[Amps] Re: VHFDX: Imepdance of twin dielectric coax.

[Dr. David Kirkby]

GeoffGrayer at aol.com wrote:
> 
> Hi Dave;
> 
> I have a length of the stuff you describe.
> 
> There may be other ways of using it, but I have assumed the
> following:
> 
> If you connect an (unbalanced) RF source between the inner and
> middle
> conductor, and presumably earth the outer conductor, it then behaves
> as
> a shielded coax. The impedance is then simply calculated as for a
> normal

No, you have misunderstood me. The structure is not as you describe,
but as can be seen here. 

http://www.microwaves101.com/encyclopedia/dualdielectric.cfm

with just one inner and one outer conductor, but two concentric
dielectrics between the two conductors. As you will see, I have found
an analytical solution to the impedance of such a coax. 

I'm not quite sure if such a coax is commercially produced - I've
never seen one. About the only point in making such a coax might be to
use a material for the inner dielectric that has a high breakdown
potential (as the E field is highest there) and a another (perhaps
less lossy) dielectric elsewhere. I'm clutching at straws a bit there,
as it probably has no practical use, but it is useful to me, as
provides one way of checking my program 'atlc' with multiple
dielectrics. Testing with single dielectrics is quite easy as there
are countless structures which can be analaysed theoretically and the
theoeritical values compared to the numerical simulations of atlc.

But testing 'atlc' with multiple dielectrics is much more difficult,
as all formula produced in books etc for microstrip or whatever, are
in themselves just approximations. This structure is interesting in
that an exact solution exists, despite the fact there is two
dielectrics. 

I'm not aware of any other structure for which the impedance can be
computed theoretically when there is more than one dielectric,
although I have been advised there might be one other.

If anyone wants it, here is a small program I wrote that computes the
impedance of this dual concentric coax. I posted this before, but for
some reason it seemed to go to /dev/null. 

Note, I cut and pasted this program from the source I wrote, but have
had to edit it a bit in the email, so I can't say for 100% certainty
this will compile, but it should do. If not, any C programmer will
soon sort it out. The following should work on a UNIX box, if you save
the file as dualcoax.c 

$ gcc dualcoax.c -lm -o dualcoax.

/*
This is file dualcoax.c

Copyright (C) 2003. Dr. David Kirkby, PhD (G8WRB).

This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License
as published by the Free Software Foundation

Dr. David Kirkby, e-mail drkirkby at ntlworld.com 

*/


/* 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 */


#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 velocity, velocity_factor, 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);
 
velocity_factor=(er1*er2*log(r2/r1))/(er1*log(r2/r1)+er2*log(r1/r0));
  velocity=1.0/(velocity_factor*sqrt(MU_0 * EPSILON_0));
  printf("Zo = %8.3f Ohms C= %8.3f pF/m L= %8.3f nH/m v= %g m/s v_f=
%8.3f\n", zo,c*1e12,l*1e9, velocity, velocity_factor);
  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,
Author of 'atlc' http://atlc.sourceforge.net/