UNIVERSIDAD DE CUENCAMAESTRIA EN TELEMATICAING. REMIGIO PILLCO PILLAJO. Msc.

Introduction

The electromagnetic waves can be guided in a given direction of propagation using different methods.

The present chapter is restricted to single-conductor (hollow-pipe) waveguides, of rectangular or circular cross section, which operate in the gigahertz (microwave) range.

These devices too support “plane waves” in the sense that the wavefronts are planes perpendicular to the direction of propagation. However, the boundary conditions at the inner surface of the pipe force the fields to vary over a wavefront.

Transverse and axial fieldsThe waveguide is positioned with the longitudinal direction along the z axis.

The guide walls have:

r

r

c

0

0

0

Perfect conductorPerfect dielectric

It is further supposed that = 0 (no free charge) in the dielectric. The dimensions for the cross section are inside dimensions.

We have the following expressions for the field vector F (which stands for either E or H), assuming wave propagation in the +z direction.

zzT

zzyyxx

jkz

yxFyxyx

yxFyxFyxFyx

eyx

aFF

aaaF

FF

),(),(),(

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

),(

Rectangular coordinates

zzT

zzrr

jkz

rFrr

rFrFrFr

er

aFF

aaaF

FF

),(),(),(

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

),(

Cylindrical coordinates

Because the dielectric is lossless ( = 0), the wave propagates without attenuation; hence the wave number :

2k

The reason for decomposing the field into a transverse vector component and an axial vector component is two fold:

a)The boundary conditions apply to ET and HT alone.b)The complete E and H fields in the waveguide are known once either cartesian component Ez or Hz is known.

Transverse components from Axial components.

yH

Kj

xE

Kjk

E

xH

Kj

yE

Kjk

E

z

c

z

cx

z

c

z

cy

22

22

yE

Kj

xH

Kjk

H

xE

Kj

yH

Kjk

H

z

c

z

cx

z

c

z

cy

22

22

222 kKc Critical wave number

TE and TM modes: Wave impedanceThe two types of waves found in the last slide are referred to as transverse electric (TE) or transverse magnetic (TM) waves, according as Ez 0 or Hz 0. When carrying such waves, the guide is said to operate in a Te or TM mode.

T

T

H

E The wave impedance (in

ohms)

TETE K

The wave impedance TE mode

The wave impedance TM mode

TMTM

K

Explicit solutions for TE modes of a rectangular guide

This mode has the axial field:

ayn

axm

HyxH mnzmn

coscos),(

The critical wave number for TEmn is:

22

ayn

am

KcTEmn

In terms of which the wave number and wave impedance for TEmn are:

22

22

cTEmn

cTEmn

K

KK

TEmn

TEmn

....)2,1,0(

....)2,1,0(

nbn

K

mam

K

y

x

byn

axm

EyxE mnzmn

sinsin),(

Explicit solutions for TM modes of a rectangular guide

This mode has the axial field:

TMTMmn

TEmncTMmn

yxcTMmn

K

KK

bn

am

KKK

22

2222

The critical wave number for TMmn is:

....)2,1,0(

....)2,1,0(

nbn

K

mam

K

y

x

Mode cutoff frequencies

In practice one deals with frequencies, not wave numbers; it is desirable to replace the concept of critical wave number (Kc ) by one of cutoff frequency (fc ).

)TMor (TE velocity Phase

/1

/1/1

/1

2

2

frecuency Operating andfrecuency Cutoff2/

2

12

mnmn2

0

202

0

2

022

22

ff

uf

ffff

ffff

uk

bn

amu

f

ff

kku

f

cmn

mnmn

cmnTMmn

cmn

TEmn

cmn

mncmn

omn

ocmn

c

cco

c

Dominant modeThe dominant mode of any waveguide is that of lowest cutoff frequency.

Both modes TM and TE

The dominant mode of a rectangular guide is invariably TE10:

Equations for for TM and TE modes for the dominant mode

Power transmitted in a lossless waveguide

The time-average power transmitted in the +z direction is calculated by integration of the z component of the complex Poynting vector over a transverse cross section of the guide.

tioncross

zTTz dSP

sec

*Re21

aHE

For the dominant mode of a lossless rectangular waveguide:

2

10

2

10

2

1010

2

1010 14

224

ff

ff

abHabaa

HP c

c

o

o

oz

Since the excitation of a guide is commonly specified through the electric field amplitude

1010

2H

aE

oo

abE

P

f

fabEP

TE

z

coz

10

2

1010

2

102

1010

4

14

Using the electric field amplitude

Problems to Study:

Chapter 16: Electromagnetics “Edminister”

Problems: 29, 33, 35, 36, 37, 39, 42, 43, 47, 48.

Bibliography:

[1] Neri Vela. R, Lineas de Transmision, McGraw-Hill, 1999.ISBN: 970-10-2546-6

[5] Edminister J, Electromagnetics, McGraw-Hill, Second Edition, 1993, ISBN: 0-07-021234-1