Roy Lewallen wrote:
A big deal is being made of the general assumption that Z0 is real.

As anyone who has studied transmission lines in any depth knows, Z0 is,
in general, complex. It's given simply as

Z0 = Sqrt((R + jwL)/(G + jwC))

where R, L, G, and C are series resistance, inductance, shunt
conductance, and capacitance per unit length respectively, and w is the
radian frequency, omega = 2*pi*f. This formula can be found in virtually
any text on transmission lines, and a glance at the formula shows that
Z0 is, in general, complex.

A good approximation to Z0 is:

Z0 = R0 sqrt(1-ja/b)

where Ro = sqrt(L/C)
a is matched loss in nepers per meter.
b is propagation constant in radians per meter.

The complex value of Z0 gives improved accuracy in
calculations of input impedance and losses of
coax lines. With Mathcad the complex value is
easily calculated and applied to the various
complex hyperbolic formulas.

Reference: QEX, August 1996

Bill W0IYH