DC bus capacitance design for SPWM inverter

Question

I am working on a 5kW grid-connected inverter where I need to find the DC bus capacitance value. I understand that the capacitor serves 2 purposes: 1) to provide a low impedance path for the high frequency currents from the device, and 2) To reduce voltage ripple in the DC bus.

Some specifications:

DC bus voltage = 800 V

Phase voltage = 240 V

So I first thought of finding out the ripple current that flows into the capacitor, so that it can be used to find the required capacitance for a given voltage ripple specification. I found the equation for the DC bus capacitor current in SPWM inverter from a paper. I have 2 questions, one mathematical and one relating to the design.

Question 1:

Here, \$ \hat{A}_{0n} \$ and \$ \hat{B}_{0n} \$ are zero (proved by the paper for SPWM inverters). The resulting equation has a dc component and a bunch of other components. Now, if I were to find the frequency spectrum of this series, will the magnitude of \$ i_C(t) \$ at frequency \$ m \omega_c + n\omega_o \$ equal to \$ \sqrt{\hat{A}_{mn}^2 + \hat{B}_{mn}^2 }\$? Given that I wanted to find the rms ripple current, I wanted to find the rms value of all the components except the DC term.

Question 2:

After finding the rms ripple current through the capacitor, I used: $$ \frac{i_{ripple, rms}}{C} = \frac{dV}{dt} $$ If the voltage ripple specification (\$ dV \$) is given, can this be rearranged as: $$ C = \frac{dV}{i_{ripple, rms} \cdot f_{sw}} $$ where \$ f_{sw} \$ is the switching frequency?

Answer 1

If the voltage ripple specification (\$ dV \$) is given, can this be rearranged as: $$ C = \frac{dV}{i_{ripple, rms} \cdot f_{sw}} $$ where \$ f_{sw} \$ is the switching frequency?

No -- you lose the harmonic information when taking RMS (the step for which was not specified, but it will be Parseval's theorem). Therefore the frequency-dependent impedance can't be factored in. Or, this works only for the fundamental (for which Parseval's is trivial). (I also assume you meant a factor of 2π in there. Or maybe not, if the waveform is square.)

Swap the steps around: use the impedance per harmonic first, and then sum over harmonics.

Which, since the harmonics are AC sinusoidal steady-state, you should use that form: \$\frac{V}{I} = \frac{1}{2 \pi F C}\$.

Since SMPS waveforms are generally square to triangle in nature, the harmonics drop off fairly quickly (1/N or 1/N²), so this is a reasonable approximation for design purposes; the actual value may be ∼10% higher.