Inrush current limiter: sizing an NTC, or using a MOSFET

An inrush current limiter is a resistance that is in the circuit when the power is switched on and out of it once the equipment is running. It is there because a supply’s bulk capacitor is discharged at the instant of switch-on and, in EPCOS’s words, the smoothing capacitors “almost produce short circuits at the moment of switching on”. The first half-cycle then drives the peak line voltage — 170 V on a 120 V line, 325 V on 230 V — into nothing but the source, wiring and bridge resistance. With half an ohm in the loop that is over 300 A for a couple of milliseconds: enough to blow a fuse rated for the running current, to exceed the bridge rectifier’s surge rating, and to pit the contacts of the switch that closed.

The usual limiter is an NTC thermistor in series with the input. Cold, it is several ohms; as the current heats it, its resistance falls “by a factor of 10 to 50” (TDK) to a few percent of that, and it stays there while the equipment runs. Choosing one takes three numbers, and both TDK and Vishay Ametherm list the same three. For Ametherm’s example — a 120 V supply with 4700 µF behind the bridge and a fuse that tolerates a 20 A peak — they are 8.5 Ω of cold resistance, 67.7 J of energy per switch-on, and the 3 A running current. The inrush current limiter calculator gives them for any supply, and the MOSFET alternative alongside. This article is where the three numbers come from and where the thermistor stops being the right answer. The NTC thermistor calculator draws the resistance-against-temperature curve that the cold and hot values sit on.

Why the first peak is so large

The charging current into a discharged capacitor is set by the voltage over the resistance in the loop, and nothing else, for as long as the capacitor voltage is still near zero. On an AC line the switch can close at any point in the cycle, and the worst case is the peak: the capacitor is presented with the full √2·V_RMS and takes whatever the loop resistance lets through. The bridge diodes’ surge rating and the fuse’s I²t are both written for that event.

Every vendor lists the same consequences. EPCOS: the currents “can damage individual components or entire subassemblies and blow fuses in error”. Ametherm gives a case: a supply with 10 A of running current and 100 A of inrush, where a 10 Ω thermistor holds the first peak to 35 A and then “continues to heat, dropping resistance to as low as 0.05 ohm where it reaches a steady state”. At 10 A that 0.05 Ω costs 5 W and half a volt. A fixed 10 Ω resistor in the same place would cost 1 kW, which is why the alternatives to an NTC are not resistors but resistors with something to short them out afterwards — a relay, a triac, or a MOSFET.

The three numbers that pick an NTC

Ametherm’s white paper walks the selection with a worked example, and TDK’s application notes describe the same inputs from the other side — “load capacitance of device to be protected”, “steady-state current (I_max) and maximum ambient temperature”, and “required reduction of inrush current to determine R₂₅”.

Cold resistance. Ametherm’s rule for the zero-power resistance is the peak voltage over the highest current the circuit tolerates:

$R_{25} \geq \frac{V_{peak}}{I_{allowed}} = \frac{120 \times \sqrt{2}}{20\ \text{A}} = \frac{169.7\ \text{V}}{20\ \text{A}} = 8.5\ \Omega$

where the allowed current is “fuse in power supply or breaker on AC line”. TDK puts it the other way round: “the higher the cold resistance (R25) of the inrush current limiter, the more the inrush current is dampened. If the current limiting effect of a component is inadequate, choose a larger model.” The resistance sets the peak; it does not set the size of the disc.

Energy. What sets the size is the energy the part absorbs on every switch-on, which is the energy the capacitor ends up holding:

$E = \tfrac{1}{2} C V_{peak}^2 = \tfrac{1}{2} \times 4700\ \mu\text{F} \times 169.7^2 = 67.7\ \text{J}$

Ametherm rates its parts in joules directly. TDK rates the same property as a “maximum permissible capacitance”, the C_test that the part can be discharged through from 375 V in the IEC 60539-1 test circuit; a capacitance at another voltage converts by equal energy, which for this example is 960 µF at 375 V. TDK’s note is blunt that this is the number that sizes the part: “the required C_test figure alone will determine the component that is needed. Within this component model the maximum steady-state current then determines the highest possible cold resistance (R25) that can be used for an application.”

Steady-state current. The disc carries the running current for as long as the equipment is on, and the datasheet I_max — “the maximum permissible continuous current (DC or RMS values for sine-shaped AC) in the temperature range 0 °C to 65 °C” — must exceed it. Above 65 °C the current is derated; Ametherm’s transformer example applies 90 % at 75 °C. The running current is input power over the minimum input voltage at the real efficiency, which Ametherm computes for its 2 kVA transformer at 90 V and 70 %:

$I_{steady} = \frac{2.0\ \text{kVA}}{0.70 \times 90\ \text{V}} = 31.75\ \text{A}$

Then round every number up — with one caution about the paper’s own example. Ametherm notes that going from 8.4 to 10 Ω “is fine when selecting an NTC thermistor inrush current limiter”, which is right. But the paper prints the energy as 67.6 J in one place and 6.65 J in another, and the part it then selects, the SL10 10003, is chosen against the mistyped figure: it rounds “8.4 Ω, 3 A, 6.65 J” up to “9 Ω, 3 A, 7 J” and asks for “a value of 7 or more” joules. The arithmetic gives 67.7 J, so the requirement is a 10 Ω, 3 A part rated above 68 J, and the SL10 is not the answer to that; the paper’s own transformer table has the 10 Ω SL22 10008 at 90 J, which is the class of part the energy calls for. The lesson is the selection order, not the part number: the energy figure picks the disc size before anything else does.

Two rules follow the numbers. If one part cannot take the energy, “two or more thermistor elements can be connected in series” (TDK). And never in parallel:

Paralleling several NTC thermistors is inadmissible, since the load will not be evenly distributed. The thermistor carrying the largest portion of current will heat up until it finally receives the entire current (which may result in destruction of the device), while the other paralleled thermistors remain cold.

— TDK Electronics, NTC Inrush Current Limiters, Application notes

What the NTC cannot do

The thermistor limits inrush by being cold. Everything it cannot do follows from that.

It needs time to cool. After switch-off “the thermistor slowly cools down. Its resistance increases steadily, but the full resistance value is only reached after 1 to 2 minutes” (TDK); EPCOS gives “30 seconds to two minutes depending on the disk size”. Switch the equipment off and on inside that window and the disc is still warm and low, and the second inrush is barely limited. EPCOS points out that for a switch-mode supply this often does not matter, “because electrolytic capacitors in the circuit usually take longer to discharge fully” — a capacitor that is still charged draws no inrush. For equipment that cycles as part of its job, Ametherm lists “welding gear or a plasma cutter” as examples, it does matter, and both vendors’ answer is to bypass the thermistor after start-up or to use a different limiter.

It depends on the ambient. Hot ambient means a lower starting resistance and less limiting. Cold ambient means a higher one, and in Ametherm’s words “the high resistance could limit all of the current and prevent the system from actually turning on, even after the initial inrush ends”. A thermistor inside a warm enclosure is limiting less than the datasheet R25 suggests every time it starts.

It gets hot, and it heats its neighbours. TDK: “the NTC thermistor may in extreme cases reach a mean temperature of up to 250 °C”, and the heat “will also be dissipated through the lead wires”. The thermistor beside the electrolytic it protects is shortening that capacitor’s life; give it clearance and airflow.

It is not a fuse. A short circuit heats the thermistor, which lowers its resistance, which passes more current — the opposite of protection. Ametherm: “high current from a short can also destroy the NTC thermistor.” The fuse stays in the circuit; the NTC is what lets the fuse be rated for the running current instead of the inrush.

Transformers: measure, do not calculate

Questions about limiting inrush for high-power transformers have a different physics behind them. A transformer’s inrush is not a capacitor charging; it is the core saturating, and whether it saturates depends on when in the cycle the switch closes. Ametherm describes both cases: “if turn-on occurs when the AC voltage wave is at its peak value, there will be no inrush current drawn by the transformer”, while “if at turn-on, the AC wave is going through its zero value, then the current drawn will be very high and exceed the saturation current”. The reason is that flux is the integral of voltage. Closing at the peak starts the flux at zero, where the steady-state waveform has it. Closing at the zero crossing starts it at zero too — but the steady-state waveform has it at its negative peak just then, so the first half-cycle drives the core to twice its normal flux, well past saturation, and the magnetising current becomes the inrush.

The size of that current is a property of the core and the winding, not of any capacitance, so Ametherm’s procedure for a transformer starts with a measurement. For a pair of 1 kVA transformers on 120 V: “peak inrush current occurs in one cycle = 564 A, as measured on the oscilloscope”. From that peak the effective inductance is X_L = 120/564 = 0.213 Ω, so L = 0.213/(2π·60) = 565 µH, and the energy the NTC must absorb is

$E = \tfrac{1}{2} L I^2 = \tfrac{1}{2} \times 565\ \mu\text{H} \times 564^2 = 90\ \text{J}$

with the steady-state current of 31.75 A computed above, derated to a part rated 36 A at 75 °C. The same three numbers, but the first two come from a scope capture rather than from the schematic. Put a current probe on the primary, switch it on at the zero crossing a few times, and take the worst peak.

When a MOSFET is the better answer

On a DC rail — a hot-swap card, a 48 V telecom input, a 12 V distribution bus — the thermistor’s weaknesses are all present at once: the ambient is warm, restarts are frequent, the efficiency loss matters, and the rail may already be live when the load connects. The answer there is a MOSFET in series with the load whose gate is ramped slowly, so the output voltage rises linearly and the load capacitance draws a constant current. TI’s SLVA156 gives the circuit for a regulator output: a FET following the output, a resistor R_T and capacitor C_T setting the ramp, and an added gate-drain capacitor C_GD that “provides a smooth, linear ramp of the output voltage”. Its equation for the timing resistor of the PMOS version is

$R_{T} \approx \frac{(V_O - V_{TH})\, t_{SS}}{C_{GD}\, V_O}$

with the conditions that the FET’s threshold is below the voltage being ramped, that C_GD is “much larger than the FET’s inherent gate to drain capacitance”, and that C_T is much larger than C_GD.

The arithmetic of the ramp is simple and the consequence is not. With the output rising linearly over t_SS, the load capacitance draws C·V/t_SS — 1.2 A for 1000 µF on 12 V over 10 ms — and no more, whatever the source could deliver. But the FET is holding the difference between the rail and the output while that current flows, so it dissipates V·I falling from 14.4 W at the start of the ramp to zero at the end. The total is ½CV², the same energy the capacitor stores, and it does not change with the ramp time; a slower ramp only spreads it out. For 1000 µF on 12 V that is 72 mJ and 7.2 W average over the 10 ms, in the FET’s linear region with the full rail across it, which is what its safe-operating-area chart is for. SLVA156’s own numbers are for a regulator output at 300 mA, where the concern is instead the drop across R_DS(on) — “approximately 59 mΩ at V_GS = −1.8 V; so, at 300-mA output current, the output voltage could be 18 mV below the nominal voltage”. At amps, the SOA is the constraint.

The MOSFET limiter has no cool-down, no ambient dependence worth speaking of, and a fraction of the NTC’s running loss; what it lacks is fault protection, which it never had — a shorted output puts the whole rail across the FET at whatever current the source gives. On a rail that needs both, that is what an eFuse or a hot-swap controller with a current limit is: the same series FET with a control loop that sets the current rather than the ramp.

Worked numbers

Ametherm’s example, as the calculator’s defaults reproduce it:

line             120 V RMS          V_peak = 169.7 V
capacitance      4700 µF            E = ½ × 4700 µF × 169.7² = 67.7 J
allowed peak     20 A (fuse)        R25 ≥ 169.7 / 20 = 8.49 Ω  → 10 Ω
running          3 A at 25 °C       I_max ≥ 3 A (no derating to 65 °C)

loop 0.5 Ω, no NTC                  peak 339 A
loop 0.5 Ω + 8.49 Ω cold            peak 18.9 A;  τ = 42 ms, 95 % at 127 ms
part                                10 Ω, 3 A, rated above 68 J — the paper's SL10 10003
                                    is picked against its mistyped 6.65 J; its own table
                                    lists the 10 Ω SL22 10008 at 90 J

MOSFET alternative on 12 V, 1000 µF, 10 ms ramp:
load current     C·V/t              1.2 A, constant during the ramp
FET energy       ½CV²               72 mJ, 7.2 W average, 14.4 W at the start

The charging curve holds the thermistor cold, which is the slow bound; the disc heats within the first cycles and the capacitor is charged sooner than 127 ms. The cool-down is the number that does not get better.