1 ENE 429 Antenna and Transmission lines Theory Lecture 2 Uniform plane waves

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ENE 429Antenna and Transmission lines Theory

Lecture 2 Uniform plane waves

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Review Wave equations

Time-Harmonics equations

where

22

2

�� E EE

t t2

22

�� H HH

t t

2 2 0 ��

s sE E

2 2 0 ��

s sH H

( ) j j

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Time-harmonic wave equationsor

where

This term is called propagation constant or we can write

= +j

where = attenuation constant (Np/m) = phase constant (rad/m)

2 2 0 ��

s sH H

( ) j j

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Transverse ElectroMagnetic wave (TEM)

http://www.edumedia.fr/a185_l2-transverse-electromagnetic-wave.html

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Solutions of Helmholtz equations The instantaneous forms of the solutions

The phasor forms of the solutions

0 0cos( ) cos( )

��z z

x xE E e t z a E e t z a

0 0cos( ) cos( )z z

y yH H e t z a H e t z a ��

0 0

z j z z j zs x xE E e e a E e e a

��

0 0

z j z z j zs y yH H e e a H e e a

��

incident wave reflected wave

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Attenuation constant

Attenuation constant determines the penetration of the wave into a medium

Attenuation constant are different for different applications

The penetration depth or skin depth, is the distance z that causes to reduce to

z = 1

z = 1/ =

E��

10E e

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Good conductor

1 1

f

At high operation frequency, skin depth decreases

A magnetic material is not suitable for signal carrier

A high conductivity material has low skin depth

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Currents in conductor

To understand a concept of sheet resistance

1L LR

A wt

1 LR

t w Rsheet () Lw

1sheetR

t sheet resistance

from

At high frequency, it will be adapted to skin effect resistance

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0

0

zx x

zx x

E E e

J E e

Therefore the current that flows through the slab at t is

;xI J dS ds dydz

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;xI J dS ds dydz

00 0

wz

xz y

I E e dydz

0

0

zxw E e

0 .xI w E A

From

Jx or current density decreases as the slab gets thicker

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0xV E L

0

0

1xskin

x

E LV L LR R

I w E w w

For distance L in x-direction

For finite thickness,

R is called skin resistanceRskin is called skin-effect resistance

0 00 0

(1 )t w

z tx x

z y

I E e dydz w E e

/

1

(1 )skin tRe

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Current is confined within a skin depth of the coaxial cable

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Ex1 A steel pipe is constructed of a material for which r = 180 and = 4106 S/m. The two radii are 5 and 7 mm, and the length is 75 m. If the total current I(t) carried by the pipe is 8cost A, where = 1200 rad/s, find:

a) The skin depth

b) The skin resistance

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c) The dc resistance

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The Poynting theorem and power transmission

2 21 1( )

2 2E H d S J E dV E dV H dV

t t

��

Poynting theorem

Total power leavingthe surface

Joule’s lawfor instantaneouspower dissipated per volume (dissi-pated by heat)

Rate of change of energy storedIn the fields

2W/mS E H ��

Instantaneous poynting vector

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Example of Poynting theorem in DC case

2 21 1( )

2 2E H d S J E dV E dV H dV

t t

��

Rate of change of energy storedIn the fields = 0

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2 z

IJ a

a

��

By using Ohm’s law,

From

2 z

J IE a

a ��

��

2 2

2 20 0 0( )

a LId d dz

a

2 22

1 LI I R

a

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E H d S��

From Ampère’s circuital law,

Verify with

H dl I��

2 aH I ��

2

IH a

a

��

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2

2 32

IS d S a d dz

a

��

2

2 2 32 2z

I I IS E H a a a

aa a

��

2 222

2 3 20 02

LI a I Ld dz I R

a a

Total power

W

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Uniform plane wave (UPW) power transmission Time-averaged power density

1Re( )

2avgP E H

��

amount of power WavgP P d S��

for lossless case, 00

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��j z j zx

avg x yx

EP E e a e a

201

2x

avg zE

P a ��

W/m2

W/m2

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Uniform plane wave (UPW) power transmission

0

z j z jxxE E e e e a

��

intrinsic impedance for lossy medium nje

0

1 1 z j z jz xxH a E a E e e e a

��

0 njz j z jxy

Ee e e e a

for lossy medium, we can write

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Uniform plane wave (UPW) power transmission

2

201Re

2jzx

zE

e e a

from1

Re( )2

avgP E H

��

2

201cos

2zx

zE

e a

W/m2

Question: Have you ever wondered why aluminum foil is not allowed inthe microwave oven?

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Polarization

UPW is characterized by its propagation direction and frequency.

Its attenuation and phase are determined by medium’s parameters.

Polarization determines the orientation of the electric field in a fixed spatial plane orthogonal to the direction of the propagation.

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Linear polarization

Consider in free space,

0( , ) cos( ) xE z t E t z a

��E��

At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave” E

��

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A pair of linearly polarized wave also produces linear polarization

Linear polarization

0 0( , ) cos( ) cos( )x yx yE z t E t z a E t z a

��

At z = 0 plane

At t = 0, both linearly polarized wavesHave their maximum values

0 0(0,0) x yx xE E a E a

��(0, ) 0

4t

E ��

0 0(0, ) cos( ) cos( )x yx yE t E t a E t a

��

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More generalized of two linearly poloraized waves,

Linear polarization occurs when two linearly polarized waves are

More generalized linear polarization

0 0( , ) cos( ) cos( )x yx x y yE z t E t z a E t z a

��

in phase 0y x

out of phase 180y x

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Super position of two linearly polarized waves that

If x = 0 and y = 45, we have

Elliptically polarized wave

0 180y x or

0 0(0, ) cos( ) cos( )

4x yx yE t E t a E t a

��

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occurs when Exo and Eyo are equal and

Right hand circularly polarized (RHCP) wave

Left hand circularly polarized (LHCP) wave

Circularly polarized wave

90y x

0 0(0, ) cos( ) cos( )


��

0 0(0, ) cos( ) cos( )


��

90y x

90y x

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Phasor forms:

for RHCP,

for LHCP,

Circularly polarized wave

0 0( 0) yx

jjx yx yE z E e a E e a

��from

0( 0) ( )x yxE z E a ja

��

0( 0) ( )x yxE z E a ja

��

Note: There are also RHEP and LHEP

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Ex2 Given

,determine the polarization of this wave

( , ) 3cos( 30 ) 8cos( 90 )x yE z t t z a t z a ��

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Ex3 The electric field of a uniform plane wave in free space is given by , determine

50100( ) j ys z xE a ja e

��

a) f

b) The magnetic field intensity sH��

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c)

d) Describe the polarization of the wave

S��

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1 ENE 429 Antenna and Transmission lines Theory Lecture 2 Uniform plane waves