ABSTRACT
In an ideal transmission line, the voltage phasor V satisfies the ordinary differential equation 2.54:
∂ ∂ + =
V
z Vβ ,
where
β ω ω2 = =2 2L C v’ ’
.
The time-harmonic equation in three dimensions (Helmholtz equation) is given by
∇ + = 2 2 E Ek 0,
where
k
v 2
= =
ω µεω2
and ∇ 2 is the Laplacian operator. If the frequency is zero ( f = 0) or low such that β2 2 0= ≈k , the Helmholtz equation can
be approximated by the Laplace equation. Static or low-frequency problems satisfy the Laplace equation
∇ = 2Φ 0, (6.1)
where Φ is called the potential. In the first undergraduate course in electromagnetics, the one-dimensional solution of the Laplace equation in various coordinate systems is discussed and illustrated through simple examples. In the following section, we list these solutions for completeness.
