David Gilligan wrote:
>
> On Wed, 15 Oct 2003, Dr. David Kirkby wrote:
>
> > Sorry if this is a little off-top, but hopefully not too much.
> >
> > Would anyone know if there is an exact analytical solution to the
> > capacitance per metre (and/or the related impedance) of a coaxial
> > cable consisting of two concentric dielectrics?
>
> <snip>
> <snip>
> >
> > I'd particulary like to analyse this twin dielectric coax.
> >
> >
> >
> Hello David
>
> Does this stuff actually exist?
> Does the spiral of ptfe-looking stuff in Andrew HSJ 4-50 count, or is this
> cable modelled as having only air diaelectric?
>
> Dave
I'm not really sure if such a cable exists or not. I don't know the
HSJ-40, but I have come across so called 'air dielectric' cables,
where there is a thin spiral spacer keeping the inner from the outer,
so 99% of the dielectric is air. I don't know if that is what you
mean, but the intention was not to compute the properies of that.
Someone from a medical company contacted me about his results whilst
using my program 'atlc'
http://atlc.sourceforge.net/
which computes the properties of arbitrary transmission lines. He was
analysing the twin coaxial structure. Whether or not it was a cable in
the way we way we normally think I don't know.
His results of modelling this in commerical Linecalc package from
Agilent differed significantly from my free program 'atlc', which
clearly worries me.
I've made a lot of effort to check atlc with structures with a single
dielectric
http://atlc.sourceforge.net/accuracy.html
and have found the accuracy very good (errors always under 1% and
usually under 0.3%), when comparing to cases like coax, offset coax,
etc, for which exact results can be computed.
Feedback from someone else who made actual measurements of the odd and
even mode impedances of coupled transmission lines, which had multiple
dielectrics, found excellent agreement with atlc too. So I'm a bit
confused and I'd like to sort out if atlc is behaving properly or not
in the case of multiple dielectrics.
When I thought about the structure, it became clear the symmetry would
probably mean an exact analytical expression was possible. After
further searching of the web, I found that this was indeed the case:
http://www.microwaves101.com/encyclopedia/dualdielectric.cfm
so the stucture does have an exact analytical expression.
I now intend to test atlc with this structure, which is the only one I
am aware which has an exact result for multiple dielectrics. Obviously
if atlc has a problem I'd like to know about it and fix the problem.
Perhaps you now understand why I wanted information about a structure
which seems odd.
The structure then got me wondering if there was any use for a cable
with such properies. I can see one possible case where it might be
desirable to make such a cable. The electric field is much higher
around the inner conductor than anywhere else, so on a high powered
line, one might conceivably use a material which can withstand a high
electric field close to the centre conductor, but another matierial,
with a lower loss, outside of this. One could imagine building a cable
where the inner conductor is covered in a high-breakdown material,
with air as an outer conductor.
Anyway, whether or not such 'cables' exists I don't know, but I'm
interested in the the analystical expression.
PS
Since this might interest others in the future and web references like
that above tend to dissapear, the following C program I wrote might be
useful to keep in the archives of amps and vhf-dx. I would add that I
copied this from a larger file that does things like checking stdlib.h
actually exist, so I don't guarantee it will compile, but if it does
not, sorting it out should be no hassle.
/* dualcoax attempts to find the properties of a coaxial cable with
two
concentric dielectrics between the inner and outer conductors, and so
can be used as a useful test for atlc
Writen by Dr. David Kirkby.
Provided under the terms of the GNU General Public license (GPL).
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define MU_0 4*M_PI*1e-7 /* By definition */
#define EPSILON_0 8.854187817e-12 /* Data taken from NPL */
int main (int argc, char **argv)
{
double r0, r1, r2, er1, er2, c, l, zo;
if (argc != 6)
{
fprintf(stderr,"Find properties of a coaxial cable with two
different concentric dielectrics.\n\n
");
fprintf(stderr,"Usage: dualcoax d0 d1 d2 Er1 Er2\n");
fprintf(stderr,"dualcoax arguments are:\n");
fprintf(stderr," d0 is the diameter of the inner
conductor\n");
fprintf(stderr," d1 is the outer diameter of the inner
dielectic\n");
fprintf(stderr," d2 is the inner diameter of the outer
conductor\n");
fprintf(stderr," Er1 is the permittivity of the inner
dielectric\n");
fprintf(stderr," Er2 is the permittivity of the outer
dielectric\n");
exit(1);
}
r0=atof(argv[1])/2.0;
r1=atof(argv[2])/2.0;
r2=atof(argv[3])/2.0;
er1=atof(argv[4]);
er2=atof(argv[5]);
if ( r0 >= r1){
fprintf(stderr,"Sorry, the diameter of the inner conductor (d0)
must be\n");
fprintf(stderr,"less than the outer diameter of the inner
dielectic (d1)\n");
exit(1);
}
if ( r1 >= r2){
fprintf(stderr,"Sorry, the diameter of the inner dielectric (d1)
must be\n");
fprintf(stderr,"less than the inner diameter of the outer
conductor (d2)\n");
exit(1);
}
if (er1 < 1.0 ){
fprintf(stderr,"Sorry, the permittivity of the inner dielectric
Er1 must be >=1");
exit(1);
}
if (er2 < 1.0 ){
fprintf(stderr,"Sorry, the permittivity of the outer dielectric
Er2 must be >=1");
exit(1);
}
c=2*M_PI*er1*er2*EPSILON_0/(er1*log(r2/r1)+er2*log(r1/r0));
l=MU_0*log(r2/r0)/(2*M_PI);
zo=sqrt(l/c);
printf("Zo = %8.3f Ohms C= %8.3f pF/m L= %8.3f nH/m \n",
zo,c*1e12,l*1e9);
return(0);
}
--
"The day Microsoft makes something that doesn't suck is probably
the day they start making vacuum cleaners." -Ernst Jan Plugge.
Dr. David Kirkby,
Senior Research Fellow,
Department of Medical Physics,
University College London,
11-20 Capper St, London, WC1E 6JA.
Website: http://www.medphys.ucl.ac.uk/~davek
Author of 'atlc' http://atlc.sourceforge.net/
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