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: 19 Jul 2008

 Quick Summary

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%;

 Further Details Test Results Results of comparison tests. Software Map An overview of the software. Examples Voltage and current distributions.