14

Best Switching Angles to Obtain Lowest THD for Multilevel DC/AC Inverters

The lowest THD of multilevel DC/AC inverters from 3-level to 81-level is derived in this chapter. This kind of multilevel DC/AC inverter can be applied in renewable energy systems, electrical vehicles, and other industrial applications [1,2].

14.1  Introduction

Multilevel DC/AC inverters have various structures and many advantages. Unfortunately, most existing inverters are unable to produce good output AC waveforms because of their poor total harmonic distortion (THD) because each level switching angle is not carefully arranged. In order to gain good power quality (PQ), we have to carefully investigate the switching angle arrangement to obtain the lowest THD.

14.2  Methods for Determination of Switching Angle

Switching angle is the moment of the level change. Referring to Figure 14.1, for an m-level (m is an odd number) waveform in the period 0°–90°, there are 2(m − 1) switching angles to be determined. We define them as α₁, α₂, … α_m−2, α_m−₁ by the time sequence. Since the sine wave is a symmetrical wave, the negative half-cycle is centrally symmetrical to its positive half cycle; the wave of the second quarter period is mirror-symmetrical to its positive half-cycle; and the wave of the second quarter period is mirror-symmetrical to the wave of its first quadrant period. We define the switching angles in the first quadrant period (i.e., 0°–90°) as main switching angles.

FIGURE 14.1
Output voltage waveform for multilevel inverter.

14.2.1  Main Switching Angles

For an m-level (m is an odd number) waveform, there are (m − 1)/2 main switching angles. Referring to Figure 14.1, we have the following relations:

1. Main switching angles in the first quadrant (i.e., 0°–90°): α₁, α₂, …, α_(m−1)/2.

2. The switching angles in the second quadrant (i.e., 90°-180°): α_(m+1)/2 = π − α_(m−1)/2, …, α_(m-1) = π − α₁.

3. The switching angles in the third quadrant (i.e., 180°-270°): = π - α α_m = π + α₁, …, α_3(m-1)/2 = π + α_(m-1)/2.

4. The switching angles in the fourth quadrant (i.e., 270°-360°): α_(3m-1)/2 = 2π − α_(m-1)/2, …, α_2(m-1) = 2π − α₁.

From the analysis, we need only determine the main switching angles. The other switching angles can be derived from the main switching angles in the first quadrant (i.e., 0°–90°):

${\text{α}}_{\text{1}},{\text{α}}_{\text{2}},\mathrm{\dots },{\text{α}}_{\text{(m}-\text{1)/2}}.$

14.2.2  Equal-Phase (EP) Method

The equal-phase (EP) method is derived from the simplest idea, to averagely distribute the switching angles in the range 0−π. The main switching angles are determined by the formula:

${\text{α}}_i=i\frac{180°}{m}wherei=1,2,\mathrm{\dots }\frac{m-1}{2}$

(14.1)

14.2.3  Half-Equal-Phase (HEP) Method

Since the multilevel waveform determined by the EP method looks very narrow, like a triangle waveform, another approach called the half-equal-phase (HEP) method to arrange the main switching angles can obtain a wider and better output waveform. The main switching angles are in the range 0–π/2, which are determined by the formula:

${\text{α}}_i=i\frac{90°}{\frac{m+1}{2}}=i\frac{180°\pm }{m+1}wherei=1,2,\mathrm{\dots }\frac{m-1}{2}$

(14.2)

14.2.4  Half-Height (HH) Method

The above two methods are able to arrange the main switching angles in a simple manner, but the output waveform is not a sine wave. According to the sine function, we established a new half-height (HH) method to determine the main switching angles. The idea is that when the function value increases to the half-height of the level, the switch angle is set and thus a better output waveform obtained. The main switching angles are determined by the formula:

${\text{α}}_i={\sin }^{-1}\left[\left(i-\frac{1}{2}\right)\frac{2}{m-1}\right]={\sin }^{-1}\left(\frac{2i-1}{m-1}\right)wherei=1,2,\mathrm{\dots }\frac{m-1}{2}$

(14.3)

14.2.5  Feed-Forward (FF) Method

Using the above three methods, we can see that there are wider gaps between the positive half-cycle and the negative half-cycle. In order to reduce the gaps, we established another new method, the feed-forward (FF) method, to determine the main switching angles by the formula:

${\text{α}}_i=\frac{1}{2}{\sin }^{-1}\left[\left(i-\frac{1}{2}\right)\frac{2}{m-1}\right]=\frac{1}{2}{\sin }^{-1}\left(\frac{2i-1}{m-1}\right)wherei=1,2,\mathrm{\dots }\frac{m-1}{2}$

(14.4)

14.2.6  Comparison of Methods in Each Level

For m = 3, we have only one main switching angle α₁. We compare them in Table 14.1.

For m = 5, we have two main switching angles α₁ and α₂ in Table 14.2.

For m = 7, we have three main switching angles α₁, α₂, and α₃ in Table 14.3.

For m = 9, we have four main switching angles α₁, α₂, α₃, and α₄ in Table 14.4.

For m = 11, we have five main switching angles α₁, α₂, α₃, α₄ and α₅ in Table 14.5.

For m = 13, we have six main switching angles α₁, α₂, α₃, α₄, α₅ and α₆ in Table 14.6.

TABLE 14.1
Comparison of Switching Angle α₁ of the Methods (m = 3)

Methods	Switching Angle α1(°)	THD
EP	60°	80.17%
HEP	45°	48.19%
HH	30°	30.9%
FF	30°	31.76%

TABLE 14.2
Comparison of Switching Angles of the Methods (m = 5)

Methods	Switching Angle α1(°)	Switching Angle α2(°)	THD
EP	36°	72°	42.77%
HEP	30°	60°	31.78%
HH	14.48°	49°	21.14%
FF	7.24°	24.5°	24.86%

TABLE 14.3
Comparison of Switching Angles of the Methods (m = 7)

Methods	α1(°)	α2(°)	α3(°)	THD
EP	25.71°	51.43°	77.14°	30.98%
HEP	22.50	45.00	67.50	31.29%
HH	9.60	30.00	56.44	11.70%
FF	4.80	15.00	28.22	22.17%

TABLE 14.4
Comparison of Switching Angles of the Methods (m = 9)

TABLE 14.5
Comparison of Switching Angle of the Methods (m = 11)

TABLE 14.6
Comparison of Switching Angles of the Methods (m = 13)

For m = 15, we have seven main switching angles α₁, α₂, α₃, α₄, α₅, α₆ and α₇ in Table 14.7.

High number of m can be listed accordingly. The different levels for each method are given in the next section.

14.2.7  Comparison of Levels for Each Method

We compare the various levels for each method to find out which method is better to obtain lower THD (Tables 14.8 to 14.11).

14.2.8  THDs of Different Methods

Comparisons of THDs for the different methods are listed in Table 14.12.

TABLE 14.7
Comparison of Switching Angles of the Methods (m = 15)

TABLE 14.8
THD of Different Levels Using EP Method with m = 35

TABLE 14.9
THD of Different Levels Using HEP Method with m = 35

TABLE 14.10
THD of Different Levels Using HH Method with m = 35

TABLE 14.11
THD of Different Levels Using FFM Method

TABLE 14.12
THD of Different Methods

14.3  Best Switching Angles

From Tables 14.1 to 14.12, we can see that THD is reduced when the number of levels (m) of the inverter increases, and the HH method is better than the other three methods. Hence, a higher level of inverter will be considered to produce output with less harmonic content.

14.3.1  Using MATLAB to Obtain Best Switching Angles

We use MATLAB software to search for the best switching angles in this section, and results (for m = 81) are shown in Table 14.13.

14.3.2  Analysis of Results of Best Switching Angles Calculation

THD values obtained using best switching angles from Table 14.13 are listed below.

From Table 14.14, the lowest THD value of a multilevel inverter with level equal to or below 81 is 0.99%. It can be easily observed that the differences between each adjacent level decrease gradually as the number of levels increase. For example, the THD value drops by 12.54% when the number of level increases from 3 to 5. However, the THD value drops by only 0.02% when the number of level increases from 79 to 81. By applying the MATLAB graph fitting tool, the relationship between lowest THD and number of levels of an inverter can be shown as the following equation:

TABLE 14.13
Best Switching Angles

TABLE 14.14
THD Value Obtained Using Best Switching Angles

Number of Levels	THD	Difference from Lower Levels
3	28.96%
5	16.42%	12.54%
7	11.53%	4.89%
9	8.90%	2.63%
11	7.26%	1.64%
13	6.13%	1.13%
15	5.31%	0.82%
17	4.68%	0.63%
19	4.19%	0.49%
21	3.79%	0.40%
23	3.46%	0.33%
25	3.18%	0.28%
27	2.95%	0.23%
29	2.74%	0.21%
31	2.57%	0.17%
33	2.41%	0.16%
35	2.28%	0.13%
37	2.15%	0.13%
39	2.04%	0.11%
41	1.94%	0.10%
43	1.85%	0.09%
45	1.77%	0.08%
47	1.70%	0.07%
49	1.63%	0.07%
51	1.56%	0.07%
53	1.51%	0.05%
55	1.45%	0.06%
57	1.40%	0.05%
59	1.35%	0.05%
61	1.31%	0.04%
63	1.27%	0.04%
65	1.23%	0.04%
67	1.19%	0.04%
69	1.16%	0.03%
71	1.13%	0.03%
73	1.10%	0.03%
75	1.07%	0.03%
77	1.04%	0.03%
79	1.01%	0.03%
81	0.99%	0.02%

FIGURE 14.2
THD versus m.

$TH{D}_{Lowest}=72.42e^{-0.4503m}+11.86e^{-0.05273m}$

(14.5)

where m is the level number of the inverter. The corresponding figure for THD versus m is shown in Figure 14.2.

14.3.3  Output Voltage Waveform for Multilevel Inverters

To verify our design, simulation results of the output voltage waveform for multilevel inverters with levels from 7 to 35 are shown in this section. See Figures 14.3 to 14.17.

FIGURE 14.3
Output voltage waveform of seven-level inverter.

FIGURE 14.4
Output voltage waveform of nine-level inverter.

FIGURE 14.5
Output voltage waveform of 11-level inverter.

FIGURE 14.6
Output voltage waveform of 13-level inverter.

FIGURE 14.7
Output voltage waveform of 15-level inverter.

FIGURE 14.8
Output voltage waveform of 17-level inverter.

FIGURE 14.9
Output voltage waveform of 19-level inverter.

FIGURE 14.10
Output voltage waveform of 21-level inverter.

FIGURE 14.11
Output voltage waveform of 23-level inverter.

FIGURE 14.12
Output voltage waveform of 25-level inverter.

FIGURE 14.13
Output voltage waveform of 27-level inverter.

FIGURE 14.14
Output Voltage Waveform of 29-level Inverter

FIGURE 14.15
Output voltage waveform of 31-level inverter.

FIGURE 14.16
Output voltage waveform of 33-level inverter.

FIGURE 14.17
Output voltage waveform of 35-level inverter.

References

1.  Luo F. L. and Ye H. 2010. Power Electronics: Advanced Conversion Technologies. Boca Raton, FL: Taylor & Francis.

2  Fang Lin Luo. 2012. Best Switching Angles to Obtain Lowest THD for Multilevel DC/AC Inverters. NTU Technical Report.