TESLA COIL DESIGN

There are many programs to calculate the physical dimensions of a Tesla Coil System (transformer, primary coil & capacitor, secondary coil and its discharge terminal; but I believe there are a lot of people who will want to calculate themselves each parameter, thus having full control of the general design process.

Here you'll find a straightforward approach (don't fool yourself thinking a program gill give better results, because inside there's nothing more than formulas, usually the same you are looking here).

• Note:

I've introduced new ideas to estimate the physical dimensions of the secondary coil as a function of the input power. This was done analyzing dozens of actually working Tesla Coils . Reports given by friends who've used this approach show Tesla coils working without problems and giving a satisfactory performance.

One of the new concepts I've introduced was "Available Transformer Power per Cubic-Inch of 2ry Coil Volume" or AVP, thus:

AVP = XFMR PWR / 2ry coil volume = PWR [watt] /(0.25 p ar D3 [inches])

where:

PWR = neon transformer power in watt
ar= 2ry coil aspect ratio (Height/Diameter)
D = 2ry coil diameter in inches

This fancy coefficient remains almost constant (aprox. 2.5 Watt/in3) for a very wide range of power (100 to 10000 watt)! It seems to me a very practical approach, because aspect ratios of 2ry Tesla coils usually fall in the 3 to 6 range.

CALCULATION SHEET

The starting input will be the neon xfmr we have at hand, thus for the given neon transformer we know:

Vx [kv] = transformer output rms voltage in kilovolt.

Ix [mA] = transformer output rms current in milliamper.

1. AVAILABLE POWER : Pd

é Pd [watt] = Vx [kv] . Ix [mA]

2. XFMR IMPEDANCE : Zx

é Zx [MOhm] = Vx [kv] / Ix [mA]

3. PRIMARY CAPACITOR MATCHING : C1

To match transformer and capacitor impedances we need:

Xc = Zx

therefore the matching capacitor will have a capacity of:

=> C1 = 1 / (2 p fL Zx )

and if AC mains line frequency is fL = 60 Hz :

é C1 [nF] = 2.653 / Zx [MOhm] primary capacitance

4. DETERMNATION OF SECONDARY COIL DIMENSIONS : D2 and ar2

These formulas were obtained from regression of dozens of tesla coils ...

Recommended Secondary diameter

é D2 * [in] = 0.323 Pd 0.4
or D2 * [cm] = 0.82 Pd 0.4 secondary coil diameter

Recommended secondary coil aspect ratio

é ar2 *= H2 / D2 = 13.8 Pd - 0.197 secondary coil aspect ratio

From these approximate (*) values we adopt D2 = @ D2 * and ar2 = @ ar2* ;
where @ means adopted value from the suggested (*) value; therefore:

secondary coil height:é H2 = ar2 . D2

5. SECONDARY NUMBER OF TURNS : N2

é SELECT AWG WIRE SIZE : For the selected wire size found in tables, the close-wound turns per inch/centimeter will be (@=adopted values):

@TPI (Turns Per Inch) and @TPC (Turns Per Centimeter)

Now we may calculate de number of turns of secondary:

é N2 = @TPI . H2 [in]
or N2 = @TPC . H2 [cm]

6. LENGTH OF SECONDARY WIRE : Lw

é Lw [ft] = p . N2 . D2 [in] / 12
or Lw [m] = p . N2 . D2 [cm] / 100

7. SECONDARY COIL INDUCTANCE : L2

é L2 [microH] = N22 . D2 [in] / (18 + 40 ar2)
or L2 [microH] = N22 . D2 [cm] / (45.72 + 101.60 ar2)

8. SECONDARY SELF RESONANCE FREQUENCY : fsr

é fsr [kHz] = 1.1750 .106 . ar2 0.2 / ( N2 . D2 [in] )
or fsr [KHz] = 307614.28 ar20.2 / Lw [ft]

é fsr [kHz] = 2.9845 .106 . ar2 0.2 / (N2 . D2 [cm] )
or fsr [KHz] = 93760.83 ar2 0.2 / Lw [m]

9. SECONDARY DISTRIBUTED CAPACITY : Csd

é Csd [pF] = 25.33 109 / ( fsr2 [kHz] L2 [microH] ) {eq #1}

é Csd [pF] = ( 0.254 ar2 + 0.81) D2 [in]

or Csd [pF] = ( 0.1 ar2 + 0.32) D2 [cm] {eq #2}

Note: Equations #1 & #2 should give similar figures.

10. SECONDARY WORKING FREQUENCY : f2

Its value is derived considering that the actual wire length Lw should be ¼ of working wavelength l2 so:

é l2 [ft] = 4 Lw [ft]

or l2 [m] = 4 Lw [m]
and as f = c / l where c = speed of light = 300000 km/s or 9.84 108 ft/s

é f2 [kHz] = 246063 / Lw [ft]

or f2 [kHz] = 75000 / Lw [m]

11. TOTAL SECONDARY CAPACITY : C2T

é C2T [pF] = 25.33 10 9 / (f22 [kHz] . L2 [microH] )
= 0.4181 Lw2 [ft] / L2 [microH]
= 4.5 Lw2 [m] / L2 [microH]

12. REQUIRED ADDITIONAL TOP TERMINAL CAPACITY IN SECONDARY : CTOP

é CTOP [pF] = C2T - Csd [pF]

we should select a sphere or toroid that has this capacity in [pF].

13. REQUIRED PRIMARY INDUCTANCE : L1

é L1 [mH] = 25.33 106 / (f2 2 [kHz] . C1 [nF] )

USEFUL FORMULAS:

ISOTROPIC CAPACITY OF A SPHERE or A TOROID:

• FOR A SPHERE OF DIAMETER d

C [pF] = k . d

where k = 1.4142 for d [in] , and k = 0.556 for d [cm]

• FOR A TOROID OF OUTER DIAMETER D AND CROSS SECTION DIAMETER d

C [pF] = k. ( 1.2781 – [d / D] ) . SQR( pd .[D-d] )

where SQR= Square Root

In both cases, k = 1.4142 for d [in] , and k = 0.556 for d [cm]

APROXIMATE NUMBER OF PRIMARY TURNS : N1

Some assumptions must be made here, i.e.: for a normal helix we define:

If D1 = 2 D2 and H1 = 4H2 then: ar1 £ (1/10) ar2 [#1]

If D1 = 3 D2 and H1 = 4H2 then: ar1 £ (1/15) ar2 [#2]

From D2 ; ar2 and adopted (#1 or #2) we obtain D1, H 1 and ar1 so:

é N12 = (18 + 40 ar1 ) . L1 [microH] / D1 [in]
or N12 = (45.72 + 101.6 ar1 ) . L1 [microH] / D1 [cm]

NUMERIC EXAMPLE

Let's use a neon transformer of 12kv / 30 mA, therefore:

Vx= 12 kv
Ix = 30 mA

1. AVAILABLE POWER : Pd

é Pd = 360 watt

2. XFMR IMPEDANCE : Zx

é Zx = 0.4 MW

3. PRIMARY CAPACITOR MATCHING :C1

é C1 = 6.63 nF

4. DETERMNATION OF SECONDARY COIL DIMENSIONS : D2 and ar2 Recommended Secondary diameter

é D2 * = 3.4 in D2 * = 8.64 cm

Recommended secondary coil aspect ratio

é ar2 *= 4.328

From these approximate (*) values we adopt

é D2 = 3.5 in = 8.89 cm
é ar2 = 4.5

therefore the secondary coil height:

é H2 = 15.75 in = 40 cm

5. SECONDARY NUMBER OF TURNS : N2

é SELECTED AWG WIRE SIZE : AWG #21 with 12.8 TPC (Turns Per Centimeter)

Now we may calculate de number of turns of secondary:
é N2 = 512 turns

6. LENGTH OF SECONDARY WIRE : Lw

é Lw = 469.1445 ft = 143 m

7. SECONDARY COIL INDUCTANCE : L2

é L2 = 4633.86 microH

8. SECONDARY SELF RESONANCE FREQUENCY : fsr

é fsr = 885.81 kHz

9. SECONDARY DISTRIBUTED CAPACITY : Csd

é Csd = 6.96 pF {eq #1}
é Csd = 6.84 pF {eq #2}

Note: Equations #1 & #2 should give similar figures: we adopt 6.9 pF

10. SECONDARY WORKING FREQUENCY : f2

é l2 = 1876.57 ft = 572 m

é f2 = 524.5 kHz

11. TOTAL SECONDARY CAPACITY : C2T

é C2T = 19.87 pF

12. REQUIRED ADDITIONAL TOP TERMINAL CAPACITY IN SECONDARY : CTOP

é CTOP = 13 pF

13. REQUIRED PRIMARY INDUCTANCE : L1

é L1 = 13.88 microH

USEFUL FORMULAS:

ISOTROPIC CAPACITY OF A SPHERE:

• FOR A SPHERE OF DIAMETER d = 9.2 in = 23 cm

C TOP = 13.9 pF

APROXIMATE NUMBER OF PRIMARY TURNS : N1

Some assumptions must be made here, i.e.: for a normal helix we define:

D1 = 3 D2 and H1 = 4H2 then: ar1 £ (1/15) ar2

threfore:

D1 = 10.5 in = 26.67 cm
H 1 = 3.15 in = 8 cm
ar1 = 0.3

so: N12 = 39.68 thus:

é N1 = 6.3 turns

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