Download ˆ with E(x,y,z,t)=Re (x,y,z)e EG

Chapter 2 Wave in an Unbounded Medium
Maxwell’s Equations
Time-harmonic form
(Real quantity)
G
G
∂B
∇× E = −
∂t
G
∇ ⋅ D = ρ
G
G G ∂D
∇× H = J +
∂t
G
∇⋅B = 0
with
Phasor form
(complex quantity)
∇ × Eˆ = − jω Bˆ
(1.11a )
∇ ⋅ Dˆ = ρ
(1.11b)
∇ × Hˆ = Jˆ + jω Dˆ
(1.11c)
∇ ⋅ Bˆ =0
(1.11d)
G
ˆ (x,y,z)e jω t
E(x,y,z,t)=Re E
{
}
Chapter 2.1 Plane Waves in a Simple, Source-Free, and
Lossless Medium
Maxwell’s Equations
G
G
∂B
∇× E = −
∂t
G
∇ ⋅ D = ρ
G
G G ∂D
∇× H = J +
∂t
G
∇⋅B = 0
(2.1a )
(2.1b)
(2.1c)
(2.1d )
2.1 Plane Wave in a Simple, Source-free, and Lossless Medium
The general equations for electric and magnetic fields:
G
2
G
∂ E
2
(2.2)
∇ E − με 2 = 0
∂t
G
2
G
∂ H
2
(2.3)
∇ H − με 2 = 0
∂t
If we restric the electric field as the function z, and the electric field
to one component E x only, equation (2.2) becomes:
∂ 2 Ex
∂ 2 Ex
− με
=0
2
2
∂z
∂t
(2.5)
the general solution of (2.5) is in the form of:
Ex =p1 (z-v p t) + p 2 (z+v p t)
(2.6)
2.1.1
G
G
The Relation between E and H
Assume we have a electric field of uniform plane wave:
Ex =p1 (z-v p t)
G
The corresponding H(z) is not independent, and must be resolved
from Maxwell equations (2.1a) and /or (2.1c):
ε
1
H y =( )p1 (z-v p t)= ( )p1 (z-v p t)
μ
η
where η = μ / ε is called "intrinsic impedance" of the medium
(in unit ohms).
2.2 Time-Harmonic Uniform Plane Waves in a Lossless
Medium
The electric field of a uniform plane wave in a lossless
medium is
{
Ex ( z , t ) = Re (C1e − j β t + C1e + j β t )e jωt )
}
or
Ex ( z , t ) = C1 cos(ωt − β z ) + C2 cos(ωt + β z )
(2.11)
(2.12)
The electric and magnetic fields of a time-harmonic wave
propagating in the + Z direction are
E x ( z , t ) = C1 cos( wt − β t )
H y ( z, t ) =
1
η
C1 cos( wt − β t )
where the "intrinsic impedance" (in unit ohms)
η = μ / ε =ωμ /β
(note:
β =ω με
).
(2.13)
Example 2-1:
AM broadcast Singnal
The instantaneous expression for the electric field component of an AM
broadcast signal propagating in air is given by
G
G
E(x,t)=z10cos(1.5π × 106t + β x)
V − m −1
(a) Determine the direction of propagation and frequency f.
(b) Determine the phase constant β and wavelength.
G
(c) Find the instantaneous expression for the corresponding H(x,t).
Example 2-2:
FM broadcast Singnal
An FM broadcast signal traveling in y direction in air has a magnetic
field given by the phasor:
G G
ˆ
A − m −1
H(y)=
2.92 × 10-3e-j0.68π y (-x+zj)
(a) Determine the frequency .
ˆ
(b) Find the corresponding E(y).
G
G
(c) Find the instantaneous expression for the E(x,t) and H(x,t).
2.3 Plane Waves in Lossy Media
The electric field of a uniform plane wave in a lossy
medium is
{
Ex ( z , t ) = Re (C1e−α z e− j β t + C2 eα z e+ j β t )e jωt )
}
or
Ex ( z , t ) = C1e −α z cos(ωt − β z ) + C2 eα z cos(ωt + β z )
(2.18)
The propagation constant γ =α +iβ :
σ 2
) − 1]1/ 2
2
ωε
με
σ
β =ω
[ 1 + ( ) 2 + 1]1/ 2
ωε
2
α =ω
με
[ 1+ (
np − m −1
rad − m −1
(2 − 19)
The waves propagate in the +Z directions are
Ex+ ( z , t ) = C1e −α z cos(ωt − β z )
+
x
H ( z, t ) =
1
ηc
C1e −α z cos(ωt − β z + φn )
(2.21)
(2.23)
where intrinsic impedance ηc is:
ηc = ηc e
jφn
μ
μ
=
=
σ
ε eff
ε−j
ω
μ
ε
=
e j (1/ 2) tan
σ
[1 + ( ) 2 ]1/ 4
ωε
−1
[σ /(ωε )]
(2 − 22)
Example 2-5
Find the complex propagation constant γ and the
intrisic impedance ηc of a microwave signal in
muscle tissue at 915 MHz (σ =1.6 s-m -1 ε r = 51)
Example 2-6
Consier distilled water at 25 GHz (ε cr = 34 − j 9.01).
Calculate the attenuation constant α , phase constant
β , penetration depth d, and the wavelength λ.
2.4 Electromagnetic Energy Flow and the Poynting Vector
Poynting Theorem:
2
G G
G G
1 G
1 G2
∂
G
∫V E ⋅ Jdv = − ∂t ∫V ( 2 μ H + 2 ε E )dv − v∫ S ( E ×H ) ⋅ ds
(2.31)
Poynting Vecor:
G G G
S = E×H
Example 2-12: Wire carrying direct current.
Consider a long cylindrical conductor of conductivity
σ and radius a, carrying a direct current I as shown
in the figure. Find the power dissipated in a portion
of the wire of length l , using
(a) the left-hand side of (2.31); (b) he right-hand side of (2.31)
Example 2-13 Power flow in a coaxial line
Consider a coaxial line delivering power to a resistor as
shown in the figure. Assume the wire to be perfect conductor,
so that there is no power dissipation in the wires, and the
electric field inside them is zero. The configurations of the
electric and magnetic field lines in the coaxial line are shown.
Find the power delivered to the resistor R.
Example 2-18 VHF/UHF broadcast radiation
A survey conducted in the United State indicated that ~50% of
the population is exposed to average power densities of 0.005
μ W-(cm)-2 due to VHF and UHF broadast radiaction. Find the
correcponding amplitudes of the electric and magnetic fields.
Example 2-19 VLF waves in the ocean.
Consider VLF wave progation in the ocean. Find the
time-average Poynting flux at the sea surface (z=0)
and at the depth of z=100m. Assume:
G
Ê(z)=x2.84e-0.218z e − j 0.218 z k V-m -1
G
Ĥ(z)=x2.84e-0.218z e − j 0.218 z e − jπ / 4 k A-m -1