[Antennas] Losses in a transmission line

[Barry L. Ornitz]

In Digest #424, there were several questions about losses in
transmission lines that I would like to address.

David Windisch asked:

> Conductors, whether in a feedline or *whatever*, dissipate
> power according to the usual formulae, not at, eg, the
> feedline's Z, right?

I am not sure I understand this question.  The loss in a
transmission line can be several things.  First the conductors
can have resistance.  This is RF resistance and not the DC
resistance.  The RF resistance will always be higher, and
sometimes significantly higher, due to the "skin effect" with
RF.  Then there can be shunt resistance usually associated
with the line insulation.  Even if there is no DC shunt
resistance, there will be dielectric losses in all real
insulators.  For mathematical purposes this can be modeled as
a simple shunt resistance, but in fact, the losses are really
in the molecules of the dielectric trying to align themselves
with the alternating electric field.  If the line is made of a
magnetic material, another series resistance term will be
added too.  Finally, the line can radiate if the fields do not
cancel.  If losses are viewed as the difference between power
in and power out of a section of a line, radiation becomes a
loss term.

The dielectric and magnetic losses are the things most hams
have difficulty understanding, so this note is my attempt at
trying to explain these.  When we talk about capacitive
effects, the term permittivity is used, and when inductive
effects are discussed, the companion term permeability is
used.

The capacitance between two conductors is the vacuum
capacitance times the relative permittivity of the insulator
between them times the permittivity of free space.  In
mathematical terms, the relative permittivity is complex, i.e.
it has both real and imaginary parts [imaginary in the
mathematical sense, j = SQRT(-1) ].  The real part is what we
normally call the dielectric constant.  The imaginary term is
normally not used directly, but instead most dielectric tables
list the loss tangent instead.

     tan (d) = e"/e' = D

where:  e" = imaginary part of the complex permittivity
        e' = real part of the complex permittivity or
             dielectric constant
        d  = loss factor
        D  = dissipation factor.

Similarly, when taking about inductors, we can say the complex
permeability is made up or real and imaginary terms.
Unfortunately the common expression for the real term is just
permeability, but it must be understood that the complex term
is always present in real materials.  These magnetic losses
are important when worrying about the power handling
capability of ferrite cores, for example.

Perhaps a numerical example might help.  Suppose we have a
capacitor whose vacuum capacitance is 1000 pF.  Now suppose
the capacitor has a vinyl dielectric (polyvinyl chloride,
PVC).  I will use data for Ultron (Monsanto) Wire Compound
UL2-4001 (because it is handy and because it is typical of
vinyl wire insulation).  This has a dielectric constant of
3.5 at 1 MHz, and a loss tangent of 0.07 at the same
frequency.  This material is PVC with plasticizers, fillers,
and stabilizers.

The actual capacitance will be 3.5 X 1000 pF = 3500 pF.  Since
e' = 3.5, we can calculate e" to be 3.5 X 0.07 = 0.245.  If
100 volts at 1 MHz is placed across this capacitor, the
reactive current will be 2.2 amps [Zc = 1/(2 X Pi X 1E6 X
3.5E-9) = -j45.47 ohms].  But the real current will be 0.539
amps.  The reactive current produces no power but the real
current does - a lot of it, almost 54 watts.  This is why
vinyl is not used as the dielectric of practical transmission
lines for radio frequencies.

The important point to note here is that while we could
consider the vinyl a perfect insulator at DC, it still
dissipates real power at RF.

> Imagine a 50-ft piece of OWL coiled on your desk bosun's
> style, dissipating 300W of r-f or whatever.
>
> Would it warm up your cuppa iced coffee?

If the line is _dissipating_ this much power, and if the line
radiation is small, it would certainly heat your coffee.  But
I think David may have wanted to ask a different question.
This is, if the line is carrying 300 watts of RF, would it
heat the coffee?

Here we have to know the losses of the line.  Let's say the
line has a nominal attenuation of 2 dB per hundred feet (which
is far, far higher than with most real open wire lines).
Let's say the line is working with a VSWR of 3 to 1.  The
matched loss will be 1 dB.  But the high VSWR adds to this
loss making the total loss to be 1.503 dB (curves for the
added loss can be found in most ARRL Antenna Books or the ARRL
Handbook).  So if the line is carrying 300 watts of power, the
power dissipated by the line will be 87.8 watts and the power
going to the load will be 212.2 watts.  That 87.8 watts is
dissipated as heat so it will still heat the coffee.

If that line had 0.02 dB per hundred feet, the total loss
would have been 0.0166 dB and the line would have dissipated
just a little over a watt.

Harvey, W4TG, noted:

> I would like to point out that the dielectric for purposes
> of impedance calculation and loss in this line and others
> like it, is mostly air. The webbing makes up only a small
> portion of the dielectric immersed in the field surrounding
> the conductors. Only when wet does the webbing cause
> any big problem.

This is exactly the point.  Just a little bit of water on the
line raises the effective dielectric constant of the webbing
considerably.  This is because the dielectric constant of the
polyethylene is about 2.5 while that of water is 80.
Additionally, the water has a high dielectric loss compared to
the polyethylene.

Finally Cletus Whitaker, WB2CPN, wrote about a buried antenna:

> Really though, the main effect of the wires being
> surrounded with dirt was that it lowered the resonant
> frequency of the dipole significantly.

Again this is because the dielectric surrounding the dipole
had a permittivity far greater than that of air.  The real
part of that permittivity is what lowered the resonant
frequency and the imaginary part of the permittivity is what
converted the RF into heating of the soil.

He added:

> Two major factors affecting loss; the dielectric loss in
> the separator and nearby material, and the ohmic resistance
> of the conductors.  Being wet doesn't matter much, and being
> covered with salt or grime only increases loss if the ohmic
> shunt resistance is increased.

The statement about the major factors is correct, but being
wet really does matter quite a bit.  Even if the water has no
conductive salts, it will increase the loss of the line
considerably.  The effect of dissolved salts on the RF
resistance of water generally obeys a reciprocal frequency
relationship.  The effect on the conductivity of water is
quite high at low frequencies but above a Megahertz the affect
is generally small.

For those interested, pure water at 77 F has the following
dielectric constant and loss tangent:

   Frequency    Dielectric Constant     Loss Tangent
    100 kHz           78.2                 0.4
      1 MHz           78.2                 0.04
     10 MHz           78.2                 0.0046
    100 MHz           78.0                 0.0050
    300 MHz           77.5                 0.0160
      3 GHz           76.7                 0.1570
     10 GHz           55                   0.5400
     25 GHz           34                   0.2650

One interesting thing to note here is that water does not have a
resonant frequency anywhere near the operating frequencies of
microwave ovens (2.45 GHz).

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