#### NPC half-bridge inverter equivalent circuit

In small and medium power applications, half-bridge circuits are widely used due to their simple structure and few power devices. As current inverters mostly use the current instantaneous value control technology, the current instantaneous value control accuracy is high, and the system dynamic response is good. However, in the current-type instantaneous value control half-bridge inverter, the problem of unbalanced DC side capacitor voltage is more serious, which will cause the midpoint voltage of the DC side voltage divider capacitor to drift, which will cause the inverter output voltage and current waveforms to be distorted, and even make the system out of control. Take the three-level midpoint registration type half-bridge inverter (3L-NPC) circuit as an example to introduce the equalization control of the DC side capacitor voltage. First, the equivalent circuit of the 3L-NPC half-bridge inverter is derived, and the relationship between the DC component of the capacitor current and the capacitor voltage difference is analyzed from the frequency domain and the time domain. Based on this, the control strategy of the capacitor midpoint voltage balance is explored.

NPC half-bridge inverter equivalent circuit

Before deriving the NPC equivalent circuit, first define a basic two-port switch network, as shown in Figure 1.

The relationship between the input voltage V1 and the output voltage V2 of the basic two-port network, the input current I1 and the output current I2 can be expressed as:

V_{2}=S_{x}×V_{1} (1.1)

I_{1}=S_{x}×I_{2} (1.2)

Among them, the switch network function S_{x} is a time-varying function that changes with the switch state.

According to the above definition, combined with specific inverter topology for specific analysis, the 3L-NPC half-bridge inverter is shown in Figure 2. In the figure, u_{1} and u_{2} are the instantaneous voltages of capacitors C_{dc1} and C_{dc2} respectively; U_{inv} is the voltage between the midpoint AO of the bridge arm; i_{1} and i_{2} are the currents of the capacitors C_{d}_{c1} and C_{dc2} respectively; i_{x} is the current flowing out of the midpoint of the capacitor bridge arm; i_{inv} is the current on the inverter side; i_{o} is the grid current after filtering, that is, the load current; Z_{eq} is the equivalent impedance at the AO end of the midpoint of the bridge arm.

In order to make the switch part of the inverter bridge arm equivalent to a two-port switch network, first define the state Si of the switch tube, the current i_{x} flowing out of the midpoint of the capacitor bridge arm, and the instantaneous voltage difference between the two capacitors v_{d}:

i_{x}=i_{1}-i_{2}_{ } (1.4)

V_{d}=u_{1}-u_{2} (1.5)

If the input voltage U_{dc} has approximately no fluctuations, and the DC side capacitance C_{dc1}=C_{dc2}=C, then there are:

U_{dc}=u_{1}+u_{2 } (1.6)

According to the working characteristics of the three levels of the 3L-NPC topology, the corresponding relationship between the output voltages uinv and ix of the bridge arm and the state of different switch combinations can be obtained, as shown in Table 1.

From Table 1:

Combining formula (1.5) and formula (1.6), we can further obtain:

According to formula (1.9), the equivalent circuit of 3L-NPC half-bridge inverter can be obtained, as shown in Figure 3, i_{inv} can be expressed as:

It should be pointed out that the inverter equivalent circuit derivation process does not consider the specific PWM modulation strategy. Although the 3L-NPC half-bridge inverter has a variety of modulation techniques, such as carrier stacking, carrier overlapping, etc., the working characteristics of the three levels (+1, 0, -1) of 3L-NPC are fixed. Therefore, the equivalent circuit based on the switching function is general, and the analysis results are applicable to various modulation techniques.