From: Paul
Date: Wed, 14 Jun 2000 12:36:18 +0100
Subject: [TSSP] In search of dispersion
This post continues a discussion by private email with Robert Jones,
on the subject of interwinding capacitance and mutual inductance.
[ Paul tries to justify a large (1000) element model in order to
explore effect of interwinding C]
> >however it does seem to provide a mechanism for dispersion. Therefore
> >lately I've been trying around 1 to 4 elements per turn so that I can
Robert Jones wrote:
> This statement is very revealing. I think you have not included
> inductive coupling in your model. Thinking about this a little more
> if you had, you would have one million simultaneous equations for a
> one thousand element model.
I believe I've included inductive coupling, with end effects. With N
sections there are N+1 currents and N+1 voltages, so 2N+2 equations,
but I algebraically eliminate the Vs to leave N+1 equations in the
N+1 currents. There are however around N^2 terms summed into the
coefficients of the currents though.
> Although I agree turn to turn coupling causes dispersion it is also
> caused by the inductive coupling a much larger affect in the average
> coil. In fact any coupling parallel to the wave causes frequency
> dependent velocity.
That might explain something, I'll address this point a little further
on.
[ Paul trying to justify use of the base input impedance as a means of
assessing correctness of model]
> >My use of the base input impedance is precisely because it is so
> >sensitive to the total losses in the resonator. Inevitably the model
> >reports an impedance lower than measurements, and this could be made
> >up by putting in an arbitrary factor to account for losses not
> >explicitly modelled, such as I^2R loss in the ground plane. However
> >I dont do this, instead I like to use the discrepancy between
> >simulated and measured impedance as a measure of the extent to which
> >the model is accounting for losses.
>
> An interesting academic exercise but not applicable to the practical
> case as the typical coil never reaches steady state.
Agreed, except in the case of CW coils, which is one of the reasons I
started looking into the resonance more closely. I wanted to know the
input impedance, and I wanted to know if the square wave from a voltage
switching driver would excite the coil at the 3rd harmonic in addition
to the fundamental.
> How close is the
> frequency and propagation impedance?
The model 1/4 wave resonance always seems to come out about 20% higher
than the measured frequency, (about 25% if I ignore inter-winding
capacitance). Here are some numbers
(I use f1 for 1/4 wave, f3 for 3/4 wave, etc)
f1 f3 f5 ratio f3:f1 ratio f5:f1
Real coil 91kHz 213kHz 320kHz 2.34 3.52
Model - no 112kHz 326kHz 539kHz 2.91 4.81
interturn C
Model with
interturn C 107kHz 244kHz 308kHz 2.28 2.88
The model shows a small amount of dispersion from mutual inductance
alone, but not enough. You can see from these figures why I'm keen on
introducing interturn capacitance. It seems at the moment the dispersion
is a little too high now.
Impedance wise, the model gives around 50 ohms at 1/4 wave resonance,
the real coil measures around 110 ohms, so only a portion of the losses
are accounted for.
[Paul asked]
> >1/ What causes the 3/4 and 5/4 resonant frequencies to be
> >significantly below the harmonics of the 1/4 wave frequency?
> Dispersion due the mutual inductive coupling being a function of
> wavelength. I guess you have not been reading my posts
This is very interesting. As I mentioned above I could not produce
the enough dispersion without invoking interwinding capacitance, so
it looks like the model is not doing justice to the mutual inductance.
This may explain why the f1 is always around 20% high in the model.
[Paul asked]
> >... what is the proportion by which energy
> >storage is divided between inter-winding capacitance and the
> >wider capacitance to ground?
>
> I had never calculated it so I did for a typical one thousand turn
> coil assuming a half sine voltage profile. The ratio is more than
> one thousand times lower in the interturn C than in the intrinsic C
> to ground. Hence it can only have 0.1% effect on the propagation
> velocity ie forget it its almost irrelevant. (except for high harmonics)
Damn, never thought to do this obvious calculation. Just repeated it
and got the same sort of answer. On the face of it interturn C
should be almost ignorable. I'll have to assume that the model or code
is faulty with respect to mutual inductance, and that the figures above
are nonsense.
Hope to release code and docs next weekend, so that we can get to the
bottom of this - I've taken a day off work today so that I can get a
bit more done.
Regards,
--
Paul Nicholson,
Manchester, UK.
--
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.