Torque Control of an Induction Motor by the Frequency Method
Abstract Generated abstract
This paper examines torque regulation of an induction motor under small speed variations, autonomous power supply current limits, and significant supply voltage fluctuations. It proposes a simplified frequency-control scheme that stabilizes the active component of stator current by regulating supply current through a two-loop voltage and current control system using a pulse-width modulated inverter and controlled-frequency generator. Analytical expressions are derived for slip and electromagnetic torque, and calculated characteristics are compared with experimental results for a representative motor. The study indicates that frequency control can produce either drooping or nearly rigid torque characteristics, with torque deviation remaining small under the tested supply-voltage variation and with operating modes selectable for slip dependent on or independent of speed.
Full Text
TORQUE CONTROL OF AN INDUCTION MOTOR BY THE FREQUENCY METHOD
V. P. Lyanzburg, G. P. Ivanova, B. P. Soustin
(Presented by the scientific and technical seminar of the Research Institute of AEM)
For some mechanisms, the main load is dynamic when the rotor speed changes within small limits. In addition, it becomes necessary to regulate (stabilize) the torque when limiting energy consumption from an autonomous power source (APS) and under significant voltage fluctuations. It is expedient to solve such a problem by frequency control.
Known methods are based either on regulation of frequency, voltage, and current [1, 2], or on stabilization of absolute slip [3]; in this case the electric-drive circuits are rather complex. To simplify the circuit and to form the necessary mechanical and energy characteristics, stabilization of the active component of the motor current by changing the frequency is considered.
If, as the inverter, a circuit with two-way conductivity is chosen when the motor windings are connected in delta, then the current from the APS is found from the expression
\[ I_{\mathrm{i}}=\frac{3\sqrt{3}}{\pi}\gamma I_{1a}+I_{1m}B_1(\gamma), \tag{1} \]
where
\[ I_{1a},\ I_{1m} \]
are the active component and the amplitude value of the phase current of the first harmonic.
The component \(I_{1m}B_1(\gamma)\) is caused by higher harmonics and is a function of the pulse duty factor
\[ \gamma=\frac{U_{\mathrm{av}}}{U_{\mathrm{aps}}}, \]
where \(U_{\mathrm{av}}\) is the voltage stabilized on average over a half-period \(U\). In [1] it is shown that for \(0 \leq \gamma \leq 1\) this component does not exceed the value \(0.1 I_{\mathrm{i}}\); consequently, with an error of no more than 10%, the active component of the stator current can be stabilized by regulating the APS current.
The control circuit in Fig. 1 has two control loops—voltage and current. The first includes an inverter I with a pulse-width modulator (PWM), and a phase-voltage rectifier with a filter (PVR); the second includes a current setpoint unit (CSU) and a controlled-frequency generator (CFG). The voltage at the output of I is set by the voltage setpoint \(U_y\), and the current by the setpoint \(I_y\).
The CFG is implemented on the basis of a magnetic-transistor multivibrator [4] in such a way that the APS current \(I_{\mathrm{i}}\) is applied to one input, and the setpoint of the stabilized current \(I_y\) to the other. The dependence of the output frequency on the controlling ampere-turns has the following nonlinear dependence (Fig. 2), which...
Fig. 1. Block diagram of the electric drive.
To exclude operation of the motor at a slip greater than the critical value. In the initial starting mode the frequency is determined by the position of point \(a\), which, as acceleration proceeds, moves to the right. During normal operation of the drive the frequency is determined by point \(b\) and, depending on the mismatch signal \(I_y - I_n\), moves upward or downward.
Fig. 2. Output characteristic of the VCO.
The SET converts the pulses of the output voltage of the VFN, having duty factor \(\gamma\), into the setting current \(I_y\).
The expression \(I_{1a}\) for the motor is written as [5]
\[ G_a = \frac{ \dfrac{\tau_1}{x_0}\left(\dfrac{r'_2}{\beta}\right)^2 - a\dfrac{r'_2}{\beta} + a^2 r_1 }{ \left(\tau_1^2 + a_1^2 a^2\right)\left(\dfrac{r'_2}{\beta}\right)^2 + 2a\dfrac{r'_2}{\beta} + a_2^2 r_1^2 + a^2\sigma^2 x_0^2 }, \tag{2} \]
where
\(r_1, r'_2, x_0, \sigma_1, \sigma_2, \sigma, a_1, a_2, \tau_1\) are motor parameters;
\(U_y\) is the stabilized inverter voltage.
For the case \(\gamma=\mathrm{const}\), and taking into account that \(\tau_1^2 \ll x_0^2\), \(a^2\alpha^2 \ll \tau_1^2\), the relative slip is determined as
\[ \beta = \pm \frac{2r'_2}{b_1 \pm \sqrt{b_1^2 - 4b_2}}, \tag{3} \]
Here
\[ b_1 = \frac{1 - 2r_1 G_a}{a_1^2 \cdot G_a \cdot \alpha}; \qquad b_2 = \frac{\sigma^2 x_0^2}{a_1^2} - \frac{a_2^2 r_1(1 - r_1G_a)}{a_1^2 G_a \cdot \alpha^2}. \]
For a motor with parameters \(r_1=0.807\,\Omega\); \(r'_2=0.862\,\Omega\); \(x_0=28\,\Omega\); \(\sigma=0.0972\); \(\sigma_1=0.044\); \(\sigma_2=0.051\), the dependences \(\beta(\alpha)\) are plotted in Fig. 3.
for the motor \(\beta>0\) and generator \(\beta<0\) modes at various \(G_a\), from which it is seen that \(\beta\) depends nonlinearly on the relative speed of rotation of the rotor \(\alpha_p=\alpha\pm\beta\).
Fig. 3. Absolute slip of the motor: 1, 4, 7, 10 —
\(G_a=0.05\); 2, 5, 8, 11 — \(G_a=0.1\); 3, 6, 9, 12 —
\(G_a=0.15\).
The electromagnetic torque of the motor is determined by the expression
\[ \frac{M}{M_0}= \frac{\pm G_a} {\dfrac{r_1 r_2^{1}}{\beta X_0}+a\pm \alpha^2\dfrac{r_1}{r_2^{1}}\beta}. \tag{4} \]
The torque characteristics calculated from (4) are given in Fig. 4 for \(G_a=0.05\) and the same motor parameters (curves 1, 2). It is seen that these characteristics are decreasing in nature and depend nonlinearly on the rotor speed. The experimental dependences (3, 4) differ only slightly from the calculated ones. Also shown here is the dependence \((M/M_0)=f(U_{\text{am}})\), curve 5, from which it is seen that when the supply voltage is varied from 23 to 33 V, the deviation of the torque from the prescribed value differs by 4.6%.
Fig. 4. Torque characteristics of the electric drive: 1, 3,
6 — motor mode; 2, 4, 7 — generator modes.
If it is possible to obtain information on the rotor speed (for example, by means of a tachogenerator), then the output voltage \(U\) is regulated in proportion to \(\alpha_p\), and the necessary correction of the current \(I_n\) when \(U_{\text{aip}}\) changes is carried out by the voltage-control circuit.
The curves calculated in this way, 7—9 and 10—12 (Fig. 3), show that the slip is constant and that its value changes when \(G_a\) changes. The motor torque (Fig. 4, curves 6 and 7) is practically independent of speed.
Conclusions
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Frequency-control circuits for an induction motor make it possible to obtain both drooping and rigid torque characteristics.
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Frequency-control systems make it possible to implement energetically most advantageous operating modes with slip dependent on, and independent of, speed.
REFERENCES
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V. A. Shubenko, I. Ya. Braslavskii, R. T. Shreiner. Induction Drive with Thyristor Control, Moscow, “Energiya,” 1967.
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D. A. Popov. On frequency starting of hydraulic motors. “Elektrichestvo,” 1968, No. 8.
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V. N. Brodovskii, E. S. Ivanov. Contactless electric drive with frequency-current control for closed-loop control systems. “Elektrichestvo,” 1969, No. 3.
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G. P. Ivanova. Magnetic-transistor amplifier. Author’s certificate No. 258391. Bulletin of Inventions, No. 1, 1970.
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V. L. Gruzov, Yu. A. Sabinin. Low-Power Induction Drives with Static Converters. Leningrad, “Energiya,” 1970.