DC-Link Capacitor Ripple: The Kolar & Round Method

Why the DC-link capacitor exists

In a three-phase inverter, the switches chop the DC bus into PWM pulses at the switching frequency. That switching draws a violently pulsating current from the DC side, but the battery (through its cabling inductance) can't supply a fast-changing current cleanly. The DC-link capacitor bridges the gap: it sources and sinks the high-frequency ripple current locally, so the bus voltage stays stiff and the pulses don't propagate back up the cable. Sizing that capacitor is really about two things — the ripple current it has to carry (which drives heating and life) and the peak voltage it must withstand.

The ripple current — and the Kolar & Round formula

The hard part is knowing how much RMS ripple current the capacitor actually sees, because it depends on the operating point: the load current, the power factor, and the modulation index (how deep the PWM is driving toward full output). Kolar & Round published a closed-form expression for exactly this (IEE Proc. Electr. Power Appl., 2006) — the RMS DC-link capacitor current for a three-phase voltage-source PWM inverter with sinusoidal output current and a constant DC-link voltage. It replaces a full switching simulation with one equation.

The key result to carry in your head: the ripple current peaks near a modulation index of M ≈ 0.6, at roughly 0.6–0.65 × the RMS phase current. So the worst case for capacitor heating isn't at full modulation — it's partway up. Sizing for full-modulation ripple alone can under-size the capacitor for the operating point that actually cooks it.

The peak voltage: ripple pushes the rating up

A film DC-link capacitor's rated voltage is a peak limit for the (non-reversing) DC waveform — datasheets state the peak voltage must not exceed the rated voltage. So the governing voltage isn't the DC bus alone, it's:

V_peak = V_bus + ½·ΔV_pp

the bus plus half the peak-to-peak ripple. The ripple pushes the required voltage rating up. Two more effects tighten it: above ~85 °C the permissible voltage derates (down to roughly 0.7× rated at 105 °C), and a fast switching turn-off adds a repetitive overshoot spike (ΔV = L_loop·di/dt) on top — which, because it repeats every cycle, also has to stay inside the rating.

What actually sizes the capacitance

The required capacitance is the larger of two constraints:

• Switching-ripple-voltage limit — enough capacitance that the ripple current at the switching frequency doesn't produce more than your allowed ripple voltage (C = I_C,rms / (2π·f_sw·V_rip,rms)).
• Source-decoupling minimum — enough capacitance to keep the resonance between the cable inductance and the capacitor below the switching frequency (C ≥ 1 / (L_cable·(2π·f_sw)²)), so the harness doesn't ring.