William E. Sabin wrote:

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


The usage of complex conjugate Z0* becomes
significant when calculating very large values of
VSWR, according to some authors. But for these
very large values of standing waves, the concept
of VSWR is a useless numbers game anyway. For
values of VSWR less that 10:1 the complex Z0 is
plenty good enough for good quality coax.

W.C. Johnson points out on page 150 that the concept:

Pload = Pforward - Preflected

is strictly correct only when Z0 is pure
resistance. But the calculations of real power
into the coax and real power into the load are
valid and the difference between the two is the
real power loss in the coax. For these
calculations the complex value Z0 for moderately
lossy coax is useful and adequate.

The preoccupation with VSWR values is unfortunate
and excruciatingly exact answers involve more
nitpicking than is sensible.

Bill W0IYH