| TSSP: Precision Resonator Model |
The mathematical equations which describe the operation of the resonating solenoid are too difficult for us to solve
analytically. We therefore rely heavily on a numerical computational model as a means of generating testable predictions of
resonator performance. This page introduces that model.
Updated 9th June 2001.
This is a precision model
of a closewound single-layer solenoid with a non-conductive core, operating perpendicular to
a ground plane - the configuration typically used as a resonating secondary in Tesla coils.
We employ a non-uniform transmission line model extended by additional current and voltage
injections to enable accurate representation of the solenoid's longitudinal coupling due to internal capacitance and mutual inductance.
A detailed capacitance matrix is obtained from the geometry of the resonator and its surroundings
by means of a boundary element method,
and the mutual inductance profile is generated by
summing contributions from elementary current filaments. The software solves the integral equations of the solenoid
in the frequency domain for steady state operation, and in the time domain for impulsed operation. The model will compute
the normal modes of single and dual resonator configurations, along with the time domain response to a given set of
starting conditions.
The model has only two arbitrary
constants which calibrate loss factors, and is otherwise constructed
from reasonable physical models, parameterised by input measurements of
only lengths and turns.
Factors taken into account by the model are:
- Distributed external capacitance between the secondary and ground.
- Distributed internal capacitance between one region of the secondary and another.
- Lumped capacitance of a toroidal topload to ground.
- Distributed capacitance between regions of the secondary and
the topload.
- Distributed capacitance between secondary and primary.
- Self inductance of each secondary turn.
- Mutual inductance between each pair of secondary turns.
- Distributed mutual inductance between each turn of primary and secondary.
- Self inductance of the primary.
As currently implemented, the software model:
-
Successfully predicts the frequencies of the quarter wave, three-quarter wave,
and five quarter-wave resonances to within a few percent for all coils less than one
secondary height above ground.
-
Predicts the complex magnitude of the voltage and current distributions along the secondary for both the forced
response and the free resonance.
-
Estimates the input impedance spectrum expected when the resonator is fed from the secondary base or via a primary coil,
series (center or anywhere) feed, or top feed.
-
Estimates Q factor, energy storage, and power dissipation.
-
Computes the equivalent reactances of the resonator to an accuracy of around 2% at the quarter wave frequency.
- Takes about 2 hours to fully model a resonator on a P2 500Mhz processor.
At present, the model does not:
- Perform well on Q factors, due to the lack of an effective physical model of winding losses.
- Handle solenoids where the base is more than one coil height above ground,
or where the ground plane is not well defined, due to the difficulty of accurately computing the capacitance matrix.
- Handle resonators which lack cylindrical symmetry.
- Handle shunt feed configurations, ie the autotransformer.
Estimate of frequency prediction error limits: 1/4 wave, +/- 4%; 3/4 wave, +/- 3%; 5/4 wave, +/- 3%;
Test Results Results of comparison tests.
Software Map, An overview of the software.
Examples Document pn1710, example voltage and current distributions, html 155k.
The software is approaching the end of a major reworking and will be available for download around about the end of
June.
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.