length rule of thumb
to go to our main page on cables
here to go to our page on quarterwave tricks
here to go to our page on transmission lines
New for January 2008!
This page is the result of a question that someone asked us about
calculating the physical length of a coax cable of from its swept
frequency response. But you can apply the math "backwards"
and use this as a poor man's method of measuring
effective dielectric constant for microstrip and stripline as
well as coax structures.
Here's a related page that shows
a technique for calculating
dielectric constant from group delay.
Calculating cable length from
Whenever you measure the S-parameters
of a coax cable, there are noticeable dips in S11 (and perhaps less
noticeable dips in S21) periodic with frequency in the response.
These occur at regular intervals, separated by a quantity we'll
call "deltaF". The dips are caused by something within
the cable that causes constructive and destructive interference.
The length can be calculated from the distance between the peaks
or the dips, however, the dips in S11 are better defined so we prefer
to use them for this purpose. Bear in mind that this is always an
approximate solution, if you want more accuracy in an electrical
length measurement, you should fit a model to measured data, or
look at group delay.
For people that don't care about
derivations, we'll present the formula right up front.
is the velocity of light in the
medium. The velocity of light is equal to "c" multiplied
by the velocity factor of the medium VF (the measure of how much
the velocity of light is slowed in the medium). For a coax cable
the velocity factor is 1/SQRT(ER), where ER is the dielectric constant
of the dielectric fill.
Two types of mismatch cause the
same effect, we'll describe both types below.
In the first type of mismatch,
the cable impedance is slightly mismatched from 50 ohms. For a 0.049"
cable for ER=2.1, an inner dielectric of 13 mils gives an impedance
of about 55 ohms (unmatched to 50 ohms). We modeled the cable in
ADS, then looked at the frequency response.
The dips in S11 are regularly
spaced at about 104 MHz, as evidenced by the markers in the plot
below. This is the "deltaF" we will enter in the equation
at the top of the page. Solving for the cable's length we obtain
0.995 meters, an error of just 0.5%!
Looking at this case on a Smith
chart we see that the input reflection coefficient traces a circle
between fifty ohms and a higher impedance. The higher impedance
occurs whenever the length is an odd number of quarter wavelengths,
in which case it acts like an impedance transformer. The dips occur
whenever the cable acts like any multiple of two quarterwavelengths.
Here's the explanation: one quarterwave transformer moves the load
to a non-fifty-ohm impedance, but the second quarterwave transformer
moves the impedance back to fifty ohms.
Guess what? You can calculate
the cable impedance from the maximum points along the return loss
curve. At these points the cable is acting like a true quarterwave
transformer. Check back later and we'll post the calculation!
Here the cable well matched to
fifty ohms (14.6 mil center conductor, outer conductor inner diameter
49 mils) but the connectors at each end have an ugly VSWRs (but
are the same at each end). We modeled this problem as a small shunt
capacitor at each end of a fifty ohm line.
In this case, we know from our
"quarterwave tricks" page
that equal mismatches can be canceled by locating them approximately
a quarterwave apart (capacitive mismatches require somewhat less
than a quarterwavelength distance to cancel).
Below is the response of this
ugly cable model. Note that the very first dip is where the cable
is less than 1/4 wavelength. From then on the dips occur when the
cable is an odd number of quarterwaves, or the distance between
each dip is caused by an additional half-wave. The distance between
the first two dips (deltaF) is 99 MHz. Plugging this into the equation
we calculate the cable length at 1.045 meters, an error of 4.5%.
When we look at the response
of this case on a Smith chart, we see the reflection coefficient
spiraling outward, but at all of the frequencies where the cable
is an odd number of quarterwavelengths, fifty ohms is achieved.
As frequency is increased, the maximum reflection coefficient spirals
out farther and farther from fifty ohms. The capacitor we modeled
the connector as looks closer and closer to an RF short circuit
as frequency is increased..
This is under construction. Check