DETERMINATION OF THE OUTPUT VOLTAGES OF

\(m\)-PHASE INVERTERS BY MEANS OF SWITCHING FUNCTIONS

V. I. IVANCHURA, B. P. SOUSTIN

(Presented by the scientific-technical seminar of the NII AEM)

The points of connection of the load lines of an \(m\)-phase inverter \(1, 2, \ldots, m\) are assumed to be symmetrical in resistance with respect to the poles of the source \(E_n(t)\) (in the ideal case the resistances are zero), and the source \(E_n(t)\) is of infinite power. The instantaneous values of the phase voltages are denoted by \(U_1, U_2, \ldots, U_m\), the line voltages by \(U_{12}, U_{23}, \ldots, U_{m1}\), the line load currents by \(i_1, i_2, \ldots, i_m\), and the potentials of points \(1, 2, \ldots, m\) with respect to the zero of the supply source by \(V_1, V_2, \ldots, V_m\). Then the phase and line voltages can be expressed through the potentials

\[ [U_{ij}] = [V_i] - [V_j], \tag{1} \]

\[ \sum_{i=1}^{m} U_i = 0, \qquad U_i = V_i - \frac{1}{m}\sum_{j=1}^{m} V_j, \tag{2} \]

where \(i, j \in (1, 2, \ldots, m)\).

Obviously, if the points \(i, j\) are connected to the positive pole of the supply source, then their potentials will be \(+0.5E_n(t)\); if to the negative pole, \(-0.5E_n(t)\). The nature of the change in the potentials of the points can be expressed by the product

\[ V_i(t) = 0.5\,E_n(t)\cdot F_i(t), \tag{3} \]

where \(F_i(t)\) is the switching function of point \(i\). In the case of connection to the positive pole of the supply source, \(F_i(t)=+1\); if to the negative pole, \(F_i(t)=-1\); in the case when both arms of the inverter are open, \(F_i(t)=0\).

Solving (1) and (2) jointly, we find the representation of the inverter voltages through the supply-source voltage and the switching functions:

\[ U_{ij}(t)=0.5\,E_n(t)[F_i(t)-F_j(t)], \tag{4} \]

\[ U_i(t)=0.5\,E_n(t)\left[F_i(t)-\frac{1}{m}\sum_{j=1}^{m}F_j(t)\right]. \tag{5} \]

An important characteristic of an \(m\)-phase system is the expression of the generalized voltage vector

\[ \overline{U}(t)=\frac{2}{m}\sum_{i=1}^{m} a^{i-1}U_i(t)=0.5\,E_{\mathrm{p}}(t)\overline{F}(t), \tag{6} \]

where \(a=e^{j\frac{2\pi}{m}}\), and

\[ \overline{F}(t)=\frac{2}{m}\sum_{i=1}^{m}\overline{a}^{\,i-1}F_i(t) \]

is the generalized switching vector.

Such a representation makes it possible to determine any characteristics of the inverter voltage through the characteristics of the power supply and the switching functions, and to find the response of any linear system under the action of an \(m\)-phase voltage as a combination of simple actions.

The switching function is determined by the nature of control of the inverter switches and, in the general case, may be arbitrary. With symmetrical control of the inverter phases, the switching functions form a symmetrical \(m\)-phase system. Thus, for inverters with 180° control, the switching function may be either \(+1\) or \(-1\); if the conduction angle, together with the angle \(\varepsilon\) determined by the reserve of reactive energy in the load, is less than 180°, the load current is discontinuous and the switching function takes the values \(\pm 1\) and \(0\). With pulse-width modulation of the inverter output voltage and equal duration of the phase pulses, or with pulse-width control according to a sinusoidal law, the switching function takes the values \(\pm 1\). The shift between adjacent potential functions under symmetrical \(m\)-phase control is

\[ \pm \frac{2\pi}{m}. \]

Fig. 1

The relations between the spectra of the potential function and the spectra of the phase and line voltages are found from expressions (1), (2). The \(n\)-th harmonic of the potential function \(V_i\) is written as

\[ V_{ni}=V_n\sin\left[n\left(\nu+\frac{2\pi}{m}i\right)+\Psi_n\right] = V_n \operatorname{Im} e^{jn\nu+\Psi_n+\frac{2\pi n}{m}i}, \tag{7} \]

the \(n\)-th harmonic of the line voltage is

\[ U_{nij}=V_n \operatorname{Im} e^{jn\nu+\Psi_n} - V_n \operatorname{Im} e^{jn\nu+\Psi_n(i-i)+\frac{2\pi n}{m}}. \tag{8} \]

Here \((j-i)<m\) is an integer.

After transformations, (8) may be written as

\[ U_{nij}=V_n k_n(j-i)\sin\left[n\nu+\Psi_n+\Psi_n(j-i)\right], \tag{9} \]

where

\[ k_n(j-i)=2\sin (j-i)\frac{\pi n}{m}; \qquad \Psi_n(j-i)=\frac{\pi}{2}-\frac{(j-i)\pi n}{m}, \]

and the \(n\)-th harmonic of the phase voltage satisfies equation (1).

\[ U_{ni}=V_{ni}-\frac{1}{m}\sum_{j=1}^{m} V_{nj}. \tag{10} \]

From the analysis of (10) it is seen that \(U_{ni}=V_{ni}\) for \(n\ne im\), and \(U_{ni}=0\) for \(n=im\).

Thus, harmonics of the potential function are present in the spectrum of the line and phase voltages. In the spectrum of the phase voltage they do not change; in the spectrum of the line voltage they change their amplitude and phase. Harmonics that are multiples of the number of phases are absent from the line and phase voltages.

The voltage spectrum of an \(m\)-phase inverter can be divided into a positive sequence \(n=mk+1\) and a negative sequence \(n=mk-1\), where \(k=0, 1, 2, 3\ldots\). Harmonics of the zero sequence \(n=mk\) are absent; however, harmonics forming various oscillatory sequences are present. In Fig. 1,a, as an example, the positive (negative) and two oscillatory sequences (Fig. 1,b) are shown for the case \(m=5\).

In connection with the new approach to the design of electric motors supplied from inverters, it is reasonable to recommend that they be made with a large number of phases, since the nearest harmonic of the negative sequence will be \(n=m-1\). In addition, the frequency of torque pulsation of the motor under nonsinusoidal voltage is a multiple of \(2m\), while the amplitude of the pulsations is inversely proportional to \(m\).