Basics Equations for Filters
Synthesis
Giuseppe Macchiarella
Polytechnic of Milan, Italy
Electronic and Information Department
Scattering parameters for lumped-elements filters
2-port
network
R0
R0
For linear, lumped-element networks the scattering
parameters can be always represented as a polynomial
ratio of the complex frequency p ( p    j , with   2 f ) :
N11 ( p )
S11 
,
D( p)
N12 ( p )
S 21  S12 
,
D( p)
Roots of N11 (N22)
Roots of N21
Roots of D
N 22 ( p )
S 22 
D( p)
: reflection zeros
: transmission zeros
: poles (natural frequencies)
The above polynomials are called characteristic polynomials
Properties of lossless networks
For synthesis purposes the 2-port network representing a
filter is assumed lossless (i.e. composed by lossless elements)
The scattering matrix of a lossless network is unitary:
S  S *  U 2
*
S11  p   S11*  p   S 21  p   S 21
 p  1
*
S 22  p   S 22
 p   S12  p   S12*  p   1
As a consequence the following properties hold for the
polynomials defining the scattering parameters:
N11 ( p )  N ( p ),
N 22 ( p )   1 N * ( p )
n
*
N ( p )  N * ( p )  N 21 ( p )  N 21
( p )  D ( p )  D* ( p )
The latter expression is know as Feldtkeller equation
Properties of characteristic polynomials




The roots of D(p) are real or conjugate pairs. The real
part must be negative (strict Hurwitz polynomial)
The roots of N(p) can be everywhere in the complex
plane (if complex they occur as conjugate pairs)
The roots of N12(p) can be on the imaginary axis
(conjugate pair) or on the real axis (pairs with opposite
values) or as a complex quad in the p plane
The coefficients of all the polynomials are real
numbers
Steps of the synthesis process




Assignment of selectivity specifications (Attenuation
mask)
Selection of a suitable approximating function and
evaluation of its parameters in order to satisfy the
specifications
Evaluation of the characteristic polynomials from the
approximating function
Synthesis of the network from the polynomials
Synthesis in a transformed frequency domain
In order to simplify the design process, the synthesis can be
performed in a normalized low-pass domain, analytically
defined by a frequency transformation. The synthesized
network is then de-normalized to the band-pass frequency
domain with a suitable circuit transformation
Specifications
Requirements in
the normalized
domain
De-normalization of
the synthesized
network
Synthesis in the
transformed
domain
Low-pass  Band-pass transformation
Let be s=+j the normalized low-pass domain where
the filter passband is defined B=-1+1.
The following equation relates the normalized domain
with a pass-band domain suitably defined:
f0  p f0 
s   
B  f0 p 
f0 and B are the frequency parameters defining the
transformation:
f0 
f p1  f p 2 ,
B  f p 2  f p1
Properties of the transformation (imaginary axis)
For s=j and p=jf it has
f0  f
f0 
   
B  f0 f 
s
-1
1
0
s

fs1
fp1
f0
fp2
f p1  f p 2  f s1  f s 2  f 02
fs2
f (>0)
Conservation of response: A(j)=A(j)
80
dB
60
40
20
0
-2
-1.5
-1
-0.5
0
Normalized
0.5
1
1.5
2
80
dB
60
B=55 MHz
f0=1000 MHz
40
20
0
950
960
970
980
990
1000
MHz
1010
1020
1030
1040
1050
Circuital property of the transformation:
Conservation of Z and Y
Band pass
Low pass
Ceq
gc
Y=s.gc
gL
f0  p f0 
s   
B  f0 p 
Leq
f 0  p 0 
1
g
p
C






 C
eq
B  0 p 
p  Leq
g
1
Ceq  C , Leq  2
2 B
0  Ceq
Y
Leq
Z=s.gL
Ceq
f 0 f 0  p 0 
1





g
p
L

 L
eq
B B  0 p 
p  Ceq
g
1
Leq  L , Ceq  2
2 B
0  Leq
Z
Normalized low-pass prototype

Ladder network synthesized in the normalized domain
assuming unitary reference load (generator side):
Lowpass
Bandpass
Bandpass de-normalization:
g1, g3, …  shunt resonators with Ck=gk/(R1.)
g2, g4, …  series resonators with Lk=R1.gk/
Specifications in the lowpass domain
Amin
Amin
Stopband
f s ,1  f s ,2  f 02
Stopband
As3
Passband
A
(dB)
Amax
fs1
fp1
fp1
fs2
Freq.
Amin
Passband
f 0  f s ,1 f 0 
a  



B  f0
f s ,1 
Stopband
Amax
1
a

Approximation of the ideal Lowpass response
General expression for attenuation in the normalized
domain:
A     1   2Cn2   
Cn is called Characteristic Function and define the
approximation of ideal filter response. The parameter
n represents the order of the characteristic function
and corresponds to the number of resonators in the
de-normalized network
Properties:
-1<Cn<1
for -1<<1
|Cn|=1
for =±1
A(±1)=1+’2
Characteristic function and S parameters
Assuming lossless condition (|S11|2+ |S21|2=1):
1
A 
S 21
2
 1   2Cn2   
1 S11  j 
C   2
  S 21  j 
 2Cn2    

2
2
n
1
S 21
2
S11
1 
S 21
1 S11  j 
 Cn  j   
  S 21  j 
’ can be expressed as function of Return Loss (RL):
1   
2
2
1
1  S11 max
2

 
2
S11 max
1  S11 max
2

10

RL
10
1  10

RL
10
2
The characteristic function in s domain
Being s=j,it has:
analytic continuation
Cn2     Cn  j   Cn   j 
 Cn2  s   Cn  s   Cn   s   Cn  s   Cn*  s 
As a function of the scattering parameters:
1 S11  s 
Cn  s  
  S 21  s 
The S parameters can be expressed as ratio of
characteristic polynomials (monic):
F (s)
,
S11  s  
E ( s)
P( s) 
S 21 
E ( s)
 F s
Cn  s  
  P s
Properties of characteristic polynomials
Roots of F(s): reflection zeros
Roots of P(s): transmission zeros
Roots of E(s): poles
For the unitary of S matrix:
P( s )  P( s )   2  F  s   F   s    E  s   E   s 
E(s) is found from the roots with negative real part of right
hand side (Hurwitz polynomial)
In conclusion: given Cn(), the transmission zeros and the
requested return loss at =±1, the polynomials P(s), F(s),
E(s) can be analitically evaluated
All-pole characteristic functions (P=cost, Cn=F)
Butterworth function:

Cn      n
Main property: maximum flatness for =0, monotonic
for |>0

Chebycheff function:
cos  n  cos 1    
 1
cosh  n  cosh 1    
 1
Cn    
Main property: oscillates between ±1 for |<1 (n peaks),
increases monotonically for |>1
Comparison between Butterworth and Chebicheff
70
n=5
60
50
Chebycheff
40
Butterworth
30
20
10
0
0
0.5
1
1.5
2
2.5
Attenuation of Chebycheff function is 6(n-1) dB larger than
Butterworth function for ||>>1
Evaluation of ’ and n

-
Assigned parameters:
Passband Return Loss in dB (RL)
Filter bandwidth (B) and center frequency (f0)
Minimum attenuation (dB) in stopband (AM)
Frequencies at the beginning of stopband fs1,fs2 with fs1.fs2=(f0)2.
Evaluation of fs1 in the lowpass domain:
Evaluation of   
m
1   m2
AM  RL
n
20 log   s 
AM  RL  6
n
20 log   s   6
s 
f 0  f s1 f 0 



B  f 0 f s1 
with  m  10 RL 20
Butterworth characteristic
Chebycheff characteristic
Synthesis of lowpass prototype

Butterworth characteristic
rn  1,

g q  2 aq  ,
n
  2q  1  
aq  sin 

2
n


Chebycheff characteristic
2
rn  1 (n odd),
rn   1   2    (n even)


4aq 1  aq
  2q  1  
2a1
g1 
, gq 
,
aq  sin 


bq 1  g q 1
2
n


 1  1  2 1  
 q 
  sinh  ln 
  , bq   2  sin 2 

 2n  1   2  1  
n





De-normalized (bandpass) network with only
series or shunt resonators
For microwave frequencies implementation is requested to
have only one type of resonator (series or shunt). This is
obtained with the introduction of impedance (admittance)
inverters
General design equation
Introduction of inverter determines additional degrees of
freedom which allows the arbitrary assignments of some
parameters (inverters or resonators parameter).
The condition to be satisfied are expressed by the
following equations:
Shunt Network
Series Network
K 01 
B 0 Ls ,1
R1
f 0 g1
B
K q ,q 1 
f0
0 Ls ,q  0 Ls ,q 1
g q  g q 1
J 01 
B 0Cs ,1
G1
f 0 g1
B
J q ,q 1 
f0
0Cs ,q  0Cs ,q 1
g q  g q 1
The bandpass network is assumed symmetric respect
the central inverter (n even) or resonator (n odd)
Universal coupling parameters
Coupling coefficient:
kq ,q 1 
K q ,q 1
0 Ls ,q  0 Ls ,q 1
External Q:
QE 

0 Ls ,1
K
2
01
R1
J q ,q 1
0Cs ,q  0Cs ,q 1

B

f0
1
g q  g q 1
0Cs ,1
J
2
01
g1

G1 B f 0
0 Ls ,1  Equivalent reactance of series resonators
0Cs ,1  Equivalent susceptance of shunt resonators

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