lecture 12 plane waves in conductor, poynting theorem, and power transmission

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ENE 325 Electromagnetic Fields and Waves. Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission. Review (1). Wave equations Time-Harmonics equations where. Review (2). where This  term is called propagation constant or we can write  = +j - PowerPoint PPT Presentation

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Lecture 12Lecture 12 Plane Waves in Plane Waves in Conductor, Poynting Theorem, and Conductor, Poynting Theorem, and Power TransmissionPower Transmission

ENE 325ENE 325ElectromagnetElectromagnetic Fields and ic Fields and WavesWaves

2

Review (1)Review (1) Wave equationsWave equations

Time-Harmonics equationsTime-Harmonics equations

wherewhere

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

3

Review (2)Review (2)

wherewhere

This This term is called term is called propagation constantpropagation constant or we or we can write can write

= = +j+j

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

( ).j j

4

Review (3)Review (3)

The instantaneous forms of the The instantaneous forms of the solutionssolutions

The phasor forms of the solutionsThe 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

5

Attenuation constant Attenuation constant

Attenuation constant determines the penetration Attenuation constant determines the penetration of the wave into a mediumof the wave into a medium

Attenuation constant are different for different Attenuation constant are different for different applicationsapplications

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

z = -1z = -1

z = -1/ z = -1/ = - = -..

E��������������

10E e

6

Good conductorGood 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.

7

Currents in conductorCurrents in conductor

To understand a concept of sheet To understand a concept of sheet resistanceresistance

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

8

Currents in conductorCurrents in conductor

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

9

Currents in conductorCurrents in conductor

;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.

10

Currents in conductorCurrents in conductor

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

11

Currents in conductorCurrents in conductor

Current is confined within a skin depth of the coaxial cable.

12

Ex1Ex1 A steel pipe is constructed of a A steel pipe is constructed of a material for which material for which rr = = 180 and 180 and = = 44101066 S/m. The two radii are 5 and 7 S/m. The two radii are 5 and 7 mm, and the length is 75 m. If the total mm, and the length is 75 m. If the total current current I(t)I(t) carried by the pipe is carried by the pipe is 8cos8costt A, where A, where = = 12001200 rad/s, find: rad/s, find: a)a) skin depthskin depth

b)b) skin resistanceskin resistance

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c) c) dc resistancedc resistance

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The Poynting theorem and The Poynting theorem and power transmissionpower 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 Example of Poynting theorem in DC caseDC 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

16

Example of Poynting theorem in Example of Poynting theorem in DC caseDC case

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

17

Example of Poynting theorem in Example of Poynting theorem in DC caseDC case

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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Example of Poynting theorem in Example of Poynting theorem in DC caseDC case

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

1Re( )2

avgP E H

������������������������������������������

amount of power avgP P d S����������������������������

for lossless case, 00

12

j z j zxavg x yx

EP E e a e a

��������������

201

2x

avg zE

P a ��������������

W

W/m2

20

Uniform plane wave (UPW) Uniform plane wave (UPW) power transmissionpower 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

21

Uniform plane wave (UPW) Uniform plane wave (UPW) power transmissionpower transmission

2

201Re2

jzxz

Ee e a

from

1Re( )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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