UEE 3201 Electromagnetics II Fall, 2014

8 Plane Electromagnetic Waves

8.2 Plane Waves in Lossless Media

1. Homogeneous wave equation in free space

2E 1c2

Et2

= 0

For a time harmonic (sinusoidal) field,

2E+ k20E = 0, (E = phasor)

where k0 = 00 = /c = free space wave number.

In Cartesian corrdinate,

( 2

x2+

2

y2+

2

z2+ k20)Ex,y,z = 0

2. wavefront: set of points/positions in space with same phase

plane wave: wave with planar wavefront perpendicular to the direction

of propagation

spherical wave: wave with spherical wavefront

uniform wave: wave with constant E and H field magnitude across the

wavefront

3. Now consider a uniform plane wave with constant Ex on the wavefront

xy plane,

General solution in phasor form is

where E+0 and E0 are complex constant to be determined from bound-

ary or initial conditions.

4. The real-time solution represented by the first phasor term is

= wave traveling in the +z direction

cYi Chiu, ECE Dept., NCTU 8-1

UEE 3201 Electromagnetics II Fall, 2014

wave velocity (phase velocity up)

=

5. Real-time solution represented by the second phasor term is

Ex (z, t) = [E0 ej(t+k0z)]

= E0 cos(t+ k0z)

= wave traveling in z direction with phase velocity up = c

- Sign of up (or k0) represents direction of wave propagation.

- If there is only one wave in the +z direction, then E0 = 0.

6. H field can be found from E and Maxwells equations:

For +z wave,

Similarly for z wave,

Hy (z) = 1

0Ex (z)

cYi Chiu, ECE Dept., NCTU 8-2

UEE 3201 Electromagnetics II Fall, 2014

7. 0 =E+x (z)H+y (z)

= intrinsic impedance of the free space

If 0 = real E+x (z) and H+y (z) are in phase

8. Real time solution of the H field

(directions of wave propagation (az), electric field (ax) and magnetic

field (ay) are mutually perpendicular.)

Ex 8-1: A uniform plane wave E = axEx propagates in +z direction,

r = 4, r = 1, = 0, f = 100MHz

(a) instantaneous expression of E

(b) instantaneous expression of H

cYi Chiu, ECE Dept., NCTU 8-3

UEE 3201 Electromagnetics II Fall, 2014

(c) positions for postitive peaks Ex at t = 108 sec

- E and H are perpendicular to each other and both are transverse

(perpendicular) to propagation direction

transverse electromagnetic wave (TEM wave)

9. Doppler effect (see the textbook)

10. For a uniform plane wave propagating in an arbitrary direction in a loss-

less medium, the solution to the wave equation 2E+k2E = 0 has the

general form (from separation of variables, let E = E0X(x)Y (y)Z(z))

E(x, y, z) = E0ejkxxejkyyejkzz

= E0ej(kxx+kyy+kzz)

= E0ejkR

where R = axx+ ayy + azz = position vector

k = axkx + ayky + azkz = wave vector

|k|2 = k2x + k2y + k2z = k2 = 2

cYi Chiu, ECE Dept., NCTU 8-4

UEE 3201 Electromagnetics II Fall, 2014

Wavefront of the wave E(x, y, z) = set of points with same phase

k = ank = wave vector

an = propagation direction

k = |k| = = 2/ = wave number

11. For a uniform plane wave in a charge free region,

E0 is transverse to the propagation direction

12. H(R) =

where = /k =/ () = intrinsic impedance of the medium,

H(R) = 1 (an E0) ejkR

H an and E, an is in the direction of EH (right hand rule)

TEM wave in the an (or k) direction

Ex 8-2: Find E(R) in terms of H(R)

()

E(R) = an H(R)

cYi Chiu, ECE Dept., NCTU 8-5

UEE 3201 Electromagnetics II Fall, 2014

13. Polarization of plane waves (, )

Polarization is the time varying behavior of the orientation of the E

field at a given point in space. For example, in Ex 8-1,

The tip of E field oscillates only along the x axis

linear polarization (linaerly polarized)

14. Now consider another wave in the +z direction,

= superposition of two orthogonal linearly polarized wave with

y polarization lagging x polarization by /2.

The instantaneous E field is

At a given position z = 0,

Locus of tip of E(0, t) is(E1(0, t)E10

)2+

(E2(0, t)E20

)2= cos2 t+ sin2 t = 1

ellipse in the counterclockwise sense

elliptically polarized if E10 = E20circularly polarized if E10 = E20

right-hand or positive circularly polarized wave

cYi Chiu, ECE Dept., NCTU 8-6

UEE 3201 Electromagnetics II Fall, 2014

15. If E2(z) leads E1(z) by 90 (/2),

E(z) = axE10ejkz + ayjE20e

jkz

E(0, t) = axE10 cost ayjE20 sint

elliptically or circularly polarized wave

= tan1 E2(0, t)E1(0, t)

= t for E20 = E10 left-hand or negative circularly polarized wave

16. If E1(z) and E2(z) are in time phase,

linearly polarized

17. In general,

elliptically polarized

Ex 8-3: Linear and circular waves

Consider a linearly polarized wave propagating in +z direction,

= superposition of a right-hand and a left-hand circular waves

Note: - AM wave: linearly polarized with E field perpendicular to ground

- TV wave: linearly polarized in the horizontal direction

- FM wave: circularly polarized

orientation of receiving antenna should be adjusted accordingly

cYi Chiu, ECE Dept., NCTU 8-7

UEE 3201 Electromagnetics II Fall, 2014

8.3 Plane Waves in Lossy Media

1. wave equation in lossy media

- kc = c = complex wave number (assume = real)

- plane wave ejkz in lossless medium becomes ejkcz in lossy medium

We can define a propagation constant

* For lossless media, = 0, = 0 = 0, = k =

The Helmholtz wave equation becomes

For a linearly polarized uniform plane wave in +z direction,

ez: attenuation factor

: attenuation constant (Np/m, 1/m)

(wave amplitude decresed by a factor of e1 after traveling a dis-

tance of 1/ meters)

ejz: phase factor

: phase constant (rad/m)

(similar to the wave number in lossless media)

8.3.1 Low-Loss Dielectrics

1. propagation constant

cYi Chiu, ECE Dept., NCTU 8-8

UEE 3201 Electromagnetics II Fall, 2014

For a good but imperfect insulator, , or / 1,

= + j = jc = j(1 j/)1/2

attenuation constant

2

phase constant

[1 + 18

(

)2]2. intrinsic impedance

c =

(1 j

)1/2

Ex and Hy are not in time phase

3. phase veolcity

up =

1

(1 18

(

)2)8.3.2 Good Conductor

1. propagation constant

= + j = j

(1 + j

)1/2

2. intrinsic impedance

3. phase velocity

up =

2

cYi Chiu, ECE Dept., NCTU 8-9

UEE 3201 Electromagnetics II Fall, 2014

4. example: copper

after a distance of = 1/ = 0.038 mm, the wave amplitude will be

attenuated by a factor of e1 = 0.368.

(At f = 10 GHz, = 0.66m high frequency EM wave is attenuated

very rapidly in a good conductor.)

5. skin depth or depth of penetration

= distance over which the wave amplitude is attenuated by a factor

of 1/e

=

=

=

E,J are distributed near the surface of a conductor at high

frequency.

(See Table 8-1)

Ex 8-4: Seawater (Summary)

cYi Chiu, ECE Dept., NCTU 8-10

UEE 3201 Electromagnetics II Fall, 2014

At f = 5 MHz,

/ = 200 1 good conductor,

attenuation constant = phase constant =f = 8.89 (Np/m,

rad/m),

wavelength in water = 2/ = 0.707 m,

skin depth = 0.112 m

(a) At 5 MHz, the wavelength in air is 60 m. It is very difficult to

communicate with a submarine due to the small penetration depth

in seawater and long wavelength in air (difficult to build efficient

tranmission antenna).

(b) After the real-time electric field Ex(z, t) is calculated, it is in-

correct to calculate the real-time magnetic field Hy(z, t) from

Hy(z, t) = Ex(z, t)/c. Since the complex impedance is defined

in the phasor form (frequency domain), the correct formula are

Hy(z) =Ex(z)

c,

Hy(z, t) = [Ex(z)

cejt

]8.3.3 Ionized Gas (plasma)

Plasma - assembly of equal numbers of positive ions and negative electrons

and possibly other neutral species

- electrons are much lighter than ions

- motion of ions can be neglected and electrons can be viewed as a

free electron gas (collision is neglected)

- neutral species, if any, are not affected by electric field

cYi Chiu, ECE Dept., NCTU 8-11

UEE 3201 Electromagnetics II Fall, 2014

1. From Newtons force law, force on an electron in a time harmonic E

field is

In phasor form,

Displacement x from the background positive ions gives rise to a dipole

moment

If there are N electrons per unit volume, the polarization vector is

Equivalent permittivity of a plasma is

- propagation constant in a plasma

- intrinsic impedance

cYi Chiu, ECE Dept., NCTU 8-12

UEE 3201 Electromagnetics II Fall, 2014

(a) f < fp , = pure real

pure attenuation of wave in plasma

p = E/H = pure imaginary

plasma is a reactive load

no power transmission (electrons can move fast enough to

screen the EM fields)

(fp 9N 0.9 9 MHz for ionosphere)

(b) f > fp , = pure imaginary

no attenuation in plasma

Ex 8-5: Communication with space ship

cYi Chiu, ECE Dept., NCTU 8-13

UEE 3201 Electromagnetics II Fall, 2014