1

Plane Wave: Introduction

According to Maxwell’s equations, a time-varying electric field produces a time-varying magnetic field and, conversely, a time-varying magnetic field produces an electric field( i.e. )

– Called electromagnetic waves

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

HEHE →→→

2

Spherical Wave and Plane Wave

When energy is emitted by a source, such as an antenna, it expands outwardly from the source in the form of spherical waves.Far away from the source, the spherical wave could be considered as a plane wave.

3

Time-Harmonic Fields

In last chapter, both electric and magnetic fields are expressed as functions of time and position.– In real applications, time signals can be expressed as

sum of sinusoidal waveforms. So it is convenient to use the phasor notation to express fields in the frequency domain.

– tjezyxtzyx ω),,(Re),,,( EE ≡

tj

tj

ezyxj

tezyx

ttzyx

ω

ω

ω ),,(Re

),,(Re),,,(

E

EE

=

∂∂

≡∂

∂

4

Time-Harmonic Fields

Therefore, the differential form of Maxwell’s equations becomes these the differential form time-harmonic Maxwell’s equations

0 0

=⋅∇⇒=⋅∇=⋅∇⇒=⋅∇

+=×∇⇒∂∂

+=×∇

−=×∇⇒∂∂

−=×∇

BBDD

DJHDJH

BEBE

vv

jt

jt

ρρ

ω

ω

Attention: A

lot of equatio

ns

D,E,B,H,J,ρv are functions of x,y,z,t

D,E,B,H,J,ρv are functions of x,y,z

5

Time-Harmonic Fields

Using the constitutive relations and , the equations becomes

0 0/

=⋅∇⇒=⋅∇=⋅∇⇒=⋅∇

+=×∇⇒+=×∇−=×∇⇒−=×∇

HBED

EJHDJHHEBE

ερρωεω

ωµω

vv

jjjj

EDHB εµ ==

6

Wave equations in Source-Free Media

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

The equation can be written as

where and it is called complex permittivity.

00

)(

=⋅∇=⋅∇

=+=×∇−=×∇

HE

EJEHHE

σωεσωµ

Qjj

EH )( ωεσ j+=×∇

EH cjωε=×∇

ωσεεεε /''' jjc −≡−≡

7

Wave equations in Source-Free Media

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

using the vector identity .

and called Laplacian

00

=⋅∇=⋅∇=×∇−=×∇

HE

EHHE

cjjωεωµ

EEEHE

)(2cjj

jωεωµ

ωµ

−=∇−⋅∇⇒

×∇−=×∇×∇

AAA 2∇−⋅∇≡×∇×∇

2

2

2

2

2

22

zyx ∂∂

+∂∂

+∂∂

≡∇

8

Wave equations in Source-Free Media

where and γ is called propagation constant The equation is called the homogeneous wave equation for E.

Similarly, we can obtain, (Try yourself)

00

=⋅∇=⋅∇=×∇−=×∇

HE

EHHE

cjjωεωµ

0

0

)(

22

22

2

=−∇⇒

=+∇⇒

−=∇−⋅∇

EEEE

EEE

γ

µεω

ωεωµ

c

cjj

cµεωγ 22 −=

022 =−∇ HH γ

9

Plane wave in lossless media

If the medium is nonconducting (σ=0),

and then

In the textbook, the wavenumber k is used which is defined as

εωσεε =−≡ /jc

βγµεωγµεωγ jjor ==−= or 22

µεω=k

10

Plane wave in lossless media

In rectangular cocodinates,

can be separated into three equations.

Suppose Ex is a function of z only and Ey=Ez=0, we have

022 =−∇ EE γ

zyxiEEzyx ii ,,02

2

2

2

2

2

2

==−

∂∂

+∂∂

+∂∂ γ

022

2

=−∂∂

xx E

zE γ

11

Comparison

Wave equation Transmission Line equations

In rectangular co-ordinates

If Ex is a function of z only and Ey=Ez=0

022 =−∇ EE γ

022 =−∇ HH γ IzI

VzV

22

2

22

2

γ

γ

=∂∂

=∂∂

022 =−∇ ii EE γ zyxi ,,=

022

2

=−∂∂

xx E

zE γ

12

Solution

This second order linear differential equation

has two independent solutions, so the solution of Ex is

022

2

=−∂∂

xx E

zE γ

zjxo

zjxox

zxo

zxox

eEeEzEor

eEeEzE

ββ

γγ

+−−+

+−−+

+=

+=

)(

)(

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

13

Parameters

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 =

14

Magnetic field

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

15

Magnetic field

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

16

Summary

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.

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

Example: M7.3

17

Example

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

18

Example

)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: