 #### Frequency domain analysis of capacitor voltage self-balance mechanism

According to the equivalent circuit of the 3L-NPC half-bridge inverter, the relationship between the capacitor current and the capacitor voltage difference is first analyzed from the perspective of frequency domain and time domain. Before the analysis, first assume that the capacity of the DC side capacitors Cdc1 and Cdc2 is large enough, the ripple amount of the capacitor instantaneous voltage difference vd is small relative to its DC component, and the rate of change of vd is much smaller than the dynamic response rate of the system. Therefore, when the system is in a steady state, vd is approximately a DC voltage, and its magnitude is recorded as Vd, then Vd=U1-U2, where U1 and U2 are the average voltages of the upper and lower capacitors on the DC side, respectively.

Frequency domain analysis

According to Figure 1, the frequency domain expression of iinv can be obtained as

According to the property of “time-domain product↔frequency-domain convolution”, the expression for converting ix to frequency domain in formula (1.2) is:

Among them, * is the convolution operation symbol in the frequency domain. According to the definition of frequency domain convolution operation, we can get:

If the 3L-NPC half-bridge inverter works stably, the voltage of the capacitor is constant, that is, the current flowing through the capacitor has no DC component, which means that the average value of the current flowing out of the midpoint of the capacitor bridge arm is ix(avg)=0 , That is, Ix(ω)|ω=0=0, then the equation (1.5) can be obtained from equation (1.4), where Sd(ω) is an even function (see Figure 2), so Sd(ξ)=Sd(-ξ).

In fact, Udc and Vd are both DC quantities when they are stable, so only the real part is concerned for , so that equation (1.5) can be equivalent to equation (1.6):