Operation of an AC Motor with an Asymmetric Rotor in Steady-State Short-Circuit Mode
Abstract Generated abstract
This paper analyzes the steady-state short-circuit operation of AC machines with asymmetric rotors, motivated by phase-wound induction motor starting and asynchronous starting of synchronous motors. Using rotating magnetic field equations under balanced stator supply and a stationary rotor condition, it derives expressions for stator currents, phase powers, total power, and electromagnetic torque as functions of rotor asymmetry and spatial rotor position. The analysis shows that individual phase currents, phase power, and wattmeter readings depend on rotor position and may become negative under specified asymmetry conditions, while the total active power and average torque remain independent of rotor position. It also finds that total instantaneous power and electromagnetic torque include components pulsating at twice the line frequency.
Full Text
OPERATION OF AN AC MOTOR WITH AN ASYMMETRICAL ROTOR IN STEADY-STATE SHORT-CIRCUIT MODE
E. V. Kononenko
(Presented by the scientific seminar of the Electromechanical Faculty)
In induction motors with a phase-wound rotor, for economic reasons starting is carried out with asymmetrical resistances of the starting rheostat inserted in the rotor circuit. Synchronous motors during asynchronous starting generally have both electrical and magnetic asymmetry.
In the literature known to us, the operation of AC motors with an asymmetrical rotor in short-circuit mode has been treated quite insufficiently.
The present paper considers the question of what influence rotor asymmetry has on the steady-state short-circuit mode, as well as the influence of the rotor position in space on the operation of the motor.
The study of the indicated operating modes can be carried out with the aid of equations obtained on the basis of the method of rotating magnetic fields [1].
In the investigation we assume that, on the stator side, the motor is supplied by a positive-sequence voltage system from a source of infinitely large power. In this case the phase voltages vary according to the law
\[ u_a = U_m \cos t, \]
\[ u_b = U_m \cos (t - 120^\circ), \tag{1} \]
\[ u_c = U_m \cos (t - 240^\circ). \]
The stator winding is symmetrical. We also assume that the multiphase rotor winding has been reduced to a two-phase winding with axes \(d\) and \(q\), and that a system of relative units has been adopted.
The mutual arrangement of the axes of the stator and rotor windings is shown in Fig. 1. The angle \(\gamma_0\) determines the position of the rotor relative to the axis of phase \(a\) of the stator winding (that is, in space).
Taking into account the adopted assumptions, the current vector \(i_s\) and the flux-linkage vector \(\Psi_s\) of the stator in the \(d, q\) axes can be represented in the form [1]:
\[ i_s = i_{s1} + i_{s2} = I_1 e^{j(t-\gamma_0-\varphi_1)} + I_2 e^{-j(t-\gamma_0-\varphi_2)}, \tag{2} \]
\[ \Psi_s = \Psi_{s1} + \Psi_{s2} = \Psi_1 e^{j(t-\gamma_0-\psi_1)} + \Psi_2 e^{-j(t-\gamma_0-\psi_2)}, \tag{3} \]
where \(I_1, I_2, \Psi_1, \Psi_2\) are the moduli of the corresponding vectors, depending only on the parameters of the machine. At \(s=1.0\) they are constant quantities;
\(\varphi_1\) and \(\psi_1\) are the phase angles between the vectors \(i_{s1}\), \(\Psi_{s1}\), and \(u_s\);
\(\varphi_2\) and \(\psi_2\) are the phase angles between the vectors \(i_{s2}\), \(\Psi_{s2}\), and \(u_s^*\)¹).
The currents in the phases of the stator winding are determined as projections of the vector \(i_s\) onto the phase axes.
Consequently, the instantaneous values of the currents in the stator phases can be determined as:
\[ \begin{aligned} i_a &= I_1 \cos(t-\varphi_1) + I_2 \cos(t - 2\gamma_0 - \varphi_2),\\ i_b &= I_1 \cos(t-\varphi_1-120^\circ) \\ &\quad + I_2 \cos(t-2\gamma_0-\varphi_2+120^\circ),\\ i_c &= I_1 \cos(t-\varphi_1-240^\circ) \\ &\quad + I_2 \cos(t-2\gamma_0-\varphi_2+240^\circ). \end{aligned} \tag{4} \]
It is known that the sum of two sinusoidal quantities of one and the same frequency is also a sinusoidal quantity. On this basis, by adding the individual components of the currents in the stator phases, we obtain the following expressions:
\[ \begin{aligned} i_a &= I_{am}\cos(t-\varphi_a),\\ i_b &= I_{bm}\cos(t-\varphi_b-120^\circ),\\ i_c &= I_{cm}\cos(t-\varphi_c-240^\circ), \end{aligned} \tag{5} \]
Fig. 1.
Here
\[ I_{am} = \sqrt{I_1^2 + I_2^2 + 2 I_1 I_2 \cos(\varphi_1-\varphi_2-2\gamma_0)}, \tag{6} \]
\[ \varphi_a = \operatorname{arc\,tg} \frac{I_1\sin\varphi_1 + I_2\sin(\varphi_2+2\gamma_0)} {I_1\cos\varphi_1 + I_2\cos(\varphi_2+2\gamma_0)}. \tag{7} \]
\(I_{bm}, I_{cm}, \varphi_b, \varphi_c\) are obtained from equations (6) and (7) by replacing \(\gamma_0\) by \((\gamma_0-120^\circ)\) and \((\gamma_0-240^\circ)\), respectively.
It follows from equations (6) and (7) that, with a stationary rotor (\(s=1.0\)), the amplitude of the stator current and the phase angle depend on the position of the rotor in space. When the angle \(\gamma_0\) changes, the amplitude of the current in a phase changes from a maximum value equal to \((I_1+I_2)\) to a minimum equal to \((I_1-I_2)\); at the same time, the power factor also changes. The dependences of the change in the current amplitude and the power factor in phase \(a\), calculated for an induction motor with a single-phase rotor, are presented in Fig. 2.
¹) Conjugate vectors are denoted by an asterisk \(*\).
It also follows from equations (5) that the currents in the phases of an induction motor with an asymmetrical rotor at \(s = 1.0\) constitute an asymmetrical system.
Knowing the nature of the variation of the current in the stator phase windings, one can determine the power consumed from the network.
The instantaneous value of the power is determined as the product of the corresponding instantaneous values of current and voltage. On this basis, the instantaneous power of phase \(a\) is equal to
\[ P_a = u_a i_a = U_m \cos t \,[ I_1 \cos (t-\varphi_1) + I_2 \cos (t - 2\gamma_0 - \varphi_2)] . \]
After transformation we obtain
\[ \begin{aligned} P_a &= \frac{U_m I_1}{2}\cos\varphi_1 + \frac{U_m I_2}{2}\cos(\varphi_2 + 2\gamma_0) + \frac{U_m I_1}{2}\cos(2t-\varphi_1) \\ &\quad + \frac{U_m I_2}{2}\cos(2t-\varphi_2-2\gamma_0). \end{aligned} \tag{8} \]
It follows from equation (8) that the active component of the power in phase \(a\) is equal to the sum of the first two terms:
Fig. 2.
\[ P_{a1} = \frac{U_m I_1}{2}\cos\varphi_1 + \frac{U_m I_2}{2}\cos(\varphi_2 + 2\gamma_0). \tag{9} \]
Considering the last equation, we see that the active power of the phase with a stationary rotor depends on the position of the rotor in space. This is explained by the fact that the first component of equation (9) is constant, while the second depends on the position of the rotor relative to the stator, i.e., on the angle \(\gamma_0\).
With a certain asymmetry of the rotor, the consumed power may even change sign to the opposite one, that is, power will be delivered to the network.
The power changes sign to the opposite one if the following condition is satisfied:
\[ I_2 > I_1 \cos \varphi_1 . \tag{10} \]
The dependence of the variation of active power in phase \(a\), calculated for an induction motor with a single-phase rotor having the parameters \(r = 0.042;\ x_d = 2.6;\ T_d = 82;\ T'_d = 4.22\), is presented in Fig. 2. From the given dependence it follows that the active power of the phase, as the position of the rotor in space changes, varies according to a sinusoidal law. Within the range of variation of the angle \(\gamma_0\) from \(32^\circ\) to \(88^\circ\), the power is negative, that is, it is generated into the network.
These phenomena can be explained in the following way. Let us represent the voltage (1) and the current (4) in phase \(a\) in the form of vectors, as shown in Fig. 3. As the position of the rotor in space changes,
Fig. 3.
the angle \(\gamma_0\) changes and, consequently, the end of the current vector \(I_2\) describes a circle. From Fig. 3 it is clearly seen how, as the angle \(\gamma_0\) changes, the amplitude of the current \(I_{am}\) and the power factor \(\cos \varphi_a\) change.
At a certain value of the angle \(\gamma_0\), the power consumed from the network may even change its sign to the opposite one. This will occur when \(\varphi_a\) becomes greater than \(90^\circ\). From Fig. 3 it is clear that this is possible only when condition (10) is satisfied.
Similarly to (8), the instantaneous power in phases \(b\) and \(c\) can be determined.
\[ P_b = u_b i_b = \frac{U_m I_1}{2}\cos \varphi_1 + \frac{U_m I_2}{2}\cos(\varphi_2 + 2\gamma_0 - 240^\circ) + \]
\[ + \frac{U_m I_1}{2}\cos(2t-\varphi_1-240^\circ) + \frac{U_m I_2}{2}\cos(2t-\varphi_2-2\gamma_0); \tag{11} \]
\[ \begin{aligned} P_c = u_c i_c &= \frac{U_m I_1}{2}\cos\varphi_1 + \frac{U_m I_2}{2}\cos(\varphi_2+2\gamma_0-120^\circ) \\ &\quad + \frac{U_m I_1}{2}\cos(2t-\varphi_1-120^\circ) + \frac{U_m I_2}{2}\cos(2t-\varphi_2-2\gamma_0). \end{aligned} \tag{12} \]
The total power consumed by the motor from the network is equal to the sum of the instantaneous values of the power in the phases
\[ P = u_a i_a + u_b i_b + u_c i_c. \tag{13} \]
Summing equations (8), (11), and (12), after transformation we obtain
\[ P = 3\frac{U_m I_1}{2}\cos\varphi_1 + 3\frac{U_m I_2}{2}\cos(2t-\varphi_2-2\gamma_0). \tag{14} \]
It follows from equation (14) that AC motors with an asymmetrical rotor constitute an unbalanced system.
The first term of equation (14) is the active power consumed by the motor from the network
\[ P_1 = 3\frac{U_m I_1}{2}\cos\varphi_1; \tag{15} \]
From equation (15) it is evident that the total active power consumed by the motor from the network is constant and does not depend on the position of the rotor in space.
The power consumed by the motor can be measured by the usual methods, using three or two wattmeters. In doing so, it must be taken into account that the readings of each wattmeter, when the position of the rotor in space (the angle \(\gamma_0\)) changes, vary and, under certain conditions, may change sign to the opposite one. Therefore, when determining the total power it is necessary to take into account the sign of the power readings of each wattmeter.
The electromagnetic torque of rotation can be determined in the following way [1]:
\[ M = \operatorname{Re}\left[j\Psi_s i_s^*\right]. \tag{16} \]
Substituting into equation (16) the value of the current (2) and of the flux linkages (3), after transformations we obtain
\[ M = M_c + M_n \sin(2t+\vartheta). \tag{17} \]
where
\[ M_c = \Psi_1 I_1 \sin(\psi_1-\varphi_1) + \Psi_2 I_2 \sin(\varphi_2-\psi_2) \]
is the average, time-constant torque;
\[ M_n = \sqrt{ (\Psi_2 I_1)^2 + (\Psi_1 I_2)^2 + 2(\Psi_1 I_2)(\Psi_2 I_1)\cos(\varphi_2-\varphi_1+\psi_2-\psi_1) } \]
is the amplitude of the alternating component of the torque;
\[ \vartheta = \operatorname{arc\,tg} \frac{ \Psi_1 I_2 \sin(\varphi_2+\psi_1+2\gamma_0) - \Psi_2 I_1 \sin(\varphi_1+\psi_2+2\gamma_0) }{ \Psi_2 I_1 \cos(\varphi_1+\psi_2+2\gamma_0) - \Psi_1 I_2 \cos(\varphi_2+\psi_1+2\gamma_0) } \]
is the initial angle.
Equation (17) indicates that the electromagnetic torque of rotation of an AC motor with an asymmetrical rotor can be represented (just like the power consumed from the network) as consisting of two components: an average torque, constant in time, and a variable torque pulsating in time at twice the line frequency.
The average torque does not depend on the position of the rotor in space.
REFERENCES
- E. V. Kononenko. Investigation of asynchronous operating modes of an AC machine with an asymmetrical rotor. Izvestiya vuzov, Elektromekhanika, No. 2, 1960.
Received by the editors
in May 1962.