CH 8. Plane Electromagnetic Waves Chap 8.1~8.3

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 )

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.

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.)

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

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.

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.

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)

8.2 Plane Waves in Lossless Media

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)

8. 2 Plane Waves in Lossless Media 8-2 8.2 Plane Waves in Lossless Media 8-2.2 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)

8. 2 Plane Waves in Lossless Media 8-2 8.2 Plane Waves in Lossless Media 8-2.2 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!

8. 2 Plane Waves in Lossless Media 8-2 8.2 Plane Waves in Lossless Media 8-2.2 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 .

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

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

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.)

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

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).

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 :

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.

Ex. 8-4, p370 – Crimson Tide E = ax100 cos(107πt) (V/m) @z=0, 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 rapidly @5 MHz. Hard to communicate with submarine. Communication with 30~300 Hz wave Movie ‘Crimson Tide’

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:

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).