ALL POLE FILTERS
SYNTHESIS AND REALIZATION
TECHNIQUES
TWO PORT PARAMETERS
Rg
Eg
Pmax 
H

2
E g2
4Rg
I1
+
V1
-
 Pinc
I2
2-PORT
NETWORK
; P2 
 power loss ratio  PLR
Eg
2V2
+
V2 RL
-
V2
2
RL
Pmax
1


2
P2
1  ( )
2
RL
Rg
1
ENEE 481 K.A.Zaki
2
IL  10 log PLR  10 log H
2


 10 log   1
( s) is the characteri stic function,
H
2
2
 1 
2
( ) is an even function of  , it can be expressed as :
2
M ( 2 )
( ) 
M ( 2 )  N ( 2 )
2
M and N are real polynomial in  2 .
M ( 2 )
PLR  1 
N ( 2 )
ENEE 481 K.A.Zaki
3
Maximally flat
2N
 
PLR  1  k   , N is the order of the filter,
 c 
 c is the cut off frequency. At the band edge PLR  1  k 2 .
2
If k  1 , the band edge is at - 3 dB.
For    c , PLR

 k 
 c
2



2N
( slope is 20N dB/decade) .
Equal ripple : Chebyshev
2 2  
PLR  1  k TN   , TN is the Chebyshev polynomial
 c 
The pass band ripples of magnitude 1  k 2
TN ( x) oscillates between  1 for x  1.
ENEE 481
K.A.Zaki
k 2 determines the passband
ripple
level.
4
LOW PASS PROTOTYPE FILTERS
AND THEIR RESPONSES
ENEE 481 K.A.Zaki
5
Maximally Flat Low Pass Prototype
1
L
C
R
Low-pass filter prototype, N=2
Let  c  1
PLR  1   4
Z in  1
R(1  jRC )
Z in  jL 
, 
2 2 2
Z in  1
1
ENEER481C
K.A.Zaki
6
Z in  1
1
2
2R
PLR 

, ( Z in  Z ) 
2
*
2 2 2
2
(
Z

Z
)
1


R C
1 
in
in
*
in

R
CR

 

Z in  1  
 1   L 
2 2 2
2 2 2 
1 R C 
1  R C
 
2
2
2
2
2
2
2

 
1 R C 
R
CR
 
 

PLR 
 1   L 
2 2 2
2 2 2 
4R
1 R C  
 
 1   R C

1
 1
(1  R) 2  ( R 2 C 2  L2  2 LCR 2 ) 2  L2 C 2 R 2 4
4R
PLR  1 at   0 and the coefficien t of  2 must vanis h 
2
2
2


LC 2
ENEE 481 K.A.Zaki
7
PROTOTYPE LOW PASS FILTER
ELEMENT VALUES:
BUTTERWORTH:
g 0  g n 1  1
 (2k  1)
; g k  2 sin 
2N

ORDER n :
INSERTION LOSS:

; k  1,2,...N

log( 10 LT  / 10  1)
N
T
2 log(
)
c
 2N
PLR  1  k ( ) , the passband is the region from   0 to  c
c
2
The maximum value of PLR in the passbans is 1  k 2 . k is typically 1.
 2N
LT  10 log( 1  k ( ) )
c
2
ENEE 481 K.A.Zaki
8
CHEBYSHEV:
4a k -1 a k
g 0  1 ; g1 
; gk 
; k  2,3,..., n

bk 1 g k 1
2a1
1
for N odd 

g n 1  
2 
 ;
coth
(
)
for
N
even


4
Lar

  ln(coth
) ;   sinh( )
17.34
2n
 (2k  1) 
a k  sin 
; k  1,2,....N

 2N 
 k 
bk   2  sin 2   ; k  1,2,..., N
N
N : the order of the filter
Lar : the maximum ripple in dB in the pass band
ENEE 481 K.A.Zaki
9
INSERTION LOSS:
PLR

 1  k T 
 c
2
2
N




TN 
 c

 is the Chebyshev polynomial of degree N


  
-1   
TN     Ncos     N cos 1  n
 c  
 c 
The power loss ratio oscillate between 1 and 1  k 2
 n : the normalized angular frequency with respect to the
cut off frequency
k  10 Lar / 10
Order n:
N
cosh 1 (10 LT  / 10  1) /( k  1)
T
ENEE 481 K.A.Zakicosh (
)
c
1
10
Impedance and Frequency Scaling
Impedance scaling :
C
L   R 0 L, C  
, RS  R0 , RL  R0 RL
R0
Frequency Scaling :
replace  by


c
 
R0 L k
Ck
PlR  PLR  , Lk 
, C k 
c
R 0 c
 c 
Low pass to high pass transform ation :
c
R0
1


  - , Ck 
, Lk 

R0 c L k
 cCk
ENEE 481 K.A.Zaki
11
MICROWAVE BANDPASS FILTER
FREQUENCY MAPPING:
 2  1
1   0 


 

; 
;  0  1 2

  0  
0
 : NORMALIZED FREQUENCY OF THE PROTOTYPE
 0 : CENTER FREQUECY OF THE BPF
1 ,  2 : BAND EDGE FREQUENCIES OF THE BPF
3 dB FOR BUTTERWORTH AND EQUAL RIPLE FOR
TCHEBYSCHEFF FILTER
:
RELATIVE BANDWIDTH OF THE BANDPASS FILTER
ENEE 481 K.A.Zaki
12
ELEMENT VALUES:
FOR SERIES ELEMENT:

1
Ck  r ( )
0 gk Z0
Lk 
;
BP
LP
gk
 0 r
gk
Lk C k
(Z 0 )
FOR SHUNT ELEMENTS:
gk
1
Ck 
( )
 0 k Z 0
;
r
Lk 
(Z 0 )
0 gk
ENEE 481 K.A.Zaki
gk
Ck
Lk
13
LOW PASS FILTER RESPONSE AND
CORRESPONDING BANDPAS FILTER RESPONSE
25
0.8
20
0.6
15
0.4
10
0.2
5
0
0
1.5
0
0.5
1
1
100
0.8
80
0.6
60
0.4
40
0.2
20
0
0
1.7
Normalized Frequency
1.8
1.9
2
2.1
2.2
Stop Band
Insertion Loss [dB]
1
Pass Band
Insertion Loss [dB]
Band Pass Filter Response
Stop Band Ins e rtion
Los s [dB]
Pas s Band Ins . Los s
[dB]
Low Pass Prototype Response
2.3
Normalized Frequency
ENEE 481 K.A.Zaki
14
MAPPING OF LPF TO BPF
BPF WITH ONE TYPE OF RESONATORS
ENEE 481 K.A.Zaki
15
DEFINITIONS OF IMPEDANCE & ADMITTANCE INVERTERS
Zb
Kk,k+1
Zin= K2k,k+1/Zb
IMPEDANCE INVERTER
Yb
Jk,k+1
Yin= J2k,k+1/Yb
ADMITTANCE INVERTER
Zb
K
Zin
Yin 
Z b  jL  1
K2
C’
jC
 jC '  1
ENEE 481 K.A.Zaki
L’
 jL 
'
Y’b
 Yb'
16
Zs()
Yp()
K=1
K=1
Impedance inverter used to convert a parallel admittance into
An equivalent series impedance.
Zs()
J=1
J=1
Yp()
Admittance inverter used to convert a series impedance into
An equivalent parallel admittance
ENEE 481 K.A.Zaki
17
Yp()
Z in  K Y p
2
1
K1
K2
Z in  K 22Y p
Zs()
Yin  J 12 Z s
J1
J2
ENEE 481 K.A.Zaki
Yin  J 22 Z s
18
REALIZATION OF IMPEDANCE INVERTERS
-C
-C
-L
-L
C
L
K=1/ C
K= L
f
Z0
K  Z 0 tan
f
Z0
X

2
K 
 Z 
X
0


;
2
Z0
K
1   
 Z0 
X
   tan 1  2 X Z 
ENEE 481 K.A.Zaki

0

19
GENERAL EQUIVALENT CIRCUIT OF AN
IMPEDANCE INVERTER
f/2
f/2
jXA
Z0
jXA
jXB
Z0
 2X B X A 
1  X A 



   tan 

 tan 

Z0 
 Z0
 Z0 
1

1 X A 


K  Z 0 tan   tan
Z0 
2
ENEE 481 K.A.Zaki
20
Filter Implementation
Richard’s transformation is used to convert lumped
elements to transmission line sections.
Kuroda’s identities can be used to separate filter elements
by using transmission line sections.
Richard' s Transforma tion :
  
  tan   tan   maps the  plane to the plane.
v 
 p
For inductor jX L  jL  jL tan 
and for a capacitor jB C  jC  jC tan 
The inductor can be replaced by a short circuited stub of lenth 
and characteri stic impedance L
The capacitor can be replaced by an open circuited stub of lenth 
and characteri stic impedance 1/C.
ENEE 481 K.A.Zaki
Unit impedance is assumed.
21
 / 8 at c
iX
L
S.C.
iX
L
L
Z0=L
 / 8 at c
iBC
C
O.C.
iBC
Z0=1/C
The inductors and capacitors of a lumped-element filter
Design can be replaced with a short-circuited and opencircuited stubs. All the length of the stubs are the same
(  / 8 at  c ) These lines are called commensurate lines
ENEE 481 K.A.Zaki
22
Kuroda’s Identities
1/Z2
/n2
Z1
Z2
Z2
n2 Z1
Z1/n2
Z1
1/n2Z2
Z1/n2
Z1
Z2
Z2/n2
1/Z2
1/n2Z2
Z1
n2 Z1
n 2 ENEE
1 481
Z 2K.A.Zaki
/ Z1
1
23
Low Pass Filter Using Stubs




Z1
n2
Z1
Z2
1
Z2
L1
n
L3
2
Z 0  L1
Z 0  L3
1
C2
1
1
1
Z0 
C2
ENEE 481 K.A.Zaki
24
ENEE 481 K.A.Zaki
25
Stepped- Impedance Low Pass Filters
Approximate Equivalent Circuits for Short Transmission Line
The Z parameter of a T.L of length  is :
A
Z11  Z 22    jZ0 cot 
C
1
Z12  Z 21    jZ0 csc 
C
 cos   1
  
Z11  Z12   jZ0 

jZ
tan
 
0

 2 
 sin  
X
  
 Z 0 tan  ,
2
 2 
1
B
sin 
Z0
If the line is short    / 4, a large characteri stic impedance 
X  Z 0 , B  0
For a small characteri stic impedance
, X  0, B  Y0 
ENEE 481 K.A.Zaki
26
SMALL SECTION OF TRANSMISSION LINE
AND ITS EQUIVALENT CIRCUIT
MICROWAVE LPF & ITS EQUIVALENT CIRCUIT
ENEE 481 K.A.Zaki
27
jX/2
jX/2
jB
LR0
 
,
Zh
CZ 
 
R0
ENEE 481 K.A.Zaki
28
CONFIGURATION OF WAVEGUIDE FILTERS
• ••
COUPLING USING RECTANGULAR SLOTS
• • •
COUPLING USING INDUCTIVE WINDOWS
ENEE 481 K.A.Zaki
29
INPUT AND OUTPUT CONFIGURATION
FIRST/LAST
RESONATOR
INPUT/OUTPUT USING PROBES
FIRST/LAST
RESONATOR
OUTSIDE WAVEGUIDE
INPUT/OUTPUT USING SLOTS AND ADAPTER
ENEE 481 K.A.Zaki
30
q0
jXA2
jXA2
jXB2
CONFIGURATION
jXA1
jXA1
jXB1
Z0
COMBINING EQUIVALENT
CIRCUITS
q 0  (f1  f2 ) / 2
f2/2
f2/2
jXA2
Z0
f1/2
jXA1
jXA2
jXB2
f1/2
Z0
Z0
jXA1
jXB1
Z0
COMBINATION OF A CAVITY AND TWO SLOTS
ENEE 481 K.A.Zaki
31
SCATTERING MATRIX
q
Z0
SLOT
CAVITY
CAVITY
jXA
Z0
SLOT
X A  2X B
A
Z0
jXA
jXB
Z0
e  jq 

0 
XA
B
Z0
 0
S    jq
e
;
 AB  1
 AB  1  j ( A  B)
S
j ( A  B) / 2
 AB  1  j ( A  B)
ENEE 481 K.A.Zaki
j ( A  B) / 2
 AB  1  j ( A  B)
 AB  1
 AB  1  j ( A  B)
32
CASCADING MULTIPOR BLOCKS
cI
a1
b1
Side 1
SI
cII
dI
dII
D
Side 1
Side 2 Side 2
a2
SII
b2
Side 1
ACCURATE FILTER RESPONSE IS COMPUTED BY CASCADING
THE GENERALIZED SCATTERING MATRICES OF SECTIONS OF
WAVEGUIDES, DISCONTINUTIES AND COUPLING SECTIONS
ENEE 481 K.A.Zaki
33
MILLIMETER WAVE SEVEN POLE FILTER EXAMPLE
ENEE 481 K.A.Zaki
34
OPTIMIZED RESPONSE OF 7-POLE FILTER
ENEE 481 K.A.Zaki
35
SENSITIVITY ANALYSIS OF 7-POLE FILTER
TO RANDOM MANUFACTURING TOLERANCES
ENEE 481 K.A.Zaki
36
MEASURED PERFORMANCE OF A MILLIMETER WAVE
DIPLEXER DESIGNED BY MODE MATCHING WITH NO TUNING
ENEE 481 K.A.Zaki
37
ENEE 481 K.A.Zaki
38
REALIZATION OF PRACTICAL FILTERS
THE ABCD MATRIX FOR A LENGTH OF
TRANSMISSION LINE IS :
A
cos q
B
=
C
D
jZ() sin q()
jY() sin q()
cos q()
FOR A COAXIAL LINE OPERATING IN THE TEM MODE ,
q () =   /(2 0 ) , Z IS CONSTANT,
q   l   l / v , 0 IS THE FREQUENCY FOR WHICH
THE LINE LENGTH IS QUARTER WAVELENGTH
ENEE 481 K.A.Zaki
39
LENGTH OF LINE:
a

b
l
FOR AN OPEN CIRCUITED LINE:
1
Yinoc =
=
C
Z 11
A
jY0 sin q
==
cos q
= jY0 tan q
FOR A SHORT CIRCUITED LINE:
1
Zinsc =
Y 11
==
B
D
==
jZ0 sin q
cos q
ENEE 481 K.A.Zaki
= jZ0 tan q
40
FOR A SMALL LENGTH OF
TRANSMISSION LINE
TAN q ~
= q
Y inoc

j Y0 q
 j Y0   /2  0
j C ‘
Z insc

jZ0q
 j Z0   /2 0
 j  L’
FOR A SERIES INDUCTORS:
A
B
1
j  L’
;
=
C
D
FOR A SHUNT CAPACITOR:
0
1
A
B
=
C
ENEE 481 K.A.Zaki
D
1
0
j  C’
1
41
MICROWAVE LOW PASS FILTER
ELEMENT VALUES
gk
gk Z0
Ck 
; Lk 
c Z0
c
TRANSMISSION LINE RELATION
HIGH IMPEDANCE LINE: SERIES INDUCTOR
1 L
l L  sin (
)
Z0L
LOW IMPEDANCE LINE: SHUNT CAPACITOR
lC  sin 1 (CZ 0C )
lL , lC : ELECTRICAL LENGTHS OF T.L. IN DEGREE
Z0C , Z0L : CHARACTERISTIC IMPEDANCES
L, C :
SERIES INDUCTOR,
SHUNT CAPACITOR 42
ENEE 481 K.A.Zaki