Capacitor Switching Inrush Formulas
Capacitor switching formulas: the capacitance from the reactance, the peak inrush current limited by the circuit inductance and the frequency of the inrush.
The formulas
| To find | Formula |
|---|---|
| Capacitance per phase from Xc | C = 1 / (2π × f × Xc) |
| Surge impedance | Z = √(L / C) |
| Peak phase voltage | Vpk = √2 × VL / √3 |
| Peak inrush current | Ipeak = Vpk / Z = Vpk × √(C / L) |
| Frequency of the inrush | fi = 1 / (2π × √(L × C)) |
L is the inductance in the circuit between the source and the capacitor in henries (the cable, the transformer and any series reactor), C the capacitance per phase in farads and Xc its reactance in ohms. The formulas are for a bank switched on when the voltage is at its peak and with no charge on it, which is the worst case, and the resistance of the circuit is ignored, so the real peak is lower. A bank switched next to a bank that is already live (back-to-back) sees a higher current at a higher frequency, because the inductance between the banks is small. The inductance used here is an example, so use the real value for your circuit.
Worked example: a 50 kVAR bank at 400 V
Xc = 3.2 Ω per phase and 10 µH of circuit inductance.
- C = 1 / (2π × 50 × 3.2) = 994.7 µF
- Z = √(10e-6 / C) = 0.1003 Ω
- Vpk = √2 × 400 / √3 = 326.6 V
- Ipeak = 326.6 / 0.1003 = 3,257 A
- fi = 1 / (2π × √(10e-6 × C)) = 1,596 Hz
Peak inrush for different inductance (same bank)
| Circuit inductance | Peak current | Frequency |
|---|---|---|
| 5 µH | 4,607 A | 2,257 Hz |
| 10 µH | 3,257 A | 1,596 Hz |
| 20 µH | 2,303 A | 1,128 Hz |
| 50 µH | 1,457 A | 713.6 Hz |
Frequently asked questions
Why is the inrush so large?
An uncharged capacitor is a short circuit at the instant of switching, so only the inductance of the circuit limits the current, and it rises very quickly.
What reduces the inrush?
A series reactor, a pre-insertion resistor, or a contactor made for capacitor duty. A detuning reactor also limits it.
Does the inrush matter for the contactor?
Yes. The contacts must be rated for the peak current and the frequency, and a standard contactor can weld, so use one made for capacitors.
What if the bank has several steps?
Switching a step next to steps that are already on gives the highest current, because the inductance between the banks is small.
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