NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 242

in radius must be calculated so that the total surface area is equal to that of the
roughness profile.

5.4 TRANSMISSION-LINE PARAMETERS FOR NONIDEAL
CONDUCTORS

As bus data rates increase and physical implementations of high-speed digi-
tal designs shrink, the transmission-line losses become more important. Con-
sequently, the engineer must have the ability to calculate the response of a
transmission line successfully and account for realistic conductor behavior. The
next two sections will (1) describe how to include ac resistance and internal
inductance in an equivalent circuit, and (2) modify telegrapher's equations to
comprehend realistic conductors.

5.4.1 Equivalent Circuit, Impedance, and Propagation Constant
In deriving the equivalent circuit for a transmission line in Section 3.2.3, which
is shown in Figure 3-9, the conductor was considered to be infinitely conductive,
meaning that all the current flows only on the surface because the skin depth
δ = 0, as shown by taking the limit of (5-10):

2
ωµσ

= =

lim σ →∞ [ δ ] lim σ →∞ 0

Furthermore, the assumption of perfect conductors did not allow the calculation
of a resistive term or an internal inductance term because there was no field pen-
etration into the conductor. As described in Section 5.1.2, physical conductors
manufactured with metals of finite (although good) conductivity behave very
similar to perfect conductors except for a small transition region where inter-
nal currents exist that are mostly confined to a few skin depths. As described
in Sections 5.2.2 and 5.2.3, the skin effect leads directly to frequency-dependent
resistance and internal inductance terms that must be comprehended in the equiv-
alent circuit.
Fortunately, the form of the equivalent circuit derived in Section 3.3 is also
applicable to a line whose conductors have finite conductivity. To begin this
derivation, the series impedance of an ideal transmission line with infinite con-
ductivity is calculated in units of ohms.

Z external

= jωL external (5-67)

The idealized parameters must be modified to include the surface impedance
(also known as the internal impedance):

Z s = R ac

+ jωL internal

(5-29)