ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 29

equations using the procedure employed in Section 2.3.1. Again, assume a
source-free, linear, homogeneous medium:

∇× ( ∇× E) =− jωµ( ∇× H)

The formula can be further simplified by using the following vector identity
(Appendix A):

∇× ( ∇× E) =∇ ( ∇· E) −∇

2

E

Since we have assumed a source-free medium, the charge density is zero ( ρ = 0)

∇· E = and Gauss's law reduces to 0, yielding

2

E + j 2 ω 2 µε E =∇

2

E − ω 2 µε E =

∇ 0 (2-38)

Substituting γ 2 = ω 2 µε yields

E − γ 2 E =

2

∇ 0 (2-39)

which is the time-harmonic plane-wave equation for the electric field, where γ
is known as the propagation constant.
If the solution is limited to plane waves propagating in the z -direction that have
an electric field component only in the x -direction, the wave equation becomes
(see Appendix A)

∂ 2 E x
∂x 2

∂ 2 E x
∂y 2

∂ 2 E x
∂z 2

∂ 2 E x
∂z 2

E =

2

− γ 2 E x = ∇ + + =

0

(2-40)

∂ 2 E x
∂z 2

− γ 2 E x = 0

which is a second-order ordinary differential equation with the general solution

E x = C 1 e − γz

+ C 2 e γz

(2-41)

where C 1 and C 2 are determined by the boundary conditions of the particular
problem.
As discussed in Chapter 3, equation (2-41) and its magnetic field equiva-
lent will prove to be particularly important for signal integrity because they
describe the propagation of a signal on a transmission line. The first term,
C 1 e − γz , describes completely the forward-traveling part of the wave propagating
in the z -direction (i.e., down the length of the transmission line), and the second
term, C 2 e + γz , describes the propagation of the rearward-traveling wave in the
− z -direction. Observing equation (2-41) allows the definition of an important
term, the propagation constant :

γ = α + jβ (2-42)