IDEAL TRANSMISSION-LINE FUNDAMENTALS - Advanced Signal Integrity for High-Speed Digital Design - page 92

Note that ln (a/b) =− ln (b/a) , which is done here as a convenience:

V
ln (b/a)

V
ln (b/a)

V
ln (b/a)

ln b
r

(r) =− ln (r) + ln (b) =

The electric field is computed using (2-65):

ln (b/a) ln b

V ∂

∂r ln (b/a) ln b

V V

r ln (b/a)
(3-41)

E =−∇ =−∇

=− a r = a r

r r

The next step is to find the total surface charge on the signal conductor using
(3-3), which describes the boundary conditions between a dielectric and a perfect
conductor:

ε E C / m 2 n · = ρ (3-3)

At r = a (the inner conductor surface), the surface charge is calculated with

εV
a ln (b/a) = ρ

C / m 2 εE = (3-42)

Multiplying (3-42) by the circumference of the inner conductor will give the
equivalent surface charge per unit length:

Q
l = ρ 2 πa =

2 πεV
ln (b/a)

C / m (3-43)

Using (2-76) we can get the capacitance per unit length of a coaxial transmission
line :

Q/ l
V

2 πε
ln (b/a)

C = = F / m (3-44)

To find the inductance per unit length, we exploit the relationship between the
magnetic and electric fields:

1
√ µ r µ 0 ε r ε 0

c
√ µ r ε r

= m / s (2-52)

ν p =

ω
β =

1
√ LC

m / s (3-31)

which allows us to write (assuming µ r = 1)

c
√ ε r =

1
√ LC (3-45)

Since µ r = 1, the dielectric properties do not influence the inductance. Conse-

quently, since the speed of light in a vacuum is constant, the inductance can be