1 Prof. David R. Jackson Dept. of ECE Notes 4 ECE 5317-6351 Microwave Engineering Fall 2011 Waveguides Part 1: General Theory.

1

Prof. David R. JacksonDept. of ECE

Notes 4

ECE 5317-6351 Microwave Engineering

Fall 2011

Waveguides Part 1:General Theory

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In general terms, a waveguide is a devise that confines electromagnetic energy and channels it from one point to another.

Examples

– Coax– Twin lead (twisted

pair)– Printed circuit lines

(e.g. microstrip)– Optical fiber

– Parallel plate waveguide– Rectangular waveguide– Circular waveguide

Waveguide Introduction

Note: In microwave engineering, the term “waveguide” is often used to mean rectangular or circular waveguide (i.e., a hollow pipe of metal).

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General Solutions for TEM, TE and TM Waves

Assume ejt time dependence and homogeneous source-free materials.

Assume wave propagation in the z direction

zjk zze e

ˆ, , , , zjk zt zE x y z e x y z e x y e

, , , ˆ , zjk zt zH x y z h x y h x y ez

,z zjk k j

z

y

x

, ,

PEC

transverse components

J E

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Helmholtz Equation

E j H

H j E J

E j H j j E J

0

vE

H

Vector Laplacian definition :

2E E E

2 2 2 2ˆ ˆ ˆx y zE x E y E z E

where

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Helmholtz Equation

2

2 2

2 2

2 2

v

v

v

E E j j E J

E E j J

E E j J

E E j E

E j H j j E J

Assume Ohm’s law holds:

J E

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2 2 vE E j E

2 2

2 2

2 2

v

vc

v

E j E

E E

E k E

Next, we examine the term on the right-hand side.

Helmholtz Equation (cont.)

2 2ck

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0

0

0v

H j E J

H j E J

H j E E

E j

E

To do this, start with Ampere’s law:

In the time-harmonic (sinusoidal steady state, there can never be any volume charge density inside of a linear, homogeneous, isotropic, source-free region that obeys Ohm’s law.


2 2 vE k E

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Helmholtz equation

2 2 0E k E

Hence, we have


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2

c

c

c

H j E J

H j E E

H j E

H j E

H j j H

H H j j H

Similarly, for the magnetic field, we have

H j E J


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2 2 0H k H

Hence, we have

Helmholtz equation


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2 2 0H k H

Summary

Helmholtz equations

2 2 0E k E


These equations are valid for a source-free homogeneous isotropic linear material.

12

Assume a guided wave with a field variation in the z direction of the form

zjk ze

Field Representation

Then all six of the field components can be expressed in terms of these two fundamental ones:

,z zE H

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Types of guided waves:

Field Representation (cont.)

TEMz: Ez = 0, Hz = 0

TMz: Ez 0, Hz = 0

TEz: Ez = 0, Hz 0

Hybrid: Ez 0, Hz 0

Microstrip

h

w

er

TEMz

TMz , TEz

Hybrid

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Assume a source-free region with a variation zjk ze

E j H

1) zz y x

Ejk E j H

y

2) zz x y

Ejk E j H

x

3) y xz

E Ej H

x y

4) zz y c x

Hjk H j E

y

5) zz x c y

Hjk H j E

x

6) y xz

H Hj E

x y

Field Representation: Proof

cH j E

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Combining 1) and 5)

2

2

2 2

2

1

( )

c

z z z zx

c c

z

z zz z x x

c

z zx c z

c

k

zc z z x

E k H kj H

y x j

E Hj jk k k H

E Hjk jk H j H

y j x

E HjH k

y x

k y x

1/22 2c zk k k

Cutoff wave number

Field Representation: Proof (cont.)

A similar derivation holds for the other three transverse field components.

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2

2

2

2

z zx c z

c

z zy c z

c

z zx z

c

z zy z

c

E HjH k

k y x

E HjH k

k x y

E HjE k

k x y

E HjE k

k y x

Summary

These equations give the transverse field components in terms of longitudinal components, Ez and Hz.


2 2ck

1/22 2c zk k k

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Therefore, we only need to solve the Helmholtz equations for the longitudinal field components (Ez and Hz).


2 2 0z zH k H

2 2 0z zE k E

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Transverse Electric (TEz) Waves

0zE

In general, Ex, Ey, Hx, Hy, Hz 0

To find the TEz field solutions (away from any sources), solve

2 2( ) 0zk H

2 2 22

2 2 22 2 ) 0( 0 zz k H

x y zk H

The electric field is “transverse” (perpendicular) to z.

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Recall that the field solutions we seek are assumed to vary as zjk ze

( , , ) ( , )jk zz

z zH x y z h x y e

2

2 22 2

2 2, 0z

c

z

k

k k h x yx y

2 2 2c zk k k

2 2

22 2

, 0c zk h x yx y

Solve subject to the appropriate boundary conditions.

2 2 22

2 2 20zk H

x y z

Transverse Electric (TEz) Waves (cont.)

2 2

22 2

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

(This is an eigenvalue problem.)2 .ckThe eigenvalue is always real

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Once the solution for Hz is obtained,

2 2

2 2

z z zx x

c c

z z zy y

c c

jk H HjH E

k x k y

jk H HjH E

k y k x

TE wave impedance

TEz

Zk

yx

y x z

EE

H H k

For a wave propagating in the positive z direction (top sign):

yx

y x z

EE

H H k


For a wave propagating in the negative z direction (bottom sign):

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Also, for a wave propagating in the positive z direction,

ˆ ˆˆ

ˆ ˆˆ

1ˆ( )

t x y

t TE x y

TE t

t t

x TE y

y TE

TE

x

z e ye xe

z e Z

e

xh yh

Z h

h z

h

e Z h

e

Z

Z

ˆ ˆ, , ,t x ye x y xe x y ye x y

Similarly, for a wave propagating in the negative z direction,

1ˆ( )t t

TE

h z eZ


1ˆ, ,t t

TE

h x y z e x yZ

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0zH

Transverse Magnetic (TMz) Waves

In general, Ex, Ey, Ez ,Hx, Hy 0

To find the TEz field solutions (away from any sources), solve

2 2( ) 0zk E

2 2 22

2 2 22 2 ) 0( 0 zz k E

x y zk E

23

2 2

22 2

, 0c zk e x yx y

solve subject to the appropriate boundary conditions

2

2 22 2

2 2, 0z

c

z

k

k k e x yx y

2 2 2c zk k k

Transverse Magnetic (TMz) Waves (cont.)

2 2

22 2

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

(Eigenvalue problem)

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2 2

2 2

c z z zx x

c c

c z z zy y

c c

j E jk EH E

k y k x

j E jk EH E

k x k y

TM wave impedance

zTM

c

kZ

yx z

y x c

EE k

H H

yx z

y x c

EE k

H H

Once the solution for Ez is obtained,

For a wave propagating in the positive z direction (top sign):

For a wave propagating in the negative z direction (bottom sign):


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Also, for a wave propagating in the positive z direction,

ˆ ˆˆ

ˆ ˆˆ

1ˆ( )

t x y

t TM x y

TM t

t t

x TM y

y TM

TM

x

z e ye xe

z e Z

e

xh yh

Z h

h z

h

e Z h

e

Z

Z

ˆ ˆ, , ,t x ye x y xe x y ye x y

Similarly, for a wave propagating in the negative z direction,

1ˆ( )t t

TM

h z eZ

1ˆ, ,t t

TM

h x y z e x yZ


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Transverse ElectroMagnetic (TEM) Waves0, 0z zE H

From the previous equations for the transverse field components, all of them are equal to zero if Ez and Hz are both zero.

Unless 2 0ck

For TEM waves 2 2 2 0c zk k k

z ck k

In general, Ex, Ey, Hx, Hy 0

(see slide 16)

Hence, we have

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In a linear, isotropic, homogeneous source-free region,

0E

0

, 0

, 0

, 0

z

z

t

jk zt t

jk zt t

t t

E

e x y e

e e x y

e x y

ˆ ˆt x yx y

In rectangular coordinates, we have

Notation:0yx zEE E

x y z

, 0t te x y

Transverse ElectroMagnetic (TEM) Waves (cont.)

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Also, for the TEMz mode we have from Faraday’s law (taking the z component):

ˆ ˆ 0zz E z j H j H

0y xE E

x y

, 0t te x y

0y xe e

x y

, 0t te x y

ˆ ˆt x yx y

Notation:

or


Taking the z component of the curl, we have

Hence

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, 0t te x y

, ,t te x y x y

, 0t te x y , 0t t x y

2 , 0t x y

Hence


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2 , 0x y

Since the potential function that describes the electric field in the cross-sectional plane is two dimensional, we can drop the “t” subscript if we wish:


Boundary Conditions:

,

,

a

b

a

b

x y V

x y V

conductor " "

conductor " "

This is enough to make the potential function unique. Hence, the potential function is the same for DC as it is for a high-frequency microwave signal.

2 , 0x y

a b

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Notes:

A TEMz mode has an electric field that has exactly the same shape as a static (DC) field. (A similar proof holds for the magnetic field.)

This implies that the C and L for the TEMz mode on a transmission line are independent of frequency.

This also implies that the voltage drop between the two conductors of a transmission line carrying a TEMz mode is path independent.

A TEMz mode requires two or more conductors (a static field cannot

be supported by a single conductor such as a hollow metal pipe.


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TEM Solution Process

A) Solve Laplace’s equation subject to appropriate B.C.s.:

B) Find the transverse electric field:

C) Find the total electric field:

D) Find the magnetic field:

2 , 0x y

1ˆ ;

TEM

H z E zZ

propagating

TEMz

Zk k

, ,te x y x y

, , , ,zjk zt zE x y z e x y e k k

1 Prof. David R. Jackson Dept. of ECE Notes 4 ECE 5317-6351 Microwave Engineering Fall 2011 Waveguides Part 1: General Theory.

Documents

te z hybrid slide

te z field solutions

proof slide

positive z direction

negative z direction

transverse electric

electric field

microstrip h w rr tem