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

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1 Lecture 12 Lecture 12 Plane Waves in Plane Waves in Conductor, Poynting Theorem, Conductor, Poynting Theorem, and Power Transmission and Power Transmission ENE 325 ENE 325 Electromagneti Electromagneti c Fields and c Fields and Waves Waves

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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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Page 1: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

1

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

Page 2: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 3: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 4: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 5: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 6: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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.

Page 7: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 8: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 9: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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.

Page 10: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 11: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

11

Currents in conductorCurrents in conductor

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

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

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

Page 13: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

13

c) c) dc resistancedc resistance

Page 14: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

14

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

Page 15: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

15

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

Page 16: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 17: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

��������������

Page 18: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

18

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

Page 19: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

19

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

Page 20: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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

Page 21: Lecture 12  Plane Waves in Conductor,  Poynting  Theorem, and Power Transmission

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?