. PPFOPTIMA, a GA based optimization tool for RF CMOS PPFs has been
implemented by using MATLAB. In order to enable the designers in design
automation, PPFOPTIMA has been developed further towards a specificationdriven automated synthesis and verification environment incorporating
dedicated worst-case and Monte Carlo simulation options.
150
Chapter 5 Summary and Outlook
5.2 Outlook
Compared to Electronic Design Automation (EDA) tools for digital circuit, those
for analog circuit, especially for RF circuit, are still in their infancy. Nowadays, RF
analog design is still a kind of ‘art’, which is done by ‘Analog Artists’. Although some
research groups, e.g. at the University of California at Los Angeles, and at the
Katholieke Universiteit Leuven, have contributed some design automation tools for
some RF components, there is still a far way to go in the research topic.
Optimization is a key step in design automation of analog and RF circuits. Though
RF circuits are comparatively simple in the sense of the number of components, taking
layout considerations into account, optimization of RF circuits is complicated because
of multiple design objectives and constraints. In this situation, genetic algorithm (GA)
has advantages over other conventional optimization techniques for its capability in
escaping from local optima. The successful application of GA in the optimization of RF
CMOS PPFs demonstrates that GA is effective and efficient in RF circuit optimization.
However, this optimization methodology has following drawbacks:
1. GA is able to escape from local optima, but it is not good at finding exact
optimum point.
2. GA is not intelligent enough to be trained, that is, for GA based optimization
tools, if modification is needed; change of fitness function and more work is
needed.
3. It is still heuristics based trials to find suitable weights in applications of GAs.
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Chapter 5 Summary and Outlook
Compared to GA, an artificial neural network (ANN) may lead to improvements
because of its self-learning mechanisms. For RF circuit synthesis and optimization, an
ANN may find a wider application in the future.
152
References
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162
Appendix 1. Publications from the Author
Appendix 1
Publications from the Author
Part 1. Publications from the Author After 2001
ZHA1. Zhang, Z.S.; Christoffers, N.; Hosticka B.J.; Kaiser, T.; Kokozinski, R.; Lin, J.: A survey of
oscillator phase-noise: analysis and simulation. Proceedings of the Second Joint Symposium on
Opto- & Microelectronic Devices and Circuits (SODC’02), pp. 322-326, Stuttgart/Germany,
March 2002. (in English)
ZHA2. Zhang, Z.S.; Kolnsberg, S.; Kokozinski, R.: Design considerations for RF CMOS polyphase
filter in low-cost digital SoC solution. On-line Proceedings of The sixth workshop on Analog
Circuit: http://www.imtek.uni-freiburg.de/workshop2004/, Freiburg/Germany, March 2004. (in
English)
ZHA3. Zhang, Z.S.; Kolnsberg, S.; Kokozinski, R.: Synthesis of RF CMOS polyphase filters with
layout considerations. Chinese Journal of Semiconductors, vol.25 (2004), No.12, pp. 1612-1617.
(in English)
Part 2. Publications from the Author Before 2001
ZHA4. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Sun, M.: Design and experiments of PHV-I portable highfrequency vibrator, Instrument Technology & Sensors, vol.36 (2000), No.10, pp. 14-18. (in
Chinese)
ZHA5. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Yu, S.B.: Development of portable high-frequency vibrator.
Automation in Petroleum Industry, vol.14 (2000), No.6, pp. 14-17. (in Chinese)
ZHA6. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Yu, S.B.: Design and implementation of portable high
frequency vibrator. Measurement & Control Technique, vol.19 (2000), No.10, pp. 10-53. (in
Chinese)
ZHA7. Zhang, Z.S.; Lin, J.; Chen, Z.B.; Yu, S.B.: An industrial PC-based measurement & control
system in portable high-frequency vibrator. Computer Automated Measurement & Control, vol.8
(2000), No.4, pp. 10-13. (in Chinese)
ZHA8. Zhang, Z.S.; Lin, J.; Jiang, H.: A novel interface between non-standard keyboard and industrial
PC in portable measurement & control instruments. Measurement & Control Technique, vol.19
(2000), No.6, pp. 10-61. (in Chinese)
ZHA9. Zhuang, J.; Lin, J.; Wu D. J.; Zhang, B.J.; Zhang, Z.S.: Analysis on amplitude characteristics of
the coupling process between PHV and the earth. Journal of Changchun University of Science &
Technology, vol.29 (1999), No.2, pp. 184-188. (in Chinese)
163
Appendix 1. Publications from the Author
ZHA10.
Yu, S.B.; Lin, J.; Zhang, Z.S.: The Study on the frequency characteristic of portable highfrequency vibrator. Proceedings of The Fourth International Conference on Electronic
Measurement & Instruments (ICEMI’99), Harbin/China, pp. 191-195, 1999. (in English)
ZHA11. Zhang, Z.S.; Lin, J.; Yu, S.B.: The optimal signal design of linear sweep in PHVS system.
Petroleum Instruments, vol.12 (1998), No.3, pp. 1-5. (in Chinese)
ZHA12. Zhang, Z.S.: Serial ADC DSP101/102 directly interfaced with DSP and its applications.
Petroleum Instruments, vol.12 (1998), No.2, pp. 19-22. (in Chinese)
ZHA13. Zhang, Z.S.; Lin, J.; Fu X.J.; Sun C.T.: An analysis on sweep signal design of vibroseis
system. Petroleum Instruments, vol.11 (1997), No.6, pp. 10-14. (in Chinese)
ZHA14. Zhang, Z.S.: Realizing digital programmable filter by using universal filter UAF42.
Electronic Technology, vol.24 (1997), No.9, pp. 13-34. (in Chinese)
164
Appendix 2 Transfer Functions of Multi-stage RF CMOS PPFs
Appendix 2
Transfer Functions of Multi-Stage RF CMOS PPFs
A2.1 Explicit Transfer Function of 2-Stage RF CMOS PPFs
The transfer function of the 2-stage RF CMOS PPF can be obtained by multiplying the chain
matrices of each stage as follows:
R1 
 Vin 
1+jω R1C1
1
 Iin  = 1-ω R1C1  2jω C1 1+jω R1C1
 


R2   Vout 
1+jω R2C2
1
.

1-ωR2C2  2jω C2 1+jω R2C2   Iout 
(A2.1)
The open circuit ( Iout = 0) complex transfer function is:
H(jω )=
(1-ω R1C1)(1-ω R2C2)
.
1+jω (R1C1+R2C2+2R1C2)-ω 2 (R1R2C1C2)
(A2.2)
Equation (A2.1) can also be expressed as:
H(jω )=H1(jω )+jH2(jω ) ,
Where
1+ω 2 (R1R2C1C2)
H1(jω )=
,
1+jω (R1C1+R2C2+2R1C2)-ω 2 (R1R2C1C2)
H2(jω )=
jω (R1C1+R2C2)
,
1+jω (R1C1+R2C2+2R1C2)-ω 2 (R1R2C1C2)
(A2.3)
(A2.4)
(A2.5)
A2.2 Explicit Transfer Function of 3-Stage RF CMOS PPFs
The transfer function of the 3-stage RF CMOS PPF can be derived by multiplying
the chain matrices of each stage as follows:
R1 
 Vin 
1+jω R1C1
1
 Iin  = 1-ω R1C1  2jω C1 1+jω R1C1
 


R2 
1+jω R2C2
1

1-ωR2C2  2jω C2 1+jω R2C2 
R3   Vout 
1+jω R3C3
1
.

1-ω R3C3  2jω C3 1+jω R3C3   Iout 
The complex transfer function is:
165
(A2.6)
Appendix 2 Transfer Functions of Multi-stage RF CMOS PPFs
H(jω )=
Where
N(jω )
,
D(jω )
N(jω )=(1-ω R1C1)(1-ω R2C2)(1-ω R3C3) ,
D(jω )=DR(ω )+jDI(ω ) ,
DR(ω )=1-ω 2 (R1R2C1C2+R1R3C1C3+R2R3C2C3
+2R1R3C2C3+2R1R2C1C3+2R1R2C2C3),
DI(ω )=ω (R1C1+R2C2+R3C3+2R1C2+2R1C3
+2R2C3)-ω 3 (R1R2R3C1C2C3).
(A2.7)
(A2.8)
(A2.9)
(A2.10)
(A2.11)
Equation (A2.7) can also be expressed as:
H(jω )=H1(jω )+jH2(jω )
Where
H1(jω )=
1+ω 2 (R1R2C1C2+R1R3C1C3+R2R3C2C3)
,
D(jω )
jω (R1C1+R3C3+R2C2)+jω 3 (R1R2R3C1C2C3)
H2(jω )=
.
D(jω )
166
(A2.12)
(A2.13)
(A2.14)
Appendix 3 Source Code of PPFANA
Appendix 3
Source Code of PPFANA
A3.1 Source Code for Analysis of Gain Mismatch Influenced by
Process Tolerances
A.3.1.1 Source Code for Analysis of Gain Mismatch Influenced by Process
Tolerances in 2-stage RF CMOS PPFs
global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global pt_r; global pt_c;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string'));
%get the value from GUI input
f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));
rt_1=findobj(gcf,'Tag','rt_1'); rt_1=str2double(get(rt_1,'string'));
rt_2=findobj(gcf,'Tag','rt_2'); rt_2=str2double(get(rt_2,'string'));
ct_1=findobj(gcf,'Tag','ct_1'); ct_1=str2double(get(ct_1,'string'));
ct_2=findobj(gcf,'Tag','ct_2'); ct_2=str2double(get(ct_2,'string'));
pt_r=findobj(gcf,'Tag','pt_r'); pt_r=str2double(get(pt_r,'string'));
if isempty(pt_r), pt_r=0.16; else pt_r= pt_r; end
pt_c=findobj(gcf,'Tag','pt_c'); pt_c=str2double(get(pt_c,'string'));
if isempty(pt_c), pt_c=0.21; else pt_c=pt_c; end
if isempty(f_1), f_1=1.44e9; else f_1=f_1; end
if isempty(f_2), f_2=3.47e9; else f_2=f_2; end
if isempty(rt_1), rt_1=223.5; else rt_1=rt_1; end
if isempty(rt_2), rt_2=265.6; else rt_2=rt_2; end
if isempty(ct_1), ct_1=205.2; else ct_1=ct_1; end
if isempty(ct_2), ct_2=416.7; else ct_2=ct_2; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);
c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15;
r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RinCin=abs(h-n); plot(w,RinCin,'g -s')
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
167
Appendix 3 Source Code of PPFANA
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RdeCde=abs(h-n); plot(w,RdeCde,'b -*')
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RinCde=abs(h-n); plot(w,RinCde,'r -^')
c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RdeCin=abs(h-n); plot(w,RdeCin,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4);
c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RinCin=abs(h-n); plot(w,RinCin,'g -')
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RdeCde=abs(h-n); plot(w,RdeCde,'b -.') % %
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
168
Appendix 3 Source Code of PPFANA
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RinCde=abs(h-n); plot(w,RinCde,'r --') % %
c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=20.*log10(a); n=20.*log10(d); RdeCin=abs(h-n); plot(w,RdeCin,'k :')
xlabel('{\itf} / GHz','FontSize', 14)
ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title('Gain mismatch influenced by process tolerances','FontSize', 14)
A.3.1.2 Source Code for Analysis of Gain Mismatch Influenced by Process
Tolerances in 3-stage RF CMOS PPFs
global f_1; global f_2; global rth_1; global rth_2; global rth_3; global cth_1; global cth_2; global cth_3;
global cm; global pt_r; global pt_c;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));
rth_1=findobj(gcf,'Tag','rth_1'); rth_1=str2double(get(rth_1,'string'));
rth_2=findobj(gcf,'Tag','rth_2'); rth_2=str2double(get(rth_2,'string'));
rth_3=findobj(gcf,'Tag','rth_3'); rth_3=str2double(get(rth_3,'string'));
cth_1=findobj(gcf,'Tag','cth_1'); cth_1=str2double(get(cth_1,'string'));
cth_2=findobj(gcf,'Tag','cth_2'); cth_2=str2double(get(cth_2,'string'));
cth_3=findobj(gcf,'Tag','cth_3'); cth_3=str2double(get(cth_3,'string'));
pt_r=findobj(gcf,'Tag','pt_r'); pt_r=str2double(get(pt_r,'string'));
if isempty(pt_r), pt_r=0.16; else pt_r= pt_r; end
pt_c=findobj(gcf,'Tag','pt_c'); pt_c=str2double(get(pt_c,'string'));
if isempty(pt_c), pt_c=0.21;else pt_c=pt_c; end
if isempty(f_1), f_1=1.44e9; else f_1=f_1; end
if isempty(f_2), f_2=3.47e9; else f_2=f_2; end
if isempty(rth_1), rth_1=263.9; else rth_1=rth_1; end
if isempty(rth_2), rth_2=293.7; else rth_2=rth_2; end
if isempty(rth_3), rth_3=315.2; else rth_3=rth_3; end
169
Appendix 3 Source Code of PPFANA
if isempty(cth_1), cth_1=173.8; else cth_1=cth_1; end
if isempty(cth_2), cth_2=221.2; else cth_2=cth_2; end
if isempty(cth_3), cth_3=350.7; else cth_3=cth_3; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);
c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15;
c3_1=cth_1.*(1+pt_c).*1e-15; r3_1=rth_1.*(1+pt_r);
r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).
^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -s')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -*')
c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15; c3_1=cth_1.*(1+pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
170
Appendix 3 Source Code of PPFANA
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'r -^')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4); c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15;
c3_1=cth_1.*(1+pt_c).*1e-15; r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
171
Appendix 3 Source Code of PPFANA
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -.')
c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15; c3_1=cth_1.*(1+pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'r --')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
172
Appendix 3 Source Code of PPFANA
r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'k :')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title('Gain mismatch influenced by process tolerances','FontSize', 14)
A.3.1.3 Source Code for Analysis of Gain Mismatch Influenced by Process
Tolerances in 4-stage RF CMOS PPFs
global f_1; global f_4; global rf_1; global rf_2; global rf_3; global rf_4; global cf_1; global cf_2;
global cf_3; global cf_4; global pt_r; global pt_c;
f_1=findobj(gcf,'Tag','f_1');
f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_4=findobj(gcf,'Tag','f_4');
f_4=str2double(get(f_4,'string'));
rf_1=findobj(gcf,'Tag','rf_1'); rf_1=str2double(get(rf_1,'string'));
rf_2=findobj(gcf,'Tag','rf_2');
rf_2=str2double(get(rf_2,'string'));
rf_3=findobj(gcf,'Tag','rf_3');
rf_3=str2double(get(rf_3,'string'));
rf_4=findobj(gcf,'Tag','rf_4');
rf_4=str2double(get(rf_4,'string'));
cf_1=findobj(gcf,'Tag','cf_1');
cf_1=str2double(get(cf_1,'string'));
cf_2=findobj(gcf,'Tag','cf_2');
cf_2=str2double(get(cf_2,'string'));
cf_3=findobj(gcf,'Tag','cf_3');
cf_3=str2double(get(cf_3,'string'));
cf_4=findobj(gcf,'Tag','cf_4');
cf_4=str2double(get(cf_4,'string'));
pt_r=findobj(gcf,'Tag','pt_r');
pt_r=str2double(get(pt_r,'string'));
if isempty(pt_r), pt_r=0.16; else pt_r= pt_r; end
pt_c=findobj(gcf,'Tag','pt_c'); pt_c=str2double(get(pt_c,'string'));
if isempty(pt_c), pt_c=0.21; else pt_c=pt_c; end
if isempty(f_1), f_1=1.44e9; else f_1=f_1; end
if isempty(f_4), f_4=3.47e9; else f_4=f_4; end
if isempty(rf_1), rf_1=289.9; else rf_1=rf_1; end
if isempty(rf_2), rf_2=329; else
rf_2=rf_2; end
173
Appendix 3 Source Code of PPFANA
if isempty(rf_3), rf_3=377.4; else rf_3=rf_3; end
if isempty(rf_4), rf_4=390; else
rf_4=rf_4; end
if isempty(cf_1), cf_1=158.2; else cf_1=cf_1; end
if isempty(cf_2), cf_2=186.9; else cf_2=cf_2; end
if isempty(cf_3), cf_3=218.5; else cf_3=cf_3; end
if isempty(cf_4), cf_4=283.4; else cf_4=cf_4; end
figure hold on
w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15;
c2=cf_2.*(1+pt_c).*1e-15; c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r);
r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RinCin=abs(m-k); plot(w,RinCin,'g -s')
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
174
Appendix 3 Source Code of PPFANA
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RdeCde=abs(m-k); plot(w,RdeCde,'b -*') %
c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15; c2=cf_2.*(1+pt_c).*1e-15;
c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4+2*r1*r2*r4*c3
*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RdeCin=abs(m-k); plot(w,RdeCin,'r -^') %
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
175
Appendix 3 Source Code of PPFANA
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RinCde=abs(m-k); plot(w,RinCde,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4); c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15;
c2=cf_2.*(1+pt_c).*1e-15; c1=cf_1.*(1+pt_c).*1e-15;
r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RinCin=abs(m-k); plot(w,RinCin,'g -')
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1-pt_r);
r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
176
Appendix 3 Source Code of PPFANA
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RdeCde=abs(m-k); plot(w,RdeCde,'b -.') %
c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15; c2=cf_2.*(1+pt_c).*1e-15;
c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RdeCin=abs(m-k); plot(w,RdeCin,'r --') %
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
d=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*
c4*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
177
Appendix 3 Source Code of PPFANA
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
a=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*
r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*
c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2
.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+
2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
m=20.*log10(d); k=20.*log10(a); RinCde=abs(m-k); plot(w,RinCde,'k :')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title('Gain mismatch influenced by process tolerances','FontSize', 14)
A3.2 Source Code for Analysis of Phase Mismatch Influenced
by Process Tolerances
A.3.2.1 Source Code for Analysis of Phase Mismatch Influenced by
Process Tolerances in 2-stage RF CMOS PPFs
global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global pt_r; global pt_c;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));
rt_1=findobj(gcf,'Tag','rt_1'); rt_1=str2double(get(rt_1,'string'));
rt_2=findobj(gcf,'Tag','rt_2'); rt_2=str2double(get(rt_2,'string'));
ct_1=findobj(gcf,'Tag','ct_1'); ct_1=str2double(get(ct_1,'string'));
ct_2=findobj(gcf,'Tag','ct_2'); ct_2=str2double(get(ct_2,'string'));
pt_r=findobj(gcf,'Tag','pt_r');
pt_r=str2double(get(pt_r,'string'));
if isempty(pt_r), pt_r=0.16; else pt_r= pt_r; end
pt_c=findobj(gcf,'Tag','pt_c'); pt_c=str2double(get(pt_c,'string'));
if isempty(pt_c), pt_c=0.21; else pt_c=pt_c; end
178
Appendix 3 Source Code of PPFANA
if isempty(f_1), f_1=1.44e9; else f_1=f_1; end
if isempty(f_2), f_2=3.47e9; else f_2=f_2; end
if isempty(rt_1), rt_1=223.5; else rt_1=rt_1; end
if isempty(rt_2), rt_2=265.6; else rt_2=rt_2; end
if isempty(ct_1), ct_1=205.2; else ct_1=ct_1; end
if isempty(ct_2), ct_2=416.7; else ct_2=ct_2; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2); c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15;
r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*
w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
h=atan(a); n=atan(d); RinCin=(h-n).*180./pi+90; plot(w,RinCin,'g -s')
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*
w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
h=atan(a); n=atan(d); RdeCde=(h-n).*180./pi+90; plot(w,RdeCde,'b -*')
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*
w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
h=atan(a); n=atan(d); RinCde=(h-n).*180./pi+90; plot(w,RinCde,'r -^')
c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
h=atan(a); n=atan(d); RdeCin=(h-n).*180./pi+90; plot(w,RdeCin,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4);
c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*
w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
h=atan(a); n=atan(d); RinCin=(h-n).*180./pi+90; plot(w,RinCin,'g -')
179
Appendix 3 Source Code of PPFANA
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*
w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
h=atan(a); n=atan(d); RdeCde=(h-n).*180./pi+90; plot(w,RdeCde,'b -.') % %
c2_2=ct_2.*(1-pt_c).*1e-15; c2_1=ct_1.*(1-pt_c).*1e-15; r2_1=rt_1.*(1+pt_r); r2_2=rt_2.*(1+pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*
w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
h=atan(a); n=atan(d); RinCde=(h-n).*180./pi+90; plot(w,RinCde,'r --') % %
c2_2=ct_2.*(1+pt_c).*1e-15; c2_1=ct_1.*(1+pt_c).*1e-15; r2_1=rt_1.*(1-pt_r); r2_2=rt_2.*(1-pt_r);
a=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
d=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*
w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
h=atan(a); n=atan(d); RdeCin=(h-n).*180./pi+90; plot(w,RdeCin,'k :')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase difference influenced by process tolerances','FontSize', 14)
A.3.2.2 Source Code for Analysis of Phase Mismatch Influenced by
Process Tolerance in 3-stage RF CMOS PPFs
global f_1; global f_2; global rth_1; global rth_2; global rth_3; global cth_1; global cth_2;
global cth_3; global cm; global pt_r; global pt_c;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));
rth_1=findobj(gcf,'Tag','rth_1'); rth_1=str2double(get(rth_1,'string'));
rth_2=findobj(gcf,'Tag','rth_2'); rth_2=str2double(get(rth_2,'string'));
rth_3=findobj(gcf,'Tag','rth_3'); rth_3=str2double(get(rth_3,'string'));
cth_1=findobj(gcf,'Tag','cth_1'); cth_1=str2double(get(cth_1,'string'));
cth_2=findobj(gcf,'Tag','cth_2'); cth_2=str2double(get(cth_2,'string'));
cth_3=findobj(gcf,'Tag','cth_3'); cth_3=str2double(get(cth_3,'string'));
pt_r=findobj(gcf,'Tag','pt_r'); pt_r=str2double(get(pt_r,'string'));
if isempty(pt_r), pt_r=0.16; else pt_r= pt_r; end
pt_c=findobj(gcf,'Tag','pt_c');
pt_c=str2double(get(pt_c,'string'));
if isempty(pt_c), pt_c=0.21; else pt_c=pt_c; end
if isempty(f_1), f_1=1.44e9; else f_1=f_1; end
if isempty(f_2), f_2=3.47e9; else f_2=f_2; end
if isempty(rth_1), rth_1=263.9; else rth_1=rth_1; end
180
Appendix 3 Source Code of PPFANA
if isempty(rth_2), rth_2=293.7; else rth_2=rth_2; end
if isempty(rth_3), rth_3=315.2; else rth_3=rth_3; end
if isempty(cth_1), cth_1=173.8; else cth_1=cth_1; end
if isempty(cth_2), cth_2=221.2; else cth_2=cth_2; end
if isempty(cth_3), cth_3=350.7; else cth_3=cth_3; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);
c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15; c3_1=cth_1.*(1+pt_c).*1e-15;
r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c
3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -s')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c
3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -*')
c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15; c3_1=cth_1.*(1+pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
181
Appendix 3 Source Code of PPFANA
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2.
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90; plot(w,RdeCin,'r -d')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_2.*(1+0.4); c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15;
c3_1=cth_1.*(1+pt_c).*1e-15; r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
182
Appendix 3 Source Code of PPFANA
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c
3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -.')
c3_3=cth_3.*(1+pt_c).*1e-15; c3_2=cth_2.*(1+pt_c).*1e-15; c3_1=cth_1.*(1+pt_c).*1e-15;
r3_1=rth_1.*(1-pt_r); r3_2=rth_2.*(1-pt_r); r3_3=rth_3.*(1-pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
t=abs(( j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c
3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90; plot(w,RdeCin,'r --')
c3_3=cth_3.*(1-pt_c).*1e-15; c3_2=cth_2.*(1-pt_c).*1e-15; c3_1=cth_1.*(1-pt_c).*1e-15;
r3_1=rth_1.*(1+pt_r); r3_2=rth_2.*(1+pt_r); r3_3=rth_3.*(1+pt_r);
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
183
Appendix 3 Source Code of PPFANA
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c
3_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'k :')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase difference influenced by process tolerances','FontSize', 14)
A.3.2.3 Source Code for Analysis of Phase Mismatch Influenced by
Process Tolerance in 4-stage RF CMOS PPFs
global f_1; global f_4; global rf_1; global rf_2; global rf_3; global rf_4; global cf_1;
global cf_2; global cf_3; global cf_4; global cm; global pt_r; global pt_c;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_4=findobj(gcf,'Tag','f_4'); f_4=str2double(get(f_4,'string'));
rf_1=findobj(gcf,'Tag','rf_1');
rf_1=str2double(get(rf_1,'string'));
rf_2=findobj(gcf,'Tag','rf_2');
rf_2=str2double(get(rf_2,'string'));
rf_3=findobj(gcf,'Tag','rf_3');
rf_3=str2double(get(rf_3,'string'));
rf_4=findobj(gcf,'Tag','rf_4');
rf_4=str2double(get(rf_4,'string'));
cf_1=findobj(gcf,'Tag','cf_1');
cf_1=str2double(get(cf_1,'string'));
cf_2=findobj(gcf,'Tag','cf_2');
cf_2=str2double(get(cf_2,'string'));
cf_3=findobj(gcf,'Tag','cf_3');
cf_3=str2double(get(cf_3,'string'));
cf_4=findobj(gcf,'Tag','cf_4');
cf_4=str2double(get(cf_4,'string'));
pt_r=findobj(gcf,'Tag','pt_r');
if isempty(pt_r), pt_r=0.16; else
pt_c=findobj(gcf,'Tag','pt_c');
pt_r=str2double(get(pt_r,'string'));
pt_r= pt_r; end
pt_c=str2double(get(pt_c,'string'));
if isempty(pt_c),
pt_c=0.21; else
pt_c=pt_c; end
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_4),
f_4=3.47e9; else
f_4=f_4; end
if isempty(rf_1),
rf_1=289.9; else
if isempty(rf_2),
rf_2=329; else
if isempty(rf_3),
rf_3=377.4; else rf_3=rf_3; end
if isempty(rf_4),
rf_4=390; else
if isempty(cf_1),
cf_1=158.2; else
cf_1=cf_1; end
if isempty(cf_2),
cf_2=186.9; else
cf_2=cf_2; end
rf_1=rf_1; end
rf_2=rf_2; end
rf_4=rf_4; end
if isempty(cf_3),
cf_3=218.5; else cf_3=cf_3; end
if isempty(cf_4),
cf_4=283.4; else
cf_4=cf_4; end
184
Appendix 3 Source Code of PPFANA
figure hold on
w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*(1+pt_c).*1e-15;
c3=cf_3.*(1+pt_c).*1e-15; c2=cf_2.*(1+pt_c).*1e-15; c1=cf_1.*(1+pt_c).*1e-15;
r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RinCin=(n-h).*180./pi+90; plot(w,RinCin,'g -s')
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
185
Appendix 3 Source Code of PPFANA
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RdeCde=(n-h).*180./pi+90; plot(w,RdeCde,'b -*') %
c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15; c2=cf_2.*(1+pt_c).*1e-15;
c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4.+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RdeCin=(n-h).*180./pi+90; plot(w,RdeCin,'r -^') %
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)
186
Appendix 3 Source Code of PPFANA
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)
-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RinCde=(n-h).*180./pi+90; plot(w,RinCde,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4); c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15;
c2=cf_2.*(1+pt_c).*1e-15; c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r);
r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RinCin=(n-h).*180./pi+90; plot(w,RinCin,'g -')
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))/(1(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*
r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*
c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2
.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2
+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r
3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c
4+2*r1*r4*r3*c3*c2*c4+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
187
Appendix 3 Source Code of PPFANA
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4+2*r1*r2*r4*c3
*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RdeCde=(n-h).*180./pi+90; plot(w,RdeCde,'b -.') %
c4=cf_4.*(1+pt_c).*1e-15; c3=cf_3.*(1+pt_c).*1e-15; c2=cf_2.*(1+pt_c).*1e-15;
c1=cf_1.*(1+pt_c).*1e-15; r1=rf_1.*(1-pt_r); r2=rf_2.*(1-pt_r); r3=rf_3.*(1-pt_r); r4=rf_4.*(1-pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)
-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RdeCin=(n-h).*180./pi+90; plot(w,RdeCin,'r --') %
c4=cf_4.*(1-pt_c).*1e-15; c3=cf_3.*(1-pt_c).*1e-15; c2=cf_2.*(1-pt_c).*1e-15;
c1=cf_1.*(1-pt_c).*1e-15; r1=rf_1.*(1+pt_r); r2=rf_2.*(1+pt_r); r3=rf_3.*(1+pt_r); r4=rf_4.*(1+pt_r);
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
188
Appendix 3 Source Code of PPFANA
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RinCde=(n-h).*180./pi+90; plot(w,RinCde,'k :')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase difference influenced by process tolerances','FontSize', 14)
A3.3 Source Code for Analysis of Gain Mismatch Influenced
by Component Mismatch
A.3.3.1 Source Code for Analysis of Gain Mismatch Influenced by Component
Mismatch in 2-stage RF CMOS PPFs
global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global cm; global pt;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));
rt_1=findobj(gcf,'Tag','rt_1'); rt_1=str2double(get(rt_1,'string')); rt_2=findobj(gcf,'Tag','rt_2');
rt_2=str2double(get(rt_2,'string')); ct_1=findobj(gcf,'Tag','ct_1'); ct_1=str2double(get(ct_1,'string'));
ct_2=findobj(gcf,'Tag','ct_2');
cm=findobj(gcf,'Tag','cm');
if isempty(cm), cm=0.01;else
ct_2=str2double(get(ct_2,'string'));
cm=str2double(get(cm,'string'));
cm=cm; end
if isempty(f_1), f_1=1.44e9; else
f_1=f_1;end
if isempty(f_2),
f_2=f_2;end
f_2=3.47e9; else
if isempty(rt_1), rt_1=223.5; else
rt_1=rt_1;end
if isempty(rt_2), rt_2=265.6; else
rt_2=rt_2;end
if isempty(ct_1), ct_1=205.2; else
ct_1=ct_1; end
if isempty(ct_2), ct_2=416.7; else
ct_2=ct_2; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2); c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15;
r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15;
189
Appendix 3 Source Code of PPFANA
r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w
)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b); k=20.*log10(c); RinCin=abs(m-k); plot(w,RinCin,'g -s')
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w
)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b); k=20.*log10(c); RdeCde=abs(m-k); plot(w,RdeCde,'b -*')
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1+cm);r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w
)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b); k=20.*log10(c); RinCde=abs(m-k); plot(w,RinCde,'r - ^')
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w
)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b); k=20.*log10(c); RdeCin=abs(m-k); plot(w,RdeCin,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2); c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w
)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b); k=20.*log10(c); RinCin=abs(m-k); plot(w,RinCin,'g -')
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w
)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b); k=20.*log10(c); RdeCde=abs(m-k); plot(w,RdeCde,'b -.') % %
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
190
Appendix 3 Source Code of PPFANA
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b);k=20.*log10(c); RinCde=abs(m-k); plot(w,RinCde,'r --')% %
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w
)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
m=20.*log10(b); k=20.*log10(c); RdeCin=abs(m-k); plot(w,RdeCin,'k :')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title('Gain mismatch influenced by component mismatch','FontSize', 14)
A.3.3.2 Source Code for Analysis of Gain Mismatch Influenced by
Component Mismatch in 3-stage RF CMOS PPFs
global f_1; global f_2; global rth_1; global rth_2; global rth_3; global cth_1;
global cth_2; global cth_3; global cm;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));
rth_1=findobj(gcf,'Tag','rth_1');
rth_1=str2double(get(rth_1,'string'));
rth_2=findobj(gcf,'Tag','rth_2');
rth_2=str2double(get(rth_2,'string'));
rth_3=findobj(gcf,'Tag','rth_3');
rth_3=str2double(get(rth_3,'string'));
cth_1=findobj(gcf,'Tag','cth_1');
cth_1=str2double(get(cth_1,'string'));
cth_2=findobj(gcf,'Tag','cth_2');
cth_2=str2double(get(cth_2,'string'));
cth_3=findobj(gcf,'Tag','cth_3');
cth_3=str2double(get(cth_3,'string'));
cm=findobj(gcf,'Tag','cm');
if isempty(cm), cm=0.01; else
cm=str2double(get(cm,'string'));
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else f_1=f_1; end
if isempty(f_2),
f_2=3.47e9; else
f_2=f_2; end
if isempty(rth_1),
rth_1=263.9; else
rth_1=rth_1; end
if isempty(rth_2),
rth_2=293.7; else
rth_2=rth_2; end
if isempty(rth_3),
rth_3=315.2; else
rth_3=rth_3; end
if isempty(cth_1),
cth_1=173.8; else
cth_1=cth_1; end
if isempty(cth_2),
cth_2=221.2; else
cth_2=cth_2; end
if isempty(cth_3),
cth_3=350.7; else
cth_3=cth_3; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2); c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15;
191
Appendix 3 Source Code of PPFANA
c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
b=abs((*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3
_2.*c3_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -s') c3_3=cth_3.*1e-15;
c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -*') c3_3=cth_3.*1e-15;
c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
192
Appendix 3 Source Code of PPFANA
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'r -^') c3_3=cth_3.*1e-15;
c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2); c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;
r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
193
Appendix 3 Source Code of PPFANA
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCin=abs(m-k); plot(w,RinCin,'g -')
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCde=abs(m-k); plot(w,RdeCde,'b -.') %
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RinCde=abs(m-k); plot(w,RinCde,'r --') %
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
194
Appendix 3 Source Code of PPFANA
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); RdeCin=abs(m-k); plot(w,RdeCin,'k :') %
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title('Gain mismatch influenced by component mismatch','FontSize', 14)
A.3.3.3 Source Code for Analysis of Gain Mismatch Influenced by Component
Mismatch in 4-stage RF CMOS PPFs
global f_1; global f_4; global rf_1; global rf_2; global rf_3; global rf_4; global cf_1; global cf_2;
global cf_3; global cf_4; global cm;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_4=findobj(gcf,'Tag','f_4'); f_4=str2double(get(f_4,'string'));
rf_1=findobj(gcf,'Tag','rf_1'); rf_1=str2double(get(rf_1,'string'));
rf_2=findobj(gcf,'Tag','rf_2'); rf_2=str2double(get(rf_2,'string'));
rf_3=findobj(gcf,'Tag','rf_3');
rf_3=str2double(get(rf_3,'string'));
rf_4=findobj(gcf,'Tag','rf_4');
rf_4=str2double(get(rf_4,'string'));
cf_1=findobj(gcf,'Tag','cf_1');
cf_1=str2double(get(cf_1,'string'));
cf_2=findobj(gcf,'Tag','cf_2');
cf_2=str2double(get(cf_2,'string'));
cf_3=findobj(gcf,'Tag','cf_3');
cf_3=str2double(get(cf_3,'string'));
cf_4=findobj(gcf,'Tag','cf_4');
cf_4=str2double(get(cf_4,'string'));
cm=findobj(gcf,'Tag','cm');
cm=str2double(get(cm,'string'));
if isempty(cm), cm=0.01; else
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_4),
f_4=3.47e9; else
f_4=f_4; end
if isempty(rf_1),
rf_1=289.9; else
if isempty(rf_2), rf_2=329; else
if isempty(rf_3),
rf_1=rf_1; end
rf_2=rf_2; end
rf_3=377.4; else
rf_3=rf_3; end
195
Appendix 3 Source Code of PPFANA
if isempty(rf_4),
rf_4=390; else
rf_4=rf_4; end
if isempty(cf_1),
cf_1=158.2; else
cf_1=cf_1; end
if isempty(cf_2),
cf_2=186.9; else
cf_2=cf_2; end
if isempty(cf_3),
cf_3=218.5; else
cf_3=cf_3; end
if isempty(cf_4),
cf_4=283.4; else
cf_4=cf_4; end
figure hold on
w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15;
c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*
r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15; c3=cf_3.*(1+cm).*1e-15; c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1+cm); r2=rf_2.*(1+cm); r3=rf_3.*(1+cm); r4=rf_4.*(1+cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4).
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RinCin=abs(l-m); plot(w,RinCin,'g -s')
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1;
r2=rf_2; r3=rf_3; r4=rf_4;
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*
r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15; c3=cf_3.*(1-cm).*1e-15; c2=cf_2.*(1-cm).*1e-15; c1=cf_1.*(1-cm).*1e-15;
196
Appendix 3 Source Code of PPFANA
r1=rf_1.*(1-cm); r2=rf_2.*(1-cm); r3=rf_3.*(1-cm); r4=rf_4.*(1-cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4) -(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RdeCde=abs(l-m); plot(w,RdeCde,'b -*') % %
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*
r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15; c3=cf_3.*(1-cm).*1e-15; c2=cf_2.*(1-cm).*1e-15; c1=cf_1.*(1-cm).*1e-15;
r1=rf_1.*(1+cm); r2=rf_2.*(1+cm); r3=rf_3.*(1+cm); r4=rf_4.*(1+cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*
r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*
c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2
.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+
2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RinCde=abs(l-m); plot(w,RinCde,'r -^') % %
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
197
Appendix 3 Source Code of PPFANA
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15; c3=cf_3.*(1+cm).*1e-15; c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1-cm); r2=rf_2.*(1-cm); r3=rf_3.*(1-cm); r4=rf_4.*(1-cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RdeCin=abs(l-m); plot(w,RdeCin,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4); c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15;
c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4).
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*
r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15; c3=cf_3.*(1+cm).*1e-15; c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1+cm); r2=rf_2.*(1+cm); r3=rf_3.*(1+cm); r4=rf_4.*(1+cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*
r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*
c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2
.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+
2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r
4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2
*r1*r4*r3*c3*c2*c4 +2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RinCin=abs(l-m); plot(w,RinCin,'g -')
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
198
Appendix 3 Source Code of PPFANA
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15; c3=cf_3.*(1-cm).*1e-15; c2=cf_2.*(1-cm).*1e-15; c1=cf_1.*(1-cm).*1e-15;
r1=rf_1.*(1-cm); r2=rf_2.*(1-cm); r3=rf_3.*(1-cm); r4=rf_4.*(1-cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RdeCde=abs(l-m); plot(w,RdeCde,'b -.') % %
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4.
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)
-(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15; c3=cf_3.*(1-cm).*1e-15; c2=cf_2.*(1-cm).*1e-15; c1=cf_1.*(1-cm).*1e-15;
r1=rf_1.*(1+cm); r2=rf_2.*(1+cm); r3=rf_3.*(1+cm); r4=rf_4.*(1+cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
199
Appendix 3 Source Code of PPFANA
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RinCde=abs(l-m); plot(w,RinCde,'r --') % %
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15; r1=rf_1;
r2=rf_2; r3=rf_3; r4=rf_4;
a=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15; c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1-cm); r2=rf_2.*(1-cm); r3=rf_3.*(1-cm); r4=rf_4.*(1-cm);
d=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))/(1(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*
r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*
c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2
.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+
2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
l=20.*log10(a); m=20.*log10(d); RdeCin=abs(l-m); plot(w,RdeCin,'k :')%
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title('Gain mismatch influenced by component mismatch','FontSize', 14)
A3.4 Source
Code for Analysis of Phase Mismatch Influenced by
Component Mismatch
A.3.4.1 Source Code for Analysis of Phase Mismatch Influenced by Component
Mismatch in 2-stage RF CMOS PPFs
global f_1; global f_2; global rt_1; global rt_2; global ct_1; global ct_2; global cm; global pt;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string'));%get the value from GUI input
f_2=findobj(gcf,'Tag','f_2'); f_2=str2double(get(f_2,'string'));
rt_1=findobj(gcf,'Tag','rt_1');
rt_1=str2double(get(rt_1,'string'));
rt_2=findobj(gcf,'Tag','rt_2'); rt_2=str2double(get(rt_2,'string'));
ct_1=findobj(gcf,'Tag','ct_1'); ct_1=str2double(get(ct_1,'string'));
200
Appendix 3 Source Code of PPFANA
ct_2=findobj(gcf,'Tag','ct_2'); ct_2=str2double(get(ct_2,'string'));
cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string'));
if isempty(cm), cm=0.01;else
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_2),
f_2=3.47e9; else
f_2=f_2; end
if isempty(rt_1),
rt_1=223.5; else
rt_1=rt_1; end
if isempty(rt_2),
rt_2=265.6; else
rt_2=rt_2; end
if isempty(ct_1),
ct_1=205.2; else
ct_1=ct_1; end
if isempty(ct_2),
ct_2=416.7; else
ct_2=ct_2; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1;r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.
*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1*c2_2)));
c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RinCin=(m-k).*180./pi+90;
plot(w,RinCin,'g -s') c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RdeCde=(m-k).*180./pi+90; plot(w,RdeCde,'b -*')
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RinCde=(m-k).*180./pi+90; plot(w,RinCde,'r -^')
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
201
Appendix 3 Source Code of PPFANA
c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RdeCin=(m-k).*180./pi+90; plot(w,RdeCin,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2);
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*(1+cm).*1e-15; c2_1=ct_1.*(1+cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RinCin=(m-k).*180./pi+90; plot(w,RinCin,'g -')
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RdeCde=(m-k).*180./pi+90; plot(w,RdeCde,'b -.') % %
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*(1-cm).*1e-15; c2_1=ct_1.*(1-cm).*1e-15; r2_1=rt_1.*(1+cm); r2_2=rt_2.*(1+cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RinCde=(m-k).*180./pi+90; plot(w,RinCde,'r --') % %
c2_2=ct_2.*1e-15; c2_1=ct_1.*1e-15; r2_1=rt_1; r2_2=rt_2;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*(1+cm).*1e-15;c2_1=ct_1.*(1+cm).*1e-15;r2_1=rt_1.*(1-cm); r2_2=rt_2.*(1-cm);
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
202
Appendix 3 Source Code of PPFANA
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); RdeCin=(m-k).*180./pi+90; plot(w,RdeCin,'k :')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase difference influenced by component mismatch','FontSize', 14)
A.3.4.2 Source Code for Analysis of Phase Mismatch Influenced by Component
Mismatch in 3-stage RF CMOS PPFs
global f_1; global f_2; global rth_1; global rth_2; global rth_3;
global cth_1; global cth_2; global cth_3; global cm;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2');
f_2=str2double(get(f_2,'string'));
rth_1=findobj(gcf,'Tag','rth_1');
rth_1=str2double(get(rth_1,'string'));
rth_2=findobj(gcf,'Tag','rth_2');
rth_2=str2double(get(rth_2,'string'));
rth_3=findobj(gcf,'Tag','rth_3');
rth_3=str2double(get(rth_3,'string'));
cth_1=findobj(gcf,'Tag','cth_1');
cth_1=str2double(get(cth_1,'string'));
cth_2=findobj(gcf,'Tag','cth_2');
cth_2=str2double(get(cth_2,'string'));
cth_3=findobj(gcf,'Tag','cth_3');
cth_3=str2double(get(cth_3,'string'));
cm=findobj(gcf,'Tag','cm');
if isempty(cm), cm=0.01; else
cm=str2double(get(cm,'string'));
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_2),
f_2=3.47e9; else
f_2=f_2; end
if isempty(rth_1),
rth_1=263.9; else
rth_1=rth_1; end
if isempty(rth_2),
rth_2=293.7; else
rth_2=rth_2; end
if isempty(rth_3),
rth_3=315.2; else
rth_3=rth_3; end
if isempty(cth_1),
cth_1=173.8; else
cth_1=cth_1; end
if isempty(cth_2),
cth_2=221.2; else
cth_2=cth_2; end
if isempty(cth_3),
cth_3=350.7; else
cth_3=cth_3; end
figure hold on
w=f_2.*(1+0.1):f_2./20:f_2.*(1+0.2);
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;
r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
203
Appendix 3 Source Code of PPFANA
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -s')
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1;
r3_2=rth_2; r3_3=rth_3;
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -*')
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;
r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
204
Appendix 3 Source Code of PPFANA
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'r -^')
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1;
r3_2=rth_2; r3_3=rth_3;
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90;
plot(w,RdeCin,'k -d') legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.2):f_1./1e3:f_2.*(1+0.2); c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15;
c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCin=(l-n).*180./pi+90; plot(w,RinCin,'g -')
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
205
Appendix 3 Source Code of PPFANA
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3
_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.
*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCde=(l-n).*180./pi+90; plot(w,RdeCde,'b -.') %
c3_3=cth_3.*1e-15;c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15;
r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1-cm).*1e-15; c3_2=cth_2.*(1-cm).*1e-15; c3_1=cth_1.*(1-cm).*1e-15;
r3_1=rth_1.*(1+cm); r3_2=rth_2.*(1+cm); r3_3=rth_3.*(1+cm);
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RinCde=(l-n).*180./pi+90; plot(w,RinCde,'r --') %
c3_3=cth_3.*1e-15; c3_2=cth_2.*1e-15; c3_1=cth_1.*1e-15; r3_1=rth_1; r3_2=rth_2; r3_3=rth_3;
h=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
206
Appendix 3 Source Code of PPFANA
c3_3=cth_3.*(1+cm).*1e-15; c3_2=cth_2.*(1+cm).*1e-15; c3_1=cth_1.*(1+cm).*1e-15;
r3_1=rth_1.*(1-cm); r3_2=rth_2.*(1-cm); r3_3=rth_3.*(1-cm);
t=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
l=atan(h); n=atan(t); RdeCin=(l-n).*180./pi+90; plot(w,RdeCin,'k :') %
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase difference influenced by component mismatch','FontSize', 14)
A.3.4.3 Source Code for Analysis of Phase Mismatch Influenced by Component
Mismatch in 4-stage RF CMOS PPFs
global f_1;global f_4;global rf_1;global rf_2;global rf_3;global rf_4;
global cf_1;global cf_2; global cf_3;global cf_4;global cm;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_4=findobj(gcf,'Tag','f_4');
f_4=str2double(get(f_4,'string'));
rf_1=findobj(gcf,'Tag','rf_1');
rf_1=str2double(get(rf_1,'string'));
rf_2=findobj(gcf,'Tag','rf_2');
rf_2=str2double(get(rf_2,'string'));
rf_3=findobj(gcf,'Tag','rf_3');
rf_3=str2double(get(rf_3,'string'));
rf_4=findobj(gcf,'Tag','rf_4');
rf_4=str2double(get(rf_4,'string'));
cf_1=findobj(gcf,'Tag','cf_1');
cf_1=str2double(get(cf_1,'string'));
cf_2=findobj(gcf,'Tag','cf_2');
cf_2=str2double(get(cf_2,'string'));
cf_3=findobj(gcf,'Tag','cf_3');
cf_3=str2double(get(cf_3,'string'));
cf_4=findobj(gcf,'Tag','cf_4');
cf_4=str2double(get(cf_4,'string'));
cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string'));
if isempty(cm),
cm=0.01; else
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_4),
f_4=3.47e9; else
f_4=f_4; end
if isempty(rf_1),
rf_1=289.9; else
if isempty(rf_2),
rf_2=329; else
if isempty(rf_3),
rf_3=377.4; else
if isempty(rf_4),
rf_4=390; else
if isempty(cf_1),
cf_1=158.2; else
cf_1=cf_1; end
if isempty(cf_2),
cf_2=186.9; else
cf_2=cf_2; end
if isempty(cf_3),
cf_3=218.5; else
cf_3=cf_3; end
if isempty(cf_4),
cf_4=283.4; else
cf_4=cf_4; end
rf_1=rf_1; end
rf_2=rf_2; end
rf_3=rf_3; end
rf_4=rf_4; end
figure hold on
207
Appendix 3 Source Code of PPFANA
w=f_4.*(1+0.1):f_4./20:f_4.*(1+0.2); c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15;
c1=cf_1.*1e-15; r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)
-(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15; c3=cf_3.*(1+cm).*1e-15; c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1+cm); r2=rf_2.*(1+cm); r3=rf_3.*(1+cm); r4=rf_4.*(1+cm);
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RinCin=(n-h).*180./pi+90; plot(w,RinCin,'g -s')
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15;
r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;c1=cf_1.*(1-cm).*1e-15;
r1=rf_1.*(1-cm);r2=rf_2.*(1-cm);r3=rf_3.*(1-cm);r4=rf_4.*(1-cm);
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
208
Appendix 3 Source Code of PPFANA
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RdeCde=(n-h).*180./pi+90; plot(w,RdeCde,'b -*') % %
c4=cf_4.*1e-15; c3=cf_3.*1e-15; c2=cf_2.*1e-15; c1=cf_1.*1e-15;
r1=rf_1; r2=rf_2; r3=rf_3; r4=rf_4;
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*
r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;
c1=cf_1.*(1-cm).*1e-15;r1=rf_1.*(1+cm);r2=rf_2.*(1+cm);r3=rf_3.*(1+cm);r4=rf_4.*(1+cm);
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4) (j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r
1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c);n=atan(b);RinCde=(n-h).*180./pi+90;plot(w,RinCde,'r -^')% %
c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;r1=rf_1;r2=rf_2; r3=rf_3;r4=rf_4;
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1
*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)
+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*
r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15;c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15;c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1-cm);r2=rf_2.*(1-cm);r3=rf_3.*(1-cm);r4=rf_4.*(1-cm);
209
Appendix 3 Source Code of PPFANA
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c);n=atan(b);RdeCin=(n-h).*180./pi+90;plot(w,RdeCin,'k -d')
legend('RinCin','RdeCde','RinCde','RdeCin')
w=f_1.*(1-0.4):f_1./1e3:f_4.*(1+0.4);
c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;
r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15;c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15;c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1+cm);r2=rf_2.*(1+cm);r3=rf_3.*(1+cm);r4=rf_4.*(1+cm);
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*
r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*
c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2
.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+
2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r
4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2
*r1*r4*r3*c3*c2*c4 +2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c);n=atan(b);RinCin=(n-h).*180./pi+90;plot(w,RinCin,'g -')
c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;
r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
210
Appendix 3 Source Code of PPFANA
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;
c1=cf_1.*(1-cm).*1e-15;r1=rf_1.*(1-cm);r2=rf_2.*(1-cm);r3=rf_3.*(1-cm);r4=rf_4.*(1-cm);
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c);n=atan(b);RdeCde=(n-h).*180./pi+90;plot(w,RdeCde,'b -.') % %
c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1-cm).*1e-15;c3=cf_3.*(1-cm).*1e-15;c2=cf_2.*(1-cm).*1e-15;c1=cf_1.*(1-cm).*1e-15;
r1=rf_1.*(1+cm);r2=rf_2.*(1+cm);r3=rf_3.*(1+cm);r4=rf_4.*(1+cm);
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c
4)-(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c); n=atan(b); RinCde=(n-h).*180./pi+90; plot(w,RinCde,'r --') % %
c4=cf_4.*1e-15;c3=cf_3.*1e-15;c2=cf_2.*1e-15;c1=cf_1.*1e-15;r1=rf_1;r2=rf_2;r3=rf_3;r4=rf_4;
211
Appendix 3 Source Code of PPFANA
c=abs((1+(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4
*c3) +(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4))./(1-(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3
+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c..
+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4
+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c
3*c4)+(j*(2.*pi.*w)).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+2*r1*c3+2*r2*c3+2*r2*c4+2*r1*
c4)(j*(2.*pi.*w)).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4.
+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2*r1*r4*r3*c3*c2*c4
+2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
c4=cf_4.*(1+cm).*1e-15;c3=cf_3.*(1+cm).*1e-15;c2=cf_2.*(1+cm).*1e-15; c1=cf_1.*(1+cm).*1e-15;
r1=rf_1.*(1-cm);r2=rf_2.*(1-cm);r3=rf_3.*(1-cm);r4=rf_4.*(1-cm);
b=abs(((2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4)+(2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c
4 +r1*r3*r4*c1*c3*c4+r3*r2*r4*c3*c2*c4))./(1(2.*pi.*w).^2.*(r1*r2*c1*c2+r1*r3*c1*c3+r2*r3*c2*c3+r1*r4*c1*c4+r2*r4*c2*c4+r4*r3*c4*c3+2*r1*
r3*c1*c4+2*r2*r3*c2*c4+2*r1*r2*c2*c4+2*r1*r2*c1*c4+2*r1*r3*c3*c4+2*r2*r3*c3*c4+2*r1*r3*c2*
c3+2*r1*r2*c2*c3+2*r1*r4*c2*c4+2*r1*r4*c3*c4+2*r2*r4*c3*c4+2*r1*r2*c1*c3+4*r1*r3*c2*c4)+(2
.*pi.*w).^4.*(r1*r2*r3*r4*c1*c2*c3*c4)+(j.*2.*pi.*w).*(r1*c1+r2*c2+r3*c3+r4*c4+2*r3*c4+2*r1*c2+
2*r1*c3+2*r2*c3+2*r2*c4+2*r1*c4)(j.*2.*pi.*w).^3.*(r1*r2*r3*c1*c2*c3+r1*r2*r4*c1*c2*c4+r1*r3*r
4*c1*c3*c4+r3*r2*r4*c3*c2*c4+2*r1*r2*r3*c1*c2*c4+2*r1*r2*r3*c3*c2*c4+2*r1*r2*r3*c1*c3*c4+2
*r1*r4*r3*c3*c2*c4 +2*r1*r2*r4*c3*c2*c4+2*r1*r2*r4*c1*c3*c4)));
h=atan(c);n=atan(b);RdeCin=(n-h).*180./pi+90;plot(w,RdeCin,'k :')%
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase difference influenced by component mismatch','FontSize', 14)
A3.5 Source
Code for Analysis of Gain Mismatch in the Worst-Case
Simulation
A.3.5.1 Source Code for Analysis of Gain Mismatch in the Worst-Case Simulation
of 2-stage RF CMOS PPFs
global f_1;global f_2;global rt_1;global rt_2;global ct_1;global ct_2;global cm; global pt;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string'));%get the value from GUI input
f_2=findobj(gcf,'Tag','f_2');
f_2=str2double(get(f_2,'string'));
rt_1=findobj(gcf,'Tag','rt_1');
rt_1=str2double(get(rt_1,'string'));
rt_2=findobj(gcf,'Tag','rt_2');
rt_2=str2double(get(rt_2,'string'));
ct_1=findobj(gcf,'Tag','ct_1');
ct_1=str2double(get(ct_1,'string'));
ct_2=findobj(gcf,'Tag','ct_2');
ct_2=str2double(get(ct_2,'string'));
cm=findobj(gcf,'Tag','cm');
if isempty(cm), cm=0.01;else
cm=str2double(get(cm,'string'));
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else
if isempty(f_2),
f_2=3.47e9; else
f_1=f_1; end
f_2=f_2; end
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Appendix 3 Source Code of PPFANA
if isempty(rt_1), rt_1=223.5; else
rt_1=rt_1; end
if isempty(rt_2),
rt_2=265.6; else
rt_2=rt_2; end
if isempty(ct_1),
ct_1=205.2; else
ct_1=ct_1; end
if isempty(ct_2),
ct_2=416.7; else
ct_2=ct_2; end
h = waitbar(0,'Simulation is processing, please wait'); for x=1:100, waitbar(x/100,h), end close(h)
rand('seed',0); % Crossover Operators xFns = 'simpleXover'; xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
% Selection Function electFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'ppfana2stage'; evalOps = []; %type gaMichEval3stage;
% Bounds on the variables b_1=f_1; b_2=f_2; bounds = [b_1 b_2];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20
startPop = initializega(20,bounds,'ppfana2stage',[],[1e-6 0]);
% Lets run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found %x; % endPop is the ending population %endPop;
% trace is a trace of the best value and average value of generations %trace;
% Lets increase the population size by running the defaults rand('seed',0); termOps=[100];
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,
selectFn,selectOps); % x is the best solution found x; fmax=x; f=f_1+(f_2-f_1)./2; ptt=fmax./f;
pt_two=1-sqrt(1./ptt); figure
w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3); c2_2=ct_2.*1e-15+ct_2.*1e-5.*pt_two;
c2_1=ct_1.*1e-15+ct_1.*1e-15.*pt_two; r2_1=rt_1+rt_1.*pt_two; r2_2=rt_2+rt_2.*pt_two;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*1e-15+ct_2.*1e-15.*(1+cm).*pt_two; c2_1=ct_1.*1e-15+ct_1.*1e-15.*(1+cm).*pt_two;
r2_1=rt_1+rt_1.*(1+cm).*pt_two; r2_2=rt_2+rt_2.*(1+cm).*pt_two;
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=20.*log10(b); k=20.*log10(c); wcga=abs(m-k); plot(w,wcga,'b -')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title(' Worst case gain mismatch of 2-stage RF CMOS PPFs','FontSize', 14)
A.3.5.2 Source Code for Analysis of Gain Mismatch in the Worst-Case Simulation
of 3-stage RF CMOS PPFs
global f_1; global f_2; global rth_1; global rth_2; global rth_3;
global cth_1; global cth_2; global cth_3; global cm;
213
Appendix 3 Source Code of PPFANA
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2');
f_2=str2double(get(f_2,'string')); rth_1=findobj(gcf,'Tag','rth_1');
rth_1=str2double(get(rth_1,'string'));
rth_2=findobj(gcf,'Tag','rth_2');
rth_2=str2double(get(rth_2,'string'));
rth_3=findobj(gcf,'Tag','rth_3');
rth_3=str2double(get(rth_3,'string'));
cth_1=findobj(gcf,'Tag','cth_1');
cth_1=str2double(get(cth_1,'string')); cth_2=findobj(gcf,'Tag','cth_2');
cth_2=str2double(get(cth_2,'string')); cth_3=findobj(gcf,'Tag','cth_3');
cth_3=str2double(get(cth_3,'string')); cm=findobj(gcf,'Tag','cm');
cm=str2double(get(cm,'string')); if isempty(cm), cm=0.01; else
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_2),
f_2=3.47e9; else
f_2=f_2; end
if isempty(rth_1),
rth_1=263.9; else
rth_1=rth_1; end
if isempty(rth_2),
rth_2=293.7; else
rth_2=rth_2; end
if isempty(rth_3),
rth_3=315.2; else
rth_3=rth_3; end
if isempty(cth_1),
cth_1=173.8; else
cth_1=cth_1; end
if isempty(cth_2),
cth_2=221.2; else
cth_2=cth_2; end
if isempty(cth_3),
cth_3=350.7; else
cth_3=cth_3; end
cm=cm; end
h = waitbar(0,'Simulation is processing, please wait');
for x=1:100,
waitbar(x/100,h),
end
close(h)
rand('seed',0); % Crossover Operators xFns = 'simpleXover'; xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
% Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'ppfana3stage'; evalOps = [];
%type gaMichEval3stage;
% Bounds on the variables
% b_1=f_1./(1-2.*pt); % b_2=f_2./(1+2.*pt); b_1=f_1; b_2=f_2; bounds = [b_1 b_2];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20 startPop = initializega(20,bounds,'ppfana3stage',[],[1e-6 0]);
% Lets run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found x;% endPop is the ending population%endPop;
% trace is a trace of the best value and average value of generations trace;
% Lets increase the population size by running the defaults
rand('seed',0); ermOps=[100]; x endPop bestPop
trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps, selectFn,selectOps);
% x is the best solution found
x; max=x; f=f_1+(f_2-f_1)./2;ptt=fmax./f;
pt_three=1-sqrt(1./ptt);figure
w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3);c3_3=cth_3.*(1+pt_three).*1e-15;
c3_2=cth_2.*(1+pt_three).*1e-15;c3_1=cth_1.*(1+pt_three).*1e-15;
214
Appendix 3 Source Code of PPFANA
r3_1=rth_1.*(1+pt_three);r3_2=rth_2.*(1+pt_three);r3_3=rth_3.*(1+pt_three);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+pt_three).*(1+cm).*1e-15;c3_2=cth_2.*(1+pt_three).*(1+cm).*1e-15;
c3_1=cth_1.*(1+pt_three).*(1+cm).*1e-15;r3_1=rth_1.*(1+pt_three).*(1+cm);
r3_2=rth_2.*(1+pt_three).*(1+cm);r3_3=rth_3.*(1+pt_three).*(1+cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=20.*log10(c); k=20.*log10(b); wcga3=abs(m-k); plot(w,wcga3,'r-')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title(' Worst case gain mismatch of 3-stage RF CMOS PPFs','FontSize', 14)
A.3.5.3 Source Code for Analysis of Gain Mismatch in the Worst-Case Simulation
of 4-stage RF CMOS PPFs
global f_1;global f_4;global rf_1;global rf_2;global rf_3;global rf_4;
global cf_1;global cf_2;global cf_3;global cf_4;global cm;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_4=findobj(gcf,'Tag','f_4');
f_4=str2double(get(f_4,'string'));
rf_1=findobj(gcf,'Tag','rf_1');
rf_1=str2double(get(rf_1,'string'));
rf_2=findobj(gcf,'Tag','rf_2');
rf_2=str2double(get(rf_2,'string'));
rf_3=findobj(gcf,'Tag','rf_3');
rf_3=str2double(get(rf_3,'string'));
rf_4=findobj(gcf,'Tag','rf_4');
rf_4=str2double(get(rf_4,'string'));
cf_1=findobj(gcf,'Tag','cf_1');
cf_1=str2double(get(cf_1,'string'));
cf_2=findobj(gcf,'Tag','cf_2');
cf_2=str2double(get(cf_2,'string'));
cf_3=findobj(gcf,'Tag','cf_3');
cf_3=str2double(get(cf_3,'string'));
cf_4=findobj(gcf,'Tag','cf_4');
cf_4=str2double(get(cf_4,'string'));
cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string'));
if isempty(cm),
cm=0.01; else
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_4),
f_4=3.47e9; else
f_4=f_4; end
if isempty(rf_1),
rf_1=289.9; else
rf_1=rf_1; end
215
Appendix 3 Source Code of PPFANA
if isempty(rf_2),
rf_2=329; else
rf_2=rf_2; end
if isempty(rf_3),
rf_3=377.4; else
if isempty(rf_4),
rf_4=390; else
if isempty(cf_1),
cf_1=158.2; else
cf_1=cf_1; end
if isempty(cf_2),
cf_2=186.9; else
cf_2=cf_2; end
if isempty(cf_3),
cf_3=218.5; else
cf_3=cf_3; end
if isempty(cf_4),
cf_4=283.4; else
cf_4=cf_4; end
rf_3=rf_3; end
rf_4=rf_4; end
h = waitbar(0,'Simulation is processing, please wait');
for x=1:100,
waitbar(x/100,h),
end
close(h)
rand('seed',0);
% Crossover Operators xFns = 'simpleXover'; xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
% Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'ppfana4stage'; evalOps = [];
%type gaMichEval3stage; % Bounds on the variables
% b_1=f_1./(1-2.*pt); % b_2=f_2./(1+2.*pt); b_1=f_1; b_2=f_4; bounds = [b_1 b_2];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20 startPop = initializega(20,bounds,'ppfana4stage',[],[1e-6 0]);
% Lets run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found %x; % endPop is the ending population %endPop; % trace is a trace of the
best value and average value of generations %trace;
% Lets increase the population size by running the defaults rand('seed',0);
termOps=[100]; [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,
selectFn,selectOps);
% x is the best solution found x; fmax=x; f=f_1+(f_4-f_1)./2; ptt=fmax./f; pt_four=1-sqrt(1./ptt);
w=f_1.*(1-0.3):f_1./1e3:f_4.*(1+0.3);
c1=cf_1.*1e-15; c2=cf_2.*1e-15; c3=cf_3.*1e-15; c4=cf_4.*1e-15; dr_1=pt_four.*rf_1;
dr_2=pt_four.*rf_2; dr_3=pt_four.*rf_3; dr_4=pt_four.*rf_4; dc_1=pt_four.*c1; dc_2=pt_four.*c2;
dc_3=pt_four.*c3; dc_4=pt_four.*c4; dr_11=cm*rf_1; dr_22=cm*rf_2; dr_33=cm*rf_3;
dr_44=cm*rf_4; dc_11=cm*c1; dc_22=cm*c2; dc_33=cm*c3; dc_44=cm*c4;
r1=rf_1+dr_1+dr_11; r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33; r4=rf_4+dr_4+dr_44;
c41=c1+dc_1+dc_11; c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;
d=F_amplitudew_in(w,r1,c41,r2,c42,r3,c43,r4,c44);
n=20*log10(d); r1=rf_1+dr_1+dr_11; r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33; r4=rf_4+dr_4+dr_44;
c41=c1+dc_1+dc_11; c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;
b=F_amplitudew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44);
m=20*log10(b); gmis=abs(m-n); figure plot(w,gmis,'m-') xlabel('{\itf} / GHz','FontSize', 14)
ylabel('{\itm}-{\itgain} / dB','FontSize', 14) title(' Worst case gain mismatch of 4-stage RF CMOS
PPFs','FontSize', 14)
216
Appendix 3 Source Code of PPFANA
A3.6 Source Code for Analysis of Phase Mismatch in the Worst-Case
Simulation
A.3.6.1 Source Code for Analysis of Phase Mismatch in the Worst-Case simulation
of 2-stage RF CMOS PPFs
global f_1; global f_2; global rt_1; global rt_2;
global ct_1; global ct_2;global cm; global pt;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_2=findobj(gcf,'Tag','f_2');
f_2=str2double(get(f_2,'string'));
rt_1=findobj(gcf,'Tag','rt_1');
rt_1=str2double(get(rt_1,'string'));
rt_2=findobj(gcf,'Tag','rt_2');
rt_2=str2double(get(rt_2,'string'));
ct_1=findobj(gcf,'Tag','ct_1');
ct_1=str2double(get(ct_1,'string'));
ct_2=findobj(gcf,'Tag','ct_2');
ct_2=str2double(get(ct_2,'string'));
cm=findobj(gcf,'Tag','cm');
if isempty(cm), cm=0.01; else
cm=str2double(get(cm,'string'));
cm=cm; end
if isempty(f_1),
f_1=1.44e9; else
f_1=f_1; end
if isempty(f_2),
f_2=3.47e9; else
f_2=f_2; end
if isempty(rt_1),
rt_1=223.5; else
rt_1=rt_1; end
if isempty(rt_2),
rt_2=265.6; else
rt_2=rt_2; end
if isempty(ct_1),
ct_1=205.2; else
ct_1=ct_1; end
if isempty(ct_2),
ct_2=416.7; else
ct_2=ct_2; end
h = waitbar(0,'Simulation is processing, please wait');
for x=1:100, waitbar(x/100,h), end close(h) rand('seed',0);
% Crossover Operators xFns = 'simpleXover'; xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
% Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'ppfana2stage'; evalOps = [];
%type gaMichEval3stage;
% Bounds on the variables
b_1=f_1; b_2=f_2; bounds = [b_1 b_2];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20 startPop = initializega(20,bounds,'ppfana2stage',[],[1e-6 0]);
% Lets run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found %x; % endPop is the ending population
%endPop; % trace is a trace of the best value and average value of generations
%trace;
% Lets increase the population size by running the defaults
217
Appendix 3 Source Code of PPFANA
rand('seed',0); termOps=[100]; [x endPop bestPop
trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,
selectFn,selectOps);
% x is the best solution found x; fmax=x; f=f_1+(f_2-f_1)./2; ptt=fmax./f;
pt_two=1-sqrt(1./ptt); figure
w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3);
c2_2=ct_2.*1e-15+ct_2.*1e-15.*pt_two;c2_1=ct_1.*1e-15+ct_1.*1e-15.*pt_two;
r2_1=rt_1+rt_1.*pt_two;r2_2=rt_2+rt_2.*pt_two;
b=abs(((1+(2.*pi.*w).^2.*r2_1*r2_2*c2_1*c2_2)./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
c2_2=ct_2.*1e-15+ct_2.*1e-15.*(1+cm).*pt_two; c2_1=ct_1.*1e-15+ct_1.*1e-15.*(1+cm).*pt_two;
r2_1=rt_1+rt_1.*(1+cm).*pt_two; r2_2=rt_2+rt_2.*(1+cm).*pt_two;
c=abs(((j*(2.*pi.*w).*(r2_1*c2_1+r2_2*c2_2))./
(1+j*(2.*pi.*w)*(r2_1*c2_1+r2_2*c2_2)+2*j*(2.*pi.*w)*r2_1*c2_2+(j*(2.*pi.*w)).^2*r2_1*r2_2*c2_1
*c2_2)));
m=atan(b); k=atan(c); wcga=(m-k).*180./pi+90; plot(w,wcga,'r -')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title(' Worst case phase mismatch of 2-stage RF CMOS PPFs','FontSize', 14)
A.3.6.2 Source Code for Analysis of Phase Mismatch in the Worst- Case
simulation of 2-stage RF CMOS PPFs
global f_1;global f_2;global rth_1;global rth_2;global rth_3;
global cth_1;global cth_2;global cth_3;global cm;
f_1=findobj(gcf,'Tag','f_1'); f_1=str2double(get(f_1,'string'));%get the value from GUI input
f_2=findobj(gcf,'Tag','f_2');
f_2=str2double(get(f_2,'string'));
rth_1=findobj(gcf,'Tag','rth_1');
rth_1=str2double(get(rth_1,'string'));
rth_2=findobj(gcf,'Tag','rth_2');
rth_2=str2double(get(rth_2,'string'));
rth_3=findobj(gcf,'Tag','rth_3');
rth_3=str2double(get(rth_3,'string'));
cth_1=findobj(gcf,'Tag','cth_1');
cth_1=str2double(get(cth_1,'string'));
cth_2=findobj(gcf,'Tag','cth_2');
cth_2=str2double(get(cth_2,'string'));
cth_3=findobj(gcf,'Tag','cth_3');
cth_3=str2double(get(cth_3,'string'));
cm=findobj(gcf,'Tag','cm');
if isempty(cm), cm=0.01;else
cm=str2double(get(cm,'string'));
cm=cm;end
if isempty(f_1),
f_1=1.44e9;else
f_1=f_1;end
if isempty(f_2),
f_2=3.47e9;else
f_2=f_2;end
if isempty(rth_1),
rth_1=263.9;else
rth_1=rth_1;end
if isempty(rth_2),
rth_2=293.7;else
rth_2=rth_2;end
if isempty(rth_3),
rth_3=315.2;else
rth_3=rth_3;end
if isempty(cth_1),
cth_1=173.8;else
cth_1=cth_1;end
218
Appendix 3 Source Code of PPFANA
if isempty(cth_2),
cth_2=221.2;else cth_2=cth_2;end
if isempty(cth_3),
cth_3=350.7;else
cth_3=cth_3;end
h = waitbar(0,'Simulation is processing, please wait');
for x=1:100,
waitbar(x/100,h),
end
close(h)
rand('seed',0);
% Crossover Operators xFns = 'simpleXover'; xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
% Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'ppfana3stage'; evalOps = [];
%type gaMichEval3stage; % Bounds on the variables
% b_1=f_1./(1-2.*pt); % b_2=f_2./(1+2.*pt); b_1=f_1; b_2=f_2; bounds = [b_1 b_2];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20 startPop = initializega(20,bounds,'ppfana3stage',[],[1e-6 0]);
% Lets run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found%x; % endPop is the ending population
%endPop; % trace is a trace of the best value and average value of generations
%trace; % Lets increase the population size by running the defaults
rand('seed',0);
termOps=[100];
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,
selectFn,selectOps);
% x is the best solution found x; fmax=x; f=f_1+(f_2-f_1)./2; ptt=fmax./f; pt_three=1-sqrt(1./ptt);
figure
w=f_1.*(1-0.3):f_1./1e3:f_2.*(1+0.3);
c3_3=cth_3.*(1+pt_three).*1e-15; c3_2=cth_2.*(1+pt_three).*1e-15;
c3_1=cth_1.*(1+pt_three).*1e-15; r3_1=rth_1.*(1+pt_three);
r3_2=rth_2.*(1+pt_three); r3_3=rth_3.*(1+pt_three);
c=abs((1+(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2))./
(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
c3_3=cth_3.*(1+pt_three).*(1+cm).*1e-15; c3_2=cth_2.*(1+pt_three).*(1+cm).*1e-15;
c3_1=cth_1.*(1+pt_three).*(1+cm).*1e-15;r3_1=rth_1.*(1+pt_three).*(1+cm);
r3_2=rth_2.*(1+pt_three).*(1+cm);r3_3=rth_3.*(1+pt_three).*(1+cm);
b=abs((j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)+(j.*2.*pi.*w).^3.*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3
_3)./(1+j.*2.*pi.*w.*(r3_1.*c3_1+r3_2.*c3_2+r3_3.*c3_3)
219
Appendix 3 Source Code of PPFANA
+2.*j.*2.*pi.*w.*(r3_1.*c3_2+r3_1.*c3_3+r3_2.*c3_3)(2.*pi.*w).^2.*(r3_1.*r3_2.*c3_1.*c3_2+r3_1.*r3_3.*c3_1.*c3_3+r3_3.*r3_2.*c3_3.*c3_2
+2.*r3_1.*r3_3.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_3.*c3_2+2.*r3_1.*r3_2.*c3_1.*c3_3)+(j.*2.*pi.*w).^3.
*r3_1.*r3_2.*r3_3.*c3_1.*c3_2.*c3_3));
m=atan(c); k=atan(b); wcph3=(m-k).*180./pi+90; plot(w,wcph3,'b-')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title(' Worst case phase mismatch of 3-stage RF CMOS PPFs','FontSize', 14)
A.3.6.3 Source Code for Analysis of Phase Mismatch in the Worst-Case simulation
of 4-stage RF CMOS PPFs
global f_1;global f_4;global rf_1;global rf_2;global rf_3; global rf_4;
global cf_1;global cf_2;global cf_3;global cf_4;global cm;
f_1=findobj(gcf,'Tag','f_1');
f_1=str2double(get(f_1,'string')); %get the value from GUI input
f_4=findobj(gcf,'Tag','f_4');
f_4=str2double(get(f_4,'string'));
rf_1=findobj(gcf,'Tag','rf_1');
rf_1=str2double(get(rf_1,'string'));
rf_2=findobj(gcf,'Tag','rf_2');
rf_2=str2double(get(rf_2,'string'));
rf_3=findobj(gcf,'Tag','rf_3');
rf_3=str2double(get(rf_3,'string'));
rf_4=findobj(gcf,'Tag','rf_4');
rf_4=str2double(get(rf_4,'string'));
cf_1=findobj(gcf,'Tag','cf_1');
cf_1=str2double(get(cf_1,'string'));
cf_2=findobj(gcf,'Tag','cf_2');
cf_2=str2double(get(cf_2,'string'));
cf_3=findobj(gcf,'Tag','cf_3');
cf_3=str2double(get(cf_3,'string'));
cf_4=findobj(gcf,'Tag','cf_4');
cf_4=str2double(get(cf_4,'string'));
cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string'));
if isempty(cm),
cm=0.01; else
cm=cm; end
if isempty(f_1),
f_1=1.44e9;else
f_1=f_1;end
if isempty(f_4),
f_4=3.47e9;else
f_4=f_4;end
if isempty(rf_1),
rf_1=289.9;else
if isempty(rf_2),
rf_2=329;else
if isempty(rf_3),
rf_3=377.4;else
if isempty(rf_4),
rf_4=390;
if isempty(cf_1),
cf_1=158.2;else
if isempty(cf_2),
cf_2=186.9;else cf_2=cf_2;end
if isempty(cf_3),
cf_3=218.5;else
cf_3=cf_3;end
if isempty(cf_4),
cf_4=283.4;else
cf_4=cf_4;end
rf_1=rf_1;end
rf_2=rf_2;end
rf_3=rf_3;end
rf_4=rf_4;end
cf_1=cf_1;end
h = waitbar(0,'Simulation is processing, please wait');
for x=1:100, waitbar(x/100,h),
end
close(h)
rand('seed',0);
% Crossover Operators xFns = 'simpleXover';xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
220
Appendix 3 Source Code of PPFANA
% Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'ppfana4stage'; evalOps = [];
%type gaMichEval3stage; % Bounds on the variables
% b_1=f_1./(1-2.*pt); % b_2=f_2./(1+2.*pt);
b_1=f_1; b_2=f_4; bounds = [b_1 b_2];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20 startPop = initializega(20,bounds,'ppfana4stage',[],[1e-6 0]);
% Lets run the GA
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found %x; % endPop is the ending population %endPop;
% trace is a trace of the best value and average value of generations %trace;
% Lets increase the population size by running the defaults
rand('seed',0); termOps=[100];
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,
selectFn,selectOps);
% x is the best solution found x; fmax=x; f=f_1+(f_4-f_1)./2; ptt=fmax./f;
pt_four=1-sqrt(1./ptt); figure w=f_1.*(1-0.3):f_1./1e3:f_4.*(1+0.3); c1=cf_1.*1e-15; c2=cf_2.*1e-15;
c3=cf_3.*1e-15; c4=cf_4.*1e-15; dr_1=pt_four.*rf_1; dr_2=pt_four.*rf_2; dr_3=pt_four.*rf_3;
dr_4=pt_four.*rf_4; dc_1=pt_four.*c1; dc_2=pt_four.*c2; dc_3=pt_four.*c3; dc_4=pt_four.*c4;
dr_11=cm*rf_1; dr_22=cm*rf_2; dr_33=cm*rf_3; dr_44=cm*rf_4; dc_11=cm*c1; dc_22=cm*c2;
dc_33=cm*c3; dc_44=cm*c4; r1=rf_1+dr_1+dr_11; r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33;
r4=rf_4+dr_4+dr_44; c41=c1+dc_1+dc_11; c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33;
c44=c4+dc_4+dc_44; a=F_phasew_in(w,r1,c41,r2,c42,r3,c43,r4,c44); r1=rf_1+dr_1+dr_11;
r2=rf_2+dr_2+dr_22; r3=rf_3+dr_3+dr_33; r4=rf_4+dr_4+dr_44; c41=c1+dc_1+dc_11;
c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;
d=F_phasew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44); wcph4=d-a+90;
plot(w,wcph4,'r-') xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\itgain} / dB','FontSize', 14)
title(' Worst case phase mismatch of 4-stage RF CMOS PPFs','FontSize', 14)
221
Appendix 4 Source Code of PPFOPTIMA
Appendix 4
Source Code of PPFOPTIMA
A4.1 Source Code for an Implementation of Genetic Algorithm
function [x,endPop,bPop,traceInfo] = ga(bounds,evalFN,evalOps,startPop,opts,...
termFN, termOps,selectFN,selectOps,xOverFNs,xOverOps,mutFNs,mutOps)
% GA runs a genetic algorithm % Output Arguments: % x - the best solution found during the course of
the run % endPop - the final population % Input Arguments: % bounds - a matrix of upper and lower
ounds on the variables % evalFN - the fitness function % evalOps - options to pass to the evaluation
function ([NULL]) % startPop - a matrix of solutions that can be initialized % from initialize.m
% termFN - name of the .m termination function (['maxGenTerm']) % termOps - options string to be
passed to the termination function % selectFN - name of the .m selection function
% selectOpts - roulette selection % xOverFNS – simpleXover % xOverOps - A matrix of options to pass
to Xover.m files with the % first column being the number of that xOver to perform % similiarly for
mutation % mutFNs – binaryMutation % mutOps - A matrix of options to pass to Xover.m files with the
% first column being the number of that xOver to perform % similiarly for mutation
n=nargin; if n<2 n==6 n==10 n==12 disp('Insufficient arguements') end
if n<3 %Default evalation opts. evalOps=[]; end if n<5 opts = [1e-6 1 0]; end
if isempty(opts) opts = [1e-6 1 0]; end
if any(evalFN<48) %Not using a .m file
e1str=['x=b2f(endPop(j,:),bounds,bits); endPop(j,xZomeLength)=',... evalFN ';']; else %Are using a .m
file e1str=['x=b2f(endPop(j,:),bounds,bits);[x v]=' evalFN ... '(x,[gen evalOps]);
ndPop(j,:)=[f2b(x,bounds,bits) v];']; end
if n<6 %Default termination information termOps=[100]; termFN='maxGenTerm'; end
if n<12 %Default muatation information mutFNs=['binaryMutation']; mutOps=[0.05]; end
if n<10 %Default crossover information xOverFNs=['simpleXover']; xOverOps=[0.6]; end
if n<9 %Default select opts only i.e. roullete wheel. selectFN=['roulette']; selectOps=[]; end
if n<6 %Default termination information termOps=[100]; termFN='maxGenTerm'; end
if n<4 %No starting population passed given startPop=[]; end if isempty(startPop) %Generate a
population at random %startPop=zeros(80,size(bounds,1)+1);
startPop=initializega(80,bounds,evalFN,evalOps,opts(1:2)); end
if opts(2)==0 %binary bits=calcbits(bounds,opts(1)); end
xOverFNs=parse(xOverFNs); mutFNs=parse(mutFNs); xZomeLength = size(startPop,2); %Length of the
xzome=numVars+fitness numVar = xZomeLength-1; %Number of variables popSize = size(startPop,1);
%Number of individuals in the pop endPop = zeros(popSize,xZomeLength); %A secondary population
atrix c1 = zeros(1,xZomeLength); %An individual c2 = zeros(1,xZomeLength); %An individual
numXOvers = size(xOverFNs,1); %Number of Crossover operators numMuts = size(mutFNs,1);
%Number of Mutation operators epsilon = opts(1); %Threshold for two fittness to differ oval =
222
Appendix 4 Source Code of PPFOPTIMA
max(startPop(:,xZomeLength)); %Best value in start pop FoundIn = 1; %Number of times best has
changed done = 0; %Done with simulated evolution gen = 1; %Current Generation Number collectTrace
= (nargout>3); %Should we collect info every gen floatGA = opts(2)==1; %Probabilistic application of
ops display = opts(3); %Display progress while(~done) %Elitist Model [bval,bindx] =
max(startPop(:,xZomeLength)); %Best of current pop best = startPop(bindx,:); if ( (abs(bval oval)>epsilon) (gen==1)) %If we have a new best sol if display fprintf(1,'\n%d %f\n',gen,bval); %Update
the display end bPop(bFoundIn,:)=[gen b2f(startPop(bindx,1:numVar),bounds,bits)...
startPop(bindx,xZomeLength)]; bFoundIn=bFoundIn+1; %Update number of changes
oval=bval; %Update the best val else if display fprintf(1,'%d ',gen); %Otherwise just update num gen
end end endPop = feval(selectFN,startPop,[gen selectOps]); %Select for i=1:numMuts,
for j=1:mutOps(i,1), a = round(rand*(popSize-1)+1);
c1 = feval(deblank(mutFNs(i,:)),endPop(a,:),bounds,[gen mutOps(i,:)]);
if c1(1:numVar)==endPop(a,(1:numVar)) c1(xZomeLength)=endPop(a,xZomeLength);
else %[c1(xZomeLength) c1] = feval(evalFN,c1,[gen evalOps]); eval(e1str); end endPop(a,:)=c1; end
end else %We are running a probabilistic model of genetic operators for i=1:numXOvers,
xN=deblank(xOverFNs(i,:)); %Get the name of crossover function
cp=find(rand(popSize,1)<xOverOps(i,1)==1); if rem(size(cp,1),2) cp=cp(1:(size(cp,1)-1)); end
cp=reshape(cp,size(cp,1)/2,2); for j=1:size(cp,1) a=cp(j,1); b=cp(j,2); [endPop(a,:) endPop(b,:)] =
feval(xN,endPop(a,:),endPop(b,:),... bounds,[gen xOverOps(i,:)]); end end
for i=1:numMuts mN=deblank(mutFNs(i,:)); for j=1:popSize endPop(j,:) =
eval(mN,endPop(j,:),bounds,[gen mutOps(i,:)]); eval(e1str); end end end gen=gen+1;
done=feval(termFN,[gen termOps],bPop,endPop); %See if the ga is done startPop=endPop; %Swap the
populations [bval,bindx] = min(startPop(:,xZomeLength)); %Keep the best solution
startPop(bindx,:) = best; %replace it with the worst end [bval,bindx] = max(startPop(:,xZomeLength));
if display fprintf(1,'\n%d %f\n',gen,bval); end x=startPop(bindx,:); if opts(2)==0 %binary
x=b2f(x,bounds,bits); bPop(bFoundIn,:)=[gen b2f(startPop(bindx,1:numVar),bounds,bits)...
startPop(bindx,xZomeLength)]; else bPop(bFoundIn,:)=[gen startPop(bindx,:)];
end
A4.2 Source Code of Monte Carlo Simulation
A4.2.1 Source Code for Monte Carlo Simulation
of 2-stage RF CMOS PPFs
% load optimization results load twostageppfopt.mat r_1 r_2 c_1 c_2;
global f; w=f; global r_1; global r_2; global c_1;
global c_2; global pt_r; global pt_c; global cm; global m2_mu;
% visiable waitbar h = waitbar(0,'Simulation is processing, please ait...');
for x=1:100, % computation here % waitbar(x/100,h) end close(h) % get the value of resistor process
tolerance from input pt_r=findobj(gcf,'Tag','pt_r'); pt_r=str2double(get(pt_r,'string'));
if isempty(pt_r), pt_r=0.16; else pt_r= pt_r; end
223
Appendix 4 Source Code of PPFOPTIMA
% get the value of capacitor process tolerance from input
pt_c=findobj(gcf,'Tag','pt_c'); pt_c=str2double(get(pt_c,'string')); if isempty(pt_c), pt_c=0.21; else
pt_c=pt_c; end % get the value of maximum component mismatch from input
cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string')); if isempty(cm), cm=0.01; else cm=cm; end
% get the number of runs of MC from input m2_mu=findobj(gcf,'Tag','m2_mu');
m2_mu=str2double(get(m2_mu,'string')); if isempty(m2_mu), m2_mu=200; else m2_mu=m2_mu; end
% caculate maximum R & C variant dc_1=c_1.*pt_c.*1e-15; dc_2=c_2.*pt_c.*1e-15; dr_1=r_1.*pt_r;
dr_2=r_2.*pt_r; % generate random numbers with truncated Gaussian distribution amm_2=0;
pmm_2=0; for i=1:m2_mu; i=i+1; N=2000; % size of random numbers M=1; mu=0; % mean value
sig=0.3; % standard deviation xlo=-1; % low truncation point xhi=1; % high truncation point
s=truncnormrnd(N,M,mu,sig,xlo,xhi); N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;
t=truncnormrnd(N,M,mu,sig,xlo,xhi); % MC simulations of gain mismatch and phase difference
r2_1=r_1+dr_1.*s; r2_2=r_2+dr_2.*s; c2_1=c_1.*1e-15+dc_1.*s; c2_2=c_2.*1e-15+dc_2.*s;
d=T_amplitudept_in(w,r2_1,c2_1,r2_2,c2_2); m=T_phasept_in(w,r2_1,c2_1,r2_2,c2_2);
r2_1=(r_1+dr_1.*s).*(1+cm.*t); r2_2=(r_2+dr_2.*s).*(1+cm.*t); c2_1=(c_1.*1e-15+dc_1.*s).*(1+cm.*t);
c2_2=(c_2.*1e-15+dc_2.*s).*(1+cm.*t); b=T_amplitudept_qu(w,r2_1,c2_1,r2_2,c2_2);
n=T_phasept_qu(w,r2_1,c2_1,r2_2,c2_2); amm_1=abs(20.*log10(b)-20.*log10(d));
t1=sort(amm_1); amm_2=amm_2+t1; pmm_1=n-m+90; t2=sort(pmm_1); pmm_2=pmm_2+t2;
end amm=amm_2./m2_mu; pmm=pmm_2./m2_mu; % caculate standard deviation kerse_1=std(amm)
kerse_1=round(kerse_1.*1000)./1000; st1=findobj(gcf,'Tag','kerse_1'); set(st1,'String',[num2str(kerse_1)])
kerse_2=std(pmm-90) kerse_2=round(kerse_2.*1000)./1000; stm=findobj(gcf,'Tag','kerse_2');
set(stm,'String',[num2str(kerse_2)]) figure subplot(2,1,1);
% plot gain mismatch histogram of gain mismatch
hist(amm) xlabel('{\itm}-{\rmgain} / dB','FontSize', 14)
title('Monte Carlo Simulation for 2-stage RF CMOS PPF','FontSize', 14)
ylabel('Number of Samples','FontSize', 14)
subplot(2,1,2); % plot phase difference histogram of gain mismatch hist(pmm)
xlabel('{\itd}-{\rmphase} / degree','FontSize', 14)
ylabel('Number of Samples','FontSize', 14)
A4.2.2 Source Code for Monte Carlo Simulation
of 3-stage RF CMOS PPFs
load threestageppfopt.mat r1 r2 r3 c1 c2 c3;
global f; w=f; global r1; global r2; global r3; global c1;global c2; global c3; global pt_r; global pt_c;
global cm; global m3_mu;
h = waitbar(0,'Simulation is processing, please wait...');
for x=1:100,
% computation here %
pt_r=findobj(gcf,'Tag','pt_r');
if isempty(pt_r),
waitbar(x/100,h)
pt_r=str2double(get(pt_r,'string'));
pt_r=0.16; else
pt_r= pt_r; end
dr_1=r1.*pt_r; dr_2=r2.*pt_r; dr_3=r3.*pt_r; pt_c=findobj(gcf,'Tag','pt_c');
224
end
close(h)
Appendix 4 Source Code of PPFOPTIMA
pt_c=str2double(get(pt_c,'string'));
if isempty(pt_c),
pt_c=0.21; else
pt_c=pt_c; end
dc_1=c1.*pt_c.*1e-15;dc_2=c2.*pt_c.*1e-15; dc_3=c3.*pt_c.*1e-15
cm=findobj(gcf,'Tag','cm');
if isempty(cm),
cm=str2double(get(cm,'string'));
cm=0.01; else
cm=cm; end;
m3_mu=findobj(gcf,'Tag','m3_mu');
m3_mu=str2double(get(m3_mu,'string'));
if isempty(m3_mu),
m3_mu=200; else
m3_mu=m3_mu; end
amm_2=0; pmm_2=0;
for i=1:m3_mu;
i=i+1; N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;
s=truncnormrnd(N,M,mu,sig,xlo,xhi); N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;
t=truncnormrnd(N,M,mu,sig,xlo,xhi);
r3_1=r1+dr_1.*s; r3_2=r2+dr_2.*s; r3_3=r3+dr_3.*s; c3_1=(c1.*1e-15)+dc_1.*s;
c3_2=(c2.*1e-15)+dc_2.*s; c3_3=(c3.*1e-15)+dc_3.*s;
d=Th_amplitudept_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
m=Th_phasept_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
r3_1=(r1+dr_1.*s).*(1+cm.*t); r3_2=(r2+dr_2.*s).*(1+cm.*t);
r3_3=(r3+dr_3.*s).*(1+cm.*t); c3_1=((c1.*1e-15)+dc_1.*s).*(1+cm.*t);
c3_2=((c2.*1e-15)+dc_2.*s).*(1+cm.*t); c3_3=((c3.*1e-15)+dc_3.*s).*(1+cm.*t);
b=Th_amplitudept_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
n=Th_phasept_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
amm_1=abs(20.*log10(d)-20.*log10(b)); t1=sort(amm_1); amm_2=amm_2+t1;
pmm_1=(m-n)+90; t2=sort(pmm_1); pmm_2=pmm_2+t2; end
amm=amm_2./m3_mu; pmm=pmm_2./m3_mu; figure subplot(2,1,1);
hist(amm) xlabel('{\itm}-{\rmgain} / dB','FontSize', 14)
title('Monte Carlo Simulation of 3-stage RF CMOS PPF','FontSize', 14)
ylabel('Number of Samples','FontSize', 14) subplot(2,1,2); hist(pmm)
xlabel('{\itd}-{\rmphase} / degrees','FontSize', 14) ylabel('Number of Samples','FontSize', 14)
A4.2.3 Source Code for Monte Carlo Simulation
of 4-stage RF CMOS PPFs
load fourstageoptrc.mat;
global f; w=f; global r4_1; global r4_2; global r4_3; global r4_4; global c4_1;
global c4_2; global c4_3; global c4_4; global pt_r; global pt_c; global cm; global m4_mu;
r_1=r4_1; r_2=r4_2; r_3=r4_3; r_4=r4_4; c_1=c4_1*1e-15;
c_2=c4_2*1e-15; c_3=c4_3*1e-15; c_4=c4_4*1e-15;
h = waitbar(0,'Simulation is processing, please wait...');
for x=1:100,
% computation here %
pt_r=findobj(gcf,'Tag','pt_r');
if isempty(pt_r),
waitbar(x/100,h)
pt_r=str2double(get(pt_r,'string'));
pt_r=0.16; else
pt_r= pt_r; end
225
end
close(h)
Appendix 4 Source Code of PPFOPTIMA
pt_c=findobj(gcf,'Tag','pt_c');
if isempty(pt_c),
pt_c=str2double(get(pt_c,'string'));
pt_c=0.21; else
pt_c=pt_c; end cm=findobj(gcf,'Tag','cm');
cm=str2double(get(cm,'string'));
if isempty(cm),
cm=0.01; else
cm=cm; end
m4_mu=findobj(gcf,'Tag','m4_mu');
if isempty(m4_mu),
m4_mu=str2double(get(m4_mu,'string'));
m4_mu=200; else
amm_2=0; pmm_2=0; for i=1:m4_mu;
m4_mu=m4_mu; end
i=i+1; N=2000; M=1;mu=0; sig=0.3; xlo=-1; xhi=1;
t=truncnormrnd(N,M,mu,sig,xlo,xhi); N=2000; M=1; mu=0; sig=0.3; xlo=-1; xhi=1;
s=truncnormrnd(N,M,mu,sig,xlo,xhi); dr4_1=r_1.*pt_r; dr4_2=r_2.*pt_r; dr4_3=r_3.*pt_r;
dr4_4=r_4.*pt_r; dc_1=c_1.*pt_c; dc_2=c_2.*pt_c; dc_3=c_3.*pt_c; dc_4=c_4.*pt_c;
c1=c_1+dc_1.*t; c2=c_2+dc_2.*t; c3=c_3+dc_3.*t; c4=c_4+dc_4.*t; r4_1=r_1+dr4_1.*t;
r4_2=r_2+dr4_2.*t; r4_3=r_3+dr4_3.*t; r4_4=r_4+dr4_4.*t;
d=F_amplitudept_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
m=F_phasept_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
c1=(c_1+dc_1.*t).*(1+cm.*s); c2=(c_2+dc_2.*t).*(1+cm.*s);
c3=(c_3+dc_3.*t).*(1+cm.*s); c4=(c_4+dc_4.*t).*(1+cm.*s);
r4_1=(r_1+dr4_1.*t).*(1+cm.*s); r4_2=(r_2+dr4_2.*t).*(1+cm.*s);
r4_3=(r_3+dr4_3.*t).*(1+cm.*s); r4_4=(r_4+dr4_4.*t).*(1+cm.*s);
b=F_amplitudept_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
n=F_phasept_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
amm_1=abs(20.*log10(d)-20.*log10(b)); t1=sort(amm_1); amm_2=amm_2+t1; pmm_1=(m-n)+90;
t2=sort(pmm_1); pmm_2=pmm_2+t2; end amm=amm_2./m4_mu; pmm=pmm_2./m4_mu; figure
subplot(2,1,1); hist(amm)
xlabel('{\itm}-{\rmgain} / dB','FontSize', 14)
title('Monte Carlo Simulation of 3-stage RF CMOS PPF','FontSize', 14)
ylabel('Number of Samples','FontSize', 14)
subplot(2,1,2); hist(pmm)
xlabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
ylabel('Number of Samples','FontSize', 14)
A4.3 Source Code of the Worst-Case Simulation
A4.3.1 Source Code for the Worst-Case Simulation
of 2-stage RF CMOS PPFs
% load optimization results load twostageppfopt.mat r_1 r_2 c_1 c_2;
global f_1; global f_2;global f; global r_1; global r_2;
global c_1; global c_2; global cm; global irrt;
% get the value of maximum component mismatch from input
cm=findobj(gcf,'Tag','cm'); cm=str2double(get(cm,'string'));
if isempty(cm), cm=0.01; else cm=cm; end % visiable waitbar
226
Appendix 4 Source Code of PPFOPTIMA
h = waitbar(0,'Simulation is processing, please wait...'); for x=1:100, % computation here %
waitbar(x/100,h), end close(h) % search worst IRR corresponding frequency point with R&C nominal
values rand('seed',0); % Crossover Operators xFns = 'simpleXover'; xOpts = [.4]; % Mutation Operators
mFns = 'binaryMutation'; mOpts = [0.005]; % Termination Operators termFns = 'maxGenTerm';
termOps = [100]; % 200 Generations % Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'gaMichEval2stage2'; evalOps = []; b_1=f_1; b_2=f_2; bounds = [b_1
b_2]; gaOpts=[1e-6 0 1]; % Generate an intialize population of size 20
startPop = initializega(20,bounds,'gaMichEval2stage2',[],[1e-6 0]); % lets run the GA
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,...
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
rand('seed',0); termOps=[100];
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,...
selectFn,selectOps);
% x is the best solution found x; fmax=x; w=fmax; % caculate worst case IRR
r2_1=r_1; r2_2=r_2; c2_1=c_1*1e-15; c2_2=c_2*1e-15;
d=T_amplitude_qu(w,r2_1,c2_1,r2_2,c2_2);
k=T_phase_qu(w,r2_1,c2_1,r2_2,c2_2);
n_1=(d); p1_1=(k); b=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2); h=T_phase_in(w,r2_1,c2_1,r2_2,c2_2);
p2_1=(h); m_1=(b); a_delta_1=abs(m_1-n_1); p_delta_1=abs((p2_1-p1_1).*pi./180);
r2_1=r_1; r2_2=r_2; c2_1=c_1*1e-15; c2_2=c_2*1e-15; d=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2);
k=T_phase_in(w,r2_1,c2_1,r2_2,c2_2); n_2=(d); p1_2=(k); dr_11=cm*r_1; dr_22=cm*r_2;
dc_11=cm*c_1*1e-15; dc_22=cm*c_2*1e-15; r2_1=r_1+dr_11; r2_2=r_2+dr_22; c2_1=c_1*1e15+dc_11; c2_2=c_2*1e-15+dc_22; b=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2);
h=T_phase_in(w,r2_1,c2_1,r2_2,c2_2); p2_2=(h); m_2=(b); a_delta_2=abs(m_2-n_2);
a=sqrt(m_2.^2+n_2.^2); p_delta_2=abs((p2_2-p1_2).*pi./180); a_delta=(a_delta_1+a_delta_2);
p_delta=(p_delta_1+p_delta_2); irrt=ImageRR(a_delta,a,p_delta); irrw=round(irrt.*10)./10;
hh=findobj(gcf,'Tag','irrw'); set(hh,'String',[num2str(irrw)]) % load IRR of design specification
load designspeci.mat irrspec; % compare with worst case IRR if irrw<irrspec,
warndlg('Two-stage PPF IRR can not satisfy the design specification, please use higher
stage PPF.','Warning') end
% caculate corresponding process tolerance of R&C ptt=fmax./f; pt_two=1-sqrt(1./ptt); % caculate R & C
variant dr_1=r_1*pt_two; dr_2=r_2*pt_two; dc_1=c_1*pt_two*1e-15; dc_2=c_2*pt_two*1e-15;
% caculate gain mismatch and phase difference r2_1=r_1+dr_1; r2_2=r_2+dr_2; c2_1=c_1*1e-15+dc_1;
c2_2=c_2*1e-15+dc_2; w=f_1*(1-0.2):f_1./1e4:f_2*(1+0.2); d=T_amplitude_in(w,r2_1,c2_1,r2_2,c2_2);
k=T_phase_in(w,r2_1,c2_1,r2_2,c2_2); n=20*log10(d); p1=(k); dr_11=cm*r_1;dr_22=cm*r_2;
dc_11=cm*c_1*1e-15;dc_22=cm*c_2*1e-15; r2_1=r_1+dr_1+dr_11; r2_2=r_2+dr_2+dr_22;
c2_1=c_1*1e-15+dc_1+dc_11; c2_2=c_2*1e-15+dc_2+dc_22;
b=T_amplitude_qu(w,r2_1,c2_1,r2_2,c2_2); h=T_phase_qu(w,r2_1,c2_1,r2_2,c2_2);
p2=(h); m=20*log10(b); gmis=abs(m-n); figure % polt gain mismatch plot(w,gmis,'m-')
xlabel('{\itf} / Hz','FontSize', 14) ylabel('{\itm}-{\rmgain} / dB','FontSize', 14) title('Gain Mismatch in 2Stage RF CMOS PPF','FontSize', 14) figure pmis=-p1+p2+90; % plot phase difference
227
Appendix 4 Source Code of PPFOPTIMA
plot(w,pmis,'m-') xlabel('{\itf} / Hz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase Difference in 2-Stage RF CMOS PPF','FontSize', 14)
A4.3.2 Source Code for the Worst-Case simulation
of 3-stage RF CMOS PPFs
load threestageppfopt.mat r1 r2 r3 c1 c2 c3 we;
global f_1; global f_2; global f; global r1; global r2; global r3; global c1;
global c2; global c3; global cm; global irrth;
cm=findobj(gcf,'Tag','cm');
if isempty(cm),
cm=str2double(get(cm,'string'));
cm=0.01; else
cm=cm; end
h = waitbar(0,'Simulation is processing, please wait...'); for x=1:100,
waitbar(x/100,h)
end
% computation here %
close(h)
rand('seed',0); % Crossover Operators xFns = 'simpleXover'; xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
% Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'gaMichEval3stage2'; evalOps = []; bounds = [f_1 f_2];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20
startPop = initializega(20,bounds,'gaMichEval3stage2',[],[1e-6 0]);
% Lets run the GA
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,...
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found %x; % endPop is the ending population %endPop;
% trace is a trace of the best value and average value of generations %trace;
% Lets increase the population size by running the defaults rand('seed',0);
termOps=[100]; [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,...
selectFn,selectOps);
% x is the best solution found x; fmax=x; w=fmax;
r3_1=r1; r3_2=r2; r3_3=r3; c3_1=c1*1e-15; c3_2=c2*1e-15; c3_3=c3*1e-15;
d=Th_amplitude_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
k=Th_phase_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
n=(d); p1=(k); b=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
h=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
p2=(h); m=(b); a_delta_1=abs(m-n);
p_delta_1=abs((p2-p1).*pi./180);
a=n; r3_1=r1; r3_2=r2; r3_3=r3; c3_1=c1*1e-15; c3_2=c2*1e-15; c3_3=c3*1e-15;
d=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
k=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3); n=(d); p1=(k);
dr_11=cm*r1; dr_22=cm*r2; dr_33=cm*r3; dc_11=cm*c1*1e-15;
228
Appendix 4 Source Code of PPFOPTIMA
dc_22=cm*c2*1e-15; dc_33=cm*c3*1e-15; r3_1=r1+dr_11;
r3_2=r2+dr_22; r3_3=r3+dr_33; c3_1=c1*1e-15+dc_11; c3_2=c2*1e-15+dc_22;
c3_3=c3*1e-15+dc_33; b=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
h=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3); p2=(h); m=(b); a_delta_2=abs(m-n);
% a=sqrt(m.^2+n.^2); p_delta_2=abs((p2-p1).*pi./180); p_delta=p_delta_1+p_delta_2;
a_delta=a_delta_1+a_delta_2; irrth=ImageRR(a_delta,a,p_delta); irrw=round(irrth.*10)./10;
hh=findobj(gcf,'Tag','irrw'); set(hh,'String',[num2str(irrw)]) %save threestageppfirr.mat irrth;
load designspeci.mat irrspec; if irrw<irrspec, warndlg('Three-stage PPF IRR can not satisfy the design
specification, please use higher stage PPF.','Warning') end ptt=fmax./f;
pt_three=1-sqrt(1./ptt); a=pt_three; w=f_1*(1-0.3):f_1./1e4:f_2*(1+0.5);
dr_1=r1*pt_three; dr_2=r2*pt_three; dr_3=r3*pt_three; dc_1=c1*pt_three*1e-15;
dc_2=c2*pt_three*1e-15; dc_3=c3*pt_three*1e-15; r3_1=r1+dr_1;
r3_2=r2+dr_2; r3_3=r3+dr_3; c3_1=c1*1e-15+dc_1; c3_2=c2*1e-15+dc_2;
c3_3=c3*1e-15+dc_3; d=Th_amplitude_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
k=Th_phase_in(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3); n=20*log10(d);
p1=(k); dr_11=cm*r1; dr_22=cm*r2; dr_33=cm*r3; dc_11=cm*c1*1e-15;
dc_22=cm*c2*1e-15; dc_33=cm*c3*1e-15; r3_1=r1+dr_1+dr_11; r3_2=r2+dr_2+dr_22;
r3_3=r3+dr_3+dr_33; c3_1=c1*1e-15+dc_1+dc_11; c3_2=c2*1e-15+dc_2+dc_22;
c3_3=c3*1e-15+dc_3+dc_33; b=Th_amplitude_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
h=Th_phase_qu(w,r3_1,c3_1,r3_2,c3_2,r3_3,c3_3);
p2=(h); m=20*log10(b); % gmis=we.*abs(m-n); gmis=abs(m-n);
figure subplot(2,1,1); plot(w,gmis,'m-')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itm}-{\rmgain} / dB','FontSize', 14)
title('Gain Mismatch in 3-Stage RF CMOS PPF','FontSize', 14)
subplot(2,1,2); pmis=-p1+p2+90; plot(w,pmis,'m-')
xlabel('{\itf} / GHz','FontSize', 14) ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase Mismatch in 3-Stage RF CMOS PPF','FontSize', 14)
A4.3.3 Source Code for the Worst-Case simulation
of 4-stage RF CMOS PPFs
load fourstageoptrc.mat;
global f_1; global f_4; global f; global r4_1; global r4_2; global r4_3; global r4_4;
global c4_1; global c4_2; global c4_3; global c4_4; global cm; global irrf;
h = waitbar(0,'Simulation is processing, please wait...');
waitbar(x/100,h)
end
cm=findobj(gcf,'Tag','cm');
if isempty(cm),
close(h)
cm=str2double(get(cm,'string'));
cm=0.01; else
f_1=findobj(gcf,'Tag','f_1');
f_4=findobj(gcf,'Tag','f_4');
f=findobj(gcf,'Tag','f');
for x=1:100,
cm=cm; end
f_1=str2double(get(f_1,'string'));
f_4=str2double(get(f_4,'string'));
f=str2double(get(f,'string')); rand('seed',0);
229
% computation here %
Appendix 4 Source Code of PPFOPTIMA
% Crossover Operators xFns = 'simpleXover'; xOpts = [.4];
% Mutation Operators mFns = 'binaryMutation'; mOpts = [0.005];
% Termination Operators termFns = 'maxGenTerm'; termOps = [100]; % 200 Generations
% Selection Function selectFn = 'roulette'; selectOps = [];
% Evaluation Function evalFn = 'gaMichEval4stage2'; evalOps = []; bounds = [f_1 f_4];
% GA Options [epsilon float/binar display] gaOpts=[1e-6 0 1];
% Generate an intialize population of size 20
startPop = initializega(20,bounds,'gaMichEval4stage2',[],[1e-6 0]);
% Lets run the GA
[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,...
termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);
% x is the best solution found %x; % endPop is the ending population %endPop;
% trace is a trace of the best value and average value of generations %trace;
% Lets increase the population size by running the defaults
rand('seed',0); termOps=[100]; [x endPop bestPop
trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,...
selectFn,selectOps);
% x is the best solution found x; fmax=x; w=fmax;
c1=c4_1*1e-15; c2=c4_2*1e-15; c3=c4_3*1e-15; c4=c4_4*1e-15;
r4_1=r4_1; r4_2=r4_2; r4_3=r4_3; r4_4=r4_4; d=F_amplitude_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
k=F_phase_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,4); n=(d); p1=(k);
b=F_amplitude_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
h=F_phase_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4); p2=(h); m=(b); a_delta_1=abs(m-n);
p_delta_1=abs((p2-p1).*pi./180); c1=c4_1*1e-15;c2=c4_2*1e-15; c3=c4_3*1e-15; c4=c4_4*1e-15;
r4_1=r4_1; r4_2=r4_2; r4_3=r4_3; r4_4=r4_4;
d=F_amplitude_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
k=F_phase_in(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
n=(d); p1=(k); dr_11=cm*r4_1; dr_22=cm*r4_2; dr_33=cm*r4_3; dr_44=cm*r4_4; dc_11=cm*c1;
dc_22=cm*c2; dc_33=cm*c3; dc_44=cm*c4; r4_1=r4_1+dr_11; r4_2=r4_2+dr_22; r4_3=r4_3+dr_33;
r4_4=r4_4+dr_44; c1=c1+dc_11; c2=c2+dc_22; c3=c3+dc_33; c4=c4+dc_44;
b=F_amplitude_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
h=F_phase_qu(w,r4_1,c1,r4_2,c2,r4_3,c3,r4_4,c4);
p2=(h); m=(b); a_delta_2=abs(m-n); p_delta_2=abs((p2-p1).*pi./180);
a=sqrt(m.^2+n.^2); a_delta=a_delta_1+a_delta_2; p_delta=p_delta_1+p_delta_2;
irrf=ImageRR(a_delta,a,p_delta); irrw=round(irrf.*10)./10; hh=findobj(gcf,'Tag','irrw');
set(hh,'String',[num2str(irrw)]) %save fourstageppfirr.mat irrf;
load designspeci.mat irrspec; if irrw<irrspec,
warndlg('Four-stage PPF IRR can not satisfy the design specification, this tool box can not help you any
more, please use Double Quadrature Transceiver architecture.','Warning') end
ptt=fmax./f; pt_four=1-sqrt(1./ptt); a=pt_four; load fourstageoptrc.mat;
w=f_1.*(1-0.30):f_1./1e4:f_4.*(1+0.30); c1=c4_1.*1e-15; c2=c4_2.*1e-15; c3=c4_3.*1e-15;
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Appendix 4 Source Code of PPFOPTIMA
c4=c4_4.*1e-15; dr_1=pt_four.*r4_1; dr_2=pt_four.*r4_2; dr_3=pt_four.*r4_3; dr_4=pt_four.*r4_4;
dc_1=pt_four.*c1; dc_2=pt_four.*c2; dc_3=pt_four.*c3; dc_4=pt_four.*c4; dr_11=cm*r4_1;
dr_22=cm*r4_2; dr_33=cm*r4_3; dr_44=cm*r4_4; dc_11=cm*c1; dc_22=cm*c2; dc_33=cm*c3;
dc_44=cm*c4; r1=r4_1+dr_1; r2=r4_2+dr_2; r3=r4_3+dr_3; r4=r4_4+dr_4; c41=c1+dc_1;
c42=c2+dc_2; c43=c3+dc_3; c44=c4+dc_4; d=F_amplitudew_in(w,r1,c41,r2,c42,r3,c43,r4,c44);
k=F_phasew_in(w,r1,c41,r2,c42,r3,c43,r4,c44);n=20*log10(d); p1=(k); r1=r4_1+dr_1+dr_11;
r2=r4_2+dr_2+dr_22; r3=r4_3+dr_3+dr_33; r4=r4_4+dr_4+dr_44; c41=c1+dc_1+dc_11;
c42=c2+dc_2+dc_22; c43=c3+dc_3+dc_33; c44=c4+dc_4+dc_44;
b=F_amplitudew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44);
h=F_phasew_qu(w,r1,c41,r2,c42,r3,c43,r4,c44); p2=(h);
m=20*log10(b); gmis=abs(m-n); figure subplot(2,1,1); plot(w,gmis,'m-')
xlabel('{\itf} / GHz','FontSize', 14)
ylabel('{\itm}-{\rmgain} / dB','FontSize', 14)
title('Gain Mismatch in 4-Stage RF CMOS PPF','FontSize', 14)
subplot(2,1,2); pmis=-p1+p2+90; plot(w,pmis,'m-')
xlabel('{\itf} / GHz','FontSize', 14)
ylabel('{\itd}-{\rmphase} / degrees','FontSize', 14)
title('Phase Mismatch in 4-Stage RF CMOS PPF','FontSize', 14)
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