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Old August 21st 03, 07:16 PM
Roy Lewallen
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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.

It turns out that R is a function of frequency because of changing skin
depth, but it increases only as the square root of frequency. jwL, the
inductive reactance per unit length, however, increases in direct
proportion to frequency. So as frequency gets higher, jwL gets larger
more rapidly. For typical transmission lines at HF and above, jwL R,
so R + jwL ~ jwL. G represents the loss in the dielectric, and again for
typical cables, it's a negligibly small amount up to at least the upper
UHF range. Furthermore, G, initially very small, tends to increase in
direct proportion to frequency for good dielectrics like the ones used
for transmission line insulation. So the ratio of jwC to G stays fairly
constant, is remains very large, at just about all frequencies. The
approximation that jwC G is therefore valid, so G + jwC ~ jwC.

Putting the simplified approximations into the complete formula, we get

Z0 ~ Sqrt(jwL/jwC) = Sqrt(L/C)

This is a familiar formula for transmission line characteristic
impedance, and results in a purely real Z0. But it's very important to
realize and not forget that it's an approximation. For ordinary
applications at HF and above, it's adequately accurate.

Having a purely real Z0 simplifies a lot of the math involving
transmission lines. To give just a couple of examples, you'll find that
the net power flowing in a transmission line is equal to the "forward
power" minus the "reverse power" only if you assume a real Z0.
Otherwise, there are Vf*Ir and Vr*If terms that have to be included in
the equation. Another is that the same load that gives mininum
reflection also absorbs the most power; this is true only if Z0 is
assumed purely real. So it's common for authors to derive this
approximation early in the book or transmission line section of the
book, then use it for further calculations. Many, of course, do not, so
in those texts you can find the full consequences of the complex nature
of Z0. One very ready reference that gives full equations is _Reference
Data for Radio Engineers_, but many good texts do a full analysis.

Quite a number of the things we "know" about transmission lines are
actually true only if the assumption is made that Z0 is purely real;
that is, they're only approximately true, and only at HF and above with
decent cable. Among them are the three I've already mentioned, the
simplified formula for Z0, the relationship between power components,
and the optimum load impedance. Yet another is that the magnitude of the
reflection coefficient is always = 1. As people mainly concerned with
RF issues, we have the luxury of being able to use the simplifying
approximation without usually introducing significant errors. But
whenever we deal with formulas or situations that have to apply outside
this range, we have to remember that it's just an approximation and
apply the full analysis instead.

Tom, Ian, Bill, and most of the others posting on this thread of course
know all this very well. We have to know it in order to do our jobs
effectively, and all of us have studied and understood the derivation
and basis for Z0 calculation. But I hope it'll be of value to some of
the readers who might be misled by statements that "authorities" claim
that Z0 is purely real.

Roy Lewallen, W7EL