Waveguide Types - University of Delawaremirotzni/ELEG413/WaveguidesI.pdf · Rectangular Waveguides....

Waveguide Types

Uniform Waveguides

We are interested in finding what electromagnetic field solutions are possible in a uniform infinite waveguide with no sources.

We can always find those solutions by solving:

or

Subject to boundary conditions.

Uniform Waveguides

This is really three PDEs


Uniform Waveguides



That’s a lot of work!Are there any shortcuts?

Uniform Waveguides

Because the cross section does not change in the z direction

Uniform Waveguides


zjt

zjt

z

z

eyxHzyxH

eyxEzyxEβ

β

−

−

=

=

),(~),,(~),(~),,(~

Uniform Waveguides


zjt

zjt

z

z

eyxHzyxH

eyxEzyxEβ

β

−

−

=

=

),(~),,(~),(~),,(~

[ ] [ ] [ ]

[ ] [ ] [ ] 0),(~),(~),(~),(~

0),(~),(~),(~),(~

22

2

2

2

2

2

22

2

2

2

2

2

=+∂∂

+∂∂

+∂∂

=+∂∂

+∂∂

+∂∂

−−−−

−−−−

zjt

zjt

zjt

zjt

zjt

zjt

zjt

zjt

zzzz

zzzz

eyxHkeyxHz

eyxHy

eyxHx

eyxEkeyxEz

eyxEy

eyxEx

ββββ

ββββ

Uniform Waveguides

zjt

zjt

z

z

eyxHzyxH

eyxEzyxEβ

β

−

−

=

=

),(~),,(~),(~),,(~

[ ] [ ] [ ]

[ ] [ ] [ ] 0),(~),(~),(~),(~

0),(~),(~),(~),(~

22

2

2

2

2

2

22

2

2

2

2

2

=+∂∂

+∂∂

+∂∂

=+∂∂

+∂∂

+∂∂

−−−−

−−−−

zjt

zjt

zjt

zjt

zjt

zjt

zjt

zjt

zzzz

zzzz

eyxHkeyxHz

eyxHy

eyxHx

eyxEkeyxEz

eyxEy

eyxEx

ββββ

ββββ

[ ] [ ]

[ ] [ ] 0),(~),(~),(~),(~

0),(~),(~),(~),(~

222

2

2

2

222

2

2

2

=+−∂∂

+∂∂

=+−∂∂

+∂∂

−−−−

−−−−

zjt

zjtzt

zjt

zj

zjt

zjtzt

zjt

zj

zzzz

zzzz

eyxHkeyxHyxHy

eyxHx

e

eyxEkeyxEyxEy

eyxEx

e

ββββ

ββββ

β

β

Uniform Waveguideszj

t

zjt

z

z

eyxHzyxH

eyxEzyxEβ

β

−

−

=

=

),(~),,(~),(~),,(~

[ ] [ ] [ ]

[ ] [ ] [ ] 0),(~),(~),(~),(~

0),(~),(~),(~),(~

22

2

2

2

2

2

22

2

2

2

2

2

=+∂∂

+∂∂

+∂∂

=+∂∂

+∂∂

+∂∂

−−−−

−−−−

zjt

zjt

zjt

zjt

zjt

zjt

zjt

zjt

zzzz

zzzz

eyxHkeyxHz

eyxHy

eyxHx

eyxEkeyxEz

eyxEy

eyxEx

ββββ

ββββ

[ ] [ ]

[ ] [ ] 0),(~),(~),(~),(~

0),(~),(~),(~),(~

222

2

2

2

222

2

2

2

=+−∂∂

+∂∂

=+−∂∂

+∂∂

−−−−

−−−−

zjt

zjtzt

zjt

zj

zjt

zjtzt

zjt

zj

zzzz

zzzz

eyxHkeyxHyxHy

eyxHx

e

eyxEkeyxEyxEy

eyxEx

e

ββββ

ββββ

β

β

[ ] [ ] ( )

[ ] [ ] ( ) 0),(~),(~),(~

0),(~),(~),(~

222

2

2

2

222

2

2

2

=−+∂∂

+∂∂

=−+∂∂

+∂∂

yxHkyxHy

yxHx

yxEkyxEy

yxEx

tztt

tztt

β

β

Uniform Waveguideszj

t

zjt

z

z

eyxHzyxH

eyxEzyxEβ

β

−

−

=

=

),(~),,(~),(~),,(~

[ ] [ ] ( )

[ ] [ ] ( ) 0),(~),(~),(~

0),(~),(~),(~

222

2

2

2

222

2

2

2

=−+∂∂

+∂∂

=−+∂∂

+∂∂

yxHkyxHy

yxHx

yxEkyxEy

yxEx

tztt

tztt

β

β

( )( ) 0),(~),(~

0),(~),(~

222

222

=−+∇

=−+∇

yxHkyxH

yxEkyxE

tztt

tztt

β

β

Uniform Waveguides

zjt

zjt

z

z

eyxHzyxH

eyxEzyxEβ

β

−

−

=

=

),(~),,(~),(~),,(~

( )( ) 0),(~),(~

0),(~),(~

222

222

=−+∇

=−+∇

yxHkyxH

yxEkyxE

tztt

tztt

β

β

222zc kk β−=LET

0),(~),(~0),(~),(~

22

22

=+∇

=+∇

yxHkyxH

yxEkyxE

tctt

tctt

Uniform Waveguides



zjt

zeyxEzyxE β−= ),(~),,(~

Uniform Waveguides



This makes it a bunch easier!

Any other short cuts?

zjt

zeyxEzyxE β−= ),(~),,(~

Uniform Waveguides

Uniform Waveguides

where

Uniform Waveguides

Uniform Waveguides

Three Cases:(1) Ez=0, Hz = 0 TE (2) Ez=0, Hz = 0 TM

Uniform Waveguides

Three Cases:

(3) Ez=0, Hz = 0 TEM

What do we do with this?Looks like all the fields are zero!

Uniform Waveguides

Three Cases:

(3) Ez=0, Hz = 0 TEM

What do we do with this?Looks like all the fields are zero!

The only way the fields are not all zero is if

Uniform Waveguides

Three Cases:

(3) Ez=0, Hz = 0 TEM

=0

Uniform Waveguides

Three Cases:

(3) Ez=0, Hz = 0 TEM

Rectangular Waveguides

TM Modes


TM Modes

Solve using separation of variables


TM Modes

function of x

function of y

constant


TM Modes


TM Modes

????

????


TM Modes


TM Modes

How many unknowns do we have?


TM Modes

What boundary conditions should we use?


TM Modes

Boundary Conditions


TM Modes

Boundary Conditions

Boundary Conditions


TM Modes

How do we find ?


TM Modes


TM Modes

Case I:

is real and the mode propagates without attenuation


TM Modes

Case II:

is imaginary and the wave decays exponentially


TM Modes

At the point

the mode changes from evanescent to propagating.

Cutoff frequency


TM Modes


TM Modes

The dominate mode is the one that has the lowest cutoff frequency(i.e. first mode to propagate)

What is the dominate mode for this waveguide?


TM Modes

The dominate mode is the one that has the lowest cutoff frequency(i.e. first mode to propagate)

What is the dominate TM mode for this waveguide?2222

11 112

12

1

+

=

+

=

babafc µε

ππµεπ

a>b


TM Modes

What is wavelength of the guided mode?


TM Modes


222

22

+

−

==

bm

an

nmz

nmb

ππµεω

πβπλ


TM Modes


µεω

ππ

µεω

π

ππµεω

πβπλ

2

22222

1

222

+

−

=

+

−

==

bm

an

bm

an

nmz

nmb


TM Modes


( )

2

2

22

2

2

22222

1

211

1

222

−

=

+

−

=

+

−

=

+

−

==

ff

fb

ma

n

f

c

bm

an

bm

an

c

o

nmz

nmb

λ

ππ

µεπ

µεω

ππ

µεω

π

ππµεω

πβπλ


TM Modes


2222 1

22

−

=

+

−

==

ff

bm

an nm

c

onmz

nmb

λ

ππµεω

πβπλ


TM Modes

What is wavelength of the impedance of the guided mode?


TM Modes


nmx

nmy

nmy

nmxnm

w HE

HEZ

−==


TM Modes


ωε

ππµεω

ωεβ

222

−

−

===b

ma

n

HEZ

nmz

nmy

nmxnm

w


TM Modes


2

222

1

−=

−

−

===f

fbm

an

HEZ

nmc

nmz

nmy

nmxnm

w εµ

ωε

ππµεω

ωεβ


TM Modes


2

1

−==

ff

HEZ

nmc

onmy

nmxnm

w η


TM Modes

What is wavelength of the phase velocity of the guided mode?


TM Modes


nmz

nmpv

βω

=


TM Modes


222

−

−

==

bm

an

v nmz

nmp

ππµεω

ωβω


TM Modes


2222 1

−

=

−

−

==

ff

c

bm

an

vnm

c

nmz

nmp

ππµεω

ωβω

Summary of TM modesPlane waves in the dielectric medium

Inside the waveguide

εµη /=PW

µεωβ =PW

2

1

−=

ffc

PWTM ηη

2

1

−

=

ffc

PWλλ

2

1

−

=

ffc

PW

p

β

ων

2

1

−=

ffc

PWββ

fc

PW =λ

cfPWp ==== µελβων /1/

Waveguide Types - University of Delawaremirotzni/ELEG413/WaveguidesI.pdf · Rectangular Waveguides....

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Transcript of Waveguide Types - University of Delawaremirotzni/ELEG413/WaveguidesI.pdf · Rectangular Waveguides....