TeslaCoilImpedance.pdf

Tesla Coil Impedance

Dr. Gary L. Johnson

Professor Emeritus

Electrical and Computer Engineering Department

Kansas State University

gjohnson@ksu.edu

Abstract

The input impedance of a Tesla coil operated as an ‘ex-

tra’ coil, or as a quarter-wave antenna above a ground

plane, is given here. Eects of coil form, wire size, wire

insulation, and humidity are discussed.

C 1

v a

v b G

L 1 L 2

C 2

1. Introduction

A classical Tesla coil contains two stages of voltage

increase. The rst is a conventional iron core trans-

former that steps up the available line voltage to a

voltage in the range of 12 to 50 kV, 60 Hz. The second

is a resonant air core transformer (the Tesla coil itself)

which steps up the voltage to the range of 200 kV to

1 MV. The high voltage output is at a frequency much

higher than 60 Hz, perhaps 500 kHz for the small units

and 80 kHz (or less) for the very large units.

iron core

air core

Figure 1: The Classical Tesla Coil

very high. The impedance during conduction depends

on the geometry of the gap and the type of gas (usually

air), and is a nonlinear function of the current density.

This impedance is not negligible. A considerable frac-

tion of the total input power goes into the production

of light, heat, and chemical products at the spark gap.

The lumped circuit model for the classical Tesla coil

is shown in Fig. 1. The primary capacitor C 1 is a low

loss ac capacitor, rated at perhaps 20 kV, and often

made from mica or polyethylene. The primary coil L 1

is usually made of 4 to 15 turns for the small coils and

1 to 5 turns for the large coils. The secondary coil

L 2 consists of perhaps 50 to 400 turns for the large

coils and as many as 400 to 1000 turns for the small

coils. The secondary capacitance C 2 is not a discrete

commercial capacitor but rather is the distributed ca-

pacitance between the windings of L 2 and the voltage

grading structure at the top of the coil (a toroid or

sphere) and ground. This capacitance changes with the

volume charge density around the secondary, increas-

ing somewhat when the sparks start. It also changes

with the surroundings of the coil, increasing as the coil

is moved closer to a metal wall.

The arc in the spark gap is similar to that of an elec-

tric arc welder in visual intensity. That is, one should

not stare at the arc because of possible damage to the

eyes. At most displays of classical Tesla coils, the spark

gap makes more noise and produces more light than the

electrical display at the top of the coil.

When the gap is not conducting, the capacitor C 1

is being charged in the circuit shown in Fig. 2, where

just the central part of Fig. 1 is shown. The inductive

reactance is much smaller than the capacitive reactance

at 60 Hz, so L 1 appears as a short at 60 Hz and the

capacitor is being charged by the iron core transformer

secondary.

A common type of iron core transformer used for

small Tesla coils is the neon sign transformer (NST).

Secondary ratings are typically 9, 12, or 15 kV and

30 or 60 mA. An NST has a large number of turns

on the secondary and a very high inductance. This

inductance will limit the current into a short circuit at

about the rated value. An operating neon sign has a

low impedance, so current limiting is important to long

The symbol G represents a spark gap, a device which

will arc over at a suciently high voltage. The simplest

version is just two metal spheres in air, separated by a

small air gap. It acts as a voltage controlled switch in

this circuit. The open circuit impedance of the gap is

C 1

by the symbol M . The coecient of coupling is well

under unity for an air cored transformer, so the ideal

transformer model used for an iron cored transformer

that electrical engineering students study in the rst

course on energy conversion does not apply here.

v b

Figure 2: C 1 Being Charged With The Gap Open

R 1 M R 2

^^^

i 1

i 2

transformer life. However, in Tesla coil use, the NST

inductance will resonate with C 1 . The NST may supply

two or three times its rated current in this application.

Overloading the NST produces longer sparks, but may

also cause premature failure.

v 1

C 1

L 1 L 2

C 2

v 2

Figure 4: Lumped Circuit Model Of A Tesla Coil, arc

on.

When the voltage across the capacitor and gap

reaches a given value, the gap arcs over, resulting in

the circuit in Fig. 3. We are not interested in eciency

in this introduction so we will model the arc as a short

circuit. The shorted gap splits the circuit into two

halves, with the iron core transformer operating at 60

Hz and the circuit to the right of the gap operating at

a frequency (or frequencies) determined by C 1 , L 1 , L 2 ,

and C 2 . It should be noted that the output voltage of

the iron core transformer drops to (approximately) zero

while the input voltage remains the same, as long as

the arc exists. The current through the transformer is

limited by the transformer equivalent series impedance

shown as R s + jX s in Fig. 3. As mentioned, this oper-

ating mode is not a problem for the NST. However, the

large Tesla coils use conventional transformers with per

unit impedances in the range of 0.05 to 0.1. A trans-

former with a per unit impedance of 0.1 will experience

a current of ten times rated while the output is shorted.

Most transformers do not survive very long under such

conditions. The solution is to include additional reac-

tance in the input circuit.

At the time the gap arcs over, all the energy is stored

in C 1 . As time increases, energy is shared among C 1 ,

L 1 , C 2 , L 2 , and M . The total energy in the circuit

decreases with time because of losses in the resistances

R 1 and R 2 . There are four energy storage devices so a

fourth order dierential equation must be solved. The

initial conditions are some initial voltage v 1 , and i 1 =

i 2 = v 2 = 0. If the arc starts again before all the

energy from the previous arc has been dissipated, then

the initial conditions must be changed appropriately.

With proper design (proper values of C 1 , L 1 , C 2 ,

L 2 , and M ) it is possible to have all the energy in C 1

transferred to the secondary at some time t 1 . That

is, at t 1 there is no voltage across C 1 and no current

through L 1 . If the gap can be opened at t 1 , then there

is no way for energy to get back into the primary. No

current can ow, so no energy can be stored in L 1 , and

without current the capacitor cannot be charged. The

secondary then becomes a separate RLC circuit with

nonzero initial conditions for both C 2 and L 2 , as shown

in Fig. 5. This circuit will then oscillate or “ring” at

a resonant frequency determined by C 2 and L 2 . With

the gap open, the Tesla coil secondary is simply an

RLC circuit, described in any text on circuit theory.

The output voltage is a damped sinusoid.

R s

X s

C 1

^^^

___

gap

shorted

L 1

L 2

C 2

R 2

^^^

Figure 3: Tesla Circuit With Gap Shorted.

i 2

v 2

L 2

C 2

The equivalent lumped circuit model of the Tesla coil

while the gap is shorted is shown in Fig. 4. R 1 and R 2

are the eective resistances of the air cored transformer

primary and secondary, respectively. The mutual in-

ductance between the primary and secondary is shown

Figure 5: Lumped Circuit Model Of A Tesla Coil, arc

L 1

___

Finding a peak value for v 2 given some initial value

for v 1 thus requires a two step solution process. We

rst solve a fourth order dierential equation to nd

i 2 and v 2 as a function of time. At some time t 1 the

circuit changes to the one shown in Fig. 5, which is

described by a second order dierential equation. The

initial conditions are the values of i 2 and v 2 determined

from the previous solution at time t 1 . The resulting

solution then gives the desired peak values for voltage

and current. The process is tedious, but can readily

be done on a computer. It yields some good insights

as to the eects of parameter variation. It helps estab-

lish a benchmark for optimum performance and also

helps identify parameter values that are at least of the

correct order of magnitude. However, there are several

limitations to the process which must be kept in mind.

above 2 cm. It requires 0.02/80 = 25 µ s for the disc

to turn this distance. This time can be shortened by

making the disc larger or by turning it at a higher rate

of speed, but in both cases we worry about the stress

limits of the disc. Nobody wants fragments of a failed

disc ying around the room. The practical lower limit

of arc length seems to be about 10 µ s. With larger coils

this may be reasonably close to the optimum value.

The third reason for concern about the above calcu-

lations is that the Tesla coil secondary has features that

cannot be precisely modeled by a lumped circuit. One

such feature is ringing at ‘harmonic’ frequencies. Nei-

ther distributed nor lumped models do a particularly

good job of predicting these frequencies. For example,

a medium sized secondary might usually ring down at

160 kHz. Sometimes, however, it will ring down at

3.5(160) = 560 kHz. A third harmonic appears in many

electrical circuits and has plausible explanations. A 3.5

‘harmonic’ is another story entirely.

First, the arc is very dicult to characterize accu-

rately in this model. The equivalent R 1 will change,

perhaps by an order of magnitude, with factors like i 1 ,

ambient humidity, and the condition, geometry, and

temperature of the electrode materials. This intro-

duces a very signicant error into the results.

2. The Extra Coil

As mentioned above, the classical Tesla coil uses two

stages of voltage increase. Some coilers get a third

stage of voltage increase by adding a magnier coil,

also called an extra coil, to their classical Tesla coil.

This is illustrated in Fig. 6.

Second, the arc is not readily turned o at a precise

instant of time. The space between electrodes must be

cleared of the hot conducting plasma (the current car-

rying ions and electrons) before the spark gap can re-

turn to its open circuit mode. Otherwise, when energy

starts to bounce back from the secondary, a voltage will

appear across the spark gap, and current will start to

ow again, after the optimum time t 1 has passed. With

xed electrodes, the plasma is dissipated by thermal

and chemical processes that require tens of microsec-

onds to function. When we consider that the optimum

t 1 may be 2 µ s, a problem is obvious. This dissipation

time can be decreased signicantly by putting a fan on

the electrodes to blow the plasma away. This also has

the benet of cooling the electrodes. For more powerful

systems, however, the most common method is a rotat-

ing spark gap. A circular disc with several electrodes

mounted on it is driven by a motor. An arc is estab-

lished when a moving electrode passes by a stationary

electrode, but the arc is immediately stretched out by

the movement of the disc. During the time around a

current zero, the resistance of the arc can increase to

where the arc cannot be reestablished by the following

increase in voltage.

C 1

v a

v b G

L 1 L 2

C 2

@ @

magnier

iron core

air core

Figure 6: The Classical Tesla Coil With Extra Coil

The extra coil and the air core transformer are not

magnetically coupled. The output (top) of the classi-

cal coil is electrically connected to the input (bottom)

of the extra coil with a section of copper water pipe

of large enough diameter that corona is not a major

problem. A separation of 2 or 3 meters is typical.

Voltage increase on the extra coil is by transmission

line action (eld theory), or by RLC resonance (cir-

cuit theory), rather than the transformer action of the

iron core transformer. Voltage increase on the air core

transformer is partly by transformer action and partly

by transmission line action. When optimized for ex-

tra coil operation, the air core transformer looks more

like a transformer (greater coupling, shorter secondary)

The rotary spark gap still has limitations on the min-

imum arc time. Suppose we consider a disc with a ra-

dius of 0.2 m and a rotational speed of 400 rad/sec

(slightly above 3600 RPM). The edge of the disc is

moving at a linear velocity of r! = 80 m/s. Suppose

also that an arc cannot be sustained with arc lengths

than when optimized for classical Tesla coil operation.

Although not shown in Fig. 6 the extra coil depends

on ground for the return path of current ow. The

capacitance from each turn of the extra coil and from

the top terminal to ground is necessary for operation.

Impedance matching from the Tesla coil secondary to

the extra coil is necessary for proper operation. If the

extra coil were fabricated with the same size coil form

and wire size as the secondary, the secondary and extra

coil tend to operate as a long secondary, probably with

inferior performance to that of the secondary alone.

There are guidelines for making the coil diameters and

wire sizes dierent for the two coils, but optimization

seems to require a signicant amount of trial and error.

i -

Driver

v i

Figure 7: Drive for Tesla Coil

higher order resonances are not exact multiples of the

fundamental frequency. That is, I apply a square wave

of voltage to the feed point, and observe a current that

looks sinusoidal at the fundamental frequency. The

lumped RLC model automatically excludes higher or-

der resonances, so if they are of signicance, we must

use a distributed model to describe them. I believe that

higher order resonances are not a problem, at least not

enough of a problem to exclude the use of the lumped

model. The input impedance of the Tesla coil is the

(fundamental) input voltage divided by the current i .

In my quest for a better description of Tesla coil

operation, I decided that the extra coil was the appro-

priate place to start. It looks like a vertical antenna

above a ground plane, so there is some prior art to draw

from. While the classical Tesla coil makes an excellent

driver to produce long sparks, it is not very good for in-

strumentation and measurement purposes. There are

just too many variables. The spark gap may be the

best high voltage switch available today, but inability

to start and stop on command, plus heating eects,

make it dicult to use when collecting data.

The lumped model used here is the series resonant

RLC circuit shown in Fig. 8.

I therefore decided to build a solid state driver.

Vacuum tube drivers have been used for many years

and several researchers have developed drivers us-

ing power MOSFETs, so this was not entirely new

territory. I used this driver to measure the input

impedance of several coils under various operating con-

ditions, and compared these results with what the-

ory I could nd. This paper describes my results.

Some data on my driver can be found on my web site,

www.eece.ksu.edu/˜gjohnson.

3. The Lumped RLC Model

There are two ways of modeling the Extra Coil: dis-

tributed (elds) and lumped (circuits). I spent con-

siderable time with the distributed model, but was un-

able to predict all the interesting features. The lumped

model is not perfect either, but may be easier to visual-

ize. The system we are attempting to describe is shown

in the next gure.

V L

V base

^^^

___

L tc

R tc

v i

C tc

V C

Figure 8: Tesla Coil with Series Resonant LC Circuit

The resonant frequency is given by

! o =

p L tc C tc

(1)

At resonance, the inductive reactance ! o L w is can-

celed by the capacitive reactance − 1 /! o C tc so the cur-

rent is

The voltage input v i at the base of the Tesla coil may

be either a sine wave or a square wave. The square wave

is composed of an innite series of cosinusoids, the fun-

damental and all odd harmonics. The harmonics of the

exciting wave will drive higher order resonances of the

Tesla coil, if these resonances are harmonically related.

It appears, at least for the coils I have built, that any

i =

v i

R tc

(2)

The magnitude of the voltage across either the in-

ductance or the capacitance is

V C = iX C = v i

R tc

! o C tc =

v i

! o C tc R tc

(3)

Books on circuit theory dene the quality factor Q

ceiling) with the coil setting on a large copper sheet.

However, most coils are operated indoors without any

copper sheet. Ground then consists of some combi-

nation of a concrete oor, electrical wiring, grounded

light xtures, and soil moisture. That is, the geometry

necessary even for a numerical solution of capacitance

is dicult to describe precisely.

Q = ! o L tc

R tc

! o C tc R tc =

R tc

L tc

C tc

(4)

so we can write

V C = Qv i

(5)

To obtain the capacitance of a sphere to ground, we

start with the capacitance of a spherical capacitor, two

concentric spheres with radii a and b ( b>a ) as shown

in Fig. 9.

We see that to get a large voltage on the toroid of

a Tesla coil that it needs to be high Q . We need R tc

small, L tc large, and C tc small. Let us look at ways of

calculating or estimating these quantities.

4. Inductance

2 a

? 2 b

An empirical expression for the low-frequency induc-

tance of a single-layer solenoid is [14, p. 55].

Figure 9: Spherical Capacitor

It is not practical to actually build capacitors this

way, but the symmetry allows an exact formula for ca-

pacitance to be calculated easily. This is done in most

introductory courses of electromagnetic theory. The

capacitance is given by [12, Page 165]

L tc =

r 2 N 2

9 r +10 `

µ H

(6)

where r is the radius of the coil and ` is its length in

inches. This formula is accurate to within one percent

for `> 0 . 8 r , that is, if the coil is not too short. It is

known in the Tesla coil community as the Wheeler for-

mula. The structure of a single-layer solenoid is almost

universally used for the extra coil, so this formula is

very important. In normal conditions (no other coils

and no signicant amounts of ferromagnetic materials

nearby) it is quite adequate for calculating resonant

frequency.

C =

1 /a − 1 /b

(7)

If the outer sphere is made larger, the capacitance

decreases, but does not go to zero. In the limit as

b !1 , the isolated or isotropic capacitance of a sphere

of radius a becomes

The classical Tesla coil has a short primary that is

magnetically coupled into a taller secondary. The pa-

per by Fawzi [5] contains the analytic expressions for

the self and mutual inductances necessary for this case.

A relatively simple numerical integration is required to

get the nal values. We can get the same results as

the Wheeler formula by this numerical integration, but

with somewhat less insight as to how inductance varies

with the number of turns, and the length and radius of

the coil.

5. Capacitance

C tc is much more dicult to calculate or to estimate

with any accuracy. The capacitance value used to de-

termine the resonant frequency of the Tesla coil is a

combination of the capacitance of the coil and the ca-

pacitance of the top load, usually a sphere or a toroid,

with respect to ground. As a practical matter, ground

will be dierent in every Tesla coil installation. Easiest

to model would be an outdoor installation (no walls or

C 1 =4 a

(8)

Assuming a mean radius of 6371 km, the isotropic

capacitance of the planet earth is 709 µ F. A sphere of

radius 0.1 m (a nice size for a small Tesla coil) would

have an isotropic capacitance of 11.1 pF.

The other type of top load used for Tesla coils is

the toroid. The dimensions of a toroid are shown in

Fig. 10.

side

view

? d

Figure 10: Toroid Dimensions

The analytic expression for the isotropic capacitance

of a toroid involves Legendre functions of the rst and

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