Calculation of a synchronous rotary spark gap for an AC powered SGTC

A Spark Gap Tesla Coil (SGTC) with either the following configuration 1 or configuration 2 is beeing considered.

The inductance L can be the leakage inductance of a neon sign transformer or/and an external inductance. The resistance R can be the copper resistance of the HV transformer or/and an external resistance. The mains frequency is denoted by f_mains. The synchronously rotating spark gap is assumed to fire exactly N times per mains period in constant time lags; thereby N>=2 is integer. The firing angle phi denotes the phase distance between a maximum of the AC HV supply voltage and a firing event. Thereby is phi>0 if the firing event is later in time than the maximum of the voltage, and phi<0 if the firing event is earlier. The relative spark duration (=closing duration) is denoted by α=alpha, thus the quotient between the duration of a spark and the time interval between the beginnings of two consecutive sparks.

If you're only interested in the RMS value of the mains current and in the real and apparent power, the following replacement model can be used approximately in both configurations (if the configuration 2 is used, L_prim of the replacement model must be set to zero and R_load must be so small that the capacitor C is discharged nearly immediately after the closing of the switch). In both configurations the power consumption of the resistor R_load corresponds to the sum of the powers beeing consumed by streamers and by losses of the primary and the secondary oscillating circuits.

If actually the configuration 1 is used, it is assumed that the capacitor C in the replacement circuit is nearly completely discharged before the switch opens, i.e. before the spark breaks off. Therefor the resistor R_load must be sufficiently small. Further L_prim is so small that after closing of the switch a HF oscillation occurs in the oscillating circuit consisting of C, R_load and L_prim. The time average of the capacitor voltage over one period of this HF oscillation is zero in a good approximation. If you're interested in the low frequency components of the current through the inductor L, you can assume in a good approximation that the capacitor C was discharged immediately after the closing of the switch and was short-circuited during the closing time interval of the switch. This simplification is the basis of the following calculations.

Input
RMS value of the AC supply voltage (U0_eff)The RMS value of the open-circuit secondary voltage of the HV transformer
Inductance LThe inductance L can be the leakage inductance of a neon sign transformer or/and an external inductance.

Capacitance C of the primary oscillating circuit
N = Ratio of firing frequency of the rotary / mains frequency (N>=2, integer)
Firing angle phi of switch resp. rotary (relative to maximum of supply voltage)
Relative closing duration (α=alpha) of the switch resp. the rotary
Mains frequency (f_mains)

In the following optimizations (buttons "Optimize C and L" and "Optimize C" and "Optimize L"; only if N=2):
If N=2, adjust phi so that the current at the opening time of the switch is zero.
Desired value of z=2 π f_mains sqrt(L C) (only if N=2) (if α=0, try 1.25 or 0.557 or others; if α>0, try 1.25 or 0.557*(1−α) or others)

Desired maximal apparent power of the voltage source

for maximal real power at R_load maintaining the given maximal apparent power (if N=2, adjust also phi)

so as to obtain exactly the given apparent power, and don't change L and the other parameters except for C

so as to obtain exactly the given apparent power, and don't change C and the other parameters except for L

Desired real power at the resistor R_load

for minimal apparent power of the source maintaining the given real power at R_load (if N=2, adjust also phi)

so as to obtain exactly the given real power at R_load, and don't change L and the other parameters except for C

so as to obtain exactly the given real power at R_load, and don't change C and the other parameters except for L

for maximal real power at R_load (only if N>=3, allowing arbitrary apparent power of the source), and don't change L and the other parameters except for C

for maximal real power at R_load (only if N>=3, allowing arbitrary apparent power of the source), and don't change C and the other parameters except for L

If N=2, the "Optimize C" resp. "Optimize L" button causes C resp. L to be calculated so that the value of z=2 π f_mains sqrt(L C) is maintained as specified above, and the firing angle phi is adjusted. If the above checkbox "If N=2, adjust phi so that the current at the opening time of the switch is zero" is checked, phi is calculated in this way. Otherwise, phi is calculated so that the real power at R_load gets maximal (irrespective of the apparent power of the source).

Results

Real power at the resistor R_load
Apparent power of the voltage source
Ratio real power at R_load / apparent power of the source
Power loss at the resistor R
Real power of the source = real power at R_load + power loss at R
RMS value of the current through the source
Maximal absolute value of an in instantaneous current through the source


Maximal absolute value of an instantaneous voltage of the capacitor C
Virtual optimal firing voltage of the capacitor CThat voltage beeing equal directly before ALL closing events of the switch, which would result in an unchanged real power at the resistor R_load. In reality, a capacitor voltage beeing equal directly before all switch closing/rotary firing events can only be achieved with N=2 or N=4 (and with N=4 only for special firing angles phi).
Ratio max. instantaneous voltage of C / virtual optimal firing voltageThis ratio is always greater or equal than 1; thereby 1 is the best value. A value >1 compels one to take a capacitor C with a higher maximal allowed voltage and therefor one has higher costs.

Maximal absolute value of an instantaneous voltage at the inductor L
Maximal absolute value of an instantaneous voltage at the series connection of L and the voltage sourceIf L is the internal leakage inductance of a neon sign transformer, this maximal absolute value of an instantaneous voltage at the series connection of L and the voltage source is the maximal absolute value of the voltage occuring between the HV output wires of the transformer.
Resonance frequency of the undamped oscillating circuit consisting of L and C
z = 2 π f_mains sqrt(L C)
Q factor = sqrt(L/C) / R of the R-L-C-series oscillating circuit
Resistance R
D = 1/(2 Q) = R/2 sqrt(C/L) : Damping ratio of Lehr of the R-L-C-series oscillating circuit

If the switch (spark gap) would not close or would not exist ...
... this would result in the following voltages and currents as in a simple series oscillating circuit consisting of R, L and C:

RMS value of the current through the voltage source
RMS value of the capacitor voltage
Maximal absolute value of an instantaneous voltage of C

Remarks

All calculated values refer to the stationary state which appears not before the transients occuring after switching on have been decayed.

In the following cases numerical problems might lead to inaccurate or useless results. In these cases, take the results with some mistrust.
- If L is small but not zero.
- If the damping ratio of Lehr D = R/2 sqrt(C/L) is close to 1.0 (e.g. 0.99 < D < 1.01)
- If z = 2 π f_mains sqrt(L C) is close to 1.0 (e.g. 0.99 < z < 1.01)
- If N = 2 and z = 2 π f_mains sqrt(L C) is close to 1.0−α (e.g. 0.99−α < z < 1.01−α). If N=2, z=1.0−α and R=0, there is resonance with infinite currents and voltages.

If you use a neon sign transformer or another leak transformer as AC HV source, L should be set to the leakage inductance (if no additional external inductance is used). If U_off is the secondary off-load-voltage of the transformer and I_short is the secondary current at secondary short-circuit, the leakage inductance can be calculated by L=U_off / (2 π f_mains I_short).
If the leakage inductance is fixed (not variable by adjusting the iron core), a maximal real power at R_load can be achieved in the following way. It is recommended that you use the following input elements in this order. Some of the elements have identical functions as the above input elements. For output take also a look at the above "Results" section.

RMS value of the secondary off-load-voltage of the HV transformer
RMS value of the secondary current of the HV transformer at secondary short-circuit

N = Ratio of firing frequency of the rotary / mains frequency (N>=2, integer)
Firing angle phi of switch resp. rotary (relative to maximum of supply voltage)
Relative closing duration (α=alpha) of the switch resp. the rotary
Mains frequency (f_mains)

of the HV leakage transformer and insert the result into the above input field for L and U0_short into the field for U0_eff

for maximal real power at R_load (only if N>=3, allowing arbitrary apparent power of the source), and don't change L and the other parameters except for C

If N=2, the "Optimize C" button causes C to be calculated so that the value of z=2 π f_mains sqrt(L C) is maintained as specified above, and the firing angle phi is adjusted. If the above checkbox "If N=2, adjust phi so that the current at the opening time of the switch is zero" is checked, phi is calculated in this way. Otherwise, phi is calculated so that the real power at R_load gets maximal (irrespective of the apparent power of the source).

	Results
(Leakage) inductance LThe inductance L can be the leakage inductance of a neon sign transformer or/and an external inductance.
Capacitance C of the primary oscillating circuit
Real power at the resistor R_load
Apparent power of the voltage source
Ratio real power at R_load / apparent power of the source
Power loss at the resistor R
Real power of the source = real power at R_load + power loss at R
RMS value of the current through the source

		for maximal real power at R_load maintaining the given maximal apparent power (if N=2, adjust also phi)
		so as to obtain exactly the given apparent power, and don't change L and the other parameters except for C
		so as to obtain exactly the given apparent power, and don't change C and the other parameters except for L

		for minimal apparent power of the source maintaining the given real power at R_load (if N=2, adjust also phi)
		so as to obtain exactly the given real power at R_load, and don't change L and the other parameters except for C
		so as to obtain exactly the given real power at R_load, and don't change C and the other parameters except for L

Calculation of a synchronous rotary spark gap for an AC powered SGTC

Input

Results

If the switch (spark gap) would not close or would not exist ...

Remarks