VOLTAGE REGULATION OF AN AUTONOMOUS INVERTER OPERATING WITH AN INDUCTION MOTOR

B. P. Soustin

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

A method is presented for setting up equations for circuits with an inverter and an induction motor, and for analyzing these equations by the frequency method.

An inverter for supplying an induction motor is a complex nonlinear element, and when the equations of the regulation process are set up, the differential equations also turn out to be nonlinear. Analysis of nonlinear equations higher than second order is difficult and requires special mathematical apparatus. Therefore, equations are set up for the steady-state regime, and the input quantity is varied in such a way that the response of all elements to this variation lies in the linear zone. For a prescribed variation of the input quantity, the quality of the system and its stability are determined. It is assumed that a system that is stable “in the small” must also be stable “in the large.”

Fig. 1

In studying the regulation system, it is assumed that the inverter frequency does not depend on the load, the motor speed, or the inverter voltage. On this basis, the inverter–induction-motor system can be considered at an unchanged frequency. In the case where the frequency varies according to some law, it can be taken into account as an independent disturbing action with a known function. The control circuit for the inverter voltage is shown in Fig. 1, and it is the principal part of the speed regulator of the induction motor.

The alternating voltage of the inverter, INV, \(U_1\), proportional to the rectified voltage \(U_0\) with frequency \(f\), specified from the frequency generator FG through the inverter control system, ICS, is applied to the induction motor in series with the current and voltage regulator \(U_1 - I_1 r_1\). The regulator \(U_1 - I_1 r_1\) is the principal feedback element, and its voltage \(U\) depends on the load on the motor shaft.

… that is, on the rotor slip. The voltage \(U\) is compared with the required set value \(U_z\) and, through the rectifier-control circuit \(UV\), acts on the rectifier \(V\) so that the voltage on the induction motor corresponds to the specified control law. To formulate the control equations, the method of [1] is used.

The inverter voltage and the motor torque can be found by expansion in a Maclaurin series (both functions have no discontinuities). In this case, terms of degree higher than the first are discarded:

\[ \Delta U=\frac{\partial U}{\partial U_0}\Delta U_0+\frac{\partial U}{\partial \beta}\Delta\beta+\varphi_u(f), \tag{1} \]

\[ \Delta M=\frac{\partial M}{\partial U_0}\Delta U_0+\frac{\partial M}{\partial \beta}\Delta\beta+\varphi_M(f). \tag{2} \]

Here \(\beta=\dfrac{f_2}{f_{1N}}\) is the absolute-slip parameter; \(\varphi_u(f)\), \(\varphi_M(f)\) are functions allowing for the change in frequency \(f\); \(M\) is the electromagnetic torque of the motor.

The linearization coefficients are found from the T-shaped equivalent circuit of the inverter and the induction motor and have the following form:

\[ \kappa_1=\frac{\partial M}{\partial U_0} =2A_xU_0\frac{r_2'c'\alpha}{z^2\beta} \left[ 1+\frac{r_1^2(x_0-z_c)^2}{z_c^2x_0^2} \right]; \tag{3} \]

\[ \kappa_2=\frac{\partial M}{\partial\beta} =M\frac{\alpha}{\beta} \left( \frac{2r_2'c'}{r_1}-1 \right) -2A_x\frac{\alpha^2}{\beta^2} \]

\[ {}\times \frac{ r_1r_2'c'(x_0-z_c)(x_0z^2-x_0r_1^2+z_cr_1^2) }{ x_0^2z_c^2z^4 }; \tag{4} \]

\[ \kappa_3=\frac{\partial U_1}{\partial U_0} =A_y\frac{1}{\cos\varphi}; \tag{5} \]

\[ \kappa_4=\frac{\partial U_1}{\partial\beta} =A_yU_0'\frac{\alpha}{\beta} \frac{r_2'c'\sin\varphi}{x_0z_cz^2} \left(x_0r_1^2-2x_0z^2+z_cr_1^2\right). \tag{6} \]

Here \(r_2'\), \(r_1\), \(x_0\), \(z\) are motor parameters; \(f_{1N}\) is the nominal stator supply frequency; \(\alpha=\dfrac{f_1}{f_{1N}}\) is the relative frequency; \(A_x\), \(A_y\) are coefficients depending on the conversion circuit; \(c'\) is the relative value of the capacitance of the commutating capacitors connected to the motor terminals; \(z_c\) is the value of the reactance determined by the method of delivering reactive power to the load.

Figure 2 shows the linearization coefficient as a function of slip \(\beta\) for different capacitances of the commutating capacitors. The other coefficients have an analogous character of variation; moreover, the range in the region of slip variation from \(\beta=0\) to \(\beta=\beta_{\mathrm{cr}}\) is especially wide. In the zone \(\beta_{\mathrm{cr}}<\beta<\alpha\), the variation occurs practically along a straight line, since the parameters of the induction motor in this range change very little. The curve in Fig. 2 is plotted for \(\alpha\ne1\) for an A-41-4 motor and an inverter with limited commutating capacitors at different commutating capacitances.

The monotonic character of the change in the transfer coefficients as a function of slip and the entire character of the process in the zone of operating slips make it possible to consider the structural diagram of the regulator with essential nonlinearities. From the theory of automatic regula-

... it is known [2] that, for the case of an analogous character of the nonlinearities of the links, with an appropriate choice of the system gain coefficient and of the parameters of the stabilizing devices, it is possible to achieve stability as a whole.

To control the rectified voltage, a circuit with a saturable reactor and uncontrolled valves is used. As the preliminary amplification stage, a low-power magnetic amplifier with gain coefficient \(k_5\) and time constant \(T_1\) is taken.

Fig. 2

Then the transfer function of the rectifier is written as

\[ W_1(p)=\frac{U_0(p)}{U_y(p)}= \frac{k_0 k_5}{(1+T_0p)(1+T_1p)}, \tag{7} \]

where \(k_0\) and \(T_0\) are the gain coefficient and time constant of the saturable reactor; \(U_y\) is the voltage at the input of the magnetic amplifier, which is found from the relation

\[ U_y=U_z-U. \tag{8} \]

Here \(U_z\) is the command voltage or a voltage varying proportionally to the frequency according to the prescribed functions \(\varphi_n(t)\) and \(\varphi_M(t)\).

The block diagram with the \(U_1-I_1r_1\) regulator for the closed-loop regulation circuit is shown in Fig. 3.

The overall transfer function of the open-loop regulating system will have the form

\[ W_{\mathrm{tot}}(p)=\frac{U(p)}{U_y(p)} = \frac{a c k_0 k_1 k_5(1+T_d' p)} {(1+T_0p)(1+T_1p)(1+T_d p)}, \tag{9} \]

where

\[ a=\frac{k_1}{k_2}k_4+k_3, \]

\[ T_d'=\frac{T_d k_2 k_3}{k_1 k_4+k_2 k_3}; \]

\(c\) is a coefficient taking into account part of the inverter output voltage;

\(T_d\) is the electromechanical time constant of the motor.

To determine the parameters of the correcting circuits and their points of insertion, the optimal logarithmic characteristic is determined.

from the condition of the specified static accuracy at \(\beta_N\). A study of the system at slip greater than the critical value cannot be carried out by this method, since the open-loop system is unstable in this case. In the case

Fig. 3

where it is necessary to determine the acceleration time of the motor up to the slip determined by the load torque, another method must be used.

Since the transmission coefficient, when the slip changes from zero to the critical value, has a very large range, a system that is stable at nominal slip may prove oscillatory at slip close to zero. To eliminate instability of the system at no load, it is necessary to introduce rigid feedbacks, which, in addition, will increase the speed of response of the system. At slip close to nominal, this correction must be disconnected, which is accomplished by introducing a voltage cutoff that is set for each frequency. The magnitude of the feedback is chosen so that the gain coefficient, when the slip decreases below nominal, remains approximately constant. In this case the required correcting devices can be determined from the amplitude characteristic in the usual way.

Figure 4 shows amplitude characteristic 1 for an overall gain coefficient \(K = 1500\), the amplitude characteristic of the series correcting circuit 2 and of the feedback circuit 3, enclosing the magnetic amplifier with characteristic 1. The characteristic of the corrected system is II, and 4 is its phase characteristic.

To determine the value of the feedback coefficient with voltage cutoff, the condition is imposed that the voltage variation be monotonic. The process will be monotonic if the roots of the characteristic equation are negative. Thus, if rigid feedback encloses the magnetic amplifier, then

\[ W_1(p) = \frac{k_0 k_5} {(1 + k_0 k_{oc})\left(1 + \frac{T_0}{1 + k_0 k_{oc}}\,p\right)(1 + T_1 p)} . \tag{10} \]

The monotonicity condition for the closed-loop system is written as

\[ \frac{T_0 + (1 + k_{oc} k_0)T_1}{1 + k_{oc} k_0} \ge 4T_1T_0 \left( 1 + \frac{k_0 k_5}{1 + k_0 k_{oc}} \right). \tag{11} \]

Hence, the overall gain coefficient of the system, with the other parameters unchanged, is found from

\[ k = k_{0} k_{5} \leq \frac{\left[T_{0} - \left(1 + k_{0} k_{\mathrm{fb}}\right)T_{1}\right]^2} {4T_{0}T_{1}} . \tag{12} \]

Nonlinear feedback must also be used in the case where it is necessary to suppress inadmissible current surges through the inverter valves. In this case current cutoff must be applied.

Fig. 4

The speed of response of the inverter voltage-regulation system depends on the inertia of the elements of the control circuit. A mode of constant slip, or variation of the slip according to a predetermined law, can be achieved by setting the regulator \(U_i - I_1 r_1\).

References

1. B. P. Soustin. Fundamentals of constructing automatic-control circuits for frequency converters. Proceedings of LIAP, issue 42, 1963.

2. M. V. Meerov. Synthesis of structures of high-accuracy automatic-control systems. Fizmatgiz, Moscow, 1959.