What is a single-phase two-level SPWM inverter?

Single-phase two-level SPWM inverter refers to an SPWM inverter whose modulation wave is sine wave, carrier wave is bipolar triangle wave, and output voltage is two-level waveform. It is the most basic of all SPWM inverters. Constituent unit, can be used as a bridge arm of a three-phase SPWM inverter.

For SPWM inverters, special attention should be paid to prevent a series of short-circuit faults in which the upper and lower switching tubes of the same bridge arm are turned on at the same time. For this reason, a “dead zone” Δt must be set between the upper and lower switching tubes. The main circuit and working waveform of the single-phase half-bridge two-level SPWM inverter with dead zone are shown in Figure 1. Figure (a) is the main circuit, and Figure (b) is the working waveform. In figure (b), u_s is a sinusoidal modulation wave, and u_c is a bipolar carrier triangle wave. Compare the sine wave with the triangle wave. In the part where the sine wave is greater than the triangle wave, the switch tube S_ap is turned on to generate a positive pulse in the two-level SPWM waveform; in the part where the sine wave is smaller than the triangle wave, the switch tube S_an is turned on, resulting in Negative pulse in a two-level SPWM waveform. The complete two-level SPWM waveform is the waveform of the voltage u_A on the load R_L, as shown in the lower part of Figure 1(b). It has two levels of +E/2 and -E/2, so it is called a two-level SPWM inverter. The black part in the figure is the dead zone △t. The switching frequency f_s of the inverter is equal to the frequency f_c of the carrier triangle wave.

SPWM can be divided into two types: synchronous and non-synchronous. In the asynchronous mode, the number of pulses and pulse patterns contained in each period of the modulation wave in SPWM modulation are not repetitive. Therefore, it cannot be based on the angular frequency of the modulation wave ω_s, and it can be decomposed into the modulation wave by a Fourier series. Harmonic of multiples of angular frequency. However, it is more appropriate to use the angular frequency ω_c of the carrier triangle wave as a reference to investigate the harmonic distribution of its sidebands. This is the double Fourier series analysis method. Such a harmonic analysis method is also applicable to the synchronous type.

For the convenience of analysis, the carrier triangle wave is represented by two piecewise linear functions, as shown in Figure 2. Their slopes are +2U_c/π and -2U_c/π, the initial values ​​are +U_c and -U_c, and U_c is the amplitude of the carrier triangle wave. ω_c is the angular frequency of the carrier triangle wave. In this way, the mathematical equation of the carrier triangle wave can be written as follows, namely

The equation of sine modulation wave is

Let modulation degree Us/Uc=M≤1, Uc is constant; carrier ratio F=ωc/ωs≥1, F is any positive integer, namely F=1, 2, 3,…

The sampling point of the two-level SPWM wave is the intersection point of the sine wave and the carrier triangle wave, and there is us=uc at the intersection point.
At sampling point α there is

At sampling point b

X=2πk+(π/2) (1 + MsinY)

The two-level SPWM wave can be described by the geometric model shown in Figure 3.

In Figure 1(b), when the lower two-level SPWM wave is obtained from the upper modulating wave, it is divided into F intervals with the amplitude point of the carrier triangle as the boundary. For example, X=ω_ct from 2π[k－(1/2)] to 2π[k＋(1/2)] in the interval x, for the sampling point a, at X≥2π(k+1)﹣[(π/2) (1+MsinY)]+Δt, get a positive pulse, when X＜2π(k+1)﹣[(π/2)(1+MsinY)], get a negative pulse; for sampling point b, when X＜2πk＋[(π /2) (1+MsinY)], a positive pulse is obtained, and a negative pulse when X≥2πk+[(π/2)(1+MsinY)]+Δt. The DC power supply of the inverter is E, then the time function u_A of the two-level SPWM wave is

In the formula, Y=ω_st-ψ; X=ω_ct; ω_c/ω_s=F; U_s/U_c=M, k=0,1,2,3,…
Assuming that m is the harmonic order relative to the carrier triangle wave, and n is the harmonic order relative to the modulating wave, the double Fourier series expression of μ_A is

The series coefficient (fundamental wave and harmonic amplitude) of this formula is

In the formula, m=1, 2, 3,… are multiples of the carrier triangle wave frequency.
Substituting formula (1) into formula (3), we get

Note that e^im2πk = 1, e^im2π(k+1) = 1, then we have

According to Bessel theory and (-1) ⁿ=e^jnπ

From this

It can be seen from this formula that when m=0 and n=0, sin(m+n)π/2=0, so the constant component A_oo=0

For the fundamental wave and the harmonics of the fundamental wave, m=0, then

For the fundamental wave, n=1, then

That is, A₀₁=0, B₀₁=ME/2.
For the harmonics of the fundamental wave, n>1, at this time there are

Indicates that there is no harmonic of the fundamental wave.
For the carrier wave and the harmonic wave of the carrier wave, n=0, by formula (4)

When m=even, sin(mπ/2)=0, that is, A_m0=0.
When m=odd, there are

For the upper and lower side frequencies of the carrier and the m-th harmonic of the carrier, by formula (4)

Since ω_c/ω_s=F is any positive integer, the double Fourier series expression of the two-level SPWM wave with dead zone At can be obtained from the above analysis results, namely A₀₀; A_0n, B_0n. ; A_m0, B_m0; A_mn, B_mn results into the formula ( 2), the double Fourier series expression of u_A can be obtained as

In addition, formula (5) can be simplified as

When Δt is converted into carrier triangle wave frequency angle, there is

It can be seen from formula (5) that 1+e^im△t increases with the increase of m and △t. When △t=0, 1+e^im△t=2, and formula (5) becomes dead zone-free uA double Fourier series expression, namely

Convert the derived sin[(m+n)π/2] os(nπ/2)=sin(mπ/2), sin[(m+n)π/2] sin(nπ/2)=cos (Mπ/2) Substitute into the above formula to get

From the u_A expression obtained above, it can be seen that the output voltage u_A of the single-phase two-level SPWM inverter includes the fundamental wave, the carrier harmonic, the m-th harmonic of the carrier, the upper, the upper and the m-th harmonic of the carrier and the carrier. The lower side frequency. The amplitude of the fundamental wave is proportional to the modulation degree M. Since M=U_s/U_c, the output voltage can be adjusted by adjusting the amplitude of the modulation wave U_s. When m is an even number, the m-th harmonic of the carrier does not exist; when m+n is an even number, the carrier and carrier The upper and lower side-frequency harmonics of the m-th harmonic of the m-th order do not exist.

At the same time, it can be seen from equations (5) and (6) that the dead zone has no effect on the fundamental wave of the two-level SPWM, but has an effect on the amplitude and phase of the carrier and carrier harmonics and their upper and lower frequencies. of. But when m and Δt are not large, this effect is also small. Therefore, when designing the inverter, it can be considered that 1+e^jm△t=2, or △t=0, that is, it is calculated based on no dead zone.