HARMONIC ANALYSIS OF THE MULTILEVEL WAVEFORM OF THE OUTPUT VOLTAGE OF A CONVERTER

A. M. AZAROV, V. R. BIRYUKOV, O. I. KHASAEV

(Presented by the joint scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

The output voltage of the simplest semiconductor voltage converters has a rectangular waveform, although it is often required that it be sinusoidal. One of the promising ways of solving this problem is the formation of a multilevel voltage.

Harmonic analysis of a stepped voltage has been considered in a number of works [1, 2, 3]. In these works the parameters of its steps are determined from the condition of eliminating the lower harmonics, which is explained by the possibility of reducing the weight of the converter output filter.

However, stepped approximation of a sinusoid makes it possible to obtain, without a filter, a voltage that satisfies many consumers. In this case the principal index characterizing its quality is the harmonic coefficient.

Let us consider the synthesis of a stepped voltage of the form shown in Fig. 1a from the condition of minimizing this coefficient. The desired parameters are the reference angles \(\alpha_1, \alpha_2, \ldots, \alpha_m\) and the amplitudes of the steps \(a_1, a_2, \ldots, a_m\). The harmonic coefficient of a voltage of any form is equal to

\[ K_{\Gamma}=\sqrt{\frac{U_{\text{д}}^{2}}{U_{\text{д}1}^{2}}-1}, \tag{1} \]

where \(U_{\text{д}}\) is the rms value of the voltage;
\(U_{\text{д}1}\) is the rms value of the voltage of the first harmonic.

The rms value of the voltage of a stepped waveform is equal to:

\[ U_{\text{д}}= \sqrt{ \frac{2}{\pi} \sum_{m=1}^{p} a_m\left(\alpha_{m+1}-\alpha_m\right) }, \tag{2} \]

where \(p\) is the number of voltage steps in a quarter period.

The amplitude of the \(n\)-th harmonic of the stepped voltage is determined by expanding it in a Fourier series:

\[ U_{nm}= \frac{4}{\pi n} \sum_{m=1}^{2p-1} a_m \sin n\frac{\alpha_{m+1}+\alpha_m}{2} \cdot \sin n\frac{\alpha_{m+1}-\alpha_m}{2}. \tag{3} \]

For convenience in the subsequent exposition, let us denote

\[ A_m= a_m \sin\frac{\alpha_{m+1}+\alpha_m}{2} \cdot \sin\frac{\alpha_{m+1}-\alpha_m}{2}. \]

Then the rms value of the voltage of the first harmonic is equal to

\[ U_{\mathrm{d}1}=\frac{4\sqrt{2}}{\pi}\sum_{m=1}^{p} A_m . \tag{4} \]

Substituting (2) and (4) into (1), we determine the dependence of the harmonic coefficient on the amplitudes and sampling angles of the steps

\[ K_{\mathrm{g}}= \frac{ \sqrt{ \frac{\pi}{16}\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m) -\sum_{m=1}^{p} A_m^2 -2\sum_{m=1}^{p-1} A_m \sum_{m=1}^{p} A_{m+1} } }{ \sum_{m=1}^{p} A_m }. \tag{5} \]

Another index characterizing the quality of the output voltage is the form factor, equal to

\[ K_{\mathrm{f}}= \frac{ \sqrt{ \frac{\pi}{2}\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m) } }{ \sum_{m=1}^{p} a_m(\alpha_{m+1}-\alpha_m) }. \tag{6} \]

To determine the parameters of the stepped voltage from the condition of a minimum of the harmonic coefficient, we find the partial derivatives of this coefficient with respect to all amplitudes and sampling angles of the steps.

The derivatives of the harmonic coefficient with respect to the step amplitudes are equal to:

\[ \begin{aligned} \frac{\partial K_{\mathrm{g}}}{\partial a_1} &=(a_2-a_1)\sum_{m=1}^{p} A_m -\frac{A_1}{a_1}\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m);\\ \frac{\partial K_{\mathrm{g}}}{\partial a_2} &=(a_3-a_2)\sum_{m=1}^{p} A_m -\frac{A_2}{a_2}\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m);\\ &\cdots\\ \frac{\partial K_{\mathrm{g}}}{\partial a_p} &=(\alpha_{p+1}-\alpha_p)\sum_{m=1}^{p} A_m -\frac{A_p}{a_p}\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m). \end{aligned} \tag{7} \]

The derivatives of the harmonic coefficient with respect to the step sampling angles are equal to:

\[ \begin{aligned} \frac{\partial K_{\mathrm{g}}}{\partial \alpha_1} &=(a_0+a_1)\sum_{m=1}^{p} A_m +2\sin\alpha_1\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m);\\ \frac{\partial K_{\mathrm{g}}}{\partial \alpha_2} &=(a_1+a_2)\sum_{m=1}^{p} A_m +2\sin\alpha_2\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m);\\ &\cdots\\ \frac{\partial K_{\mathrm{g}}}{\partial \alpha_p} &=(a_{p-1}+a_p)\sum_{m=1}^{p} A_m +2\sin\alpha_p\sum_{m=1}^{p} a_m^2(\alpha_{m+1}-\alpha_m), \end{aligned} \tag{8} \]

where

\[ \alpha_{p+1}=\frac{\pi}{2}. \]

Equating the obtained derivatives to zero, we find a system of equations, the total number of which is \(2p\),

\[ \left. \begin{aligned} a_1(a_2-a_1)&=C(\cos\alpha_1-\cos\alpha_2);\\ a_2(a_3-a_2)&=C(\cos\alpha_2-\cos\alpha_3);\\ &\cdots\\ a_p(a_{p+1}-a_p)&=C(\cos\alpha_p-\cos\alpha_{p+1}). \end{aligned} \right\} \tag{9} \]

\[ \left. \begin{aligned} a_1 &= 2C\sin \alpha_1;\\ a_2+a_1 &= 2C\sin \alpha_2;\\ &\ldots\\ a_p+a_{p-1} &= 2C\sin \alpha_p, \end{aligned} \right\} \tag{10} \]

where

\[ C=\frac{2\sqrt{2}\,U_{\mathrm{d}}^{2}}{U_{\mathrm{d}1}}. \]

Eliminating \(a_1, a_2, \ldots, a_p\) from (9) and (10), we obtain a new system of equations in which the number of unknowns is equal to the number of steps, \(p\):

\[ \begin{aligned} 2(\alpha_2-\alpha_1)\sin\alpha_1-\cos\alpha_1&=0;\\ 2(\alpha_3-\alpha_2)(\sin\alpha_2-\sin\alpha_1)-\cos\alpha_2+\cos\alpha_3&=0;\\ &\ldots\\ 2(\alpha_{p+1}-\alpha_p)\bigl[\sin\alpha_p-\sin\alpha_{p-1}+\sin\alpha_{p-2} - \ldots +(-1)^{p+1}\sin\alpha_1\bigr]-\cos\alpha_p+\cos\alpha_{p+1}&=0. \end{aligned} \tag{11} \]

Solving this system makes it possible to determine the sampling angles of all the steps. Determination of the parameters of the curve without the zero step (Fig. 1b) is carried out by (11) with \(a_1=0\).

Fig. 1

As follows from (11), the sampling angles cannot be found by analytical methods, since they are linked by a system of transcendental equations. This system was solved on a digital computer for different numbers of steps. The results of the solutions for \(p=1\div6\) are presented in Table 1 with a zero step and in Table 2 without a zero step.

Using the angles found, from (9) and (10) one obtains the relative value of the amplitude of any voltage step:

\[ \frac{a_m}{a_p} = -\frac{\left(\dfrac{\pi}{2}-\alpha_p\right)(\cos\alpha_m-\cos\alpha_{m+1})} {\cos\alpha_p(\alpha_{m+1}-\alpha_m)}. \tag{12} \]

Expression (12) is also valid in the case where the sampling angles are specified and have an arbitrary relation to one another.

Let us consider a special case of formula (12) for the four most characteristic forms of stepped voltage: with a zero step and without it, with step durations equal over a half-period or over a quarter-period. Such durations are adopted when finding the voltage parameters from the condition of eliminating the lower harmonics. In this case, from (12) we obtain one general expression for determining the amplitudes of the steps of the indicated forms

\[ \frac{a_m}{a_p}=U_{m0}\sin\left(\frac{\pi}{k}m-\gamma\right). \]

where \(U_{m0}=1\) or \(U_{m0}=\sec \frac{\pi}{2k}\) for voltages with equal step durations over a half-period or over a quarter-period, respectively,

\[ \gamma = \begin{cases} \dfrac{\pi}{2k}, & \text{for voltages without a zero step,}\\[6pt] 0, & \text{for voltages with a zero step.} \end{cases} \]

Comparison of (12) with the analogous formula found from the condition of eliminating the lower harmonics [2, 3] shows that they coincide completely. Thus, the condition considered is the most general one, from which, as a special case, expressions can be obtained for determining the step parameters under which elimination of the lower harmonics is ensured.

Using the parameters found, Figs. 2, 3, and 4 plot, respectively, the dependences of the harmonic coefficient, the amplitude of the first harmonic, and the form factor as functions of the number of voltage steps with a zero step (curve 1) and without it (curve 2). For comparison, analogous dependences are given for voltages whose step parameters are determined from the condition of eliminating the lower harmonics (curves 3, 4). It can be seen that a voltage with optimal parameters has a smaller value of the harmonic coefficient, a larger value of the useful component, and that the form factor approaches its limiting value, equal to 1.11, more rapidly.

Fig. 2

Fig. 3

Fig. 4

Figure 5 shows a diagram of the amplitudes of the higher harmonics \(U_{mn}\) for three-step voltages without a zero step, with the step parameters found (solid lines) and known (dashed lines). It follows from it that the amplitudes of like-numbered harmonics of the two voltages are different, and for the former they are always smaller.

Usually the number of steps is counted from zero, at which the voltage level is not always defined (voltage without a zero step). It is more correct to take as the starting point the instant \(t = \pi/4\), at whi-

Fig. 5

in which the voltage always has a maximum value. In this case, the number of steps in a half-period for a voltage with a zero step is always equal to the even number \(K=2p\), and without it to the odd number \(K=2p-1\).

Figure 6 shows the dependences \(K_{\mathrm{r}}=f(k)\) for a voltage with optimal parameters (curve 1) and for a voltage whose step parameters are determined from the condition of eliminating the lower harmonics (curve—

Fig. 6

Table 1

Table 2

2). The graphs show that the curves of \(K_{\mathrm{g}}\) for the voltage with a zero step (light dots) and without it (dark dots) coincide in both cases.

Conclusions

1. In the general case, the parameters of the stepped voltage determined from the conditions of eliminating the lower harmonics and of minimizing the harmonic coefficient turn out to be different; however, for multiple angles of counting off the voltage steps, these parameters are obtained as identical, i.e., the condition for eliminating the lower harmonics is a special case of the condition for the minimum harmonic coefficient.

2. The harmonic coefficient of the output voltage with parameters determined from the condition of the minimum value of this coefficient is better than for a voltage with parameters determined from the condition of eliminating the lower harmonics by 2.1% for a single-step voltage and by 1% for a six-step voltage.

3. The curves with a zero step and without it correspond to voltages with an even and an odd number of steps.

References

1. I. A. Voitovich, O. I. Khasaev. Transistor converters with a multistep form of the output voltage. Avtomatika i telemekhanika, 1967, No. 8.

2. S. S. Tonkal. On the harmonic analysis of a stepped function approximating a sinusoidal voltage. Problemy tekhnicheskoi elektrodinamiki, 1970, No. 24.

3. V. G. Konstantinov. Multiphase converters on transistors. Moscow, Energiya, 1972.