TIME VARYING FIELD - GUCeee.guc.edu.eg/Courses/Communications/COMM402... · Faraday’s law in...

ELECTROMAGNETIC PROF. A.M.ALLAM

5/2/2015 LECTURES 1

TIME VARYING FIELD

EMF

Michael Faraday

(1791–1867

John H Poynting

1884


2

Static E&M fields

E &H are independent

We've learned

Now we are going to

Time varying current Electromagnetic waves (E & H)

E &H are interdependent

Time-varying E(t) produces time varying H(t)

Time-varying H(t) produces time varying E(t)

Introduction-1

Stationary charges Electrostatic fields (E)

Steady current Magnetostatic field (H)


In 1820 C.H. Oersted demonstrated that an electric current

affected a compass needle

After this, Faraday professed his belief that if a current could

produce a magnetic effect, then the magnetic effect should be

able to produce a current (magnetism)

In 1831, the electric induction phenomenon was discovered as

a results of Faraday’s experiments

If two separate coils are wound on an iron

ring. One of them is connected through a

switch to DC battery

Faraday's law of induction -2

It was observed that whenever the current

was changed, an induced current would

flow in the other coil

•Faraday’s first experiment:


4

-If a magnet moves near a coil, an induced

current will be produced in the galvanometer

•Faraday’s second experiment:

-Generally, for any closed path C in space

linked by a changing magnetic field

the induced voltage; electromagnetic force

(emf) around this path is produced and is

equal to the negative time rate of change of

the total magnetic flux through the closed

path

This process is called electromagnetic induction

t

tVfme ind

)( ..

The minus sign means that the induced voltage is in such

direction that it resists the original change ( Lenz’s law)

Transformer emf if time varying B(t) links a stationary loop

Motional emf if a moving loop changes its area with time relative to normal B

This is Faraday's law of induction

The change of magnetic flux with time produces an induced EMF

( electric field ) in any closed circuit surrounding that flux =1 +2 +…

different in each

turn

N-turns

= N

same in each turn

(t)

N-turns


5/2/2015 LECTURES 5

Faraday’s law in integral form:

tVind

Faraday’s law in differential form:

SC

SdBt

dE

..

Notes:

The electric field has two sources (charges and time varying magnetic field)

If there is no time variation ( / t =0), gives (Static case)

The induced electric field is not conservative (rotational)

0Eor 0d.EC

S

SdBt

.

t

BE

)(

Stock’s Th.

C

dE

. S

SdE

).(

Maxwell’s equation in time

varying field, Faraday’s law E in time varying field is not conservative i.e., the work

done in moving a charge along a closed path is due to

the energy from time varying B


Ampere’s law for magnetostatic field says:

There is an identity div-curl =curl-grade =0:

=0 The conduction

current

But for the time varying charge:

To satisfy these two conditions we must add another term, such that:

Hence, =0

Displacement current-3

The displacement

current

Ampere’s law for

time varying field


Differential form Integral form

)1t

BE

. )3 D

t

DJH

)2

0. )4 B

tJ

.

SC

S.dB .dE

t

VS

dvSdD

.

SSC

SdDt

SdJdH

...

0.S

SdB

VS

dvt

SdJ

.

Constitutive relations:

HB ; EJ ;

ED

where , and are the medium

parameters.

J

Jimp

Jind

Note:

Jcond = E

Jconv = v

s equations’Maxswell-4

Faraday’s law of induction

Ampere’s circuital law

Gauss flux theorem

Continuity of B lines

Continuity equation


8

In free-space:

[H/m]. 104

]. [F/m 10854.8

7

o

12

o

V/m]. ;er [Volts/met intensity field .Electric.......... E

A/m]. ; [Amperes/m intensity field .Magnetic.......... H

T]. ; Teslaor wb/m; [Webers/m density flux .Magnetic.......... B 22

]C/m ; /m[Coulombsdensity current)ent (Displacemflux .Electric.......... D 22

].A/m ; [Amperes/mdensity current ..Electric.......... J 22

].C/m ; m[Coulombs/ density charge ..Electric.......... 33

H/m]. ; [Henery/mty permeabili .Magnetic..........

F/m]. ; [Farad/my permitivit ic..Dielectr..........

/m].; [Moh/mty conductivi .Electric..........


5-Complex representation of field quantities

1. Scalars:

]Re[)( otj

oe

]e Re[ tj

)cos()( oo tt

]e eRe[ tjj

oo

e oj

o

o

o

+1

+j

Complex phasor form which is represented by a point in complex domain

e ˆ ˆ ˆ Re tj

z

j

ozy

j

oyx

j

ox aeEaeEaeE zyx

tj

zzyyxx e a E a E a E Re

zyˆ )cos( ˆ )cos(ˆ )cos(),( atEatEatEtrE zozyoyxxox

2. Vectors:

zzyyxxa E a E a E E

)cos()( oo tt ]e Re[ tj

]e )(- Re[)(

]e )(j Re[)(

tj2

2

2

tj

t

t

t

t

)(

)(

2

2

2

t

jt

3. Derivatives:

The phasor form


6- Source-free wave equation in complex form

HjE

0. E

)( EjEjEH

0. H

Let us define the following complex quantities: jY & jZ

Thus, Maxwell’s equations will be;

(1 ) ..... HZE

(3) ..... 0 . E

(2 ) ..... EYH

(4) ..... 0 . H

Taking the curl of both sides of eqn. (1) :

)( HZE

)(Z ).( 2 EYEE

0

0 Z 2 EYE

0 Z 2 HYH

Similarly;

Generally;

0 22

H

E

numb er veco mp lex w a the.....is Z Y

where,

jjj ))((

Phase shift const. Attenuation const.

Helmholtz’s

equations

jLet

The solution region does not include any sources 0 & 0 impJ

General lossy media ( σ ≠ 0 )

Is the complex propagation coefficient


0 22 EE

One of the solutions of this differential equation has the forward solution as

where, r ra za ya xr zyx

)r . k( j-)r . k( - 1

eeAE Attenuation Phase angle

Solution of source-free wave equation in complex form

The general solution has a forward wave and backward wave

Generally the backward wave is not significant in unbounded media

r

Is the position vector

k

Is the unit vector in direction of propagation vector

)r . k()r . k( j-21

j

eAeAE


12

y

z

ox

z

o azteE

tzHazteEtzE ˆ )cos(),( , ˆ )cos(),(

Example:

A plane wave propagating in z-direction and its electric field intensity has an

x- components only. Draw a sketch for the electric field as a function of z Solution:

Amplitude Attenuation

coefficient

Phase

coefficient

Direction of

propagation Field

direction


13

x

y

z

k

r

co n strk

.ˆ

Note: “ for any source-free wave equation, the solution is always a plane wave ”

Source

Plane wave

Equation of wave front

r ra za ya xr zyx

If

then:

is the equation of a constant phase surface

perpendicular to the direction of propagation

and is called the wave front. If this equation

represents a plane and all of these planes are

parallel, it is called plane wave

k

)r . k( j-)r . k( - 1

eeAE

1.ˆ crk

2.ˆ crk

is the radius vector from the origin,


14

Plane wave is a class of waves having both electric

and magnetic fields contained in one plane called the

wave front which is perpendicular to the direction of

propagation

7- Properties of plane wave

(1) Intrinsic impedance (η)

||

Y

Z

EkH

ˆ 1

kHE ˆ

oooj

377 12036/10

104/j/Y/Z

9

7

oo

Example: In free space jY , )0 a nd ,( oo oo jZ

||/ HE


15

|| and A A , ˆ

ooj

x

Z eaeAEExample: Let j , then find

t).(z,H and t)(z,E ,

H

x

zj-ja e e o z

o eAE

EkH

ˆ 1

y

j-zj-ja e e e

||o

zo e

AH

xo a )z-tco s( ),( z

o eAtzE

yo a )z-tcos( ||

),(

Zo e

AtzH

]e Re[),( tjEtzE

similarly,

kEH ˆ (2)

k

E

H

The velocity of points in the wave moving at constant

phase or ; it is the velocity of equiphase plane wave front

c o nsta nt)z-t( o

)-(0dt

dz

Differentiate w.r.t. time ,

dt

dzv p

pv

z

f(z,t)

:) Phase velocity ( Vp )3(


5/2/2015 LECTURES 16

(4) Wavelength ( ):

(5) Depth of penetration ( ) : (skin depth)

The distance between two successive points having the same phase

-z-t o

2

2or,

The distance that the wave travel through the medium until its value decreased by

(1/e=36.78%)

1

|E|

e

Ao

z

Ao

z

o eAE || 1 eAeA oo

2-)z(-t o


5/2/2015 LECTURES 17

(6) Electric and magnetic energy densities (we and wm):

a) Time form:

]/.......[|H| 2

1H.

2

1H.H

2

1 32 mJBwm

]/.........[|E| 2

1E.

2

1E.E

2

1 32 mJDwe

The average values (denoted by < >) of the above quantities are given by:

dtwT

w

T

m 0

m z)(t,1

dtwT

w

T

e 0

e z)(t,1

b) Complex form:

][J/m ....... 2

1 3*HHwm

][J/m ........ 2

1 3*EEwe

. 4

1 *HHwm

*. 4

1 EEwe

The ( * ) stands for the complex conjugate


John H Poynting

1884

Since e.m.w are used to transport information from a point to another either in

guided or wireless medium, it should be associated with power and energy

k S HES

The quantity that describes the power associated with e.m.w is the instantaneous

Poynting vector S

S is the power density [W/m2]

(7) The Poynting vector ( ) : S

The total power that flow through or intercepted by an aperture of area A

dAnSPA

ˆ.

n

A For a uniform plane wave propagating in direction k

making an angle θ with n then

c o sS AP

kθ

average the it is often desirable to find time varying fieldFor application of

integrating the instantaneous Poynting which is obtained by power density

vector over one period and dividing by the period

dtztST

S

T

0

),(1

this integration results in two terms:fields, complex and the instantaneousUse the relation between the

1- time average Poynting vector which is independent on time (the average

power density or radiation density which is the real part) )Re(

2

1 *HES av

sdHEPs

rad

).(Re

2

1 * and the total power crossing a closed surface is obtained by

integrating the normal component over the entire surface

2- time varying term of double the operating frequency (the stored power density which is the imaginary part)


19

Poynting theorem states that the net power flowing out of a given volume is

equal to the time rate of decrease in the energy stored within the volume

minus the conduction losses

Losses due

to σ

Stored

electrical

energy

Stored

Magnetic

energy

Power input

Power output

V

B J

v

*dvE.E2

dv H.H2v

*

v

* dv E.E2

v

m dvHH W *.4

1

v

*e dvE.E

4

1 W


jZ jY

Z jY

pv

2

1

||/ YZ

j

8- Plane wave in different media

Complex wave number Propagation constant Phase shift constant [Rad/m]

Attenuation constant [Np/m]

Phase velocity [m/s] Wavelength [m]

Skin or penetration depth [m/s] Intrinsic impedance [Ω]

Summary of wave parameters


(1)Free space 0 ) σ a n d μ , (ε oo

jY ; o ojZ

cY oo / Z /c ; 0

; / cv p ; /2

2

/

22

f

c

cfc

/1

120// ooYZ 0 ; 1 2 0||

E & H lies in a plane transverse (orthogonal)

to the direction of propagation

No E & H components along the direction of

propagation ( uniform plane wave )

(2)Lossless dielectric 0 ) σ a n d μ ,( jY ; jZ

vY / Z /v ; 0

; /rr

p

cvv

;

/

22

f

v

f

v

v

p

// YZ 0 ; /||

/1 H

E


(3)Lossy dielectric 0 ) σ a n d μ ,(

The lossy dielectric is considered as a partially conductor and the wave is losing

energy while traveling along that medium

Is the ratio of the magnitude of the conduction

current to that of the displacement current

•The loss tangent of lossy medium

'

j- ''' c

Loss angle

0.1)( 1

Slightly lossy material

10)( 1

Highly lossy material (good conductor)

Perfect conductor

''

tan'

''

Ej

E

J

J

d

c

EjJ d

EJ c

θ

TIME VARYING FIELD - GUCeee.guc.edu.eg/Courses/Communications/COMM402... · Faraday’s law in...

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