IDEAL TRANSMISSION-LINE FUNDAMENTALS - Advanced Signal Integrity for High-Speed Digital Design - page 93

calculated from (3-45) if the capacitance is calculated with ε r = 1:

1
c 2 C ε r = 1

L = (3-46)

where c is the speed of light in a vacuum and C ε r = 1 is the capacitance with the
relative dielectric permittivity ε r

set to 1. Substituting (3-44) into (3-46) with

ε r = 1 gives the value of the inductance per unit length for a coaxial line :

ln (b/a)
c 2 2 πε 0

L = H / m (3-47)

3.4.3 Transmission-Line Parameters for a Microstrip

To derive the impedance of a microstrip transmission line, we begin with the
solution to Laplace's equation. To begin the derivation, consider Figure 3-18,
which shows the boundary conditions that we must satisfy when solving the
differential equations. Note that the microstrip is surrounded by a box with con-
ducting walls at . This is a valid placement of

the boundary conditions only if d h , where h is the height of the dielectric.
In this case, the partial differential equations are solved using a method called
separation of variables , as described by Jackson [1999]. The rectangular nature
of the microstrip geometry allows us to remain in rectangular coordinates, which
gives the two-dimensional Laplace equation the form

x =± d/ 2, y = 0, and y =∞

∂ 2
∂x 2

∂ 2
∂y 2

2

∇ = + = 0 (3-48)

y

∞

e 0

w w

−

2 2

h

e r

d d

−

2 2

x

Figure 3-18 Dimensions used to derive microstrip transmission-line parameters from
Laplace's equation.