Periodic Structuresand

Filter Design by the Image

Parameter Method

ECE531: Microwave Circuit Design I

Pozar Chapter 8, Sections 8.1 & 8.2

Josh Ottosen

2/24/2011

Microwave Filters(Chapter Eight)

• “A microwave filter is a two-port network used to

control the frequency response at a certain point

in a microwave system by providing

transmission at frequencies within the passband

of the filter and attenuation in the stopband of

the filter. Typical frequency responses include

low-pass, high-pass, bandpass, and band-reject

characteristics. Applications can be found in

virtually any type of microwave communication,

radar, or test and measurement system.”

Periodic Structures(Section 8.1)

“An infinite transmission line or waveguide

periodically loaded with reactive elements

is referred to as a periodic structure...

Periodic structures can take various forms,

depending on the transmission line media

being used. Often the loading elements are

formed as discontinuities in the line, but in

any case they can be modeled as lumped

reactances across a transmission line...

Periodic structures… have passband and

stopband characteristics similar to those of

filters; they find applications in traveling-

wave tubes, masers, phase shifters, and

antennas.”

3

Unit cell“Each unit cell of this line consists of a

length d of transmission line with a

shunt susceptance across the

midpoint of the line; the susceptance b

is normalized to the characteristic

impedance, Zo. If we consider the

infinite line as being composed of a

cascade of identical two-port networks,

we can relate the voltages and

currents on either side of the nth unit

cell using the ABCD matrix:”

=

+

+

1n

1n

n

n

I

V

I

V

DC

BA

4

Table 4.1 (or inside

cover of Pozar)

If a network is reciprocal,

AD – BC = 1

Reciprocal Networks: “A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port

1 is the same as the voltage appearing at port 1 when the same current is applied to port 2. Exchanging voltage and

current results in an equivalent definition of reciprocity. In general, a network will be reciprocal if it consists entirely of

linear passive components (that is, resistors, capacitors and inductors). In general, it will not be reciprocal if it contains

active components such as generators.”

p.311 , Mahmood Nahvi, Joseph Edminister, Schaum's outline of theory and problems of electric circuits, McGraw-Hill

Professional, 2002

*

*

5

221

221

IVI

IVV

DC

BA

+=

+=

=

+

+

1n

1n

n

n

I

V

I

V

DC

BA

(note direction of I2)

⇒(8.1)

:

/

examplerefresher

tionmultiplica

matrix

6

7

=

2cos

2sin

2sin

2cos

1

01

2cos

2sin

2sin

2cos

θθ

θθ

θθ

θθ

j

j

jbj

j

DC

BA

22

θβ =⋅=⋅ l

dk

22

θβ =⋅=⋅ l

dk

8

=

2cos

2sin

2sin

2cos

1

01

2cos

2sin

2sin

2cos

θθ

θθ

θθ

θθ

j

j

jbj

j

DC

BA

−++

−+−=

)sin2

(cos)2

cos2

(sin

)2

cos2

(sin)sin2

(cos

θθθθ

θθθθ

bbbj

bbj

b

DC

BA

c

(8.2) 9

( ) ( )( ) ( ) z

z

eIzI

eVzV

γ

γ

−

−

=

=

0

0For a wave propagating in the +z direction,

Since the structure is infinitely long, the voltage and current at the nth

terminals can differ from the voltage and current at the n+1 terminals only

by the propagating factor, de γ−

d

nn

d

nn

eII

eVV

γ

γ

−+

−+

=

=

1

1⇒

=

=

+

+

+

+

d

n

d

n

n

n

n

n

eI

eV

I

V

DC

BA

I

Vγ

γ

1

1

1

1

(8.3)

(8.4)

01

1 =

−

−

+

+d

n

d

n

d

d

eI

eV

eDC

BeAγ

γ

γ

γ

From (8.1),

⇒

For a nontrivial solution, the determinant of the above matrix must vanish:

0)(2 =−+−+ BCeDAeAD dd γγ

Since AD – BC =1,

0)(1 2 =+−+ dd eDAe γγ

(8.5)

(8.6)

=

+

+

+

+

+

+

+

d

n

d

n

n

n

n

n

eI

eV

DI

BI

CV

AVγ

γ

1

1

1

1

1

1⇒

10

0)(1 2 =+−+ dd eDAe γγ

)( DAee dd +=+− γγ

θθγγγ

sin2

cos2

)(

2cosh

bDAeed

dd

−=+

=+

=−

βαγ j+=&

θθβαβαγ sin2

cossinsinhcoscoshcoshb

ddjddd −=+=⇒

From (8.2),

(8.7)

(8.8)

11

2cosh

dd eed

γγ

γ+

=−

Hyperbolic Function Refresher

12

θθβαβαγ sin2

cossinsinhcoscoshcoshb

ddjddd −=+=

Since the right-hand side of (8.8) is purely real, or0=α 0=β

Case#1: Propagating, Non-Attenuating => PASSBAND

θθβ sin2

coscosb

d −=

Case#2: Attenuating, Non-Propagating => STOPBAND

0

0

≠

=

βα

1sin2

coscosh ≥−= θθαb

d

πβα

,0

0

=

≠

⇒ Depending on frequency and normalized susceptance, the periodically

loaded line will exhibit either passbands or stopbands and therefore act as

a filter.

Remember that the equations are for V & I waves defined at terminals of

unit cells and don’t necessarily describe conditions at other points along

the line. These are similar to Bloch waves.

(8.9b)

(8.9a)

13

Bloch gives his name to the characteristic impedance of these waves:

1

10

+

+⋅=n

nB

I

VZZ

( ) 011 =+− ++ nn

d BIVeA γFrom (8.5),

dBeA

BZZ γ−

−⋅=⇒ 0

From (8.6),

( ) 42

2

2

0

−+−−

−=⇒ ±

DADAA

BZZ B

m

( ) ( )2

42 −+±+

=DADA

e dγ

0)(1 2 =+−+ dd eDAe γγSo we can solve for

deγ

⇒

So we can solve for the two solutions of the Bloch impedance:

12

0

−

±=⇒ ±

A

BZZ BSince the unit cell is symmetrical, A=D

(8.10)

(8.11)

(8.12)

14

If (passband), then, for symmetrical networks: 0,0 ≠= βα

1sin2

coscos ≤=−= Ab

d θθβ

+−= θθ cos22

sinbb

jB

θθβαβαγ sin2

cossinsinhcoscoshcoshb

ddjddd −=+=

θθγ sin2

cos2

)(cosh

bA

DAd −==

+=

From (8.2) we see that B is always purely imaginary.

12

0

−

±=±

A

BZZ B(8.12) shows that ZB will be real.

If (stopband), then, for symmetrical networks: 0,0 =≠ βα

1sin2

coscoshcosh ≥=−== Ab

dd θθαγ

12

0

−

±=±

A

BZZ B(8.12) shows that ZB will be imaginary.

This situation is similar to that for the wave impedance of a

waveguide, which is real for propagating modes and imaginary for

cutoff, or evanescent, modes.

⇒

⇒

15

We earlier assumed that the structure was infinitely long, but to implement

this filter we will need to terminate the line. If the load impedance doesn’t

match our Bloch impedance, there will be reflections, which will invalidate

our earlier work.

ndjndj

n

ndjndj

n

eIeII

eVeVV

ββ

ββ

−−+

−−+

+=

+=

00

00

(8.4)

BL

BL

ZZ

ZZ

+−

=Γ

d

nn

d

nn

eII

eVV

γ

γ

−+

−+

=

=

1

1

⇒

To avoid reflections, ZL must match ZB, which is real for a lossless structure

operating in a passband. If necessary, a quarter-wave transformer can be

used between the periodically loaded line and the load.

16

ββω k

cvp ==

ββω

d

dkc

d

dvg ==

(Brillouin diagram)

diagramsk −β

ckk −= 2β

ckk < βFor , there is no solution for

(Waveguide)

17

θθβ sin2

coscosb

d −= (8.9)

diagramk −β (Periodically Loaded Line Example)

18

Figure 8.7 shows an arbitrary, reciprocal two-port

network with image impedances defined as follows:

Zi1 = input impedance at 1 when 2 is terminated with Zi2

Zi2 = input impedance at 2 when 1 is terminated with Zi1

DCZ

BAZ

DICV

BIAV

I

VZ

i

iin +

+=

++

==2

2

22

22

1

11

221

221

IVI

IVV

DC

BA

+=

+=

112

112

IVI

IVV

AC

BD

+−=

−=

Since AD – BC = 1

ACZ

BDZ

AICV

BIDV

I

VZ

i

iin +

+=

+−−

=−

=1

1

11

11

2

22

11 iin ZZ =

22 iin ZZ = CD

ABZi =1

AC

BDZi =2We want ⇒ and

Image Parameter Method of Filter Design(Section 8.2)

21 ii ZZ =⇒If symmetric, A=D

19

1

1

112 VIVV

−=−=

iZ

BDBD

( )BCADA

D

AB

CDBD

Z

BD

i

−=−=

−=

11

2

V

V

( )BCADD

AACZA

I

VC i −=+−=+−= 1

1

1

1

2

I

I

20

Two important types of two-port

networks are the T and π

circuits, which can be made in

symmetric form. Table 8.1 list

the image impedances and

propagation factors, along with

other useful parameters, for

these two networks.

21

122

12

2

22

2

−+−=ccc

eωω

ωω

ωωγ

41

1,

2

21

LC

C

LZ

CjZLjZ iT

ωω

ω −=⇒==

kC

LR

LCc === 0,

2ω

2

2

0 1c

iT RZωω

−=⇒

0RZ iT =0=ω ⇒when

22

kC

LR

LCc === 0,2

1ω

kC

LR

LCc === 0,

2ω

There are only two parameters to choose (L and C), which are deteremined by the cutoff

frequency and the image impedance at zero frequency.

These results are only valid when the filter section is terminated in its image impedance,

which is a function of frequency and is not likely to mach a given source or load impedance.

Its attenuation isn’t very good in the stopband.

23

To improve our design from the constant-k filter, we

are going to try the m-derived filter.

Replace Z1 with Z’1 and Z2 with Z’2 where Z’1=mZ1

Choose Z’2 to keep ZiT the same:

4'

4

'''

4

2

121

2

121

2

121

mZZmZ

ZZZ

ZZZZ iT +=+=+=

( )1

2

21122

4

1

44' Z

m

m

m

ZmZ

m

Z

m

ZZ

−+=−+=

24

CjZLjZ

ωω

1, 21 ==

For a low-pass filter,

( )Lj

m

m

CmjZLmjZ ω

ωω

4

11','

2

21

−+==

So the m-derived impedances will be:

+++=

2

1

2

1

2

1

'4

'1

'

'

'2

'1

Z

Z

Z

Z

Z

Zeγ

( ) ( )2

2

2

22

1

11

2

411'

'

−−

−

=

−+

=

c

c

m

m

mmLj

Cmj

Lmj

Z

Z

ωω

ωω

ωω

ωLC

c

2=ω

( )2

2

2

2

1

11

1

'4

'1

−−

−

=+⇒

c

c

mZ

Z

ωω

ωω

25

+++=

2

1

2

1

2

1

'4

'1

'

'

'2

'1

Z

Z

Z

Z

Z

Zeγ

( )2

2

2

2

1

11

1

'4

'1

−−

−

=+

c

c

mZ

Z

ωω

ωω

If we restrict 0 < m < 1, then these results show that is real and >1 for >

Thus the stopband begins at , as for the constant-k section.

However, when , where ,

ωcωω =

∞=ωω21 m

c

−=∞

ωω

γe γe

γe

cω

becomes infinite.

The m-derived section has a very

sharp cutoff but then the attenuation

decreases as

To have infinite attenuation as

we can cascade it with a constant-k

section.

∞→ωω∞→ωω

26

The m-derived T-section was designed so

that its image impedance was identical to

that of the constant-k section (independent

of m), so we still have the problem of a

nonconstant image impedance. But a π-

section’s image impedance does depend on

m. By adjusting m as needed, we can use

this to optimize our match.

( )0

2

2

2

1

11

R

m

Z

c

c

i

−

−−

=

ωω

ωω

π

27

To benefit from the π-section’s ability to

keep a relatively constant image impedance

but still match up with constant-k or sharp

cutoff T-section, we will bisect a π-section.

28

29

30

31

Backup Slides

=

2cos

2sin

2sin

2cos

1

01

2cos

2sin

2sin

2cos

θθ

θθ

θθ

θθ

j

j

jbj

j

DC

BA

+

=

2cos

2sin

2sin

2cos

2cos

2cos

2sin

2sin

02

sin

02

cos 2

θθ

θθ

θθ

θθ

θ

θ

j

j

jb

jbj

jDC

BA

+

−=

2cos

2sin

2sin

2cos

2cos

2cos

2sin

2sin

2sin

2cos

θθ

θθ

θθθ

θθθ

j

j

bj

jb

DC

BA

+

+

+

−−=

2cos

2sin2

cos

2sin2

cos2

sin

2sin

2sin2

cos2

cos2

sin2

cos

2sin

2sin2

cos2

sin2

cos2

cos

2

22

22

22

θθθ

θθθ

θθθθθθ

θθθθθθ

j

jj

jjbjbj

jbjb

DC

BA

−++

−+−=

)sin2

(cos)2

cos2

(sin

)2

cos2

(sin)sin2

(cos

θθθθ

θθθθ

bbbj

bbj

b

DC

BA

+

+

+

−−=

2cos

2sin2

cos

2sin2

cos2

sin

2sin

2sin2

cos2

cos2

sin2

cos

2sin

2sin2

cos2

sin2

cos2

cos

2

22

22

22

θθθ

θθθ

θθθθθθ

θθθθθθ

j

jj

jjbjbj

jbjb

DC

BA

θθθθθθθθθθθθ

sin2

cossin22

sin212

sin2

cos222

sin2

cos2

sin2

sin2

cos2

cos 222222 bbbjbA −=

−

−=

−

−=

+

−=⇒

+−=

−−=

+

−=⇒ θθθθθθθθθθ

cos22

sin2

cos1

2sin2

cos22

sin2

cos2

sin2

sin2

cos 2 bbjbjjjbjB

++=

++=+

+=⇒ θθθθθθθθθθ

cos22

sin2

cos1

2sin2

cos22

sin2

cos2

cos2

sin2

cos 2 bbjbjjbjC

θθθθθθθθθθθθθ

sin2

cos2

sin2

cos222

sin212

sin2

cos2

sin2

cos2

cos2

sin2

sin2

cos 22222 bbbjjbjD −=

−

−=

−

−=+

+=⇒