ece 344 microwave fundamentals engineering/833...insulating medium (rectangular, circular)...

Prepared By

Dr. Sherif Hekal

ECE 344

Microwave Fundamentals

Spring 2017

Lecture 04:

Rectangular Waveguides

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Introduction

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Introduction

• Waveguides, like transmission lines, are structures used to

guide electromagnetic waves from point to point.

• However, the fundamental characteristics of waveguide and

transmission line waves (modes) are quite different.

• The differences in these modes result from the basic

differences in geometry for a transmission line and a

waveguide.

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Introduction

Waveguides can be generally classified as either

Metal waveguides

Dielectric waveguides

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Introduction

enclosed conducting metal pipe

The dielectric waveguide consists of dielectrics only

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Introduction

Comparison of Waveguide and Transmission Line Characteristics

T.L W.G

Two or more conductors separated by some insulating medium (two-wire, coaxial, microstrip, etc.).

Metal waveguides are typically one enclosed conductor filled with an insulating medium (rectangular, circular) Dielectric waveguide consists of multiple dielectrics.

Normal operating mode is the TEM or quasi-TEM mode

Operating modes are TE or TM modes (cannot support a TEM mode).

No cutoff frequency for the TEM mode. Must operate the waveguide at a frequency above the respective TE or TM mode cutoff frequency for that mode to propagate.

Significant signal attenuation at high frequencies due to conductor and dielectric losses.

Lower signal attenuation at high frequencies than transmission lines.

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Introduction Comparison of Waveguide and Transmission Line Characteristics

T.L W.G

Small cross-section transmission lines (like coaxial cables) can only transmit low power levels

Metal waveguides can transmit high power levels.

Large cross-section transmission lines (like power transmission lines) can transmit high power levels.

Large cross-section (low frequency) waveguides are impractical due to large size and high cost.

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One of the earliest waveguides.

Still common for high power and low-

loss microwave / millimeter-wave

applications.

Rectangular Waveguide

It is essentially an electromagnetic pipe

with a rectangular cross-section.

Single conductor No TEM mode

For convenience

a b.

The long dimension lies

along x.

8

y

z a

b

x

,m,s

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We will now generalize our discussion of transmission lines by

considering EM waveguides. These are “pipes” that guide EM

waves


y

z a

b

x

,m,s

PEC

Proceeding from the Maxwell curl

equations

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However, the spatial variation in z is known so that

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Consequently, these curl equations simplify to

(1)

(2)

(3)

(4)

(5)

(6)

Similar expansion of Ampere’s equation 𝛻 × 𝐻 = 𝑗𝜔𝜀𝐸

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Substituting for Ey from (5) we find

(7)

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Rectangular Waveguide Similarly, we can show that

(8)

(9)

(10)

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2 2

2

2 2, 0c zk h x y

x y

, , , zjk zz zH x y z h x y ewhere

0 0zxH

Ey

Subject to B.C.’s:

0 0zyH

Ex

@ 0,y b

@ 0,x a

TEz Modes

1/2

2 2

c zk k k

14

2

2

z zx z

c

z zy z

c

j E HE k

k x y

j E HE k

k y x

m

m

From Eq. (9), (10):

y

z a

b

x

,m,s

PEC Hence, in (7)-(10) with Ez = 0, the

transverse components 𝐸 and 𝐻 are known once we find a solution for only Hz. Hz

must satisfy the reduced wave equation of

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2 2

2

2 2, ,z c zh x y k h x y

x y

Using separation of variables, let ,zh x y X x Y y

2 22

2 2 c

d X d YY X k XY

dx dy

2 22

2 2

1 1c

d X d Yk

X dx Y dy

2 22 2

2 2

1 1x y

d X d Yk k

X dx Y dy and

Must be a constant

where Separation equation

TEz Modes (cont.)

2 2 2

x y ck k k 15 3/12/2017

( ) ( )

, ( cos sin )( cos sin )

X x Y y

z x x y yh x y A k x B k x C k y D k y

Boundary Conditions:

0zh

y

0zh

x

@ 0,y b

@ 0,x a

0 0,1,2,...yn

D k nb

and

0 0,1,2,...xm

B k ma

and

2 2

2, cos cosz mn cm x n y m n

h x y A ka b a b

and

B

A

A

B

TEz Modes (cont.)

16

The general solution for hz can then be written as

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Therefore,

cos cos zjk z

z mn

m nH A x y e

a b

2 2

2 2

2

z ck k k

m nk

a b

2

2

2

2

cos sin

sin cos

sin cos

cos sin

z

z

z

z

jk z

x mn

c

jk z

y mn

c

jk zzx mn

c

jk zzy mn

c

j n m nE A x y e

k b a b

j m m nE A x y e

k a a b

jk m m nH A x y e

k a a b

jk n m nH A x y e

k b a b

m

m

From the previous field-representation equations,

we can obtain the following:

But m = n = 0

is not allowed!

(non-physical solution)

Note:

m = 0,1,2,…

n = 0,1,2,…

00ˆ ; 0jkzH z A e H

17

TEz Modes (cont.)

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2 2

22 2mn mn

z c

m nk k k k

a b

TEmn mode is at cutoff when mn

ck k

Lossless case

2 21

2

mn

c

m nf

a bm

For (a > b), Lowest cutoff frequency is for TE10 mode

10 1

2cf

a m Dominant TE mode

(lowest fc)

We will

revisit this

mode later.

c

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TEz Modes (cont.)

The mode with the lowest cutoff frequency is called the dominant mode;

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At the cutoff frequency of the TE10 mode (lossless waveguide):

10 1

2cf

a m

102

1

2

d d dd

c

c c ca

f f

a

m

/ 2c

df fa

For a given frequency (with f > fc) , the dimension a must be at least d / 2 in

order for the TE10 mode to propagate.

Example: Air-filled waveguide, f = 10 GHz. We have that a > 3.0 cm/2 = 1.5 cm.

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TEz Modes (cont.)

so

Minimum dimension of a for TE10 mode

to propagate

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Recall:

, , , zjk zz zE x y z e x y e

where

2 2

2

2 2, ,z c ze x y k e x y

x y

1/22 2

c zk k k

Subject to B.C.’s: @ 0,x a

@ 0,y b

Thus, following same procedure as before, we have the following result:

TMz Modes

20

y

z a

b

x

,m,s

PEC

0yE

0xE

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( ) ( )

, ( cos sin )( cos sin )

X x Y y

z x x y ye x y A k x B k x C k y D k y

Boundary Conditions: @ 0,y b

@ 0,x a

0 0,1,2,...yn

C k nb

and

0 0,1,2,...xm

A k ma

and

2 2

2sin sinz mn cm n m n

e B x y ka b a b

and

B

A

A

B

TMz Modes (cont.)

21

0

y

Ez

0

x

Ez

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Therefore

sin sin zjk z

z mn

m nE B x y e

a b

2 2

2

2

z ck k k

m nk

a b

2

2

2

2

sin cos

cos sin

cos sin

sin cos

z

z

z

z

jk zcx mn

c

jk zcy mn

c

jk zzx mn

c

jk zzy mn

c

j n m nH B x y e

k b a b

j m m nH B x y e

k a a b

jk m m nE B x y e

k a a b

jk n m nE B x y e

k b a b

m =1,2,3,…

n =1,2,3,…

From the previous field-representation equations,

we can obtain the following:

Note: If either m or n is zero, the field becomes a trivial

one in the TMz case.

22

TMz Modes (cont.)

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2 21

2

mn

c

m nf

a bm

The lowest cutoff frequency is obtained for the TM11 mode

2 2

11 1 1 1

2cf

a bm

(same as for

TE modes)

Lossless case c

2 2

22 2mn mn

z c

m nk k k k

a b

Dominant TM mode

(lowest fc)

23

TMz Modes (cont.)

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Note that the TE10 and TE01 modes are degenerate modes

(modes with the same cutoff frequency) for a square

waveguide.

The rectangular waveguide allows one to operate at a

frequency above the cutoff of the dominant TE10 mode

but below that of the next highest mode to achieve single

mode operation.

A waveguide operating at a frequency where more than

one mode propagates is said to be overmoded.

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The maximum band for single mode operation is fc

10 [Hz] (67% bandwidth).

/ 2b a

10TE 01TE 11TE

11TM

10TE 20TE11TE

11TM

b < a / 2

b > a / 2

f

f

20TE10TE

Single mode operation

Single mode operation

10TE 01TE

a > b

Mode Chart

Two cases are considered:

2 21

2

mn

c

m nf

a bm

25

Lossless case c y

z a

b

x

,m,s

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Dominant Mode: TE10 Mode

For this mode we have

10 coszjk z

zH A x ea

10 sinzjk zz

x

k aH j A x e

a

10

10 sinzjk z

y

E

j aE A x e

a

m

2

10 2

z zk k ka

10 10A Ej a

m

0x z yE E H

1, 0, cm n ka

Hence we have

10 sinzjk z

yE E x ea

26

y

z a

b

x

,m,s

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Top view

E

H

End view Side view

Field Plots for TE10 Mode

27

xz a

y

xa

b

z

y

b

y

z a

b

x

,m,s

PEC

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TE10 Mode

Mode with lowest cutoff frequency is dominant mode

Single mode propagation is

highly desirable to reduce

dispersion

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E-field phase animation inside a rectangular waveguide at 10 GHz.

TE10 Mode

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The velocity of propagation for a TEM wave (plane wave or transmission line wave) is referred to as the phase velocity (the velocity at which a point of constant phase moves).

2

1

f

fk mn

c

mn

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22

2

b

n

a

mmnc

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Guided wavelength

Note that the product of the TE and TM wave impedances is equal to the square of the TEM wave impedance = intrinsic impedance of the material filling the waveguide.

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Waveguide Group Velocity and Phase Velocity

The phase velocity of a TEM wave traveling in a lossless medium characterized by (μ, ε) is given by

For the general TEmn of TMmn waves,

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The group velocity Is the velocity at which the energy travels.

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22

'

22

2

11

mnmn

mn

c

p

c

c

mnu

kk

Waveguide Group Velocity and Phase Velocity

''

2

' 1 pg

c

pg uf

fuu

mn

mn

mn

2'

pgp uuu mnmn

2

2

2 2

zk ka

k

m

Phase velocity:

Group velocity: gd

vd

1

m

10

c

pv slope

pv

Dispersion Diagram for TE10 Mode

Lossless case c

cf f

(“Light line”)

Velocities are slopes on

the dispersion plot.

gv slope

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Solved Problems

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Solved Problems

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Solved Problems

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Solved Problems

An attenuator can be made using a section of waveguide operating below cutoff, as shown in the accompanying figure. If a = 2.286 cm and the operating frequency is 12 GHz, determine the required length of the below-cutoff section of waveguide to achieve an attenuation of 100 dB between the input and output guides. Ignore the effect of reflections at the step discontinuities.

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Solved Problems

Solution: In the section of WG of width a/2, the TE10 mode is below cutoff (evanescent),with an attenuation constant α

neper/m 3.111)3.251(02286.0

2

2/

m 3.25122

2

2

2

2

1-

ka

fck

m 103.03.111

5.11

e10

)20log(edB100

-5-

-

l

l

l

To obtain 100 dB attenuation (ignoring reflection)

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Solved Problems

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Solved Problems

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Solved Problems

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ece 344 microwave fundamentals engineering/833...insulating medium (rectangular, circular)...

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