Multilevel DC/AC Inverters
9.2 Proof of Greatest Number of Output Voltage Levels
9.2.3 Modulation Strategies for THMI
9.2.3.1 Step Modulation Strategy
Figure 9.3 shows a general quarter-wave symmetric stepped voltage waveform synthesized by a THMI where E indicates the unit voltage of DC source. Consider that ς is the number of switching angles in a quarter wave of van and σ is the num-ber of positive and negative levels of van. In the step modulation strategy,
ς = σ (9.17) By applying Fourier series analysis, the amplitude of any odd jth harmonic of van can be expressed as where j is an odd harmonic order and θi is the ith switching angle. The ampli-tudes of all even harmonics are zero. According to Figure 9.3, θ1 to θς must satisfy
< θ < θ < < θ < π
Step modulation strategy of THMI.
where MR is the relative modulation index and is expressed as
where |van|1 is the amplitude of fundamental component of the output volt-age of the inverter.
The switching angles controlled by step modulation technique are derived from Equation (9.20). Up to (ς − 1) harmonic contents can be removed from the voltage waveform and the amplitude of fundamental component can be controlled.
The equation sets (Equation 9.20) from which the switching angles can be derived are nonlinear and transcendental. For example, in a two-HB THMI, with the step modulation technique, the equations set is expressed as Equation (9.22) when the relative modulation index is 0.83. The correct solu-tion must satisfy the inequality shown in Equasolu-tion (9.19).
The constrained optimization approach can be used to solve the non-linear and transcendental equations sets. Each equation is regarded as an equational constraint. However, the computational problems of constrained optimization do not converge easily. Since in the actual electric system there are always mismatches and parameter tolerances, lower-order harmonics will be small but not exactly zero. This gives rise to the idea of transform-ing the constraint optimization model to a nonconstraint one. The noncon-straint optimization is expected to have a better convergence property.
The target function of the new scheme of optimization without equational constraints can be written as
∑ ∑
∑
p1 ∼ pς are penalty factors. The penalty factors were selected as= −
p i 4
i 2 1 (9.24)
Thus, the penalty factors put more weight on elimination of lower-order har-monics. Function fmincon in the MATLAB® optimization toolbox was used to solve this minimization problem.
The two-HB THMI can synthesize a nine-level output voltage. Figures 9.4 and 9.5 show the typical synthesized waveform of the phase leg voltage, line-to-line voltage waveform and their frequency spectrums, as MR is equal to 0.83. The switching angles are 0.1478, 0.3232, 0.5738, and 0.9970. According to Equation (9.20), the fifth, seventh, and eleventh harmonics of the phase leg voltage can be eliminated in the two-HB THMI as shown in Figure 9.4. The THD of phase leg voltage is 9.66%. The triple-order harmonic components do not exist in the to-line voltage as shown in Figure 9.5. The THD of line-to-line voltage is 5.91%.
According to Equation (9.20), all switching angles must satisfy the con-straint (Equation 9.19). If switching angles do not satisfy the concon-straint, this scheme no longer exists. The theoretical maximum amplitude of the funda-mental component is 4ςE/π, which occurs as θ1–θh equal zero. Because of the internal restriction of switching angles, the relative modulation index has upper and lower limitations. The limitation of the relative modulation index can be explained using Figures 9.6 and 9.7.
As shown in Figure 9.6, as the relative modulation index is less than a certain value, denoted by MR (min), θς approaches π/2 and the limitation of
4
0
–4
0 0.005 0.01
Time (seconds) 0.015 0.02
Voltage (E)
Multiple of Fundamental Frequency 29
Synthesized phase leg voltage waveform and frequency spectrum of a two-HB THMI with step modulation technique.
minimum modulation index occurs. Similarly, when the relative modulation index is greater than MR (max), θ1 approaches zero and the limitation of the maximum modulation index occurs as shown in Figure 9.7.
For a THMI with h HBs, the maximum number of levels of the phase leg voltage is m, which equals 3h. The maximum number of the positive/negative phase leg voltage levels is σmax, which equals (m−1)/2. As mentioned above, the relative modulation index MR has limitations. To extend it to the smaller
8
0
–8
0 0.005 0.01
Time (seconds) 0.015 0.02
Voltage (E)
1
0.4 0.2 0.8
0 1
Multiple of Fundamental Frequency 29 27 25 23 21 19 17 15 13 11 9 7 5 3
Voltage (E)
0.6
FIGURE 9.5
Synthesized line-to-line voltage waveform and frequency spectrum of a two-HB THMI with step modulation technique.
σE (σ – 1)E 2E E van
θ1 θ2 θζ
π/2
π
2π ωt
FIGURE 9.6
Limitation to the minimum MR in the step modulation.
ranges of the modulation index, the inverter will output fewer voltage levels.
Consequently, the number of positive/negative voltage levels that the inverter outputs, σ, is smaller than the maximum number of the positive/negative lev-els, σmax. In the step modulation strategy, the number of switching angles in the quarter wave of van, ς, equals σ. The definition of the relative modulation index, MR, is based on σ as shown in Equation (9.21). This definition is easily included in Equation (9.20) to express the nonlinear transcendental equation sets that are used to calculate the switching angles. In practice, the modula-tion index, M, is used. M is based on the σmax and can be expressed as
= πσ
M v
E
| | 4
an1 max
(9.25)
The relationship between MR and M can be expressed as
= σσ M
MR max
(9.26)
In the two-HB THMI, according to Equation (9.20), the maximum MR is cal-culated as 0.86 and the minimum MR is 0.55 as the levels of output volt-age are nine in number. The range of M is also from 0.55 to 0.86 with the nine-level output voltage. To extend the lower modulation index, fewer out-put voltage levels are synthesized. The range of MR is 0.46–0.83 when the number of output voltage levels is seven. According to Equation (9.26), the range of M is 0.34–0.62 when there are seven output voltage levels. Thus, the modulation range is extended to 0.34 by decreasing levels of output voltage.
Table 9.6 shows the relative modulation index and the modulation index with different output voltage levels in the two-HB THMI. First, the mini-mum and maximini-mum MR is calculated by the optimization method. Second, the minimum and maximum M is calculated by Equation (9.26). It is prefer-able to use more output voltage levels. The last column of Tprefer-able 9.6 shows the arrangement of M with different output voltage levels. In addition, the upper
σE (σ – 1)E 2E E van
θ1θ2 θζ π/2 π
2π ωt
FIGURE 9.7
Limitation to the maximum MR in the step modulation
limit of M can reach 0.94 independent of the elimination of the 11th harmonic as shown in the last row of Table 9.6.
The scheme of switching angles of the two-HB THMI is shown in Figure 9.8.
When the modulation index reaches the lower limit, such as 0.34, the third switching angle is close to π/2, which corroborates Figure 9.6. When the mod-ulation index reaches the maximum value 0.86 or 0.94, the first angle is close to zero, which corroborates Figure 9.7.