3B. RF/Microwave Filters
November 06
 2006 Fabian Kung Wai Lee
1
References
•
•
•
•
•
•
•
[1] R. E. Collin, “Foundations for microwave engineering”, 2nd Edition 1992,
McGraw-Hill.
[2] D. M. Pozar, “Microwave engineering”, 2nd Edition 1998, John Wiley &
Sons.* (3rd Edition 2005, John-Wiley & Sons is now available)
Other more advanced references:
[3] W. Chen (editor), “The circuits and filters handbook”, 1995, CRC Press.*
[4] I. Hunter, “Theory and design of microwave filters”, 2001, The Instutitution
of Electrical Engineers.*
[5] G. Matthaei, L. Young, E.M.T. Jones, “Microwave filters, impedancematching networks, and coupling structures”, 1980, Artech House.*
[6] F. F. Kuo, “Network analysis and synthesis”, 2nd edition 1966, John-Wiley
& Sons.
* Recommended
November 06
 2006 Fabian Kung Wai Lee
2
1
1.0 Basic Filter Theory
November 06
 2006 Fabian Kung Wai Lee
3
Introduction
•
•
An ideal filter is a linear 2-port network that provides perfect transmission
of signal for frequencies in a certain passband region, infinite attenuation
for frequencies in the stopband region and a linear phase response in the
passband (to reduce signal distortion).
The goal of filter design is to approximate the ideal requirements within
acceptable tolerance with circuits or systems consisting of real
components.
November 06
 2006 Fabian Kung Wai Lee
4
2
Categorization of Filters
•
•
•
•
•
Low-pass filter (LPF), High-pass filter (HPF), Bandpass filter (BPF),
Bandstop filter (BSF), arbitrary type etc.
In each category, the filter can be further divided into active and passive
types.
In active filter, there can be amplification of the of the signal power in
the passband region, passive filter do not provide power amplification in
the passband.
Filter used in electronics can be constructed from resistors, inductors,
capacitors, transmission line sections and resonating structures (e.g.
piezoelectric crystal, Surface Acoustic Wave (SAW) devices, and also
mechanical resonators etc.).
Filter
Active filter may contain transistor, FET and Op-amp.
LPF
November 06
Active
 2006 Fabian Kung Wai Lee
Passive
HPF
BPF
Active
Passive
5
Filter’s Frequency Response (1)
•
•
Frequency response implies the behavior of the filter with respect to
steady-state sinusoidal excitation (e.g. energizing the filter with sine
voltage or current source and observing its output).
There are various approaches to displaying the frequency response:
– Transfer function H(ω) (the traditional approach).
– Attenuation factor A(ω).
– S-parameters, e.g. s21(ω) .
– Others, such as ABCD parameters etc.
November 06
 2006 Fabian Kung Wai Lee
6
3
Filter Frequency Response (2)
•
|H(ω)|
Low-pass filter (passive).
Transfer
function
1
A Filter
H(ω)
V1(ω)
V2(ω)
ZL
V (ω )
H (ω ) = 2
(1.1a)
V1 (ω )
Complex value
Arg(H(ω))
ω
ωc
A(ω)/dB
50
40
ω
ωc
Real value
 V (ω ) 

Attenuation A = −20 Log10  2
 V (ω ) 
1


30
20
10
3
0
(1.1b)
ω
ωc
 2006 Fabian Kung Wai Lee
November 06
7
Filter Frequency Response (3)
•
•
Low-pass filter (passive) continued...
For impedance matched system, using s21 to observe the filter response
is more convenient, as this can be easily measured using Vector
Network Analyzer (VNA).
a1
Vs
Zc
Zc
Zc
Zc
20log|s21(ω)|
Zc
Filter
Zc
Arg(s21(ω))
Transmission line
is optional
0dB
ωc
November 06
ω
ω
 2006 Fabian Kung Wai Lee
b2
b
b
s11 = 1
s21 = 2
a1 a =0
a1 a =0
2
2
Complex value
8
4
Filter Frequency Response (4)
•
Low-pass filter (passive) continued...
A(ω)/dB
Passband
Transition band
50
40
30
20
10
3
Stopband
0
ω
ωc
Cut-off frequency (3dB)
V1(ω)
A Filter
H(ω)
V2(ω)
ZL
 2006 Fabian Kung Wai Lee
November 06
9
Filter Frequency Response (5)
•
High-pass filter (passive).
|H(ω)|
A(ω)/dB
Transfer
function
Passband
50
40
1
ω
ωc
30
20
10
3
0
ωc
ω
Stopband
November 06
 2006 Fabian Kung Wai Lee
10
5
Filter Frequency Response (6)
•
Band-pass filter (passive).
Band-stop filter.
A(ω)/dB
A(ω)/dB
40
40
30
30
20
20
10
3
0
ω1
10
3
0
ω
ωo ω2
|H(ω)|
1
ω1
|H(ω)|
Transfer
function
ωo
ω2
ω
Transfer
function
1
ω
ω1
ω1 ωo ω2
ω2
ωo
ω
 2006 Fabian Kung Wai Lee
November 06
11
Basic Filter Synthesis Approaches (1)
•
Image Parameter Method (See [4] and [2]). • Consider a filter to be a
Filter
Zo
ω
Zo
Zo
Zo
Zo
H1(ω)
cascade of linear 2-port
networks.
• Synthesize or realize each
2-port network, so that
the combine effect give the
required frequency response
• The ‘image’ impedance seen
and the input and output of
each network is maintained.
Zo
H2(ω)
Response of
a single
network
Zo
Hn(ω)
Zo
ω
November 06
 2006 Fabian Kung Wai Lee
The combine
response
12
6
Basic Filter Synthesis Approaches (2)
•
Insertion Loss Method (See [2]).
Approximate ideal filter response
With polynomial function:
|H(ω)|
Filter
Ideal
Approximate with rational polynomial
function
sn +a
s n −1 + + a s + a
Zo
ω
H (s ) = K
n −1
L
L
1
o
s n + bn −1s n −1 + + b1s + bo
We can also use Attenuation Factor or s21
for this.
Use RCLM circuit synthesis theorem ([3], [6])
to come up with a resistive terminated
LC network that can produce the
Z
approximate response.
o
Zo
November 06
 2006 Fabian Kung Wai Lee
13
Our Scope
•
•
Only concentrate on passive LC and stripline filters.
Filter synthesis using the Insertion Loss Method (ILM). The Image
Parameter Method (IPM) is more efficient and suitable for simple filter
designs, but has the disadvantage that arbitrary frequency response
cannot be incorporated into the design.
November 06
 2006 Fabian Kung Wai Lee
14
7
2.0 Passive LC Filter
Synthesis Using Insertion
Loss Method
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 2006 Fabian Kung Wai Lee
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Insertion Loss Method (ILM)
•
•
•
•
•
•
The insertion loss method (ILM) allows a systematic way to design and
synthesize a filter with various frequency response.
ILM method also allows filter performance to be improved in a
straightforward manner, at the expense of a ‘higher order’ filter.
A rational polynomial function is used to approximate the ideal |H(ω)|,
A(ω) or |s21(ω)|.
Phase information is totally ignored.
Ignoring phase simplified the actual synthesis method. An LC network
is then derived that will produce this approximated response.
Here we will use A(ω) following [2]. The attenuation A(ω) can be cast
into power attenuation ratio, called the Power Loss Ratio, PLR, which is
related to A(ω)2.
November 06
 2006 Fabian Kung Wai Lee
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8
More on ILM
tra
Ex
•
•
There is a historical reason why phase information is ignored. Original
filter synthesis methods are developed in the 1920s-60s, for voice
communication. Human ear is insensitive to phase distortion, thus only
magnitude response (e.g. |H(ω)|, A(ω)) is considered.
Modern filter synthesis can optimize a circuit to meet both magnitude
and phase requirements. This is usually done using computer
optimization procedures with goal functions.
 2006 Fabian Kung Wai Lee
November 06
17
Power Loss Ratio (PLR)
Zs
Lossless
2-port network
Vs
PA
ZL
PL
Pin
Γ1(ω)
PLR = Power available from source network
P
= inc =
PLoad
Power delivered to Load
PA
1
=
2
2


1
−
Γ
PA 1− Γ1 (ω )
1 (ω )



(2.1a)
PLR large, high attenuation
PLR close to 1, low attenuation
For example, a low-pass
filter response is shown
below:
PLR(f)
High
attenuation

1
Low-Pass filter PLR
November 06
 2006 Fabian Kung Wai Lee
0
Low
attenuation
f
fc
18
9
PLR and s21
•
In terms of incident and reflected waves, assuming ZL=Zs = ZC.
b1
a1
b2
Zc
Lossless
2-port network
Vs
PA
Pin
1
a1
P
PLR = A = 2
PL
PLR =
Zc
PL
2
1b 2
2 2
1
s21 2
a
= 1
2
b2
(2.1b)
 2006 Fabian Kung Wai Lee
November 06
19
PLR for Low-Pass Filter
•
Since |Γ1(ω)|2 is an even function of ω, it can be written in terms of ω2 as:
Γ(ω ) =
2
•
(2.2)
PLR can be expressed as:
PLR =
•
( )
( ) ( )
M ω2
M ω2 +N ω2
1
1− Γ1 (ω ) 2
=
1
1−
M  ω 2 


=1+
M  ω 2 


N  ω 2 


M  ω 2  + N  ω 2 




This is also known
as Characteristic Polynomial
PLR = 1 + [P (ω )]2
[P(ω )]2 = M (ω2 )
2
( )
Nω
(2.3a)
(2.3b)
Various type of polynomial functions in ω can be used for P(ω)2. Among the
classical functions are:
The characteristics we need
– Maximally flat or Butterworth functions.
from P(ω):
– Equal ripple or Chebyshev functions.
• P(ω) → 0 for ω < ωc
– Elliptic function.
• P(ω) >> 1 for ω >> ωc
– Many, many more.
November 06
 2006 Fabian Kung Wai Lee
20
10
Characteristic Polynomial Functions
•
Maximally flat or Butterworth:
•
Equal ripple or Chebyshev:
P (ω ) =  ωω


 c
N
N = order of the
polynomial
(2.4a)
P(ω ) = εC N (ω ) , ε = ripple factor
C0 (ω ) = 1

C N (ω ) = C1 (ω ) = ω
C (ω ) = 2ωC (ω ) − C (ω ) , n ≥ 2
n −1
n−2
 n
•
Bessel [6] or linear phase:
For other types of
polynomial functions,
please refer to
reference [3] and [6].
(2.4b)
[P(ω )]2 = B( jω )B(− jω ) − 1
 B0 (s ) = 1

(2.4c)

BN (s ) =  B1 (s ) = s + 1

2

 Bn (s ) = (2 s − 1)Bn −1 (s ) + s Bn−2 (s ) , n ≥ 2
 2006 Fabian Kung Wai Lee
November 06
21
Examples of PLR for Low-Pass Filter (1)
•
PLR of low pass filter using 4th order polynomial functions (N=4) Butterworth, Chebyshev (ripple factor =1) and Bessel. Normalized to ωc
= 1 rad/s, k=1.
2

PLR (chebyshev ) = 1 + k 2 8 ωω
Ideal
1 .10
4
1 .10
3



 c
2

− 4 ωω  + 1
 c

Chebyshev
4

 
 c



PLR ( Butterwort h) = 1 + k 2  ωω
PLRbt ( ω )
PLRcbP
( ωLR
) 100
2
Butterworth
PLRbs ( ω )
10
1
Bessel
0
0.5
1
ω
November 06
1.5
2
PLR ( Bessel ) = 1 + k 2 [B ( jω )B (− jω ) − 1]
4

 s 
+ 10 s

105  ω c 
 ωc
B(s ) = 1
 2006 Fabian Kung Wai Lee




3
+ 45 s
 ωc



2

+ 105 s  + 105
 ωc 


22
11
Examples of PLR for Low-Pass Filter (2)
•
PLR of low pass filter using Butterworth characteristic polynomial,
N 2

normalized to ωc = 1 rad/s, k=1.
2  ω 
PLR ( Butterworth) = 1 + k
1 .10
5
PLR( ω , 2) . 4
1 10
N=7
PLR( ω , 3)
N=6
PLR( ω , 4)1 .10
N=5
PLR( ω , 5)
N=4
3
PLR( ω , 6)
PLR( ω , 7)
100
N=3
N=2


 ωc 





Conclusion:
The type of
polynomial
function and
the order
determine the
Attenuation rate
in the stopband.
10
1
0
0.5
1
1.5
2
ω
November 06
 2006 Fabian Kung Wai Lee
23
Characteristics of Low-Pass Filters
Using Various Polynomial Functions
•
•
•
Butterworth: Moderately linear phase response, slow cut-off, smooth
attenuation in passband.
Chebyshev: Bad phase response, rapid cut-off for similar order,
contains ripple in passband. May have impedance mismatch for N
even.
Bessel: Good phase response, linear. Very slow cut-off. Smooth
amplitude response in passband.
November 06
 2006 Fabian Kung Wai Lee
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12
Low-Pass Prototype Design (1)
•
•
•
•
A lossless linear, passive, reciprocal network that can produce the
insertion loss profile for Low-Pass Filter is the LC ladder network.
Many researchers have tabulated the values for the L and C for the
Low-Pass Filter with cut-off frequency ωc = 1 Rad/s, that works with
source and load impedance Zs = ZL = 1 Ohm.
This Low-Pass Filter is known as the Low-Pass Prototype (LPP).
As the order N of the polynomial P increases, the required element also
increases. The no. of elements = N.
1
L1=g2
C1=g1
L2=g4
C2=g
RL= gN+1
3
L1=g1
g0= 1
November 06
L2=g3
C1=g2
C2=g4
RL= gN+1
Dual of each
other
 2006 Fabian Kung Wai Lee
25
Low-Pass Prototype Design (2)
•
•
•
•
The LPP is the ‘building block’ from which real filters may be
constructed.
Various transformations may be used to convert it into a high-pass,
band-pass or other filter of arbitrary center frequency and bandwidth.
The following slides show some sample tables for designing LPP for
Butterworth and Chebyshev amplitude response of PLR.
See Chapter 3 of Hunter [4], on how the LPP circuits and the tables can
be derived.
November 06
 2006 Fabian Kung Wai Lee
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13
Table for Butterworth LPP Design
N
g1
g2
g3
g4
g5
g6
g7
1
2
3
4
5
6
2.0000
1.4142
1.0000
0.7654
0.6180
0.5176
1.0000
1.4142
2.0000
1.8478
1.6180
1.4142
g8
1.0000
1.0000
1.8478
2.0000
1.9318
1.0000
0.7654
1.6180
1.9318
1.0000
0.6180
1.4142
1.0000
0.5176
1.0000
7
0.4450
1.2470
1.8019
2.0000
1.8019
1.2470
0.4450
1.0000
8
0.3902
1.1111
1.6629
1.9615
1.9615
1.6629
1.1111
0.3902
g9
1.0000
Taken from Chapter 8, Pozar [2].
See Example 2.1 in the following slides on how the constant values g1, g2, g3…etc.
are obtained.
 2006 Fabian Kung Wai Lee
November 06
27
Table for Chebyshev LPP Design
•
Ripple factor 20log10ε = 0.5dB
N
1
2
3
4
5
6
•
g1
0.6986
1.4029
1.5963
1.6703
1.7058
1.7254
g2
1.0000
0.7071
1.0967
1.1926
1.2296
1.2479
g3
g4
g5
g6
g7
1.9841
1.5963
2.3661
2.5408
2.6064
1.0000
0.8419
1.2296
1.3137
1.9841
1.7058
2.4578
1.0000
0.8696
1.9841
Ripple factor 20log10ε = 3.0dB
N
1
2
3
4
5
6
g1
1.9953
3.1013
3.3487
3.4389
3.4817
3.5045
November 06
g2
1.0000
0.5339
0.7117
0.7483
0.7618
0.7685
g3
g4
g5
g6
g7
5.8095
3.3487
4.3471
4.5381
4.6061
1.0000
0.5920
0.7618
0.7929
5.8095
3.4817
4.4641
1.0000
0.6033
5.8095
 2006 Fabian Kung Wai Lee
28
14
Table for Maximally-Flat Time Delay
LPP Design
N
g1
g2
g3
g4
g5
g6
g7
1
2
3
4
5
6
2.0000
1.5774
1.2550
1.0598
0.9303
0.8377
1.0000
0.4226
0.5528
0.5116
0.4577
0.4116
g8
1.0000
0.1922
0.3181
0.3312
0.3158
1.0000
0.1104
0.2090
0.2364
1.0000
0.0718
0.1480
1.0000
0.0505
1.0000
7
0.7677
0.3744
0.2944
0.2378
0.1778
0.1104
0.0375
1.0000
8
0.7125
0.3446
0.2735
0.2297
0.1867
0.1387
0.0855
0.0289
g9
1.0000
Taken from Chapter 8, Pozar [2].
 2006 Fabian Kung Wai Lee
November 06
tra
Ex
29
Example 2.1 - Finding the Constants
for LPP Design (1)
Consider a simple case of 2nd order Low-Pass Filter:
R
R
L
Vs
R
C
V1 (ω ) =
R V
1+ jωRC s
R
R + jωL + 1+ jω
RC
Thus
jωL
Vs
RV
R V1
1/jωC
RV
s
= R + (R + jωL )(s1+ jωRC ) =
2 R −ω 2 RLC + jω (L + R 2C )
2
PL (ω ) = 21R V1 (ω ) =
2
Vs R

2

(2−ω LC ) R +ω (L+ R C )
2
2
2
2
2
and
2


PA = 81R Vs
2
Therefore we can compute the power loss ratio as:
2
PLR (ω ) = P (Aω ) =
L
P
Vs
8R

(
)
2
(
2  2−ω 2 LC R 2 +ω 2 L + R 2C
(
)

( )
2
2 4


= 1 +  1 2 L + R 2C − LC ω 2 + LC
ω 
2
 4 R
November 06

(
) (
)
2
2


= 1 2  2 R 2 2 − ω 2 LC + 2 L + R 2C ω 2 
2
Vs R

)
8R 

2


[P(ω)]2
 2006 Fabian Kung Wai Lee
30
15
tra
Ex
Example 2.1 - Finding the Constants
for LPP Design (2)
PLR can be written in terms of polynomial of ω2:
(
)
( )
[
2
2 4


PLR (ω ) = 1 +  1 2 L + R 2C − LC ω 2 + LC
ω  = 1 + a1ω 2 + a2ω 4
2
 4 R


]
(E1.1)
For Butterworth response with k=1, ωc = 1:
[ ] =1+ ω
PLR ( Butterwort h) = 1 + ω 2
2
(E1.2)
4
Comparing equation (E1.1) and (E1.2):
a2 = 1
⇒ LC2 = 1 ⇒ LC = 2
⇒ 4 R1 (L + R 2C )2 − LC = 0
⇒ LC = 21R (L + R 2C ) (E1.4)
a1 = 0
(E1.3)
Setting R=1 for Low-Pass Prototype (LPP):
R = 1 Thus from equation (E1.4):
LC = 12 (L + C )2
⇒ (L − C )
⇒L=C
2
=0
⇒ L2 + C 2 − 2LC = 0
⇒C =
⇒ C2 = 2
2 ≅ 1.4142
L = C ≅ 1.4142
Using (E1.3)
Compare this result with
N=2 in the table for LPP
Butterworth response.
This direct ‘brute force’
approach can be
extended to N=3, 4, 5…
 2006 Fabian Kung Wai Lee
November 06
tra
Ex
LC = 2
2
31
Example 2.1 – Verification (1)
AC
AC
AC1
Start=0.01 Hz
Stop=2.0 Hz
Step=0.01 Hz
R
R2
R=1 Ohm
V_AC
SRC1
Vac=polar(1,0) V
Freq=freq
November 06
Vin
L
L1
L=1.4142 H
R=
 2006 Fabian Kung Wai Lee
Vout
C
C1
C=1.4142 F
R
R1
R=1 Ohm
32
16
Example 2.1 – Verification (2)
tra
Ex
Eqn PA=1/8 Eqn PL=0.5*mag(Vout)*mag(Vout)
m1
5
0
Eqn PLR=PA/PL
m1
freq=160.0mHz
m1=-3.056
-10
)
5.
0/
t -20
ou
V
(
dB -30
2.5E4
2.0E4
The power loss ratio
versus frequency
1.5E4
R
L
P 1.0E4
-40
-50
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
freq, Hz
5.0E3
0.0
-3dB at 160mHz (miliHertz!!),
which is equivalent to 1 rad/s
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
freq, Hz
November 06
 2006 Fabian Kung Wai Lee
33
Impedance Denormalization and
Frequency Transformation of LPP (1)
•
•
•
•
•
Once the LPP filter is designed, the cut-off frequency ωc can be
transformed to other frequencies.
Furthermore the LPP can be mapped to other filter types such as highpass, bandpass and bandstop (see [2] and [3] for the derivation and
theories).
This frequency scaling and transformation entails changing the value
and configuration of the elements of the LPP.
Finally the impedance presented by the filter at the operating frequency
can also be scaled, from unity to other values, this is called impedance
denormalization.
Let Zo be the new system impedance value. The following slide
summarizes the various transformation from the LPP filter.
November 06
 2006 Fabian Kung Wai Lee
34
17
Impedance Denormalization and
Frequency Transformation of LPP (2)
LPP to
High-Pass
LPP to
Low-Pass
LPP to
Bandpass
LPP to
Bandstop
LZ o
L
Zo L
C
C
ωo∆
1
ω c LZ o
ωc
∆
ω o LZ o
Zo
L∆Z o
ωo
Zo
∆Z o
ω oC
C
ω o ∆Z o
ω cC
Z oω c
1
ω o L∆Z o
ω o C∆
C∆
ωo Z o
ωo =
ω1 +ω 2
2
or ω1ω 2
(2.5a)
ω −ω1
∆ = 2ω
o
(2.5b)
Note that inductor always
multiply with Zo while
capacitor divide with Zo
 2006 Fabian Kung Wai Lee
November 06
35
Summary of Passive LC Filter Design
Flow Using ILM Method (1)
•
Step 1 - From the requirements, determine the order and type of
approximation functions to used.
–
–
–
–
–
•
Insertion loss (dB) in passband ?
Attenuation (dB) in stopband ?
Cut-off rate (dB/decade) in transition band ?
Tolerable ripple?
Linearity of phase?
Step 2 - Design the normalized low-pass prototype (LPP) using L and C
elements.
|H(ω)|
1
L1=g2
C1=g1
L2=g4
C2=g
3
1
RL= gN+1
0
November 06
 2006 Fabian Kung Wai Lee
1
ω
36
18
Summary of Passive Filter Design Flow
Using ILM Method (2)
•
Step 3 - Perform frequency scaling and denormalize the impedance.
|H(ω)|
50
79.58nH
0.1414pF
1
Vs
15.916pF
0.7072nH
15.916pF
RL
50
0.7072nH
0
•
November 06
tra
Ex
•
•
•
ω
ω2
Step 4 - Choose suitable lumped components, or transform the lumped
circuit design into distributed realization.
All uses microstrip
stripline circuit
•
ω1
See Ref. [4]
See Ref. [2]
See Ref. [3]
 2006 Fabian Kung Wai Lee
37
Filter vs Impedance Transformation
Network
If we ponder carefully, the sharp observer will notice that the filter can
be considered as a class of impedance transformation network.
In the passband, the load is matched to the source network, much like a
filter.
In the stopband, the load impedance is highly mismatched from the
source impedance.
However, the procedure described here only applies to the case when
both load and source impedance are equal and real.
November 06
 2006 Fabian Kung Wai Lee
38
19
Example 2.2A – LPF Design:
Butterworth Response
•
Design a 4th order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc =
1.5GHz.
Step 1&2: LPP
L1=0.7654H
g0= 1
L2=1.8478H
C1=1.8478F C2=0.7654F
Step 3: Frequency scaling
and impedance denormalization L =4.061nH
1
g0=1/50
ω c = 2π (1.5GHz ) = 9.4248 × 109 rad/s
Zo = 50
RL= 1
L
L = Zo n
ωc
L2=9.803nH
Cn
C=
Z oω c
C1=3.921pF C2=1.624pF
RL= 50
 2006 Fabian Kung Wai Lee
November 06
R = Z o Rn
39
Example 2.2B – LPF Design:
Chebyshev Response
•
Design a 4th order Chebyshev Low-Pass Filter, 0.5dB ripple factor. Rs
= 50Ohm, fc = 1.5GHz.
Step 1&2: LPP
L1=1.6703H
g0= 1
L2=2.3661H
C1=1.1926F C2=0.8419F
Step 3: Frequency scaling
and impedance denormalization L =8.861nH
1
g0=1/50
November 06
ω c = 2π (1.5GHz ) = 9.4248 × 109 rad/s
Zo = 50
RL=
1.9841
R = Z o Rn
L
L = Zo n
ωc
L2=12.55nH
C1=2.531pF C2=1.787pF
 2006 Fabian Kung Wai Lee
Cn
C=
Z oω c
RL=
99.2
40
20
Example 2.2 Cont...
Ripple is roughly
0.5dB
5
))
,12
(
.S
.h
rto ))1
,2
rw
ett ((S
|s21
ub B|
_d
F
P
(L
B
d
0
Butterworth
Computer simulation result
Using AC analysis (ADS2003C)
-10
-20
Chebyshev
-30
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Better phase
Linearity for Butterworth
LPF in the passband
0
-50
freq, GHz
Note: Equation used in Data Display of ADS2003C
to obtain continuous phase display with built-in
function phase( ).
rtho evh -100
ys
rw
tteu ebh -150
b _c -200
_
Arg(s
es se 21)
ah ah -250
PP
-300
Butterworth
Chebyshev
-350
Eqn Phase_chebyshev = if (phase(S(2,1))<0) then phase(S(2,1)) else (phase(S(2,1))-360)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
freq, GHz
November 06
 2006 Fabian Kung Wai Lee
41
Example 2.3: BPF Design
•
•
•
•
Design a bandpass filter with Butterworth (maximally flat) response.
N = 3.
Center frequency fo = 1.5GHz.
3dB Bandwidth = 200MHz or f1=1.4GHz, f2=1.6GHz.
November 06
 2006 Fabian Kung Wai Lee
42
21
Example 2.3 Cont…
•
From table, design the Low-Pass prototype (LPP) for 3rd order
Butterworth response, ωc=1.
g2
2.000H
Zo=1
Step 1&2: LPP
g1
1.000F
2<0o
Simulated result
using PSPICE
g4
1
g3
1.000F
ω c = 2πf c = 1
⇒ fc = 21π = 0.1592 Hz
Voltage across g4
 2006 Fabian Kung Wai Lee
November 06
43
Example 2.3 Cont…
•
•
ω1 = 2π (1.4GHz )
ω 2 = 2π (1.6GHz )
LPP to bandpass transformation.
Impedance denormalization.
Step 3: Frequency scaling
and impedance denormalization
fo =
LZ o
ωo ∆
C
∆
ω o LZ o
50
Vs
November 06
f1 f 2 = 1.497GHz
ω −ω
∆ = 2ω 1 = 0.133
o
ω o ∆Z o
79.58nH
∆Z o
ω oC
0.1414pF
RL
15.916pF
0.7072nH
15.916pF
 2006 Fabian Kung Wai Lee
50
0.7072nH
44
22
Example 2.3 Cont…
•
Simulated result using PSPICE:
Voltage across RL
 2006 Fabian Kung Wai Lee
November 06
All Pass Filter
tra
Ex
•
•
45
There is also another class of filter known as All-Pass Filter (APF).
This type of filter does not produce any attenuation in the magnitude
response, but provides phase response in the band of interest.
APF is often used in conjunction with LPF, BPF, HPF etc to
compensate for phase distortion.
•
Example of APF response
|H(f)|
Arg(H(f))
|H(f)|
Nonlinear
phase in
passband
1
0
f
f
Arg(H(f))
1
0
f
f
|H(f)|
BPF
APF
Zo
1
0
November 06
 2006 Fabian Kung Wai Lee
Arg(H(f))
Linear
phase in
passband
f
f
46
23
Example 2.4 - Practical RF BPF Design
Using SMD Discrete Components
CPWSub
C
Ct3
C=Ct_value2 pF
L
Lt1
L=Lt_value nH
R=
Term
Term1
Num=1
Z=50 Ohm
C
Ct1
C=Ct_value pF
CPWSUB
CPWSub1
H=62.0 mil
Er=4.6
Mur=1
Cond=5.8E+7
T =1.38 mil
T anD=0.02
Rough=0.0 mil
S-PARAMETERS
Var
Eqn
S_Param
SP1
Start=0.1 GHz
Stop=3.0 GHz
Step=1.0 MHz
INDQ
1_0pF_NPO_0603
CPWG
L4
C1
CPW1
L=15.0 nH
Q=90.0
Subst="CPWSub1"
b82496c3229j000
4_7pF_NPO_0603
F=800.0 MHz
W=50.0 mil
L2
C2
G=10.0 mil
Mode=proportional to freq
param=SIMID 0603-C (2.2 nH +-5%)
L=28.0 mm
Rdc=0.1 Ohm
November 06
VAR
VAR1
Lt_value=4.8
Ct_value=3.5
Ct_value2=2.9
C
Ct2
C=Ct_value pF
CPWG
CPW2
b82496c3229j000
Subst="CPWSub1"
L3
W=50.0 mil
4_7pF_NPO_0603
param=SIMID 0603-C (2.2
nH +-5%)
C3 G=10.0 mil
L=28.0 mm
 2006 Fabian Kung Wai Lee
L
Lt2
L=Lt_value nH
R=
C
Ct45
C=Ct_value2 pF
T erm
T erm2
Num=2
Z=50 Ohm
47
Example 2.4 Cont…
BPF synthesis
using synthesis
tool E-syn
of ADS2003C
November 06
 2006 Fabian Kung Wai Lee
48
24
Example 2.4 Cont…
|s21|/dB
Measured
Simulated
0
))1
,2
(
.S
.d
er ) -20
us )1
ae ,(2
S
m
(
_B
Fd
P -40
_B
F
(R
B
d
Measurement is performed with
Agilent 8753ES Vector Network
Analyzer, using Full OSL calibration
-60
0.0 0.2 0.4 0.6
0.8 1.0 1.2 1.4 1.6 1.8 2.0
freq, GHz
2.2 2.4 2.6 2.8
3.0
))1 200
,2
(
S
..d
100
e
r ))
u
sa ,1
e (2
m(S 0
_
F se
a
Ph
B
_p
F -100
R
(
e
s
a
h -200
p
Arg(s21)/degree
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
freq, GHz
November 06
 2006 Fabian Kung Wai Lee
49
3.0 Microwave Filter
Realization Using Stripline
Structures
November 06
 2006 Fabian Kung Wai Lee
50
25
3.1 Basic Approach
November 06
 2006 Fabian Kung Wai Lee
51
Filter Realization Using Distributed
Circuit Elements (1)
•
•
•
•
•
Lumped-element filter realization using surface mounted inductors and
capacitors generally works well at lower frequency (at UHF, say < 3
GHz).
At higher frequencies, the practical inductors and capacitors loses their
intrinsic characteristics.
Also a limited range of component values are available from
manufacturer.
Therefore for microwave frequencies (> 3 GHz), passive filter is usually
realized using distributed circuit elements such as transmission line
sections.
Here we will focus on stripline microwave circuits.
November 06
 2006 Fabian Kung Wai Lee
52
26
Filter Realization Using Distributed
Circuit Elements (2)
•
•
Recall in the study of Terminated Transmission Line Circuit that a
length of terminated Tline can be used to approximate an inductor and
capacitor.
This concept forms the basis of transforming the LC passive filter into
distributed circuit elements.
Zc , β
l
Zc , β
≅
L
≅
C
Zo
Zo
Zo
≅
Zc , β
Zc , β
Zo
November 06
Zc , β
l
 2006 Fabian Kung Wai Lee
53
Filter Realization Using Distributed
Circuit Elements (3)
•
•
This approach is only approximate. There will be deviation between the
actual LC filter response and those implemented with terminated Tline.
Also the frequency response of distributed circuit filter is periodic.
Other issues are shown below.
How do we implement series Tline
connection ? (only practical for
certain Tline configuration)
 2006 Fabian Kung Wai Lee
Zc , β
November 06
Connection physical
length cannot be
ignored at
microwave region,
comparable to λ
Zo
Zc , β
Thus some theorems are used to
facilitate the transformation of LC
circuit into stripline microwave circuits.
Chief among these are the Kuroda’s
Identities (See Appendix)
Zo
Zc , β
•
54
27
More on Approximating L and C with
Terminated Tline: Richard’s Transformation
Z in = jZ c tan (β l ) = jωL = jLω
tan (βl ) = ω
l
≅
Zin
L
Zc , β
l
≅
Zin
(3.1.1a)
Zc = L
C
Zc , β
Yin = jYc tan (βl ) = jωC = jCω
tan (βl ) = ω
(3.1.1b)
Yc = 1 = C
Zc
For LPP design, a further requirment is
that:
Wavelength at
cut-off frequency
λ
⇒ tan 2π l  = 1 ⇒ l = c (3.1.1c)
λ
8
 c 
tan (βl ) = ωc = 1
 2006 Fabian Kung Wai Lee
November 06
55
Example 3.1 – LPF Design Using
Stripline
•
Design a 3rd order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc =
1.5GHz.
g1
1.000H
g3
1.000H
Zc=1.000
Zo=1
g4
1
g2
2.000F
1
1
Step 3: Convert to Tlines
November 06
Length = λc/8
for all Tlines
at ω = 1 rad/s
 2006 Fabian Kung Wai Lee
Zc=0.500
1 = 0.500
2.000
Zc=1.000
Step 1 & 2: LPP
56
28
Example 3.1 Cont…
Step 4: Add extra Tline on the series
connection and apply Kuroda’s
2nd Identity.
Z1 = 1.0
Z2=1
β
Yc
1
Extra Tline
1
Zc=1.000
Zc=1.000
l
n 2Z 2
= 0 .5
n2Z1=2 β
Extra Tline
n2 = 1 + 1 = 2
1
Zc=1.0
Zc=1.0
Zc=0.500
Similar operation is
performed here
1
Length = λc/8
for all Tlines
at ω = 1 rad/s
 2006 Fabian Kung Wai Lee
November 06
57
Example 3.1 Cont…
After applying Kuroda’s 2nd Identity.
1
Zc=2.0
Zc=2.0
November 06
Zc=2.000
Zc=0.500
Zc=2.000
Length = λc/8
for all Tlines
at ω = 1 rad/s
1
Since all Tlines have similar physical
length, this approach to stripline filter
implementation is also known as
Commensurate Line Approach.
 2006 Fabian Kung Wai Lee
58
29
Example 3.1 Cont…
Step 5: Impedance and frequency denormalization.
50
Zc=100
Zc=100
Zc=100
Zc=25
Zc=100
Length = λc/8
for all Tlines at
f = fc = 1.5GHz
50
Microstrip line using double-sided FR4 PCB (εr = 4.6, H=1.57mm)
Zc/Ω
50
25
100
λ/8 @ 1.5GHz /mm
13.45
12.77
14.23
W /mm
2.85
8.00
0.61
 2006 Fabian Kung Wai Lee
November 06
59
Example 3.1 Cont…
Step 6: The layout (top view)
November 06
 2006 Fabian Kung Wai Lee
60
30
Example 3.1 Cont…
Simulated results
S-PARAMETERS
MSub
MSUB
MSub1
H=1.57 mm
Er=4.6
Mur=1
Cond=1.0E+50
Hu=3.9e+034 mil
T=0.036 mm
TanD=0.02
Rough=0 mil
S_Param
SP1
Start=0.2 GHz
Stop=4.0 GHz
Step=5 MHz
MTEE
Tee1
Subst="MSub1"
W1=2.85 mm
W2=0.61 mm
W3=0.61 mm
MLIN
TL1
Subst="MSub1"
W=2.85 mm
L=25.0 mm
Term
Term1
Num=1
Z=50 Ohm
Term
Term1
Num=1
Z=50 Ohm
MTEE
Tee3
Subst="MSub1"
W1=0.61 mm
W2=0.61 mm
W3=8.00 mm
MLIN
TL3
Subst="MSub1"
W=0.61 mm
L=14.23 mm
MLOC
TL6
Subst="MSub1"
W=0.61 mm
L=14.23 mm
MTEE
Tee2
Subst="MSub1"
W1=0.61 mm
W2=2.85 mm
W3=0.61 mm
MLIN
TL4
Subst="MSub1"
W=0.61 mm
L=14.23 mm
MLOC
TL5
Subst="MSub1"
W=8.0 mm
L=12.77 mm
MLIN
TL2
Subst="MSub1"
W=2.85 mm
L=25.0 mm
MLOC
TL7
Subst="MSub1"
W=0.61 mm
L=14.23 mm
L
L1
L=5.305 nH
R=
L
L2
C
L=5.305 nH
C1
R=
C=4.244 pF
Term
Term2
Num=2
Z=50 Ohm
m1
freq=1.500GHz
m1=-6.092
Term
Term2
Num=2
Z=50 Ohm
))
1
,
2
(
S
.. )
C)
L ,1
_
F (2
PS
(
LB
_
re d
ttu
(B
B
d
0
m1
-10
-20
-30
-40
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
freq, GHz
November 06
 2006 Fabian Kung Wai Lee
61
Conclusions for Section 3.1
•
•
•
Further tuning is needed to optimize the frequency response.
The method just illustrated is good for Low-Pass and Band-Stop filter
implementation.
For High-Pass and Band-Pass, other approaches are needed.
November 06
 2006 Fabian Kung Wai Lee
62
31
3.2 Further Implementations
November 06
 2006 Fabian Kung Wai Lee
63
Realization of LPF Using StepImpedance Approach
•
•
•
•
A relatively easy way to implement LPF using stripline components.
Using alternating sections of high and low characteristic impedance
tlines to approximate the alternating L and C elements in a LPF.
Performance of this approach is marginal as it is an approximation,
where sharp cutoff is not required.
As usual beware of parasitic passbands !!!
November 06
 2006 Fabian Kung Wai Lee
64
32
Equivalent Circuit of a Transmission
Line Section
T-network equivalent circuit
Z11 - Z12
Ideal lossless Tline
Z11 - Z12
l
Z12
Zc
β
Z11 = Z 22 = − jZ c cot (β l ) (3.2.1a)
Z12 = Z 21 = − jZ c cosec(β l ) (3.2.1b)
(3.2.1c)
β ≅ ω µ oε eε o = ε e k o
 2006 Fabian Kung Wai Lee
November 06
65
Approximation for High and Low ZC (1)
•
When βl < π/2, the series element can be thought of as inductor and the
shunt element can be considered a capacitor.
1
1
X
βl
=B=
sin (β l )
Z11 − Z12 = = Z c tan 

Z
Z
12
c
2
2

•
•

For βl < π/4 and Zc=ZH >> 1:
For βl < π/4 and Zc=ZL → 1:
Z11 - Z12
B≅0
X ≅0
1
βl
ZL
B≅
Z11 - Z12
When Zc >> 1
βl < π/4
Z12
jX/2
X ≈ ZH βl
jX/2
jB
November 06
X ≅ ZH β l
When Zc → 1
βl < π/4
B ≈ YLβl
 2006 Fabian Kung Wai Lee
66
33
Approximation for High and Low ZC (2)
•
Note that βl < π/2 implies a physically short Tline. Thus a short Tline
with high Zc (e.g. ZH) approximates an inductor.
lL =
•
ωc L
ZH β
A short Tline with low Zc (e.g. ZL) approximates a capacitor.
ω CZ
lC = c L
β
•
(3.2.2a)
(3.2.2b)
The ratio of ZH/ZL should be as high as possible. Typical values: ZH =
100 to 150Ω, ZL = 10 to 15Ω.
November 06
 2006 Fabian Kung Wai Lee
67
Example 3.2 - Mapping LPF Circuit into
Step Impedance Tline Network
•
•
For instance consider the LPF Design Example 2.2A (Butterworth).
Let us use microstrip line. Since a microstrip tline with low Zc is wide
and a tline with high Zc is narrow, the transformation from circuit to
physical layout would be as follows:
L1=4.061nH
g0=1/50
November 06
L2=9.803nH
C1=3.921pF
 2006 Fabian Kung Wai Lee
C2=1.624pF
RL= 50
68
34
Example 3.2 - Physical Realization of
LPF
•
Using microstrip line, with εr = 4.2, d = 1.5mm:
W/d
10.0
2.0
0.36
Zc = 15Ω
Zc = 50Ω
Zc = 110Ω
d/mm
1.5
1.5
1.5
W/mm
15.0
3.0
0.6
εe
3.68
3.21
2.83
β L = ε eL k o = ε eL × 2πf c × 3.3356 × 10 −9 = 60.307 s −1
β H = ε eH ko = ε eH × 2πf c × 3.3356 × 10 −9 = 53.258s −1
•
L1=4.061nH, L2=9.083nH, C1=3.921pF, C2=1.624pF.
 2006 Fabian Kung Wai Lee
November 06
69
Example 3.2 - Physical Realization of
LPF Cont…
l1 =
ω c L1
= 6.5mm
ZH βH
Verification:
β H l1 = 0.392 < π4 = 0.7854
β Ll2 = 0.490 < π4 = 0.7854
ω CZ
l2 = c 1 L = 9.2mm
βL
β H l3 = 0.905 > π4 = 0.7854
l3 = 15.0mm
l4 = 3.8mm
β Ll4 = 0.202 < π4 = 0.7854
l1
l2
l3
Nevertheless we still
proceed with the implementation. It will be seen
that this will affect the
accuracy of the -3dB cutoff
point of the filter.
l4
3.0mm
50Ω line
50Ω line
To 50 Ω
Load
15.0mm
November 06
0.6mm
 2006 Fabian Kung Wai Lee
70
35
Example 3.2 - Step Impedance LPF
Simulation With ADS Software (1)
•
Transferring the microstrip line design to ADS:
Microstrip line substrate model
Microstrip line model
Microstrip step junction
model
 2006 Fabian Kung Wai Lee
November 06
71
Example 3.2 - Step Impedance LPF
Simulation With ADS Software (2)
m1
freq=1.410GHz
dB(S(2,1))=-3.051
0
m1
-5
))1 -10
,2
(
S
(
dB -15
-20
-25
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
freq, GHz
November 06
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36
Example 3.2 - Step Impedance LPF
Simulation With ADS Software (3)
•
However if we extent the stop frequency for the S-parameter simulation
to 9GHz...
Parasitic passbands,
artifacts due to using
m1
freq=1.410GHz
transmission lines.
dB(S(2,1))=-3.051
m1
0
-5
))
1
,2
(
S
(
B
d
-15
-25
0
1
2
3
4
5
6
7
8
9
freq, GHz
November 06
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Example 3.2 - Verification with
Measurement
The -3dB point is around 1.417GHz!
The actual LPF constructed in year
2000. Agilent 8720D Vector Network
Analyzer is used to perform the
S-parameters measurement.
November 06
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37
Example 3.3 - Realization of BPF Using
Coupled StripLine (1)
tra
Ex
•
Based on the BPF design of Example 2.3:
79.58nH
50
Vs
15.916pF
To source
network
λo
0.7072nH
Admittance
inverter
tline
4
J1
-90o
0.1414pF
An equivalent circuit
model for coupled tlines
with open circuit at
2 ends.
RL
50
15.916pF
0.7072nH
See appendix (using Richard’s transformation
And Kuroda’s identities)
J2
-90o
J3
-90o
To RL
J4
-90o
An Array of coupled
microstrip line
λo
4
Section 1
Section 2
Section 3
λo = wavelength at ωo
Section 4
 2006 Fabian Kung Wai Lee
November 06
75
Example 3.3 - Realization of BPF Using
Coupled StripLine (2)
tra
Ex
•
Each section of the coupled stripline contains three parameters: S, W,
d. These parameters can be determined from the values of the odd and
even mode impedance (Zoo & Zoe) of each coupled line.
W
S
W
d
•
•
Zoo and Zee are in turn depends on the “gain” of the corresponding
admittance inverter J.
From Example 2.3
And each Jn is given by: ω1 = 2π (1.4GHz)
Z = Z 1 + JZ + ( JZ )2
ω 2 = 2π (1.6GHz)
fo = f1 f 2 = 1.497GHz
J1 =
π∆
1
Zo
ω −ω
∆ = 2ω 1 = 0.133
o
2 g1
π∆
J n = 2 1Z
for n = 2,3,4
g n −1g n
o
J N +1 = Z1
o
November 06
π∆
2 g N g N +1
LN
( o o )
Z oo = Z o (1 − JZ o + ( JZ o )2 )
oe
o
For derivation see chapter 8,
Pozar [2].
 2006 Fabian Kung Wai Lee
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38
Example 3.3 - Realization of BPF Using
Coupled StripLine (3)
tra
Ex
Section 1:
(
)
Z oo1 = Z o (1 − J1Z o + (J1Z o )2 ) = 37.588
Z oe1 = Z o 1 + J1Z o + (J1Z o )2 = 83.403
π∆ = 0.009163
J1 = Z1
2 g1
o
Section 2:
J 2 = 2 1Z π∆ = 0.002969
g1g 2
o
Section 3:
J 3 = 2 1Z
o
π∆
g 2 g3
= 0.002969
(
)
2
Z oo 2 = Z o (1 − J 2 Z o + (J 2 Z o ) ) = 43.680
Z oe 2 = Z o 1 + J 2 Z o + (J 2 Z o )2 = 58.523
Z oe3 = 83.403
Z oo3 = 37.588
Section 4:
J 4 = Z1
o
π∆
= 0.009163
2 g3g 4
Z oe 4 = 58.523
Z oo 4 = 43.680
Note:
g1=1.0000
g2=2.0000
g3=1.0000
g4=1.0000
 2006 Fabian Kung Wai Lee
November 06
77
Example 3.3 - Realization of BPF Using
Coupled StripLine (4)
tra
Ex
•
•
•
In this example, edge-coupled stripline is used instead of microstrip line.
Stripline does not suffers from dispersion and its propagation mode is
pure TEM mode. Hence it is the preferred structured for coupled-line
filter.
From the design data (next slide) for edge-coupled stripline, the
parameters W, S and d for each section are obtained.
Length of each section is l.
vp =
v
1
ε r ε o µo
= 1.463 × 108
ε r = 4.2
l = 4 fp = 1.463×109 = 0.024 or 24.0mm
o
4⋅1.5×10
November 06
8
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39
Example 3.3 - Realization of BPF Using
Coupled StripLine (5)
tra
Ex
Section 1 and 4:
S/b = 0.07, W/b = 0.3
Section 2 and 3:
S/b = 0.25, W/b = 0.4
By choosing a suitable b, the W and
S can be computed.
W
S
b
November 06
tra
Ex
•
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Example 3.3 - Coupled Line BPF
Simulation With ADS Software (1)
Using ideal transmission line elements:
Ideal open circuit
Ideal coupled tline
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40
Example 3.3 - Coupled Line BPF
Simulation With ADS Software (2)
tra
Ex
Parasitic passbands. Artifacts due to using distributed
elements, these are not present if lumped components
are used.
1.0
0.8
)1 0.6
,2
S((
ga
m 0.4
0.2
0.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
freq, GHz
2fo
 2006 Fabian Kung Wai Lee
November 06
tra
Ex
•
81
Example 3.3 - Coupled Line BPF
Simulation With ADS Software (3)
Using practical stripline model:
Stripline substrate model
Coupled stripline model
Open circuit
model
November 06
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41
tra
Ex
Example 3.3 - Coupled Line BPF
Simulation With ADS Software (4)
Attenuation due to losses in the conductor and dielectric
1.0
0.9
0.8
0.7
) 0.6
,12
(S 0.5
g(a
m 0.4
0.3
0.2
0.1
0.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
freq, GHz
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Things You Should Self-Study
•
•
•
•
•
Network analysis and realizability theory ([3] and [6]).
Synthesis of terminated RLCM one-port circuits ([3] and [6]).
Ideal impedance and admittance inverters and practical implementation.
Periodic structures theory ([1] and [2]).
Filter design by Image Parameter Method (IPM) (Chapter 8, [2]).
November 06
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42
Other Types of Stripline Filters (1)
•
LPF
•
HPF:
For these delightfully simple
approaches see Chapter 43 of
[3]
BPF:
SMD capacitor
 2006 Fabian Kung Wai Lee
November 06
85
Other Types of Stripline Filters (2)
•
More BPF:
•
BSF:
More information can be obtained from [2], [3], [4] and the book:
J. Helszajn, “Microwave planar passive circuits and filters”, 1994,
John-Wiley & Sons.
November 06
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43
Appendix 1 – Kuroda’s
Identities
 2006 Fabian Kung Wai Lee
November 06
87
Kuroda’s Identities
•
As taken from [2].
l
1
Z2
β
Z1
Note: The inductor represents
shorted Tline while the capacitor
represents open-circuit Tline.
Z
n2 = 1 + 2
Z1
l
Z2/n2
β
Z2
n2Z1
β
Z2
Z2/n2
November 06
Z1
n 2Z 2
1: n2
Z1
β
n2
l
l
1
Z2
1
β
l
l
Z1
Z1
n
l
l
Z1
β
β
n2: 1
1
n2Z1
 2006 Fabian Kung Wai Lee
β
n 2Z 2
88
44
THE END
November 06
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45