Page 315 - Electromagnetics

For TE polarization we have from (4.213)
˜
1 ∂ H z (ρ, ω)
˜
E φ (ρ, ω) =− (4.328)
c
jω˜ (ω) ∂ρ
c ˜
where ˜ = ˜ + ˜σ/jω is the complex permittivityintroduced in § 4.4.1. Since E =
ˆ ˜
˜
˜
ˆ ˜
˜
φE φ + ˆ zE z and H = φH φ + ˆ zH z , we can always decompose a cylindrical electromagnetic
wave into cases of electric and magnetic polarization. In each case the resulting field is
˜
˜
TEM ρ since E, H, and ˆρ are mutuallyorthogonal.
˜
˜
Wave equations for E z in the electric polarization case and for H z in the magnetic
polarization case can be derived bysubstituting (4.210) into (4.208):
∂ 1 ∂ 2 E z
2 ˜
+ + k ˜ = 0.
∂ρ 2 ρ ∂ρ H z
Thus the electric field must obeythe ordinarydifferential equation
2 ˜ ˜
d E z 1 dE z 2 ˜
+ + k Ez = 0. (4.329)
dρ 2 ρ dρ
This is merelyBessel’s equation (A.124). It is a second-order equation with two inde-
pendent solutions chosen from the list
(1) (2)
J 0 (kρ), Y 0 (kρ), H (kρ), H (kρ).
0 0
We find that J 0 (kρ) and Y 0 (kρ) are useful for describing standing waves between bound-
(1) (2)
aries, while H (kρ) and H (kρ) are useful for describing waves propagating in the
0 0
ρ-direction. Of these, H (1) (kρ) represents waves traveling inward while H (2) (kρ) repre-
0 0
sents waves traveling outward. At this point we are interested in studying the behavior
of outward propagating waves and so we choose
j (2)
˜
˜
E z (ρ, ω) =− E z0 (ω)H 0 (kρ). (4.330)
4
˜
As explained in § 2.10.7, E z0 (ω) is the amplitude spectrum of the wave, while the term
− j/4 is included to make the conversion to the time domain more convenient. By(4.327)
we have
˜ 1 ∂
j
˜ 1 ∂E z ˜ (2)
H φ = = − E z0 H 0 (kρ) . (4.331)
jω ˜µ ∂ρ jω ˜µ ∂ρ 4
(2) (2)
Using dH (x)/dx =−H (x) we find that
0 1
˜
1 E z0
˜ (2)
H φ = H 1 (kρ) (4.332)
Z TM 4
where
ω ˜µ
Z TM =
k
is called the TM wave impedance.
˜
For the case of magnetic polarization, the field H z must satisfyBessel’s equation
(4.329). Thus we choose
j (2)
˜
˜
H z (ρ, ω) =− H z0 (ω)H 0 (kρ). (4.333)
4
© 2001 by CRC Press LLC