Welcome to the Discontinuity
Calculator. This site will calculate the equivalent
capacitor for a
step discontinuity in a
coaxial transmission line.
When you join two coaxial
cables having
different diameters, you create a discontinuity in the transmission
line that will result in a reflection of the transmitted signal. This
reflection is characterised by the impedances of the two
cables, and also by the profile at the junction.
The program that you can run in this page will
calculate the aditional effect due to the steps at the junction. This
effect can be described as an extra capacitance on the line. It is due to a
spurious component of
the electric field along the propagation axis. The
accurate evaluation of this capacitance requires to
calculate the electromagnetic field distribution near the junction.
Using an analytical model, and assuming perfect cylindrical
symmetry, the links on this page can evaluate this capacitance
for
any parameters such as dimensions and frequency that you will specify.
To use the calculator is very simple:
First, you
select the model that you would like to use. Then,
you enter the dimensions of the lines and signal frequency. Finally,
you press the SUBMIT
button. The returned
page will give the equivalent capacitance and a few
other results coresponding to your problem.
Thank-you and come back again to
the Discontinuity Calculator.
[1] A. Michaud, 'Discontinuity
Calculator Equations', available ON LINE. (in progress!)
[2] A. Jurkus, 'Computation
of Step Discontinuities in Coaxial Line', IEEE Transactions on Microwave
Theory and Techniques, Vol. MTT-20, No. 10, pp.
708-9, October 1972.
[3] A. Jurkus, Private
communication.
Single/Double
Step Discontinuities
The four models
(cases) shown below cover all possibilities. The step
on outer conductor and step
on inner conductor cases are special cases, while
the shrink
and mismatch
cases are more general.
Each one of the four calculation shown below is
independant form the others. Chose the one that corresponds to your
problem and just ignore the other three. The form on the right of the
schematic, already contains an
example that
works. If you would like to calculate a different
case,
then just change the numbers in apropriate fields. Come back
to this
page by clicking the BACK
button of your browser. Experiment: try a
different
frequency (DC), etc... Have fun!
Notes:
The equivalent capacitance is located at the same
position as the discontinuity on the transmission line (z=0). The main effect of this capacitance is to change the phase of the reflection (or transmission).
The line is assumed very long on both sides so that all the
evanescent modes do not extend to the generator or
termination.
The fundamental physical constants used for the
calculations are the latest CODATAvalues.
Only the speed of light and vacuum permeability are used. Vacuum
permitivity, vacuum impedance, are deduced from the previous in the
program.
Although the cases 3 and 4 will are not supposed not work when the outer or
inner radiusses are equal, the program is very robust and will give an accurate
answer with even a very small difference. It is therefore
possible to compare the results from case-3 , -4 to the limit case-1 , -2.
At this time (version 0.2) the program calculates 150
× 3 modes and the response time is about 5 seconds. Future
versions will allow to change the accuracy/response time tradoff.
At this time (version 0.2) we have performed several tests
to verify the validity of the equations/code. For example it is easy to
compare the the results from case-1 to those from case-3 or case-4. We have found a perfect agreement. The only exception is when we compare the results from case-2 to case-3 or case-4. We think that this is due to an error in one equation of the case-3 model !#@!%! We are presently working on this, and a future version of this page will correct this problem.
At this time (version 0.2), the reference [1] that gives the
details about those calculations is still in writing. The latest
version is posted regularly.
Model 1:
Step on the inner conductor
The inner diameter of the outer
conductor
is the same on both sides of the discontinuity (the radius is
labeled R3).
The inner conductor has a radius R1 on
the right but changes to R2
at the plane of reference (z=0)
towards the left.
Model 2: Step on outer
conductor
The inner conductor
has the same on both sides of the discontinuity (the radius is
labeled R1).
The outer conductor has a radius R3 on
the right, but changes to R2
at the plane of reference (z=0)
towards the left.
Model 3: Steps on both
conductors, 'match'
This is a very common case where both
the inner and the outer conductors are smaller on one side.
This is the case when two cables have the same characteristic
impedance. The outer conductor has a radius R3 on
the right but changes to R1
at the plane of reference (z=0)
towards the left, while the inner goes from R2 to R0.
Model 4: Steps on both
conductors, 'mismatch'
This case [Jurkus, 1972] is often
present when the characteristic impedances of the two cables are
very different. The gap between the two conductors is larger on the
right side.
Equivalent
electrical model
The equivalent circuit for all the models above is shown
here:
Zl
C(f)
Zr
Zl,
(Zr)
is the characteristic impedance of the left, (right) transmission
lines, given by the well known:
Z
= ( µ0c / 2
¶ ) ln (Rout
/ Rin).
The
constant in the bracket is approximatively equal to 60
(The exact values are used in
the program).
C(f) is the
equivalent capacity of the of the discontinuity returned by
the program is in femtofarad, f F (10-15 F). The equivalent
capacitance is located on the same plane as the junction.
