all pole filters synthesis and realization techniques

ALL POLE FILTERS

SYNTHESIS AND REALIZATION TECHNIQUES

ENEE 481 K.A.Zaki 2

TWO PORT PARAMETERS

Rg I2 I1

V1 V2 RLEg 2-PORT

NETWORK

+

-

+

-

12

)(1

1ratio losspower

; P4

2

2

22

max2

2

22inc

2

max

g

Lg

LR

Lg

g

R

R

V

E

P

PPH

R

VP

R

EP

ENEE 481 K.A.Zaki 3

)(

)(1

.in polynomial real are and

)()(

)()(

:as expressed becan it , offunction even an is )(

2

2

2

22

22

2

N

MP

NM

NM

M

LR

22

22

1 function, sticcharacteri theis )(

1log10log10log10

Hs

HPIL LR

ENEE 481 K.A.Zaki 4

Maximally flat

level. ripple passband thedetermines k

.1xfor 1between oscillates )(

k1 magnitude of ripples band pass The

polynomial Chebyshev theis T , 1

:ripple Equal

.dB/decade) 20N is slope ( ,For

dB. 3-at is edge band the, 1k If

.1 edge band At the frequency. offcut theis

filter, theoforder theis N , 1

2

2

N22

2

2c

2c

2

2

xT

TkP

kP

kP

kP

N

cNLR

N

cLR

LR

N

cLR

Chebyshev

ENEE 481 K.A.Zaki 5

LOW PASS PROTOTYPE FILTERSAND THEIR RESPONSES

ENEE 481 K.A.Zaki 6

Maximally Flat Low Pass Prototype1 L

C R

Low-pass filter prototype, N=2

1

1 ,

1

)1(

1

1Let

222

4

c

in

inin

LR

Z

Z

CR

RCjRLjZ

P

ENEE 481 K.A.Zaki 7

2CL

hmust vanis oft coefficien theand 0at 1

)2()1(4

11

1

CR1

1

R

4

1

1

CR1

1

R1

1

2R)( ,

)(2

1

1

1

2

4222222222

2

222

22

222

222

2

222

22

222

2

222*

*

2

2

LR

LR

in

inininin

inLR

P

RCLLCRLCRRR

CRL

CRR

CRP

CRL

CRZ

CRZZ

ZZ

ZP

ENEE 481 K.A.Zaki 8

PROTOTYPE LOW PASS FILTERELEMENT VALUES: BUTTERWORTH:

NkN

kgg n ,...2,1;

2

)12(sin2g ; 1 k10

ORDER n :

)log(2

)110log( 10/

c

T

LT

N

INSERTION LOSS:

))(1log(10

1. typicallyis .1 is passbans in the of valuemaximum The

to0 fromregion theis passband the,)(1

22

2

c22

N

cT

LR

N

cLR

kL

kkP

kP

ENEE 481 K.A.Zaki 9

CHEBYSHEV:

band pass in the dBin ripple maximum the:

filter theoforder the:

,...,2,1 ; sin

,....2,1 ; 2

)12(sin

)2

sinh( ; )34.17

ln(coth

; even Nfor )4

(coth

odd Nfor 1

,...,3,2 ; 4a

; 2

; 1

22

21

11

1-k110

ar

k

k

ar

n

kk

kk

L

N

NkN

kb

NkN

ka

n

L

g

nkgb

ag

agg

ENEE 481 K.A.Zaki 10

INSERTION LOSS:

2

11-

22

1 and 1between oscillate ratio losspower The

cosNcos

N degree of polynomial Chebyshev theis

1

k

NT

T

TkP

ncc

N

cN

cNLR

10/10

frequency offcut

therespect towith frequency angular normalized the:

arL

n

k

)(cosh

)1/()110(cosh

1

10/1

c

T

L kN

T

Order n:


Impedance and Frequency Scaling

kckck

c

kk

c

kLRlR

LLS

C

R

LRC

R

CC

LRPP

RRRRRR

CCL

0k

0

c

0

0k

c

c

000

0

L ,1

,-

:ation transformpasshigh topass Low

,L ,

by replace

:ScalingFrequency

,, ,RL

:scaling Impedance


MICROWAVE BANDPASS FILTER

FREQUENCY MAPPING:

:

: ,

:

:

; ; 1

2 1

0

2100

120

0

NORMALIZED FREQUENCY OF THE PROTOTYPE

CENTER FREQUECY OF THE BPF

BAND EDGE FREQUENCIES OF THE BPF

3 dB FOR BUTTERWORTH AND EQUAL RIPLE FORTCHEBYSCHEFF FILTER

RELATIVE BANDWIDTH OF THE BANDPASS FILTER


ELEMENT VALUES:

FOR SERIES ELEMENT:

FOR SHUNT ELEMENTS:

)( ; )1

( 0000

Zg

LZg

Cr

kk

k

rk

)( ; )1

( 0000

Zg

LZ

gC

k

rk

k

kk

LP BP

kg

kg

kL

kLkC

kC


Low Pass Prototype Response

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

Normalized Frequency

Pass

Ban

d In

s. L

oss

[dB]

0

5

10

15

20

25

Stop

Ban

d In

serti

on

Loss

[dB]

Band Pass Filter Response

0

0.2

0.4

0.6

0.8

1

1.7 1.8 1.9 2 2.1 2.2 2.3

Normalized Frequency

Pass

Band

Ins

ertio

n Los

s [dB

]

0

20

40

60

80

100

Stop

Ban

d Ins

ertio

n Los

s [dB

]

LOW PASS FILTER RESPONSE AND CORRESPONDING BANDPAS FILTER RESPONSE


MAPPING OF LPF TO BPF

BPF WITH ONE TYPE OF RESONATORS


DEFINITIONS OF IMPEDANCE & ADMITTANCE INVERTERS

ZbKk,k+1Zin= K2k,k+1/Zb

IMPEDANCE INVERTER

YbJk,k+1Yin= J2k,k+1/Yb

ADMITTANCE INVERTER

Y’bL’

C’

ZbKZin

'

''

21

1

b

b

in YLj

CjK

CjLjZY


K=1K=1

Yp()Zs()

Impedance inverter used to convert a parallel admittance intoAn equivalent series impedance.

J=1J=1

Zs()

Yp()

Admittance inverter used to convert a series impedance into An equivalent parallel admittance


K1

Yp()

K2pin YKZ 2

1 pin YKZ 22

J2J1

Zs()

sin ZJY 21

sin ZJY 22


REALIZATION OF IMPEDANCE INVERTERS

-L -L -C -C

C

K=L K=1/C

Z0

XZ0

2tan0

ZK

0

1 2tan ZX

2

0

0

01

ZK

ZK

Z

X;

L

X


GENERAL EQUIVALENT CIRCUIT OF AN IMPEDANCE INVERTER

jXA jXA

jXB

Z0Z0

0

10

0

1

00

1

tan2

tan

tan2

tan

Z

XZK

Z

X

Z

X

Z

X

A

AAB


Filter ImplementationRichard’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.

assumed. is impedanceUnit

1/C. impedance sticcharacteri and

lenth of stub circuitedopen an by replaced becan capacitor The

L impedance sticcharacteri and

lenth of stub circuitedshort aby replaced becan inductor The

tanjBcapacitor afor and

tanjXinductor For

plane. the toplane themaps v

tantan

:tionTransforma sRichard'

C

L

p

jCCj

jLLj


iXLS.C.

cat 8/

iBC

L

C O.C.

cat 8/

Z0=L

Z0=1/C

iXL

iBC

The inductors and capacitors of a lumped-element filterDesign can be replaced with a short-circuited and open-circuited stubs. All the length of the stubs are the same( ) These lines are called commensurate linescat 8/


Kuroda’s Identities

Z1

1/n2Z2

Z2/n2

Z1

Z2

Z2

Z1

1

Z1

Z2/n2

n2

n2

Z1/n2

Z1/n2

Z1

Z1

1/Z2

1/Z2

1/n2Z2

122 /1 ZZn


Low Pass Filter Using Stubs

22

nZ

21

nZ

1Z

2Z

1

1

L1 L3

C21

110 LZ

30 LZ

20

1

CZ


Stepped- Impedance Low Pass FiltersApproximate Equivalent Circuits for Short Transmission Line

0

0

00

001211

02112

02211

YB0,X , impedance sticcharacteri small aFor

0B ,

impedance sticcharacteri large a ,4/short is line theIf

sinZ

1B ,

2tan

2

2tan

sin

1cos

csc1

cotZ

:is length of T.L a ofparameter ZThe

ZX

ZX

jZjZZZ

jZC

ZZ

jZC

AZ


SMALL SECTION OF TRANSMISSION LINE AND ITS EQUIVALENT CIRCUIT

MICROWAVE LPF & ITS EQUIVALENT CIRCUIT


jX/2 jX/2

jB

,0

0

R

CZ

Z

LR

h


CONFIGURATION OF WAVEGUIDE FILTERS

• • •

• • •

COUPLING USING RECTANGULAR SLOTS

COUPLING USING INDUCTIVE WINDOWS


INPUT AND OUTPUT CONFIGURATION

INPUT/OUTPUT USING PROBES

FIRST/LAST RESONATOR

OUTSIDE WAVEGUIDEFIRST/LAST RESONATOR

INPUT/OUTPUT USING SLOTS AND ADAPTER


jXA2

jXB2jXB1

jXA1jXA2jXA1

Z0

CONFIGURATION COMBINING EQUIVALENTCIRCUITS

COMBINATION OF A CAVITY AND TWO SLOTS

Z0

jXA2

Z0

Z0

Z0jXB2 jXB1

jXA2 jXA1jXA1

2/)( 210


SCATTERING MATRIX

CAVITY

Z0

0

0

j

j

e

eS

SLOT

Z0

jXA

jXB

jXA

Z0

CAVITY

)(1

1

)(1

2/)()(1

2/)(

)(1

1

; 2

00

BAjAB

AB

BAjAB

BAjBAjAB

BAj

BAjAB

AB

S

Z

XB

Z

XXA ABA

SLOT


D

Side 1 Side 1 Side 1Side 2 Side 2

dI

cI

SI

cII

dII SII

CASCADING MULTIPOR BLOCKS

a1

b1

a2

b2

ACCURATE FILTER RESPONSE IS COMPUTED BY CASCADINGTHE GENERALIZED SCATTERING MATRICES OF SECTIONS OF WAVEGUIDES, DISCONTINUTIES AND COUPLING SECTIONS


MILLIMETER WAVE SEVEN POLE FILTER EXAMPLE


OPTIMIZED RESPONSE OF 7-POLE FILTER


SENSITIVITY ANALYSIS OF 7-POLE FILTERTO RANDOM MANUFACTURING TOLERANCES


MEASURED PERFORMANCE OF A MILLIMETER WAVEDIPLEXER DESIGNED BY MODE MATCHING WITH NO TUNING


THE ABCD MATRIX FOR A LENGTH OF TRANSMISSION LINE IS :

A B cos jZ() sin () = C D jY() sin () cos ()

FOR A COAXIAL LINE OPERATING IN THE TEM MODE , () = /(2 0 ) , Z IS CONSTANT,ll / v , 0 IS THE FREQUENCY FOR WHICH THE LINE LENGTH IS QUARTER WAVELENGTH

REALIZATION OF PRACTICAL FILTERS


LENGTH OF LINE:

a b

l

Yinoc = 1

Z 11A

=jY0 sin

cos = jY0 tan

FOR A SHORT CIRCUITED LINE:

Zinsc = 1

Y 11

= =jZ0 sin

cos = jZ0 tan

B

D

FOR AN OPEN CIRCUITED LINE:

C=

=

=

=


FOR A SMALL LENGTH OF TRANSMISSION LINE

TAN ~

Y inoc j Y0 j Y0 j C ‘

Z inscj Z 0 j Z0 j L’

FOR A SHUNT CAPACITOR:

A B 1 0 =C D j C’ 1

FOR A SERIES INDUCTORS:

A B 1 j L’ =C D 0 1

;

=


MICROWAVE LOW PASS FILTER

ELEMENT VALUES

c

kk

c

kk

ZgL

Z

gC

0

0

;

TRANSMISSION LINE RELATION

HIGH IMPEDANCE LINE: SERIES INDUCTOR

LOW IMPEDANCE LINE: SHUNT CAPACITOR

)(sin0

1

LL Z

Ll

)(sin 01

CC CZl

: ,

: Z, Z

: ,

0L0C

CL

ll CL ELECTRICAL LENGTHS OF T.L. IN DEGREE

CHARACTERISTIC IMPEDANCES

SERIES INDUCTOR, SHUNT CAPACITOR

all pole filters synthesis and realization techniques

Documents

responses slide

x slide

zaki7 slide

zaki3 slide

zaki25 slide

bandpass filter slide

equivalent circuit slide

frequency scaling slide