Re: THOR resonance freq. measurement results

["Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> (by way of Terry Fritz <twftesla-at-uswest-dot-net>)]

Tesla list wrote:
> 
> Original poster: "Marco Denicolai" <Marco.Denicolai-at-tellabs.fi>

> - what "TEM" stands for?

Transversal Electric and Magnetic fields. This results in several
simplifications and models quite well coaxial cables and other usual
transmission lines. Applies only reasonably well for a long coil
above ground. 

> - can I model this behaviour e.g. with MicroSim or have you got some material
> (equations) about this (your web site seems to be down right now)?

You can model this behavior with a simulator, using transmission lines,
or using a long LC ladder instead (recently some models were
described in the list).

Following is some theory, that I wanted to derive anyway, and answering
your question was a good opportunity:

The lossless case is not difficult to analyze, and results in the
following transfer function in Laplace transform, between the output
voltage at the terminal capacitor and the input voltage at the other
side, where you connect the signal generator:

Vout/Vin(s)=2*e^(s*T)/(e^(2*s*T)*(s*C*Z0+1)-s*C*Z0+1)

where C is the terminal capacitance, Z0 is the characteristic
impedance of the line, and T its delay.

The resonance frequencies in rad/s are the solutions for w of the 
equation below. No analytical solution appears to be possible, but
it's simple to plot the left side and look for the zeros:

((C*w*Z0)^2-1)*cos(2*w*T)+2*C*w*Z0*sin(2*w*T)-(C*w*Z0)^2-1=0

To compute Z0 and T for a coil, use:

T=(pi/2)*sqrt(Cm*L)
Z0=(2/pi)*sqrt(L/Cm))

where L is the inductance of the coil and Cm is it's self-capacitance,
and pi=3.1416.
This makes the resonance of the lumped model of the coil without
terminal:
w=1/sqrt(L*Cm) rad/s
coincident with the first solution of the equation above when C=0:
w=pi/(2*T) rad/s
and is also consequence that at low frequency the coil is just an
inductor with inductance L, what adds the relation 1/(T*Z0)=L.
 
Antonio Carlos M. de Queiroz