IDEAL TRANSMISSION-LINE FUNDAMENTALS - Advanced Signal Integrity for High-Speed Digital Design - page 75

where C is normalized to length with units of farads per meter. As a result, for
a specific transmission-line geometry, we can write the left-hand side of (2-30)
in terms of the capacitance:

ε ∂E x
∂t

→ C ∂v
∂t

(3-15)

To derive the circuit equivalent of the right side of (2-30), recall from equation
(2-79) that current is the rate of charge flow per second:

dQ
dt

i = (3-16)

Substituting Q = Cv into equation (3-16) produces the current in terms of the
voltage and capacitance:

i = C dv
dt

(3-17)

which is equal to equation (3-15). Therefore, (2-30) can be rewritten in terms of
circuit parameters:

∂i(z,t)
∂z

=− C ∂v(z,t)
∂t

(3-18)

Note that (3-18) is the classic response of a capacitor from circuit theory.
Equations (3-12) and (3-18) are the loss-free forms of the telegrapher's
equations, which describe the electrical characteristics of a transmission line .

3.2.3 Equivalent Circuit for the Loss-Free Case

Although signal integrity is largely a study in electromagnetic theory, application
of the discipline is performed almost entirely using circuit parameters because
they are more intuitive to most engineers. Consequently, it is necessary to derive
a model for the transmission line in terms of the equivalent inductance L and
the capacitance C per unit length. In this section the equivalent circuit of a
transmission line is developed for the loss-free case. The model is refined in
Chapters 5 and 6 to include loss from nonperfect dielectrics and finite conductivity
conductors.
To begin, consider a differential element of transmission line with a length
of z as shown in Figure 3-8, which represents a section of a transmission line
with a signal conductor and a reference conductor. If we assume that current
is traveling down the signal conductor and returning on the reference conductor
(remember from circuit theory that current must always complete a loop along
a return path), it can be represented by a series of differential current elements,