1. Inverter nonlinearity

Here we will introduce the inverter nonlinearity in two parts: the first part is the nonlinearity caused by the tube voltage drop, and the second part is the nonlinearity caused by the dead time.

1.1 Nonlinearity of tube pressure drop

In the inverter, various devices do not have ideal characteristics. The nonlinearity caused by the tube voltage drop, that is, the conduction voltage drop of the switching device and the conduction voltage drop of the reverse parallel diode, causes the output voltage to be inconsistent with the command voltage. Take the A-phase bridge arm as an example for analysis. Figure 1 shows the situation when the A-phase bridge arm in the three-phase inverter outputs current to the motor winding, and the current is defined as the positive direction at this time.

When the current of phase A is positive, there are two situations for the conduction of the bridge arm, namely, the upper tube of the bridge arm of phase A is turned on (Sa = 1) and the upper tube is turned off (Sa = 0), as shown in Figure 1 (a) and Figure 1 (b) respectively. In Figure 1 (a), the upper tube is turned on (Sa = 1), at this time, the current flows from the positive end of the power supply through the switch tube to the winding, and the voltage drop through the switch tube S1 is Vce.

Therefore, when the current is positive and the upper tube is turned on, the voltage at point a is Va0 = Vdc /2- Vce. Similarly, in Figure 1 (b), the upper tube is turned off (Sa = 0), and the current flows from the negative end of the power supply through the anti-parallel diode to the winding, and the voltage drop when passing through the diode next to S1 is - Vd. Therefore, when the upper tube is turned off, the voltage at point a is Va0 = - Vdc /2- Vd.

Fig.1 Nonlinear analysis when phase A current is positive

Figure 2 Nonlinear analysis when phase A current is negative

Figure 2 shows the situation when the current is negative. Consistent with the theory when the current is positive, it can be concluded that when Sa = 1, Va0 = Vdc /2 + Vd; when Sa = 0, Va0 = - Vdc /2 + Vce.

Summarizing the above four situations, we can get the formula for the A-phase terminal voltage shown in Table 1.

Table 1 Calculation of inverter nonlinear terminal voltage

Since the voltage drop of the power device is due to the conduction characteristics and a part of the resistive voltage drop, the voltage drop of the switch device and the diode can usually be expressed as

(Formula 1)

Among them, Vce0 and Vd0 are the threshold voltages of the power device, and rce and rd are the equivalent resistances of the power device.

Therefore, the voltage at the phase A terminal can be expressed as (Equation 2)

It should be noted that only the nonlinearity caused by the tube pressure drop is considered in the formula in Table 1. In the next section, the nonlinearity caused by the dead time will be discussed.

1.2 Dead time nonlinearity

Figure 3 shows the relationship between the reference on time and the actual on time during PWM modulation. Figure 3 (a) shows the reference on time calculated by the reference voltage. Since the switch tube is not an ideal switch device, there is a delay in the on and off actions. In order to prevent the upper and lower tubes of the same bridge arm from being turned on at the same time, a dead time needs to be set to avoid the upper and lower tubes of the same bridge arm being in the on or off process at the same time, as shown in Figure 3 (b). In order to avoid the upper tube being turned on when the lower tube is not completely turned off or the lower tube being turned on when the upper tube is not completely turned off, a dead time Tdead is set to avoid the two switch tubes being turned on at the same time.

Assuming that the on-time of the upper tube of the bridge arm of phase A should be Ta_ref, after the dead time adjustment, the on-time becomes Ta_ref - Tdead. In addition, when the PWM signal of Figure 3 (b) is applied to the switch tube, as mentioned earlier, since the on-and-off process of the switch tube is not ideal, the on-and-off delay time is Ton and Toff respectively.

Figure 3 Switching timing considering dead time

Finally, combined with Figure 3 (c) and Figure 1 (a), when the current is positive, the actual conduction time of the A phase upper tube is

(Formula 3)

Similarly, combined with Figure 3 (c) and Figure 2 (b), when the current is negative, the actual conduction time of the lower tube of phase A is

(Formula 4)

In summary,

(Formula 5)

(Formula 6)

In addition, the terminal voltage calculation formula for phase B and phase C is

(Formula 7)

(Formula 8)

1.3 Phase voltage calculation

The calculation of nonlinear three-phase terminal voltage considering tube voltage drop and dead time was analyzed above. As shown in the figure, the terminal voltage is derived from the voltage from point abc to point 0. However, in actual analysis, what we need is the phase voltage, that is, the voltage from point abc to point N respectively. Therefore, the analysis results in the previous section need to be further derived.

Figure 4 Schematic diagram of three-phase inverter

According to Figure 4,

(Formula 9)

and

(Formula 10)

Combining (Equation 9) and (Equation 10), the calculation formula for the three-phase voltage is:

(Formula 11)

Furthermore, by combining (Equation 6), (Equation 7) and (Equation 11), the expression of the nonlinear phase voltage is

(Formula 12)

From (Equation 11), we can also get the expression of phase voltage without considering nonlinearity (ideal phase voltage):

(Formula 13)

Taking phase A as an example, by combining (Equation 12) and (13), we can get the nonlinear error caused by the voltage drop of the power device tube and the dead zone effect on the phase voltage:

(Formula 14)

Where (Formula 15)

Similarly, phases B and C can also be derived, and the expression of the three-phase nonlinear voltage error is summarized as follows:

(Formula 16)

By performing Park transformation on (Equation 16), the error formula of dq axis voltage can be obtained.

(Formula 17)

Before we start the next section, it is important to explain that the measurement error of the DC bus voltage is not considered here. If this is taken into account, the formula will become more complicated, but adding this part of the analysis does not help much with the main purpose of this article. Therefore, in order to simplify the analysis, it is assumed that the bus voltage is accurate.

2. Nonlinear compensation

Through analysis and deduction, the expression of nonlinear phase voltage and its error, dq axis error voltage (Equation 17) was obtained in the previous chapter. According to Equation 17, the schematic diagram of dq axis error voltage can be obtained as shown in Figure 5. In one electrical cycle, the dq axis error voltage fluctuates for 6 cycles. That is to say, the frequency of the dq axis error voltage is 6 times the fundamental frequency. The 6-fold frequency fluctuation in the rotating coordinate system will cause 5/7-fold fluctuation in the three-phase stationary coordinate system. In other words, the 5/7 harmonic in the three-phase voltage is manifested as a 6-fold frequency fluctuation in the rotating coordinate system. If there are readers who do not understand this conclusion, they can get the answer in the multi-level rotating coordinate system theory in the next chapter.

Figure 5 dq axis error voltage

As shown in Figure 6 (a), the waveform of the nonlinear phase voltage is shown. The effect of the inverter nonlinearity on the phase voltage waveform can also be clearly seen from the phase voltage waveform. At the same time, the phase voltage in Figure 6 (a) is subjected to harmonic analysis, as shown in Figure 6 (b). It can be seen that the 5/7th harmonic is the main factor in the phase voltage. Therefore, the harmonic brought by the inverter nonlinearity in the phase current is also 5/7th.

Figure 6 Phase voltage waveform considering inverter nonlinearity (a) Phase voltage (b) Harmonic analysis

Figure 7. Comparison of dq axis nonlinear voltage

Figure 7 shows the phase current waveform obtained by simulation after considering nonlinear factors and its corresponding harmonic analysis. It can be seen that the two are quite close.

Figure 8 Phase current and its harmonic analysis

As shown in Figure 8, the phase current waveform and harmonic analysis after considering the nonlinear factors of the inverter, it can be seen that the 5th/7th harmonics are the main harmonic components. The THD is 14.33%.

The most direct method for low-frequency harmonics caused by inverter nonlinearity is to use the error voltage derived above to compensate the command voltage. Figure 9 is a block diagram of nonlinear voltage compensation. Based on the calculated nonlinear dq axis voltage, compensation is performed after the reference voltage output by the current loop.

Figure 9 Nonlinear voltage compensation

Figure 10 shows the phase current and its harmonic analysis before and after voltage compensation. Nonlinear voltage compensation is performed at the red dotted line. Before compensation, the THD of the phase current is 14.33%, and after compensation, the THD drops to 6.63%. It can be seen that nonlinear voltage compensation has a significant current harmonic suppression effect.

Figure 10 Current waveform and harmonic analysis before and after nonlinear voltage compensation

When the harmonic content is low, the inverter nonlinear compensation can suppress the harmonics. However, when the harmonic content is high, the calculated nonlinear voltage error jitter increases due to the jitter of the phase current at the zero crossing point. This part of the jitter will have a certain impact on the current harmonics in the loop, and the purpose of compensating the nonlinear voltage cannot be achieved.