The objective of this is to identify optimum designs for coaxial cable. Particular designs lead
to particular characteristic impedances. Why are certain characteristic impedances very
common (e.g., 50 ohms and 75 ohms) whereas and other values seem to be less popular or
non-existent? There are many myths out there that attempt to explain this (my personal
favorite myth: That somehow this is related to the impedances of popular antennas -- it
really isn't). The real reasons have to with power handling and attenuation rate.
Your report must consist of the following sections, having the following titles. (Numbers in
square brackets are points.)
1. "Introduction"
-- [3] introduce your report and present a problem statement. Be sure to include a figure
and define all relevant parameters. You may assume all metals are non-magnetic.
2. "Characteristic Impedance of a Low-Loss Coaxial Cable"
-- [3] Starting with the general expression for Z_0 in terms of R', G', L', C' and frequency,
show that a simpler expression that depends only on L' and C' is possible if R' and G' are
sufficiently small.
-- [3] State exactly how what you mean by "sufficiently small" and explain what this means
in terms of frequency range for which this simplified expression is valid.
-- [3] Confirm that the simplified expression is valid using the values of L' and C' obtained in
class for RG-59, and calculate the frequency range for which this is expected to be valid.
-- [3] Using expressions for L' and C' of coaxial cable, develop an version of the low-loss
expression for Z_0 in terms of the materials parameters and the cable geometry. You'll need
this in later sections, so make sure you get it right.
3. "Optimization for Power Handling": "Power handling" refers to maximum power that can
be safely transferred by the cable. This power is limited because when the electric field
within the cable becomes too large, then dielectric breakdown becomes possible. In this
section you will work out how to design the cable to maximize the power that can be
delivered by the cable without exceeding the peak electric field.
-- [3] First, using Laplace's equation, work out an expression for potential as a function of
radial position "rho" within the cable. Hint: This expression should depend only on rho, the
voltage applied between the conductors and the radii of the conductors.
-- [3] From that, develop an expression for the magnitude of the peak electric field intensity
within the cable.
-- [3] Next, write an expression for the power delivered by the cable assuming the source
and load are impedance-matched. Make substitutions to obtain an expression in terms of
the peak electric field intensity, conductor radii, and relevant materials parameters.
-- [3] Find design parameters which maximize this power. Do this by taking the derivative
with respect to the radius of the inner conductor, and setting this to zero.
-- [3] Using the result from the previous step, find the value(s) of Z_0 which optimize power
handling.
4. "Resistance above DC" : The DC expressions/values we were using for R' previously are
not suitable for use at higher frequencies, because skin effect becomes important and is
frequency-dependent.
-- [3] Develop the appropriate expression for R', accounting for both the inner and outer
conductors.
-- [3] Develop a simpler expression for the special case in which the conductivity of the
outer conductor >> the conductivity of the inner conductor. Let us call this "Resistance Case
I"
-- [3] Develop a simpler expression for the special case in which the conductivity of the
outer conductor = the conductivity of the inner conductor. Let us call this "Resistance Case
II"
With expressions for R' for these two special cases in hand, we will be able to make quitebroad assessments without having to know the specific values of the conductivities.
5. "A Simplified Expression for the Attenuation" : Optimization of attenuation is done by
minimizing "alpha"; i.e., the real part of the propagation constant "gamma". The easy way to
do this is to begin with a simplified approximate expression for alpha, which looks like this:
alpha = alpha_R + alpha_G
where
alpha_R = K_R * R' / Z_0
alpha_G = K_G * G' * Z_0
where K_R and K_G are real-valued constants to be determined.
-- [3] The above three expressions together are known as an "ansatz"; which means "an
educated guess at a solution". Please justify this ansatz using what you know about how
about (a) how alpha appears in the expression for a propagating wave, and (b) dimensional
analysis. You do not have to prove this ansatz, you merely need convince the reader that it's
plausible.
-- [3] Show that the ansatz gives a very accurate approximation of alpha as a function of
frequency. Do this by plotting the ansatz value of alpha along with the value that you obtain
by taking the real part of gamma computed from R', L', G', C' and frequency (remember to
use the expression for R' from Section 4!). Do this for the RG-59 example from class. You
will need to figure out what K_R and K_G are, but you can easily do this using the plot, once
you have it working.
-- [3] Include your source code AS A SEPARATE TEXT FILE.
6. "Optimization for Attenuation"
-- [3] Using the (now verified) ansatz for alpha from the previous section, figure out the
design parameters that minimize alpha. As in Section 3, do this by taking the derivative
with respect to the radius of the inner conductor and setting the result to zero. Please do
this for both "Resistance Case I" and "Resistance Case II".
-- [3] Using the result from the previous step, find the value(s) of Z_0 which minimize
attenuation.
7. "Conclusions"
-- [2] Summarize the values of material and geometrical parameters which optimize power
handling.
-- [2] Summarize the values of material and geometrical parameters which minimize
attenuation.
-- [1] Speculate why 50 ohms has emerged as a standard impedance for coaxial cable. For
what applications is it best suited?
-- [1] Speculate why 75 ohms is also quite popular. For what applications is it best suited;
i.e., when might you prefer this over 50 ohm cable?