EE 41139Microwave Technique1 Lecture 8 Periodic Structures Image Parameter Method Insertion Loss...

EE 41139 Microwave Technique 1

Lecture 8

Periodic Structures

Image Parameter Method

Insertion Loss Method

Filter Transformation


Periodic Structures

periodic structures have passband and stopband characteristics and can be employed as filters


Periodic Structures

consider a microstrip transmission line periodically loaded with a shunt susceptance b normalized to the characteristic impedance Zo:

jb

d In In+1

Vn Vn+1

+

Zo=1

unit cell


Periodic Structures

the ABCD matrix is composed by cascading three matrices, two for the transmission lines of length d/2 each and one for the shunt susceptance,

V

I

A B

C D

V

In

n

n

n

1

1


Periodic Structures

i.e.

A B

C D

j

j jb

j

j

A B

C D

bj

b b

jb b b

cos sin

sin cos

cos sin

sin cos

cos sin (sin cos )

(sin cos ) cos sin

2 2

2 2

1 0

12 2

2 2

2 2 2 2

2 2 2 2


Periodic Structures

= kd, and k is the propagation constant of the unloaded line

AD-BC = 1 for reciprocal networks

assuming the the propagation constant of the loaded line is denoted by , then

V z

I z

V

Ie z( )

( )

( )

( )

0

0


Periodic Structures

therefore,

or

V

I

A B

C D

V

IV e

I e

n

n

n

n

nd

nd

1

1

1

1

A e B

C D e

V

I

d

dn

n

1

10


Periodic Structures

for a nontrivial solution, the determinant of the matrix must vanish leading to

recall that AD-CB = 0 for a reciprocal network, then

Or

AD e A D e CBd d 2 0 ( )

1 02 e A D ed d ( )

cosh cos sin dA D b

2 2


Periodic Structures

Knowing that, the above equation can be written as

since the right-hand side is always real, therefore, either or is zero, but not both

j

cosh cosh cos sinh sin cos sin d d d j d db 2

cosh cosh cos sinh sin cos sin d d d j d db 2


Periodic Structures

if =0, we have a passband, can be obtained from the solution to

if the the magnitude of the rhs is less than 1

cos cos sin db 2


Periodic Structures

if =0, we have a stopband, can be obtained from the solution to

as cosh function is always larger than 1, is positive for forward going wave and is negative for the backward going wave

cosh |cos sin | db 2

1


Periodic Structures

therefore, depending on the frequency, the periodic structure will exhibit either a passband or a stopband


Periodic Structures

the characteristic impedance of the load line is given by

, + for forward wave and - for backward wavehere the unit cell is symmetric so that A = D

ZB is real for the passband and imaginary

for the stopband

Z ZV

I

BZ

AB o

n

n

o

1

1 2 1


Periodic Structures

when the periodic structure is terminated with a load ZL , the reflection coefficient

at the load can be determined easily

Z Lunitcell

unitcell

I N

VN+


Periodic Structures

Which is the usual result

V V V Z I ZV

Z

V

Z

Z

ZV VN N N L N L

N

B

N

B

L

BN N

V

V

Z Z

Z ZN

N

L B

L B


Periodic Structures

it is useful to look at the k- diagram (Brillouin) of the periodic structure

k k=

cutoff

propagation

vp

vgc

c


Periodic Structures

in the region where < k, it is a slow wave structure, the phase velocity is slow down in certain device so that microwave signal can interacts with electron beam more efficiently

when = k, we have a TEM line


Filter Design by the Image Parameter Method

let us first define image impedance by considering the following two-port network

A BC DZi1 Zi2

V11

V2

+ +

I1 I2

Zin1 Zin2



if Port 2 is terminated with Zi2, the input impedance at Port 1 is Zi1

if Port 1 is terminated with Zi1, the input impedance at Port 2 is Zi2

both ports are terminated with matched loads



at Port 1, the port voltage and current are related as

the input impedance at Port 1, with Port 2 terminated in , is

V AV BI I CV DI1 2 2 1 2 2 ,

ZV

I

AV BI

CV DI

AZ B

CZ Dini

i1

1

1

2 2

2 2

2

2



similarly, at Port 2, we have

these are obtained by taking the inverse of the ABCD matrix knowing that AB-CD=1

the input impedance at Port 2, with Port 1 terminated in , is

V DV BI I CV AI2 1 1 2 1 1 ,

ZV

I

DV BI

CV AI

DZ B

CZ Aini

i2

2

2

1 1

1 1

1

1



Given and , we have

, ,

if the network is symmetric, i.e., A = D, then

Z Zin i1 1 Z Zin i2 2

Z CZ D AZ B Z CZ A DZ Bi i i i i1 2 2 2 1 1( ) , ( )

ZAB

CDi1 ZBD

ACi2 ZDZ

Aii

21

Z Zi i1 2



if the two-port network is driven by a voltage source

A BC D

Zi1

Zi2V11

V2

+ +

I1 I2

Zin1 Zin2

2V1



Similarly we have, , A = D for symmetric network

Define ,

I

I

A

DAD BC2

1

e AD BC

e AD BC AD BC AD BC AD BC 1 / ( ) ( ) / ( )

e AD BC AD BC AD BC AD BC 1 / ( ) ( ) / ( )

cosh AD



consider the low-pass filter

L/2 L/2

C

Z1/2 Z1/2

Z2



the series inductors and shunt capacitor will block high-frequency signals

a high-pass filter can be obtained by replacing L/2 by 2C and C by L in T-network



the ABCD matrix is given by

Image impedance

A Z Z B Z Z Z C Z D Z Z 1 2 4 1 1 21 2 1 12

2 2 1 2/ , / , / , /

A Z Z B Z Z Z C Z D Z Z 1 2 4 1 1 21 2 1 12

2 2 1 2/ , / , / , /

Z AB CD Z Z Z ZiT / /1 2 1 21 4



Propagation constant

For the above T-network,

e Z Z Z Z Z Z 1 2 41 2 1 2 12

22/ ( / ) ( / )

Z j L Z j C1 2 1 , /

ZL

C

LCiT 1

4

2



Define a cutoff frequency as,

a nominal characteristic impedance Ro

, k is a constant

c LC 2

RL

Cko



the image impedance is then written as

the propagation factor is given as

Z RiT oc

12

2

ec c c

1

2 21

2

2

2

2



For , is real and which imply a passband

For , is imaginary and we have a stopband

c Z iT | |e 1

c Z iT



this is a constant-k low pass filter, there are two parameters to choose (L and C) which are determined by c and Ro

when , the attenuation is slow, furthermore, the image impedance is not a constant when frequency changes



the m-derived filter section is designed to alleviate these difficulties

let us replace the impedances Z1 with Z mZ1 1

'



we choose Z2 so that ZiT remains the same

therefore, Z2 is given by

Z Z ZZ

Z ZZ

mZ Zm Z

iT 1 212

1 212

1 2

212

4 4 4' '

''

ZZ Z m Z

mZ

Z

m

m

mZ2

1 22

12

1

22

11 4 1

4' ( ) /



recall that Z1 = jL and Z2 = 1/jC, the

m-derived components are Z j Lm Z

j Cm

m

mj L1 2

21 1

4' ',

Z

Z

j Lm

j Cm m j L m

m LC

m LC

1

22

2 2

2 21 1 4 1 1 4

'

' / ( ) / ( ) /

Z

Z

m

mLCc

cc

1

2

2

2 22

1 12

'

'( / )

( )( / ), /



the propagation factor for the m-derived section is

14

4 4 1 2

4 4 1

1

1 1

1

2

2 2 2

2 2

2

2 2

Z

Z

m m

m m

c c

c

c

c

'

'( )( / ) ( / )

( )( / )

( / )

( )( / )

eZ

Z

Z

Z

Z

Z

12

14

1

2

1

2

1

2

'

'

'

'

'

'( )

14

4 4 1 2

4 4 1

1

1 1

1

2

2 2 2

2 2

2

2 2

Z

Z

m m

m m

c c

c

c

c

'

'( )( / ) ( / )

( )( / )

( / )

( )( / )



if we restrict 0 < m < 1, is real and

>1 , for > the stopband begins at =as for the constant-k sectionWhen = , where

e becomes infinity and the filter has an infinite attenuation

e

| |e c c

c m/ 1 2



when > , the attenuation will be reduced; in order to have an infinite attenuation when , we can cascade a the m-derived section with a constant-k section to give the following response



the image impedance method cannot incorporate arbitrary frequency response; filter design by the insertion loss method allows a high degree of control over the passband and stopband amplitude and phase characteristics


Filter Design by the Insertion Loss Method

if a minimum insertion loss is most important, a binomial response can be usedif a sharp cutoff is needed, a Chebyshev response is betterin the insertion loss method a filter response is defined by its insertion loss or power loss ratio



, IL = 10 log

, , M and N are real polynomials

PP

PLRinc

Load

1

1 2| ( | PLR

| ( )|( )

( ) ( )

2

2

2 2

M

M N

PM

NLR 1

2

2( )

( )



for a filter to be physically realizable, its power loss ratio must be of the form shown above

maximally flat (binomial or Butterworth response) provides the flattest possible passband response for a given filter order N

P kLRc

N 1 2 2( )



The passband goes from to , beyond , the

attenuation increases with frequency

the first (2N-1) derivatives are zero for

and for , the insertion loss increases at a rate of 20N dB/decade

0

c c

c

0



equal ripple can be achieved by using a Chebyshev polynomial to specify the insertion loss of an N-order low-pass filter as

P k TLR Nc

1 2 2



a sharper cutoff will result; (x) oscillates between -1 and 1 for |x| < 1, the passband response will have a ripple of 1+ in the amplitude

For large x, and therefore for

TN

k 2

T x xNN( ) ( ) / 2 2

Pk

LR cN

22

42( / )

c



therefore, the insertion loss of the Chebyshev case is times of the binomial response for

linear phase response is sometime necessary to avoid signal distortion, there is usually a tradeoff between the sharp-cutoff response and linear phase response

( ) /2 42N

c



a linear phase characteristic can be achieved with the phase response

( )

A pc

N

12



a group delay is given by

this is also a maximally flat function, therefore, signal distortion is reduced in the passband

d

c

Nd

dA p N

1 2 12

( )



it is convenient to design the filter prototypes which are normalized in terms of impedance and frequency the designed prototypes will be scaled in frequency and impedancelumped-elements will be replaced by distributive elements for microwave frequency operations



consider the low-pass filter prototype, N=2

RC

L1

Z in



assume a source impedance of 1 and a cutoff frequency

the input impedance is given by

c 1

PLR 1 4

Z j LR

j RCj L

R j RC

R Cin

1

1

1 2 2 2( )



the reflection coefficient at the source impedance is given by

the power loss ratio is given by

Z

Zin

in

1

1

PR

R R C L LCR L C RLR

1

11

1

41 2

22 2 2 2 2 2 2 2 2 4

| |( ) ( )

PR

R R C L LCR L C RLR

1

11

1

41 2

22 2 2 2 2 2 2 2 2 4

| |( ) ( )



compare this equation with the maximally flat equation, we have R=1, which implies C = L as R = 1 which implies C = L =

( )RC L 2 0

( ) /LC 2 4 1 2



for equal-ripple prototype, we have the power loss ratio

Since

Compare this with

P k TLR 1 222 ( )

T x x P kLR22 2 4 22 1 1 4 4 1( ) , ( )

PR

R R C L LCR L C RLR 11

41 22 2 2 2 2 2 2 2 2 4( ) ( )

PR

R R C L LCR L C RLR 11

41 22 2 2 2 2 2 2 2 2 4( ) ( )



we have or

note that R is not unity, a mismatch will result if the load is R=1; a quarter-wave transformer can be used to match the load

k R R2 21 4 ( ) / ( )

R k k k 1 2 2 12 2

4 2 42 2 2 2 2k R C L LCR R( ) /

4 42 2 2 2k L C R R /



if N is odd, R = 1 as there is a unity power loss ratio at = 0 of N being oddTable 9.4 can be used for equal-ripple low-pass filter prototypesTable 9.5 can be used for maximally flat time delay low-pass filter prototypesafter the filter prototypes have been designed, we need to perform impedance and frequency scaling


Filter Transformations

impedance and frequency scaling

the source impedance is , the impedance scaled quantities are:

L' R L C C R R R R R Ro o s o L o L , ' / , ,' '


Filter Transformations

both impedance and frequency scaling

low-pass to high-pass transformation

,

L R L C C Rk o k c k k o c' '/ , / ( )

cC L L R Ck c k k o c k

' '/ , / 1


Filter Transformations Bandpass transmission

As a series indicator , is transformed into a series LC with element values

A shunt capacitor, , is transformed into a shunt LC with element values

1 2 1

1 2

o

o

oo, ,

Lk

L L C Lk k o k k o' '/ , / ( )

Ck

L C C Ck k o k k o' '/ , / ( )


Filter Transformationsbandstop transformation

A series indicator, , is transformed into a parallel LC with element values

A shunt capacitor, , is transformed into a series LC with element values

o

o

oo

12 1

1 2, ,

Lk

L L C Lk k o k k o' '/ , / ( ) 1

Ck

L C C Ck k o k k o' '/ , / 1


Filter Implementation

we need to replace lumped-elements by distributive elements:

LjXLSC

at c

Zo = L

CjBcOC

at c

Zo = 1/C



there are four Kuroda identities to perform any of the following operations: physically separate transmission line stubs transform series stubs into shunt stubs, or

vice versa change impractical characteristic

impedances into more realizable ones



let us concentrate on the first two

a shunt capacitor can be converted to a series inductor

YoY1 Yo+Y1

Yo(1+Yo/Y1)

Zo(1+Zo/Z1)

Zo

Z1

Zo+Z1

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EE 41139Microwave Technique1 Lecture 8 Periodic Structures Image Parameter Method Insertion Loss...

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