Calculation of Starting Processes in Asynchronous Machines by the Method of Successive Intervals

G. A. Sipailov

Introduction

The analytical investigation of transient processes in asynchronous machines presents certain difficulties.

The initial voltage-balance equations contain variable coefficients, since the mutual inductance between the stator and rotor windings changes periodically as the rotor rotates. In addition, whereas in the theory of synchronous machines the principal difficulty is the magnetic asymmetry generally present in the rotor, in asynchronous machines the difficulty lies in the need to take account of the asynchronous speed of rotation of the rotor.

In order to simplify the consideration of transient processes in synchronous and asynchronous machines, a number of methods have been developed in the modern theory of electrical machines; these include, above all:

1. The use of per-unit quantities, which makes it possible to free the equations from a large number of coefficients.
2. The consideration of asymmetrical regimes as a combination of symmetrical ones, by means of the method of symmetrical components.
3. The use of the operational-calculus method.
4. The reduction of a three-phase, or in the general case a multiphase, system to an equivalent two-phase system.
5. The transformation of variables to a single coordinate system.

However, the application of the above-listed methods of analysis is possible only under certain assumptions. In all works that examine transient processes in asynchronous machines on the basis of modern methods of analysis, the following assumptions and approximations are, as a rule, adopted:

1. The applied voltages and currents are sinusoidal.
2. Only the first harmonics of the m.m.f. and fluxes are taken into account.
3. Saturation is taken into account only by choosing appropriate values of the inductances.
4. Losses in the stator steel are not taken into account.
5. The change in the active resistance of the windings due to heating is not taken into account.
6. All stator and rotor phases are assumed to be arranged symmetrically, and the air gap along the circumference of the stator is taken as constant.
7. The machine is connected to a network of infinite power.
8. The mutual inductances between the stator and rotor windings vary according to a sinusoidal law.
9. The investigations are carried out without taking zero-sequence currents into account.

But even with all the assumptions listed above, the analytical expressions characterizing the required quantities during the transient process have

has a rather complicated form, and reducing the number of assumptions leads to still greater complication of the general equations, so that the solution of practical problems by means of them becomes difficult and not always expedient.

Under operating conditions of an electric drive, some coefficients entering into the differential equations are not constant; therefore a mathematical analysis based on the assumption that these coefficients are constant may give significant deviations from the quantities characterizing the actual processes.

Such nonlinear coefficients include, above all, inductance. Because the coefficients are not constant, the equations cease to be linear; therefore one has to resort to an approximate solution of the differential equations.

V. T. Kas’yanov [2] has developed a method for solving such equations for calculating processes of regulation of magnetic fields, calculation of starting, braking, reversal, and regulation of the speed and load of DC motors.

In the present work, the method of successive intervals is proposed for calculating transient processes in asynchronous machines. This method does not claim to replace the method of modern mathematical analysis; on the contrary, it is proposed as a supplement to it.

Basic equations for calculating the switching-on processes of a single-phase asynchronous machine

The question of considering transient processes in a single-phase asynchronous motor with a single-phase rotor is of great practical and mathematical interest. Indeed, from the standpoint of mathematical analysis, a single-phase asynchronous motor presents the greatest difficulties in investigating the transient processes of asynchronous machines. To this day no general analytical solution has been given for determining currents and flux linkages under sudden disturbances of the normal regime (during switching on, switching off, short circuits, etc.).

The difficulty of solving these problems lies in the fact that, in a rotating motor, periodic coefficients appear in the general voltage equilibrium equations; because of the single-axis nature of the single-phase machine, these coefficients cannot be eliminated by any of the techniques widely used in multiphase machines. Therefore, in analytical investigations, in accordance with the physics of the phenomenon, it is necessary to consider a pulsating field consisting of two fields rotating with half amplitudes and in opposite directions.

Thus, the stator of a single-phase asynchronous motor is replaced by two three-phase stators connected in series, with positive and negative phase sequence. The positive field, rotating in the direction of rotation of the rotor, induces slip-frequency currents in the rotor conductors. These currents create a rotating magnetic field which, moving in space with the speed of the positive field, forms, by its interaction with it, a rotating torque constant in magnitude. Whereas the negative synchronous field will induce in the rotor currents of double frequency, which, in turn, will cause currents of triple frequency to appear in the stator, and so on.

Therefore a single-phase asynchronous motor with a single-phase rotor, when studied analytically using modern mathematical analysis, has to be considered as a combination of a whole series of systems, symmetrical in electrical and magnetic respects, with different frequencies.

By considering equations with periodic coefficients, it is possible to avoid the piling up of individual systems one upon another; however, another difficulty then arises—the complexity of solving equations with variable coefficients.

However, with the application of the method of successive intervals, this problem is solved comparatively easily.

The voltage-balance equations for the stator and rotor of a single-phase asynchronous machine with a single-phase rotor, in the general case, will be:

\[ u_1=r_1 i_1+L_1\frac{di_1}{dt}+\frac{d}{dt}(Mi_2\cos\Theta), \tag{1} \]

\[ u_2=r_2 i_2+L_2\frac{di_2}{dt}+\frac{d}{dt}(Mi_1\cos\Theta). \]

Under the condition of constant speed,

\[ \Theta=\int_0^t \omega\,dt+\alpha_0=\omega t+\alpha_0, \tag{2} \]

where \(\Theta\) is the angle of rotation of the axis of the rotor winding relative to the axis of the stator winding,

\(\alpha_0\) is the initial angle (except in specially stipulated cases, \(\alpha_0\) is taken equal to zero, i.e., at \(t=0\), \(\alpha_0=0\)).

With a constant amplitude value of the mutual inductance \(M\) and an unchanged speed \(\dfrac{d\Theta}{dt}=\omega\), equations (1) may be written in the following form:

\[ u_1=r_1 i_1+L_1\frac{di_1}{dt}+M\frac{di_2}{dt}\cos\omega t-M\omega i_2\sin\omega t, \]

\[ u_2=r_2 i_2+L_2\frac{di_2}{dt}+M\frac{di_1}{dt}\cos\omega t-M\omega i_1\sin\omega t. \tag{3} \]

Equations (3) are the initial equations for the investigation of transient processes of a single-phase asynchronous motor with a single-phase rotor.

In order to present more clearly the essence of calculation by the method of successive intervals, one may consider the simplest case, for example, the switching on of a single-phase asynchronous motor with a stationary and open-circuited single-phase rotor.

In this case, equations (3) take the following form:

\[ u_1=r_1 i_1+L_1\frac{di_1}{dt}, \]

\[ E_2=M\frac{di_1}{dt}. \tag{4} \]

Here \(u_1=U\sin 2\pi f t\) is the instantaneous value of the sinusoidal supply voltage,

\[ L_1=L_{s1}+M=\frac{x_1+x_m}{2\pi f} \]

is the total inductance of the stator,

\(L_{s1}\) is the leakage inductance of the stator.

To determine the stator current and the rotor e.m.f. by the method of successive intervals, it is necessary to write the system of differential equations (4) in the form of finite increments. Denoting by the subscript \(m\) the mean values of the variable quantities over the time interval \(\Delta t\), we obtain:

\[ u_{1m}=r_1 i_{1m}+L_1 \frac{\Delta i_1}{\Delta t}, \qquad \tag{5} \]

\[ E_{2m}=M \frac{\Delta i_1}{\Delta t}. \]

These mean values can be expressed in terms of the initial and final values of the quantities in the time interval \(\Delta t\) under consideration:

\[ i_{1m}=\frac{i'_1+i''_1}{2}=i'_1+\frac{\Delta i_1}{2}; \qquad E_{2m}=\frac{E'_2+E''_2}{2}=E'_2+\frac{\Delta E_2}{2}, \tag{6} \]

where \(i'_1\) and \(E'_2\) are the instantaneous values of the stator current \(i_1\) and the rotor e.m.f. \(E_2\) at the beginning of the time interval \(\Delta t\),

\(i''_1\) and \(E''_2\) are the instantaneous values of \(i_1\) and \(E_2\) at the end of the time interval \(\Delta t\),

\(\Delta i_1=i''_1-i'_1\); \(\Delta E_2=E''_2-E'_2\) are the increments of the current and the e.m.f. over the time \(\Delta t\).

For practical calculations, the time interval \(\Delta t\) may be taken, depending on the required accuracy, as equal to \(1/12\) or \(1/18\) of a period, which at a frequency of 50 cps corresponds to \(1/600\) or, respectively, \(1/900\) sec.

Substituting from equations (6) the mean values of the stator current \(i_{1m}\) and the rotor e.m.f. \(E_{2m}\) into equations (5), we obtain

\[ u_{1m}=\left(i'_1+\frac{\Delta i_1}{2}\right)r_1+L_1\frac{\Delta i_1}{\Delta t}; \qquad E'_2+\frac{\Delta E_2}{2}=M\frac{\Delta i_1}{\Delta t}. \tag{7} \]

Solving the system of equations (7) with respect to \(\Delta i_1\) and \(\Delta E_2\), we find the calculation equations:

\[ \Delta i_1= \frac{u_{1m}-r_1 i'_1} {\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}}, \qquad \Delta E_2=2\frac{M}{\Delta t}\Delta i_1-2E'_2. \tag{8} \]

Having the initial conditions (the values \(U'_1\), \(i'_1\), and \(E'_2\) at \(t=0\)), which are usually entered in the first row of the calculation table, and substituting them into the first of equations (8), we obtain the current increment over the selected time interval \(\Delta t\); after this, substituting the value \(\Delta i_1\) into the second of equations (8), we can determine the increment of the rotor e.m.f. \(\Delta E_2\) over the same time interval \(\Delta t\).

The values of the stator current and the rotor e.m.f. at the end of the first interval \(\Delta t\) are found from the following equations:

\[ i_1=i'_1+\Delta i_1; \qquad E_2=E'_2+\Delta E_2. \tag{9} \]

After entering the obtained values of the stator current \(i_1\) and the rotor e.m.f. \(E_2\) in the second row of the calculation table, one may proceed to the calculation of the next, third row, for which the initial conditions will be only the data found in the second row, and so on.

The table used in calculations by the method of successive intervals usually has a simple form.

Table 1

and so on.

When calculating with allowance for saturation, it is advisable to add one more column to calculation Table 1, reflecting the change in mutual inductance in the transition from one interval to another.

The calculated values are obtained closer to the actual (experimental) ones if the mean value $u_{1m}$ is taken not as the half-sum of the voltage values at the beginning and at the end of the interval, but as that voltage which actually occurs in the middle of the interval. This voltage corresponds to the time instant $t+\dfrac{\Delta t}{2}$ and is found from the mean angle. For example, for the first interval, at $\Delta t=\dfrac{1}{600}\ \text{sec}$, instead of

\[ u_{1m}=\frac{U\sin 0+U\sin 30^\circ}{2}=0{,}25\,U \]

we obtain

\[ u_{1m}=U\sin\left(\frac{0+30^\circ}{2}\right)=0{,}259\,U. \]

For the second interval, instead of

\[ u_{1m}=\frac{U\sin 30^\circ+U\sin 60^\circ}{2}=0{,}683\,U \]

we obtain

\[ u_{1m}=U\sin\left(\frac{30^\circ+60^\circ}{2}\right)=0{,}707\,U \]

and so on.

In an analogous manner, equations are derived for calculating the processes of switching on, switching off, and short-circuiting a single-phase asynchronous motor with a stationary and rotating, open-circuited and short-circuited, single-phase and three-phase rotor.

Thus, for example, to determine the stator and rotor currents when switching on a single-phase stator with a short-circuited rotating single-phase rotor, it is necessary to substitute $u_2=0$ in equation (3). Then equations (3), written in the form of finite increments, after substitution of equations (6), take the form:

\[ \begin{aligned} \left(\frac{r_1}{2}+\frac{L_1}{\Delta t}\right)\Delta i_1 &+\left(\frac{M}{\Delta t}\cos\omega t-\frac{M\omega}{2}\sin\omega t\right)\Delta i_2 \\ &=u_{1m}-r_1 i'_1+M\omega i'_2\sin\omega t, \\[6pt] \left(\frac{M}{\Delta t}\cos\omega t-\frac{M\omega}{2}\sin\omega t\right)\Delta i_1 &+\left(\frac{r_2}{2}+\frac{L_2}{\Delta t}\right)\Delta i_2 \\ &=-r_2 i'_2+M\omega i'_1\sin\omega t. \end{aligned} \tag{10} \]

Solving the system of equations (10) with respect to \(\Delta i_1\) and \(\Delta i_2\) gives:

\[ \Delta i_1 = \frac{ \left(u_{1m}-r_1 i'_1+M\omega i'_2\sin\omega t\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) + \left(r_2 i'_2-M\omega i'_1\sin\omega t\right) \left(\dfrac{M}{\Delta t}\cos\omega t-\dfrac{M\omega}{2}\sin\omega t\right) }{ \left(\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) - \left(\dfrac{M}{\Delta t}\cos\omega t-\dfrac{M\omega}{2}\sin\omega t\right)^2 }, \tag{11} \]

\[ \Delta i_2 = \frac{ \left(u_{1m}-r_1 i'_1+M\omega i'_2\sin\omega t\right) \left(\dfrac{M}{\Delta t}\cos\omega t-\dfrac{M\omega}{2}\sin\omega t\right) + \left(r_2 i'_2-M\omega i'_1\sin\omega t\right) \left(\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}\right) }{ \left(\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) - \left(\dfrac{M}{\Delta t}\cos\omega t-\dfrac{M\omega}{2}\sin\omega t\right)^2 }. \]

If, in the machine under investigation, the stator parameters and the referred rotor parameters may be taken as equal, i.e.

\[ \frac{r_1}{2}+\frac{L_1}{\Delta t} = \frac{r_2}{2}+\frac{L_2}{\Delta t} = \frac{r}{2}+\frac{L}{\Delta t}, \]

then the calculation equations (11) can be reduced to the following form:

\[ \Delta i_1 = \frac{ \left(u_{1m}-r_1 i'_1+M\omega i'_2\sin\omega t\right) + \left(r_2 i'_2-M\omega i'_1\sin\omega t\right)A\sin(\gamma-\omega t) }{ B\left[C-\sin^2(\gamma-\omega t)\right] }, \tag{12} \]

\[ \Delta i_2 = \frac{ \left(u_{1m}-r_1 i'_1+M\omega i'_2\sin\omega t\right)A\sin(\gamma-\omega t) - \left(r_2 i'_2-M\omega i'_1\sin\omega t\right) }{ B\left[C-\sin^2(\gamma-\omega t)\right] }, \]

where

\[ A= \frac{ \sqrt{\left(\dfrac{M}{\Delta t}\right)^2+\left(\dfrac{M\omega}{2}\right)^2} }{ \dfrac{r}{2}+\dfrac{L}{\Delta t} }, \qquad B= \frac{ \left(\dfrac{M}{\Delta t}\right)^2+\left(\dfrac{M\omega}{2}\right)^2 }{ \dfrac{r}{2}+\dfrac{L}{\Delta t} }, \]

\[ C= \frac{ \left(\dfrac{r}{2}+\dfrac{L}{\Delta t}\right)^2 }{ \left(\dfrac{M}{\Delta t}\right)^2+\left(\dfrac{M\omega}{2}\right)^2 } = \frac{1}{A^2}, \qquad \gamma=\operatorname{arctg}\frac{2}{\Delta t\cdot\omega}. \]

Here the coefficient \(C\), which usually has a value only slightly greater than unity, must be calculated with particular rigor. Therefore, if the value of \(C\) is determined inaccurately, for values of \(\sin^2(\gamma-\omega t)\) close to unity, it may turn out that the current increment tends to infinity.

The results of the calculation are entered in a table, which in this case has the following form:

Table 2

Here \(\sin \omega t_m\) and \(\sin(\gamma-\omega t)_m\) are the mean values of the coefficients in the time interval \(\Delta t\) under consideration.

To speed up the calculation, it is advisable, before beginning the calculation, to fill in the first five columns of Table 2. Such an operation takes comparatively little time.

Substituting \(\omega = 0\) in (11), one can obtain equations for calculating the switching-on process of a single-phase motor with a stationary closed single-phase rotor:

\[ \Delta i_1 = \frac{ \left(u_{1m}-r_1 i'_1\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) + \dfrac{M}{\Delta t} r_2 i'_2 }{ \left(\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) - \left(\dfrac{M}{\Delta t}\right)^2 }, \]

\[ \Delta i_2 = - \frac{ \left(u_{1m}-r_1 i'_1\right)\dfrac{M}{\Delta t} + \left(\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}\right) r_2 i'_2 }{ \left(\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) - \left(\dfrac{M}{\Delta t}\right)^2 }. \tag{13} \]

When carrying out numerical calculations with allowance for saturation, equations (13) are transformed in such a way that, first, the operations of dividing the variable mutual inductance \(M\) by the quantity \(\Delta t\) are not repeated in each interval and, second, so that the calculation equations do not contain the square of the variable quantity \(M\):

\[ \Delta i_1 = K \frac{ \left(u_{1m}-r_1 i'_1\right) \left(\dfrac{r_2\Delta t}{2}+L_{S2}+M_m\right) + M_m r_2 i'_2 }{ C+M_m }, \]

\[ \Delta i_2 = -K \frac{ \left(u_{1m}-r_1 i'_1\right)M_m + \left(\dfrac{r_1\Delta t}{2}+L_{S1}+M_m\right) r_2 i'_2 }{ C+M_m }, \tag{14} \]

where \(K\) and \(C\) are constant coefficients, namely

\[ K = \frac{ \Delta t }{ \dfrac{r_1\Delta t}{2}+L_{S1} + \dfrac{r_2\Delta t}{2}+L_{S2} }, \tag{15} \]

\[ C = \frac{ \left(\dfrac{r_1\Delta t}{2}+L_{S1}\right) \left(\dfrac{r_2\Delta t}{2}+L_{S2}\right) }{ \dfrac{r_1\Delta t}{2}+L_{S1} + \dfrac{r_2\Delta t}{2}+L_{S2} }. \]

\[ M_m=\frac{M'+M''}{2} \]

—the mean value of the variable mutual inductance in the interval under consideration,

\[ L_1=L_{s1}+M \quad \text{and} \quad L_2=L_{s2}+M \]

—the total inductances of the stator and rotor,

\(L_{s1}\) and \(L_{s2}\)—the leakage inductances of the stator and rotor.

Using equations (14), the calculation can be carried out both without taking saturation into account and with it taken into account. In calculating with saturation taken into account, for each interval under consideration one has to find, from the curve \(M=f(i)\), its own value of the mutual inductance, which makes the calculation more laborious. However, the number of computational operations is substantially reduced if the mean value of the variable mutual inductance is found not by successive approximations, but by extrapolation. In all other respects, the calculation with saturation taken into account proceeds in the same order as without taking it into account.

Basic equations for calculating the processes of switching on a two-phase asynchronous machine

A two-phase asynchronous machine is the simplest system in all respects. Its simplicity is due to the presence, both on the stator and on the rotor, of only two windings, arranged in space at an angle of 90 electrical degrees. This eliminates mutual induction between the stator phases, and also between the rotor phases, which, in turn, leads to a simplification of the general voltage-balance equations, in which, of all the e.m.f.’s of mutual induction, it is necessary to take into account only the e.m.f.’s due to mutual induction between the stator and rotor windings.

The analysis of a two-phase machine is of great interest, since, in essence, any polyphase machine can be reduced to a two-phase one; therefore almost all authors, when studying transient processes in three-phase or, in the general case, polyphase machines, have as a rule investigated the processes occurring in a two-phase machine, subsequently generalizing the results of the investigation to the case of a polyphase machine.

For a symmetrical two-phase machine (Fig. 1)

\[ L_a=L_b=L_1; \qquad r_a=r_b=r_1; \]

\[ L_\alpha=L_\beta=L_2; \qquad r_\alpha=r_\beta=r_2; \]

\[ M_{a\alpha\max}=M_{a\beta\max}=M_{b\alpha\max}=M_{b\beta\max}=M. \tag{16} \]

Fig. 1

The voltage-balance equations for the stator are

\[ u_a=r_1 i_a+L_1\frac{di_a}{dt} +\frac{d}{dt}(Mi_\alpha \cos\Theta) -\frac{d}{dt}(Mi_\beta \sin\Theta), \]

\[ u_b=r_1 i_b+L_1\frac{di_b}{dt} +\frac{d}{dt}(Mi_\alpha \sin\Theta) +\frac{d}{dt}(Mi_\beta \cos\Theta). \tag{17} \]

For the rotor

\[ \begin{aligned} u_{\alpha}&=r_2 i_{\alpha}+L_2\frac{di_{\alpha}}{dt} +\frac{d}{dt}(Mi_a\cos\Theta)+\frac{d}{dt}(Mi_b\sin\Theta),\\ u_{\beta}&=r_2 i_{\beta}+L_2\frac{di_{\beta}}{dt} -\frac{d}{dt}(Mi_a\sin\Theta)+\frac{d}{dt}(Mi_b\cos\Theta). \end{aligned} \tag{18} \]

Here the indices \(a\) and \(b\) denote, respectively, the first and second phases of the stator; the indices \(\alpha\) and \(\beta\)—the first and second phases of the rotor.

One of the methods of modern mathematical analysis that makes it possible to simplify substantially the solution of a number of technical problems is the method of coordinate transformation. This method makes it possible to pass from considering equations with periodic coefficients to equations with constant coefficients. The essence of the method consists in passing from the study of a real electrical machine to the study of a transformed model whose circuit equations are more readily amenable to mathematical analysis. L. N. Gruzov gave a simple method for deriving equations in the transformed system. This method is based on a vector representation of the physical quantities participating in the physical picture of the phenomena taking place.

Mathematically, the coordinate transformation reduces to replacing the variables in the equations under consideration. As the new variables, the projections of the old variables on the coordinate axes are taken. These axes are chosen in such a way that the law of variation of the projections of the quantities under consideration is as simple as possible.

Physically, the coordinate transformation is equivalent to replacing all electrical circuits, or part of them, by transformed circuits rotating in space with the speed of rotation of the chosen coordinate axes. In this case, for the transformed circuits such parameters are used and such voltages are applied to the circuits that the power of the machine and its torque remain unchanged.

In the present work, a system of coordinate axes fixed in space is adopted. The axes of the chosen coordinate system coincide with the axes \(a\) and \(b\) of the stator.

Since the stator windings are stationary, they remain untransformed, while all rotor quantities are projected onto two mutually perpendicular stator axes. Such a transformation is equivalent to reducing the rotor quantities to the stator frequency.

The equations for the rotor circuit in the transformed system take the form (for distinction, quantities relating to the rotor in the transformed coordinate system are denoted by a “prime”; for example, the projections of the rotor quantities on the axis \(a\) are denoted \(U'_{\alpha}, i'_{\alpha}\), and on the axis \(b\)—\(U'_{\beta}, i'_{\beta}\), etc.):

\[ \begin{aligned} u'_{\alpha}&=u_{\alpha}\cos\Theta-u_{\beta}\sin\Theta,\\ u'_{\beta}&=u_{\alpha}\sin\Theta+u_{\beta}\cos\Theta. \end{aligned} \tag{19} \]

Correspondingly,

\[ \begin{aligned} i'_{\alpha}&=i_{\alpha}\cos\Theta-i_{\beta}\sin\Theta,\\ i'_{\beta}&=i_{\alpha}\sin\Theta+i_{\beta}\cos\Theta. \end{aligned} \tag{20} \]

After simple mathematical transformations we obtain the voltage-balance equations of the stator and rotor windings in the transformed coordinate system:

\[ u_a=r_1 i_a+L_1\frac{di_a}{dt}+M\frac{di'_{\alpha}}{dt}, \]

\[ \begin{gathered} u_b=r_1 i_b+L_1\frac{di_b}{dt}+M\frac{di'_\beta}{dt},\\ u'_\alpha=r_2 i'_\alpha+L_2\frac{di'_\alpha}{dt}+M\frac{di_a}{dt}+L_2\omega i'_\beta+M\omega i_b,\\ u'_\beta=r_2 i'_\beta+L_2\frac{di'_\beta}{dt}+M\frac{di_b}{dt}-L_2\omega i'_\alpha-M\omega i_a. \end{gathered} \tag{21} \]

As follows from the system of equations (21), equations with constant coefficients have been obtained.

Since the subsequent operations are performed only with the transformed values of the quantities relating to the rotor circuit, for convenience of notation the prime marks are omitted, in order to use them in the calculation by the method of successive intervals to denote the quantities at the beginning of the interval under consideration. It should be borne in mind, however, that now the indices \(\alpha\) and \(\beta\) correspond to the projections of the rotor quantities respectively on the \(a\) and \(b\) axes of the stator.

Passing to finite increments and replacing the mean values of the variable quantities in the same way as was done for the single-phase machine (equation 6), we arrange the terms of the equation so that it is convenient to solve the system by means of determinants

\[ \begin{gathered} \left(\frac{r_1}{2}+\frac{L_1}{\Delta t}\right)\Delta i_a+0+\frac{M}{\Delta t}\Delta i_\alpha+0 =u_{am}-r_1 i'_a,\\ 0+\left(\frac{r_1}{2}+\frac{L_1}{\Delta t}\right)\Delta i_b+0+\frac{M}{\Delta t}\Delta i_\beta =u_{bm}-r_1 i'_b,\\ \frac{M}{\Delta t}\Delta i_a+\frac{M\omega}{2}\Delta i_b+ \left(\frac{r_2}{2}+\frac{L_2}{\Delta t}\right)\Delta i_\alpha+ \frac{L_2\omega}{2}\Delta i_\beta=\\ =u_{\alpha m}-r_2 i'_\alpha-L_2\omega i'_\beta-M\omega i'_b,\\ -\frac{M\omega}{2}\Delta i_a+\frac{M}{\Delta t}\Delta i_b- \frac{L_2\omega}{2}\Delta i_\alpha+ \left(\frac{r_2}{2}+\frac{L_2}{\Delta t}\right)\Delta i_\beta=\\ =u_{\beta m}-r_2 i'_\beta+L_2\omega i'_\alpha+M\omega i'_a. \end{gathered} \tag{22} \]

Here \(i'_a,\ i'_b,\ i'_\alpha\) and \(i'_\beta\) are the instantaneous values of the corresponding currents at the beginning of the time interval \(\Delta t\) under consideration,

\[ \Delta i_a,\ \Delta i_b,\ \Delta i_\alpha \ \text{and}\ \Delta i_\beta \]

are the increments of the currents during the time \(\Delta t\).

To shorten the notation in solving the system of equations (22), the following designations are introduced:

\[ \begin{gathered} A=u_{am}-r_1 i'_a;\qquad C=u_{\alpha m}-r_2 i'_\alpha-L_2\omega i'_\beta-M\omega i'_b;\\ B=u_{bm}-r_1 i'_b;\qquad D=u_{\beta m}-r_2 i'_\beta+L_2\omega i'_\alpha+M\omega i'_a; \end{gathered} \tag{23} \]

\[ x=\frac{r_1}{2}+\frac{L_1}{\Delta t};\qquad y=\frac{M}{\Delta t};\qquad z=\frac{r_2}{2}+\frac{L_2}{\Delta t}; \]

\[ u=\frac{M\omega}{2};\qquad v=\frac{L_2\omega}{2}. \]

Expanding the determinants of system (22), we find the general equations for determining the current increments \(\Delta i_a\), \(\Delta i_b\), \(\Delta i_\alpha\), and \(\Delta i_\beta\) of the transient process:

\[ \Delta i_a=\frac{AK_1+BK_2-CK_3+DK_4}{K_9}, \]

\[ \Delta i_b=\frac{-AK_2+BK_1-CK_4-DK_3}{K_9}, \]

\[ \Delta i_\alpha=-\frac{AK_5+BK_6-CK_7+DK_8}{K_9}, \tag{24} \]

\[ \Delta i_\beta=-\frac{-AK_6+BK_5-CK_8-DK_7}{K_9}. \]

Here

\[ \begin{aligned} K_1&=x(z^2+v^2)-y(uv+zy),& K_5&=x(yz+uv)-y(y^2+u^2),\\ K_2&=y(uz-yv),& K_6&=x(uz-yv),\\ K_3&=y(xz-y^2),& K_7&=x(xz-y^2),\\ K_4&=y(xv-uy),& K_8&=x(xv-uy),\\ K_9&=(xz-y^2)^2+(xv-yu)^2. \end{aligned} \tag{25} \]

Under the condition of constant speed and without taking saturation into account, these coefficients will be constant, with \(K_2 \simeq 0\) and \(K_6 \simeq 0\); then the entire calculation reduces to computing the values of \(A\), \(B\), \(C\), and \(D\) for each interval being calculated.

On the basis of equations (22) and (24), the abbreviated notations (23), and the coefficients (25), equations are derived for calculating the processes of switching on, switching off, and short-circuiting a two-phase asynchronous motor with a stationary and rotating open-circuited and short-circuited two-phase rotor.

For example, to determine the currents in the stator phases and the e.m.f. in the rotor phases when a two-phase stator is switched on with an open-circuited rotating rotor, we obtain

\[ \Delta i_a= \frac{u_{am}-r_1 i'_a} {\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}}, \]

\[ \Delta i_b= \frac{u_{bm}-r_1 i'_b} {\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}}, \tag{26} \]

\[ \Delta E_\alpha= 2\frac{M}{\Delta t}\Delta i_a +2M\omega\left(i'_b+\frac{\Delta i_b}{2}\right)-2E'_\alpha, \]

\[ \Delta E_\beta= 2\frac{M}{\Delta t}\Delta i_b -2M\omega\left(i'_a+\frac{\Delta i_a}{2}\right)-2E'_\beta. \]

From equations (26), the projections of the e.m.f. \(E_\alpha\) and \(E_\beta\) on the stator axes are determined. To determine the actual rotor e.m.f.’s, it is necessary to perform

the inverse transition from the projections of the rotor e.m.f. to their actual values. The equations for the inverse transition can be obtained by solving equations (19) or directly from Fig. 1. They have the form:

\[ \begin{aligned} u_\alpha &= u'_\alpha \cos\Theta + u'_\beta \sin\Theta,\\ u_\beta &= u'_\beta \cos\Theta - u'_\alpha \sin\Theta . \end{aligned} \tag{27} \]

To determine the stator and rotor currents when a two-phase stator is switched on with a stationary short-circuited rotor, when the axes of the stator and rotor windings coincide, we obtain equations entirely analogous to the equations for a single-phase machine,

\[ \Delta i_\alpha = \frac{ \left(u_{am} - r_1 i'_\alpha\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) + \dfrac{M}{\Delta t} r_2 i'_\alpha }{ \left(\dfrac{r_1}{2}+\dfrac{L_1}{\Delta t}\right) \left(\dfrac{r_2}{2}+\dfrac{L_2}{\Delta t}\right) - \left(\dfrac{M}{\Delta t}\right)^2 } \tag{28} \]

and so on.

Having calculated the currents in the stator and rotor phases, one can determine the magnitude of the electromagnetic torque from the known equation

\[ T=\frac{3}{2}x_m(i_\beta i_a-i_\alpha i_b). \tag{29} \]

In equation (29) all quantities are given in relative units.

Basic equations of the switching-on processes of a three-phase asynchronous machine

The difference between a three-phase, or, in the general case, a multiphase system and a two-phase one consists in the need to take into account the mutual induction between the windings on the stator and on the rotor. This leads to an increase in the number of voltage-balance equations, and the equations themselves become more complicated.

In the present work, two methods are considered for investigating transient processes in three-phase asynchronous machines.

Method of investigation using equations in the untransformed system of coordinate axes

In this system the axes of the windings are taken as the coordinate axes. A characteristic feature of the system is that all frequencies of the currents and voltages and the speeds of rotation of the windings and magnetic fields are real physical quantities corresponding to the physical picture of the machine operation.

In a three-phase asynchronous machine with a three-phase rotor there are six electrical circuits: three on the stator and three on the rotor. The voltage-balance equations for each of these circuits can be written in the following form:

\[ u=ri+\frac{d}{dt}(Li)+\sum_{k=1}^{n}\frac{d}{dt}(M_k i_k). \tag{30} \]

Mutual induction exists between the windings on the stator and on the rotor, as well as between the stator and rotor windings. Neglecting saturation, it may be assumed that the mutual inductance between two stator windings or between two rotor windings remains unchanged as a result—

...the effect of the constant air gap, while between the stator and rotor windings it varies in proportion to the cosine of the angle of rotation of the rotor relative to the stator. The self-inductance of the individual stator and rotor windings may be assumed constant.

The general equation derived on the basis of the assumptions listed for calculating the currents of the transient process proved to be very cumbersome, and therefore of little use for practical calculations (for this reason it is not given here). However, it can be used for certain particular cases of calculating switching-on, switching-off, and short-circuit processes.

For example, in order to determine the starting current when a three-phase asynchronous motor with a three-phase rotor is switched on, when the stator and rotor windings are connected in star and the axes of the windings coincide, the following calculation equations are obtained:

\[ \Delta i_{s1}= \frac{(u_{23m}+2u_{12m}-3r_1 i'_{s1})z+3r_2 i'_{r1}y} {3(xz-y^2)}, \]

\[ \Delta i_{s2}= \frac{(u_{23m}-u_{12m}-3r_1 i'_{s2})z+3r_2 i'_{r2}y} {3(xz-y^2)}, \]

\[ \Delta i_{r1}=- \frac{(u_{23m}+2u_{12m}-3r_1 i'_{s1})y+3r_2 i'_{r1}x} {3(xz-y^2)}, \tag{31} \]

\[ \Delta i_{r2}=- \frac{(u_{23m}-u_{12m}-3r_1 i'_{s2})y+3r_2 i'_{r2}x} {3(xz-y^2)}. \]

Here

\(u_{12m}\) and \(u_{23m}\) are the instantaneous values of the line voltages at the middle of the interval under consideration,

\[ x=\frac{r_1}{2}+\frac{L_{01}}{\Delta t}; \qquad y=\frac{3M}{2\Delta t}; \qquad z=\frac{r_2}{2}+\frac{L_{02}}{\Delta t}, \]

\[ L_{01}=L_{s1}+\frac{3}{2}M \]

is the total inductance of the three-phase winding.

The calculation equations become still simpler if one starts from the equilibrium equations of the phase voltages.

The advantage of studying this transient process in an untransformed system of coordinate axes is that, as a result of calculation by formulas (31), one obtains directly the actual currents in the stator and rotor phases, which can be checked against oscillograms taken at the moment of switching on.

Transformation method using equations in the “transformed” system of coordinate axes

With the aid of the method of “two-phase components,” a three-phase, or in the general case a multiphase, machine is reduced to an equivalent two-phase one. The essence of the transformation is that the resultant m.m.f. is represented not as the result of the addition of three m.m.f. waves corresponding to the three phases of the winding, but as the result of the addition of two equivalent m.m.f. waves arranged at an angle of 90 electrical degrees.

The replacement of three vectors arranged in space at an angle of 120° to one another by two vectors displaced relative to one another by an angle of 90° (Fig. 2) is carried out as follows:

Usually the axes of the new system are arranged so that the axis \(a\) coincides with axis 1, while the axis \(b\) is perpendicular to the axis \(a\) and is located between the axes of phases 1 and 2; then the projections of the current components along the new axes onto the old ones, in the general case when the sum of the instantaneous values is not equal to zero, will be:

Fig. 2

\[ i_1=i_a+i_0, \]

\[ i_2=-\frac{1}{2}i_a+\frac{\sqrt{3}}{2}i_b+i_0, \tag{32} \]

\[ i_3=-\frac{1}{2}i_a-\frac{\sqrt{3}}{2}i_b+i_0. \]

The corresponding currents of the equivalent two-phase system can be obtained by solving the system of equations (32)

\[ i_a=\frac{2}{3}\left(i_1-\frac{1}{2}i_2-\frac{1}{2}i_3\right), \]

\[ i_b=\frac{2}{3}\left(\frac{\sqrt{3}}{2}i_2-\frac{\sqrt{3}}{2}i_3\right), \tag{33} \]

\[ i_0=\frac{1}{3}(i_1+i_2+i_3). \]

The equations obtained after transforming a three-phase system into a two-phase one are valid for any three-phase system of windings arranged in space at an angle of \(120^\circ\), irrespective of whether these windings rotate or are stationary. In this case the axes of the new windings \(a\) and \(b\) are stationary relative to the axes of windings 1, 2, and 3.

The equations obtained after transforming a three-phase system into a two-phase one are transformed to a single system of coordinate axes. Then the calculation of the transient processes of a three-phase asynchronous machine can be carried out using the equations derived for a two-phase machine.

Appendix

Example 1. Determine the stator current and the rotor e.m.f. when a single-phase stator is switched on, with an open and stationary single-phase rotor. It is known that the axes of the stator and rotor windings coincide, the voltage at the instant of switching passes through zero, the magnetic circuit is saturated, \(U=150\ \text{V}\).

Motor parameters: \(r_1=0.32\ \Omega,\quad x_1=0.65\ \Omega,\quad x_m=f(i)\) (Fig. 3),

\[ U_n=120\ \text{V},\quad I_n=19.2\ \text{A},\quad n=1435\ \text{rpm}. \]

Taking into account that

\[ L_1=L_{s1}+M=\frac{x_1+x_m}{2\pi f}, \]

equations (8) may be written in the following form:

\[ \begin{aligned} \Delta i_1&=2\pi f\Delta t\, \frac{u_{1m}-r_1 i'_1}{r_1\pi f\Delta t+x_1+x_m},\\ \Delta E_2&=\frac{x_m}{\pi f\Delta t}\,\Delta i_1-2E'_2, \end{aligned} \tag{34} \]

where

\(x_1\)—the leakage reactance,
\(x_m\)—the reactance due to the main flux.

Fig. 3

Substituting into equation (34) the parameters of the machine under investigation and the selected time interval \(\Delta t=\dfrac{1}{12}\) of a period \(\left(\dfrac{1}{600}\ \text{sec}\right)\), we obtain

\[ \begin{aligned} \Delta i&=0.523\,\frac{u_{1m}-0.32\,i'_1}{0.7+x_{m\,\text{av}}},\\ \Delta E_2&=3.82\,x_{m\,\text{av}}\Delta i-2E'_2. \end{aligned} \tag{35} \]

Here

\[ x_{m\,\text{av}}=\frac{x'_m+x''_m}{2},\qquad u_{1m}=U\sin 2\pi f t_m. \]

The initial conditions of the process will be:

\[ u'_1=0;\qquad i'_1=0;\qquad E'_2=0;\qquad x'_m=20\ \Omega. \]

For the calculation, which will be carried out in named units, Table 3 is compiled.

In the first column of the table, the time is entered in intervals of \(1/600\) sec; in the second column, the average values of the applied voltage. Thus, with the selected time interval, taking into account the sinusoidal variation of the voltage, the first two columns can be filled in at once, knowing the phase of the voltage at the instant of switching on.

Having established the law of variation \(x_m=f(t)\) during the first complete period, in further calculation, with some experience, one can find the average values \(x_{m\,\mathrm{av.}}\) without continuing the curve \(x_m=f(t)\).

Table 3

etc.

From the data of Table 3, Fig. 4 gives the curves of current and voltage as functions of time. For comparison, experimental data are also given in the same figure.

In Fig. 4: 1—calculated curve,
2—experimental curve.

Example 2. Determine the stator and rotor currents when a single-phase stator is switched on with a closed single-phase rotor rotating at constant speed.

It is known that switching occurs at the instant when the voltage passes through zero, i.e., at \(t=0\), \(u'_1=50\sin(2\pi f t+\alpha_0)=0\); the magnetic circuit is unsaturated, \(\omega=0,8\cdot 2\pi f\).

The parameters of the machine are

\[ r_1=0,32\ \text{ohm},\qquad r_2=0,4\ \text{ohm},\qquad x_1=0,65\ \text{ohm}, \]

\[ x_2=0,72\ \text{ohm},\qquad x_m=20,0\ \text{ohm}. \]

For the chosen time interval

\[ \Delta t=\frac{1}{600}\ \text{sec}, \]

we obtain

\[ \frac{r_1}{2}+\frac{L_1}{\Delta t} =0,16+\frac{0,65+20}{2\pi\cdot 50}\cdot 600 =39,6\ \text{ohm}. \]

\[ \frac{r_2}{2}+\frac{L_2}{\Delta t} =0,2+\frac{0,72+20}{2\pi\cdot 50}\cdot 600 =39,8\ \text{ohm}. \]

\[ \frac{M}{\Delta t} =\frac{20}{2\pi\cdot 50}\cdot 600 =38,2\ \text{ohm}; \qquad \gamma=\operatorname{arctg}\left(\frac{2}{0,8\cdot 2\pi\cdot 50}\cdot 600\right)=78^\circ \]

If we take

\[ \frac{r_1}{2}+\frac{L_1}{\Delta t} = \frac{r_2}{2}+\frac{L_2}{\Delta t} =39,7\ \text{ohm}, \]

then the equations (12) may be used. Then, after substituting the calculated parameters into these equations, we obtain the following calculation formulas:

\[ \Delta i_1= \frac{(u_{1m}-0,32\,i'_1+16\sin\omega t_m\,i'_2)+(0,4i'_2-16\sin\omega t_m\,i'_1)\,0,983\sin(78-\omega t)_m} {38,4\,[1,035-\sin^2(78-\omega t)_m]}, \]

\[ \Delta i_2= -\frac{(u_{1m}-0,32\,i'_1+16\sin\omega t_m\,i'_2)\,0,983\sin(78-\omega t)_m+(0,4i'_2-16\sin\omega t_m\,i'_1)} {38,4\,[1,035-\sin^2(78-\omega t)_m]}. \tag{36} \]

Resultant magnetizing current

\[ i_1 = 3.65 + 3.65 = 7.3\ \text{A}. \]

To this current, according to the curve \(x_m = f(i)\) in Fig. 3, there corresponds \(x''_m = 16.4\ \Omega\). Consequently,

\[ x_{m\,\text{av.}}=\frac{20+16.4}{2}=18.2\ \Omega. \]

Substituting this value into formula (35), we find, in the second approximation,

\[ \Delta i_1 = 3.98\ \text{A}, \]

\[ E_2 = 18\ \text{V}. \]

Consequently,

\[ i_1 = 3.65 + 3.98 = 7.63\ \text{A}, \]

\[ E_2 = 129 + 18 = 147\ \text{V}, \]

to which, according to the curve of Fig. 3, there corresponds \(x''_m = 16.3\ \Omega\). Since the discrepancies with the quantities found in the first approximation are not very large, the calculation in the third approximation may be omitted.

Therefore the quantities found in the second approximation are entered in Table 3 as final.

Before proceeding to the calculation of the fifth row, in Fig. 4, from the data of Table 3, the initial portion of the curve \(x_m = f(t)\) is constructed. Extrapolating this curve over the time interval \(\dfrac{\Delta t}{2}\), we obtain an approximate value of \(x_{m\,\text{av.}}\) for calculating the fifth row.

Initial conditions for calculating the fifth row:

\[ i'_1 = 7.63\ \text{A}, \qquad E'_2 = 147\ \text{V}; \qquad u_{1m}=145\ \text{V}; \]

\[ x_m = 14.4\ \Omega \quad \text{(according to the curve of Fig. 4)}. \]

Substituting these values into formula (35), we obtain

\[ \Delta i_1 = 4.97\ \text{A}, \qquad \Delta E_2 = -22\ \text{V}. \]

Consequently,

\[ i_1 = 12.60\ \text{A}; \qquad E_2 = 125\ \text{V}. \]

Fig. 4

To the current value \(i_1 = 12.6\ \text{A}\), according to the curve of Fig. 3, there corresponds \(x''_m = 12.3\ \Omega\). Checking the correctness of the average value \(x_{m\,\text{av.}}\) found by extrapolation, we obtain:

\[ x_{m\,\text{av.}}=\frac{x'_m+x''_m}{2}=\frac{16.3+12.3}{2}=14.3\ \Omega. \]

The discrepancy proved insignificant, and therefore the computed quantities may be entered in the fifth row of the table as final.

Further, continuing the curve \(x_m = f(t)\), one may proceed to the calculation of the sixth row, which is carried out analogously to the calculation of the fifth row, etc.

of these deviations, it is meaningful to compute the coefficients substituted into the calculation formulas as an average between the arithmetic mean and the actual mean value.

Calculations for any specified operating condition are carried out in an analogous manner.

The calculations, performed by the method described, for currents and electromotive forces during switching on and switching off of two-phase and three-phase machines with an open-circuited and a closed rotating and stationary rotor gave very

Fig. 5

close results in comparison with those obtained experimentally; for example, for the starting current of a three-phase motor with a three-phase rotor, the greatest deviation of the calculated curve from the oscillogram did not exceed 5%.

Conclusions

The investigations presented of the processes of switching on, switching off, and short-circuiting asynchronous machines have shown that the application of the method of successive intervals makes it possible, in a simple form accessible for direct computation, to obtain the basic equations for calculating the transient processes under consideration. The derivation of these equations is reduced to simple arithmetic and algebraic operations, and their solution presents no mathematical difficulties. The accuracy of the solution of these equations depends, apart from the assumptions adopted, on the size of the interval chosen.

Analyzing the equations obtained in the present work and their application to the solution of practical problems, the following conclusions may be drawn.

1. Existing theories, when considering transient processes, operate with transient parameters, the calculation or experimental determination of which in individual cases presents considerable difficulties and introduces additional errors.

In solving these problems by means of the method of successive intervals, it is necessary to operate only with the basic parameters: the active resistances of the stator and rotor windings \(r_1\) and \(r_2\), the leakage inductances of the stator and rotor windings \(L_{s1}\) and \(L_{s2}\), and the inductance due to the main flux \(M\).

1. A merit of the method of successive intervals is the possibility of substantially reducing the number of assumptions and approximations, which makes it possible to reduce the error of the calculation, in particular:

a) the calculation can be performed taking saturation into account; moreover, the analytical expressions of the equations for calculating transient processes by the method of successive intervals have one and the same form both without allowance for saturation and with allowance for it;

b) in calculating transient processes, the voltage drop in the active resistance of the stator winding is often neglected, whereas it has a substantial influence on the character of the course of the transient process, especially in machines of small power, on the order of several kilowatts. In calculation by the method of successive intervals, allowance for the indicated resistance does not introduce any substantial complications into the calculation.

1. The problem of calculating the transient processes of a single-phase asynchronous motor has up to now had no general solution. Using the method of successive intervals, this problem can be solved; moreover, as the calculations performed have shown, the result is a picture very close to reality.

2. Calculation by the method of successive intervals requires the performance of a significantly smaller number of simpler computational operations in comparison with other analytical methods of calculation.

3. As a result of replacing complex mathematical operations by simple arithmetical and algebraic operations, the method of successive intervals is a sufficiently convenient engineering method of calculation.

LITERATURE

1. Kazovskii E. Ya.
   a) Generalized consideration of transient regimes in asynchronous and synchronous machines, Elektrosila No. 2–3, 1945.
   b) Transient processes in asynchronous machines during switching-on and short circuits, Elektrichestvo No. 6, 1947.
   c) Transient processes in asynchronous machines during switching-on and load application, VEP No. 2, 1949.

2. Kas’yanov V. T.
   a) Allowance for the influence of the transverse armature reaction on transient processes in uncompensated direct-current machines, Elektrosila No. 5, 1948.
   b) On the calculation of a sudden short circuit of a compensated direct-current machine, Tekhotchet, 1949.
   c) Asynchronous machines at variable frequency, Elektrichestvo No. 2, 1949.
   d) Calculation of transient processes of regulation of the magnetic flux of electrical machines with allowance for saturation, Tekhotchet, 1949.
   e) On the calculated determination of the rate of rise of the exciter voltage, Tekhotchet, 1950.