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).
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
è Pd [watt] = Vx [kv] . Ix [mA]
2. XFMR IMPEDANCE : Zx
è Zx [MW] = 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 :
èC1 [nF] = 2.653 / Zx [MW] 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
è D2 * [in] = 0.323 Pd 0.4
or D2 * [cm] = 0.82 Pd 0.4 secondary coil diameter
Recommended secondary coil aspect ratio
è
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:è H2 = ar2 . D2
5. SECONDARY NUMBER OF TURNS : N2
è 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:è N2 = @TPI . H2 [in]
or N2 = @TPC . H2 [cm]
6. LENGTH OF SECONDARY WIRE : Lw
è Lw [ft] = p . N2 . D2 [in] / 12
or Lw [m] = p . N2 . D2 [cm] / 100
7. SECONDARY COIL INDUCTANCE : L2
è L2 [mH] = N22 . D2 [in] / (18 + 40 ar2)
or L2 [mH] = N22 . D2 [cm] / (45.72 + 101.60 ar2)
8. SECONDARY SELF RESONANCE FREQUENCY : fsr
è fsr [kHz] = 1.1750 .106 . ar2 0.2 / ( N2 . D2 [in] )
or fsr [KHz] = 307614.28 ar20.2 / Lw [ft]
è 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
è Csd [pF] = 25.33 109 / ( fsr2 [kHz] L2 [mH] ) {eq #1}
è 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:
è 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
è f2 [kHz] = 246063 / Lw [ft]
or f2 [kHz] = 75000 / Lw [m]
11. TOTAL SECONDARY CAPACITY : C2T
èC2T [pF] = 25.33 10 9 / (f22 [kHz] . L2 [mH] )
= 0.4181 Lw2 [ft] / L2 [mH]
= 4.5 Lw2 [m] / L2 [mH]
12. REQUIRED ADDITIONAL TOP TERMINAL CAPACITY IN SECONDARY : CTOP
èCTOP [pF] = C2T - Csd [pF]
we should select a sphere or toroid that has this capacity in [pF].
13. REQUIRED PRIMARY INDUCTANCE : L1
èL1 [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:
è N12 = (18 + 40 ar1 ) . L1 [mH] / D1 [in]
or N12 = (45.72 + 101.6 ar1 ) . L1 [mH] / 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
è Pd = 360 watt
2. XFMR IMPEDANCE : Zx
è Zx = 0.4 MW
3. PRIMARY CAPACITOR MATCHING :C1
è C1 = 6.63 nF
4. DETERMNATION OF SECONDARY COIL DIMENSIONS : D2 and ar2
Recommended Secondary diameter
è D2 * = 3.4 in D2 * = 8.64 cm
Recommended secondary coil aspect ratio
è
ar2 *= 4.328
From these approximate (*) values we adopt
è D2 = 3.5 in = 8.89 cm
è ar2 = 4.5
therefore the secondary coil height:
è H2 = 15.75 in = 40 cm
5. SECONDARY NUMBER OF TURNS : N2
è SELECTED AWG WIRE SIZE : AWG #21 with 12.8 TPC (Turns Per Centimeter)
Now we may calculate de number of turns of secondary:
è N2 = 512 turns
6. LENGTH OF SECONDARY WIRE : Lw
è Lw = 469.1445 ft = 143 m
7. SECONDARY COIL INDUCTANCE : L2
è L2 = 4633.86 mH
8. SECONDARY SELF RESONANCE FREQUENCY : fsr
è fsr = 885.81 kHz
9. SECONDARY DISTRIBUTED CAPACITY : Csd
è Csd = 6.96 pF {eq #1}
è Csd = 6.84 pF {eq #2}
Note: Equations #1 & #2 should give similar figures: we adopt 6.9 pF
10. SECONDARY WORKING FREQUENCY : f2
è l2 = 1876.57 ft = 572 m
è f2 = 524.5 kHz
11. TOTAL SECONDARY CAPACITY : C2T
è C2T = 19.87 pF
12. REQUIRED ADDITIONAL TOP TERMINAL CAPACITY IN SECONDARY : CTOP
èCTOP = 13 pF
13. REQUIRED PRIMARY INDUCTANCE : L1
è L1 = 13.88 mH
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:
è N1 = 6.3 turns
