EELE 3332 Electromagnetic II Chapter 12 Waveguidessite.iugaza.edu.ps/tskaik/files/EMII_Chapter_12.pdf · 2012. 5. 18. · EELE 3332 – Electromagnetic II Chapter 12 Waveguides Islamic

EELE 3332 – Electromagnetic II

Chapter 12

Waveguides

Islamic University of Gaza

Electrical Engineering Department

Dr. Talal Skaik

2012 1

2

Waveguides

Waveguides are used at high frequencies since they have larger

bandwidth and lower signal attenuation than transmission lines.

Waveguides are used at high power applications.

Waveguides can operate above certain frequencies. (act as high

pass filter).

Normally circular or rectangular.

3

Waveguides

4

Waveguides

Dr. Talal Skaik 2012 IUG

Rectangular waveguide Waveguide to coax adapter

E-tee Waveguide bends

5

Waveguide Filter

6

Transmission Lines, Waveguides - Comparison

7

Transmission Lines, Waveguides - Comparison

8

12.2 Rectangular Waveguides

Assume a rectangular waveguide filled with lossless dielectric

material and walls of perfect conductor, Maxwell equations in

phasor form become,

2 2

2 2

E E 0 where

H H 0

s s

s s

kk

k

9

Rectangular Waveguides

2 2

2 2 22

2 2 2

2

Applying on z-component:

0

0

Solving by method of Separation of Variables:

( , , ) ( ) ( ) ( )

from where we obtain:

zs zs

zs zs zszs

z

'' '' ''

E k E

E E Ek E

x y z

E x y z X x Y y Z z

X Y Zk

X Y Z

1 2

3 4

cos sin

cos sin

( , , ) ( ) ( ) ( ) ( )

x x

y y

zs

X(x) c k x c k x

Y(y) c k y c k y

E x y z X x Y y Z z Z z c

5 6

1 2 3 4 5 6

1 2 3 4

1

cos sin cos sin

Assume wave propagates along waveguide in direction:

cos sin cos sin

Similarly for the magnetic field,

cos

z z

z z

zs x x y y

z

zs x x y y

zs

e c e

E c k x c k x c k y c k y c e c e

z

E A k x A k x A k y A k y e

H B

2 3 4sin cos sin zx x y yk x B k x B k y B k y e 10

Rectangular Waveguides

11

Other Components

From Maxwell’s equations, we can determine the other components Ex , Ey , Hx , Hy .

2 2

2 2

2 2

2 2

2 2 2 2 2

zs zsxs

zs zsys

zs zsxs

zs zsys

x y

E HjE

h x h y

E HjE

h y h x

E HjH

h y h x

E HjH

h x h y

where

h k k k

*So once we know

Ez and Hz, we can

find all the other

fields.

From these equations we notice that there are different field patterns,

each of these field patterns is called a mode.

• Ezs=Hzs=0 (TEM mode): transverse electromagnetic mode. Both E

and H are transverse to the direction of propagation. From previous

equations we notice that all field components vanish for Ezs=Hzs=0.

→Rectangular waveguide can’t support TEM mode.

• Ezs=0, Hzs≠0 (TE modes) transverse electric

The electric field is transverse to the direction of propagation.

• Ezs ≠ 0, Hzs= 0 (TM modes) transverse magnetic

The magnetic field is transverse to the direction of propagation.

• Ezs ≠ 0, Hzs ≠ 0 (HE modes) hybrid modes

All components exist. 12

Modes of Propagation

13

Transverse Magnetic (TM) mode

1 2 3 40, cos sin cos sin

0 at 0 (bottom and top walls)

0 at 0 (left and right walls)

Applying boundary conditio

Boundar

ns at ( 0 and

y

Conditio s

0

n

z

z zs x x y y

zs

zs

H E A k x A k x A k y A k y e

E y ,b

E x ,a

y x

zs 1 3

0 0 2 4

zs

) to E 0

sin sin ( )

Applying boundary conditions at ( and ) to E

sin 0, sin 0 , This implies that :

, 1, 2,3,...

,

z

zs x y

x y

x

y

A A

E E k x k y e E A A

y b x a

k a k b

k a m m

k b n n

0

1,2,3,...

,

sin sin

x y

z

zs

m nor k k

a b

m x n yE E e

a b

Tangential components are continuous

14


• Other components are

2

2

2

2

zsx

zsy

zsx

zsy

EE

h x

EE

h y

EjH

h y

EjH

h x

sin sin , 0zzs o zsm n

E E x y e Ha b

2

2

2

2

cos sin

sin cos

sin cos

cos sin

z

xs o

z

ys o

z

xs o

z

ys o

m m x n yE E e

h a a b

n m x n yE E e

h b a b

j n m x n yH E e

h b a b

j m m x n yH E e

h a a b

2 2

2 2 2 x ym n

where h k ka b

15


2 2

2 2

2

2 2

2

Propagation constant: ,

,

h k

m nh k

a b

m n

a b

•Each set of integers m and n gives a different field pattern or mode.

•Integer m equals the number of half cycle variations in the x-direction.

•Integer n is the number of half cycle variations in the y-direction.

•Note that for the TM mode, if n or m is zero, all fields are zero. Hence,

TM11 is the lowest order mode of all the TMmn modes.

16

Example: Field configuration for TM21 mode

17

Transverse Magnetic (TM) mode 2 2

2m n

a b

2 2

2

2 2

,

then 0

1 1or

2

No propagation takes pla

The cuttoff occurs when

ce at this frequenc

:

y

c

cm n

m nj

a b

m nf

a b

2 2

2When and 0

No wave propagation at all. (everything is attenuated)

So when

Evanescent m

, all field components will decay exponantially

odes :

.c

m n

a b

f f

18


2 2

2 and 0

This is the case we are interested since is when the wave

is allowed to travel t

Propaga

hrough the guide.

, at a given operating fre

tion occurs

que

whe

nc

n

m nj

a b

So

y f, only those modes with

will propagate.f fc

fc,mn

attenuation Propagation

of mode mn

The cutoff frequency is the

frequency below which attenuation

occurs and above which propagation

takes place. (High Pass)

2 2

,

2 2

2 2

2

:

1 1

2

' 1 , where '

2

T can be written in term

The cutoff Frequency is

he phase co s of asnst t :an

cm n

cmn

c

m nf

a b

u m nor f u

a b

f

m n

a b

2 2

2

2

1

' 1 , where ' / 'c

m n

a b

fk u

f

19


20


2 2

2 2

2 2 2 ' ', but ' , '

'' 1 1

:- (varies with freq

The gu

uency)

1 ' 1

ide wavelength is:

Intrinsic Impedance

, w

g g

c c

x c cTM TM

y

u

ff f

f f

E f f

H f f

' /

', ', ', and ' are parameters for unguided wave propagating

in the same dielectric medium ( , ) unbounded by the waveguide.

(i.e. waveguide removed and entire space is filled with diele

here

u

ctric.)

21

Transverse Electric (TE) modes

1 2 3 4

2

0, cos sin cos sin

0 at 0 (bottom and top walls)

0 at 0 (le

Bounda

ft and right walls)

0 at

ry

Conditions

0

z

z zs x x y y

xs

ys

zs zsxs

E H B k x B k x B k y B k y e

E y ,b

E x ,a

j H HE y ,b

h y y

2

0 1 3

0 at 0

From this we conclude

cos cos ( = )

zs zsys

z

zs o

j H HE x ,a

h x x

m x nH H y e H B B

a b

Tangential components are continuous

Other components are

22

Transverse Electric (TE) modes

2

2

2

2

cos sin

sin cos

sin cos

cos sin

z

xs o

z

ys o

z

xs o

z

ys o

j n m x n yE H e

h b a b

j m m x n yE H e

h a a b

j m m x n yH H e

h a a b

j n m x n yH H e

h b a b

0, cos cos zz zs om n

E H H x y ea b

2

2

2

2

zxs

zys

zxs

zys

HjE

h y

HjE

h x

HH

h x

HH

h y

23

Example: Field configuration for TE32 mode

• The cutoff frequency is the same expression as for the TM

mode

• For TE modes, (m,n) may be (0,1) or (1,0) but not (0,0). Both

m and n cannot be zero at the same time because this will force

the field components to vanish.

• Hence, the lowest mode can be TE10 or TE01 depending on the

values of a and b.

• It is standard practice to have a>b, thus TE10 is the lowest

mode.

24

22

2

'

b

n

a

muf

mnc

TE modes - Cuttoff

10 '/ 2cf u a

25

TE modes

• The dominant mode is the mode with lowest cutoff frequency.

The cutoff frequency of the TE10 mode is lower than that of TM11

mode. Hence, TE10 is the dominant mode.

If more than one mode is propagating, the waveguide is overmoded.

Single mode propagation is highly desirable to reduce dispersion.

This occurs between cutoff frequency for TE10 mode and cuttoff

frequency of next higher mode.

26

2

2 2

he phase constant is the same as TM mode:

The intrinsic impedance of the TE mod

T

' 1 , where ' / '

1 ' , w ' /

1 1

e is:

c

xTE TE

yc c

fu

f

Ehere

H f f

f f

2

2Note that ' ' 1 cTE TM TMf

f

TE modes

10

For TE mode, cos cos

For TE mode, cos

In the time domain: =Re

cos cos

,

j z

zs o

j z

zs o

j t

z zs

z o

y

m nH H x y e

a b

xH H e

a

H H e

or

xH H t z

a

Similarly

E

0

0

sin sin

sin sin

0

x

z x y

a xH t z

a

a xH H t z

a

E E H

27

TE10 mode

Variation of the field components with x for TE10 mode.

28

TE10 mode

Field lines for TE10 mode

29

TE/TM modes Wave in the dielectric medium Inside the waveguide

/'

'/' u

2

1

'

f

fc

TE

2

'

1 cf

f

/

1'

2

f

f

u

c

p

2

1'

f

fc

fu /''

/1'/' fu

2

, ' 1 cTMf

f

2 2

2 2 2 2

2 2

The cutoff frequency is given by:

' 1 , u'=

2 2

Hence

4 4 2.5 10 1 10

cmn

r r

cmn

u m n c cf

a b

c m n c m nf

a b

30

Example 12.1

• Example: A rectangular waveguide with dimensions a=2.5 cm,

b=1 cm is to operate below 15.1 GHz. How many TE and TM

modes can the waveguide transmit if the guide is filled with a

medium characterized by σ=0, ε=4 ε0, µr=1? Calculate the

cutoff frequencies of the modes.

2 2

2 2

01 01

02 02

03 03

10 10

20 10

4 2.5 10 1 10

For TE mode ( =0, =1), 7.5 GHz

For TE 15 GHz

For TE 22.5 GHz

For TE 3 GHz

For TE 6 GH

cmn

c m nf

m n fc

fc

fc

fc

fc

30 10

40 10

50 10

60 10

z

For TE 9 GHz

For TE 12 GHz

For TE 15 GHz

For TE 18 GHz

fc

fc

fc

fc

31

Example 12.1 - solution

2 2

2 2

11 11 11

21 21 21

31 31 31

4 2.5 10 1 10

For TE , TM modes , 8.078 GHz



cmn

c m nf

fc

fc

fc

41 41 41

12 12 12

Modes with cutoff frequencies les


s than or equal 15.1 GHz

will be tran

For TE

smitt

, TM modes , 15

ed. (11 TE modes

.3

an

GHz

d 4 TM

modes)

fc

fc

32


33


Cutoff frequencies of rectangular waveguide with a 2.5b; for Example 12.1.

34

Example 12.3

• Example: in a rectangular waveguide for which a=1.5 cm,

b=0.8 cm, σ=0, µ=µ0, ε=4ε0.

• Determine:

• (a) the mode of operation.

• (b) the cutoff frequency

• (c) the phase constant β.

• (d) the propagation constant γ.

• (e) the intrinsic wave impedance η.

113

2sin cos sin 10 A/mxx y

H t za b

13 13 13

2 2

2 2

13 2 2

( ) the guide is operating at TM or TE . Suppose we choose TM .

' 1( ) , '

2 2

1 3 28.57 GHz

4 1.5 10 0.8 10

(c) ' 1

cmn

r r

c

c

a

u m n c cb f u

a b

cHence f

f

f

13

2 2 2

11

211

8

2 2

TM

1 1

2 10 50 GHz

10 4 28.571 1718.81 rad/m

3 10 50

( ) = 1718.81/ m

377 28.57( ) ' 1 1 154.7

50

rc c

c

r

f f

f c f

f f

d j j

fe

f

35

Example 12.3 - Solution

Documents

EELE 3332 Electromagnetic II Chapter 12 Waveguidessite.iugaza.edu.ps/tskaik/files/EMII_Chapter_12.pdf · 2012. 5. 18. · EELE 3332 – Electromagnetic II Chapter 12 Waveguides Islamic