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Fundamental WaveguideTheory

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UCF

Types of Waveguides

1. TEM and quasi-TEM waveguides

2. Metallic waveguides (TE and TM modes)

3. Dielectric waveguides (TE, TM, TEM or

hybrid modes)

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UCF TEM and Quasi-TEM Waveguides

TEM:

Quasi TEM:

Coaxial Strip Line Rectangular Coaxial

r r r

Microstrip Line Slot Line Coplanar Line

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UCF Metallic WaveguidesRectangular waveguide

Circular waveguide

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UCF Dielectric Waveguides

Ridge waveguide

1r

2r

FiberPlanar dielectric

waveguide

r

1r

2r

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UCF Modes in Waveguides (1)

Modes: certain field patterns that can

propagate independently

TEM mode: Transverse Electromagneticmode. All the fields are in the cross

section or there are no Ez and Hz

components

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UCF Modes in Waveguides (2)TE modes: transverse electric modes

Electric fields are in the cross section (orno Ez component).

Only Hz component in the longitudinal

direction. Also called H modes

TM modes: transverse magnetic modes

Magnetic fields are in the cross section (orno Hz component).

Only Ez component in the longitudinal

direction. Also called E modes

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UCF

Conditions for the Existence of

TEM Modes At least two perfect electric conductors

Dielectric distribution in the crosssection is homogeneous

Note: TEM line can have higher order

TE and TM modes

Yes Yes No

No

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UCF Quasi-TEM Line Some planar waveguide structure with

inhomogeneous dielectric distribution., But ,0zE 0zH tzE E tzH H

r r

Microstrip Line Slot Line Coplanar Line

r

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UCF Basic Waveguide Theory (1)

zzt EaEE

zzt HaHH

Equivalent

Generalized

Fourier Transform

)(),( zVyx mm

tmt eE

)(),( zIyx m

m

tmt hH

with

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UCF Basic Waveguide Theory (2)

)(

)(0 zIZjkdz

zdV

mmzm

m

)()(

0 zVYjkdz

zdImmzm

m

m

mY

Z0

0

1 isthecharacteristicimpedanceofthemthmode

zmk isthepropagationconstantofthemthmode

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UCF Basic Waveguide Theory (3)zjk

m

zjk

mmzmzm eBeAzV )(

ForwardWave

BackwardWave

effzmzm

eff kkk

k

2

0

22

0

)(

EffectiveRelativeDielectricConstantofthemthmode

Themodewiththelargest iscalledthedominantmode.eff

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UCF TEM Mode (1)

zyx zyx

aaa

zt

Let

Maxwells Equations

0

0

t

t

tt

tt

j

j

H

E

EH

HE

0

0

H

E

EH

HE

j

j0

zE0

zH

TEM mode

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UCF TEM Mode (2)0)( ttttzt j EHE

ttz jz

HEa )(t

t

z jz

HE

a

)(

0)( ttttzt j HEH

ttz j

z

EHa )(t

t

z j

z

EH

a

)(

0

0

tt

tt

H

E

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UCF TEM Mode (3)

From

The fields in the cross-section are similar to

2-D electrostatic & 2-D magnetostatic fields

for TEM mode even if actual operating

frequency can be very high.

0

0

0

0

tt

tt

tt

tt

H

H

E

E

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UCF TEM Mode (4)

ttE

2

112 lE dV Forvoltage

lH dI Forcurrent

02 t Laplace Equation

Let

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UCF TEM Mode (5))(

1)(

zjj

z

t

ztt

t

z

EaHH

Ea

)(

t

t

z jz

EH

a tt

zz jzzj

EE

aa

)(1

t

t

zz z E

E

aa 2

2

2

)(

tzz

tt

zzzz

EaaEE

aa 22

2

2

2

)()(

0)( 2

2

z

t

zz

Eaa

022

2

t

t kz

EE

22 k

02

2

2

t

t kz

HH

where

where

Likewise

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UCF TEM Mode (6)

zjk

ttzeyx

),( eE

zjk

ttzeyx

),( hHzjk

z

02

2

2

t

t

kz E

E

0)(

22

tzkk e

0te22 zkk

0222 zc kkk

222

yxc kkk

For +z direction propagation mode, we have

Since For TEM mode

From

Or:

where

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UCF TEM Mode (7)For +z direction propagating TEM mode: jkzeyx ),( eE

jkzeyx ),( hH

Vte

0 2 Vt

)(1

zj

t

zt

EaH

From:

ea

hEaH

z

zj

jk

wave impedance of the media

inside the TEM waveguide

k

jkz

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UCF TE, TM and Hybrid Modes (1)From

0

0

H

EEH

HE

j

j

00

22

22

HHEE

kk

For z component:

0

0

22

22

zczt

zczt

HkH

EkE

since2222

2

2

, zcz kkkk

z

00

22

22

zz

zz

HkHEkE

or

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UCF

TE, TM and Hybrid Modes (2)

From HE j

z

xy

y

z

xz

xyz

z

Hjy

Ex

E

Hjx

EEjk

HjEjk

y

E

EH j

z

xy

y

z

xz

xyz

z

Ejy

Hx

H

Ejx

HHjk

EjHjk

y

H

For a mode propagating in +z direction:zjk

z

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UCF

TE, TM and Hybrid Modes (3)

y

H

k

jk

x

E

k

jH

x

H

k

jk

y

E

k

jH

xH

kj

yE

kjkE

y

H

k

j

x

E

k

jkE

z

c

zz

c

y

z

c

zz

c

x

z

c

z

c

z

y

z

c

z

c

z

x

22

22

22

22

Or

zt

c

z

ztz

c

ztz

c

zt

c

z

Hk

jkE

k

j

Hk

jE

k

jk

22

22

aH

aE

t

t

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UCF

Auxiliary Potential Approach

y

Fk

x

AH

x

Fk

y

AH

xF

yAkE

y

F

x

AkE

zzz

y

zzz

x

zzzy

zzzx

1

1

1

1

Actually:

0,0 2222 zcztzczt FkFAkA

z

c

zz

c

z F

k

jHA

k

jE

22

,

In Balaniss book:

Balaniss is my k.

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UCF

TE Modes (1)

0

0

1

1

22

2

zczt

z

c

z

z

zz

y

zz

x

z

y

z

x

FkF

Ak

jH

E

y

FkH

x

FkH

x

FE

y

FE

0

0

22

2

2

2

2

zczt

z

z

c

z

y

z

c

z

x

z

cy

z

c

x

HkH

E

y

H

k

jk

H

x

H

k

jkH

x

H

k

j

E

y

H

k

jE

zt

c

z

z

z

ztz

c

Hkjk

k

Hk

j

2

2

t

t

t

H

Ha

aE

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UCF

TE Modes (2)

y

h

k

jkh

x

h

k

jkh

x

h

k

je

y

h

k

je

hkh

z

c

z

y

z

c

z

x

z

c

y

z

c

x

zczt

2

2

2

2

22 0

ttt

t

hahae

h

zTEz

z

ztc

z

Zk

hk

jk

2

zjkzeyx ),( eEzjkzeyx

),( hH

For a mode propagating

in +z direction:

zTE k

Z characteristic impedance

of TE modes

+ Boundary Conditions

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UCF

TM Modes (1)

0

0

1

1

22

2

zczt

z

c

z

z

z

y

z

x

zz

y

zz

x

AkA

Ak

jE

H

x

AH

y

AH

y

AkE

x

AkE

0

0

22

2

2

2

2

zczt

z

z

c

y

z

c

x

z

c

z

y

z

c

z

x

EkE

H

x

E

k

j

H

y

E

k

jH

y

E

k

jk

E

x

E

k

jkE

t

t

t

Ea

aH

E

z

z

ztz

c

zt

c

z

k

Ek

j

Ekjk

2

2

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UCF

TM Modes (2)

zjkzeyx ),( eEzjkzeyx

),( hH

For a mode propagating

in +z direction:

x

e

k

jh

y

e

k

jh

y

e

k

jke

x

e

k

jke

eke

z

c

y

z

c

x

z

c

z

y

z

c

z

x

zczt

2

2

2

2

22 0

ttt

t

EaEah

e

z

TM

z

z

zt

c

z

Zk

e

k

jk

1

2

z

TM

kZ characteristic impedance

of TM modes

+ Boundary Conditions

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UCF

Hybrid Modes

zjkzeyx ),( eEzjkzeyx

),( hH

For a mode propagating

in +z direction:

y

h

k

jk

x

e

k

jh

x

h

k

jk

y

e

k

jh

x

h

k

j

y

e

k

jke

y

h

k

j

x

e

k

jke

z

c

zz

c

y

z

c

zz

c

x

z

c

z

c

z

y

z

c

z

c

z

x

22

22

22

22

zt

c

zztz

c

ztz

c

zt

c

z

hk

jke

k

j

hk

je

k

jk

22

22

ah

ae

t

t

0

022

22

zczt

zczt

hkh

eke+ Boundary Conditions

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UCF

Phase Velocity and Group Velocity (1)

For a single frequency: zjkze

)cos( zkt z

Phase velocity: the velocity of constant phase plane (t-kzz=const)

zp kv

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UCFPhase Velocity and Group Velocity (3)

m

m

z deAFts zktj

0

0

])(Re[2

1)( )(0

Expand

)()(

)()()(

00

00

0

0

d

dkk

d

dkkk

z

z

z

zz

Let

00

00

'

)(

d

dkk

kk

z

z

zz

)()( 0'

00 zzz kkk

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UCFPhase Velocity and Group Velocity (4)

)cos()(

)(Re

])(Re[2

])(Re[2

1)(

00

'

0

)('

0

)(

)(

0

00

'000

0

0

0'00

zktzktAf

ezktfA

dpepFeA

deeAFts

zz

zktj

z

pzktjzktj

zkkjtj

z

m

m

zz

m

m

zz

Information with group velocity Carrier withphase velocity

0

1

'

1

)(

0

00

d

dkkv

kk

zz

g

zz

0

0

z

pk

v

0p

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UCF

)cos()( 00'

0 zktzktf zz

Phase Velocity and Group Velocity (5)

zt

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UCF

Cutoff Frequency in

Metallic Waveguide

0222 cz kkk

ck

Find cutoff frequency

cc

fk 2

zjkze

For

When ,02

z

k the mode is an evanescent mode.

2

c

c

kf

From

is cutoff wavenumber.

,cutoff frequency

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