## Tesla Coil Calculations

On this page of my site, I will explain how to design a Tesla coil to achieve the highest voltage gain possible with maximum efficiency. Computing all the component values for optimum performance is the most critical part of building a Tesla coil. A Tesla coil is a finely tuned resonant circuit. Simply slapping all the components together with random values will not get you anywhere. Get out your calculator, because there is a lot of math involved in this.

There are two main ways to design a Tesla coil. The first is to start with the transformer, and the second is to start with the secondary coil. Both methods involve some guess and check work if you are designing the Tesla coil around a specific transformer to power it. If you have already bought a transformer then you should use the first method. Find the maximum capacitance value for the tank capacitor and then design the primary and secondary coils. You will have to design a primary coil and calculate the resonant frequency of the tank circuit. Then design a properly coupled secondary coil to match the primary coil.
The second method is to start with the secondary coil and work down. If for some reason you want your Tesla coil to operate at a specific frequency, then you can use this method. First design the secondary coil to the desired frequency and design a primary coil to match it. Next, based on the inductance of the primary you designed, calculate the necessary capacitance to make the tank circuit resonate at the same frequency you have chosen for the secondary coil. Then, find the specifications for a primary transformer that is capable of charging the tank capacitor. If you want a certain power level for the primary transformer, then adjust your secondary coil design until the tank capacitor matches the transformer you want.
Since these calculations can be rather time-consuming, several computer programs have been developed by other coilers to help you in this process. They are available on my download page. I have also written Java applets for the necessary electrical formulas and they are on my Java Calculator page.

### Primary Transformer Calculations

I suggest that you actually obtain your transformer before you start building the other components, unless you are absolutely sure that you can get a specific type of transformer. If you build your Tesla coil to be powered by a certain transformer, and then when you try to buy it you can't find the right one, you could have quite a problem on your hands.
There are two important things you need to know about your transformer. The first is the impedance of the secondary coil in the transformer. The second is the amount of current your transformer will draw from the 120/240 volt line source. This is only an important factor if you are using a high-power transformer such as a pole pig.
Different types of transformers will tell you different electrical characteristics. Most transformers, such as pole pigs will have the voltage, and VA (volts multiplied by amps) ratings written on them. Neon sign transformers will often give the voltage and current values of the secondary winding (the high-voltage coil of the transformer).

If you have a power transformer with a VA and voltage rating, then use the formula below. First, divide the transformer's VA rating by V (voltage rating) to find the current rating I.

Next, use Ohm's law to find the impedance of the transformer's secondary coil (the high-voltage windings). Ohm's law states that E = IZ, where E = volts, I = amps, and Z = impedance in ohms. Divide the transformer's output voltage by its current rating to find Z.

You will use this impedance value for the capacitor calculations in the next section of this page. The impedance of a transformer's secondary coil limits the amount of current it is capable of supplying.
If your transformer gives you its voltage and current ratings, then use the same Ohm's law formula above to solve for impedance.

If you are using a pole pig, calculate the current draw of its primary coil by dividing the VA rating by the input voltage, 120 or 240. It is usually better to power a pole pig with 240 volts instead of 120. Most houses have a pair of 240 volt wires at 100 amps. The two legs of the 240, plus the neutral line make two sets of 120 at 100 amps each. For example, a 10,000VA pole pig would draw 41.7 amps from 240V, but would draw 83.3 amps at l20V. At 240 volts, the transformer draws half the current. Having less current decreases the thickness of wire you need to power it, and you won't have to use a huge variac. Note that these current calculations assume a maximum load across the output of the transformer.
After you know how much current your transformer will draw, you need to use AC cable that is rated for that much current. If you want to be able to plug your transformer into the wall, you may have to use a heavy-duty appliance outlet with a high enough current rating. Your local hardware store will carry them. If you have a load that exceeds the wire rating, you will start a fire.

### Tank Capacitor Calculations

Now that you know the impedance of your transformer, you can calculate the amount of capacitance you need for the tank capacitor. Below is the capacitive reactance formula.

 XC = capacitive reactance in ohms Z = impedance of the transformer in ohms F = frequency in hertz C = capacitance in farads

Capacitive reactance is the capacitor's resistance to the AC waveform. The dielectric in a capacitor isn't conductive, but a capacitor does resist the changes in the AC waveform, and thus it is a load. When the capacitor is being charged, and after it is fully charged and the AC voltage begins to drop below the capacitor's voltage, current flows. Whenever the input voltage across the capacitor changes from its last instantaneous value, current flows through it.
This formula converts the load that a capacitor imposes on an AC circuit into standard ohms. Capacitive reactance is based on the capacitance value, and the frequency of the waveform it is being charged with. The higher capacitance value a capacitor has, the less impedance it has, the more current it requires to charge it, and the more energy it can store. The constant 2pi in the formula is the circumference of the unit circle and is multiplied by the frequency F to find the angular velocity of the waveform. The angular velocity of a waveform is the ratio of the rate of change in angle (radians/sec), and is related to the rate of change in the magnitude of the waveform.
The idea here is to find the highest capacitance value possible that will not overload the transformer. By substituting the impedance of the transformer (Z) for the capacitive reactance (XC) in the formula, the capacitor will impose a load of Z ohms, which is equal to the maximum current capabilities of the transformer. If the capacitance is too high, the load on the transformer will be too great which causes excess stress on the transformer, and the amount of power transferred to the capacitor will be decreased. However, the capacitance may be less than the calculated maximum value. If the capacitance is less, you won't get as much of a bang and you won't be utilizing the full capabilities of your transformer (the capacitor will not draw as much current as the transformer is capable of delivering).
For the variable F, plug in 60, because the AC line source operates at 60 hertz. Plug in your transformer's impedance for variable (XC), and solve for C (this is shown in the formula on the right).

If you have decided to use a lower capacitance value, then you can plug in your capacitance value, and solve for (XC). You can use this impedance value to calculate how much current your capacitor will draw from the secondary coil of the transformer by using Ohm's law. Divide the transformer's voltage by the capacitive reactance to find the current flow. From here, you can also calculate approximately how much current the primary coil of the transformer will draw based on the load imposed on its secondary. This is not really necessary, unless you are just curious or need to know the current draw of the transformer. If a load across the output of a transformer consumes W watts of power, then the primary of the transformer consumes W watts of power. In other words, power in equals power out. Ohm's law also states that P = IE where P = power in watts, I = amps, and E = volts. Thus, the following formula is derived:

EPIP = ESIS

EP = primary voltage
IP = primary current in amps
ES = secondary voltage
IS = secondary current in amps

### Primary Coil Calculations

The next calculations will be for designing the primary coil. The size of the primary is an important factor. You should generally have about 10 to 15 turns in your primary coil, and tapped at a slightly lower number. This usually gives the best output for most coils, but needs to be varied by experiment to suit your design. You should use an adjustable tap for your primary coil to fine tune it. For example, when calculating the inductance of your primary, plug in a number of 12 turns, but actually build your primary with 14 or 15 turns to allow for tuning. The primary coil should be wound from thick conductor with a large surface area, such as copper tubing. The spacing between turns in the primary coil must be large enough to prevent arcing between turns. Spacing should usually not exceed 1/2 inch between turns.

There are 4 basic types of primary coil designs.

1. The single turn primary. It is a single turn at the base of the secondary coil, and usually has a large diameter
2. The helix primary. It is a single-layer, cylinder-shaped coil, just like the secondary coil, but much shorter.
3. The inverse conical section, or saucer coil. It is shaped like a V, and usually rises at a 30° angle.
4. The Archimedes spiral, or flat pancake coil. It is a flat coil shaped in a spiral.

I do not recommend the first two coil designs, based on what I have read. They are not as efficient as the second two designs. The saucer coil is good for small to medium power Tesla coils. I would not use it on a high-power coil, because the sparks from the toroid are more prone to strike it since it is upraised. For high-power coils, the flat spiral is the best type of primary. The diameter of these two types of coils is usually about equal to the length of the secondary winding.
Here are the primary coil inductance formulas. You can use them to find the inductance of the primary you are designing, or to design the dimensions of a primary to have a specific inductance. You will need to know the inductance of the primary coil to calculate the resonant frequency of the tank circuit.

Archimedes Spiral

 L = inductance of coil in microhenrys (µH) R = average radius of the coil in inches N = number of turns W = width of the coil in inches

Helical Coil

 L = inductance of coil in microhenrys (µH) N = number of turns R = radius of coil in inches (Measure from the center of the coil to the middle of the wire.) H = height of coil in inches

Inverse Conical Coil

 L = inductance of coil in microhenrys (µH) L1 = helix factor L2 = spiral factor N = number of turns R = average radius of coil in inches H = effective height of the coil in inches W = effective width of the coil in inches X = rise angle of the coil in degrees

In some formulas, you will have to input the inductance value in henrys, not microhenrys. To convert microhenrys to henrys, divide by 1,000,000 or 106, and multiply henrys by 1,000,000 to find microhenrys.

As discussed on the How a Tesla Coil Works page, a resonant circuit is in tune when the inductive reactance (XL) equals the capacitive reactance (XC) at the same frequency. Here are the inductive and capacitive reactance formulas:

Inductive Reactance

XL = inductive reactance in ohms
F = frequency in hertz
L = inductance in henrys

Capacitive Reactance

XC = Capacitive reactance in ohms
F = frequency in hertz

The circuit will resonate at the frequency F for which XL = XC. So, by the combination of these two formulas the resonant circuit formula is derived.

XL = XC

F = frequency in hertz
L = inductance in henrys

If you want to solve for F, use the simplified formula below.

The resonant circuit formula can be used to find the frequency of a given LC circuit, or to find the necessary inductance and capacitance to make the circuit resonate at a frequency F. Depending on how you are designing your Tesla coil, you will use this formula to calculate the missing variable in the tank circuit.

Next, I will explain secondary coil design. As mentioned before, you may have to make these calculations several times to make everything fit.

### Secondary Coil Design

The next step is to design the secondary coil. The secondary coil must resonate at the same frequency as the primary coil in the tank circuit. One of the factors to consider in designing the secondary coil is the aspect ratio, which is the ratio of the length of the secondary coil to the diameter of the secondary coil. The chart below shows the basic guidelines that typically produce the best output. You do not have to follow this exactly.

#### Secondary coil aspect ratios:

All dimensions are in inches.

 Coil Diameter Aspect Ratio Length of Coil 3 6:1 18 4 5:1 20 5 4.5:1 22.5 6 4:1 24 7 3.5:1 24.5 8 3.1 24 8+ 3:1 multiply by 3 to find length

The general guideline for the type of wire to use for the secondary coil, is 22 gauge or thicker. Really thin wire is a bad idea. It causes too much resistance and too much self capacitance. Use a high-quality enameled magnet wire. Don't use regular insulated hookup wire, unless you want to effectively space out the turns. Below is a list of wire types compiled by Richard Quick.

 ```AWG D.C. OHMS WIRE DIAM APPROX. TURNS PER FEET PER SIZE PER 1000 FT INCHES INCH, SOLID ENAMEL POUND COVERED 1 .1264 .2893 X 3.947 2 .1593 .2576 X 4.977 3 .2009 .2294 X 6.276 4 .2533 .2043 X 7.914 5 .3195 .1819 X 9.980 6 .4028 .1620 X 12.58 7 .5080 .1443 X 15.87 8 .6405 .1286 7.6 20.01 9 .8077 .1144 8.6 25.23 10 1.018 .1019 9.6 31.82 11 1.284 .0907 10.7 40.12 12 1.619 .0808 12.0 50.59 13 2.042 .0720 13.5 63.80 14 2.524 .0641 15 80.44 15 3.181 .0571 16.8 101.40 16 4.018 .0508 18.9 127.90 17 5.054 .0453 21.2 161.3 18 6.386 .0403 23.6 203.4 19 8.046 .0359 26.4 256.5 20 10.13 .0320 29.4 323.4 21 12.77 .0285 33.1 407.8 22 16.20 .0253 37.0 514.2 23 20.30 .0226 41.3 648.4 24 25.67 .0201 46.3 817.7 25 32.37 .0179 51.7 1031 26 41.02 .0159 58.0 1300 27 51.44 .0142 64.9 1639 28 65.31 .0126 72.7 2067 29 81.21 .0113 81.6 2607 30 103.7 .0100 90.5 3287 31 130.9 .0089 101 4145 32 162.0 .0080 113 5227 33 205.7 .0071 127 6591 34 261.3 .0063 143 8310 35 330.7 .0056 158 10480 36 414.8 .0050 175 13210 37 512.1 .0045 198 16660 38 648.2 .0040 224 21010 39 846.6 .0036 248 26500 40 1079 .0031 282 33410 41 1323 .0028 42 1659 .0025 43 2143 .0022 44 2593 .0020 45 3348 .00176 46 4207 .00157 47 5291 .00140 ```
"For winding Tesla secondary coils the general consensus is to use number 22 AWG, or larger diameter, double Formvar Magnet wire. Often surplus partial spools of odd wire sizes are found, so I took time to post a more complete chart of AWG numbers and fractional diameters than is usually available. All information above is approximate despite the decimal places. Wire diameter, turns per inch, resistance, feet per pound, etc. all vary slightly from one manufacturer to another. Insulation thickness will vary depending upon type and the supplier. Turns per inch will also vary with the quality of the winding. In practice, Tesla coil windings are not perfect (nor do they need to be) so expect some slight variations from the specification table above."
Richard Quick

Choosing the coil form is the next step in designing the secondary coil. Your coil should usually not be much less than 4 inches in diameter. Large Tesla coils can have coil forms over 12 inches in diameter. The type of coil form used can be important, so try to use a low loss material. PVC is commonly used for Tesla coils because it is cheap, but it is a very high loss material. Materials such as polyethylene, polystyrene, polypropylene, lexan, and Plexiglas are more suitable. The coil form should be as thin as possible, such as 1/8 inch. Unless you really care about that extra inch or two of spark length, use whatever is cheapest.

Once you have chosen a wire gauge and coil diameter, you need to calculate the number of turns and length of the coil. Use the formula below to make these calculations. Secondary coils should generally have 800 to 1000 total turns.

 T = AH

L = length of wire in feet
D = outer diameter of coil form
H = height of coil in inches
A = number of turns per inch
T = total number of turns

To find the approximate number of turns per inch, use the wire gauge chart to find the average thickness of the wire you chose.

A = turns per inch
B = thickness of wire in inches

These coil calculations are approximate and will have to be made again when the secondary coil construction is complete. Wire thickness for the same gauge wire will vary for each manufacturer. The same is also true for coil forms. Even though a piece of pipe may be labeled as being 5 inches, it may actually have an outer diameter of 5.25 inches.

After the dimensions of the secondary coil are designed, you need to find the inductance of the coil. Use the helical coil inductance formula below.

 L = inductance of coil in microhenrys (µH) N = number of turns R = radius of coil in inches H = height of coil in inches

This inductance value will be used to find the necessary toroid capacitance.

### Toroid Calculations

The toroid (discharge terminal) is just as critical as every other component in a Tesla coil. The toroid is what tunes the resonant frequency of the secondary circuit. The toroid is one plate of a virtual capacitor. The dielectric is air, and the second plate is the earth itself.
The inductive reactance of the secondary and the capacitive reactance of the toroid need to be near the designed operating frequency. The primary coil is then tapped at the appropriate number of turns to exactly match the secondary frequency. The larger the surface area of the toroid, the more capacitance it has. There is also another factor in determining how much toroid capacitance you need, and that is the virtual capacitance of the secondary coil itself. At these frequency levels just about everything becomes a significant capacitor, and the slightest amount of capacitance can affect the frequency of the secondary resonant circuit. The capacitance of the secondary coil needs to be considered in finding the required toroid capacitance value.
The capacitance of the secondary coil is found by the Medhurst formula. This formula is only a good approximation, and is fairly accurate for most coils. There may be a difference of 1 or 2 picofarads from the real capacitance, but this is easily overcome by adjusting the primary coil tap. This formula is inaccurate for very small coils with a large number of turns. The Medhurst formula is shown below.

C = self capacitance in picofarads
R = radius of coil in inches
L = length of coil in inches

The capacitance of the secondary coil depends on many variables such as the dielectric constant of the wire enamel, wire thickness and number of turns, proximity to ground, and is difficult to accurately calculate. Terry Fritz has written an excellent program for accurately measuring the capacitance of a coil with or without a toroid. His ETesla5 is available on his site. You can also measure it. My oscilloscope has been a very helpful tool in designing and tuning my Tesla coil. If you do not have an oscilloscope, then I suggest that you get one. If you can't afford one, then borrow it. It is an extremely useful tool, and will save you a lot of time.
To measure the capacitance of the secondary coil, you will first have to measure its resonant frequency when it is operating without any toroid or extra capacitance. Connect a small wire to the probe of your oscilloscope as an antenna. Run your Tesla coil without any toroid or discharge terminal, and look at the graph on your scope to find the frequency it is resonating at. Simply putting the scope in the same room with a small Tesla coil will give you a good reading, but keep the scope's wire well out of spark range of the coil, or you'll fry it. Then, use your frequency measurement, and the inductance of the secondary coil to calculate its self-capacitance. Use the resonant circuit formula.

The total capacitance of the secondary circuit will be a little less than the sum of coil capacitance and toroid capacitance. Their close proximity causes the toroid to shield some of the secondary coil surface area from ground. Below is the capacitance formula for a toroid.

D1 = outside diameter of toroid in inches
D2 = diameter of cross section of toroid in inches

Capacitance of a sphere:

I do not recommend using a sphere as the secondary capacitor, because it is not as space efficient as a toroid, and a toroid is also much more impressive-looking. A toroid works better than a sphere in reducing the intensity of the electric field near the top of the secondary, thus preventing sparks from breaking out of the top secondary coil turns. Below is the formula for a sphere anyway.