Chapter 2: Transmission lines and waveguides

2.1 Generation solution for TEM, TE and TM waves2.2 Parallel plate waveguide2.3 Rectangular waveguide2.4 Circular waveguide2.5 Coaxial line2.6 Surface waves on a grounded dielectric slab2.7 Stripline2.8 Microstrip2.9 Wave velocities and dispersion2.10 Summary of transmission lines and waveguids

Transmission lines and waveguides

2.1 Generation solution for TEM, TE and TM waves

Two-conductor TL

Closed waveguide

Electromagnetic fields (time harmonic eiωt and propagate along z axis):

where e(x,y) and h(x,y) represent the transverse (x,y) E and H components, while ez and hz are the longitudinal E and H components.

zjz

zjz

eyxhzyxhzyxH

eyxezyxezyxE

)],(),([),,(

,)],(),([),,(

In the case without source, the harmonic Maxwell’s equations can be written as:

With the e-iβz dependence, the above vector equations can be divided into six component equations and then solve the transverse fields in terms of the longitudinal components Ez amd Hz:

Wavevector in xy plane: (kc

2=kx2 + ky

2)

(1) TEM waves (Ez = Hz = 0)

Propagation constant:

k

The Helmholtz equation for Ex:

0)( 2222

222

xEkzyx

For a e-jz dependence, and the above equation can be simplified

xxx EkEEz 2222 )/(

0)(22

22

xEyx

Similarly, 0)(22

22

yEyx

0),(2 yxet

0),(2 yxht Transverse magnetic field:

Laplace equations, equal to static fields

(kc = 0, no cutoff)

0

0

Note:

1.Wave impedance, Z relates transverse field components and is dependent only on the material constant for TEM wave. For TEM wave

2.Characteristic impedance of a transmission line, Z0:relates an incident voltage and current and is a function of the line geometry as well as the material filling the line.For TEM wave: Z0= V/I

),(1

),( yxezZ

yxhTEM

(V incident wave voltage I incident wave current)

Wave impedance:

x

y

y

xTEM H

E

H

EZ

0

0

k

(2) TE waves (Ez = 0 and Hz 0) The field components can be simplified as:

is a function of frequency and TL/WG structure

Solve Hz from the Helmholtz equation

0)( 2222

222

zHkzyx

Because , then

zjzz eyxhzyxH ),(),,(

0)( 222

22

zc hkyx

TE wave impedance:

where . Boundaries conditions will be used to solve the above equation.

222 kkc

k

H

E

H

EZ

x

y

y

xTE

Solve Hz first and then obtain Hx, Hy, Ex, Ey

Solve Hz first and then obtain Hx, Hy, Ex, Ey

(3) TM waves (Hz = 0 and Ez 0) The field components can simplified:

is a function of frequency and TL/WG structure

Solve Ez from Helmholtz equation:

0)( 2222

222

zEkzyx

Because , then

zjzz eyxezyxE ),(),,(

0)( 222

22

zc ekyx

TM wave impedance:

kH

E

H

EZ

x

y

y

xTM

where . Boundaries conditions will be used to solve the above equation.

222 kkc

k

H

E

H

EZ

x

y

y

xTE

Solve Ez first and then obtain Hx, Hy, Ex, Ey

Solve Ez first and then obtain Hx, Hy, Ex, Ey

(4) Attenuation due to dielectric loss

Total attenuation constant in TL or WG = c + d.

c: due to conductive loss; calculated using the perturbation method; must be evaluated separately for each type.

d: due to the dielectric loss; calculated from the propagation constant.

Taylor expansion (tan << 1)

ModeDefinitio

nPropagation

constantWave

impedanceSolve the fields

TEMEz = Hz =

0

TE (H wave)

Hz 0 andEz = 0

TM(E wave)

Ez 0 and Hz = 0

0),(2 yxet

0),(2 yxht

0)( 222

22

zc ekyx

Ez Hx, Hy, Ex, Ey

Hz Hx, Hy, Ex, Ey

kH

E

H

EZ

x

y

y

xTM

x

y

y

xTEM H

E

H

EZ

w >> d (fringing fields and any x variation could be ignored)

Formed from two flat plates or

strips

Probably the simplest type of

guide

Support TEM, TE and TM

modes

Important for practical reasons.

2.2 Parallel plate waveguide

(a) TEM modes (Ez = Hz = 0)

Laplace equation for the electric potential (x,y)

The transverse field , so that we have

dyWxyxt 0,0,0),(2for

Characteristic impedance:

1

pv

Boundary conditions:

0),(0)0,( Vdxandx

dyVyx /),( 0

Phase velocity:

The transverse ez(x,y) satisfies

0),(22

2

yxeky zc )( 222 kkc

(b) TMn modes (Hz = 0)

General solutions:

Boundary condition 1: Bn = 0

Boundary condition 2:

Solutions of TMn modes:

The components of TMn:

Example: TM1

(b) TMn mode (Hz = 0) Propagation constant:

2222 )(d

nkkk c

k > kc Traveling wavek = kc Tunneling? k < kc Evanescent waveCutoff frequency:

@ (k = kc)

Power flow:

Frequency and geometry dependent

The TMn mode cannot propagate at f < fc!

(c) TEn mode (Ez = 0)

The transverse hz(x,y) satisfies

And boundary Ex(x,y) = 0 at y = 0, d.

0),(22

2

yxhky zc

zjn

cy

zjn

cx

zjnz

ed

ynB

k

jH

ed

ynB

k

jE

ed

ynByxH

sin

sin

cos),(

• Cutoff frequency :

where the propagation constant

2222 )/( dnkkk c

d

nkf cc

22

k

H

EZ

y

xTE

• Wave impedance:

An = 0 TE1

Parallel plate waveguide

TEM TM1 TE1

Homework:1. Derive the field solutions of TE1 mode for a parallel-plate metallic waveguide and plot the field pattern of each component roughly if possible.

Require: start from the first two Maxwell’s equations

Substitute Eq. (1) into (2) by eliminating H and we have

(no source, isotropic)

In the case without source, the harmonic Maxwell’s equations can be written as:

With the e-iβz dependence, the above vector equations can be divided into six component equations and then solve the transverse fields in terms of the longitudinal components Ez amd Hz:

Wavevector in xy plane: (kc

2=kx2 + ky

2)