1

Prof. David R. JacksonDept. of ECE

Notes 5

ECE 5317-6351 Microwave Engineering

Fall 2011

Waveguides Part 2:Parallel Plate Waveguide

2

2

2

2

2

z zx c z

c

z zy c z

c

z zx z

c

z zy z

c

E HjH k

k y x

E HjH k

k x y

E HjE k

k x y

E HjE k

k y x

Summary

2 2ck

1/22 2c zk k k

Field Equations (from Notes 4)

These equations will be useful to us in the present discussion.

3

Parallel-Plate Waveguide

Both plates assumed PEC w >> d,

0x

Neglect x variation,edge effects

The parallel-plate stricture is a good 1ST order model for a microstrip line.

y

z w

d

x

, , s

d,

w

,

4

Parallel-plate waveguide2 conductors 1 TEM mode

To solve for TEM mode:

2 0 0

0t x w

y d

for

Boundary conditions:

0( ,0) 0 ; ( , )x x d V

2 22

2 20t x y

TEM Mode

z ck j k k jk

c js

k

k

y

z w

d

x

, , s

5

where

0

( , )

( , ) ; 0

0

Vx y

x y

y x w

y

A y

dd

B

0ˆ( , , ) ( , ) jkz jkzt

Vx y z e x y e y eE

d

0

@ 0

0

@

y

A

y d

VB

d

2

20

y

0,0 0 & ,x x d V

ˆ ˆ, ot t

Ve x y y y

y d

z ck k

TEM Mode (cont.)

c js

6

Recall

0ˆ, , jkzVH x y z x e

d

For a wave prop. in + z direction

Time-ave. power flow in + z direction:

2

*

2

0 2* 2

0 0

22

0 *

1ˆRe ( )

2

1ˆ ˆRe

1 1 1Re

)

2

2

s

w dk z

k z

P E H z dS

Vz z

V w

e dy

dd

dxd

e

1ˆ( )H z E

y

x

V0

EH, , s

0ˆ( , , ) jkzVE x y z y e

d

2 20 *

1 1Re

2k zw

P V ed

TEM Mode (cont.)y

z w

d

x

, , s

7

Transmission line voltage0

0

ˆ( )

( ) j

d

z

kz

ck k

V z

V z E y d

V e

y

Transmission line current

0

0

0

( )

ˆ( ) , ,w

k

I

j zI z

I z H x d z x d

Ve

x

w

d

Characteristic Impedance

00

0

jkz

jkz

V eZ

I e

Phase Velocity (lossless case)

p

r r

cv

c = 2.99792458 108 m/s

x

d

I

I+

-V

y

z

C , , s

w

(Assume + z wave)

0

dZ

w

TEM Mode (cont.)

ˆs

sz z

J n H

J H

PEC :Note:

8

For wave propagating in + z direction

Time-ave. power flow in +z direction: *

200

2 20 *

1 1

1Re *

2

1

e

e2

2

R

R

k z

k zwP V e

d

P VI

V wV e

d

Recall that we found from the fields that:

2 20 *

1 1Re

2k zw

P V ed

same

TEM Mode (cont.)

(calculated using the voltage and current)

This is expected, since a TEM mode is a transmission-line type of mode, which is described by voltage and current.

9

TEM Mode (cont.)We can view the TEM mode in a parallel-plate waveguide as a “piece” of a plane wave.

The PEC and PMS walls do not disturb the fields of the plane wave.

ˆ 0n E PEC : ˆ 0n H PMC :

y

PEC

PEC

PMCPMC , , s

x

E

H

10

Recall

, sin( ) cos( )

@ 0 0

@ 0,1,2,.... c

z c c

c

e x y A k y B k y

y B

y d k d nn

kd

n

where12 2

2 2 2 22 2

0, [ ]c z c zk e k k kx y

subject to B.C.’s Ez = 0 @ y = 0, d

( , , ) ( , ) zjk zz zE x y z e x y e

TMz Modes (Hz = 0)

y

z w

d

x

, , s

11

, sin 0,1,2,...z

ne x y A y n

d

sin zjk zz n

nE A y e

d

Recall:

2 2

2 2

cos

cos

0 0 0

z

z

jk zc czx n

c c

jk zz z zy n

c c

x y z

j jE n nH A y e

k y k d d

jk E jk n nE A y e

k y k d d

E H H

2 2

22

z ck k k

nk

d

2 2ck

TMz Modes (cont.)

No x variation

12

sin zjk zz n

nE A y e

d

Summary

22

2 2

cos

cos

0

; 0,1,2,...

z

z

jk zzy n

c

jk zcx n

c

x y z

c

z

c

jk nE A y e

k d

j nH A y e

k d

E H H

nk n

d

nk k

d

k

Each value of n corresponds to a unique TM field solution or “mode.”

TMn mode

Note:0

0

TEMzn k k

TM

TMz Modes (cont.)y

z w

d

x

, , s

(In this case, we absorb the An coefficient with the kc term.)

13

21

2

22

12 2 2

c

z

c

k

nk k

d

k k

0,1,2,...n

2 2

2

2

2 2 2

2

z

z c z c

c

j z

c

k z

k k k k j k k j

e

k k k k

e

propagating mode

for for

Fields decay exponentially evanescent fields “cutoff” mode

Lossless Casec

2 2k

TMz Modes (cont.)

y

z w

d

x

, , s

14

Frequency that defines border between cutoff and propagation (lossless case): fc cutoff frequency

@ cnf fc cn

nk k

d

1

2cn

nf

d cutoff frequency for TMn mode

prop.

cuttoff

TEM TM1 TM2 TM3

singlemodeprop.

2 modes prop

3 mode prop.

0

f

….

3cf1cf 2cf

TMz Modes (cont.)

c

15

Time average power flow in z direction (lossless case):

*

0 0

*

0 0

2 22

0

1ˆRe

2

1Re

2

Re{ } cos2

w d

TMn

w d

y x

d

z nc

P E H z dydx

E H dydx

nk A w y dy

k d

2

2

; 0Re{ } 2

2; 0

0,1,2,...

TMn z nc

dn

P k A wk

d n

n

Real for f > fc

Imaginary for f < fc

TMz Modes (cont.)

y

z w

d

x

, , s

c

16

Recall ( , , ) ( , ) zjk zz zH x y z h x y e

where

12 2

2 2 2 22 2

, 0, [ ]c z c zk h x y k k kx y

subject to B.C.’s Ex = 0 @ y=0, d

1 yzx

c

HHE

j y z

sin( ) cos( )

@ 0 0

@ , 1,2,3,...

z c c

c c

h A k y B k y

y A

y d k d n nn

kd

TEz Modes

ˆ 0H n PEC :

y

z w

d

x

, , s

17

, cos 1,2,3,

co

.

s

..

zjk zz

n

n

z

nH

nh x y B y n

d

B y ed

Recall:

2 2

2 2

sin

sin

0 0 0

z

z

jk zzx n

c c

jk zz z zy n

c c

x y z

Hj j n nE B y e

k y k d d

jk H jk n nH B y e

k y k d d

H E E

2 2

22

z ck k k

nk

d

2 2ck

TEz Modes (cont.)

No x variation

18

Summary

cos zjk zz n

nH B y e

d

TEn mode

Cutoff frequency1

2cn

nf

d

Each value of n corresponds to a unique TE field solution or “mode.”

22

2 2

sin

sin

0

; 1,2,...

z

z

jk zx n

c

jk zzy n

c

x y z

c

z

c

j nE B y e

k d

jk nH B y e

k d

H E E

nk n

d

nk k

d

k

TEz Modes (cont.)y

z w

d

x

, , s

19

For all the modes of a parallel-plate waveguide, we have

1

2cn

nf

d

The mode with lowest cutoff frequency is called the “dominant” mode of the wave guide.

prop.

cuttoff

TEM TM1 TM2 TM3

singlemodeprop.

3 modes prop

5 mode prop.

0

f

….

3cf1cf 2cf

TE3TE2TE1

All Modes

c

20

*

0 0

*

0 0

2 22

0

1ˆRe

2

1Re

2

Re{ } sin2

w d

TEn

w d

x y

d

z nc

P E H z dydx

E H dydx

nk B W y dy

k d

2

2Re

4TEn z nc

P k B Wdk

n = 1,2,…..

Power in TEz ModeTime average power flow in z direction (lossless case):

Real for f > fc

Imaginary for f < fc

y

z w

d

x

, , s

c

21

TEM

TM1

TE1

y

y

y

x

x

x

Field Plots