**(1) Characteristics of SPWM modulation wave**The rectifier and inverter in the UPS adopt SPWM control, and the obtained SPWM modulated wave has the following four characteristics.

①The SPWM modulation waveform can be turned into a numerically stable quasi-sine wave waveform

②By moving the phase of the sinusoidal modulation wave us, the phase of the SPWM modulation wave can be moved. Subtracting two SPWM modulation waveforms with different phases can make the output SPWM modulation waveform further sinusoidal. This feature is the theoretical basis of the SPWM inverter multiple superposition method to eliminate harmonics. The multiple superposition of the SPWM modulation waveform of the sine modulation wave phase shifting 1800 is shown in Figure 1. Two-level voltage u

_{pa1}can be obtained by comparing u

_{s}with u

_{c}, and two-level voltage u

_{pa2}can be obtained by comparing -us with u

_{c}. U

_{pa2}is 180° behind upa1, and the voltage u

_{pa}obtained by u

_{pa1}+u

_{pa2}is a three-level voltage.

③By moving the phase of the carrier triangle wave, the phase of the SPWM modulation wave can also be moved. Adding two SPWM modulation waveforms of different phases can also make the output SPWM modulation waveform further sinusoidal. This feature will derive a carrier triangle wave phase-shifted 1800 SPWM modulation waveform cascade superposition (PSC-PWM), as shown in the figure 2 shown. In the figure, the two-level voltage upa1 generated by the comparison between us and u_{c} and the two-level voltage u_{pa2} generated by the comparison between us and -u_{c} are added, and the resulting voltage u_{pa1} +u_{pa2}=u_{pa} is a three-level voltage; in addition, this feature will also Derive the unipolar SPWM modulation, the voltage upa obtained by comparing the us in Figure 2 with the unipolar carrier triangle wave synthesized by u_{c} and -u_{c} with the same polarity as us is also a three-level voltage. This reveals the relationship between the unipolar SPWM and bipolar SPWM waveforms, as shown in Figure 3. Since the frequency f’_{c} of the carrier triangle wave u’_{c} of the unipolar SPWM is twice as high as the frequency fc of the bipolar SPWM carrier triangle wave u_{c}, that is, f’_{c}=2f_{c}, the chopping ratio of the unipolar SPWM is F’= (f’_{c}/f_{s}), which is twice as large as the carrier of bipolar SPWM than F=(f_{c}/f_{s}), that is, F’=2F. Therefore, m’of unipolar SPWM should be half smaller than m of bipolar SPWM, that is, m’=(m/2).

④ Both the basic SPWM and the derived SPWM can use double Fourier series for harmonic analysis. According to the difference between the angular frequency of the carrier triangle wave and the sinusoidal modulation wave, it is divided into synchronous SPWM modulation and non-synchronous SPWM modulation.

Synchronous SPWM modulation is a modulation method that synchronizes the carrier angular frequency ω_{c} (determining the switching operating frequency) and the modulating wave angular frequency ω_{s}, (equivalent to the output frequency or the fundamental frequency), so the modulation wave includes half a cycle The number of carrier pulses is a fixed value (carrier ratio F=(ω_{c}/ω_{s}) is a constant value): Asynchronous SPWM modulation refers to a modulation method in which the carrier angular frequency ω_{c} and the modulation wave frequency ω_{s} are asynchronous, that is, the modulation wave is half The number of carrier pulses contained in a period is not a fixed value (that is, the carrier ratio F=(ω_{c}/ω_{s}) is not a constant value). Generally speaking, the carrier angular frequency ω_{c} is kept constant, but sometimes it changes according to the working conditions.

In synchronous SPWM modulation, when the frequency of the modulating wave is low, the carrier frequency will be reduced accordingly, so it is difficult to achieve smooth control. In order to overcome this shortcoming, a method of changing the number of carrier pulses can be used. In asynchronous SPWM modulation, although there is no need to change the number of carrier pulses, when the carrier ratio F=(ω_{c}/ω_{s}) is smaller (the output frequency is close to the switching frequency), the output frequency fs and the carrier frequency fc will be lower than the frequency The speech wave is very close to ws and jumps, which makes the characteristics significantly worse and cannot be used (so the asynchronous SPWM modulation hopes to increase the carrier ratio by using fast switching devices, such as F≥15, to avoid the lower side frequency and output of the carrier frequency The frequency is close). At this time, a synchronous modulation method that is slightly more complicated than the asynchronous SPWM modulation can be used. The non-synchronous triangle wave-sine wave comparison method is generally used in online control, while the synchronous method can be used online or offline. There are also inverters that use asynchronous mode for low-frequency output and synchronous mode for high-frequency output. This method of use is called synchronous-non-synchronous alternating mode. No matter it is synchronous, asynchronous, or synchronous-non-synchronous alternating mode, the inverter circuit is unchanged.

(2) SPWM pulse width modulation switching loss

Figure 4 shows the typical waveforms of the voltage, current and power of the IGBT switch. The time of the four working states in the figure are cut-off time T_{OFF}, turn-on time T_{TON}, turn-on time T_{ON}, and turn-off time T_{TOFF}. The corresponding average power consumption sequence is P’_{OFF}, P’_{TON}, P’ON and P’ _{TOFF}. This figure describes the turn-off transient process with the most serious average power consumption, during which the maximum instantaneous power consumption point P_{Cmax} in the entire cycle often appears. When its value exceeds the critical power, a secondary breakdown will occur.

The relevant formulas for analyzing switching losses are explained as follows.

In the formula, P_{c} is the switching loss in general, which is equal to the average value of the instantaneous power consumption in the entire cycle; I(t) and U(t) are the collector current and collector-emitter voltage of the IGBT, respectively.

P_{c }can be divided into segments of integrals to show the relationship of the four states, namely

Because power is energy consumed per unit time, the integral value of power over a period of time is the energy consumed during this period of time. Therefore, equation (1) can also be written as the relational equation of energy consumption, namely

With a slight change in equation (2), the relationship between the switching loss and the four average losses can also be obtained, as

Among them, the switching loss caused by transient transition is

Because formula (5) is difficult to calculate accurately, the following approximate calculation formula is usually used to estimate the switching loss of IGBT, namely

In the formula, tr and tf are the rise and fall times of the collector current, the size of the switching loss, and the sum of the rise and fall times of the collector current of the IGBT is related to the size of t_{r}+t_{f}. For some high-frequency low-loss ICBT, because its t_{r}+t_{f} is small, its switching loss is also small.