Intro.1

Reference

Wave Equation (2 Week)6.5 Time-Harmonic fields7.1 Overview7.2 Plane Waves in Lossless Media7.3 Plane Waves in Lossy Media7.5 Flow of Electromagnetic Power and the Poynting Vector7.6 Normal Incidence of Plane Waves at Plane Boundaries

Intro.2

PLANE ELECTROMAGNETIC WAVES

Wireless applications are possible because electromagnetic fields can propagate in free space without any guiding structures.

Plane waves are good approximations of electromagnetic waves in engineering problems after they propagate a short distance from the source.

Intro.3

Time-Harmonic Electromagnetics

In past chapter, field quantities are expressed as functions of time and position. In real applications, time signals can be expressed assuperimposition of sinusoidal waveforms. So it is convenient to use the phasor notation to express fields in the frequency domain, just as in a.c. circuits.

tjezyxtzyx ω),,(Re),,,( EE ≡

tj

tj

ezyxj

tezyx

ttzyx

ω

ω

ω ),,(Re

),,(Re),,,(

E

EE

=

∂∂

=∂

∂

Intro.4

Time-Harmonic Electromagnetics

Differential form of Maxwell’s equations

Time-harmonic Maxwell’s equations

0=⋅∇=⋅∇

∂∂

+=×∇

∂∂

−=×∇

BD

DJH

BE

v

t

t

ρ

0=⋅∇=⋅∇

+=×∇−=×∇

BD

DJHBE

v

jj

ρω

ω

Intro.5

Wave equations in Source-Free Media

In a source-free media (i.e.charge density ρv=0), Maxwell’s equation becomes:

00

)(

=⋅∇=⋅∇

+=×∇−=×∇

HE

EHHEωεσ

ωµj

j

We will derive a differential equation involving E or H alone. First take the curl of both sides of the 1st equation:

EEEHE

)(2 ωεσωµ

ωµ

jjj

+−=∇−⋅∇

×∇−=×∇×∇

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Since in source-free media, we obtain

where

0=⋅∇ E

022 =−∇ EE γ

)(2 ωεσωµγ jj +=

The above equation is called Helmholtz equation, and γ is called the propagation constant. In rectangular coordinates, Helmholtz equation can be decomposed into three scalar equations:

022 =−∇ ii EE γ zyxi ,,=

Similarly for the magnetic field,

022 =−∇ HH γ

Intro.7

Helmoltz Equation and Transmission Line equations

Helmoltz equation Transmission Line equations

In rectangular co-ordinates

022 =−∇ EE γ

022 =−∇ HH γI

zI

Vz

V

22

2

22

2

γ

γ

=∂∂

=∂∂

022 =−∇ ii EE γ zyxi ,,=

Intro.8

Plane waves in lossless media

In a lossless medium, the conductive current is zero so σ is equal to zero; there is no energy loss and µ, ε are both real numbers. Therefore in lossless media

µεωγ j=

Plane wave is a particular solution of the Maxwell’s equation, where both the electric field and magnetic field are perpendicular to the propagation direction of the wave. Let the wave propagate along the z-direction, and the electric field is along the x-direction.

Intro.9

E- and H-field vectors for a plane wave propagating in z-direction

Intro.10

xaE )(zEx=

Substituting this into Holmholtz equation, and since

, we obtain

022

2

=− xx E

dzEd γ

This second order linear differential equation has two independent solutions, so the solution of Ex is

zxo

zxox eEeEzE γγ +−−+ +=)(

The 1st term is a wave travelling in the +z direction, and the 2nd term is a wave travelling in the -z direction.

0=∂∂

=∂∂

yE

xE xx

Intro.11

γ is called the propagation constant. In general, γ is complex

where α, β are the attenuation and phase constants. But in lossless media,

βαγ j+=

µεωβ

α

=

= 0

So the electric field in lossless media is given by:

( ) xaE zjxo

zjxo eEeE ββ +−−+ +=

(µ, ε are real)

Intro.12

Let up be the phase velocity with which either the forward or backward wave is travelling, and λ is the wavelength.

Since , therefore

βπλ

µεβω

2

1

=

==pu

In free space (vacuum), ε = εo, µ = µo and the phase velocity is the same as the velocity of light in vacuum (3×108 m/s)

.

µεωβ =

Intro.13

Similarly for the magnetic field, H can be found by solving

under the conditions

We obtain

ωµj−×∇

=EH

[ ] yaH zjxo

zjxo eEeE

zjββ

ωµ+−−+ +

∂∂−

=1

[ ] yazjxo

zjxo eEeE ββ

ωµβ +−−+ −=

0=∂∂

=∂∂

yE

xE xx

Intro.14

We notice that the magnetic field is in the y-direction, i.e. perpendicular to the electric field, and the ratio of magnitudesof electric and magnetic fields is:

εµ

βωµη

η

==

== −

−

+

+

xo

xo

xo

xo

HE

HE

η is called the intrinsic (or wave) impedance of the medium. In free space η has a value approximately equal to 377Ω.

Intro.15

The above plane wave is called TEM (transverse electromagnetic) wave, because both the electric and magnetic fields only exist in the transverse directions perpendicular to the propagation direction.

Other waves (e.g. in waveguides) can be also:

(a) TE (transverse electric) wave - electric field exists only in transverse direction, i.e. zero electric field in the propagation direction.

(b) TM (transverse magnetic) wave - magnetic field exists only in transverse direction, i.e. zero magnetic field in the propagation direction.

Intro.16

Example: A uniform plane wave with E=Exax propagates in a lossless medium (εr=4, µr=1, σ=0) in the z-direction. If Ex has a frequency of 100MHz 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 and H, b) determine the locations where Ex is a positive maximum when t=10-8s.

Solution:

jj3

44103102

8

8 ππγ =××

=

rrroro cjj εµωεεµµωγ ==

since velocity of light 81031×==

oo

cεµ

Intro.17

)3

4102cos(10 84 φππ +−= − ztxaE(a)

Setting Ex=10-4 at t=0 and z=1/8,

6πφ == kz

πεεµµη

η

60==

==

or

or

xy

EH yy aaH

(b) At t=10-8, for Ex to be maximum,

nz

nz

m

m

23

813

281

34)10(102 88

±=

±=

−−− πππ

,....2,1,0=n

The H-field:

Intro.18

7.4 Plane Waves in Lossy Media

The electric field in the x-direction is given by:

zxo

zxox eEeEzE γγ +−−+ +=)(

where γ is in general complex for lossy media

βαγωεσωµγ

jjj

+=

+= )(

So E-field can be written as:

xx aaE zjzxo

zjzxo eeEeeE βαβα ++−−−+ +=

Intro.19

Substitute into Maxwell’s equations, the magnetic field is:

[ ] yaH zxo

zxo eEeE γγ

η+−−+ −=

1

where the intrinsic impedance η is complex.

ωεσωµ

γωµη

jjj+

==

The loss can be due to (i) conduction loss where σ is non-zero, (ii) dielectric loss where ε is complex,

''' εεε j−=

Intro.20

Example: The E-field of a plane wave propagating in z-direction in sea water is E=ax100 cos(107πt) V/m at z=0. The parameters of sea water are εr=72, µr=1, σ=4 S/m. (a) Determine the attenuation constant, phase constant, intrinsic impedance,and phase velocity. (b) Find the distance where the amplitude of E is 1% of its value at z=0.

Solution: (a))( ωεσωµγ jj +=

In our case σ/ωε =200>>1 so that α, β are approximately

89.8=== µρπβα f

βα j+=

Intro.21

σωµ

ωεσωµη j

jj

≈+

=

4/76

4)104)(105()1()1( ππππ

σµπη jejfj =

××+=+=

−

Phase velocity: m/s67

1053.389.8

10×===

πβωu

(b) Distance z1 where wave amplitude decreases to 1%:

518.089.8

605.4100ln101.0

1

1

===

=−

α

α

z

e z

Intro.22

7.5 Polarization: Polarization describes how the E-field vector varies with time as the wave propagates. Let Ex, Ey be the components of E-field in the x and y directions.

• Linear polarization: The E-field vector is always in the same direction. Ex, Ey are either in phase or out of phase.

yx

yx

aaE

aaE

)cos()cos()(

)cos()cos()(

21

21

ztEztEt

ztEztEt

mm

mm

βωβω

βωβω

−−−=

−+−=

Intro.23

• Circular polarization: The E-field vector rotates around the axis of propagation and has constant amplitude. Ex, Ey have equal magnitudes and phase angle difference of π/2.

• Elliptic polarization: The E-field vector rotates around the axis of propagation but has time-varying amplitude. Ex, Ey can have any values of magnitude and phase angle difference.

yx aaE )2/cos()cos()( πβωβω ±−+−= ztEztEt mm

yx aaE )cos()cos()( 21 δβωβω +−+−= ztEztEt mm

Intro.24

7.6 Power transmission - Poynting’s Theorem

Power is transmitted by an electromagnetic wave in the direction of propagation. It can be derived from Maxwell’s equation in the time domain that:

dvHEt

dvdVVS ∫∫∫

+

∂∂

+⋅=⋅×−22

22 µεJESHE

On the right hand side, the 1st term represents the instantaneous ohmic power loss, the 2nd and 3rd term represent the rate of increase of energies stored in the electric and magnetic fields respectively. So the left hand side represents the net instantaneous power supplied to the enclosed surface, or

Intro.25

)()()( ttt

dWSout

HEP

SH.E

×=

×= ∫

The instantaneous Poynting vector P (in watt per sq.m) represents the direction and density of power flow at a point.

In time-varying fields, it is more important to find the average power. We define the average Poynting vector for periodic signals as Pavg

∫ ∫ ×==T T

dtttT

dtT 0 0

)()(11 HEPPavg

Using complex phasor notation of E, H,

( )*Re21 HEPavg ×=

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For a uniform plane wave, E & H field vectors are given by:

xaE zjzxo eeE βα −−=

[ ] yaH zjzxo eeE βα

η−−=

1

ηθηη je=

Substituting in equ. of Pavg gives

zavg aP ηα θ

ηcos

22

2zxo eE −=

Intro.27

7.7 Reflection of Plane Waves at Normal Incidence

x

z

Medium 2 (σ2,ε2,µ2)Medium 1(σ1,ε1,µ1)

(incident wave)

(reflected wave)

(transmitted wave)

Ei

ak

ak

ak

Et

Er

Hi

Hr

Ht

ak – propagation direction

y direction

-y direction

Intro.28

Incident wave:

yzio

yz

ioi

xz

ioi

eEeH

eE

aaH

aE

11

1

1

γγ

γ

η−−

−

==

=

Reflected wave:

yzro

yz

ror

xz

ror

eEeH

eE

aaH

aE

11

1

1

)( γγ

γ

η−=−=

=

Transmitted wave:

yzto

yz

tot

xz

tot

eEeH

eE

aaH

aE

22

2

2

γγ

γ

η−−

−

==

=

Intro.29

Boundary conditions at the interface (z=0):

Tangential components of E and H are continuous in the absence of current sources at the interface (refer to tutorials), so that

( )21

1ηη

toroio

toroio

toroio

EEE

HHHEEE

=−

=+=+

Reflection coefficient12

12

ηηηη

+−

==Γio

ro

EE

Transmission coefficient21

22ηη

ητ+

==io

to

EE

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Note that Γ and τ may be complex, and

101

≤Γ≤

=Γ+ τ

Similar expressions may be derived for the magnetic field.

In medium 1, a standing wave is formed due to the superimposition of the incident and reflected waves. Standing wave ratio can be defined as in transmission lines.

Intro.31

END