INVERTER WITH FREQUENCY-PULSE MODULATION OF THE OUTPUT VOLTAGE

B. P. SOUSTIN, S. I. KOROLEV, A. N. BARKOVSKII

(Presented by the scientific and technical seminar of the Department of Electrical Equipment)

The main bottleneck at the present stage in the development of frequency-controlled AC drives is the inverter. Moreover, in the inverter itself it is necessary to solve the problems of regulating the output voltage, reliable commutation of valves, and generation of reactive power. The last two issues are solved by introducing capacitors into the circuit and by additionally complicating the inverter circuit. Despite the topicality of these issues, they are not considered in this article, although they await their researchers.

Usually, to obtain regulated frequency and voltage, transverter circuits are used, in which a controlled rectifier is used as the voltage regulator. There are circuits in which the DC source is uncontrolled, while regulation of the output voltage is carried out by controlling the relative duration of switching on the next thyristor [1]. However, the number of controlled valves here is no smaller than in a transverter. Circuit [2] has only 8 thyristors for obtaining three-phase controlled voltage; however, it requires an additional DC voltage source for commutation of the valves. In addition, in both circuits the time provided for restoration of the controlling properties of the thyristors varies as a function of the load current.

In the proposed inverter circuit supplied from an unregulated DC source, control of the output voltage is carried out by a pulse-width method. An inverter of this type is sometimes called a pulse inverter; it possesses practically no inertia with respect to frequency and voltage and reliably commutates current under any permissible loads. A distinctive feature of the circuit in Fig. 1, a is the presence of an interrupter, which at certain time intervals closes and opens the DC circuit. The operating frequency of the interrupter exceeds the inverter output frequency several times and may be either independent or rigidly linked to it.

Regulation of the output voltage in this circuit is carried out either by a width method or by a pulse-width method, which will hereafter be called frequency-pulse, since, in addition to changing the amplitude of the output voltage, the frequency also changes. With the width method of changing the output voltage and with coupled regulation, the frequency of the interrupter exceeds the frequency of the output volta-

...voltage for a bridge three-phase inverter by a factor of 6; with frequency-pulse regulation, by a factor of \(6n\), where \(n\) is the number of switch operations during \(2\pi/3\) of the output-voltage period. The latter method of regulation is of particular interest, since, in addition to other advantages, with an active-inductive load the introduction of special devices or feedbacks makes it possible to obtain a load-current waveform close to the prescribed one, in particular a sinusoidal one.

Fig. 1

In studying the circuit it is assumed that the resistance of a thyristor in the open state is zero and in the closed state is infinity, current commutation occurs instantaneously, and the resistance of the supply source is zero. For the interrupter, a circuit also using thyristors is adopted [3], Fig. 1, b. The circuit is studied only during \(T/6\) of the period, since during each other \(T/6\) the pattern is repeated, but for other load phases. This time interval is divided into four segments: \(t'_1\), with valves I, 1, 2 and diode D1 open, the capacitor \(C_k\), which had been charged to the voltage \(U_{\mathrm{p}}\) and with the polarity indicated in Fig. 1, is recharged through the inductance \(L\) and diode D1; the time \(t''_1\), during which a current determined by the parameters of the supply source and load flows through thyristors I, 1, and 2; the time \(t_2\), when thyristor II opens and capacitor \(C_k\) is recharged through the load, turning off all thyristors; and the pause time \(t_3\), during which there are no currents in the power elements of the inverter. With an active load, simple differential equations are solved for the time instant \(t_1\). The values of the voltages and currents in the circuit elements are:

\[ i_c(t)=i_L(t)=U_{\mathrm{p}}\omega_0 C\sin \omega_0 t, \]

\[ U_c(t)=-U_L(t)=U_{\mathrm{p}}\cos \omega_0 t, \]

\[ i_{\mathrm{p}}(t)=\frac{U_{\mathrm{p}}}{R}, \tag{1} \]

\[ U_{\mathrm{n}}(t)=U_{\mathrm{p}}, \]

where

\[ \omega_0=\frac{1}{\sqrt{LC}};\qquad R=2R_\phi. \]

A necessary condition for the next time segment is

\[ t_1=t'_1+t''_1\geq \frac{\pi}{\omega_0}=\pi\sqrt{LC}. \]

If necessary, in order to satisfy this inequality, the frequency \(\omega_0\) can be increased; however, according to [1], this will lead to an increase in the amplitude of the charging current. Moreover, the increase in frequency is limited by the duration of the control-current pulse of thyristor 1.

For

$$ \begin{aligned} i_c(t)&=i_L(i)=0,\\ U_c(t)&=-U_{\mathrm{p}},\\ U_L(t)&=0,\\ i_{\mathrm{n}}&=\frac{U_{\mathrm{p}}}{R},\\ U_{\mathrm{n}}&=U_{\mathrm{p}}. \end{aligned} \tag{2} $$

For the next time interval \(t_2\), it is assumed that thyristor 1 has turned off instantaneously; then the voltage will have the form

$$ U_c(t)=U_{\mathrm{p}}\left(1-2e^{-\frac{t}{RC}}\right), $$

$$ i_c(t)=\frac{2U_{\mathrm{p}}}{R}e^{-\frac{t}{t_c}} . \tag{3} $$

To determine the parameters \(C\) and \(L\), it is necessary to know the recovery time of the control properties of the thyristor, \(t_{\mathrm{v}}\). Specifying the time allotted for recovery of the control properties as \(t_0 \geq (2 \div 3)t_{\mathrm{v}}\) from [3], for \(U_c(t)=0\) and \(t=t_0\) one obtains

$$ C=\frac{t_0}{R\ln 2}, $$

$$ L=\frac{R\ln 2}{1.2\omega_0^2\max\, t_0}. \tag{4} $$

Amplitude values of the currents:
a) through thyristor 1

$$ I_{mI}=\frac{U_{\mathrm{p}}}{R}(1+\omega_0RC), \tag{5} $$

b) through the remaining thyristors and the load

$$ I_{mII}=I_{mIII}=I_{m1}=\frac{2U_{\mathrm{p}}}{R}. \tag{6} $$

Average values of the currents

$$ I_{\mathrm{av}\,I}=\frac{6U_{\mathrm{p}}}{TR}(t_1+2RC), $$

$$ I_{\mathrm{av}\,II}=\frac{6U_{\mathrm{p}}C}{T}\left(1-e^{-\frac{t_2}{RC}}\right), $$

$$ I_{\mathrm{av}1}=\frac{2U_{\mathrm{p}}}{TR} \left[t_1+RC\left(3-e^{-\frac{t_2}{RC}}\right)\right], \tag{7} $$

$$ I_{\mathrm{av}\,D_1}=\frac{12U_{\mathrm{p}}C}{T}. $$

When operating into an active-inductive load, the processes in the chopper remain unchanged; however, in this case the load-current expression has, at \(t=t_1\), the form

$$ i_{\mathrm{n}}(t)=\frac{U_{\mathrm{p}}}{R}\left(1-e^{-\frac{t}{T}}\right), \tag{8} $$

where

\[ T=\frac{L}{R}, \qquad L_{\phi 1}+L_{\phi 2}=L. \]

If the load inductance is such that during the time \(t_2\) the capacitor recharge current may be regarded as constant, then the voltage on the capacitor varies linearly and will have the following form:

\[ U_c(t)=\frac{I_{mu}t}{C}-U_{\mathrm{p}}. \tag{9} \]

Taking into account expression [3],

\[ C=\frac{I_{mu}}{2U_{\mathrm{p}}}\,t_0, \]

since it is assumed that the capacitor had been charged to \(U_{\mathrm{p}}\). Since the interrupter closed instantaneously, i.e., \(t_2 \ll t_1\), the energy stored in the load inductances must be delivered to some circuit other than the interrupter. In Fig. 1 the reactive energy of the load is transferred to an additional source \(U_d\). If there is only one DC source, it is necessary to connect decoupling inductances, as in ordinary inverters.

With frequency-pulse voltage regulation, the choice of the carrier frequency plays a major role, since when an asynchronous motor is supplied, the losses in it caused by the presence of higher harmonics, and the commutation losses, increase. In [4] it is recommended to choose, from the condition of equality of the 1st, 5th, and 7th harmonics under continuous and frequency-pulse regulation; it is assumed that this condition is satisfied when

Fig. 2

\[ \frac{t_0}{t}>12. \]

If any series of pulses of equal duration in Fig. 2, \(a\), is expanded in a Fourier series, then the output voltage is

\[ U=\sum_{n=1}^{\infty} b_n \sin \omega t, \tag{10} \]

where

\[ b_n=\frac{4U_{\mathrm{p}}}{\pi n}\sum_{m=1}^{k}\sin n\omega \alpha_m \sin n\omega \frac{\tau_m}{2}. \]

Here \(n\) is the harmonic number, \(m\) is the pulse number, \(\tau_m\) is the width of the \(m\)-th pulse, and \(\alpha_m\) is the angular coordinate of the next pulse.

With coupled control and variation of \(f\), \(f_0\) changes proportionally, so that \(\dfrac{f}{f_0}=\mathrm{const}\), and the content of higher harmonics remains unchanged. When \(K=\mathrm{var}\), the resulting curve shape may contain pulsating harmonics and a constant component. This fact gives preference to coupled control. In addition, with coupled control, by varying \(\tau\) in a definite way as a function of \(\alpha\), suppression of the higher harmonics is possible.

Thus, if the \(m\)-th pulse has a relative duration proportional to the integral of the sinusoid (Fig. 2, b), that is,

\[ \tau_m=\frac{\tau_{\max}}{2}\int\limits_{\frac{m\pi}{K}}^{\frac{(m+1)\pi}{K}}\sin\Theta\,d\Theta =\frac{\tau_{\max}}{2}\left[\cos\frac{(m+1)\pi}{K}-\cos\frac{m\pi}{K}\right], \tag{11} \]

then in the output voltage of the inverter under study there will be harmonics of order \((i_K+1)\), where \(i\) is an integer. The losses in a motor supplied from an inverter with PWM, with such a control system, practically do not differ from the losses when supplied from a source with a sinusoidal output voltage. This method of reducing higher harmonics without special suppression circuits gives a substantial effect even with a single-phase output and is very promising for the development of a frequency-controlled drive.

REFERENCES

1. Silicon controlled valves—thyristors. Technical handbook. Edited by V. A. Labuntsov and A. F. Sviridov. Moscow—Leningrad, “Energiya,” 1964.

2. N. W. Mapham. The Classification of SCR inverter circuits. “IEEE Internat. Convent. Rec.”, 1964, 12, No. 4.

3. L. Abraham, K. Heumann, F. Koppelmann. Wechselrichter fur Drehzahlsteuerung von Käfigläufermotoren. AEG—Mitt., 1964, 54, N 1—2, S 89—106.

4. T. A. Glazenko. Pulse regulation of the rotational speed of an asynchronous motor. Moscow, “Energiya,” 1965.