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Re: Measuring self-capacitance directly (Re: flat secondary)



Original poster: "Paul Nicholson by way of Terry Fritz <twftesla-at-qwest-dot-net>" <paul-at-abelian.demon.co.uk>

[Definition of equivalent energy capacitance]

I used the assumption that any two voltages on the coil at
resonance differed in phase by an integer multiple of pi radians.

Antonio wrote:
> I see a problem in this derivation: If the coil follows the
> lclclc...  model shown, there is no point in the RF cycle where
> all the energy is stored in the capacitors.

In general, yes this is true, although it doesn't invalidate the
argument, merely complicates the expression for Cee.  The tssp
software sums all the stored energy when computing Lee and Cee,
which involves the terms CxVx^2 and LxIx^2, as well as other terms
due to the energy stored in the mutual inductance and mutual
capacitance. 

So continue with the derivation but using the full expression for
stored energy to define a Cee which represents that energy in a
simple LC circuit having the same topvolts.

In practice the standing waves do have almost uniform phase and
discarding either the L or C energy achieves a very good
approximation.

The most noticeable departure from uniform phase of the voltage
occurs with base-driven CW steady state at resonance.  The
voltage at the base is in phase with the current, but the phase
rapidly advances to +90 deg over the lower fraction 1/sqrt(Q) of
the length of the coil.  The voltage at any point is a mixture of
the in-phase drive voltage and the quadrature induced voltage due
to the coil current. By 1/sqrt(Q) of the coil length, the latter
dominates and for the remainder of the coil the the V is leading I
by virtually 90 deg.  The currents share a common phase (modulo pi
for the higher overtones) to within 1 or 2 deg all along the coil.

The free resonance solutions for the isolated resonator approach
the uniform phase condition much more closely than the forced
responses, even when coupled to a primary resonator with normal k
factors, and even when the Q is low.  There's probably a deep 
reason for the fact that neither the non-uniformity of the self
L and C, nor the longitudinal coupling via the mutual L and C,
appear to distort the phase uniformity of the normal modes, and so
far it appears that the standing waves of all the normal modes
display uniform phase (modulo pi) just like a uniform line.  
This is not something I expected in advance, so the software is
coded not to assume it.  This feature emerges, but I'm unable to
supply a proof that it should always be so.

[Measuring the Zbase pole frequencies in grounded base config]

I wrote:
> Feed a signal into the top of the coil and find the frequency in
> between F1 and F2 at which the impedance dips to a minimum.

Antonio wrote:
> Something strange here.

You're quite right, my method was incorrect, for the reasons you
give below.

>  Zin is the base impedance with the top end open, or the z11
> open-circuit impedance parameter of the network.

Yes, ok.

> The impedance seen at the top end with the base grounded is the
> inverse of the y22 short-circuit admittance parameter of the
> network.

agreed,

> ...and that the zeros of z11 are also zeros of y22.

indeed, and I had just assumed that this was also true of the poles
as well, but then you said:

> Some calculation shows that the poles of z11 don't appear
> anywhere in y22 (the networks are different, and can't have the
> same natural frequencies *), ...

Yes, indeed you're quite right. I hadn't realised that the z11 poles
didn't line up with the y22 poles, I took it for granted without
bothering to check, but I see now why they differ.

> This is more complicated to demonstrate than appears to be...

Ok, I'll try...

If we relate the impedance and admittance parameters to the 
cascade matrix, defined as usual by 

           base                top   
                 +-----------+  
   I1-->  o------|   coil    |------o  <-- I2
         V1      +-----------+      V2 
          o-------------------------o


     | V1 |     | a11    a12| |  V2 |
     |    |  =  |           | |     |
     | I1 |     | a21    a22| | -I2 |

then we have 

   z11 = a11/a21
   y22 = a11/a12

For the kind of networks we're talking about here, the elements
of |a| are rational polynomials in w with denominator unity [*],
ie the elements a11, a12, a21, a22 have zeros but no poles. 

We see that the zeros of a11 become the shared (and only) zeros of
z11 and y22.  The poles of z11 are the zeros of a21, and the poles
of y22 are the zeros of a12.  Therefore, unless the network has the
property roots(a12) = roots(a21), we cannot expect the poles to
coincide.

The distributed model demonstrates the point nicely, as in

 http://www.abelian.demon.co.uk/tmp/z11xy22pn1.gif

Both the z11 poles and the y22 poles represent half-wave (or any
even-wave) resonances, but the z11 and y22 poles involve different
boundary conditions on the resonator, and the resulting voltage
profiles are radically different:

 http://www.abelian.demon.co.uk/tmp/z11xy22pn2.gif

In the z11 pole measurement, the impedance is high at both ends and
the end capacitances of the coil are very involved.  Conversely,
when measuring y22, both ends are at low impedance and the end
capacitances are virtually decoupled.  In this latter measurement,
the rather smaller central capacitance of the coil is mostly
responsible for determining the frequency, hence the much higher
frequency of the y22 poles.

This makes y22 poles easy to measure (immune from instrument C),
and z11 poles hard.  Unfortunately it's the z11 pole(s) that we
want...

> Anyway, to measure at the output, with the base open, is a valid
> idea, 

Ok, we can invert the cascade matrix to model this situation. Let 

     | V2 |     | a'11    a'12| |  V1 |
     |    |  =  |             | |     |
     | I2 |     | a'21    a'22| | -I1 |

in which the relevant inverted matrix elements are

  a'11 = a22/det(a)
  a'21 = a21/det(a)

Then we have the top-end impedance (with base open circuit) given
by

  z'11 = a'11/a'21 = a22/a21

We see that the poles of z'11 match those of z11 since they both
come from the zeros of the polynomial a21, so the method is correct
for determining z11 poles.  But unless a22 = a11, the zeros will
differ.  How nice.  I should alter the tssp model to test this
point.

Many thanks for drawing my attention to this behaviour.
--
[*] since the cascade matrix is a product of the form

|  1    0 |   | 1-w^2L1C1   jwL1 |         | 1-w^2LnCn   jwLn |
|         | * |                  | * ... * |                  |
| jwC0  1 |   |  jwC1         1  |         |  jwCn         1  |

for some n.
--
Paul Nicholson
--