IDEAL TRANSMISSION-LINE FUNDAMENTALS - Advanced Signal Integrity for High-Speed Digital Design - page 100

in the behavior of the fields only close to the edge, the details of the reference
conductor far away are not considered.
We begin the derivation with Laplace's equation in cylindrical coordinates:

r ∂
∂r r 2 ∂ 2

1
r

∂
∂r

1

2

∇ = + = 0 (3-75)

∂φ 2

Similar to the solution of the microstrip transmission line, the solution to this
partial differential equation is found using separation of variables. The problem
is reduced to two ordinary differential equations, assuming that the potential can
be represented by the product of a function for each coordinate [Jackson, 1999]:

(r, φ) = R(r)Y(φ) (3-76)

R(r)Y(φ)/r 2 Substitution of (3-76) into (3-75) and dividing each side by

produces

r ∂RY
∂r

r ∂R
∂r

r 2
RYr RYr 2 ∂ 2 RY

r 2 ∂ 2 Y

∂φ 2

∂
∂r

r
R

∂
∂r

1
Y

+ = + = 0 (3-77)

∂φ 2

Since the sum of the ordinary differential equations must equal zero, we can
solve each one separately:

r dR
dr

d
dr

= β 2 R r (3-78)

d 2 Y
dφ 2

=− β 2 Y (3-79)

For the solution of (3-78), try R = kr l :

r d(kr l )
dr

d
dr

d
dr (rklr l − 1 ) = r

d
dr klr l

= kl 2 r l r = r

Setting the solution above to β 2 R yields

β 2 R = β 2 (kr l ) = l 2 (kr l )
l =± β

Hence, the solution to (3-78) is shown:

R(r) = ar β + br − β (3-80)

The solution to (3-79) is found by solving the roots of the characteristic equation

Y(D 2

+ β 2 ) =

0