Research
Presentation
#2
Academic
Year
2012--2013
Insertion Loss Characteristics
Of Commonly Used Passive
Filters
John Choma
Ming Hsieh Department of Electrical Engineering
Universityy of Southern California
Los Angeles, California 90089-0271
(213) 740-4692 [Office]
(818) 384-1552 [Cell]
jjohnc@usc.edu
@
[[E-Mail]]
www.jcatsc.com [Course Notes & Technical Reports]
Overview Of Presentation

Two Port Filter Network

Filter Metrics
 Power Loss Ratio
 Insertion
I
ti Loss
L
 Passband Focus
Lowpass
Bandpass
Filter Networks
 Butterworth
 Chebyshev
 Bessel-Thomson

Research #2-- 2012-2013

Inductor Quality Factor
 Loss Implications
 Filter Pre-Distortion
 Engineering Assessment

Future Work - Sensitivity Models
Insertion Loss Of Lossless Filters
2
Insertion Loss Definition

Zout(s)
Signal
S
g a Power
o e Att Load
oad Port
ot
 Load Power If Signal Is
Coupled Directly To Load

Represent This Power
As Poo
Expression
2
Voo
 Rl
Vs
2
Rs
Vi
Pi

Linear
Two-Port
Network
Filter

Rl
Po
2
 Vs
 
Poo 

Rl
R

R
Rl
s
 l
 Load Power With Filter Inserted As Shown
Represent This Load Power As Po
Expression
Vo
2
Rs
Voo

Vs
2
Poo

Filter
Vo
2 Vs
Bypassed
Po 
 H(jω)
Insertion
se t o Loss
oss Metrics
et cs
Rl
Rl
 Power Loss
2
2


P
V
R
1
H(jω) 
l
Ratio, plr
p  oo  oo  

lr
2
P
V
R
R

o
o
s  H(jω)
 l
(j )
 Insertion Loss,,
IL, Is Decibel Value Of plr
IL  10 log10  plr   20 log10 Voo
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
Rl
Vo
Vs
Vo
3
Comments On Insertion/Power Loss
2
 Rl 
 

R

R
s
 l
2

Power
o e Loss/Insertion
oss/ se t o Loss
oss

Comments
2
H(jω)
 IL Measures Losses
Voo
Incurred By Insertion Of
IL  10 log
l 10  plr   20 log
l 10
Filter Between I/O Ports
Vo
Losses Are Aggravated By
plr 
Poo
V
 oo
Pi
Vo
1
Lossy Branch Elements Embedded In Filter Architecture
Lowpass Filters
 Losses Are Aggravated Outside Passband Where Transfer Function Falls,
Possibly Dramatically, As A Function Of Signal Frequency; These “Losses”
Are Desirable, For They Help To Highlight Response Nature Of Stopband
 Lossless, Lowpass, Passive Filters
 Have H(0) = Rl /(Rl + Rs)
 Power Loss Ratio Is Inverse Of Filter Transfer Function Normalized To
Zero Frequency Value Of Gain
2

plr Is Even Function Of Signal Frequency
Poles And Zeros
Poles Of Transfer Function Relate To Zeros
off Power
P
Loss
L
R ti
Ratio
Zeros Of Transfer Function Relate To Poles
of Power Loss Ratio
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
 Rl 
plr  

R

R
s
 l

1
H(jω)
2
1
H(jω) H(0)
2
4
Power Loss Ratio For Lossless Filter

Filter
te Abstraction
bst act o
 Average Incident Input
Signal Power Is Pio
Maximum Available
Zout(s)
Rs

Vi
Pio
Lossless,
Linear
Two-Port
Two
Port
Network
Filter
Vo
Rl
Pi
Po
Vs
Power From Signal
Pr

Source Vs
Actual Average Input
Signal Power At
Frequency ω If Rs = Zin(jω) [Conjugate Match Between Source And Input
Impedances At Frequency ω]
2
 Average Reflected Input Signal Power Is Pr
Pr  ρi (jω) Pio
i
ρi(ω) Is Input Port Reflection Coefficient Measured
Z (jω)  Rs
With Respect To A Reference Resistance Of Rs
ρi (jω)  in
Observed Average Input Signal Power Is Pi
Zin (jω)  Rs
Observed Average Output Signal Power Is
Po  Pi  Pio  Pr
Po = Pi Because Of Lossless Filter Architecture
2
 1  ρi (jω) Pio
 Power Loss Ratio, plr




Measures Power Reduction Due To
P
1
p  io 
Impedance Mismatch
lr
2
Po
1

ρ
(jω)
i
Unity Loss Ratio (No Loss) When Zin(jω) = Rs
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
5
Properties Of The Reflection Coefficient


Z (jω)
(j )
Normalized
o a ed Input
put Impedance
peda ce
zn (jω)
(j )  in
 rn ((ω))  jx
j n (ω)
( )
Rs
 rn(ω) = rn(−ω) For Physically
Realizable Driving Point Input Resistance (Even Frequency Function)
 xn((−ω)
ω) = −x
xn(ω) For Physically Realizable Driving Point Input
Reactance (Odd Frequency Function)
Z (jω)  Rs
r (ω)  1  jxn (ω)
Reflection Coefficient
ρi (jω)  in
 n
Zin
rn (ω)  1  jxn (ω)
 Magnitude Squared
i (jω)  Rs
r (ω)  1  jxn (ω) rn (ω)  1  jxn (ω)
2
ρi (jω)  ρi (jω)ρi (-jω)  n

rn (ω)  1  jxn (ω) rn (ω)  1  jxn (ω)


2
Alternative Expression
M (ω)   rn (ω)  1  xn2 (ω)
r (ω)  1  jxn (ω) rn (ω)  1  jxn (ω)
M (ω)
2
ρi (jω)  n


j n (ω)
( ) rn ((ω))  1  jx
j n (ω)
( )
M (ω)
( )  4rn (ω)
( )
rn ((ω))  1  jx
Power Loss Ratio In Terms Of Reflection/Impedance Parameters
p
R
h #2 2012 2013
2
rn (ω)  1  xn2 (ω)

M (ω)

 1
 1
2
4rn (ω)
( )
4rn ((ω))
1  ρ(jω)
1
l
lr
I
ti
L
Of L
l
Filt
6
Example: Lowpass Butterworth Filter


2
nth Order Lowpass Transfer Relationship
H(jω)

 Maximally Flat Magnitude Butterworth
H(0)
Frequency Response
1
 B Designates 3-dB Bandwidth Of Filter
 H(0) Represents Zero Frequency Value Of Transfer Function
ω
 
B
2n
Power Loss Ratio (plr) Of nth Order, Lowpass Butterworth Filter
ω
1  
plr 
2
B
H(jω) H(0)
1

1
2n
Insertion Loss (IL) Of nth Order, Lowpass Butterworth Filter
2n

ω 
IL  10 log10  plr   10 log10 1    
 B  

Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
7
Insertion Loss Frequency Response (LP)
Lowpass Butterworth Filter
80
Insertio
on Loss In D
Decibels
70
60
50
40
30
20
10
0
0.10
0.16
0.25
0.40
0.63
1.00
1.58
2.51
3.98
6.31
10.00
N
Normalized
li d Signal
Si
l Frequency,
F
( /B)
(ω/B)
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
8
Expanded Insertion Loss Response (LP)
Lowpass Butterworth Filter
3
Insertio
on Loss In Decibels
25
2.5
2
1.5
1
0.5
0
0.10
0.16
0.25
0.40
0.63
1.00
Normalized Signal
g
Frequency,
q
y, ((ω/B))
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
9
Comments LP On Insertion Loss Responses

General Comments
 Response Plots Applicable Only To Lowpass Butterworth Filters
 Frequency Scale Is Normalized To 3-dB Bandwidth Of Filters

Design
D
i
Ob
Observations
ti
 Insertion Losses Are Less Than 3 Decibels Within Filter Passband,
Independent of Filter Order
 Passband
P
b d Extends
E t d From
F
Zero
Z
Frequency
F
-ToT 3-dB
3 dB Bandwidth,
B d idth B
 Insertion Losses
Dramatically Changes With Order n Outside Passband (Which Is Termed
The Stopband)
 To Be Expected In That Large n Encourages Rapid Frequency Response
Rolloff In Stopband
 Insertion Loss Approaches 20n dB For High Signal Frequencies
Insertion Losses Are Reduced For Increased Filter Order Within Filter
Passband
 Higher Order Implies Filter Frequency Response Is Flatter Over Wider
Passband Frequencies
 Very
V
High
Hi h O
Order
d L
Lowpass B
Butterworth
tt
th Filt
Filter A
Approximates
i t Id
Idealized
li d “Brick
“B i k
Wall” Filter
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
10
Example: Butterworth Bandpass Filter


 s
Lowpass
o pass To
o Bandpass
a dpass Transformation
a s o at o
ω 
c

p

BQ

 p Is Complex Frequency For Lowpass Prototype
cω
s 
 c

 B Represents Bandwidth Of Lowpass Prototype
ω
 ωc Is Tuned Center Frequency Of Bandpass Realization
Bc  c
Q
 Qc Is Quality Factor Of Bandpass Realization
c
 Bc Represents 3-dB Bandwidth Of Bandpass Filter
Transformed Transfer Function
H(jω)
H(0)

2
1

2n
Lowpass
p
To Bandpass
H(jω)
H(jωc )
2

1
  ω ωc  
ω
1  Qc 

1  

ω
ω

B
  c
Insertion Loss (IL) Of Bandpass Butterworth Filter
  ω ωc  
 1  Qc 

plr 

2
ω
ω

H(jω) H(jωc )
  c
1


IL  10 log10  plr   10 log10 1 

Research #2-- 2012-2013
2n
  ω ωc  

Qc 

ω
ω

  c
Insertion Loss Of Lossless Filters
2n
2n 



11
Insertion Loss Frequency Response (BP)
Bandpass Butterworth Filter--Constant
Filter Constant Q
140
Qc = 5
Insertion
n Loss In De
ecibels
120
100
80
60
40
20
0
0.10
0.16
0.25
0.40
0.63
1.00
1.58
2.51
3.98
6.31
10.00
Normalized Signal Frequency, (ω/ωc )
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
12
Expanded Insertion Loss Response (BP)
Bandpass Butterworth Filter--Constant
Filter Constant n
14
n= 1
Insertion Loss In Dec
cibels
12
10
8
6
4
2
0
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
Normalized Signal Frequency, (ω/ωc )
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
13
Comments On BP Insertion Loss Responses

Power
o e Loss
oss Ratio
at o
 Neighborhood Of Tuned Center Frequency
ω = ωc + Δω
 ω  ωc  ω  ωc   2Δω  1  Δω   2Δω , Δω  2ω
ω ωc




c
ωc ω
ωωc
ωc 
2 c
2ω
ωc


2n
2n
  ω ωc  
 2Qc Δω 



Q
1
 c



ω
ω
ω
c 


  c
 10 
IL  10 log10  plr   
 ln  plr   4.343 ln  plr 
 Resultant
 ln 10 
2
2n
22n
I
Insertion
ti

 2Qc Δω  


2n Δω

Loss

IL  4.343 ln 1  
4.343
2Q





c
ω
ω


c 

 c
Observations


 No Insertion Loss At, And Very Near, Tuned Center
C
Frequency,
Which Implies Unity Center Frequency Gain
 Insertion Loss Increases As (2n)th Power Of Twice The Filter Quality
F t
Factor
Resultant Power Loss
plr  1 
Ratio
Δω  ωc
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
14
Lowpass Chebyshev Filter

2
Squa ed Transfer
Squared
a s e Relationship
e at o s p
H(jω)
1

 ε Defines Allowable Response Ripple Within H( 0 )
1  ε 2Cn2 (ω)
Filter Passband
Ripple (r) In Decibels
r  20 log
g10 1  ε 2  10 log
g10 1  ε 2
Filter Passband Extends To


Radial Frequency Of ωp

 Passband Frequency Is Generally Smaller Than Filter 3-dB Bandwidth
 Special Case Is That ωp Is Exactly The 3
3-dB
dB Bandwidth If r = 3 dB
Response In
Passband
 At Zero Frequency:
 At Passband Edge:


Chebyshev
Polynomial, Cn(ω)
Squared Sinusoidal Cn (ω) 


Cn2 (ω p )  1
B
Bracketed
k dR
Response Wi
Within
hi Passband
P
b d
Research #2-- 2012-2013

cos  n cos 1 ω ω p  , 0  ω ω p  1


coshh  n cosh
h 1 ω ω p  , ω ω p  1


1 , n even
Cn2 (0) 
0 , n odd
Insertion Loss Of Lossless Filters
1
1 ε2

H(jω)
(j )
 1
H( 0 )
15
Comments On Chebyshev Response


H(jω)

H( 0 )
General
Ge
e a Transfer
a s e Function
u ct o
Within Filter Passband, 0 ≤ ω/ωp ≤ 1
1  ε 2Cn2 (ω)
 Frequency Response Modulates Because
1
H(jω)

 1
Of Sinusoidal Nature Of Chebyshev
H( 0 )
1 ε2
0 ≤ ω/ωp ≤ 1
Polynomial
H(jω p )
 At Passband Edge, Frequency Response Is
1

Down By An Amount Related To Ripple
H( 0 )
1 ε2
H(jω)
1

H( 0 )
1  ε 2 cos 2  n cos 1 ω ω p  r  ripple  10 log10 1  ε 2


Outside Filter Passband, ω/ωp ≥ 1
 Frequency Response Attenuates Sharply Because Of Hyperbolic
Nature Of Chebyshev
y
Characteristic Polynomial
y
 Response Outside Passband Is Continuous With Passband
Response At Passband
H(jω)
1

Edge
g ((ω = ωp)
H( 0 )
1  ε 2 cosh 2  n cosh 1 ω ω p 




1

Research #2-- 2012-2013
Insertion Loss Of Lossless Filters




16
Example: Lowpass Chebyshev Filter

Power
o e Loss
oss Ratio
at o
plr 


1
2



1  ε 2 cos 2  n cos 1 ω ω p  0  ω ω p  1




1  ε 2 cosh 2  n cosh 1 ω ω p  ω ω p  1


Insertion Loss IL  10 log10  plr 
r  ripple  10 log10 1  ε 2
Plots (Next Two Slides)
 Stopband (ω > ωp)
Increased Insertion Loss With Increasing Filter Order
Larger Rate Of Insertion Loss Increase Than With Butterworth Filter
 Passband (ω ≤ ωp)
Insertion Loss Exacerbated By Response Ripple
Peak Passband Insertion Loss Identical To Filter Ripple
Insertion Loss Identical To Response Ripple At Passband Edge
Filter Order
H(jω) H(0)


 For Odd Order Filter, Insertion Loss Approaches Zero Decibels At Very Low
Signal Frequencies
 For Even Order Filter
Filter, Insertion Loss Approaches Response Ripple At Very
Low Signal Frequencies
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
17
Insertion Loss Frequency Response (LP)
Lowpass Ripple Chebyshev Filter--Constant Ripple
120
ripple = 2 dB
Insertion
n Loss In De
ecibels
100
80
60
40
20
0
0 10
0.10
0 16
0.16
0 25
0.25
0 40
0.40
0 63
0.63
1 00
1.00
1 58
1.58
2 51
2.51
3 98
3.98
6 31
6.31
10 00
10.00
Normalized Signal Frequency, (ω/ωp )
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
18
Passband Insertion Loss Response (LP)
Lowpass Chebyshev Filter--Constant
Filter Constant n
Insertion L
Loss In Deciibels
2.0
n=3
1.5
1.0
0.5
0.0
0.10
0.13
0.16
0.20
0.25
0.32
0.40
0.50
0.63
0.79
1.00
Normalized Signal Frequency, (ω/ωp )
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
19
Chebyshev Bandpass Filter

Frequency Transformation
 s
ω  B  ωc  ω
p
 Q 
 c c
Q
cω
c
s 
p
 c

p
Power Loss Ratio And Insertion
Loss
r  10 log10 1  ε 2
  ω ωc  
2
2
1 
1  ε cos  n cos Qc 

  passband
  ωc ω  

plr 
  ω ωc  
2
2
1 
1  ε cosh  n cosh Qc 

  stopband
IL  10 log10  plr 
  ωc ω  





Insertion Loss Plots
 Metrics
Fourth Order (n = 4)
Quality Factor (Qc = 7)
Ripple (r = 0.25 dB, 0.50 dB, 1 dB)
 Plot On Next Slide
 Comments Subsequent To Next Slide
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
20
Passband Chebyshev Insertion Loss (BP)
Bandpass Chebyshev Filter
Filter--Constant
Constant n & Qc
Insertion Loss In Dec
cibels
2.0
n=4
Qc = 7
1.5
1.0
0.5
0.0
0.93
0.95
0.97
0.99
1.01
1.03
1.05
1.07
Normalized Signal Frequency, (ω/ωc )
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
21
Comments On Chebyshev Insertion Loss
Passband
assba d Response
espo se
 Metrics
Shown For 4th Order Bandpass Filter
Shown For Quality Factor, Qc, Of 7
Radial Center Frequency Is ωc
 Plotted Only For Passband Response
Passband Width Is ωp, Which Is Not
3 dB Bandwidth,
3-dB
Bandwidth B
3-dB Bandwidth, B

Insertion
n Loss In Decibels

2.0
n=4
Qc = 7
1.5
1.0
0.5
0.0
0.93
0.95
0.97
1.01
1.03
1.05


1
1


 cosh   cosh 1 
 n 

r 10
p
10
1


B
Comments
 Insertion Loss Is Sensitive To Passband Ripple
Maximum Insertion Loss In Passband Is The Ripple
Maximum Insertion Loss

0.99
1.07
N
Normalized
li d Signal
Si
l Frequency,
F
( / c)
(ω/ω



 
 Occurs At Tuned Center Frequency For Even Integer n
 Occurs At Band Edges For Odd Integer n
Small Ripple Case
Chebyshev
Ch b h IL Less
L
D
Dramatic
ti Than
Th Butterworth
B tt
th Filter
Filt IL Within
Withi P
Passband
b d
Insertion Loss Exceeds That Of Butterworth In Neighborhood Of Tuned
Center Frequency
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
22
Chebyshev 3-dB Bandwidth
33-dB
dB Bandwidth Converges To Passband
Bandwidth For Large Filter Order
Norm
malized 3-dB
B Bandwidth. (B/ωp )
2.00
1.50
5th Order (n = 5)
1.00
0.10
0.40
0.70
1.00
1.30
1.60
1.90
2.20
2.50
2.80
Passband Ripple (r) In Decibels
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
23
Lowpass Bessel-Thomson Filter

()
1
Normalized
o a ed H(s)

2
3
4
Transfer
H(0)
1   sTdo   a2  sTdo   a3  sTdo   a4  sTdo   
Function
 Groupp Delayy At Zero Signal
g
Frequency
q
y Is Tdo
 Coefficients For nth Order Filter
2  n  1 n  2 
n 1
a2 
a3 
2n  1
3  2n  1 2n  2 
 n  1 n  2  n  3 
3  2n  1 2n  2  2n  3 

 2 m    2n  m  !   n! 
a4 
am  

 m!    2n  !    n  m  ! 


Squared Fourth Order Transfer Function Relationships
H(jω )
H(0)
plr 
Research #2-- 2012-2013
2

1
2
1  a  ωT  2  a  ωT  4    ωT  2 1  a  ωT 2 
2
do
4
do 
do 
3
do 




1
H(jω) H(0)
2
2
IL  10 log10  plr 
Insertion Loss Of Lossless Filters
24
Lowpass Bessel Filter Insertion Loss
Lowpass Bessel
Bessel-Thomson
Thomson Delay Equalizer
40
Insertion Loss In De
ecibels
35
30
Order = 4 (n = 4)
25
Order = 3 (n = 3)
20
Order = 2 (n = 2)
15
10
5
0
0.10
0.16
0.25
0.40
0.63
1.00
1.58
2.51
3.98
6.31 10.00
Normalized Signal Frequency, (ωTdo )
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
25
Comments On Bessel Thomson Performance

Bessel Thomson Filter
 Commonly Known As Bessel Thomson Delay Equalizer
 Provides Maximally Flat Group Delay Over Filter Passband
 Amount Of Maximally Flat Delay Is Zero Frequency Delay
Delay, Tdo

Insertion Loss Characteristics
 Minimal Insertion Loss Within Filter Passband; Slightly Better Than
B tt
Butterworth
th Filter
Filt
 Insertion Loss Within Filter Passband Actually Improves Slightly With
Filter Order

3-dB Bandwidth
 Daunting Challenge To Quantify In Closed Form
 3-dB Bandwidth Estimates, B
n = 2  B = 0.7913/Tdo = 2.374a2 /Tdo
n = 3  B = 0.7855/Tdo = 1.960a2 /Tdo
n = 4  B = 0.7855/Tdo = 1.833a2 /Tdo

Bandpass Insertion Loss Follows Insertion Footprints Of
Butterworth and Chebyshev Filters
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
26
Example: Lowpass Impedance Converter
L

Circuit
C
cu t Sc
Schematic
e at c Diagram
ag a

Filter Metrics
R
 Self Resonant Frequency Is ωo
C
R
 Circuit Quality Factor Is Q

 Transfer Function If
V
FILTER
RL
Filter Is Removed, KL: K L 

RL  RS
 Static Effect Of
RS  RL
Inductor Quality
KR 
RL
RS  RL  R
Factor, KR:
H(0) 
 KL KR
RS  RL  R
 Zero Frequency Gain Is H(0)
K z  RS RL  1
 Impedance Conversion Factor Is Kz:
H(0)
KL KR
I/O Transfer Function H(s)  Vo 

2
2
Vs
 Self-Resonant
Self Resonant




s
s
s
s



1
1



Frequency
Qωo  ωo 
Qωo  ωo 
R
Vi
Vo
S
L
S

ωo 

1
H(0)K z LC
Circuit Quality Factor
Derives From:
Research #2-- 2012-2013
1
1

Q
RL
Q = 1
Insertion Loss Of Lossless Filters
L H(0)
C
  RL  R 
H(0)K z
C Kz
L
2 for maximally flat magnitude
27
Insertion Loss Analysis

Power
o e Loss
oss Ratio
at o
 Voo /Vs Is Transfer Function With Signal
Source Coupled Directly To Load
 Vo /Vs Is Transfer Function With Filter
Inserted Between Source And Load
Terminations
V
H(jω )  o 
Vs


KL KR
 ω 
jω
1 


ω
Qωo
 o
Resultant Power
Loss Ratio
Resultant Insertion
Loss
Research #2-- 2012-2013
2
p
lr
 H(jω )
p
Voo
Vs
2

lr
2
Voo

Vo
2
Voo Vs

Vo Vs
2
 RL 
2
 KL  

R

R
S
 L
2
 K L K R 2
2
2
2
 



1   ω     ω 
Qωo 
Q
  ωo  



2
4






1
ω
ω



1
2




2
2  ω 

ω
Q  o 
V V
 o

 oo s 
2
Vo Vs
KR
V
IL  10 log10  plr   20 log10 oo
Vs
Insertion Loss Of Lossless Filters
Vo
Vs
28
Converter Insertion Loss
Insertion Loss In Decibels
40
KR = 0.9
26
Maximally
Flat Magnitude
12
-2
0.10
0.16
0.25
0.40
0.63
1.00
1.58
2.51
3.98
6.31 10.00
Normalized Signal Frequency, (ω/ωo )
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
29
Comments On Converter Insertion Loss

Insertion
se t o Loss
oss Plot
ot

General Observations
 Low Frequency Insertion Loss
Determined Largely By Metric KR
KR Is Intimately Related To



Resistance R, Which Accounts For
Finite Inductor Quality Factor
KR = 1 For Ideal Inductor (R
( = 0))
Insertion Los
ss In Decibels
40
KR = 0.9
26
Maximally
Flat Magnitude
12
-2
0.10
0.16
0.25
0.40
0.63
1.00
1.58
2.51
3.98
6.31 10.00
Normalized Signal Frequency, (ω/ωo )
Maximally Flat Magnitude Case
Gives Small, Monotonically Increasing Passband Insertion Loss
L
Large
Ci
Circuit
it Q
Corresponds to Small Second Order Damping Factor
Yields Magnitude Response Peaking In Passband
Correspondingly delivers Non-Monotonic
Non Monotonic Insertion Loss Response
Small Circuit Q
Corresponds To Second Order Overdamping
Gives Monotonically Increasing Insertion Loss Response Because
Frequency Response Is Characteristically Monotonic
Terrible Passband Insertion Loss Response Due To A Passband That Is
Constrained By Small Bandwidth Arising From Overdamping
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
30
Modeling And Metrics Of Inductance Loss


Ideal
dea Inductance
ducta ce
 Simple Inductive Impedance
 No Winding Or Other Losses
I(s)
V(s) I(s)  Z(s)  sL
Practical Inductance
 Choose A Normalizing Frequency, ωα
Can Be Bandwidth Of Lowpass Filter Or




I(s)
()
L
L

V(s)
RLL

V(s)
V(s) I(s)  Z(s)  sL  RLL
q
y Of Bandpass
p
Filter
Center Frequency
Can Be Self-Resonant Frequency Of Filter Network
Choose A Normalizing Impedance, Zα
Can Be Signal Source or Terminating Load Resistance Of Network
Can
C B
Be Ch
Characteristic
i i IImpedance
d
Of C
Considered
id d S
Structure
 Normalized Frequency, p: p  s ωα
ωα L
Q

L
 Quality Factor, QL, Of Inductor
RLL
Normalizing Circuit Parameters
 Inductance, Lα: Lα  Zα ωα
 Capacitance, Cα: Cα  1 ωα Zα
 Normalized Variables (Impedance Zn, Inductance Ln, Capacitance
Cn, Frequency, p)
Z n  Z Z α Ln  L Lα Cn  C Cα p  s ωα
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
31
Inductance Loss Mathematical Model
L
RLL

Practical
act ca Inductance
ducta ce

Analysis Of Net Inductor Impedance
V(s)
 Impedance
 s  RLL 
 s 
ωα L 



 ωα L  RLL
ω
ω
L
ω
sL  RLL
α 
 α 
  α

Z nL (s) 
Zα
Zα
Zα
 Normalized Impedance Of

L 
1 
1 
Z
(p)
p
p







 Ln
nL
Practical Inductor
Lα 
QL 
QL 

Approximation
pp
“Practical”

I(s)

Inductor Invokes Only Frequency Invariant, Constant Series Resistance
Analytical Procedure
 Replace “Practical” Inductance With Normalized Inductance, Ln = L/Lα
 Write
W it Conventional
C
ti
l Ki
Kirchhoff
hh ff E
Equilibrium
ilib i
E
Equations,
ti
B
Butt U
Using
i a complex
l
Frequency Variable In

1   s
1   s
RLL 
Inductor Impedance Of:


  

p
  
Q
ω
Q
ω
ω
L
 Invoke This Procedure For 
L
L
α 
 α
 α
E h IInductance
Each
d
IIn Network
N
kU
Undergoing
d
i S
Scrutiny
i
 Procedure Gives Transfer Function With All Inductance Qs Considered
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
32
Modeling And Metrics Of Capacitance Loss


Ideall Capacitance
Id
C
it
 Simple Capacitive Admittance
 No Dielectric Or Other Losses

I(s) V(s)  Y(s)  sC
I( )
I(s)
C

V(s)
C
Practical Capacitance
I(s)


R
 Choose A Normalizing Frequency, ωα, And A
Normalizing Admittance, Yα, As In Inductor Case
V(s)
 Normalized Frequency, p: p  s ωα I(s) V(s)  Y(s)  sC  1 R 
CC
 Quality Factor, QC, Of Capacitor
QC  ωα RCC C
Capacitive Quality Factor Generally Much Larger Than
CC
Inductor Quality Factor
Capacitors Are Far Less Lossy Than Are Inductors

Normalizing
g Circuit Parameters
 Inductance, Lα: Lα  Zα ωα
 Capacitance, Cα: Cα  1 ωα Zα
 Normalized Variables (Admittance Yn, Inductance Ln, Capacitance
Cn, Frequency, p)
Yn  Y Yα  Z αY Ln  L Lα Cn  C Cα p  s ωα
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
33
Capacitance Loss Mathematical Model

C
Practical
act ca Capac
Capacitor
to


I(s)
RCC

Analysis Of Net Capacitor Admittance
V(s)
 Normalized Admittance
 s 
1
1
ω
C

sC 

 α
 s 

ω
RCC
RCC
1
α

YnC (s) 

 ωα Z αC 


Yα
Yα
ω
ω
R
C
α CC 
 α 
 Normalized Admittance Of

C 
1 
1 
Y
(s)
p
p







 Cn
nC
Practical Capacitor
Cα 
QC 
QC 

Approximation Of “Practical”
Practical
Capacitor Invokes Only Frequency Invariant, Constant Shunt Resistance
Analytical Procedure
 Replace
p
“Practical” Capacitance
p
With Normalized Capacitance,
p
, Cn = C/Cα
 Write Conventional Kirchhoff Equilibrium Equations, But Using a complex
Frequency Variable In Capacitor Admittance 
1   s
1
Of:
p




 
Q
 Invoke This Procedure For Each Capacitance 
C
 ωα QC
In Network Undergoing Scrutiny
 Procedure Gives Transfer Function With All Capacitance Qs Considered
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters



34
Approximate System Level Analysis

First
st O
Order
de Analysis
a ys s And
d Design
es g
 Replace Normalized Complex Frequency By

s
1 
 p
Q-Corrected Normalized Frequency:

ωα
Q
L

 Leads To Conservative Estimate Of Effect Of Losses
Quality Factors Of Capacitors Are Relatively Unimportant Because They
Are Generally Much Larger Than Inductive Counterparts
Pre-distortion: Close Examination Of Lossy Result Leads To Guidelines
For Modifying
Modif ing Val
Values
es Of Energ
Energy Storage And Loss
Lossy Branch Elements

Example
 Return To Lowpass Impedance Converter Considered Earlier
 Execute Analysis With Utilized Inductance Presumed Ideal
Resistance R Is Set To Zero (No Loss In Circuit Inductance)
Let Resultant Self-Resonant Frequency Be Designated As ωx
Let Resultant Circuit Quality Factor Be Designated As Qx
 Analysis: Normalized Frequency Is (s/ωx)
Replace (s/ωx) By (p + 1/QL)
Implication Of This Simplifying Approximation Is That All Energy Storage
Elements Exude Identical Qs
Result Is Approximate (Potentially Significantly Conservative) Result
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
35
Filter Analysis
L

Filter
te With
t Ideal
dea Inductance
ducta ce

I/O Transfer Relationship, H(s)
R
Vo
H(0)

H(s) 
2
Vs
C
R
 s 
s
M ti
Metrics

LOSSLESS
1

 V
FILTER
Qx ωx  ωx 
 Self
Resonant Frequency, ωx
RL  RS
1
 Quality ωx 

RS LC
H(0)K z LC
Factor
Normalized Frequency Is
1
1 L H(0)
( )
C
(s/ωx)

 RL
H(0)K z
Qx
RL C K z
L
 Replace By (p + 1/QL)
 Large QL Presumption


RS
LC
C

 RL

  p  1 Q  2  p 2   2 p Q 
RS  RL  RS 
L  RL  RS  
L 
L
 Comments
ωx Is A Measure Of 3-dB Bandwidth Of Filter
1/Qxωx Measures The Group I/O Delay Of Filter At Low Frequencies
Qx Is A Measure Of Relative Stability And Settling Time Of Filter
S

Vo
R=0
Vi
L
S
•
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
36
Approximate Impact Of Inductor Q

Revised
e sed Transfer
a s e H(s) 
Relationship
1
(Approximate)

Alternative Form
H(0)
H(0)

2
p   1 QL 
2  2p
 s 
s


1
p


Qx
QL
Qx ωx  ωx 
H(0)
1
1
Qx QL

H(0)
H(p) 

p   1 QL 

2Qx 
2 2p
1
p 
 1

Qx
QL
p 
QL 
p2
1

1
1
Qx  1 
 1 1
ω
L
ΓL  1 
x

Qx QL 
Qx QL
Qx QL QL  R

Comments
 Infinitely Large QL Reduces Alternative Transfer Function Form To
Normalized Form Of Original Network Transfer Function
 Resultant Self-Resonant Frequency
q
y Increases By
y Square
q
Root Of ΓL
 Quality Factor (Qx) Increases If 2Qx2 > 1
 Resultant Zero Frequency Gain Reduced By ΓL
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
37
Comments On Inductor Loss Effects

1
General
Ge
e a Co
Comments
e ts Γ  1
ωx L
1
L
Q

Qx QL L
 Procedure
R
Assumes That All Energy Storage Elements
Exude The Same Quality
y Factor
Capacitors Have Larger Qs
Mitigation


 Increase Calculated QL By
Nominally A Factor Of Two
 Simulate “DC” Transfer
Function
 Compare To Calculated
Value Of “DC”
DC Transfer
 Extract Effective Value
Of QL
L
R
Vi
Vo
RS

VS
C
RL
FILTER

H(0)
( )
ΓL
H(p) 
p
1
Qx
Zero Frequency Group Delay (Tdo) Increases
If 2Qx2 > 1 And
A d ΓL ≠ 1
 1  2Q 2  Γ  1  p 2
x L


ΓL
ΓL


1  2Qx2  ΓL  1
Tdo 
ωx Qx ΓL
Design
 “DC” Gain, H(0), Self-Resonant Frequency, ωx, & Circuit Qx Affected
 Quantification Of These Effects Leads To Optimization Via Branch
Element “Pre-Distortion” Adjustments
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
38
Future Work

Qua ty Factors
Quality
acto s O
Of Energy
e gy Sto
Storage
age Elements
e e ts
 Examine Efficient, Realistic Means Of Considering Capacitive Losses
 Examine Effects Of Skin Effect and Self-Resonance In Inductances

Pole Positions
 Discern Perturbation In Real And Imaginary Parts Of Poles As A
Result Of Energy Storage Element Losses
 Generally,
Generally Pole Locations Move Closer To Imaginary Axis In
Complex Frequency Plane By an Amount Of 1/QL

Circuit Models
 Development Of Models That Project Sensitivity To Energy Storage
Losses
 Possible Use Of Network Adjoint Topologies
Available To Ascertain Performance Sensitivity To Branch Element
Parameters
Already Embedded In Cadence Sceptre For Sensitivity Analysis
Trick Is To Apply, Realistically, Adjoint Sensitivity Methods To Insertion
Loss Frequency Response
Research #2-- 2012-2013
Insertion Loss Of Lossless Filters
39