How to Limit Fault Current When Sparks Are Unwanted,

and How to Generate Energetic Sparks Safely When They Are Wanted

Sizing the Series Current-Limiting Resistor and the Parallel Storage Capacitor

Analog Technologies, Inc. | Technical Note ATNHVPS-08 | May 9, 2022

1. Introduction

High-voltage power supplies - those operating at hundreds or thousands of volts - face a problem that low-voltage designers rarely think about: sparking at the output. When the output voltage exceeds the breakdown strength of the air gap or dielectric between the conductors at the load, a spark forms. The output suddenly looks like a near-zero-resistance path, and the supply tries to deliver whatever current the rest of the circuit will allow. Without protection, that current is limited only by the supply's output impedance and any parasitic inductance - often well above the safe operating current of the supply or the load.

Designers actually face this scenario in two opposite ways:

• Sparking is undesirable - for example, in photomultiplier tube bias supplies, electrostatic precipitators, X-ray tube anodes, capacitor-charging circuits, or any HV bus where an arc would damage the load or destroy the supply. Here we want to limit the fault current to a safe value if a spark ever occurs.
• Sparking is the intended function - ignition systems, spark gaps, plasma generators, electrical-discharge machining (EDM), photoflash circuits, gas-laser triggers, and many scientific instruments need a high-current arc on purpose. Here we want to deliver a huge peak current into the load for a brief moment, then recharge for the next pulse.

Both cases are solved with simple passive networks. This note derives the formulas and sizes the components for each.

2. The Two Protection Topologies

Figure 1 shows the two basic protection topologies that handle the two scenarios above.

Panel A - protective mode. A series resistor R_s between the supply output and the load. If a spark forms across the load, the resistor caps the fault current at V_PSU / R_s, protecting both the load and the supply. In normal operation it drops a small voltage proportional to I_load.

Panel B - energetic-spark mode. The same series resistor, plus a storage capacitor C in parallel with the load. The capacitor charges through R_s between sparks. When the load fires, the capacitor dumps its full stored charge into the spark in microseconds - providing a peak current far larger than the supply alone could deliver - and then R_s controls the recharge rate for the next pulse.

Figure 1. Two protection topologies for a high-voltage power-supply output. Panel A: series Rs alone caps fault current to VPSU / Rs (use whenever sparking is unwanted). Panel B: same Rs plus parallel storage capacitor C dumps a huge peak current into the load when it fires, then recharges through Rs (use when energetic sparks are wanted).

3. An Intuitive Analogy - A Boxing Match, a Slingshot, and Some Boxing Gloves

Before deriving the design formulas, here is a picture that captures the entire story in one figure. Imagine the high-voltage supply as a strong fighter throwing a punch into the spark load. Three things can happen, depending on what is between the fighter's fist and the wall:

Figure 2. The three cases - same HVPS, same spark load, three very different outcomes depending on what you put in between.

Panel A - bare fist. Direct wires from the HVPS to the spark load. When the load fires, the load becomes a near-short circuit and the full V_PSU appears across whatever tiny internal impedance the supply has - drawing a fault current limited only by ohmic loss in the output filter, parasitic inductance, and the source impedance of the rectifier. That is usually a few amps into a supply that was sized for a few milliamps, and the result is unfortunate. Output capacitors rupture, filter resistors burn, output transistors short. The HVPS gives a single "Aaargh!" and stops working.

Panel B - boxing gloves. Add a series resistor R_s between the supply and the load. Now the fighter is wearing gloves: the punch still lands, but the impact peak is cushioned. Even when the load is shorted, the current is hard-limited to V_PSU / R_s - by design, never above the supply's safe I_max rating. The spark itself is correspondingly modest (a tame, well-behaved arc), and the HVPS keeps running indefinitely. This is the topology to choose whenever a spark would be an accident rather than a feature.

Panel C - slingshot. Add a parallel capacitor C across the load (still with R_s in series with the supply). Now the supply slowly stretches a slingshot band - pumping charge into C at the gentle trickle rate V_PSU / R_s. When the load fires, C releases all the stored energy in a single microsecond-scale flick: peak currents of hundreds or thousands of amps, far beyond anything the supply itself could deliver. Then the slingshot must restretch (recharge time tau = R_s * C), which sets the maximum repetition rate. Choose this topology whenever you need the spark to actually do something - ignite a fuel-air mixture, initiate a plasma, machine a hole.

The remaining sections turn each panel into design equations. Panel A is the warning. Panel B is sized in Section 4. Panel C is sized in Section 5.

4. Mode A - Protective Series Resistor

The series resistor R_s has exactly two design constraints: it must hold the fault current below the maximum safe value when the load sparks, and it must drop an acceptably small voltage during normal operation.

Fault-current limit. When the load sparks, its resistance drops to near zero. The full supply voltage appears across R_s and Ohm's law gives the fault current:

I_spark = V_PSU / R_s (1)

Setting this less than or equal to the maximum allowed current I_max (set by the supply rating, the wiring, or the load itself, whichever is smaller):

R_s >= V_PSU / I_max (2)

Normal-operation voltage drop. During normal operation the load draws current I_load and the resistor drops:

V_load = V_PSU - I_load * R_s (3)

Choose R_s small enough that this drop is negligible compared with V_load (a 1 % drop is a typical target - that is, I_load * R_s <= 0.01 * V_PSU). Combined with Equation (2):

V_PSU / I_max <= R_s <= 0.01 * V_PSU / I_load (4)

Resistor power dissipation. In normal operation the resistor dissipates P_R = I_load^2 * R_s, but during a sustained short the dissipation jumps to P_R = V_PSU^2 / R_s - often hundreds of times higher. Specify the resistor's power rating, voltage rating, and pulse withstand for the fault case, not the normal case.

Example - 2 kV PMT bias supply. Limit fault current to 1 mA into a tube that draws 10 uA normally. From (2): R_s >= 2000 V / 1 mA = 2 M Ohm. Normal-operation drop = 10 uA x 2 M Ohm = 20 V (1 % of 2000 V, fine). Normal-operation power = (10 uA)^2 x 2 M Ohm = 0.2 mW; fault-condition power = (2000 V)^2 / 2 M Ohm = 2 W (briefly). A 2 M Ohm, 5 W, 3 kV-rated metal-oxide resistor handles both cases.

5. Mode B - Parallel Storage Capacitor for Energetic Sparks

When the application needs a high-current arc - much higher than the supply can deliver continuously - add a storage capacitor C in parallel with the load. The capacitor charges slowly through R_s, then dumps all of its stored charge into the load in a single microsecond-scale pulse when the spark fires. Four formulas govern the design.

Stored spark energy. Between sparks, the capacitor charges to V_PSU and stores energy:

E = 0.5 * C * V_PSU^2 (5)

Pick C from the required spark energy - typically set by what the load needs (ignition, plasma initiation, EDM crater, etc.):

C = 2 * E / V_PSU^2 (6)

Peak spark current. Once the spark forms, the capacitor sees a short circuit through the spark resistance R_spark (plus the capacitor's own ESR and lead inductance - usually small):

I_peak approx V_PSU / (R_spark + R_ESR) (7)

A typical air-arc has R_spark in the range 0.1-10 Ohm once ionized, so peak currents of hundreds to thousands of amps are routine - even from a small supply.

Recharge time and repetition rate. After the spark, C is empty and must recharge through R_s. The recharge follows the familiar RC exponential with time constant tau = R_s * C. The capacitor reaches 99 % of V_PSU after:

t_99% approx 4.6 * R_s * C (8)

which sets the maximum spark repetition rate:

f_max approx 1 / (4.6 * R_s * C) (9)

Average power from the supply. Each spark drains energy E from C; at rate f the supply must deliver an average power of:

P_avg = E * f = 0.5 * C * V_PSU^2 * f (10)

Essentially all of this power is dissipated in R_s (heat from charging the capacitor - a fundamental RC-charging loss equal to the energy stored). Size R_s with at least 2x margin on this average dissipation. The capacitor voltage rating should be at least 1.5 x V_PSU (preferably 2x) to handle voltage overshoot during recharge and to tolerate aging.

Example - 5 kV / 100 mJ plasma igniter at 50 Hz. From (6): C = 2 * 0.1 J / (5000 V)^2 = 8 nF. From (9) with f = 50 Hz: t_99% = 20 ms, so R_s = 20 ms / (4.6 * 8 nF) = 543 k Ohm (pick a stock 560 k Ohm). From (10): P_avg = 0.1 J * 50 Hz = 5 W - so R_s needs a 10 W rating with at least 7 kV pulse withstand. Capacitor: 8 nF / 10 kV polypropylene. From (7) with R_spark approx 1 Ohm: peak spark current approx 5 kA for ~ R_spark * C approx 8 ns - a sharp, energetic pulse.

6. Practical Implementation Notes

• Resistor selection. Use HV-rated film, wirewound (non-inductive variants only - standard wirewound has too much inductance for fast-transient survival), or thick-film HV resistors. Common parts in the 1-100 M Ohm range with 5-25 kV ratings are available from manufacturers including Vishay, Caddock, Ohmite, and Stackpole. For Mode B applications also check the single-pulse energy rating, not just the average power.
• Capacitor selection. For energetic-spark service, use a film capacitor with low ESR and low inductance (high dV / dt withstand) - polypropylene or polyester pulse-discharge types, not ceramic (which can fail short-circuit) and not electrolytic (polarized, and the lifetime under repeated full discharge is short). Voltage rating >= 1.5 x V_PSU minimum.
• Wiring inductance. At peak currents of hundreds of amps, every centimeter of lead adds nanohenries of inductance and tens of nanoseconds of risetime. Mount C directly across the spark load with the shortest possible loop; route the return current right next to the forward current.
• Safety interlock. A charged storage capacitor remains lethal long after the supply is turned off. Always include a bleeder resistor (~10 x R_s) in parallel with C to drain it within a few seconds of power-off, plus a hard-grounding interlock on the enclosure access door.
• When both modes are needed. Some applications want a current-limited normal output and an energetic spark on demand. Use the Mode B topology with R_s sized for the normal current limit (Equation 2) and C sized for the spark energy (Equation 6); confirm that f_max from Equation (9) is high enough for the application.

7. Summary

Output sparking is a hazard for every high-voltage supply, and two simple passive networks handle the two scenarios that designers encounter:

1. To prevent spark damage, insert a series resistor R_s >= V_PSU / I_max. It caps any fault current at I_max while costing only a small voltage drop in normal operation.
2. To generate energetic sparks on purpose, add a parallel capacitor C = 2 * E / V_PSU^2 across the load. The capacitor stores E = 0.5 * C * V_PSU^2 between pulses and dumps a peak current of order V_PSU / R_spark (typically kiloamps) when the load fires. R_s then controls the recharge to set the maximum repetition rate f_max approx 1 / (4.6 * R_s * C).

Both topologies are inexpensive, passive, and predictable. They protect the supply, protect the load, and let the designer choose deliberately whether the output should never spark, or should spark with maximum energy when commanded. Analog Technologies' HV power-supply product line (see and) offers fully-assembled HV modules with output ratings from 1 kV to 60 kV; the protection components above can be added externally for the spark scenarios described here.

References

[1] F. M. Bruce, "Calibration of Uniform-Field Spark-Gaps for High-Voltage Measurement at Power Frequencies," Proc. IEE, vol. 94, pt. II, no. 38, pp. 138-149, 1947.

[2] E. Kuffel, W. S. Zaengl, J. Kuffel, High Voltage Engineering - Fundamentals, 2nd ed., Newnes / Butterworth-Heinemann, 2000.

[3] Analog Technologies, Inc., High-Voltage Power-Supply Modules, https://www.analogtechnologies.com/High_Voltage_Power_Supply.html

[4] Analog Technologies, Inc., HV Module Application Notes Index, https://www.analogtechnologies.com/High_Voltage_Power_Supply.html