Re: Magnifer vs. Tesla Coil

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Tesla list wrote:
 >
 > Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk>

 >...
 > We need to resolve the issue of just where the voltage peaks and
 > zeroes of the 3rd mode sit with respect to the coils involved, and
 > I can try to come up with some animations to show what I think is
 > happening, based on a distributed model.  This must tie up exactly
 > with the lumped model too, and I don't think we're too far off.

My program mrn6 can calculate the three components. The programs
puts the origin of time at the instant when the output voltage is
maximum. The three components are then cosinusoids, and the three
points (top, transmission line, primary) can be directly observed.
The voltages are normalized to add to 1.
Some results:
vi = Ai1 cos(w1t) + Ai2 cos(w2t) + Ai3 cos(w3t)
v1: primary, v2:secondary, v3:tertiaty.
Mode 1:2:3
v3: A31=   0.3125000000; A32=   0.5000000000 A33=   0.1875000000
v2: A21=   0.2343750000; A22=   0.0000000000 A23=  -0.2343750000
v1: A11=  -0.3125000000; A12=   0.5000000000 A13=  -0.1875000000
Mode 2:3:4
v3: A31=   0.2916666667; A32=   0.5000000000 A33=   0.2083333333
v2: A21=   0.1620370370; A22=   0.0000000000 A23=  -0.1620370370
v1: A11=  -0.2916666667; A12=   0.5000000000 A13=  -0.2083333333
Mode 3:4:5
v3: A31=   0.2812500000; A32=   0.5000000000 A33=   0.2187500000
v2: A21=   0.1230468750; A22=   0.0000000000 A23=  -0.1230468750
v1: A11=  -0.2812500000; A12=   0.5000000000 A13=  -0.2187500000
Mode 3:4:9
v3: A31=   0.4513888889; A32=   0.5000000000 A33=   0.0486111111
v2: A21=   0.1974826389; A22=   0.0000000000 A23=  -0.1974826389
v1: A11=  -0.4513888889; A12=   0.5000000000 A13=  -0.0486111111
Mode 3:8:9
v3: A31=   0.1180555556; A32=   0.5000000000 A33=   0.3819444444
v2: A21=   0.1014539931; A22=   0.0000000000 A23=  -0.1014539931
v1: A11=  -0.1180555556; A12=   0.5000000000 A13=  -0.3819444444

Of interest here are the voltages v3 and v2, that are the observable
points of the secondary-tertiary system. v3 is the top voltage and
v2 is the voltage at the transmission line. We can say that the voltages
inside the coils very linearly between the listed values, that is what
the lumped model assumes.
The values can be interperted as:
The low-frequency component rises continuously along the coils (1/4
wave).
The middle-frequency component is zero at the transmission line and
accounts always for one half of the output voltage (3/4 wave with zero
at the transmission line, or a degenerate 1/4 wave mode that starts
at the transmission line).
The high-frequency component is always negative at the transmission
line, to cancel the first component there, and positive at the top.
(3/4 wave too, with a zero somewhere at the third coil).

 > I don't see how the lumped model can predict or require a volts
 > peak inside any of the coils.  Along the secondary-tertiary, you
 > only have three voltage 'nodes' to consider: ground, transmission
 > line, and topload.

I was thinking that if for some mode (the central) there is zero voltage
at the ground, zero at the transmission line, and something at the
top, in a real system there is some voltage around the middle of the
secondary (caused by the middle frequency swinging to negative before
crossing zero at the transmission line). A question of complicating
the model to see if this really happens.

 > For each mode to be a distinct normal mode, there must be a
 > qualitatively different electrical motion of the three voltage nodes
 > of the secondary-tertiary part of the network.  Fixing ground as the
 > reference, you then only have two choices for normal mode
 > oscillations:
 >
 > a) t-line and topload oscillate up and down together.
 > b) t-line and topload oscillate up and down in antiphase.
 >
 > (a) obviously describes the 1/4 wave modes, which are distinguished
 > from each other by primary voltage polarity. This leaves (b) as the
 > only remaining option for the 3/4 wave.

Ok, but the analysis shows one 1/4 wave mode (low), one 3/4 wave mode
(high), and a "degenerate" mode (middle).

 > Thus in the lumped model,
 > the t-line and topload can only be in antiphase in mode 3.

They are.

 > Indeed,
 > it seems you can't force the t-line to be a zero of any mode, without
 > introducing another lumped voltage node between the t-line and
 > ground, ie by splitting L2.   There seems to be no alternative for
 > the lumped model, and the t-line voltage of the 3rd mode must be in
 > antiphase to the topload, ie a 1/2 wave apart, and is therefore a
 > voltage max rather than a zero.
 >
 > I'm sure this can be confirmed by looking at the complex amplitudes
 > of mode 3 in the lumped model, comparing the phase of the t-line
 > volts with the top volts.

The simulations above.

 > If I'm wrong, the amplitude of mode 3
 > on the t-line will either be identically zero (ie a zero of the mode,
 > which I think is impossible in the lumped model), or if I'm still
 > wrong, they will be in phase.  But if it agrees with the distributed
 > model they will be exactly in antiphase in a lossless model.  This
 > condition, applying in isolation to the 3rd mode, should be true
 > independently of tuning.

At least with perfect tuning, the low and high frequencies are
really in antiphase at the transmission line.

 > Here's a set of graphs showing the voltage distributions along the
 > two coils of a magnifier,
 >
 >   http://www.abelian.demon.co.uk/tmp/mag1.gif

Seems consistent with the behavior of the low-frequency and the
high-frequency components. What happens with the middle-frequency
component?

 > I'll have to work on animating this set, meanwhile you'll just have
 > to imagine them waggling up and down!  The two coils here have same
 > h/d, the tertiary is twice as long, and has twice the turns, of the
 > secondary.  The relative voltages between the two modes is not
 > significant in this set of graphs.   There's no additional C2.

And so no third mode?

 > Someone will ask what precisely happens at the instant of peak
 > topvolts, at which the t-line is momentarily at zero volts.  The
 > distributed model says this:  All three modes reach their voltage
 > peak with no current anywhere.  The two 1/4 wave signals are of the
 > same polarity and add to some value on the t-line, but the 3/4 wave
 > is peaking at the opposite polarity at that instant, and so all three
 > together sum, momentarily, to zero (with the correct tuning, that is).

What is missing is that one of the 1/4 wave components is degenerate,
as it is zero at the transmission line. This apparently happens in
all the possible modes.

 > Not really.  Define [V] and [I] to be complex column vectors
 > representing the coil volts and current distributions.  Then
 > construct a square matrix [L] such that [V] = [L][I], where the
 > elements of [L] come from the inductance matrix of the coil and will
 > contain factors of j*omega.  Then use the capacitance matrix (the one
 > we've been talking about in the other thread) to produce another
 > equation [I] = [C][V].

A problem (changing the notation a bit): in the usual equations for
a group of inductors, V'=jw[L]I', V' are the voltages over the
inductors,
and I' are currents in parallel with them, but in I=jw[C]V, V are
nodal voltages and I are currents between the ground and the nodes.
So, I and V are not I' and V', although it's easy to obtain ones from
the others. So I suppose that you are arranging the matrix [L] so
V=jw[L]I.

 > Put the two together to get [V] = [L][C][V].

Or V=jw[L]jw[C]V, V=-w^2[L][C]V

 > Then you see that the [V] which satisfy this equation are the
 > eigenvectors of the square matric [L][C].   Look for the omegas which
 > make the determinant |[L][C] - [1]| = zero, and find the corresponding
 > [V] for each one. If [V] is N dimensions, there are N modes.

Ok. you are computing the natural frequencies of the network.

 > If you
 > run through this with the N=3 lumped model to illustrate the idea, then
 > your magnifier equations should pop out.

The analysis equations.

 > The distributed model is
 > just large N, and the distributed real world turns the matrices [L] and
 > [C] into integral operators.

Ok.

Antonio Carlos M. de Queiroz