chapter 5 waveguides and resonators - university of...

5-1

Chapter 5 Chapter 5 Waveguides and ResonatorsWaveguides and Resonators

Dr. Stuart Long

5-2

What is a “waveguide”(or transmission line)?

Structure that transmits electromagnetic waves in such a way that the wave intensity is limited to a

finite cross-sectional area

In this chapter we will focus on three types of waveguides:

Parallel-Plate Waveguides.

Rectangular Waveguides.

Coaxial Lines.

5-3

http://cal-crete.physics.uoc.gr/VLF-sprites/images/waveguide.jpg

5-4

Parallel Plate Waveguide

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5-5

Assume both plates to be perfect conductors.

Assume waveguide to be very large in direction (w>>λ) .

Field vectors have no y-dependence .

Neglect any fringing fields at y=0 and y=w .

Propagation along the direction.

y

ˆ+z

0y

∂=

∂



5-6

y x z z

y x z z

E H H TE E 0

From Maxwell Equations

( , , ) (Transverse Electric)

or

( , , ) (Transverse Magnetic)

H E E TM H 0

=

=


(Sect. 5.1)

5-7

TE Waves in Parallel Plate Waveguides

( )y x z yB.C. at and

The wave

Remem

equation for the case derived from Maxwell's Equations is

T

r: 0

E

be

E H H 0 E 0

, ,

y

x x a ⇒

∂=

∂

= = =

2 22

2 2

E 0

yx zω με

⎛ ⎞∂ ∂+ + =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠


5-8

y

y 0

2 2 2 2

For ppg. in direction

(satisfies B.C a )

with

To satisfy B.C. ( at

( is any

ˆ+

t 0

E 0)

E sin( )

zjk zx

x z

x

x

x a

m

E k x e

k k k

k a m

ω μ ε

π

− =

= =

=

+ = =

=

z

exin ceteger pt 0)


(5.5)

(5.6)

(5.7)


5-9

y 0

1 12 22 2

2

The Electric field can be expressed in another form bysubstituting in for

with

E sin

12

z

x

jk z

z

k

mE x ea

m mka a

π

π λω μ ε ω μ ε

−⎛ ⎞= ⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= − = −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

whe e 2

r kπ

λ =⎛ ⎞⎜ ⎟⎝ ⎠



5-10

12 2

The

guided wave propagates with the phase velocity

is imaginary for

If bec

Note:

1 12

2

or or

z

z

pz

x

k

k

m vk a

m ak ka m

ω λμ ε

πω μ ε λ

−⎡ ⎤⎛ ⎞= = −⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

< < >

omes imaginary, the wave will attenuate exponentially,

and the velocity for the guided wave will also become undefined.



5-11

( )m

of modeTE

frequency at which

2 2z

c

cc

k

ma

mfa

πωμ ε

ωπ μ ε

=

= = cutoff frequency

( )m

exponential attenuation when

of modeTE

becomes imaginary

2

c

f fc

am

λ

<

= cutoff wavelength wavelength at which

becomes imaginary zk



5-12

Physical Interpretation TE mode

x

z

y 0

0y

E sin

E2

z

x z x z

jk z

jk x jk z jk x jk z

mE x ea

jE e e

π −

− − −

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎡ ⎤= −⎣ ⎦(5.8)

0

wave trav. in and dir.

coef

ˆ

ˆ

2f jE⎛ ⎞

⎜ ⎟⎝ ⎠

+ x+ z

0

wave trav. in - and dir.

coe

ˆˆ

2ff - jE⎛ ⎞

⎜ ⎟⎝ ⎠

x+ z

5-13

Physical Interpretation of Guided Waves

xmkaπ

=

aπ

2aπ

3aπ

zk

xk

(fig 5.3)

z

For this case

so modes will ppg. but for

3.5

1,2,3 4 , is imaginary .

ka

m m k

π

= ≥

12 2

2z

mk kaπ⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦(5.9)

k

5-14

( ) -y x z y 0

z y z

z 0,

can find

For B.C. at ( )

H , E , E H cos

E ~ H E ~ sin

E

0

zjk zx

x

x a

x

H k xe

k xx

x a

mkaπ

=

=

∂∂

= =

=

⇒

TM Waves in Parallel Plate Waveguides


5-15

( )

z

y 0

2 2 2 2

For ppg. in direction

(satisfies B.C a )

with

To satisfy B.C.

ˆ+

t 0

E 0 ( ) at

(

H cos

zjk zx

x z

x

x

x a

m

H k x e

k k k

mka

ω μ ε

π

− =

= =

=

+ = =

=

z

includinis any integer 0 )

g



5-16

y 0

12 22

2

The field solutions can be expressed in another form bysubstituting in for

wi th

H cos

12

z

x

jk z

z

k

mH x ea

m mka a

π

π λω μ ε ω μ ε

−⎛ ⎞= ⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= − = −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦

12

where 2 kπ

λ =

⎥

⎛ ⎞⎜ ⎟⎝ ⎠



5-17

z y

z 0

x 0

E ~ H

E sin

E cos

since

z

z

jk zx

jk zz

x

k mH x ej a

k mH x ea

πωε

πωε

−

−

∂∂

⎛ ⎞= − ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠



5-18

0

Remember that for waves we can now have .

For

For this case and are both perpendicular to the direction of propagation ( ).

TM

We refer to this case as

0

TM

ˆ

0.0

x

z

m

mk

k kω μ ε

=

⎧ =⎪

=⎨⎪ = =⎩

E Hz

mode (Transverse Electroma T gneEM tic)



5-19

0

y 0

x 0

z

The field solutions for the or mode are given by

or equivalently

TM TEM

H

E

E 0

jk z

jk z

z

z

H e

H eη

−

−

=

=

=

x 0

0y

E

H

jk z

jk z

z

z

E e

E eη

−

−

=

=

TM0 mode


(5.13a)

(5.13b)

5-20

Physical Interpretation TEM mode

x

z

0 0x =0

0

ˆ ˆ ˆ ˆ= = =

ˆ ˆ

(on bottom plate)

=

=

jkz jkz

jkz

E Ee e

E e

η η

ρ ε ε

− −

−

× ×

=

s

s

J H

D Ei i

n x y z

n x

(5.13c)

0

0

ˆ ˆ- = -

(on top plat

e )

jkz

jkzs

En e

E eη

ρ ε

−

−

=

= −

s Jx z

(5.11d)

n

n

s J

s J

5-21

Microstrip Lines

ground plane

dielectricsubstrate

microstrip line


5-22

Microstrip Lines

- 00

20

1 = Re2

1 ˆ ˆ

The time-average Poynting power densit

= R

y can b

e ( )2

ˆ =2

e found b

y:

jkz jkzEE e e

E

η

η

∗

+

⎡ ⎤×⎣ ⎦

⎡ ⎤×⎢ ⎥

⎣ ⎦

S E H

S

S

x y

z

5-23

Microstrip Lines

0

quasi TEM

-fields

k

ε ε

ω με

= 0

ε ε>

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5-24

Rectangular Waveguide

y

z

xa

b

5-25


2 2

2 2

2 2 22z z z

z2 2 2

0

MAX EQN HELMHOLTZ EQN (wave equation)

6 EQNS like

0

H H 0 H H x y z

ω με

ω με

ω με

⇒

+ =

+ =

∂ ∂ ∂+ + + =

∂ ∂ ∂

E E

H H

∇

∇

5-26


2 2 22

z 2 2 2

2 2 2 2

2 22 2

2 2

( , , ) ( ) ( ) ( )1 X 1 Y 1 ZH X Y ZX

For a se

Y Z

- - -

1 X X- X 0

parable solution assume

each term is a constant

X

x y z

x x

x y z x y zx y z

k k k

k kx x

ω με

ω με

∂ ∂ ∂= + + = −

∂ ∂ ∂

= −

∂ ∂= + =

∂

⇒

⇒∂

5-27


1 2

3 4

5 6

z

x y

X sin cos

Y sin cos

Z

H X Y

Using Max. eqns. can solve for transverse fields (E ,

Z

E

x x

y y

jk z jk zz z

c k x c k x

c k y c k y

c e c e− +

= +

= +

= +

=

x y

z z

, , )

in terms of longtitudinal on

H H

(Ee ,Hs )

5-28


z z z zx y2 2 2 2

z z z zx y2 2 2 2

2 2 2 2 2

;

;

wher

E H E HE E

H E H EH H

e

z z

c c c c

z z

c c c c

c x y z

jk j jk jx y y xk k k k

jk j jk jx y y xk k k k

k k k k

ωμ ωμ

ωε ωε

ω με

− ∂ ∂ − ∂ ∂= − = +

∂ ∂ ∂ ∂

− ∂ ∂ − ∂ ∂= + = −

∂ ∂ ∂ ∂

= + = −

5-29


z z

z z

z z

Special cases

1) when ; Transverse Electric mode ( )

2) when ; Transverse Magnetic mode ( )

If all comp. Transverse Electric

E 0 H 0 TE

H 0 E 0 TM

E H 0M

0

= ≠

= ≠

= = =

⇒

⇒

⇒agnetic mode ( )

can not TEM

exist

5-30

Rectangular Waveguide TE-Case (Ez=0)

y

zxa

b

zy

y 1 2 1

zx

x 3 4

tangential 0 0, y 0,

HE ~ 0 0, )

B.C.

E ~ cos sin 0

at and

Side walls (at

and

Top and bot

HE ~ 0 0, )

E ~ c

tom walls (

os si

at

n

x x x x x

y y y

x a b

x ax

mk c k x k c k x c ka

y by

k c k y k c

π

= =

∂= =

∂

− = =

∂= =

∂

−

→

⇒

E

3

5 2 4 0 z

z 0

0

H X Y Z

and

H cos

Let

cos z

y y

jk z

nk y c kb

c c c H

m x n yH ea b

π

π π −

= =

= =

=

⇒

⇒

5-31y

zxa

b

2 2

2 22

2

2 22

-

=

Propagation constant

where

substituing in for

propagation ( real) for

ex

- -

Note

ponen

:

t

z c

c

c

z

z c

k k k

m nka b

k

m nka b

k k k

π π

π πω με

=

⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

>

ial attenuation ( imaginary) for z ck k k<


(5.19)

5-32y

zxa

b

( )mn

12 2 2

of mT odE e

frequency at which

1 2 2

z

cc

k

m nfa b

ω π ππ π μ ε

⎡ ⎤⎛ ⎞ ⎛ ⎞= = +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

cutoff frequency

2 2

exponential attenuation when

becomes imaginary

2

2 = -

c

gz c

f f

k k k

π πλ

<

= guide wav( )mnof modeTE

wavelength at which

zk

elength

mn

becomes imaginary

defines a doubly infinite set of modes TE


(5.21)

(5.20)

5-33y

zxa

b

z 0

11

10

Same , , , as

For ; and lowest order (smallest ) is

The lowest ord

E sin sin

TE

TM m 0 n 0 TM

er of all is called for rectangular wavegui

Ed

Tes

zjk z

z c c g

c

m x n yE ea b

k k f

f

π π

λ

−=

≠ ≠

DOMINANT MODE ( possible)

(there exist a frequency range were only it can propaTE

go M n

ate)

Rectangular Waveguide TM-Case (Hz=0)

(5.18)

5-34

z 0

z zx y 02 2

z zx 0 y2 2

H cos

H HE =0 E sin

H HH sin H 0

;

;

z

z

z

jk z

jk z

cc c

jk zz z z

cc c

xH ea

j j j xH ey x k ak k

jk jk x jkH ex k a yk k

π

ωμ ωμ ωμ π

π

−

−

−

=

∂ ∂= − = = −

∂ ∂

− ∂ − ∂= = = =

∂ ∂

Rectangular Waveguide TE10 mode (m=1 , n=0)

y

zxa

b

5-35

Rectangular Waveguide TE10 mode (m=1 , n=0)

22

2

2

12 ; ; 2

2

12

z

c c c

gz

k ka

a f kaa

k

a

π

πλμε

π λλλ

⎛ ⎞= − ⎜ ⎟⎝ ⎠

= = =

= =⎛ ⎞− ⎜ ⎟⎝ ⎠

Note: these are the eqns. found in slide 5-32 and 5-33 with m=1 and n=0

y

zxa

b

5-36

Physical InterpretationTE10 mode (m=1 , n=0)

y

zxa

b

5-37

8 1 0 0 0 0 0 0

0 .2 7 5

k z

ω

10cω

11cω

θ

zk ω με=TE10

TM11TE11

2 2z ck k k= −

tanpz

vkω θ= = (function of frequency )

where

ω versus kz graph

gz

vkω∂

=∂

and

5-38y

zxa

b

Design of Practical Rectangular Waveguide

(Example 5.4)

TE10

TE01

TE20

12 2 21

2

1 12

1 12

1 22

c

c

c

c

m nfa b

fa

fb

fa

π ππ μ ε

με

με

με

⎡ ⎤⎛ ⎞ ⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

5-39y

zxa

b


(Cont. e.g 5.4)

for for 2 2a ab b> <

a

b ab

1 2 3

2for ab =

10TE11

11

TETM

01

20

TETM

TE10

c

c

ff

10TE

01TE

20TM

01TE

10TE 20TM

5-40

[ ]10

220

0 0

20

TE

1 = Re2

ˆ

Transmitting Power in wavegui

= sin

de

2

ˆ

mode

4

z

a b

z

E k xa

P dxdy

E abkP

πωμ

ωμ

∗⎡ ⎤×⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

=

=

∫ ∫ i

S E H

S

S

z

z

Time-average Poynting vector

y

zxa

b


( Example 5.3)

Total transmitted power

5-41

204

zE abkPωμ

=

: Do not want for bandwith

Do not want small for power handling

Usual case

Not

2

2

e ab

b

ab

>

y

zxa

b


(Cont. e.g 5.3)

5-42

TE10

TE20

TE01

6.

X-Band Waveguide (8-12 GHz)

GHz

GHz

G

55

H

13.1

1 . z4 7

c

c

c

f

f

f

=

=

=

y

zxa

b

0.9a ′′=

0.4b ′′=

a


5-43y

zxa

b

Rectangular Waveguide TE10 mode

z 0

y 0

x 0

H cos

E sin

H sin

z

z

z

jk z

jk z

jk zz

xH ea

j xH ea a

jk xH ea a

π

ωμ ππ

ππ

−

−

−

=

= −

=

5-44y

zxa

b

y 0

0 0

x 0

0z

or

whe e

r

E sin

H sin

H cos

z

z

z

jk z

jk zz

jk z

xE ea

jE Ha

k xE ea

E xa ej a

π

ωμπ

πωμ

ππ

ωμ

−

−

−

=

= −

−=

=−

(5.23)

Rectangular Waveguide TE10 mode

5-45

Example (Example 5.5 & 5.6)

8 8

8 89

2

3 10 3

Design a Rectangul

10 <

ar waveguide with 10 GHz ( =3cm) at Mid-Band and

le

2

1 3 10 3 1010 10 2.252 2

t

1.1252

cm

c

BW

m

ab

fa a

aa a

ab b

λ =

× ×<

⎡ ⎤× ×× = + =⎢

⎣

= =⇒

⎦

⇒

⎥ ⇒

5-46

(Cont. e.g 5.5 & 5.6)

[ ]

( )[ ]

20

TE

TE

6

5max

10 7

TE 22

4

2 10

2 10

where

Vm

Vtake (safety fact

(2 10 )(4 10 )

or of 10)m

Max power handl

505

ing

.8

z

BDair

E abPZ

Zk

E

E

Zk a

ωμ

π π

π

−

=

=

= ×

= ×

× ×

⇒

Ω

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎢⎣

=−

⎥⎦

= Ω

Example

5-47

( ) ( )

[ ]

-1 -

2

5

max

1

2

2

2 1 2 2 1 3 4.53

1.561 156.1

note:

(2 10 ) (0.0225)(0.01125) 5.004 4(50

cm m

KW5.8)

z

z

ak

k

P

π λ πλ

⎡ ⎤ ⎡ ⎤⎣ ⎦

− −= =

= =

×=

⎣ ⎦

=

(Cont. e.g 5.5 & 5.6)Example

5-48

Coaxial Transmission Lines

b

aε

5-49

Cylindrical Coordinates

φ

z

y

x

ρ

z

ρ

φz

ˆˆ ˆ×ρ φ = z

( fig. 5.16)

5-50

Coaxial Transmission Lines

b

aε

Maxwell Eqns. V ˆ

V ˆj

z

kz

jke

eηρ

ρ

−

−⇒

=

=

0

0 H

E ρ

φ

k ω με

μηε

=

=

( 5.48a)

( 5.48b)

( 5.49a)

( 5.49b)

5-51

The “Del” Operator in Cylindrical Coordinates

( ) ( )1 1 zA A A

zρ φρ ρ

ρ ρ ρ φ

∂ ∂ ∂= + +

∂ ∂ ∂Ai∇

( )1 1ˆˆ ˆ- z zAA A AA A

z zφφ ρ ρρ

ρ φ ρ ρ ρ φ

⎛ ⎞∂∂ ∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂⎜ ⎟× − + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

A = z∇ ρ φ

φ

z

y

x

ρ

z

ρ

ˆφz

ˆˆ ˆ×ρ φ = z

( f i g . 5 .1 6 )

(5.44)

(5.45)

5-52

Currents on Coaxial Lines

b

aε

( 5.50a)

( 5.50b)

=a0 0

=a

20

0 =a

ˆˆ ˆ = =

2

ˆ

I =

= jkz jkz

jkz

V Ve ea

Va e

ρρ

π

ρ

ηρ η

πφη

− −

−

×

∂ =

×

∫

s

sJ

HJ zn ρ φ

n sJ

H

5-53

Short-Circuited Coax

(Fig. 5.18)

( 5.52a)0 1

0 1

1 0

for 0

both Incident Fields and Reflected Fields exist

B.C at

ˆ=

ˆ=

0 tangential 0 0

jkz jkz

jkz jkz

z

V Ve e

V Ve e

z E V Vρ

ρ ρ

ρη ρη

− +

− +

<

⎛ ⎞+⎜ ⎟

⎝ ⎠

⎛ ⎞−⎜ ⎟

⎝ ⎠

⇒ ⇒ ⇒= = = = −

E

H

E

ρ

φ ( 5.52b)

0z =

z

5-54

Short-Circuited Coax

(Fig. 5.18)( 5.53a) ( )0 0

0

2ˆ ˆ= - sin

Similarl

2ˆ= c

y

os

jkz jkzV jVe e kz

V kz

ρ ρ

ηρ

− + ⎛ ⎞− = ⎜ ⎟

⎝ ⎠ E

H

ρ ρ

φ( 5.53b)

0z =

z

To measure standing wave pattern cut longitudinal slot in outer conductor for moveable proble.

: totally in direction so totally i

ˆ

ˆNoten direction

sH Jz

φ

5-55

Example (Example 5.8)

20

2

Calculate the transmitted power along a coaxial line.

V ˆ

V ˆ

1 1 V V Vˆˆ ˆ= Re = R

= 2

e

2

jkz

jkz

jkz jkz

e

e

e e

ρ

ηρ

ρ ηρ ηρ

−

−

∗ −

=

=

⎧ ⎫⎡ ⎤× ×⎨ ⎬⎣ ⎦ ⎩ ⎭

0

0

0 0

E

H

S E H z

ρ

φ

ρ φ

20 V ln bP

aπ

η⎛ ⎞= ⎜ ⎟⎝ ⎠

b

aε

TEM mode in a coaxial line

5-56

Transmission Lines

b

aε

2 or more conductors , (TEM possible ) (along with TE and TM) Coaxial

two-wire (shielded)

5-57

Transmission Lines

2 or more conductors , (TEM possible ) (along with TE and TM) Parallel

Plate

microstrip


stripline

Conductor

Dielectric

5-58

Waveguides

Elliptical

Ridged

Rectangular

Optical Fiber

Dielectric Slab

Circular

http://optics.org

chapter 5 waveguides and resonators - university of...

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