Fundamental Waveguide Theory

Fundamental Waveguide Theory

UCFTypes of Waveguides

1. TEM and quasi-TEM waveguides2. Metallic waveguides (TE and TM modes)3. Dielectric waveguides (TE, TM, TEM or

hybrid modes)

UCFTEM and Quasi-TEM Waveguides

TEM:

Quasi TEM:Coaxial Strip Line Rectangular Coaxial

r r rMicrostrip Line Slot Line Coplanar Line

UCF Metallic Waveguides

Rectangular waveguide

Circular waveguide

UCF Dielectric Waveguides

Ridge waveguide

1r2r

FiberPlanar dielectricwaveguide

r

1r

2r

UCF Modes in Waveguides (1)

Modes: certain field patterns that can propagate independently

TEM mode: Transverse Electromagnetic mode. All the fields are in the cross section or there are no Ez and Hzcomponents

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

Electric fields are in the cross section (or no Ez component).

Only Hz component in the longitudinal direction. Also called H modes

TM modes: transverse magnetic modesMagnetic fields are in the cross section (or no Hz component).

Only Ez component in the longitudinal direction. Also called E modes

UCF

Conditions for the Existence of TEM Modes

At least two perfect electric conductors Dielectric distribution in the cross

section is homogeneous

Note: TEM line can have higher order TE and TM modes

Yes Yes No

No

UCF Quasi-TEM Line Some planar waveguide structure with

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

r rMicrostrip Line Slot Line Coplanar Line

r

UCF Basic Waveguide Theory (1)

zzt EaEE

zzt HaHH

Equivalent

GeneralizedFourier Transform

)(),( zVyx mm

tmt eE

)(),( zIyx mm

tmt hH

with


)()(0 zIZjk

dzzdV

mmzmm

)()(0 zVYjk

dzzdI

mmzmm

mm Y

Z0

01

is the characteristic impedance of the mth mode

zmk is the propagation constant of the mth mode


zjkm

zjkmm

zmzm eBeAzV )(

Forward Wave

Backward Wave

effzmzm

eff kkkk 2

022

0

)(

Effective Relative Dielectric Constant of the mth mode

The mode with the largest is called the dominant mode. eff

UCF TEM Mode (1)

zyx zyx

aaa

zt

Let

Maxwell’s Equations

00

t

t

tt

tt

jj

HE

EHHE

00

HE

EHHE

jj 0zE

0zH

TEM mode

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

ttz jz

HEa )( tt

z jz

HE

a

)(

0)( ttttzt j HEH

ttz jz

EHa )( tt

z jz

EH

a

)(

00

tt

tt

HE

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.

00

00

tt

tt

tt

tt

HH

EE

UCF TEM Mode (4)

ttE

2

112 lE dV For voltage

lH dI For current

0 2 t Laplace Equation

Let

UCF TEM Mode (5))(1 )(

zjj

zt

zttt

z

E

aHHE

a

)(

tt

z jz

EH

a tt

zz jzzj

EE

aa

)(1

tt

zz zE

Eaa 2

2

2

)(

tzztt

zz zzEaa

EEaa 2

2

2

2

2

)()(

0)( 2

2

zt

zzE

aa

022

2

tt k

zEE

22 k

022

2

tt k

zHH

where

where

Likewise

UCF TEM Mode (6)

zjktt

zeyx ),( eE

zjktt

zeyx ),( hHzjk

z

022

2

tt k

zEE 0)( 22 tzkk e

0te22 zkk

0222 zc kkk

222yxc kkk

For +z direction propagation mode, we have

Since For TEM mode

From

Or:

where

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

jkzeyx ),( hH

Vte

0 2 Vt

)(1 zj

tzt

EaH

From:

eahEaH

zzj

jk

wave impedance of the media inside the TEM waveguide

k

jkz

UCF TE, TM and Hybrid Modes (1)From

00

HE

EHHE

jj

00

22

22

HHEE

kk

For z component:

0

022

22

zczt

zczt

HkH

EkE

since 22222

2

, zcz kkkkz

0

022

22

zz

zz

HkH

EkE

or

UCFTE, TM and Hybrid Modes (2)

From HE j

zxy

yz

xz

xyzz

Hjy

Ex

E

Hjx

EEjk

HjEjky

E

EH j

zxy

yz

xz

xyzz

Ejy

Hx

H

Ejx

HHjk

EjHjky

H

For a mode propagating in +z direction: zjkz

UCFTE, TM and Hybrid Modes (3)

yH

kjk

xE

kjH

xH

kjk

yE

kjH

xH

kj

yE

kjkE

yH

kj

xE

kjkE

z

c

zz

cy

z

c

zz

cx

z

c

z

c

zy

z

c

z

c

zx

22

22

22

22

Or

ztc

zztz

c

ztzc

ztc

z

HkjkE

kj

HkjE

kjk

22

22

aH

aE

t

t

UCFAuxiliary Potential Approach

yFk

xAH

xFk

yAH

xF

yAkE

yF

xAkE

zzzy

zzzx

zzzy

zzzx

1

1

1

1

Actually:

0 ,0 2222 zcztzczt FkFAkA

zc

zzc

z Fk

jHAk

jE

22

,

In Balanis’s book:

Balanis’s is my k .

UCFTE Modes (1)

0

0

1

1

22

2

zczt

zc

z

z

zzy

zzx

zy

zx

FkF

Ak

jH

Ey

FkH

xFk

H

xFE

yFE

0

022

2

2

2

2

zczt

z

z

c

zy

z

c

zx

z

cy

z

cx

HkH

Ey

Hkjk

H

xH

kjk

H

xH

kjE

yH

kjE

ztc

z

zz

ztzc

Hkjkk

Hkj

2

2

t

t

t

H

Ha

aE

UCFTE Modes (2)

yh

kjk

h

xh

kjkh

xh

kje

yh

kje

hkh

z

c

zy

z

c

zx

z

cy

z

cx

zczt

2

2

2

2

22 0

ttt

t

hahae

h

zTEzz

ztc

z

Zk

hkjk

2

zjk zeyx ),( eEzjk zeyx ),( hH

For a mode propagating in +z direction:

zTE k

Z characteristic impedance

of TE modes

+ Boundary Conditions

UCFTM Modes (1)

0

0

1

1

22

2

zczt

zc

z

z

zy

zx

zzy

zzx

AkA

Ak

jE

Hx

AH

yA

H

yAkE

xAk

E

0

022

2

2

2

2

zczt

z

z

cy

z

cx

z

c

zy

z

c

zx

EkE

Hx

EkjH

yE

kjH

yE

kjkE

xE

kjkE

t

t

t

Ea

aH

E

zz

ztzc

ztc

z

k

Ekj

Ekjk

2

2

UCFTM Modes (2)



xe

kjh

ye

kjh

ye

kjke

xe

kjke

eke

z

cy

z

cx

z

c

zy

z

c

zx

zczt

2

2

2

2

22 0

ttt

t

EaEah

e

zTM

zz

ztc

z

Zk

ekjk

1

2

z

TMk

Z characteristic impedanceof TM modes


UCFHybrid Modes



yh

kjk

xe

kjh

xh

kjk

ye

kjh

xh

kj

ye

kjk

e

yh

kj

xe

kjk

e

z

c

zz

cy

z

c

zz

cx

z

c

z

c

zy

z

c

z

c

zx

22

22

22

22

ztc

zztz

c

ztzc

ztc

z

hkjke

kj

hkje

kjk

22

22

ah

ae

t

t

00

22

22

zczt

zczt

hkheke


UCFPhase Velocity and Group Velocity (1)

For a single frequency: zjk ze

)cos( zkt z

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

zp k

v


Waveguide

zjk zAe

tjetf

ttf0)(Re

)cos()( 0

)(ts

)( 0 F )()( Ftf F

zjkzAeFS )()( 0

m

m

z

m

m

deAF

deS

deSts

zktj

tj

tj

0

0

0

0

])(Re[21

])(Re[21

)(Re21)(

)(0

m m

)(F


m

m

z deAFts zktj

0

0

])(Re[21)( )(

0

Expand

)()(

)()()(

00

00

0

0

ddk

k

ddk

kk

zz

zzz

Let

00

00

'

)(

ddkk

kk

zz

zz

)()( 0'00 zzz kkk


)cos()(

)(Re

])(Re[2

])(Re[21)(

00'0

)('0

)(

)(0

00

'000

0

0

0'00

zktzktAf

ezktfA

dpepFeA

deeAFts

zz

zktjz

pzktjzktj

zkkjtj

z

m

m

zz

m

m

zz

Information with group velocity Carrier with phase velocity

0

1'1

)(

0

00

ddkk

v

kk

zzg

zz

0

0

zp k

v

0p

UCF

)cos()( 00'0 zktzktf zz

Phase Velocity and Group Velocity (5)

zt

UCF

Cutoff Frequency in Metallic Waveguide

0222 cz kkk

ck

Find cutoff frequency

cc fk 2

zjk ze For When ,02 zk the mode is an evanescent mode.

2c

ck

f

From

is cutoff wavenumber.

,cutoff frequency

UCF Degenerate Modes

If two or more modes have the same eigenvalue (propagation constant kz) but different eigenvectors (field patterns), they are called degenerate modes.

Documents

Fundamental Waveguide Theory