Lecture Notes Waveguides

8/13/2019 Lecture Notes Waveguides

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Hon Tat Hui Waveguides

NUS/ECE EE4101

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General Field Expression inside a Waveguide

( ) zeyx = ,~EE

EE 22

2

=

z

Then

Helmholtzs equations:

022 =+ EE k

=k

022 =+ HH k

Transverse directions: (x,y) or (r, )

Longitudinal direction: z (propagation direction)

In general


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Method of Solution:

Express transverse field componentsEx,Ey in

terms of longitudinal field componentEz

Obtain solution for the longitudinal field

Ezfrom the wave equation

ObtainEx,Ey fromEz

Step 1

Step 2

Step 3


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In rectangular coordinates:

022

2

2

2

2

2

=

+

+

+

Ekzyx

022

22 =

+

+ Ek

z

xy

0222 =++ EE kxy

0222 =++ Ekxy

Similarly,

0222 =++ HH kxy

(1a)

(1b)

022 =+ EE k


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HE j=

H

zyx

j

EEE zyx zyx

=

xyz

x

yz HjE

y

EHj

z

E

y

E ~~~

=+

=

yz

xyzx Hj

x

EEHj

x

E

z

E ~~

~ =

=

zxy

zxy Hj

y

E

x

EHj

y

E

x

E ~~~

=

=

(2a)

(2b)

(2c)

Note that:( ) ( )

( ) ( )

zyxi

eyxHzyxH

eyxEzyxE

z

ii

z

ii

,,

,~

,,

,~,,

=

=

=


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Similarly from

EH j=

xyz EjH

y

H ~~~

=+

(3a)

yz

x Ejx

HH

~~~=

zxy Ej

yH

xH ~

~~

=

(3b)

(3c)

We have


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Finally from equation sets in (2) and (3), we have:

+

=y

Ej

x

H

kH zzx

~~1~

22 (4a)

+

+

=x

Ej

y

H

kH zzy

~~1~22

(4b)

+

+= yH

jx

E

kE zzx

~~1~

22 (4c)

+

=

x

Hj

y

E

k

E zzy

~~1~

22 (4d)

Hence, we can solve the scalar Helmholtzs equations forEzand

Hz, and use the above formulas to determine the other components.

~~


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Waveguide Mode Classification

It is convenient to first classify waveguide modes as to

whetherEz

orHz

exists according to:

TEM: Ez= 0 Hz= 0

TE: Ez= 0 Hz 0TM: Ez 0 Hz= 0

TEM = Transverse ElectroMagneticTE = Transverse Electric

TM = Transverse Magnetic


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(1) TEM Modes:

0~~

==== zzzz HEHE

From the equations in (4), for the existence of non-trivial solutions,the denominators must be zero also. That is,

022 =+ k

jjk ==

1==k

upPhase velocity:

Propagation constants:


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From the equations in (4), the field components take an indefinitemathematical form of 0/0, whose definite values have to be

determined by boundary conditions. In general, we can write:

jkz

xxxx eEEEE

== 00 ,~

jkz

yyyy eEEEE == 00 ,

~

jkzxxxx eHHHH

== 00 ,~

jkz

yyyy eHHHH == 00 ,

~

The relations between Ex, Ey, Hx, and Hy can be further obtained

from the equations in (2) and (3), as shown below.

2 0xy =E

2 0xy =H


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Wave impedance:

=====

j

H

E

H

EZ

y

x

y

xTEM ~

~

Wave impedance = Intrinsic impedance of the medium

(5a)

TEM

x

y

x

yZ

j

H

E

H

E====

~

~

(5b)

Combining (5a) & (5b),

yZ

Ex

Z

EyHxH

TEM

x

TEM

y

yx +=+

EzH = 1

TEMZ

Therefore


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(2) TE and TM Modes:

TE and TM modes in general exist in hollow waveguides such as

rectangular waveguides and circular waveguides. They will be

studied in the context of these waveguides.

TEM modes can only exist in two-conductor waveguides such as

two-wire transmission lines, co-axial lines, parallel-plate

waveguides, etc, but not in single-conductor waveguides such as

rectangular waveguides and circular waveguides. This is because

either longitudinal field components or longitudinal currents arerequired to support the transverse magnetic field components HxandHy which form close loops in the transverse plane. There are

no longitudinal currents (not longitudinal surface currents) inside

hollow waveguides and hence hollow waveguides cannot supportTEM modes. But they can support TE and TM modes.


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Rectangular Waveguide(A) TM Modes:

yzx

a

b

0~

== zz HH

( ) ( ) zzz eyxEzyxE,

~,, =

( ) 0~22

2

2

2

=

+

+

x,yEhyx

z

From (1a), the equation for theEzfield is:

222 kh +=

We first find the longitudinal fieldEz


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function ofx only function ofy only

( ) ( ) ( )yYxXyxEz =,~

Let

Then

The above equation can be satisfied for all values of x

andy inside the waveguide only when both terms on theleft-hand side being equal to a constant.

( )

( )

( )

( ) 22

2

2

2 11

hdy

yYd

yYdx

xXd

xX =+


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Hence let

where222

hkk yx =+

Boundary conditions:( ) 00~ =,yEz( ) 0~ =a,yEz( ) 00~ =x,Ez( ) 0~ =x,bEz

( ) xkCxX xsin1= with am

kx= (m = 1, 2, 3, )

( ) ykCyY ysin2= with

b

nky= (n = 1, 2, 3, )

Solutioins:

C1 andC2 are

constants to be

determined bythe boundary

conditions along

thezdirection.

( )( ) 22

2

1xk

dxxXd

xX= ( ) ( )

22

2

1 ykdy

yYdyY

=


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( )

= yb

n

xa

m

Ex,yEz sinsin

~0

22

222

+

=+=b

n

a

mkkh yx

( )

= yb

nx

a

mE

a

m

h

x,yEx sincos

~02

( )

= yb

nx

a

mE

b

n

h

x,yEy cossin

~02

( )

= ybn

xa

m

Eb

n

h

j

x,yHx cossin

~02

( )

= yb

nx

a

mE

a

m

h

jx,yHy sincos

~02

(m = 1, 2, 3,)(n = 1, 2, 3, )

E0 is a contant

equal to C1C2and is to bedetermined by

the excitation

condition of the

waveguide.


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( )

222

2

22

22

2

:constantnPropagatio

fb

n

a

m

b

n

a

m

kh

+

=

+

=

=

Every combination of the integers m and n defines a possible TM

mode that may be designated as a TMmn mode. Hence there are

infinite number of TM mode that can exist inside the waveguide.

The frequency at which = 0 is called the cutoff frequencyfc.

( )22

2

1

+

=b

n

a

mf

mnc

( )22

21

+

==

b

n

a

mfcmnc

Note that the cutoff

frequency for a TEM

mode is zero (i.e., DC).


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(a) Whenf >fc, the propagation constant is an imaginary number and

the mode can travel inside the waveguide.

2

22

2

2

1

===

f

f

jkhkjj c

2

1

=

f

fjk c

2

1

=

ffk c


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Guided wavelength:

2

=g 21

2

=

f

fk c

2

1

=

f

fc

fk

12 ==

where is the wavelength of a plane wave with a frequencyf.

Note that . >g

2

2

2

1

=

c

g

2

22

1gc

=

222 111

cg +=


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Phase velocity: 2

1

==

f

fk

uc

p

2

1

=

f

fc

2

1

=

f

f

uu

c

p

The phase velocity is frequency dependent.

A rectangular waveguide is a dispersive device.

=

1u


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Group velocity:2

11

===

f

fu

ddd

du cg

uug


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Patslope=gu

linestraight

thisofslope=pu

Pc

Group velocity ug is the signalpropagation velocity if we assume the signal

composed of a narrow band of frequencies centered around f. Phasevelocity up is the speed of a constant-phase point of a particular mode.

Group velocity is also the speed of energy flow inside the waveguide. (See

Ref. 5, Section 8.5, for more details.)

-curve for waveguide TE and TM modes

-curve for TEM modes

Graphical Interpretation of up

and ug


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(b) When f < fc, the propagation constant is a real number and the

mode is non-propagating. The amplitude of the mode becomes

smaller (with the e-z) along the zdirection. This mode is called an

evanescent mode.

2

2

1constantnattenuatiohkh ===

2

1

=

cf

fh

Note that the energy of an evanescent mode is not lost but only

transferred back to the excitation source. That is, an evanescent mode

is constantly exchanging energy with the excitation source.

imaginary1

2

====

f

fj

j

jH

EZ c

y

xTM


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( )

= yb

x

a

E

b

h

jx,yHx cossin

~02

( )

= yb

x

a

E

a

h

jx,yHy sincos

~02

( ) 0~

=x,yHz

( ) ( ) ( ) zyxieyxEeyxEzyxE zjiz

ii ,,,,~,~,, ===

( ) ( ) ( ) zyxieyxHeyxHzyxH zjiz

ii ,,,,~

,~

,, ===

Instantaneous field expressions:( ) ( ){ } zyxiezyxEtzyxE tjii ,,,,,Re;,, ==

( ) ( ){ } zyxiezyxHtzyxH tjii ,,,,,Re;,, ==

For propagation

modes: = j


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( ) ( )ztyb

xa

Etzx,yEz

= cossinsin;, 0

( ) ( )ztyb

x

a

E

a

htzx,yEx

= sinsincos;, 02

( ) ( )ztyb

x

a

E

b

htzx,yEy

= sincossin;, 02

( ) ( )ztyb

x

a

E

b

h

tzx,yHx

= sincossin;, 02

( ) ( )ztyb

x

a

E

a

h

tzx,yHy

= sinsincos;, 02

( ) 0;, =tzx,yHz


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TM11

mode has the lowest cutoff frequency among

all the TM modes. Its field lines are shown below.

Solid lines: E field, dash lines: H field


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(B) TE Modes: 0~

== zz

EE

Using a similar analysis as for the TM modes, we can obtain field

expressions for TE modes as:

( ) = y

bnx

amHyxHz coscos,~ 0

( )

= yb

nx

a

mH

b

n

h

jyxEx

sincos,

~02

( )

= yb

nx

a

mH

a

m

h

jyxEy

cossin,

~02

( )

= ybn

xa

m

Ha

m

hyxHx

cossin,

~02

( )

= yb

nx

a

mH

b

n

hyxHy

sincos,

~02

(m = 0,1, 2, )

(n = 0,1, 2, )m & n cannot

be both equal

to zero

H0 is a constant

to be determined

by the excitation

condition of thewaveguide.


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( )22

2

1

+

=b

n

a

mf

mnc

( )22

21

+

==

bn

amfc

mnc

Cutoff frequency:

Cutoff wavelength:

2

1

=

f

fk c

Propagation constant:


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2

1

=

f

fc

g

Guided wavelength:

2

1

=

f

f

uu

c

p

Phased velocity:

2

1

=

f

fuu cg

Group velocity:


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2

1

=

f

f

Z

c

TE

Wave impedance:

2

1

==

cf

fh

Attenuation constant for evanescent modes:


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1 2 3

b/a=1/2

11

11

TM

TE

20

01

TETE

10TE

( )10TEcc

ff /

Note that in TE mode propagation, the lowest order mode is TE10 which

also has the lowest cutoff frequency among all the propation modes in arectangular waveguide. The cutoff frequencies of the different modes

are shown below for two cases of waveguide dimensions.

1 2

b/a=1

11

11

TM

TE

20

02

TE

TE

10

01

TE

TE

( )10TEcc

ff /

Case 1:

Case 2:


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TE10 Mode - Rectangular Waveguide

TE10 is the dominant mode in a rectangular waveguide with lowest

cutoff frequency (when a > b).

TE10

H field: dash lines

Surface current

E field: solid lines

(Picture form)

(Schematic form)


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Field expression of TE10 mode (m = 1 & n = 0):

( ) zjzjzz exa

HeyxHH 0 cos,

~

==

0=== yxz HEE

( ) zjzjyy exaHajeyxEE

0 sin2,~

==

( ) 0, sinj z j z

x x

aH H x y e j H x e

a

= =

( )a

fc2

110TE =

Cutoff frequency:


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Excitation of the Rectangular Waveguide

Excitation of a rectangular waveguide by a coaxial line.

Cross-section at x = a/2

Coaxial lineProbe


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A Note on the Propagating Modes inside

the Rectangular WaveguideNote that in a rectangular waveguide with an excitation source

frequency f= fi, all those TM and TE modes with a cutoff frequency

lower than fi

can propagate inside the waveguide. Whether they will

actually appear inside the waveguide depends on the excitation method.

The excitation method, for example the orientation of the coaxial

probe, can be chosen to excite certain modes while suppress other

modes. Those modes with a cutoff frequency higher than fi cannotpropagate inside the waveguide no matter what excitation method

chosen to excite them.

However, in the most general case, an EM wave inside the rectangularwaveguide is a linear combination of all those TE and TM modes

whose cutoff frequencies being lower than the excitation frequency.

Hence the rectangular waveguide is a high-pass filter.


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Example 2

A standard rectangular waveguide WG-16 is to be designed for the X-

band (8-12.4 GHz) radar application. The dimensions are a = 2.29 cm

and b = 1.02 cm. If only the lowest mode TE10 mode is to propagate

inside the waveguide and that the operating frequency be at least 25%above the cutoff frequency of the TE10 mode but no higher than 95% of

the next higher cutoff frequency, what is the allowable operating-

frequency range of this waveguide?

Solution

a = 2.29 cm b = 1.02 cm

( ) ( )Hz1055.60229.02

103

2

1 98TE10

=

==a

fc

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( ) ( )Hz1010.130229.01031 9

8

0,2TE20 =

==== af nm

c

( )22

TE2

1

+

=

bn

amf

mnc

( ) ( ) ( )2001 TE

98

1,0TE

Hz1071.140102.02

103

2

1c

nmc f

b

f >=

==

==

Hence the allowable operating-frequency range is:

( ) ( )2010 TETE

%95%125cc

fff

GHz45.12GHz19.8 f

That is:

NUS/ECE EE4101


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References:

1. David K. Cheng, Field and Wave Electromagnetic, Addison-

Wesley Pub. Co., New York, 1989.

2. David M. Pozar, Microwave Engineering, John Wiley & Sons,

Inc., New Jersey, 2005.3. Fawwaz T. Ulaby, Applied Electromagnetics, Prentice-Hall, Inc.,

New Jersey, 2007.

4. Robert E. Collin, Field theory of guided waves, IEEE Press, New

York, 1991.

5. J. D. Jackson, Classical Electrodynamics, John Wiley & Sons,

Inc., New York, 1975, Chapter 8, Section 8.5.

6. Joseph A. Edminister, Schaums Outline of Theory and Problemsof Electromagnetics, McGraw-Hill, Singapore, 1993.

7. Yung-kuo Lim (Editor), Problems and solutions on

electromagnetism, World Scientific, Singapore, 1993.

Documents

Lecture Notes Waveguides