1. Tesla Coil
High Frequency Resonant Transformers and Wireless Transmission of Power

2. Tesla coils are named after their creator Nikola Tesla.
Born in 1856, he developed numerous concepts including the fundamentals of wireless technology, and much of the alternating current power systems used today. His most important invention though, is of course the Tesla coil.
He built numerous Tesla Coils including one so large he thought he was going to send free power to everyone around the world with it.

3. Primary Circuit The primary circuit consists of:
The high voltage transformer
The spark gap
The tank capacitors
The primary inductor
The capacitors, primary inductor, and spark gap will form an LC circuit

4. How the Primary Circuit Works
First, current enters the transformer from the mains (wall outlet)
The current outputted by the transformer charges the capacitor bank
When the capacitors are charged, the voltage across them is so low that almost all of the voltage from the transformer ends up on the electrodes of the spark gap
At this point spark gap becomes conductive, forming a short circuit
This creates the LC circuit. Current sloshes back and forth from one side of the capacitors to the other in an attempt to neutralize, at radio frequency. To do this it must travel across the spark gap and the primary inductor
Energy is eventually lost to the secondary coil and heat and the spark gap ceases to be conductive
At this point, the cycle starts over

5. Secondary Circuit
The secondary circuit consists of just the secondary coil, the toroid topload, and the ground.
The secondary coil is an inductor which forms a transformer together with the primary inductor
The current oscillating at radio frequency through the primary creates a rapidly changing magnetic flux in the secondary coil
This induces an EMF and a current in the secondary coil
The charge accumulates on the toroid, which is directly connected to the secondary coil
The bottom of the secondary coil is connected to ground in order for the secondary circuit to have capacitance relative to the ground

6. Power Source
For the high voltage power source needed by the primary circuit we used a neon sign transformer
The neon sign transformer steps up the voltage of the mains AC to 12kV and outputs 60 milliamps of alternating current

7. Peak Voltage vs. RMS Voltage
It is important to note that the 12kV transformer actually has a much higher peak voltage
Since the voltage it puts out is AC, it is actually a sine function that alternates between and –6000 volts
However, these values are RMS, or Root-Mean-Square
This means they are essentially an average; the 12kV is the area under the curve
At the peak of the wave, the voltage is actually √2 ∙ RMS
Therefore, our transformer has a peak voltage output of √2 ∙ or nearly 17kV

8. Capacitors
The first thing we had to do was find our target capacitance.
Ideally, the capacitors should charge in the time the transformer takes to put out its peak voltage, which is ¼ its period
We therefore solve the system of equations Q=CV=tI
Here, t is the amount of time the current flows, which we want to be ¼ the period T of the voltage function
Therefore, Q=CV=(T/4)I
Solving for the capacitance C, we get C=TI/(4V)
T is the period of the AC which is 1/60 since it operates at 60 Hertz, and we shall use the RMS voltage of 12kV

9. More on Capacitors
We use the RMS value because it will yield a slightly larger capacitance, which is generally preferable
Substituting into the equation, C=(1/60 s)(.06A)/(4∙12000V)=20.8 nF
In addition to achieving this capacitance, our capacitors had to withstand 17kV with the current oscillating at radio frequency. Since capacitors are not rated for that kind of use, we had to get capacitors rated for about twice that much voltage
We ended up buying 30 2kV 150 nF capacitors. We had two strings of 15 in series, in which the voltages add to 30kV and the capacitance drops to 10 nF according to the formula Ceq=1/(1/C1+1/C2+…+1/Cn)
The two strings of series capacitors were then hooked up in parallel, in which the capacitances add to 20 nF and the voltage capability remains at 30 kV
There are also 1.5MΩ resistors across the capacitors. This is because each capacitor has a little internal resistance, and if it is not the same for all capacitors, they could get an unequal voltage distribution. The huge resistors prevent this from happening by making the internal resistances insignificant

10. Spark Gap
The spark gap serves the purpose of creating a short circuit when the capacitors charge, forming an LC circuit
It can be thought of as an open switch which closes when the circuit reaches a certain voltage
The voltage difference from the transformer accumulates on the diodes and makes the air between them conductive
The breakdown of air for this design is roughly 10kV per cm
Our spark gap consists of two steel bolts facing each other in an ABS T-fitting. There are two fans blowing air through the tube to quench and cool the gap

11. Primary Inductor
The primary inductor is commonly called simply the primary
Its purpose is to create a rapidly changing magnetic field in the center in order to induce a huge EMF and a small current in the secondary coil
Since the primary is a part of the LC circuit, the current is constantly oscillating through it, thus creating the changing magnetic field
The frequency of oscillation in the LC circuit is very important as we shall see, and is given by the equation ƒ=1/[2π√(LC)] where L is the inductance of the primary and C is the capacitance of the capacitors
In our case, ƒ=290kHz, which is radio frequency but well below the AM band
Our Primary consists of about seven turns of 1/8 inch copper tubing.

12. Self Inductance of the Primary
The formula for the inductance of a spiral coil is difficult to derive, therefore we shall not do it here
However, the formula is L=(NR)²/(8R+11W) where N is the number of turns, R is the average radius of the coil in inches, W is the width of the coil in inches, and L is the inductance in microhenrys

13. Resonance, Impedance and Reactance
Before proceeding to the secondary coil, it is important to have a basic understanding of resonance, impedance, and reactance
Resonance is a form of constructive wave interference. That is, waves collide at exactly the right time as to reinforce each other
A good analogy is that of holding a slinky by the ends and pushing down on it in the middle every time it’s on the way down. The slinky will begin to stretch more and more because you are inputting energy at the exact right moment each time
Impedance is similar to resistance, except it accounts for possible phase offsets in the current
It has an imaginary component for this phase offset. Impedance Z is given by Z=R+jX, where R is the resistance
X is the imaginary component of impedance, which is called reactance. There are two kinds of reactance: inductive and capacitive. Inductive reactance has to do with the fact that inductors resist changes in current. Therefore the inductive reactance XL is proportional to the frequency and is given by XL=2πƒL. Capacitive reactance has to do with the fact that while electrons cannot pass through a capacitor, AC current effectively does, so the capacitive reactance XC is proportional to 1/ƒ and is given by XC=1/(2πƒC)

14. Secondary Coil
The secondary coil/inductor, generally called simply the secondary, gets an induced EMF and current due to the changing magnetic flux at its location
Unlike most transformers, this secondary has air inside of it instead of iron. This is because instead of stepping up the voltage with the iron core, it uses the high frequency of oscillation and resonance.
For this reason the Tesla coil’s proper name is a high frequency resonant transformer.
Our secondary consists of almost exactly 1000 turns of 24 gauge magnet wire, hand-wound around 3.5” diameter ABS pipe

15. Self Inductance of the Secondary
The equation for the inductance of a helical coil is much like that of the spiral coil
It is not the formula for an ideal solenoid because it is not quite ideal. It isn’t infinitely long and does not have an infinite turn density
It is given by L=(NR)²/(9R+10H) where L is the inductance in microhenrys, N is the number of turns, R is the radius of the coil in inches, and H is the height of the coil in inches

16. Capacitance in the Secondary
The secondary coil and toroid form capacitance relative to the ground, since the bottom of the secondary is grounded
The secondary has self-capacitance according to the equation C=.29L+.41R+1.94√(R³/L) where C is the self-capacitance in picofarads, L is the length of the secondary coil in inches, and R is the radius of the coil in inches
The toroid has capacitance equal to C=1.4( D2/D1)√[πD2(D1-D2)], where C is in picofarads, D1 is the outer diameter of the toroid in inches and D2 is the diameter of a cross-section of the toroid in inches
This capacitance is necessary for the charge to arc towards grounded objects and to create streamers

17. Resonance in the Secondary
In order to function properly, the secondary has to have the same resonant frequency as the primary circuit.
This is so that the electromagnetic waves emitted creating the EMF will reinforce each other instead of canceling out.
Everything has a natural resonant frequency. For example when you hit a pitch that coincides with the natural resonant frequency of a wine glass, it shatters.
The natural resonant frequency of a coil for electromagnetic purposes occurs when its inductive reactance has equal magnitude to its capacitive reactance. Setting the equations equal, we get XC=1/(2πƒC)= XL=2πƒL.
Solving this for ƒ, we get ƒ=1/[2π√(LC)].
Note that the formula itself is the same as the formula for the frequency of oscillation in the primary circuit. This was not derived for us when we learned about LC circuits, but it is for the same reasons.

18. Streamers
The Tesla coil puts beautiful branching arcs of electricity just out into the air. It is able to do this because of the rapidly shifting charges inside the secondary, which are capable of actually pushing charge out into the air, and pulling it back in again at radio frequency.