# [time-nuts] Thermal effects on cables

Dr. David Kirkby (Kirkby Microwave Ltd) drkirkby at kirkbymicrowave.co.uk
Tue Jan 24 05:22:07 EST 2017

```  oI On 23 Jan 2017 17:02, "REEVES Paul" <Paul.Reeves at uk.thalesgroup.com>
wrote:
>
> Hi David,

Hi Paul

> Surely the impedance of the cable is only affected by the ratio of the
inner conductor and outer conductor diameters modified by the internal
dielectric constant, nothing to do with the frequency of operation.

No, that is incorrect.

There are two complications with coax, which mean that you can't trust the
"normal" equations at very low or very high frequencies.

1) LOW FREQUENCY

An accurate equation for the impedance of the cable is

Zo=sqrt ((R+j w L)/(G+j w C))

w = 2 Pi f
R = Resistance per metre (ohms/metre)
C = Capacitance per metre (Farads/metre)
G = Conductance per metre (Mho/metre)

If the frequency is high enough (above a few MHz),

j w L >> R and
j w C >>  G

so the R & G terms are insignificant and one gets the usual equation we know

Zo=sqrt (L/C)

At low frequencies the assumption that the resistive losses (R&G) are
insignificant is no longer valid, so impedance rises at frequency
decreases.

At DC a bit of coax is just a capacitor..

2) HIGH FREQUENCY

At high frequencies!the best of9 higher order modes can propagate.  The
reasons for this are more complicated to explain,  but are a result of the
breakdown of the assumption of the boundary conditions used to arrive at
the impedance.

The maths of this effort is more complicated,  needing Bessel functions.

> You might well have problems converting the larger diameters down to a
suitable size for the connectors at the higher frequencies though....

Yes, but small connectors are used for the very same reason small cables
must be used.

> I thought that the HP cabling for the 8510 series VNAs was air spaced but
I might well be wrong - I just tried not to damage them :-)

You might well be right about the air spacing.  Of course there needs to be
some support for the centre conductor,  but it might be  beads like in 3.5
mm connectors. I really don't know.

> Regards,
>
> Paul Reeves

Dave, G8WRB.
```