#### 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 C_{dc1} and C_{dc2} is large enough, the ripple amount of the capacitor instantaneous voltage difference v_{d} is small relative to its DC component, and the rate of change of v_{d} is much smaller than the dynamic response rate of the system. Therefore, when the system is in a steady state, v_{d} is approximately a DC voltage, and its magnitude is recorded as V_{d}, then V_{d}=U_{1}-U_{2}, where U_{1} and U_{2} 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 i_{inv} can be obtained as

According to the property of “time-domain product↔frequency-domain convolution”, the expression for converting i_{x} 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 i_{x(avg)}=0 , That is, I_{x}(ω)|_{ω=0}=0, then the equation (1.5) can be obtained from equation (1.4), where S_{d}(ω) is an even function (see Figure 2), so S_{d}(ξ)=S_{d}(-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):

From the formula (1.6), it can be seen that there are two main factors that affect the balance of the capacitor voltage: first, the size of the equivalent impedance Z_{eq} viewed from the midpoint of the bridge arm, that is, the size of the output filter and the load. If the equivalent impedance is purely capacitive, the real part is 0, so that the capacitor voltage difference V_{d} will be unbalanced when it is stable; secondly, the degree of spectrum overlap between S_{t }and S_{d} will also affect the equalization ability of the capacitor voltage. Since the frequency spectrum of S_{t} and S_{d} is related to the modulation strategy, this formula can be used to evaluate whether a modulation strategy has the ability to balance the capacitor voltage, and it also provides an idea of the capacitor voltage balance control.

Usually 3L-NPC adopts a unipolar SPWM modulation strategy, where the carrier frequency is much larger than the modulation wave frequency, which can effectively avoid spectrum overlap. According to the expressions of S_{t} and S_{d} in formula (1.7), the time-domain waveforms and spectrograms of S_{t} and S_{d} can be made, as shown in Figure 2, where f_{s} is the switching frequency and T_{o} is the fundamental wave period.

It can be seen from Figure 2 that S_{t} is an odd function and the half-wave is symmetrical, so its spectrum contains only odd-order sine quantities, and there is no DC component and even-order component; S_{d} is an even harmonic function (even function and half-wave overlap), which only contains a DC component and an even-order cosine. So this means that the spectra of S_{t} and S_{d} do not overlap, |S_{t}(ω)||S_{d}(ω)|=0, according to formula (1.6), V_{d}/U_{dc}=0, which indicates that the unipolar SPWM modulation strategy can make the capacitor voltage tend to be balanced when the switching frequency is much greater than the frequency of the modulation wave. This phenomenon is called self-balance characteristic.

From the perspective of frequency domain analysis, it is possible to judge whether a certain modulation strategy can make the 3L-NPC topology have capacitor voltage self-balancing characteristics simply by the degree of spectrum overlap of the switching function and the equivalent impedance of the inverter bridge arm. However, the frequency domain is relatively abstract, and it is impossible to clarify the working principle of the capacitor voltage to achieve self-balance. In fact, the frequency domain and the time domain correspond to each other, and the self-balancing characteristics of the 3L-NPC topology will be analyzed from the perspective of the time domain.