The article Checkout of SimSmith v16.3 – spot check of transmission line database raises an issue with SimSmith’s modelling of transmission lines.

The case chosen was Belden 8216, a RG174 type line with silver clad steel stranded inner conductor.

## Fully developed skin effect

Most practical transmission lines used for HF and above have fully developed skin effect above some frequency, and are well represented by the loss model MLL=k1*f^0.5+k2*f. For an RLGC model, the R is given by the first term and with fully developed skin effect, it is proportional to square root of f. The loss of good dielectrics is usually simply proportional to f and indicated by the second term.

Under this model, L and C are independent of frequency.

Many calculators use this model, and it works fine where skin effect is fully, or even well developed. The model coefficients are commonly discovered by performing a regression on measured matched loss at a range of frequencies, and the quality of the regression fit is a good indicator of the quality of the model for that particular line.

## DC / low frequency

At DC, current flows evenly across the section of a homogenous conductor so its resistance is proportional to resistivity and cross section area. Because current flows ‘inside’ the conductor, it has ‘internal’ inductance which for a non-magnetic round conductor is 50nH per metre, more if relative permeability is greater than unity. The combination of these is Zint. Under this model, L includes the frequency dependent internal inductance, C remains independent of frequency.

There must be a transition between the DC (low frequency) and high frequency characteristic, and prediction of behavior is quite challenging. It is even more challenging for certain types of constructions (eg the silver clad steel stranded conductor used in the subject coax).

## Transition region

Dan Maguire incorporated a model from Johnson. Johnson sets out the maths of a model of the internal impedance (resistance and inductance of a homogenous conductor, and it is a challenge to compute and requires the physical conductor dimensions and material characteristics.

Johnson notice a characteristic of Zint that at high frequencies it had a slope and phase that he uses to interpolate Zint for low frequencies from Rint at a frequency where skin effect is fully developed and where known that MLL is mainly due to Rint, and from the DC resistance at the low end. Well, more an extrapolation because Lint at DC is unknown, and when you consider that Lint at HF is leveraged from Rint at HF, then projected to DC, possibly a wild extrapolation when the conductor is a non-magnetic cladding effective at HF and magnetic core effective at low frequencies. Whilst Johnson does not mention whether or not it is suitable to non-homogenous conductors (eg silver clad steel in this case) the formulas used depend on it and it is my opinion that it is unlikely to be well suited.

Neverthess, it is used in a sense of a more complicated model must be better, mustn’t it?

One of the problems I have discovered is that the calculators are unaware of whether there are one or two conductors subject to Lint, and Lint can be silently calculated greater than the 50nH limit for round conductor mentioned earlier.

## Maguire’s implementation

Maguire retrofits this transition modelling to the fully developed skin effect model mentioned earlier. The new model needs one more data value, the DC resistance which Maguire specifies in a bit of a round about way, a notional loss per 100 ft due to the DC resistance.

Explanations are scarce, but k0 is described a modelling the DC resistance as if it did not play into HF results. In fact k0 and k1 both play into Zint, and results at HF are sensitive to the value of k0.

As far as I see it, you MUST supply a valid value for k0, no matter what frequency you wish to use.

It would be clever if you could opt out of this scheme by specifying k0=0 and that was used by the software to calculate Zint by the old method. The situation with most if not all tools implementing Magtuires code is that you have no choice.

You might be aware that SimSmith implements its ‘simple model’ with k0=k2=0. I think that is a mistake, k0=0 still plays into the Zint calculation and gives rise to a higher Lint that would occur if the correct DC resistance was applied.

You might ask “what is the correct DC resistance when using a copper clad steel conductor… it is the actual, should it be for a copper conductor, or are non of these actually correct?”

The issue is that as frequency decreases and significant current flows further into the steel core, its resistance is higher but most importantly, it has high permeability so a large quantum of flux develops in the steel core, much more than would have occurred in a plain copper conductor.

## Belden 8216 problem

A contribution to the problem mentioned with SimSmith v16.3 modelling of Belden 8216 that k0=0 (not easy to find), and this causes insane Zint at low frequencies.

## Is this simply a Belden 8216 problem?

… and who uses it anyway?

Well, the model question is more general applying to non-homogenous conductors and some are very popular: copper clad steel windowed ladder line, especially the 19 strand type, most RG6, many RG174 / RG316 type cables to name a few.