The main circuit and working waveform of the single-phase three-level SPWM inverter are shown in Figure 1. The figure (a) is the main circuit, and the figure (b) is the working waveform. In figure (b), the sine wave is the modulating wave, and the unipolar triangle wave is the carrier. Compare the sine wave with the carrier triangle wave. For the positive half cycle of the sine modulation wave, when the sine wave is larger than the triangle wave, the switch tubes S_{ap} and S_{bn} are turned on to get the positive pulse of the positive half-level SPWM wave; when the sine wave is smaller than the triangle wave In the part, the switch tubes S_{ap} and S_{bn} are turned off, and the zero voltage of the positive half-wave level SPWM wave is obtained. For the negative half cycle of the modulation wave, in the part where the sine wave is smaller than the triangle wave, the switching tubes S_{bp} and S_{an }are turned on, and the negative pulse of the negative half-wave level SPWM wave is obtained; in the part where the sine wave is greater than the triangle wave, the switching tubes S_{bp} and S_{an} are turned off , Get the zero voltage of the negative half-wave level SPWM wave. The complete three-level SPWM waveform is the waveform of the voltage u_{A} on the load R_{L}, as shown in Figure 1(b). It has three levels of +E, 0 and -E, so the inverter is called a three-level SPWM inverter. Its operating frequency f_{s} is equal to the unipolar carrier triangular wave frequency f_{c}.

The three-level SPWM method can also be used for synchronous and asynchronous. When used in asynchronous mode, the number of pulses and pulse patterns contained in each cycle of the SPWM modulated wave are not repetitive. Therefore, it is more appropriate to investigate the distribution of the sidebands based on the angular frequency of the carrier triangle wave. It is to use the double Fourier series analysis method to analyze.

For the convenience of analysis, the dead zone is not considered here, and the unipolar carrier triangle wave is represented by the two piecewise linear functions shown in Figure 2. In this way, the slopes of the two functions are +Uc/π and -Uc/π, and the initial values are 0 and +Uc, respectively. The mathematical expression of the carrier triangle wave can be written as

The equation of sine modulation wave is

Let modulation ratio M=U_{s}/U_{c}-<1, carrier ratio F=ω_{c}/ω_{s}>>1

The sampling point of the three-level SPWM wave is the intersection point of the sine wave and the carrier triangle wave, and there is u_{c}=u_{s} at the intersection point.

At sampling point a

make

but

At sampling point b

For the three-level SPWM waveform shown in Figure 1(b), X=ω_{c}t is in the range of 2πk to 2π(k+1), and the positive pulse of the three-level SPWM waveform can be obtained between point a and point b, so Obtain the time function of the three-level SPWM wave, that is, the expression of the voltage u_{A} on the load, that is

Since the three-level SPWM waveform shown in Figure 1(b) is symmetrical to the origin, u_{A} is an odd function, so its double Fourier series expression will only contain sine terms, not constant components and cosine terms. Also, since the u_{A} waveform is mirror-symmetrical, it only contains the odd harmonics in the sine term. From this

Note that e^{im2πk}=1, then

Known from Bessel Theory

So there is

When n is 0 or an even number, 1-e^{jnπ}=0, so A_{mn}+jB_{mn}=0.

When n is an odd number, 1-e^{jnπ}=2, so

Since sin(mπ)=0, A_{mn}=0,

When m=0, there is

Because u_{A} is an odd function, we have

When n=1, B_{01}=ME; when n≠1, B_{0n}=0.

The three-level SPWM wave is thus obtained, that is, the Fourier series expression of the output voltage u_{A} is

From formula (16), the general frequency spectrum value when M=0.5 or 1, as shown in Figure 3.

The output voltage spectrum of the single-phase three-level SPWM inverter and the relationship curve between A_{mn}/A_{01} and M are shown in Figure 4.

Comparing the frequency spectrum of Figure 4 and Figure 5, it can be seen that the harmonic content of the three-level SPWM waveform with a unipolar triangular wave as the carrier is much smaller than that of the two-level SPWM waveform with a bipolar triangular wave as the carrier. In addition, it can be found that when the carrier ratio of the two-level SPWM wave is half of the carrier ratio of the three-level SPWM wave, there is a very interesting rule in their spectrum, that is, m in the two-level SPWM wave spectrum is 2, 4 The upper and lower side frequency components of the carrier harmonics of, 6, … are completely consistent with the corresponding components of the three-level SPWM wave with m being 1, 2, 3, …; and the m is 1 in the two-level SPWM wave spectrum. 3, 5, … The carrier, carrier harmonics and their upper and lower side frequency components are zero in the three-level SPWM wave. In addition, it can be seen that when m=1, for the same value of M, the harmonic amplitude of the three-level SPWM wave is obviously smaller than the amplitude of the two-level SPWM wave. Therefore, the three-level SPWM wave has a better ability to eliminate harmonics.