ELECTRICAL ASYMMETRY OF STATOR CIRCUITS OF BRAKE ELECTRIC MOTORS

V. S. Grinberg, E. M. Guselnikov

Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering

In various branches of industry, asynchronous electric motors with built-in spring brakes are used for the rapid stopping of mechanisms. The release of the shaft of an asynchronous motor when it is connected to the supply network is accomplished here by means of a special electromagnet fed through a rectifier. A number of control circuits (Fig. 1) for brake electric motors provide for the presence of common electrical circuits in the supply system of the asynchronous motor and the releasing electromagnet [1 et al.].

Fig. 1. Circuit arrangements used for connecting brake motors

The use of such control circuits causes a disturbance of the symmetry of the connection of the stator circuits of the asynchronous motor. If the power consumed by the releasing electromagnet is significantly less than the power of the asynchronous motor, then it is possible to neglect

under the influence of the electromagnet and to consider the electromagnetic processes in the brake electric motor as in an ordinary asynchronous motor with a symmetrical winding. However, if the indicated powers become commensurate, then such neglect may lead to significant errors.

Below, several cases are considered of an analytical investigation of the influence of electrical asymmetry of the stator circuits, for the purpose of a qualitative and quantitative assessment of this influence on the electromagnetic processes in an asynchronous brake electric motor. In the investigation we neglect higher harmonics of the spatial distribution of the m.m.f. and of the magnetic fields, saturation of the steel, and adopt other generally accepted assumptions [2 et al.].

If, in addition, it is assumed that the rectifier valves are ideal, then for the first half-period of variation of the supply voltage after switching on the brake electric motor, the circuits shown in Fig. 2 will be equivalent to the corresponding circuits in Fig. 1. At a supply frequency \(f = 50\) Hz, the operating time of the releasing electromagnet amounts to several periods of the supply; therefore, in the first half-period of the supply the armature of the electromagnet is stationary, and the inductance of the electromagnet is a constant quantity.

Fig. 2. Equivalent circuits taking asymmetry into account

As is known, in order to get rid of periodic coefficients, the voltage-equilibrium equations of an asynchronous motor are transformed to orthogonal axes. If a symmetrical asynchronous motor is considered, then any system of axes may be adopted. In the case where the asymmetry of the stator circuits is taken into account, the transformed equations with constant coefficients can be obtained only in the \(\alpha, \beta\) axes, fixed relative to the stator.

For a symmetrical asynchronous motor, to the terminals of whose phase windings a symmetrical system of line voltages is applied,

\[ \begin{aligned} U_{ab} &= U \cdot \sqrt{3}\cdot \sin(\omega_0 \cdot t + \alpha_0 + 120^\circ);\\ U_{bc} &= U \cdot \sqrt{3}\cdot \sin(\omega_0 \cdot t + \alpha_0);\\ U_{ca} &= U \cdot \sqrt{3}\cdot \sin(\omega_0 \cdot t + \alpha_0 - 120^\circ), \end{aligned} \tag{1} \]

the known voltage-balance equations along the \(\alpha,\beta\) axes have the form

\[ \left. \begin{aligned} U\cdot \cos(\omega_0\cdot t+\alpha_0) &= r_1\cdot i_\alpha+\frac{d\psi_\alpha}{dt};\\ U\cdot \sin(\omega_0\cdot t+\alpha_0) &= r_1\cdot i_\beta+\frac{d\psi_\beta}{dt}; \end{aligned} \right\} \tag{2} \]

\[ \left. \begin{aligned} 0 &= r_2\cdot i_{2\alpha}+\frac{d\psi_{2\alpha}}{dt}+\omega\cdot \psi_{2\beta};\\ 0 &= r_2\cdot i_{2\beta}+\frac{d\psi_{2\beta}}{dt}-\omega\cdot \psi_{2\alpha}, \end{aligned} \right\} \tag{3} \]

where

\[ \left. \begin{aligned} \psi_\alpha &= (L_1-M_1)\cdot i_\alpha+M_{12}\cdot i_{2\alpha};& \psi_{2\alpha} &= L_2\cdot i_{2\alpha}+\frac{3}{2}\cdot M_{12}\cdot i_\alpha;\\ \psi_\beta &= (L_1-M_1)\cdot i_\beta+M_{12}\cdot i_{2\beta};& \psi_{2\beta} &= L_2\cdot i_{2\beta}+\frac{3}{2}\cdot M_{12}\cdot i_\beta; \end{aligned} \right\} \tag{4} \]

\[ \left. \begin{aligned} L_1-M_1 &= \frac{x_s}{\omega_0};& r_2 &= \frac{r'_2}{m_U\cdot m_I};\\ L_2 &= \frac{L'_2}{m_U\cdot m_I} = \frac{x_r}{\omega_0\cdot m_U\cdot m_I};& M_{12} &= \frac{M'_{12}}{m_I} = \frac{x_m}{\omega_0\cdot m_I}. \end{aligned} \right\} \tag{5} \]

In equations (1)—(5) the following notation has been used:

\(U\) — amplitude of the phase voltage of a symmetrical asynchronous motor;
\(\omega_0\) — angular frequency of the supply network;
\(\omega\) — angular speed of rotor rotation;
\(\alpha_0\) — initial phase of the voltage;
\(r_1\) and \(r_2\) — active resistances of the stator phase winding and of the equivalent rotor winding;
\(i_\alpha, i_\beta, i_{2\alpha}\) and \(i_{2\beta}\) — currents in the equivalent stator and rotor windings along the \(\alpha\) and \(\beta\) axes;
\(L_1\) and \(L_2\) — self-inductance of the stator phase and of the equivalent rotor winding;
\(M_1\) and \(M_{12}\) — respectively, the mutual inductance between any two phase windings of a symmetrical asynchronous motor and the maximum value of the mutual inductance between any stator phase winding and any equivalent rotor winding;
\(x_s\) and \(x_r\) — synchronous inductive reactances of the stator and rotor of the asynchronous motor;
\(x_m\) — inductive reactance of mutual induction between the stator and the rotor;
\(m_U\) and \(m_I\) — voltage and current reduction coefficients.

The primed quantities in (5) are referred to the stator winding.

In the general case the electromagnetic torque of an asynchronous motor is expressed through the currents along the \(\alpha,\beta\) axes in the following form:

\[ M=\frac{3}{2}M_{12}\cdot p\cdot (i_{2\alpha}\cdot i_\beta-i_\alpha\cdot i_{2\beta}), \tag{6} \]

and the stator losses \(P_{M1}\) and rotor losses \(P_{M2}\) are written as follows:

\[ \left. \begin{aligned} P_{M1} &= r_1\cdot\left[\frac{3}{2}\cdot(i_\alpha^2+i_\beta^2)+3\cdot i_0^2\right];\\ P_{M2} &= r_2\cdot(i_{2\alpha}^2+i_{2\beta}^2), \end{aligned} \right\} \tag{7} \]

where

\[ i_0=\frac{i_a+i_b+i_c}{3} \]

is the zero component of the stator current;

\(i_a, i_b, i_c\) are the instantaneous values of the phase currents of the asynchronous motor.

At a constant rotor angular velocity \(\omega\), the voltage-balance equations of type (2)—(4) are solved independently of the drive-motion equation. The expressions for the currents \(i_\alpha, i_\beta, i_{2\alpha}\), and \(i_{2\beta}\), obtained from the solution of these equations, are substituted into (6) and (7) to determine the electromagnetic torque and losses. In the first expression (7) for the stator losses there is an additional term due to the zero component of the current \(i_0\). The latter is determined from the specific conditions of asymmetry in the connection of the stator circuits.

For the circuit shown in Fig. 1,a, the zero component \(i_0\) is equal to zero. As in the case of a symmetrical asynchronous motor, equations with constant coefficients can be obtained here. However, in the case under consideration, the voltage-balance equations of the stator circuits in the \(\alpha,\beta\) axes differ from the analogous equations (2) and have the form

\[ \left. \begin{aligned} U\cdot \cos(\omega_0\cdot t+\alpha_0) &= \left(r_1+\frac{2}{3}\cdot r_{T1}\right)\cdot i_\alpha + \frac{d\left(\psi_\alpha+\frac{2}{3}\cdot \psi_{T1}\right)}{dt}; \\[1.2em] U\cdot \sin(\omega_0\cdot t+\alpha_0) &= r_1\cdot i_\beta+\frac{d\psi_\beta}{dt}; \\[1.2em] 0 &= (r_{T2}+R)\cdot i+\frac{d(\psi_{T2}+\psi)}{dt}, \end{aligned} \right\} \tag{8} \]

where

\[ \psi_{T1}=L_{T1}\cdot i_\alpha+M\cdot i;\quad \psi_{T2}=L_{T2}\cdot i+M\cdot i_\alpha;\quad \psi=L\cdot i; \tag{9} \]

\(r_{T1}, r_{T2}, R\) are the active resistances, respectively, of the primary and secondary windings of the transformer and of the electromagnet coil;

\(L_{T1}, L_{T2}\), and \(L\) are their self-inductances;

\(M\) is the mutual inductance between the transformer windings;

\(i\) is the current at the input of the rectifier.

Equations (8) and (3), taking (4) and (9) into account, constitute a system of differential voltage-balance equations for the braking asynchronous motor connected according to the circuit shown in Fig. 1,a. This system contains 5 equations with five unknown currents flowing both in the electric motor itself and in the electromagnet. That is, the motor currents \(i_\alpha, i_\beta, i_{2\alpha}, i_{2\beta}\) and the electromagnet current \(i\) depend on one another. Consequently, the electromagnetic torque and the losses in the stator and in the rotor of the asynchronous motor are determined not only by the parameters of the electric motor itself, but also by those of the electromagnet. Conversely, the electromagnetic processes in the electromagnet are interrelated with the electromagnetic processes occurring in the electric motor. The characteristic equation of the system (8), (4), (3), (9) is of fifth order; therefore, obtaining exact analytical expressions for the transient quantities in this case is fundamentally impossible [3 et al.]. The solution of such a system can be obtained by numerical methods, which is most easily implemented when using computing machines.

In the circuit shown in Fig. 1,b, the zero component of the current \(i_0\) is not equal to zero:

\[ i_0=\frac{i_a+i_b+i_c}{3}=\frac{i}{3}. \tag{10} \]

The currents \(i_\alpha,\ i_\beta,\ i_{2\alpha}\) and \(i_{2\beta}\) in this case are determined from the known system of equations (2)—(4) and are substituted into the expressions for the torque (6) and losses (7). In this case the torque of the brake electric motor connected according to the circuit in Fig. 1,b does not differ from the torque of an asynchronous motor with a symmetrical connection, since the zero component of the current \(i_0\) does not affect the resultant magnetic field of the electric motor or the torque developed by it. However, this component increases the losses \(P_{M1}\) in the stator winding.

The current \(i\) in (10) is determined from an additional equation that accounts for the asymmetry of the circuit in Fig. 1,b:

\[ -\,U\cdot \cos(\omega_0\cdot t+\alpha_0) = \left(R+\frac{r_1}{3}\right)\cdot i + \left(L+\frac{L_0}{3}\right)\cdot \frac{di}{dt}, \tag{11} \]

where

\[ L_0=L_1+2\cdot M_1=\frac{x_0}{\omega_0}\approx \frac{x_1}{\omega_0} \]

is the so-called [4 et al.] inductance of the zero component;

\(x_0\) and \(x_1\) are the inductive reactances, respectively, of the zero sequence and of stator leakage.

Equation (11), in form, is an equation describing the connection of an active-inductive load to a sinusoidal voltage source. The parameters of the indicated active-inductive load are determined not only by the parameters of the electromagnet coil, but also by the parameters of the electric motor.

The solution of equation (11) has the form [5]

\[ i=\frac{U}{z_{\mathrm{n}}}\cdot \left[ \sin(\omega_0\cdot t+\alpha_{\mathrm{n}}) - \sin\alpha_{\mathrm{n}}\cdot e^{-\frac{t}{T_{\mathrm{n}}}} \right], \tag{12} \]

where

\[ z_{\mathrm{n}}=\sqrt{r_{\mathrm{n}}^{2}+\omega_0^{2}\cdot L_{\mathrm{n}}^{2}}; \qquad r_{\mathrm{n}}=R+\frac{r_1}{3}; \qquad L_{\mathrm{n}}=L+\frac{L_0}{3}; \]

\[ T_{\mathrm{n}}=\frac{L_{\mathrm{n}}}{r_{\mathrm{n}}}; \qquad \alpha_{\mathrm{n}}=\alpha_0-90^\circ-\varphi_{\mathrm{n}}; \qquad \varphi_{\mathrm{n}}=\operatorname{arc\,tg}\frac{\omega_0\cdot L_{\mathrm{n}}}{r_{\mathrm{n}}}. \tag{13} \]

Proceeding from (12) and (13), the voltage \(U_{\mathrm{v}}\) at the rectifier input when connected according to the circuit in Fig. 1b can be represented as:

\[ U_{\mathrm{v}} = U\cdot \frac{z}{z_{\mathrm{n}}}\cdot \sin(\omega_0\cdot t+\alpha_{\mathrm{v}}) + \frac{U}{z_{\mathrm{n}}}\cdot \left( \frac{L}{L_{\mathrm{n}}}\cdot r_{\mathrm{n}}-R \right) \cdot \sin\alpha_{\mathrm{n}}\cdot e^{-\frac{t}{T_{\mathrm{n}}}}, \tag{14} \]

where

\[ z=\sqrt{R^2+\omega_0^2\cdot L^2}; \qquad \operatorname{tg}\alpha_{\mathrm{v}} = \frac{R\cdot \sin\alpha_{\mathrm{n}}-\omega_0\cdot L\cdot \cos\alpha_{\mathrm{n}}} {R\cdot \cos\alpha_{\mathrm{n}}-\omega_0\cdot L\cdot \sin\alpha_{\mathrm{n}}}. \tag{15} \]

Expression (14) shows that the voltage at the rectifier input, and consequently also at the phase winding of the electric motor, has, in addition to the steady-state component, an aperiodic free component. The magnitude of the steady-state component of \(U_{\mathrm{v}}\) is smaller than the magnitude of the voltage of the phase winding of a symmetrical asynchronous motor, the larger the electric-motor parameters \(r_1\) and \(L_0\) are.

Switching on the brake electric motor according to the circuit in Fig. 1,c, as well as according to the circuit in Fig. 1,b, is not accompanied by distortion of the electromagnetic torque produced by the induction motor. However, to determine the zero-sequence component of the current \(i_0\), the following equations must be used:

\[ \begin{aligned} U \cdot \sin(\omega_0 \cdot t + \alpha_0 - 90^\circ) &= \left(r_{\mathrm{T}1} + \frac{r_1}{3}\right) \cdot i_{\mathrm{T}1} + \left(L_{\mathrm{T}1} + \frac{L_0}{3}\right) \cdot \frac{di_{\mathrm{T}1}}{dt} + M \cdot \frac{di}{dt}; \\ 0 &= M \cdot \frac{di_{\mathrm{T}1}}{dt} + (r_{\mathrm{T}2} + R) \cdot i + (L_{\mathrm{T}2} + L) \cdot \frac{di}{dt}. \end{aligned} \tag{16} \]

Equations (16) correspond to the equations describing the transient process when a short-circuited transformer is switched on to a sinusoidal voltage. The parameters of this transformer differ from those of a real transformer in that they take into account the influence of electromagnetic processes in the electric motor and in the electromagnet. The expression found with the aid of equations (16) for the current \(i_{\mathrm{T}1}\) is equal to three times the value of the zero-sequence component \(i_0\) in the circuit in Fig. 1,c.

Equations (8), (11), and (16), which take into account the asymmetrical connections of the stator circuits of the induction motor, were obtained by composing the initial voltage-balance equations in the phase axes, followed by transformation to the \(\alpha,\beta\) axes. Verification showed that equations (11) and (16) are more simply obtained if the equivalent-generator method [6, 7] is applied. The latter method is also applicable for determining the current \(i\) when switching according to the circuit in Fig. 1,d.

Above, the electromagnetic processes were considered for the first half-period of the change in the supply voltage after the electric motor was switched on. However, the expressions obtained in this way can also be used to consider the established operating regime \((t = \infty)\) of the electric motor. For this, in the differential equations it is sufficient to replace the differentiation operator \(d/dt\) with respect to time by \(j \cdot \omega_0\), and, instead of the real values of the electromagnet parameters \(R\) and \(L\), to substitute the equivalent parameters [8, 9] of the rectifier circuit.

For the circuits in Fig. 1, the equivalent parameters of the rectifier circuit in the sinusoidal-voltage regime are determined as follows:

\[ R_{\mathrm{e}} = R;\qquad z_{\mathrm{e}} = \frac{z}{B_1 \cdot \sqrt{2}};\qquad L_{\mathrm{e}} = \frac{z_{\mathrm{e}} \cdot \sin \varphi_{\mathrm{e}}}{\omega_0},\qquad \cos \varphi_{\mathrm{e}} = \frac{R_{\mathrm{e}}}{z_{\mathrm{e}}}; \tag{17} \]

\[ \sin \varphi = \frac{\omega_0 \cdot L}{z};\qquad \tg \varphi = \frac{\sin \varphi}{\cos \varphi} = \frac{\omega_0 \cdot L}{R}; \]

\[ B_1 = \sqrt{ 0.5 + \frac{2}{\pi} \cdot \sin^2 \varphi \cdot \tg \varphi \cdot \frac{1 + e^{-\frac{\pi}{\tg \varphi}}}{1 - e^{-\frac{\pi}{\tg \varphi}}} }. \]

In the established regime the armature of the electromagnet is attracted to its core, whereas during the first half-period of the change in the supply voltage after the brake electric motor is switched on, there is an initial air gap between the armature and the core. Therefore, in calculating a specific regime it is necessary to select the corresponding value of the electromagnet inductance.

Thus, in comparison with an ordinary induction motor, the influence of the electromagnet in all cases leads to a change in the currents,

and losses in the motor. When connected according to the circuit of Fig. 1a, the motor torque also changes. The voltage at the rectifier input (and, consequently, the tractive force of the electromagnet) is determined both by the parameters of the rectifier circuit and by the parameters of the electric motor. With the aid of the expressions given above, it is possible to quantitatively estimate the change in the quantity caused by the asymmetrical connection of the stator circuits of a brake electric motor.

REFERENCES

1. V. S. Grinberg, E. V. Kononenko, E. M. Guselnikov. Investigation of brake electric motors. Report on the research work of TPI, State Reg. No. 72045000, Tomsk, 1973.

2. I. I. Treshchev. Methods for Studying Electromagnetic Processes in Alternating-Current Machines. “Energiya,” 1969.

3. P. G. Kurosh. Course of Higher Algebra. Moscow, “Nauka,” 1968.

4. A. I. Bazhanov. Fundamentals of the Theory of Transient Processes in a Synchronous Machine. Moscow–Leningrad, GEI, 1960.

5. S. G. Ginzburg. Methods for Solving Problems on Transient Processes in Electrical Circuits. Moscow, “Vysshaya shkola,” 1967.

6. L. R. Neiman, K. S. Demirchyan. Theoretical Foundations of Electrical Engineering. Vol. 1. “Energiya,” 1966.

7. M. N. Golodnov. Experimental Determination of the Parameters of an Equivalent Generator in Linear Alternating-Current Circuits. “Elektrichestvo,” 1958, No. 4.

8. A. G. Zdrok, A. A. Salyutin. On the calculation of circuits with single-phase rectifiers and active-inductive load. Izvestiya vuzov, “Elektromekhanika,” 1966, No. 12.

9. A. G. Zdrok, A. A. Salyutin. Self-Excitation Systems of Synchronous Generators with Magnetic Amplifiers. “Energiya,” 1968.