Chapter 7 Plane Electromagnetic Waves
A uniform plane wave is characterized by E and H that have uniform properties at all points across an infinite plane perpendicular to the direction of propagation.
If x-y is the infinite plane, then
7.2 Plane Wave in Lossless Media 7.3 Plane Wave in Lossy Media 7.4 Group Velocity 7.6 Flow of Electromagnetic Power and the Poynting Vector 7.6 Normal Incidence of Plane Waves at Plane Boundaries
7-2 Plane Wave In Lossless Media
What does “Lossless” mean? If the medium is noconducting (s = 0 ), then the wave doesn’t suffer any attenuation as it travels through the medium, and hence the medium is said to be lossless.
Homogeneous vector Wave equations for E, H: (Source-free medium)
Phase velocity:
Homogeneous vector Helmholtz’ equations for phasor Es, Hs:
Wavenumber:
1 7-2 Plane Wave In Lossless Media
What are the solutions?
Write vector Helmholtz’ equations to scalar Helmholtz’ equations for x component:
Uniform plane wave:
Simplified equation:
Solution of scalar Helmholtz’ equations:
Possible solutions:
Wave traveling in: positive z, negative z directions
7-2 Plane Wave In Lossless Media
Phasors E, H for a wave traveling in + z direction:
Phasor E: using
Phasor H:
Instantaneous expression for E, H:
2 7-2 Plane Wave In Lossless Media
Characteristics of a traveling wave:
o Phase velocity:
A constant phase
The velocity of propagation of an equiphase front for single-frequency plane wave
o Relation between k and wavelength l :
o Intrinsic impedance of medium: (W)
In free space (air): (W)
7-2 Plane Wave In Lossless Media
Example 7-1: p277
A uniform plane wave with E = ax Ex propagates in a lossless simple medium (er = 4, mr = 1, s = 0) in the +z-direction. Assume that Ex is sinusoidal with a frequency 100 MHz 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 for any t and z. (b) Write the instantaneous expression for H -8 (c) Determine the locations where Ex is a positive maximum when t = +10 (s)
3 7-2.1 Doppler Effect
Doppler effect is a shift in the frequency of a wave caused by the motion of the transmitting source, or the receiving system
If the transmitter with a velocity u at an angle q relative to the direct line to a stationary receiver, then the frequency perceived by the observer is:
u T q R
The frequency of a wave detected by a receiver is higher (lower) than that emitted by a transmitter if the transmitter moves toward (away from) the receiver
4 7-2.2 Transverse Electromagnetic Waves
So far, we’ve seen: E along x-direction, and H along y-direction, and both transverse to the direction of propagation (z-direction). TEM wave
We discussed the case with k along z-direction, if E along x-direction, then H along y-direction. What if k along any arbitrary direction, how about the general relation between E, and H ?
Wavenumber Vector k for arbitrary propagation direction:
Using Eqs.
General relation between E and H:
7-2.2 Transverse Electromagnetic Waves
Example (not from our textbook)
A 10-MHz uniform plane wave is traveling in a nonmagnetic medium with
m = mo and er = 9. Find (a) the phase velocity, (b) the wavenumber, (c) the wavelength in the medium, and (d) the intrinsic impedance of the medium.
5 7-2.2 Transverse Electromagnetic Waves
Example (not from our textbook)
The electric field phasor of a uniform plane wave traveling in a lossless medium with an intrinsic impedance of 188.5 W is given by (mV/m) Determine: (a) the associated magnetic field phasor
(b) the instantaneous expression for E(y,t) the medium is nonmagnetic (m = mo)
7-2.3 Polarization of Plane wave
Linearly polarized Elliptically polarized (pls ignore) Circular polarized (pls ignore)
Linearly polarized: The E vector of a plane wave is fixed in one direction, such a wave is said to be linearly polarized.
For example:
x-polarized uniform plane wave: E = axEx
y-polarized uniform plane wave: E = ayEy
6 7-3 Plane Waves in Lossy Media
So far, we consider case of “Source-free lossless simple media”: r v=0 and J = 0 Question: What about a lossy medium (i.e., conducting medium : s 0) ?
Introduce Complex Permittivity of lossy medium:
Note: in lossless medium, we have: where k is wavenumber.
Introduce Loss tangent , and Loss angle:
Introduce Propagation Constant g : Attenuation Constant a : Phase Constant b :
7-3 Plane Waves in Lossy Media
Question: Can we find attenuation constant and phase constant (a, b) in terms of e’, e”, where e’ = e, e”= s/w ?
Two extreme cases: Low-loss medium; Good conductor
Low-loss medium: Good conductor:
7 7-3 Plane Waves in Lossy Media
Two extreme cases: Low-loss medium; Good conductor
Low-loss medium:
Attenuation constant: (Np/m)
Phase constant: (rad/m)
Intrinsic impedance: (W)
Phase velocity: (m/s)
Wavelength: (m)
7-3 Plane Waves in Lossy Media
Two extreme cases: Low-loss medium; Good conductor
Good conductor:
Attenuation constant: (Np/m) Phase constant: (rad/m)
Intrinsic impedance: (W) (H lags behind E by 45o) Phase velocity: (m/s)
Wavelength: (m)
Skin depth: (m)
(The distance d through which wave magnitude decreases by a factor of e-1.)
8 7-3 Plane Waves in Lossy Media
Skin depth: (m)
Flow of electromagnetic power and the Poynting vector (p298):
P = E X H (W/m2) P H Poynting vector represents the density and the direction E of the power flow.
Current flow in a good conductor :
Skin Effect : (m)
When AC-case, at very high frequencies most of the current flows through a thin outer layer of the wire.
7-3 Plane Waves in Lossy Media
For good conductor: 1. Attenuation constant and phase constant are numerically equal: 2. The intrinsic impedance of a good conductor has a phase angle of 45 3. Skin depth is a measure of how far the wave penetrates into the conductor:
Example 7-4: p293 The electric field intensity of a linearly polarized uniform plane wave propagating in the +z-direction in seawater is: (V/m) at z = 0. The constitutive parameters of seawater are and (S/m).
(a) Determine the attenuation constant, phase constant, intrinsic impedance, phase velocity, wavelength, and skin depth. (b) Find the distance at which the amplitude of E is 1% of its value at z=0. (c) Write the expressions for E(z,t) and H(z,t) at z = 0.8 (m) as functions of t.
9 Example 7-4: p293 The electric field intensity of a linearly polarized uniform plane wave propagating in the +z-direction in seawater is: (V/m) at z = 0. The constitutive parameters of seawater are and (S/m).
(a) Determine the attenuation constant, phase constant, intrinsic impedance, phase velocity, wavelength, and skin depth. (b) Find the distance at which the amplitude of E is 1% of its value at z=0. (c) Write the expressions for E(z,t) and H(z,t) at z = 0.8 (m) as functions of t.
7-4 Group Velocity
Sum of two time-harmonic traveling waves of equal amplitude and slightly different frequencies at a given time t
Phase velocity: the velocity of Group velocity: the velocity of propagation of an equiphase wavefront propagation of the envelope (of a group (of a single frequency) of frequencies)
For plane waves in a lossless medium, up is a constant; however, in lossy dielectric, waves with different frequencies will propagate with different up, which will cause a signal distortion.
The phenomenon of signal distortion caused by a dependency of up on frequency is called dispersion. A lossy dielectric is a dispersive medium.
10 Chapter 7 Plane Electromagnetic Waves
7-5 Flow of Electromagnetic Power and the Poynting Vector
Electromagnetic waves carry with them electromagnetic power
Maxwell Eqs.
Identity of Vector Operation:
Ohmic Power density (W/m3) Poynting Vector P Energy density u: 2 P (W/m ) (J/m3)
Electric Magnetic
7-5 Flow of Electromagnetic Power and the Poynting Vector
Poynting Theorem:
The net power flowing into a closed surface S, is equal to the sum of the rates of increase of the stored electric and magnetic energies and the ohmic power dissipated within the enclosed volume V
The Poynting vector is in a direction normal to both E and H
Poynting Vector: P (W/m2)
Poynting’s theorem is a manifestation of the principle of conservation of energy
11 7-5 Flow of Electromagnetic Power and the Poynting Vector
Poynting Theorem:
Example 7-5: P300 Find the Poynting vector on the surface of a long, straight conducting wire (of radius b and conductivity s ) that carriers a direct current I. Verify Poynting’s theorem.
7-5.1 Instantaneous and Average Power Densities
Phasor: Instantaneous expression:
For a plane wave, associated H in a lossy medium: Instantaneous expression:
Instantaneous expression for the Poynting vector:
P
Average power density transmitted by a uniform plane wave in z-direction: Special case: lossless medium:
Pav P Pav
General expression for Average power density in a propagating wave:
Pav
12 7-5.1 Instantaneous and Average Power Densities
Example 7-6: P303 The phasor expression of the far field at a distance R from a short vertical current element Idl located in free space at the origin of a spherical coordinate system are
and
Where l = 2 p/b is the wavelength. a) Write the expression for instantaneous Poynting vector. b) Find the total average power radiated by the current element.
Example 7-6: P303 The phasor expression of the far field at a distance R from a short vertical current element Idl located in free space at the origin of a spherical coordinate system are
and
Where l = 2 p/b is the wavelength. a) Write the expression for instantaneous Poynting vector. b) Find the total average power radiated by the current element.
13 Chapter 7 Plane Electromagnetic Waves
7-6 Normal Incidence of Plane Waves at Plane Boundaries
x Ei Et Incident wave
ki kt Hi Ht
Er y z
kr Hr
Medium 2 ( , ) Medium 1 (e1, m1) e2 m2
Question: How do we find the reflection coefficient and transmission coefficient?
7-6 Normal Incidence of Plane Waves at Plane Boundaries
Parameters: b1, b2, h1, h2 ; Eio, Ero, Eto
Step 1: write H, E for reflected and transmitted wave
Reflected wave Transmitted wave
Step 2: Simplify above equations according to boundary conditions: Boundary conditions for electrostatics and magnetostatics remain valid for time-varying field as well.
14 7-6 Normal Incidence of Plane Waves at Plane Boundaries
Step 3: Solve Equations:
Reflection coefficient G : (Normal incidence)
Transmission coefficient t :
Relation between reflection and transmission coefficients for normal incidence:
7-6 Normal Incidence of Plane Waves at Plane Boundaries
Incident wave + reflected wave Standing wave
E, and H in medium 1
Standing-wave ratio (SWR) :
SWR in dB :
15 7-6 Normal Incidence of Plane Waves at Plane Boundaries
E, and H in medium 1 E, and H in medium 2
Exercise 7.11: p309 Given in air that impinges
normally on a lossless medium with er2 = 2.25, mr2 = 1 in the z >= 0 region. Find
(a) b, G, S, t;
(b) Er (z,t); (c) E2 (z,t); (d) H2 (z,t)
7-6.1 Normal incidence on a good conductor
Good conductor s/ we >> 1, such as metallic reflector and waveguide, we may use perfect approximation (s )
Incident wave Reflected wave
E, and H in medium 1 (Phasor)
E, and H in medium 1 (Instantaneous expression)
16 7-6.1 Normal incidence on a good conductor
Example 7-8: p310
A y-polarized uniform plane wave (Ei, Hi) with a frequency 100 MHz propagates in air in the +x direction and impinges normally on a perfectly conducting plane at
x=0. Assuming the amplitude of Ei, to be 6 (mV/m), write the phasor and instantaneous expressions for
(a) Ei, Hi of the incident wave (b) Er, Hr of the reflected wave (c) E1, H1 of the total wave in air.
17 Chapter 7 Plane Electromagnetic Waves Problems 7-20: p333 A uniform sinusoidal plane wave in air with the following phasor expression for electric intensity is incident on a perfectly conducting plane at z = 0 a) Find the frequency and wavelength of the wave b) Write the instantaneous expressions for Ei (x,z;t) and Hi (x,z;t). c) Determine the angle of incidence. d) Find Er (x,z) and Hr (x,z) of the reflected wave. e) Find E1 (x,z) and H1 (x,z) of the total field in air.
18 Chapter 7 Plane Electromagnetic Waves
Summary: Examined the behavior of uniform plane waves in both lossless and lossy media Explained Doppler effect when there is relative motion between a time-harmonic source and a receiver Explained the significant of a complex wavenumber and a complex propagation constant in lossy medium Studied the skin effect in conductors and obtained the formula for skin depth Introduced the concept of signal dispersion and explained the difference between phase and group velocities Discussed the flow of electromagnetic power and Poynting’s theorem Studied the reflection and refraction of electromagnetic waves at plane boundaries for normal incidence
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