[Antennas] Complex Characteristic Impedance of Cables

[Barry L. Ornitz]

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James Duffer, WD4AIR, is still somewhat confused that the
characteristic impedance of real coaxial cable and other real
transmission lines can be complex (have both real and
quadrature [imaginary, SQRT(-1)] parts.

A transmission line can be described as a combination of
distributed series elements and distributed shunt elements.
All that "distributed" means is that the values are per unit
length.  The differential model usually found is:


     -----R--L--o----o-----   where: R = resistance/meter
                |    |               L = inductance/meter
                C    G               C = capacitance/meter
                |    |               G = conductance/meter
     -----------o----o-----


You can derive the characteristic impedance from this as:

    Zo = SQRT((R+jwL)/(G+jwC))

At very high frequencies, w = 2*pi*f, the imaginary terms are
much larger than the real terms so the characteristic
impedance approaches SQRT(L/C).

Now in real cables, especially at low frequencies, the series
resistance term is several orders of magnitude higher than the
shunt conductance term and both can be large compared to the
reactive terms.  This makes real coaxial cables have a
strongly capacitive effect at low frequencies.

If it is possible to arrange for the elements to be in the
right propportions, i.e. if G/C = R/L, the cable velocity will
be constant with frequency.  There will still be loss,
however, but the cable will not distort a complex waveform as
it passes down the cable.  In practical cables, this often
means loading down the cable with shunt conductances and
increasing the cable losses greatly.

In real cables, losses increase greatly with frequency.
Furthermore, the conductors are subject to the skin effect
making the effective series resistance increase with
frequency.  Likewise the capacitance, which is a function of
the dielectric constant of the insulator, also changes with
frequency.  [The term dielectric constant is a poor one.  It
is hardly constant being a strong function of temperature and
frequency in virtually all materials.]  The result is that the
assumption of a lossless, constant characteristic impedance
cable is a poor one if operation is expected over a broad
frequency range.

Jim also wrote:

> Refer to the formulas for Zo of parallel or coaxial lines
> (Zo= 276 Log 2S/d where S is center to center distance
> between the conducior and d is diameter of conductor[same
> units], Zo= 138 log (b/a) where b is inside diameter of
> outer conductor and a is outside diameter of inner
> conductor.

In the case of parallel transmission lines, the above equation
is only an approximation.  A more accurate equation is:

     Zo = 120 arc-hyperbolic-cosine(S/d)

and this applies only for vacuum insulated lines (since the
dielectric constant of air is approximately unity, it works
for air too).  For example when the spacing is 1.5 times the
conductor diameter, Jim's equation predicts the characteristic
impedance to be 14 percent too high, and it gets worse with
even closer spacing.

        73,  Dr. Barry L. Ornitz     WA4VZQ     ornitz@tricon.net

P.S.  A good website to visit is:

http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/audio/part7/page1.html