Chapter 5 Waveguides and Resonators - University of Houstoncourses.egr.uh.edu/ECE/ECE3317-02053/docs/3317Chap5Lect... · 2014-08-20 · Waveguides and Resonators ECE 3317. Dr. Stuart

Chapter 5 Waveguides and Resonators

ECE 3317

Dr. Stuart Long

5-1

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:

1. Parallel-Plate Waveguides

2. Rectangular Waveguides

3. Coaxial Lines

What is a “waveguide" (or

transmission line) ?

5-2

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

5-3

www.amanogawa.com/transmission.html

Parallel Plate Waveguide

5-4


ˆ (w>> As

)sume both

plates to be perfect conductors Assume waveguide to be very large in direction

Fi 0

eld vectors have no -dependence,

Neglect any fringing fields at and Pr

y

y

y = 0 y = w

•

•

∂• =

∂

•

•

y

λ

opagation along the direct ˆ ionz+


5-5

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)



5-6

( )y x z yB.C. at and

0 Remember:

, ,

The wave equation for the case derived from Maxwell's Equations is

E H H 0 E 0

TE

y

x x a ⇒

∂=

∂

= = =

2 22

2 2

E 0

yx zω µε

∂ ∂+ + = ∂ ∂


TE Waves in Parallel Plate Waveguides

5-7

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)



5-8

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 =



5-9

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.



5-10

( )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



5-11

Physical Interpretation TE mode

x

z

(5.8)

0

wave trav. in and dir.

coef

ˆ

ˆ

2f jE

+ x+ z

0

wave trav. in - and dir.

coe

ˆˆ

2

ff - jE

x+ z

5-12

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

M=1

M=3

M=2

5-13

( ) -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

5-14

( )

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

includin is any integer 0 )

g



5-15

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 =



5-16

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

−

−

∂∂

= −

=



5-17

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)



5-18

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

−

−

=

=

TM0Mode


(5.13a)

(5.13b)

5-19

Physical Interpretation TEM mode

x

z

0 0x=0

0

ˆ ˆ ˆ ˆ= = =

ˆ ˆ

(on bottom p

= =

late) jkz jkz

jkz

E Ee e

E e

η η

ρ ε ε

− −

−

× ×

=

s

s

J H

D E

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

for TE and TM modes has both and components.

5-20

dielectricsubstrate

ground plane

microstrip line


Microstrip Lines

http://3.bp.blogspot.com/_QOpBRAPC2rY/TGW9OMdLRnI/AAAAAAAAAC8/2y2RGiyuRQ0/s1600/Picture2.jpg

substrate

5-21

- 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


Microstrip Lines

5-22

Microstrip Lines

0

quasi TEM

-fields

k

ε ε

ω µε

=

0

ε ε>

www.amanogawa.com/transmission.html www.amanogawa.com/transmission.html

5-23

y

z

xa

b

Rectangular Waveguide

http://t1.gstatic.com/images?q=tbn:ANd9GcShyNLomTqQz3wl_U9PM7jxslFsPkjS4624DvinKOuUhHrSO37h

5-24

y

zxa

b

2 2

2 2

2 2 22z z z

z2 2 2

(wave equation)MAX equations HELMHOLTZ equation

6 equation like

0

0

H H 0 H Hx y z

ω µε

ω µε

ω µε

+ =

+ =

∂ ∂ ∂+ + + =

∂

⇒

∂ ∂

E E

H H

∇

∇


5-25

y

zxa

b

z2 2 2

22 2 2

2 2 2 2

22

2

22

2

( , , ) ( ) ( ) ( )H X Y Z1 X 1 Y 1 ZX

For a se

Y Z

- - -

1 X -X

X X 0

parable solution assume

each term is a constant x y z

x

x

x y z x y z

x y z

k k k

kx

kx

ω µε

ω µε

=

∂ ∂ ∂+ + = −

∂ ∂ ∂

= −

∂=

∂∂

+

⇒

⇒ =∂


5-26

y

zxa

b

x y x y

z z

Using Maxell's equations we can solve for the

transverse fields ( , , , )in terms of the

longtitudinal fields

E E H H

E ,H( )


1 2

3 4

5 6

z

X sin cos

Y sin cos

Z

H X Y Z

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− +

= +

= +

= +

=

5-27

y

zxa

b

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

− ∂ ∂ − ∂ ∂= − = +

∂ ∂ ∂ ∂

− ∂ ∂ − ∂ ∂= + = −

∂ ∂ ∂ ∂

= + = −


5-28

Rectangular Waveguide y

zxa

b

z

zz

z z

z when ; Transverse Electric mode ( )

when ; Transverse Magnetic mode ( )

Special cases

If all comp. Transverse

:

1)

E

E 0

2

l

H 0

E 0

ectric

0

E 0 0

) H

H

≠

≠

=

⇒=

= ⇒

=

=

⇒

TE

TM

Magnetic mode ( ) can not

TEMexist

5-29

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 −

= =

= =

=

⇒

⇒

5-30

y

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)


Case (Ez=0)

5-31

y

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( )mn of modeTE

wavelength at which

zk

elength

mn

becomes imaginary

defines a doubly infinite set of modes TE

(5.21)

(5.20)


Case (Ez=0)

5-32

y

zxa

b

(5.18)

Rectangular Waveguide TM:

Case (Hz=0)

5-33

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 jkxH ex k a yk k

π

ωµ ωµ ωµ π

π

−

−

−

=

∂ ∂= − = = −

∂ ∂

− ∂ − ∂= = = =

∂ ∂

Rectangular Waveguide TE10

mode (m=1 , n=0)

y

zxa

b

5-34

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

z

xa

b


mode (m=1 , n=0)

5-35

Physical Interpretation TE10mode (m=1 , n=0)

y

zxa

b

5-36

81000000

0.275

k z

ω

10cω

11cω zk ω µε=TE10

TM11TE11

2 2z ck k k= −

(function of frequency )

where

ω versus kz graph

gz

vkω∂

=∂and

5-37

y

z

xa

b

Design of Practical Rectangular Waveguide

Example 5.4

TE10

TE01

TE20

1 12

1 12

1 22

c

c

c

fa

fb

fa

µε

µε

µε

=

=

=

1/ 22 2

c1 mπ nπ f = +

a b2π μ ε

5-38

y

zxa

b

(Cont. e.g 5.4)

for for 2 2a ab b> <

a

b ab

1 2 3

2for ab =

10TE

TE10

c

c

ff

10TE

01TE 01TE

10TE


5-39

Time-average Poynting vector

y

zxa

b

( Example 5.3)

Total transmitted power


5-40

20

Do not want for bandwith

Do not want small for power handling

Usual

No

case

te:

2

4

z

ab

E abkb Pωµ

>

=

ab =2

y

zxa

b

(Cont. e.g 5.3)


5-41

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

0.4′′

0.9′′

0.4′′

a


Historical standard design

5-42

y

z

xa

b


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

−

−

−

=

= −

=

HZ

HX

Ey

5-43

y

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)


mode

5-44

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 =

× ×<

× ×× = + =

= =⇒

⇒

⇒

5-45

(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 5-46

( ) ( )

[ ]

-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 5-47

Rectangular Waveguide Simulation

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

φ

z

y

x

ρ

z

ρ

φz

ˆˆ ˆ×ρ φ = z

( fig. 5.16)

(5.44)

(5.45)

5-52

Currents on Coaxial Lines

b

aε

( 5.50a)

( 5.50b)

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

Negative sign comes from same place as in reflected H-field in Chapter 4

5-54

Short-Circuited Coax

(Fig. 5.18)

( 5.53a)( )0 0

0Similarly,

2ˆ ˆ= - sin

2ˆ= cos

jkz jkzV jVe e kz

V kz

ρ ρ

ηρ

− + − =

E

H

ρ ρ

φ ( 5.53b)

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

: totally in direction so totally i

ˆˆ

Noten direction

sH Jz

φ

0z =

z

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

Coaxial

two-wire (shielded

)

three-wire (shielded)

2 or more conductors TEM possible

along with TE and TM

5-57

Transmission Lines

2 or more conductors TEM possible

along with TE and TMParallel Plate

Microstrip Stripline

Conductor Dielectric

5-58

Waveguides

Elliptical

Ridged

Rectangular

Optical FiberDielectric Slab

Circular

http://optics.org

5-59

CircularElliptical

Rectangular Bends

5-60

Documents

Chapter 5 Waveguides and Resonators - University of Houstoncourses.egr.uh.edu/ECE/ECE3317-02053/docs/3317Chap5Lect... · 2014-08-20 · Waveguides and Resonators ECE 3317. Dr. Stuart