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Prof. David R. Jackson

Notes 19

Waveguiding Structures

ECE 3317

Spring 2013

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Waveguiding Structures

A waveguiding structure is one that carries a signal (or power) from one point to another.

There are three common types:

Transmission lines Fiber-optic guides Waveguides

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Transmission lines

j R j L G j C

zk j j R j L G j C

j

zk LC k Lossless:

Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss May or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz)

Properties

(always real: = 0)

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Fiber-Optic Guide

Properties

Has a single dielectric rod Can propagate a signal at any frequency (in theory) Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields (“hybrid mode”)

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Fiber-Optic Guide (cont.)

Two types of fiber-optic guides:

1) Single-mode fiber

2) Multi-mode fiber

Carries a single mode, as with the mode on a waveguide. Requires the fiber diameter to be small relative to a wavelength.

Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).

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Multi-Mode Fiber

http://en.wikipedia.org/wiki/Optical_fiber

Higher index core region

max

max

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Multi-Mode Fiber (cont.)

maxsin sin2air rod cn n

maxsin cosrod cn

c

/ 2 c maxrodn

At left end of rod:

Assume cladding is air

Higher index core region

max

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Multi-Mode Fiber (cont.)

max

2

2

sin cos

1 sin

1 1 /

rod c

rod c

rod rod

n

n

n n

osin sin 90 1rod c airn n

2sin 1max rodn

sin 1 /c rodn

At top boundary with air:

Higher index core region

max

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Waveguide

Has a single hollow metal pipe Can propagate a signal only at high frequency: > c

The width must be at least one-half of a wavelength

Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz)

Properties

http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)

Inside microwave oven

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Waveguides (cont.)

Cutoff frequency property (derived later)

1/22 2z ck k k

k (wavenumber of material inside waveguide)

c ck (definition of cutoff frequency)

2 2:c z ck k k real

2 2:c z ck j k k imaginary

(propagation)

(evanescent decay)

In a waveguide: ck constant

We can write

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Field Expressions of a Guided Wave

All six field components of a guided wave can be expressed in terms of the two fundamental field components Ez and Hz.

Assumption:

0

0

E , , E ,

H , , H ,

z

z

jk z

jk z

x y z x y e

x y z x y e

(This is the definition of a guided wave.)

A proof of this statement is given next.

Statement:

"Guided-wave theorem"

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Field Expressions (cont.)

Proof (illustrated for Ey)

or

H E

HH1E xz

y

j

j x z

Now solve for Hx :

E Hj

H1E Hz

y z xjkj x

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Field Expressions (cont.)

E Hj

EE1H

E1E

yzx

zz y

j y z

jkj y

Substituting this into the equation for Ey yields the result

Next, multiply by

H E1 1E Ez z

y z z yjk jkj x j y

2j j k

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Field Expressions (cont.)

2 2H EE Ez z

y z z yk j jk kx y

2 2 2 2

H EE z z z

yz z

jkj

k k x k k y

Solving for Ey, we have:

This gives us

The other three components Ex, Hx, Hy may be found similarly.

2 2 H EE z z

z y zk k j jkx y

or

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Field Expressions (cont.)

2 2 2 2

H EE z z z

xz z

jkj

k k y k k x

2 2 2 2

H EE z z z

yz z

jkj

k k x k k y

2 2 2 2

E HH z z z

xz z

jkj

k k y k k x

2 2 2 2

E HH z z z

yz z

jkj

k k x k k y

Summary of Fields

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TEMz Wave

E 0

H 0z

z

2 2 0zk k To avoid having a completely zero field,

Assume a TEMz wave:

zk kTEMz

Hence,

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TEMz Wave (cont.)

Examples of TEMz waves:

In each case the fields do not have a z component!

A wave in a transmission line (no conductor loss) A plane wave

E

H

Coax x

y

z

EH

S Plane wave

zk k

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TEMz Wave (cont.)

Wave Impedance Property of TEMz Mode

E Hj

EEHyz

xjy z

Faraday's Law:

Take the x component of both sides:

E Hy xjk j

0E , , E , jkzy yx y z x y eThe field varies as

Hence,

Therefore, we haveE

Hy

x k

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TEMz Wave (cont.)

Now take the y component of both sides:

Hence,

Therefore, we have

E Hj

EEHxz

yjx z

E Hx yjk j

E

Hx

y k

Hence,E

Hx

y

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TEMz Wave (cont.)

These two equations may be written as a single vector equation:

ˆE Hz

The electric and magnetic fields of a TEMz wave are perpendicular to

each other, and the amplitudes of them are related by .

Summary:E

Hy

x

E

Hx

y

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TEMz Wave (cont.)

Examples

E

H

Coax x

y

z

EH

S Plane wave

Microstrip

The fields look like a plane wave in the central region.

EH

“Quasi-TEM” (TEM-like at low frequency)

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Waveguide

In a waveguide, the fields cannot be TEMz.

0B

A

E dr (property of flux line)

ˆ 0z

C S S

BBE dr n dS dS

t t

(Faraday's law in integral form)

0C

E dr contradiction!

Proof:

Assume a TEMz field

y

x

waveguide

E

PEC boundary

A

B

Cˆ ˆn z

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Waveguide (cont.)

In a waveguide (hollow pipe of metal), there are two types of fields:

TMz: Hz = 0, Ez 0

TEz: Ez = 0, Hz 0