1. CH 8. Plane Electromagnetic Waves
Chap 8.1~8.3

2. 8.1 Introduction
The main concern of this chapter
- The source-free wave equation for :
- The study of the behavior of plane waves.
Specification of study
The propagation of time-harmonic plane wave fields in an unbounded homogeneous medium. ( intrinsic impedance, attenuation constant, phase constant )
The meaning of skin depth, Poynting vector and power flux density.
Behavior of a plane wave incident normally on a plane boundary. ( reflection and refraction of plane wave)
Uniform plane wave : A particular solution of Maxwell’s equations with assuming the same direction, same magnitude and same phase in infinite planes perpendicular to the direction of propagation. ( similarly for )

3. Then, 7-7.3 SOURCE-FREE FIELDS IN SIMPLE MEDIA
If the wave is in a simple nonconducting medium with ε and μ (J=0,ρ=0, σ=0),
Then,
wavenumber
Homogeneous vector Helmholtz’s equations.

4. 8.2 Plane Waves in Lossless Media
In nonconducting source free medium :
- the source-free wave equation for free space becomes a Helmholtz’s equation. (from time-harmonic Maxwell’s equations)
For the component Ex (in Cartesian coordinates)
For a uniform plane wave
(uniform magnitude, constant phase,
in the plane surfaces perpendicular to z.)

5. 8.2 Plane Waves in Lossless Media
- the solution :
The real time representation of the first phasor term on the right side of the solution. (using )
Wave traveling in positive z direction.
E(V/m)
Z(m) A -A O
* The red point is looked
like that point is moving. ∙ T0 ∙ T1 ∙ T2 ∙ T3 ∙ T4 ∙ T5

6. 8.2 Plane Waves in Lossless Media
(Phase velocity in free space)
The definition of wavenumber.
- the number of wavelengths in a complete cycle.
- this equation are valid without the subscript 0 if the medium is a lossless material such as a perfect dielectric, instead of free space.
In the solution of Helmholtz’s equation, Eq. (8-7),
- the second phasor term on the right side represents a sinusoidal wave traveling in the –z direction with the same velocity c.

7. 8.2 Plane Waves in Lossless Media
The magnetic field can be found from Eq. (7-104a),
Intrinsic impedance of the free space,
- η0 is a real number, is in phase with
The instantaneous expression for H
- for a uniform plane wave the ratio of the magnitudes of E and H is the intrinsic impedance of the medium.
- H is perpendicular to E and both are normal to the direction of propagation.

8. Ex.8-1 p.358 Sol) (a) Sol) (b) Sol) (c)
E = axEx propagates in a lossless simple media (εr =4, μr =1, σ=0) in the +z direction. Assume sinusoidal 100 MHz and peak 104 (V/m) at t =0 and z = 1/8.
Instantaneous expression for E
Instantaneous expression for H
Location where Ex is a positive maximum when t = 10-8 (s)
Sol) (a)
Sol) (b)
Sol) (c)

9. 8.2 Plane Waves in Lossless Media

10. 8.2 Plane Waves in Lossless Media 8-2.1 Doppler Effect
When there is relative motion between a time-harmonic source and a receiver, the frequency detected by the receiver tends to be different from that emitted by the source. T` u u
uΔt r` T θ θ r0 r0 R R
At t=0
At t=Δt
- Let us assume that the source T of a time-harmonic wave of a frequency f moves with a velocity u at an angle θ.
- the maximum value of electromagnetic wave will reach R at t1=r0/c.
- new position T`, the next maximum value of wave emitted by T` at Δt will reach R at t2. (f = 1/Δt)
- the time elapsed at R,
RF speed gun, red-shift of star
(Taylor expansion)

11. 8. 2 Plane Waves in Lossless Media 8-2
8.2 Plane Waves in Lossless Media Transverse Electromagnetic Waves
TEM wave : the E and H are perpendicular to each other, and both are transverse to the direction of propagation.
The phasor electric field intensity for a uniform plane wave propagating in arbitrary direction.
- to satisfy the homogeneous Helmholtz’s equation,
Wavenumber vector
In this case, is E perpendicular to the propagation direction?
- Radius (position) vector at any point on the plane x y z P
Plane of constant phase (phase front)

12. 8. 2 Plane Waves in Lossless Media 8-2
8.2 Plane Waves in Lossless Media Transverse Electromagnetic Waves
- the equation of a plane normal to an , the direction of propagation. x y z P
= A constant
- In a charge-free region,
which requires
is transverse to the direction of propagation!

13. 8. 2 Plane Waves in Lossless Media 8-2
8.2 Plane Waves in Lossless Media Transverse Electromagnetic Waves
- The magnetic field associated with may be obtained from Eq.(7-104a) or
where
Finally,
by Eq. (8-26) & Eq. (8-29)
It is now clear that a uniform plane wave propagating in an arbitrary direction, , is a TEM wave with and that both and are normal to

14. 8.2 Plane Waves in Lossless Media 8-2.3 Polarization of Plane Waves
When the E vector of the plane wave is fixed in the x-direction , the wave is said to be linearly polarized in the x-direction.
Consider the superposition of two linearly polarized waves: one polarized in the x-direction, and the other polarized in the y-direction and lagging 90˚ in time phase.
- Phasor notation
where E10 and E20 are the amplitudes of the two linearly polarized waves.
- Instantaneous expression

15. 8.2 Plane Waves in Lossless Media 8-2.3 Polarization of Plane Waves
- Set z=0,
- Analytically, y
- Which leads to the following equation for an ellipse:
E(0,t) ω E2 α x E1
- Circularly polarized if
<Circular polarization>
- Elliptically polarized if

16. 8.2 Plane Waves in Lossless Media 8-2.3 Polarization of Plane Waves
- When ,
( E rotates at a uniform rate with an angular velocity ω in a counterclockwise direction.)
- When E2(z) leads E1(z) by 90˚ in time phase,
(E will rotate with an angular velocity ω in a clockwise direction : left-hand or negative circularly polarized wave.)

17. 8.2 Plane Waves in Lossless Media 8-2.3 Polarization of Plane Waves
If two waves are in space quadrature,
but in time phase. y
E20 x
E10
<Linear polarization>
P366 AM FM, TV antennas

18. 8.3 Plane Waves in Lossy Media
In a source-free lossy medium, the homogeneous vector Helmholtz’s equation :
- complex wavenumber
- Propagation constant :
- using Eq. (7-110), Eq. (7-114),
Helmholtz’s equation using propagation constant :
α : attenuation constant (Np/m).
β : phase constant (rad/m).

19. 8.3 Plane Waves in Lossy Media 8-3.1 Low-Loss Dielectrics
- imperfect insulator with nonzero equivalent conductivity
- by using the binomial expansion:
: Approximately proportional to the frequency.
: deviates very slightly from the value for a perfect (lossless) dielectric.
- Intrinsic impedance :
- The electric and magnetic field intensities in a lossy dielectric are thus not in time phase, as they are in a lossless medium.
- Phase velocity :

20. 8.3 Plane Waves in Lossy Media 8-3.2 Good Conductors
A medium for which
- Intrinsic impedance :
phase angle of 45˚. Hence the magnetic field intensity lags behind the electric field intensity by 45˚.
- Phase velocity in a good conductor
- Wavelength in a good conductor
- Skin depth : due to e-αz, the amplitude of a wave will be attenuated by e –1 =0.368 when it travels a distance δ=1/α (skin depth) in good conductor and at high frequency.

21. Ex. 8-4, p370 – Crimson Tide
E = ax100 cos(107πt) propagating + z direction in seawater (εr = 72, μr = 1, σ = 4 (S/m))
(a) Attenuation constant, phase constant, intrinsic impedance, phase velocity, wavelength, skin depth
(b) Distance at which the amplitude of E is 1% of its value at z = 0.
(c) Expression for E(0.8, t) and H(0.8, t).
EM Field attenuates very MHz.
Hard to communicate with submarine.
Communication with 30~300 Hz wave
Movie ‘Crimson Tide’

22. 8.3 Plane Waves in Lossy Media 8-3.3 Ionized Gases
Ionosphere : from 50 to 500 (km) in altitude, layers of ionized gases. This layers consist of free electrons and positive ions.
- Ionized gases with equal electron and ion densities are called plasma.
- In ionosphere, the electrons are accelerated more by the electric fields of electromagnetic waves for communication.
Troposphere:~20 km, Stratosphere: 20~50 Km
An electron of charge –e in a time-harmonic electric field (mass m, in x-direction, angular frequency ω.)
- Such a displacement gives rise to an electric dipole moment:
- If there are N electrons per unit volume, the polarization vector:
- Plasma frequency:

23. 8.3 Plane Waves in Lossy Media 8-3.3 Ionized Gases
- The equivalent permittivity of the ionosphere or plasma,
- The propagation constant:
- The intrinsic impedance:
If , becomes purely real. (attenuation without propagation)
becomes purely imaginary. (reactive load with no transmission of power)
( : cutoff frequency.)
If f >fp, γ becomes purely imaginary. (electromagnetic waves propagate unattenuated in the plasma.
Using the value of :
- The electron density of ionosphere range from 1010/m3 to 1010/m3.
- For communication beyond the ionosphere, we must use frequencies much higher than 9 (MHz).
- The signal with frequencies lower than 0.9 (MHz) propagate very far around the earth by reflections at the ionosphere’s boundary and the earth’s surface (HAM).