Coax
loss calculations
Updated December
4, 2012
Click
here to go to a more exact metal loss calculation for coax!
Click
here to review the derivation for coax loss due to loss tangent
Click here
to go to our main coax page
Click
here to go to our main page on transmission line losses
Click
here to go to our skin depth page
Click
here to go to our Why Fifty Ohms? page
Be sure to visit our "more exact solution" page on the loss of coax due to metal. On this page we will review
the simple math for calculating the RF losses of coaxial transmission
lines over frequency. There are actually three loss mechanisms that
can occur within coax which are each described below:
Calculating
coax loss due to metal
RG6 CATV
coax cable: an example
Calculating
coax loss due to dielectric loss tangent
Calculating
loss due to dielectric conductivity (now a separate page)
For reference, the figure below
defines two important dimensions of coax, D and d. Note that "D"
is the inner diameter of the outer conductor, not its outer diameter!
Calculating
coax loss due to metal
Update September
2006: we've added a new page that gives a more
exact calculation of coax metal loss, which is needed in the
case that the skin depth is NOT small compared to to the coax cross
section dimensions. In 99.99% of loss calculations, this is probably
just an esoteric exercise, but we'll keep it here for you in any
case.
Below is the "classic"
loss due to metal calculation that you will find in microwave
textbooks. Note: this analysis assumes that the skin depth is
much smaller than the diameter of the coax center conductor or the
thickness of its outer conductor.
Transmission
line loss due to metal calculation can be done in a simple threestep
process:
 Calculate RF sheet resistance
of conductors.
 Calculate ohms/length of
the geometry.
 Calculate the loss/length.
All of these quantities are functions of frequency.
Step one: calculate
RF sheet resistance. This is a function of the metal's permeability
and conductivity (in addition to frequency):
Note that you could easily substitute
resistivity instead of conductivity into the equation. Click
here to look up resistivity of a variety of metals. Don't assume
that the inner and outer conductors are made of the same metal,
often they are not!
Step two: calculate the resistance
per unit length. We recommend you use meters for length to cut down
on confusion, but you can use
Potrzebies for all we care (pronounced poSHEByas). For coax
that has inner conductor diameter "d" and outer conductor
insidediameter "D", you need to integrate the sheet resistance
over the "widths" of both cylindrical surfaces:
Resistance/length =(RF sheet
resistance)/[(1/d)+(1/D)]
Relax, that integration didn't
take more than 8th grade math! The "width" of the inner
conductor surface is d
and the outside conductor surface is D.
This solution assumes that five skin depths of metal are available
on both inner and outer conductor, and each is comprised of a single
metal within those five skin depths, which is usually the case.
Here's what you get when you plug in R_{RFSH }and simplify:
Check it out, our equation will
let you use different metals for inner and outer conductors, we've
never seen such a beautiful sight in any textbook! Or in Agilent's
ADS!
The final step is to divide the
resistance/length by (2Z_{0}) to arrive at loss/length (units
are Nepers/length. To convert to dB, multiply
Nepers by 8.686.
The whole ball of wax boils down
into the following totally cool closedform equation:
Note: Roger pointed out a slight
mistake in the above equation in October 2009. Thanks! And sorry
for the inconvenience to everyone else!
Now you can calculate the loss/length
of any coax, using different metals for inside and outside conductors.
Sweet! You can skip the three steps we proposed, and cut to the
chase.
RG6 CATV
coax cable: an example
Note: this example examines the
only metal loss, not dielectric loss tangent loss...
Let's use RG6 coax cable (which
you can buy at Home Depot for wiring up your satellite dish etc.)
as an example and see if the math works out close to measured data.
We downloaded a data sheet on this cable to use as a reference for
the calculations. You can see
it here. We hope CommScope doesn't mind! According to this supplier,
the dimensions of RG6 are:
Inside conductor is 18 gauge
copperclad steel (40.4 mils on the AWG scale)
Outside of dielectric is
180 mils diameter
Characteristic impedance
is 75 ohms nominal
Outside conductor is aluminum
foil.
The dielectric material is "foam
polyethylene". Polyethylene
by itself has _{R}=2.25,
but in this case it is shot full of air to make it easier to bend,
so we don't know off hand what the correct _{R}
is (it must be somewhere between 1 and 2.25, right?
Let's calculate it. We know that
the the impedance of RG6 is 75 ohms, and it must obey the coax equation:
The 75 ohm solution is that foam
polyethylene _{R}
is 1.43.
Now we calculate the RF sheet
resistance of the inner and outer conductors. We did this in a spreadsheet,
as a function of frequency. We looked up the conductivities of copper
and aluminum here and here.
RG6 is supposed to work up to 3 GHz, so we analyzed it up to this
frequency.
Note that the inner conductor
has lower sheet resistance. That's because it benefits from the
superior conductivity of copper, while the outer jacket is aluminum.
We are assuming that the coppercladding on the steel center conductor
is at least 5 skin depths across this frequency band... that assumption
might be optimistic at MHz frequencies, but CommScope doesn't provide
plating thickness on their data sheet.
Now we calculate the resistance
per unit length (ohms per meter). Here the inner conductor has the
most resistance (even though it has worse conductivity), because
its surface area is much smaller than the outer conductor.
Last, we convert to dB/length.
We scaled the length units to 100 feet, because most U.S. cable
vendors quote RG6 cable loss this way:
So how did the calculation check
with real life data? We calculate 5.85 dB per 100 feet at 1 GHz,
the supplier says 6.15 dB. That's close enough for government work!
Part of the disparity could be that we neglected to look at the
effect of dielectric loss tangent loss. But our downloadable
coax spreadsheet will do this for you!
A possible shortcut...
This info was provided to us
by Jorge who works for a large defense contractor. We haven't tried
it yet. It relies on obsolete military specification MILC17.
Instead of calculating the conductor
and dielectric attenuation using the given formulas on the coax
page, you can get the results using:
a = K1 x sqrt(F) + K2 x F (dB/100
feet)
Where K1 is the resistive loss
constant
K2 is the dielectric loss constant
F frequency in MHz
Just divide the results by 1200,
the number of inches in 100 ft., to get dB/inch. K1 and K2 are available
on MILC17 Attenuation and Power Handling tables.
Times Microwave references this
calculation on their web site, but they seem to be the only cable
supplier that uses it. We couldn't find K1 and K2 for RG6, to compare
with the loss calculation we performed.
Good luck!
Calculating
coax loses due to dielectric loss tangent
New for October 2006: the formulas
for coax loss due to loss tangent are derived on this
separate page.
Loss due to dielectric loss tangent
is a pretty easy calculation. The permittivity
of a material is actually a complex number, so "epsilon"
is made up of two parts:
Thanks to Paolo for correcting
this formula, epsilon doubleprime goes in the numerator, not the
denominator! Epsilon singleprime is the number we usually
deal with, and causes no loss, and in most daytoday engineering
you don't see the prime notation. The imaginary epsilon doubleprime
is the culprit. Microwave engineers usually deal with the ratio
between the two, which is called tangent delta, or tanD (say
"tandee"), for short. If tanD is zero, there is
no loss due to dielectric. If it is nonzero, it creates a loss
that is proportional to frequency (inversely proportional to wavelength),
shown in the equation below:
Note that in its academic form
(above), "natural" units of Nepers/meter
result. Also, whatever units you prefer to use for wavelength will
be preserved in the loss calculation, so if you enter wavelength
in centimeters, the loss will come out in Nepers/centimeter.
In order to arrive at the "more
engineeringly" units of decibels/meter, we note that one Neper
is 8.68588 dB (exactly 20/ln(10)), and pi=3.14159, then we simplify:
In practice, you can round off
the constant to 27.3 and not lose sleep. Here's perhaps a handier
formula for coax loss due to loss tangent, where you plug in frequency
in GHz instead of wavelength:
By the way, this calculation
holds for any TEM transmission line, so it is the same for stripline
and rectax! But it won't work for microstrip...
Notice that dielectric loss is
proportional to frequency, whereas conductor loss only goes up by
the square root of frequency. This goes back to scaling  dielectric
loss scales over size and frequency, but conductor loss doesn't.
The loss equation also says that you can't reduce dielectric loss
by changing the cable geometry, like you can with conductor loss.
The only way you can reduce it is to use a dielectric with a very
low loss tangent, or low dielectric constant. For example, teflon
(a.k.a. PTFE), which is commonly
used as a dielectric, has a loss tangent of 0.0004. To illustrate
the proportions of conductor and dielectric loss, with a 0.141"
diameter semirigid cable with copper conductors and Teflon dielectric,
the conductor loss is higher than the dielectric loss all the way
up to the cutoff frequency, although the two kinds of losses become
about equal at the cutoff frequency in this example.
Loss
due to dielectric conduction is constant over frequency, and
is geometry independent!
