STUDY OF RATIONAL METHODS FOR FREQUENCY CONTROL OF THE SPEED OF ASYNCHRONOUS MOTORS

I. F. Kalinin, V. I. Panteleev, B. P. Soustin

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

Rational control of an asynchronous machine consists in regulating the rotational speed of the motor rotor while ensuring the required quality of the transient process of starting, reversing, and braking. The advantages of controlled asynchronous electric drives with frequency control are well known. At the same time, the choice of a rational law of frequency control for an asynchronous machine is a subject of special consideration for each operating mode of a production mechanism. To this end, we shall write the expression for the electromagnetic torque in a form convenient for evaluating various cases of frequency control:

\[ M=\frac{m_1(c_1 f_{1N}\Phi)^2}{\omega_{1N}\left(\dfrac{r'_2}{\beta}+\dfrac{\beta}{r'_2}x_2'^2\right)} = \frac{m_1(c_1 f_{1N}I_2)^2}{\omega_{1N} f_2^2\left(\dfrac{r'_2}{\beta}+\dfrac{\beta}{r'_2}x_2'^2\right)}. \tag{1} \]

According to equation (1), a great variety of control methods can be distinguished. Thus, for the most effective operating mode of an asynchronous machine, with \(\Phi=\mathrm{const}\), the control parameters \(\Phi-I_1\), \(\Phi=f_2\), \(I_1-f_1\), \(I_1-f_2\), \(I_2-f_2\), and others may be used.

The main difficulty in implementing the mode \(\Phi=\mathrm{const}\) lies in measuring the flux. At present there are no simple and reliable methods for measuring the magnetic flux, as well as the emf and electromagnetic torque; therefore, systems constructed on the basis of the frequency-current method of control are of interest. It follows from expression (1) that if, in the control process, the values of the stator current \(I_1\) and the rotor current frequency \(f_2\) are kept constant, then the values of the magnetizing current and, consequently, the flux will also be constant. By assigning values of \(I_1\) and \(f_2\), it is possible to maintain a certain optimal (from the standpoint of losses) flux \(\Phi\) of the motor [1]. In studying the dynamics of the motor with allowance for electromagnetic processes, an analog computer of the MN-14 type was used.

The differential equations of the electromagnetic processes in an asynchronous motor in a rotating coordinate system have the following form:

\[ \left. \begin{aligned} U_c &= i_c R_c+\frac{d}{dt}(L_c i_c+L_m i_p)+j\omega_k(i_c L_c+i_p L_m),\\ U_p &= i_p R_p+\frac{d}{dt}(i_p L_p+i_c L_m)+j\Delta\omega(i_c L_m+i_p L_p),\\ M_g &= \operatorname{Re}\,[jL_m(i_p^{*}i_c)],\\ J\frac{d\omega}{dt} &= M_g-M_c. \end{aligned} \right\} \tag{2} \]

The simultaneous solution of the second and third equations of system (2), for \(\Delta\omega=\text{const.}\), \(i_c=\text{const.}\), gives the following expression for the torque:

\[ M= \frac{\Delta\omega L_m^2 i_c^2 R_p}{R_p^2+\Delta\omega^2 L_p^2} + \frac{\Delta\omega^2 L_m^2 i_c^2 L_p}{R_p^2+\Delta\omega^2 L_p^2} \sin\Delta\omega t\, \exp\left(-\frac{t}{T_p}\right) - \frac{\Delta\omega L_m^2 i_c^2 R_p}{R_p^2+\Delta\omega^2 L_p^2} \cos\Delta\omega t\, \exp\left(-\frac{t}{T_p}\right). \tag{3} \]

It is obvious that the electromagnetic torque is produced both by the parameter \((L_c^2)\) and by the parameter \((\Delta\omega)\), which makes it possible to obtain a considerable overload torque (when starting the motor) by controlling the parameter \((\Delta\omega)\), when a further increase of the stator current is inadvisable because of saturation of the machine. Of interest is the investigation of processes during the starting of an asynchronous motor with variable load torque. In this case, during starting with stable slip and stator-current parameters, the electromagnetic torque does not remain constant.

In a fast-acting drive (at maximum torque) this dependence is weakly manifested [3].

However, in a drive with limited power of the supply source, the motor does not develop maximum torque. In such a system, a change in the load torque has a noticeable effect on the dynamic characteristic of the motor. In the mode of limited stator current, the absolute slip must also be limited. Moreover, limiting the absolute slip is necessary both for reducing losses and for increasing speed of response.

Fig. 1

The optimum value of the absolute slip (Fig. 1) is found from the conditions for obtaining the maximum total torque of the motor, which is defined as the integral of expression (3):

\[ \operatorname{Max}(\Sigma M)= \int_0^\infty \frac{\Delta\omega_{\mathrm{opt}} L_m^2 i_c^2 R_p} {R_p^2+\Delta\omega_{\mathrm{opt}}^2 L_p^2}\,dt + \int_0^\infty \frac{\Delta\omega_{\mathrm{opt}} L_m^2 i_c^2 L_p} {R_p^2+L_p^2\Delta\omega_{\mathrm{opt}}^2} \sin t\Delta\omega_{\mathrm{opt}}\, \exp \]

\[ \times\left(-\frac{t}{T_p}\right)dt - \int_0^\infty \frac{L_m^2 i_c^2 R_p\Delta\omega_{\mathrm{opt}}} {R_p^2+L_p^2\Delta\omega_{\mathrm{opt}}^2} \cos t\Delta\omega_{\mathrm{opt}}\, \exp\left(-\frac{t}{T_p}\right)dt. \tag{4} \]

Specific optimization of electromagnetic processes under frequency-current control was carried out on a digital computer for a DAT-250-8 motor.

(see Appendix). The starting of the motor was investigated under constant, linear, and exponential time laws of variation of the voltage and frequency \(f_1\), for a constant load torque, and also for a load torque depending linearly and quadratically on the rotational speed (Fig. 2a–d).

From a comparison of these control laws it is evident that the shorter starting time and better formation of the transient process (with small overshoot) correspond to the exponential-in-time law of regulation of the voltage and supply frequency.

Fig. 2a, b, c, d

Fig. 2e

In a closed-loop control system for stator current and absolute slip, dynamic characteristics were obtained with an unchanged (specified) motor torque (Fig. 2e). The experimentally found (on an AVM) optimal value of the rotor frequency \((f_2)\) is 2.5 times less than the critical one with stator current limited (at the level of \(2 I_{\text{nom}}\)).

It should be borne in mind that the structural diagram of the motor model, when controlling stator current and rotor frequency, can be substantially simplified (Fig. 3b), since the current and slip may be regarded as independent control parameters. In this case, the study of transient processes in an asynchronous motor can be carried out using the equations describing the processes in the rotor and the equation for torque. Such a scheme makes it possible to investigate the dynamics

a)

\[ \begin{aligned} A08 &= \alpha'_s/2 = 0.148,\\ A02 &= k_r\alpha'_s/10 = 0.028,\\ A03 &= \alpha'_s = 0.295,\\ A04 &= k_r\alpha'_s/2 = 0.139,\\ A05 &= k_s\alpha'_r = 0.277,\\ A06 &= \alpha'_r/2 = 0.149,\\ A07 &= k_s\alpha'_r = 0.277,\\ A08 &= \alpha'_r/2 = 0.149,\\ A09 &= k_r/105X_s = 0.472,\\ A10 &= k_r/56X_s = 0.945,\\ A11 &= 16/J = 0.282,\\ A12 &= 16/J_\partial = 0.282,\\ A14 &= \frac{1}{25}\,\frac{1}{6X_s} = 0.2,\\ A15 &= \frac{1}{10}\,\frac{k_r}{6X_s} = 0.472,\\ A16 &= \frac{1}{25}\,\frac{1}{6X_s} = 0.2,\\ A17 &= \frac{1}{10}\,\frac{k_r}{6X_s} = 0.472,\\ A20 &= 0.05,\\ A21 &= 0.123. \end{aligned} \]

\[ \omega_0,\quad \omega,\quad \omega_k,\quad \Delta\omega,\quad M_c,\quad i_{\alpha s},\quad i_{\beta s},\quad \psi_{\alpha s},\quad \psi_{\beta s},\quad \psi_{\alpha r},\quad \psi_{\beta r} \]

\[ (\omega_k-\omega)\psi_{\beta r},\qquad -(\omega_k-\omega)\psi_{\alpha r} \]

\[ \psi_{\alpha r}/L_r,\qquad \psi_{\beta r}/L_r,\qquad L_m/L_r,\qquad r_s \]

\[ -i_{\alpha r},\qquad i_{\alpha r},\qquad -i_{\beta r},\qquad i_{\beta r} \]

\[ \text{BP1: }(\omega_k-\omega),\ \psi_{\beta r}\ \longrightarrow\ (\omega_k-\omega)\psi_{\beta r} \]

\[ \text{BP2: }-(\omega_k-\omega),\ \psi_{\alpha r}\ \longrightarrow\ -(\omega_k-\omega)\psi_{\alpha r} \]

\[ \text{BP3: }-i_{\alpha r},\ \psi_{\beta r}\ \longrightarrow\ -i_{\alpha r}\psi_{\beta r} \]

\[ \text{BP4: }i_{\beta r},\ \psi_{\alpha r}\ \longrightarrow\ i_{\beta r}\psi_{\alpha r} \]

\[ M,\qquad \frac{1}{J_\partial},\qquad +\omega_k,\qquad -\omega,\qquad -(\omega_k-\omega),\qquad (\omega_k-\omega) \]

closed-loop system for regulating the speed and torque of the motor on small analog computers.

A block diagram of a closed-loop motor-starting system with independent control of \(\Delta\omega\) and a specified stator current is shown in Fig. 4a. The formation of the stator frequency in the circuit is carried out by adding a frequency proportional to the rotational speed of the motor shaft \((f\omega)\) and a frequency \((f_\beta)\) proportional to the absolute slip. These operations are performed by a contactless selsyn. The power section of the frequency converter is implemented on the basis of a thyristor inverter supplied from a controlled rectifier. The closed-loop voltage-regulation system is constructed according to the principle of delayed stabilization of the specified motor current. The delayed feedback, ZOS, with current cutoff makes it possible to force the process of stator-current rise at the instant the motor starts moving \((U_y = 0\) in Fig. 4b) and to limit the current at a specified level.

Fig. 4

Appendix

The analog-computer equations for the DAT-250-8 motor have the form:

\[ -\frac{d\psi_{\alpha s}}{dt} = -U_{\alpha s} + \alpha_s' \psi_{\alpha s} - \omega_k \psi_{\beta s} - 0.277\psi_{\alpha r} \qquad \alpha_s' = 0.295 \]

\[ -\frac{d\psi_{\beta s}}{dt} = 0.295\psi_{\beta s} + \omega_k \psi_{\alpha s} - 0.277\psi_{\beta r} \]

\[ \frac{d\psi_{\alpha r}}{dt} = -0.299\psi_{\alpha r} + (\omega_k-\omega)\psi_{\beta r} + 0.277\psi_{\alpha s} \]

\[ \frac{d\psi_{\beta r}}{dt} = -0.299\psi_{\beta r} - (\omega_k-\omega)\psi_{\alpha r} + 0.277\psi_{\beta s} \]

\[ M_g = 4.72(\psi_{\beta s}\psi_{\alpha r} - \psi_{\alpha s}\psi_{\beta r}) \]

\[ \frac{d\omega}{dt} = 0.282(M_g-M_C) \]

\[ i_{\alpha s} = 5.02\psi_{\alpha s} - 4.72\psi_{\alpha r}, \qquad i_{\beta s} = 5.02\psi_{\beta s} - 4.72\psi_{\beta r}. \]

The scales of the variables were taken as

\[ m_{U_s}=1:4; \qquad m_M=1:16; \qquad m_I=1:5; \qquad m_t=1:1. \]

The model circuit of the closed-loop system for regulating the speed and torque of the IM is shown in Fig. 3a.

The optimal values of the absolute slip were determined by search modeling. It was determined that, at the optimal value of the absolute slip, the system is unstable (Fig. 5).

In this case the speed reaches its nominal value more quickly, but thereafter unstable oscillations of the torque and speed begin.

Fig. 5

With an increase in the steepness of the characteristic of the delayed current feedback, a slowing of torque formation was observed; starting was prolonged at a constant minimum voltage. The most acceptable steepness of the characteristic of the dead zone proved to be

\[ \frac{U_{\text{out}}}{U_{\text{in}}}=6. \]

The response of the system to a two- and threefold application of load at different motor rotational speeds was investigated. When load was applied at speeds \((0.1 \div 0.4)\omega_N\), a sharp increase in the electromagnetic torque was observed, which passes into unstable oscillations when the speed reaches approximately the nominal value. When load was applied at speeds of 0.6 and more of nominal, an insignificant increase in electromagnetic torque was observed and starting was somewhat slowed.

Conclusions

1. The investigations carried out showed that, when constant values of \(f_2\) and \(I_1\) are maintained, the electromagnetic torque of the motor in dynamic modes remains constant only for the case of an unchanged resistance torque (not depending on speed). With a resistance torque depending on speed, in the mode \(I_1=\text{const}\), in order to obtain a constant electromagnetic torque in dynamic modes, the rotor frequency varies according to a definite law when optimizing the system for various types of motor loads.

2. By automatic regulation of the current and rotor frequency, it is possible to obtain any mechanical characteristics, including characteristics inherent in direct-current motors.

3. For each value \(I_1=\text{const}\) there is an optimal absolute slip; when its value is maintained constant the system becomes unstable; therefore it is recommended to start the motor at values of absolute slip somewhat greater than the optimal one.

LITERATURE

1. I. F. Kalinin, B. P. Soustin. Optimal acceleration, in terms of speed of response, of an asynchronous hydraulic motor. Collection of Reports of the 5th Scientific-Technical Conference on Problems of Production Automation, TSU Publishing House, Tomsk, 1967.

2. M. M. Sokolov et al. Electromagnetic Transient Processes in an Asynchronous Drive. Energiya, Moscow, 1967.

3. V. I. Rabukha. Processes in an asynchronous motor under frequency control of its rotation speed according to a prescribed law. Proceedings of the USSR Academy of Sciences, “Power Engineering and Transport”, Leningrad, No. 2, 1966.

4. Yu. F. Bersenev. Problems of the dynamics of an asynchronous motor under partial speed control. Automation of Production Processes, Issue 5, Novosibirsk, 1967.

5. L. D. Gildebrand, N. I. Zenkin, V. M. Kirpichnikov. Selection of the structural diagram of an analog model of an induction motor. Proceedings of Higher Educational Institutions, “Electromechanics”, Moscow, No. 2, 1967.