University of Canterbury

Department of Electrical and Computer Engineering

ENEL427 - Project

Developing a New Design Procedure for the Tesla Coil using Finite

Element Method

Final Report

Student Name: P.N. Rue

Student I.D: 53936797

Student Email: pnr22@uclive.ac.nz

Project Supervisor: Professor Pat Bodger

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Abstract

Though the Tesla coil has been around for over a one hundred years, the fundamental design

process has not significantly changed. This is a result because the Tesla coil is rarely studied

by academics and is usually more of a hobbyist activity. The hobbyist who build the Tesla

coil do not fully understand the electrical principles behind the Tesla coil and hence why the

design process has not changed.

One major problem in the design process is determining the effective secondary capacitance.

The secondary capacitance is determined by the shape of the top toroid and there is no

accurate equation at can determine the secondary capacitance based on the toroid shape.

This report describes a new design procedure for the Tesla coil using Finite Element Method.

Finite Element Method is an accurate tool that can accurately solve non-linear problems

including inductances, mutual inductance and capacitances. To prove that Finite Element

Method is accurate enough a existing Tesla coil was modified. As a results of the model and

measured results, Finite Element Method can confidently be used in the design process.

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Table of Contents

1. Introduction ........................................................................................................................ 1

2. Tesla Coil Principles & Operating Theory ......................................................................... 2

2.1 Layout of a Basic Tesla Coil ..................................................................................................... 2

2.2 Operating Theory of a Basic Tesla Coil .................................................................................... 2

3. Finite Element Principles ...................................................................................................... 5

4. New Tesla Coil Design & Testing Procedure ....................................................................... 6

4.1 Overview ....................................................................................................................................... 6

4.2 Traditional Calculations................................................................................................................. 7

4.3 FEM Modeling ............................................................................................................................... 7

4.3.1 MagNet Simulation ................................................................................................................ 7

4.3.1 ElecNet Simulations ............................................................................................................... 9

4.4 PSCAD Simulation........................................................................................................................ 10

4.5 Testing Procedure ....................................................................................................................... 11

5. Design Example of a Tesla Coil ......................................................................................... 12

6. Discussion ........................................................................................................................... 14

7. Conclusion ........................................................................................................................... 15

8. References’ ......................................................................................................................... 16

P.N Rue Page 1 of 16 3/10/2011

1. Introduction

The Tesla Coil is essentially a High Frequency Air Core Resonate Transformer that can

transform mains AC voltage into hundreds of kilovolts on the secondary side. The Tesla coil

which was invented by Nikola Tesla in the 19th Century has today evolved into a hobbyist

activity with thousands of people around the world (especially in the USA) regularly meeting

up and showing their designs. The main feature of the Tesla Coil that attracts thousands of

people to them is the high frequency and high voltage sparks that can be seen coming off the

top.

Though the Tesla Coil has been studied and designed by many people, the fundamental of the

design process has not changed over the past one hundred years. This is due mainly for the

fact that people who build them come from a hobbyist approach and do not fully understand

the electrical principles behind the Tesla Coil. One major problem in the design process is

determining the secondary capacitance. The secondary capacitance is created by the shape of

the top toroid relative to other objects and ground. There is no concrete and easy way to

determine the secondary capacitance based on the shape and the process involved with

determining the value is more of an art than engineering.

This report describes a new method using Finite Element method that will allow the designer

of a Tesla Coil to draw in the physical dimensions of their design and accurately determine

the electrical parameters including primary and secondary inductance, mutual inductance

between the two coils and the effective secondary capacitance based on the shape of the

toroid.

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2. Tesla Coil Principles & Operating Theory

2.1 Layout of a Basic Tesla Coil

Figure 1 show the typical arrangement for a Tesla Coil. The typical Tesla Coil must consists

of the following[1]

• A voltage supply usually at mains voltage (230 Vac RMS)

• A step up transformer with an output voltage usually between 10 kV and 15 kV. The

output must have a current limit usually between 30 mA and 60 mA.

• A spark gap usually connected across the primary to the step up transformer

• A Primary Circuit consisting of

o A high voltage capacitor

o A primary inductor

• A Secondary Circuit consisting of

o A secondary inductor

o A secondary toroid with the shape determining the effective secondary

capacitance

2.2 Operating Theory of a Basic Tesla Coil

Figure 2 shows the different stages of operation of a Tesla Coil which can be broken down

into four different stages.

Figure 1 General Layout & Circuit Diagram of a Tesla Coil

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In the first stage, the source sees the inductor as a short circuit. The reason for this is because

the impedance of the inductor given at 50 Hz is very low compared to the resonate frequency

which is chosen by the designer. As a result of the very low impedance, the HV Capacitor on

the primary circuit is charged up to nearly the same potential as the HV side of the step up

transformer. At the point where the potential of the HV Capacitor is higher than the break

down voltage of the spark gap, the spark gap will break down causing a short circuit across

the HV side of the step up transformer. As a result of this short circuit, the supply circuit and

primary circuit are electrically decoupled.

In the second stage, as a result of the short circuit, the primary circuit can be modeled as an

RLC resonant series circuit. As a result of the decoupling, the energy stored in the primary

capacitor is dissipated into the primary inductor. Provided that the primary RLC circuit is

under damped, then energy will start to oscillate between the primary inductor and HV

Capacitor at the circuit resonate frequency.

In the third stage, the secondary circuit can also be modeled as an RLC resonant series circuit

as it consists of a secondary inductor, a toroid which determines the effective secondary

Figure 2 Operation Theory of a Tesla Coil

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capacitance and winding and air resistance. Provided that the calculated resonant frequency

of the secondary circuit is similar to the primary, and the secondary circuit is also under

damped and the coupling coefficient of the primary inductor to the secondary inductor is

between 0.05 and 0.2 then power from the primary inductor will be transferred to the

secondary inductor. As a result, energy from the secondary inductor will start to oscillate

between the secondary inductor and the effective secondary capacitance at the secondary

circuit resonate frequency resulting the high frequency and high voltage sparks seen from the

top.

In the fourth stage, once the energy in the primary circuit dies away, the power being

transferred to the secondary circuit will stop resulting in the spark gap to open up again. As a

result, the source will see the inductor as a short circuit, and charge the capacitor again,

repeating the process.

This cycle of events happens once every half cycle of a 50 Hz mains AC voltage. And hence

the spark gap shorts 100 times every second[1]

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3. Finite Element Principles Finite Element Method (FEM) is a numerical method that can solve non-linear problems.

FEM has number of applications including Mechanical and Electrical projects. For this

project, MagNet and ElecNet were used. Magnet is used to solve the magnetic fields involved

with a Tesla coil. As a result, the inductance of the primary and secondary circuit and the

mutual inductance between the two coils can accurately be determined. ElecNet is used to

solve the electric fields involved with a Tesla coil. As a result, the effective secondary

capacitance and self capacitance can be solved[3][4].

Figure 3 shows an example of how FEM works. FEM works by the user drawing in an

outline on an object. The user lists the material of the object and selects which part of the

object will see the high voltage and what part sees the ground. Then FEM will break the

drawing up into small grids. FEM knows based on the users data that the high voltage will

have an equipotential point of 100% and the ground plane will have an equipotential point of

0%. The program iteratively goes through each point on the grid and finds the average of the

sum from the surrounding points. The program does this for each point until all the points

have been calculated. The program goes over all the points again and again until the

difference between the previous value is small that there is very little change.[3][4].

Figure 3 FEM Example of Electric Fields

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4. New Tesla Coil Design & Testing Procedure

4.1 Overview

Figure 4 show an overview of the new design procedure. The Procedure is as follows:

The first step is to calculate the traditional values of a Tesla Coil based on the desired inputs.

The next step is to draw a model of the Tesla coil in MagNet[3] and ElecNet[4] based on the

physical and electrical values determined in the first step. Based on this model, the

inductances of the primary can secondary circuit can be determined and compared to first

step. The mutual inductance can be determined and the effective secondary capacitance can

be determined can compared to the estimated values.

If the model corresponds well to the first stage than the third step which involves modeling

the design using PSCAD can be carried out. If the model produces the expected waveforms,

then the next step of building the Tesla coil can be carried out. If not, the First step needs to

be carried out again until the model is right.

Once the Tesla Coil is built and tuned, testing needs to be carried out to check whether or not

the physical values match to the calculated. If so then the Tesla coil should work. If not

change the design parameters so see if Tesla coil will work.

Figure 4 Overview of New Design Procedure for a Tesla Coil

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4.2 Traditional Calculations

.

When designing a Tesla coil, there are five key inputs that needed to be known, they are:

• The desired secondary voltage

• The desired resonate frequency (same for primary and secondary circuit)

• The primary capacitor

• The secondary coil radius

• The estimate of the secondary terminal capacitance (capacitance created by the toroid)

These inputs are important because the desired resonate frequency, and capacitor will

determine the value of the primary inductor and hence shape. The value of the secondary coil

radius and resonant frequency and terminal capacitance will determine the secondary

inductance and shape[1].

When selecting the power supply for the Tesla coil, it is important to note that the size of the

output current must be able to supply enough charge to charge a capacitor up with in 20ms

otherwise the Tesla coil will not work properly.

For a full set of design equations, refer to The Ultimate Tesla Coil Design and

Construction[1].

4.3 FEM Modeling

MagNet[3] and ElecNet[4] both come in 2D and 3D versions. The 2D version of the FEM

software can be used to model a 3D version of an object provided that the object being

modeled is symmetric around a point. For the case of this project, the Tesla coil is symmetric

about a point so when modeling it, only the cross sectional part through the center needs to be

modeled.

For a full set of instructions please refer to the infolytica [3][4] website.

4.3.1 MagNet Simulation

MagNet[3] is a software package that can model magnetic fields of object. Figure 5 shows a

typical model of a Tesla coil in MagNet. It is noted that there are only 4 blocks which

consists of a primary and secondary inductor. Though in real life, the blocks are actually wire

or copper tubing, to make the complexity of the model simple, blocks have replace copper

P.N Rue Page 8 of 16 3/10/2011

tubing and wire. This is because there is very little difference in results between using

modeling wire tubing as blocks and circles.

From this model, the primary inductance, secondary inductance and mutual inductance can be

calculated.

To calculate the primary inductance, the current per turn on the secondary inductor must be

zero and the current per turn on the primary coil must be one. Based on the output results

after the model is solved, the primary inductor can be calculate to be

�������� = �,� �

Where �������� is the Primary Inductance, �,�is the Flux Linkage on the Primary Coil

created by the Primary Current and �is the Primary Current.

To calculate the secondary inductance, the current per turn in the primary must be set to zero

and set the current per turn in the secondary inductor must be set to one. Base on the output

results, the secondary inductor can be calculate to be

���������� = �,� �

Figure 5 MagNet Model of a Tesla Coil

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Where ����������is the Secondary Inductance, �,�is the Flux linkage on the secondary coil

created by the Secondary Current and �is the Secondary Current

To determine the mutual inductance between the primary and secondary coil:

The primary current per turn is set to one and secondary current per turn is set to zero

or

The secondary current per turn is set to one and primary current per turn is set to zero

Then the model is solved and hence the mutual inductance can be calculate by

� =�,� �=�,� �

Where �is the Mutual Inductance, �,�is the Flux linkage on the primary coil created by the

Secondary Current and �is the Secondary Current, �,�is the Flux linkage on the secondary

coil created by the primary current and �is the primary current.

The reason why a current per turn of one has been used instead of the actual current is

because the inductance is created by the shape of the wire, not how much current is going

through a wire hence by charging the current, and the flux linkage will change

proportionally[3].

4.3.1 ElecNet Simulations

ElecNet[4] is a software package that can model electric fields of an object. Figure 6 shows a

typical model of a Tesla Coil in ElecNet. For the same reasons as described in section 4.3.1

MagNet Simulation, Blocks have been used to show the primary and secondary inductor.

When selecting which block becomes what type of electrode, it is important to remember that

the secondary circuit will see the primary inductor as a ground plane. Hence for a basic

model to determine the effective secondary capacitance, the top toroid should be made a

positive electrode at 1 V RMS, and the primary inductor a ground electrode. The secondary

inductor is not made an electrode, and as a result, the FEM program will not see the

secondary inductor. Normally, the secondary inductor will have a small impact on the

secondary capacitance, but to reduce the complexity of the model, it has been ignored as is

has little effective on what the actual secondary capacitance is.

Once the model has been set up and solved, the total secondary capacitance can be calculated

by:

���������� =��

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where ����������is the effective secondary capacitance, Q is the modeled charge on the top

toroid and V is the applied voltage to the top of the toroid[4].

4.4 PSCAD Simulation

Once the electrical parameters of the design have been determine, the Tesla coil can be

modeled using PSCAD[5]. PSCAD is normally uses to model power systems but has been

chosen because it is relatively simple to use. Figure 7 shows a basic model in PSCAD of a

Tesla coil. The main difference between the PSCAD model and the actual Tesla coil are the

following:

Instead of modeling a step up transformer with mains supply, a source that simulates the

output of a transformer has been used. The main reason for this is it reduces the complexity

of the model.

The spark gap has been modeled using two power electronic switches. The switches are

connected to signal generators which turn the switches on when the waveform reaches the

peak and dip of the cycle as it normally would in a Tesla coil. The resaons why power

electronic switched have been used is because PSCAD does not have a simple spark gap

model.

The couple inductor has been modeled using a couple transmission line model. The reason

for this is because though the couple inductor of a Tesla coil is effectively a transformer but

all the transformer models in PSCAD are give in terms of leakage and magnetising

Figure 6 ElecNet Model of a Tesla Coil

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reactance’s not in terms of mutual inductances and inductance which have been determined

be the Tesla coil model.

4.5 Testing Procedure

Once the Tesla Coil has been designed, and built, there are three electrical tests that need to

be carried out. They are:

Measure the Primary, Secondary and mutual Inductance.

Measure the Resonate Frequency of both the primary and secondary circuit.

Determine the secondary Capacitance of the Secondary Circuit.

These tests are important because if the primary and secondary circuit of the Tesla coil tuned

with a similar resonate frequency, then the Tesla coil will not work.

It is also important to note that the secondary circuit should have a slightly lower resonate

frequency than the secondary circuit. This is because as soon as the spark breaks out of the

top toroid, the effective capacitance will slightly decrease and hence the resonate frequency

of the secondary circuit will increase.

The testing procedure used was taken from the Classical Tesla Coil Design Report[6].

Figure 7 PSCAD Model of a Tesla Coil

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5. Design Example of a Tesla Coil An existing Tesla Coil that was manufactured from a Third Pro Project in 1996 was taken and

had a second toroid added at the bottom. The second toroid was added to the secondard

circuit to prove the concept that FEM is accurate and can be applied to the Tesla Coil.

The reason why a new Tesla coil from scratch was not designed was because the Tesla coil

principle has been around for over one hundred years. But the principle of using Finite

Element method to help design a Tesla coil has not been done before. The ideas of this

project it to prove that the FEM will be accurate enough to help aid the Tesla coil design.

To make the construction of the secondary toroid easier, the secondary toroid had been

constructed so the primary inductor, and secondary inductor would sit on top of the

secondary inductor.

Figure 8 shows the model of the Tesla coil. The figure on the left shows the Tesla coil

without the second toroid and the right figure shows the Tesla coil with the second toroid.

The results from the model are as follows:

Without secondary toroid With second toroid

Applied Voltage 1 Volt RMS Applied Voltage 1 Volt RMS

Measured Charge 16.5 x 10��� C Measured Charge 17.4 x 10��� C

Capacitance = 16.5 x 10��� F Capacitance = 17.4 x 10��� F

Figure 8 Tesla Coil Design Example

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After the secondary toroid was added to the Tesla coil the secondary inductance was

measured and the secondary circuit resonant frequency was determined before the toroid was

added and after. Once the inductance and resonate frequency of the secondary circuit is

known, the effective secondary capacitance can be calculated by the following formula

���������� = ��� !

"#$%&'()*+

The following are the results from the design,

Without Second Troid With Second Troid

Inductance 36.4 mH Inductance 36.4 mH

Resonate Frequency 202.6 kHz Resonate Frequency 202.1 kHz

Measured Capacitance 16.9 pF Measured Capacitance 17.0 pF

Modeled Capacitance 16.5pF Modeled Capacitance 17.4pF

Based on the results of the modeled capacitance and measured capacitance, Finite Element

Method can accurately be used in the design process of the Tesla coil.

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6. Discussion Though the Tesla coil has been around for over one hundred years, the general design

procedure has not changed. This is mainly due to the fact that the Tesla coil is mainly a

hobbyist activity and the people that design them do not have a full understanding of the

electrical principles behind the Tesla Coil. As a result, when determining the secondary

capacitance, it has been more of an art that science.

FEM by the results of the Tesla coil design example can be used as a tool in the design

process. Though electromagnetic specialist can solve for non linear capacitances, the process

involved using special equations and special rules mean that only a select few engineers can

solve the secondary capacitances, when compared to FEM software where anyone who has a

basic knowledge in electrical engineering can use the software and get similar results. FEM is

an easy software program that can be used to solve non-linear problems such as the Tesla

coil.

PSCAD is a highly recognized power system modeling program. A simple model of the Tesla

Coil was developed using PSCAD using a power electronic switch for the spark gap,

transmission line coupled inductances for the primary and secondary inductor and a source

which simulated the output of the step up transformer with a current limit. As a result, the

model was accurate enough to determine whether or not the calculated electrical values

would make the Tesla coil work.

An existing Tesla Coil that was manufactured and designed in a third pro project in 1996 was

taken and a bottom toroid added to the secondary circuit. The purpose of this was to prove

that FEM is accurate enough to be used for the design process of a Tesla Coil. The model of

the Tesla coil in ElecNet showed that the effective secondary capacitance was 16.5 pF

without the second toroid added and 17.4 pF will the second toroid added. The modeled

values compare well with the measured value of 16.9 pF without the second toroid and 17.0

pF with the second toroid. As a result, FEM can be confidently used in the Tesla coil process.

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7. Conclusion

The Tesla coil has been around for over 100 years and is mainly a hobbyist activity. As a

result, the design procedure has not changed significantly as the hobbyists who build them do

not fully understand the electrical principles behind the Tesla coil. As a result, determining

the effective secondary capacitance has been more of an art that science. As a result of the

project, Finite Element Method which is a technique for solving non linear problems can be

used to solve the effective secondary capacitance of a Tesla coil. An existing Tesla coil that

was constructed in 1996 was taken and modified by adding a second toroid to the secondary

circuit. This was done to prove that FEM could be used in the design process. The modeled

result and measured result of the secondary capacitance of the Tesla coil was similar proving

that Finite Element Method can be used with confidents in the design process.

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8. References’

[1] Tilbury, M, The Ultimate Tesla Coil Design Guide and Construction, McGraw Hill 2008

[2] Couture, J H, Tesla HandBook, JHC Engineering – San Diego CA, 1988

[3] infolytica, “MagNet”, http://www.infolytica.com/en/products/magnet/, accessed 3/10/11

[4] infolytica, “ElecNet”, http://www.infolytica.com/en/products/elecnet/, accessed 3/10/11

[5] Manitoba HVDC Research Center, “PSCAD”, https://pscad.com/index.cfm?, accessed

3/10/11

[6] Chapman, R, Clasical Tesla Coil Design, University of Canterbury, 1996