EMT Week 4 Wave Equations

Week 4Wave Equations

Plane waves in lossless and lossy medium

PowerPoint Slides

by Dr. Chow Li Sze

1Last updated: 28 September 2014


Wave Equations

Wave Equations for Potentials

Let consider the solution of the nonhomogeneous wave

equation for scalar electric potential V.

First, find the solution for an elemental point charge at

time t, (t)v, located at the origin of the coordinates and

then by summing the effects of all the charge elements in a

given region.

For a point charge at the origin it is most convenient to use

spherical coordinates, because of spherical symmetry, V

depends only on R and t (not on or ).




Except at the origin, V satisfies the following homogenous equation:

0)(1

2

22

2

t

V

R

VR

RR Let ),(

1),( tRU

RtRV

0][1

0][1

0)]1

([1

2

2

2

2

2

2

2

2

2

2

2

2

2

t

U

RR

U

R

UR

R

U

R

t

U

RR

URU

RR

t

U

RR

U

RR

UR

RR

02

2

2

2

t

U

R

U

dx

duv

dx

dvuuv

dx

d)(Recall

1-dimensional homogenous wave equation



The solutions for 02

2

2

2

t

U

R

U

is twice-differentiable function of

)(or)( RtRt

)( Rt does not correspond to a physically useful solution. So

)(),( RtftRU which represents a wave travelling in the positive R direction with a velocity

A function U(R+R) at a later time (t+t) is

)(

])([),(

Rtf

RRttfttRRU

if uRRt /

where /1u

Velocity of propagation

/1

Wave Equations for Scalar Potentials


),(1

),( tRUR

tRV )/(1

),( uRtfR

tRV

R

qV

o4Recall

For a static point charge (t)v at the origin: R

vtRV

4

')()(

4

')/()/(

vuRtuRtf

compare

The potential due to a charge distribution over a volume V is then

')/(

4

1

'vd

R

uRtV(R,t)

V

(V)

Retarded scalar potential

It depends on the value of the charge density at an earlier time (t R/u).

Wave Equations for Vector Potentials

The solution of the nonhomogeneous wave equation for vector

magnitude potential A can be obtained in the same way as for V.

The electric and magnetic fields derived from A and V by

differentiation will obviously also be functions of (t- R/u) and therefore

retarded in time.

It takes time for electromagnetic waves to travel and for the effects of

time-varying charges and currents to be felt at distant points.


Retarded vector potential

J

A ')/(

4 'vd

R

uRt(R,t)

V

(Wb/m)

Source-free Wave Equations

In problems of wave propagation we are concerned with the behavior of an electromagnetic wave in a source-free region where and J are both zero.


0

0

H

E

EH

HE

t

t

If the wave is in a simple (linear, isotropic, and homogeneous) nonconducting medium characterized by and ( = 0), Maxwells equations reduce to:

Eq.7

Take the curl of Eq.7 and use Eq.8:

2

2

)(tt

EHE

Eq.8

Eq.9

Eq.10

E

EEE

2

2)(

02

22

t

EE

according to Eq.9

Source-free Wave Equations


since /1u

02

22

t

EE 0

12

2

2

2

tu

EE

In the same way, we can also obtain an equation in H: 0

12

2

2

2

tu

HH

Homogeneous vector wave equations

uk

wavenumber


Time-Harmonic Fields

Use of Phasors

The instantaneous (time-dependent) expression of a sinusoidal scalar quantity can be written as either a cosine or a sine function.

The specification of a sinusoidal quantity requires the knowledge of 3 parameters: amplitude, frequency, and phase.

For example:

i(t) = I cos (t + )

where I = amplitude

= angular frequency (rad/s) =2f

f = frequency in hertz (Hz)

= phase referred to the cosine function


Use of Phasors

Instantaneous expression is more complicated to work with.

It is much simpler to use exponential functions by writing the applied voltage and current as:

e(t) = E cos t = Re [(Eej0) ejt] = Re (Esejt)

i(t) = I cos (t+ ) = Re [(Iej) ejt] = Re (Isejt)

where Re means the real part of

Es = Eej0 = E

Is = Iej

are (scalar) phasors that contain amplitude and phase information but are independent of t. The phasor Es with zero phase angle is the reference phasor.


Use of Phasors


)()( tjseIti

ReUsing

)(

)(

tjs

tj

s

ej

Iidt

eIjdt

di

Re

Re

)()( tjseEte

Re

Example (Tutorial Q.1)


Express 3 cos t 4 sin t as:

(a) A1 cos (t+1)

(b) A2 sin(t+2)

Time-Harmonic Electromagnetics

Field vectors that vary with space coordinates and are sinusoidal functions of time can be represented by vector phasors that depend on space coordinates but not on time.

For example, a time-harmonic E field referring to cos t:

E(x, y, z, t) = Re [ E(x, y, z)ejt ]

where E(x, y, z) is a vector phasor that contains information on direction, magnitude, and phase.

E(x, y, z, t)/t and E(x, y, z, t)dt would be represented by vector phasor jE(x, y, z) and E(x, y, z)/ j, respectively.

Phasors are, in general, complex quantities.


Time-Harmonic Electromagnetics


0

/

H

E

EJH

HE

j

jTime-harmonic Maxwells equations in

terms of vector field phasors (E, H) and

source phasors (, J) in a simple (linear,

isotropic, and homogeneous) medium are:

Phasor quantities are not functions of t.

Any quantity containing j must necessarily be a phasor.

Source-free Fields in Simple Media


0

0

H

E

EH

HE

j

jIn simple, nonconducting source-free medium characterized by = 0, J = 0, = 0, the time-harmonic Maxwells equations become:

which can be combined to yield second-order partial differential equations in E and H.

01

2

2

2

2

tu

EE

01

2

2

2

2

tu

HH

Homogeneous vector Helmholtzs equations

022 EE k

022 HH k

uk

wavenumber:



Show that if (E, H) are solutions of source-free Maxwells equations in

a simple medium characterized by and , then so also are (E, H),

where

E = H

H = -E/

In the above equations, is called the intrinsic impedance

of the medium.

Principle of duality, which is a consequence of the symmetry of source-free Maxwells equations.

/

Loss Tangent


Loss tangent measure of the power loss in the medium.

ctan

c = loss angle = conductivity (S/m) = 2f = angular frequency = permittivity (F/m)

Good conductor if >>

Good insulator if >>

For example: A moist ground has a dielectric constant r = 10 and conductivity = 10-2 S/m. At 1kHz, the loss tangent /() = 1.8x104 which is a relatively good conductor. At 10GHz, /() = 1.8x10-3 which behaves like an insulator.

A material may be a good conductor at low frequencies but may have the properties of a lossy dielectric at very high frequencies.



A sinusoidal electric intensity of amplitude 250 (V/m) and frequency 1

(GHz) exists in a lossy dielectric medium that has a relative

permittivity of 2.5 and a loss tangent 0.001. Find the average power

dissipated in the medium per cubic meter.


Plane Waves in Lossless Media



01 2

2

2

2

EE

tc

Source-free wave equation for free space

becomes a homogeneous vector Helmholtzs equation: 0

22 EE ok

where ko is the free-space wavenumber

ck ooo

(rad/m)

Eq.1



In Cartesian coordinates, Eq.1 is equivalent to 3 scalar Helmholtzs equations:

0)( 2222

xo Ek

zyx 222

Consider a uniform plane wave characterized by a uniform Ex (uniform magnitude and constant phase) over plane surfaces perpendicular to z:

02

2x

Ex 02

2y

Exand

Eq.2

Eq.2 022

xo

x Ekz

E2

Its solution is:

zjk

o

zjk

o

xxx

oo eEeE

zEzEzE

)()()( and are arbitrary (complex) constant.

Eq.3

oE

oE



Lets examine the first phasor term on the right side of Eq.3:

[

[

]

])(),(

)( zktj

o

tj

xx

oeE

ezEtzE

Re

Re

(V/m)

At t = 0, Ex+(z,0) = Eo

+ cos koz is a cosine curve with an amplitude Eo+. At

successive times the curve effectively travels in the positive z direction.

Traveling wave.

Wave travelling, with several values of t.

)cos(),( zktEtzE oox



For a particular point on the wave, set cos (t koz) = constant or

phaseconstant

phaseconstant

zk

t

zkt

o

o

c

kdt

dzu

ooo

p

1

Velocity of propagation of an equiphase front (the phase velocity) in free space is equal to the velocity of light (3 x 108m/s in free space)

c

f

cko

2Wavenumber

o

ok

2 (rad/m)

o

ok

2 (m)



zjk

o

zjk

o

xxx

oo eEeE

zEzEzE

)()()(

The second phasor term on the right side of Eq.3, , represents

a cosinusoidal wave traveling in the z direction with the same

velocity c. If we are concerned only with the wave traveling in the +z

direction,

However, if there are discontinuities in the medium, reflected waves

traveling in the opposite direction must be considered.

Repeat Eq.3

zjk

ooeE

0oE



Use it to find the associated magnetic field H.

HE jRecall

aaa

aaa

E )(

00)(

00

zzyyxxo

x

zyx

HHHj

zEz

which leads to

0

)(1

0

z

x

o

y

x

H

z

zE

jH

H

for source free field.



Since )()()(

zEjkeEzz

zExo

zjk

ox o

z

zE

jH x

o

y

)(1

Then )(

1)( zEzE

kH x

o

x

o

oy

(A/m)

377120

o

oo ()

Intrinsic impedance of the free space

Instantaneous expression for H is:

])([),( tjyyyy ezHtzH(z,t) ReaaH

)cos( zktE

(z,t) oo

oy

a H


For a uniform plane wave, the ratio of the magnitudes of E

and H is the intrinsic impedance o of the medium.

H is perpendicular to E and that both are normal to the

direction of propagation.


o||

||

H

E



A uniform plane wave with E = axEx propagates in a lossless simple medium

(r = 4, r = 1, = 0) in the +z-direction. Assume that Ex is sinusoidal with a

frequency 100 (MHz) and has a maximum value of +10-4 (V/m) at t = 0 and z

= 1/8 (m).

(a) Write the instantaneous expression for E for any t and z.

(b) Write the instantaneous expression for H.

(c) Determine the locations where Ex is a positive maximum when t = 10-8 s.


Plane Waves in Lossly Media

Review


Documents

EMT Week 4 Wave Equations