TSSP: Tesla Secondary Simulation Project

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An Internet collaboration exploring the physics of Tesla resonators

This project was started in May 2000 with the aim of exploring the fascinating physics of self-resonant single-layer solenoids as used for Tesla transformer secondaries.

We maintain a precision software model of the solenoid which is based on our best theoretical understanding of the physics. This model provides a platform for theoretical work and virtual experiments.

Updated: 19 Jul 2008

  Site Contents

Model Modeling software.
Software Map A summary of the project's software.
Theory Notes Secondary Basics - pn2511.
More Theory Notes Resonator Theory - pn1401.
Examples Voltage and current distributions.
Virtual secondary database A database of simulated secondary performance.
Various Formulae
Time domain modeling
Mode analyser A program for analysing TC scope waveforms.
Stress Factors Secondary voltage stress factors.
Equivalent Series Inductances Tabulation by secondary shape.
Gallery Simulations, animations, and sounds.
Coil models A close look at the tuning of some coils.
Q Variations Experiments looking at variation of secondary Q-factors.
GeoTC A Javascript library for TC reactance calculations.


Tesla coils have been in use for over a century for high voltage testing, particle accelerators, and other applications requiring upwards of 100 kV at low to moderate radio frequencies. They are also capable of producing entertaining displays of electrical discharges and their construction has been developed to a fine art by several generations of dedicated coil building enthusiasts. Despite this, there are many aspects of Tesla coil design and operation which still appear to present a mystery, and in some cases a great deal of poorly supported folklore has built up around them.

This project is an attempt to gather reliable information about just one of these topics - the physics of Tesla secondary resonators.

Consisting of a close wound single layer air cored solenoid resonating at its lowest self-resonant frequency in conjunction with a ground plane and charge storing topload, the coil is excited either by a continuous wave source, or by a power compression circuit usually consisting of the spark triggered discharge of a storage capacitance. Each type of operation places its own particular requirements on the secondary coil design, and one of the aims of this project is to clarify these requirements by examining and understanding the detailed behaviour of the resonator in both of these modes of operation.

Some of the difficulty with understanding Tesla coil operation follows as a result of the poor understanding of the resonant behaviour of close wound solenoids. This research project aims to correct this deficiency, by putting together a comprehensive quantitative theory of resonant solenoids.

  Progress and Conclusions

We have established a set of differential equations which describe the operation of a solenoid and we have demonstrated that the solutions for small signal CW operation adequately predict the outcome of careful measurements on Tesla resonators. From these fundamental equations of the coil, we can put forward unambiguous definitions of the effective secondary capacitance and other useful equivalent reactances, from which a number of interesting and informative relationships are derived [pn2511], including precise expressions for the input, output, and transfer impedances.

A detailed computational model has been set up for the precision simulation of Tesla resonators. [model]. This frequency domain, small signal model accurately predicts the spectrum of resonant frequencies - better than 1% accuracy can be achieved up to the 9/4 wave resonance and the model provides qualitative information for higher frequencies.

The model has been applied to the task of mapping out the performance of secondary resonators in small signal CW operation [vsd], from which semi-empirical formulae have been obtained for the resonant frequencies and effective inductances [formulae].

Descriptions of the secondary resonator, and dual resonator Tesla coil, have been derived in terms of integral operators. The integral equations turn out to have the same mathematical form as those of the elementary lumped approximation, except that the components of the lumped circuit are replaced by the corresponding integral operators. These lead to an eigenequation for the normal modes, the solution of which provides the basis functions from which the time domain response of the resonator is computed.

We have found that, contrary to commonly stated opinion, the longitudinal capacitive coupling of the solenoid is a significant factor in determining its behaviour, accounting for the bulk of the capacitive energy storage at the higher modes. We find that longitudinal mutual capacitance works to reduce the higher mode wave velocity and is the major influence on dispersion. We also find that mutual inductive coupling acts to raise the effective velocity at higher frequencies. Only when both factors are taken into account can accurate predictions of the higher mode frequencies be obtained.

The inability to predict Q factor persists and highlights both the lack of an effective model of winding losses at high frequencies, and the difficulty of quantifying all the other contributions to the energy loss budget of the resonator.

  Current Work

After a couple of years of studying the linear behaviour of the TC, the main thrust of the project now addresses the interaction of the coil with its breakout loading. The main target is to model the formation of inhibited leaders sufficiently well to predict breakout voltages and leader lengths. Examination of the response of the system to transients induced by arc discharges from the topload is also of interest.

For more detail, there is a Things-to-do List.


If you have found anything useful in these TSSP web pages, then it is due to the efforts of the following Tesla coil enthusiasts.

and also to several others who, by keeping an eye on things through the mailing list and occasionally chipping in, help us to feel that these efforts are relevant and worthwhile to the Tesla coil building community.

Maintainer Paul Nicholson, tssp0807@abelian.org.