EMLAB

1

Waveguides

EMLAB

2Waves in cylinder

ztzttt

ztzttt

tt

ztttzt

tt

ztttzt

ztztt

ztztt

ztt

ztt

EjHzz

HjEzz

z

jz

Hjz

H

jz

Ejz

E

EjHz

HjEz

Ej

Hj

jj

zHH

zEE

EH

zzEHzz

HE

zzHEzz

zEzHz

zHzEz

zH

zE

EHHE

ˆ

ˆ

,ˆsizeby both gMultiplyin

ˆˆˆˆ

ˆˆˆˆ

)ˆ()ˆ(ˆ

)ˆ()ˆ(ˆ

)1(ˆ

ˆ

.)EqsMaxwell(,

22

2

22

2

)(0

0ˆ)()ˆ(

ˆ)()ˆ()ˆ(

ˆ)(ˆ

ˆ)(ˆ

]ˆ[)(

)1(

)3(ˆ)(

)2(ˆ)(

),(

22222

222

222

22

22

22

22

22

22

2

0

kkHkH

HkH

HkHH

HkH

HkjHj

HjEjk

EjHjk

HjEjk

zj

z

eyxE

czczt

zzt

zztztt

zztt

zztt

ztztttt

ztztt

ztztt

zj

zz

zzz

zz

zz

zE

zH

zE

EE

EE

E

넣으면에

가정하면라고

)3(ˆ

)2(ˆ

using and obtain Then

.or for

0or0

eqs. following theSolving

2

2

2222

ztzttc

ztzttc

tt

zz

zcztzczt

EjHjk

HjEjk

EH

EkEHkH

zH

zE

HE

In a waveguide whose cross-section is uniform along z-axis, variation of E, H is proportional to exp(-j z).

In a waveguide, transverse E and H can be obtained from Ez and Hz.

EMLAB

3

TM, TE mode

)ˆ(0,ˆ1

)ˆ(1

,1

ˆˆ

0

22

22

2

2

2

22

ztc

ztc

zc

zcztc

ztzttc

zttc

z

zczt

Ek

jEj

k

Ek

EkEjk

EjEjk

Ejk

E

EkE

zzH

zE

zzH

E

TM mode : Electromagnetic wave modes with Hz=0 are called TM modes (Transverse Magnetic). Those modes can be found from Ez.

)ˆ(0,ˆ1

)ˆ(1

,1

)3(

)2(ˆ

0

22

22

2

2

2

22

ztc

ztc

zc

zcztc

zttc

zttc

z

zczt

Hk

jHj

k

Hk

HkHjk

Hjk

Hjk

E

HkH

zzE

zH

H

zE

TE mode : Electromagnetic wave modes with Ez=0 are called TE modes (Transverse Electric). Those modes can be found from Hz.

EMLAB

4

)ˆ(0,ˆ1

)ˆ(1

,1

2,

2,

2,

2,2

,

nzt

nc

nzt

nc

TMn

nz

nc

nznc

nztn

nc

TMn

Ek

jEj

k

Ek

EkEjk

zzh

ze

n

TEn

zjn

zjn

n

TMn

zjn

zjn

n

TEMn

jkzn

jkzn

nnnn ececebebeaea eeeE )()()(

)ˆ(0,ˆ1

)ˆ(1

,1

2,

2,

2,

2,2

,

nzt

nc

nzt

nc

TEn

nz

nc

nznc

nztn

nc

TEn

Hk

jHj

k

Hk

HkHjk

zze

zh

Eigen-function expansions in waveguides

02,

2 nznc

nzt EkE 02

,2 n

zncnzt HkH

02 n

TM modes TE modes

TEM modes

jkzTEMi e e

EMLAB

5Rectangular waveguide (TE mode)

022

2

2

2

zc Hkyx

)()( yYxXH z

011 2

2

2

2

2

ckdy

Yd

Ydx

Xd

X

222

coscos

b

n

a

m

eb

yn

a

xmH zj

z

22

2

b

n

a

mkc

2xk

2yk

0222 cyx kkk

0,0 22

22

2

2

Ykdy

YdXk

dx

Xdyx

axatx

H z ,0,0

byaty

H z ,0,0

Boundary con-dition

a

cf

ak cc 210,

EMLAB

6

Field lines for some of the lower order modes of a rectangular waveguide. Reprinted from Fields and Waves in Communication Electronics, Ramo et al, © Wiley, 1965)

EMLAB

7

EMLAB

8

222

coscos

b

n

a

m

eb

yn

a

xmH zj

z

WR-90 X-band 8.2GHz~12.4GHz (2.286cm × 1.016cm)

Example : WR-90

]GHz[56.610286.22

103

2 2

8

a

fc

5GHz input signal

][9.882 122

22

mjff

a c

Attenuation by a 3cm-long evanescent waveguide

]dB[15.23434.01039.8820log20)(log20 21010 elje lj

EMLAB

9Cavity resonator222

,, 2

D

l

B

n

A

mcf lnm

A

B

D

22

2

b

n

a

mcfex

ab

]/[103,2

8 smca

cfex

Minimum frequencyinE

f

EMLAB

10Cylindrical waveguide

0..,022

z

zczt

HCBHkH

zjz eyxE ),(

011 2

2

2

2

ck

)()(),( R

01

2

22

R

nk

d

dR

d

dc

By separation of variables

022

2

nd

d

nBnA nn sincos

)()()( cnncnn kNBkJAR

a

EMLAB

11

a

0)()( akJaR cn

0)(,,

nmnnm

ic pJa

pk

22

a

pk nm

Cylindrical waveguide mode

zjcnnnz ekJnBnAH )()sincos(

zjcnnn

c

ekJnBnAk

jE )()sincos(

a

cf

ak c 2

841.1841.1

1,1

EMLAB

12

Cutoff frequencies of the first few TE and TM modes of a circular waveguide, relative to the cutoff frequency of the dominant TE11 mode.