Well my ITT reference data for radio engineers 4th ed.
says that:
Voltage gradient in a coaxial line
At the voltage standing wave maximum:
(gradient at surface of inner conductor) =
5.37/d * sqrt( SP_kw / (Z_0 * epsilon) )
d = diameter of inner conductor
SP_kw = power in killowatts at the crest of the modulation
cycle, thus if the carrier is 1kW and modulation 100% set
SP_kw = 4kW.
Z_0 = characteristic impedance of the coaxial line in ohms.
epsilon = effective relative permittivity of the coaxial line;
air = 1.0.
sqrt(x) = x^0.5 = square root.
The same page cites the breakdown strength of air
at atmospheric pressure as 29000 peak Volts/cm
(experimental value before derating).
The same source also cites the equation:
delta_E / delta_r =
0.434*E / (r * log_10(D/d)
== which also ==
0.059*E*C / (r * epsilon)
== which also ==
60*E / (r * Z_0 * sqrt(epsilon) )
C = capacitance between coaxial line conductors in
picofarads per foot
D = diameter of inner surface of outer conductor in
same units as d.
d = diameter of inner conductor.
E = total voltage across line (E and delta_E both
RMS or both peak)
r = radius from cable center to inner surface of outer
conductor (r and delta_r both in same units)
Now my personal comments ----
You'll have to derate the dielectric breakdown strength
to be much less as either temperature increases or as
frequency increases. There are some empirical data
points to suggest how a certain material's RF breakdown
strength may be derated vs. temperature and frequency,
but in general you'll never find the data for just exactly
the frequency and more importantly the material blend /
characteristics you have, so this is likely something to
be empirically measured at failure and then very
conservatively derated for safety / variance margins.
The actual steady state heat dissipation of the cable will
depend in a complex way on its construction and
dielectric core and jacketing materials. It's probably
easier just to use experiments of both DC and RF power
through a reasonable length (several feet?) of the cable
and use an IR temperature measurement camera and/or
small thermocouple probes on the cable's surface to
measure the temperature rise of the cable versus time and
to determine the steady state "free still air" equilibrium
temperature value that is reached. Of course the interior
dielectric temperature would be much higher, and perhaps
with a DC test you could insert a tiny thermocouple into
the middle of the cable to get a sense of the
temperature gradient between the cable's jacket and the
cable's core. Doing a resistance change measurement
across the cable's core wire may help deduce the
core wire temperatue too if the measurement can be
accurate enough.
Depending on the ambient temperature of environmental
operation during the test, the cable materials,
and the frequencies and powers involved you'll either
reach a point where parts of the cable (e.g. plastics)
start to melt, or a point where there's RF parameter
degradation/instability and ultimately an RF arcing type
of fault in the cable. Low frequencies will tend to arc less
readily than higher ones according to conventional wisdom.
It's perhaps possible to have the cable be fairly
"melty" inside even before you get a detected arc failure,
though as long as the dielectric keeps insulating the cable
even in a semi-molten state.
So the end story is that you'll have to
determine/specify certain thermal conductivity,
thermal gradient, and thermal rise wrt. ambient temperature parameters to help determine
what temperatur related factors are limitations relative
to your cables materials softening, melting,
catching fire, etc. That's about the same DC as RF.
Relevant to RF, you'll have to determine over the
frequencies and powers of interest at what point the
dielectic strength and RF impedance of the cable may
start to degrade such that it no longer can safely or
effectively transport the signal without arcing or deforming.
You should be able to test the thing pretty easily with
a high power VHF / UHF source like a magnetron or
powrer oscillator tube and a variable power coupling
arrangement or something like that, and maybe use a
circulator or digital sampling o-scope or something to look
for reflection / distortion type events that'd indicate arcing
or degradation of the cable's RF characteristic.
You could use a simple 2D heat transfer FEM model
of the cross section of the cable containing regions for
core wire, dielectic, shield, jacketing to get some idea of
the heat transport parameters of your cable but I wouldn't
be surprised if those were inaccurate by 2:1 or 3:1
depending on the fine details of your materials heat
conductivities and smoothness and so on. Measuring
the heat dissipation and conductivity with an
DC experimental setup will give you empirical data that's
a lot better in many ways than what a generic
equation / model will provide. Then you can fit a curve
or polynomial to the empirical data to generate an
equation that may more accurately extrapolate / interpolate
relative to other similar cable variants you may have.
With respect to equations, models, standards,
test specifications, or tables of empirical
data for other coaxial cables, check MIL-SPECs,
ITU, NEC, and 1915 - 1970 era publications from places
like Bell Telephone Laboratories, MIT, Bell System
Technical Journal / Technical reports, and so on.
The basic problem is that usually things are specified
*very* conservatively, so the end number you generate
as a power handling rating will be just some semi-arbitrary
fraction of the empirically determined numbers at which
the cable goes into thermal runaway (temperature
increases without limit due to insufficient free air
convective dissipation), melts or readily arcs at RF.
e.g. You could specify an free air environmental
operating temperature range of -20 to +70C and
say that 100C is the limiting specified operational
temperature for the interior of the cable, so that'd give
room for a 30C rise from free air to cable core, then you
could see what kinds of powers generate such a rise
in the cable. Maybe your dielectic softens considerably
at 130C, so that'd be your structural oriented safety
margin assuming that the RF dielectic strength didn't
seriously degrade at the maximum operating frequency
much before the mechanical softening point of the
dielectic...
Good luck; it's good to see more / better attention being
paid to specifiying the characteristics of products; a lot
of the time I'd swear that they're just making up the
numbers and that there's no chance in heck of a product
actually *working* under its rated operating conditions.
You might also ask around over at
http://www.microwaves101.com
there are probably some old timers who could rattle
off test specifications / standards, empirical figures,
and approximation equations to you from memory.
