Calculation of Transient Processes in Asynchronous Machines with a Specified Speed Variation

G. A. Sipaylov

The wide application of alternating-current machines, in particular asynchronous machines, in complex regulated installations, and the introduction of alternating current into the electric drives of shipboard and other special installations, has made it necessary to study transient processes in asynchronous machines.

The entire, relatively short, course of the fruitful development of the theory of alternating-current machines may conventionally be divided into two periods.

The first period (approximately up to 1935–40) is characterized by the fact that the authors of works, when considering questions of various methods of starting and braking asynchronous motors and other questions of nonstationary processes in asynchronous machines, proceeded, as a rule, from the equation of motion of the electric drive and used a simplified or complete equation for torque in the steady-state regime. In taking into account the transient processes of the electric drive, they limited themselves to considering only one mechanical inertia, without at all considering the electromechanical transient processes in which, in addition to the equation of motion, the equations of balance of e.m.f. and voltage drops in the electrical circuits must be investigated. A brief account of the principal works of this period is given in [6].

The second period is characterized by a deeper analysis of transient processes in asynchronous machines, namely:

1. Consideration of electromagnetic transient processes caused by commutation processes in the stator circuit (switching on, switching over, short circuit, etc.) under the condition of constant rotor speed.

2. Investigation of the influence of a specified change in speed on the character of the course of electromagnetic transient processes.

3. Consideration of the dynamics of an electric drive with allowance for transient electromagnetic processes in asynchronous machines.

4. Investigation of the simultaneous course of transient electromagnetic and mechanical processes (for example, during the starting of a squirrel-cage motor from a generator of power commensurable with it).

In such an analysis, the whole modern mathematical apparatus is used: operational calculus, the method of symmetrical components, tensor analysis, matrix calculus, special functions, etc.

Among the first works of the second period may be included the article by R. M. Kantor [1], in which new methods are considered for the investigation of transient-

processes in asynchronous machines. The author uses equations for the resultant magnetomotive forces of the stator and rotor and, applying to them the displacement theorem of operational calculus, obtains transformed equations, by solving which he determines the transient currents, power, and torques. The author’s inaccuracy consists in confusing the method of complex operational equations with the method of symmetrical components.

A major step forward in the study of transient processes in asynchronous machines was the article by E. Ya. Kazovskii [3]. In it, as well as in his subsequent works [4; 5], the author developed a method for determining currents, flux linkages, and electromagnetic torque under various transient processes: during switching-on, load application, short circuits, and other processes of asynchronous machines, by means of complex operational equations.

In these works the author made the first attempts at an analytical study of the influence of speed variation on the character of the transient process.

As in the case of constant speed, the author determines the torques developed by the motor under a linear change of speed in a transient process caused by an abrupt change of speed, using the complex representation of variables.

In a number of practical problems, despite the fact that the method of complex operational equations facilitates the solution and writing of differential equations, the computational work nevertheless becomes so cumbersome that numerical calculation still proves to be simpler.

Comparatively recently, a transition has been taking shape from analytical methods of analysis and calculation to graphical ones.

The use of the circle diagram for studying transient processes in asynchronous machines was proposed by Soviet engineers [4; 7].

The graphical method for calculating transient processes caused by an abrupt change of speed makes it possible to make full use of the advantages of the complex representation. With a symmetrical applied voltage, the calculation is based on the ordinary circle diagram.

When investigating a continuous change of speed, in a number of cases it is replaced by a stepwise one, which makes it possible, for each section, to use the regularities valid at constant speed [7].

From the first works employing the graphical method it may be noted that, for obtaining results in a specific problem, this method is more visual and requires less time, and therefore, in those cases where special accuracy is not required, it is more expedient. Therefore at the present time many authors, in studying transient processes in asynchronous machines, resort to the use of the graphical method.

Of the analytical calculation methods, the simplest is the method of successive intervals.

The present article gives a procedure for calculating the electromagnetic torque for a prescribed speed variation and the electromagnetic torque of an asynchronous machine operating in braking mode, on the basis of the equations set forth in the author’s work [9].

The assumptions made in the calculation are analogous to those adopted in [9].

In order to judge the laboriousness of the computational work in calculating transient processes in asynchronous machines by the method of successive intervals in comparison with other analytical methods [3], the course of the calculation is accompanied by numerical examples.

Calculation of the Electromagnetic Rotating Torque of an Asynchronous Motor under Rotor Accelerations

The influence of a change in speed on the nature of electromagnetic transient processes is of substantial practical interest. Acceleration of the rotor gives rise to significant rotating torques, which can cause an ordinary asynchronous machine to attain, for a time, rotational speeds above synchronous speed.

In recent years this question has received considerable attention from specialists in electrical machines [3—8].

However, owing to the complexity of the problem, all authors [3—8], when investigating the influence of acceleration, in addition to generally accepted assumptions, as a rule neglect the active resistance of the stator winding, whereas it has a substantial effect on the nature of the transient process in asynchronous machines, especially in machines of small and medium power.

The law of variation of speed or slip is usually taken to be rectilinear, on the grounds that the mechanical inertia of the rotor smooths out slip oscillations. This assumption is valid in many operating conditions (starting, allowance for speed variation when voltage is applied, etc.) over a comparatively large part of the process, at least over a certain interval of time [3].

When calculating the processes considered by the method of successive intervals, taking account of the active resistance of the stator winding does not cause any noticeable complications.

To take into account the influence of the change in speed, one may, on the basis of the foregoing, assume the acceleration to be constant, or find an approximate expression for the slip as a function of time \(s=f(t)\) from the approximate characteristic \(M_e=f(s)\), by the method known from the general course of electrical machines.

In the article by E. Ya. Kazovskii [3], a numerical example is given for calculating the electromagnetic rotating torque in the presence of constant rotor acceleration by the method of complex operational equations. The course of the calculation is illustrated by tables.

For a more illustrative comparison with respect to the laboriousness of the computational operations in calculating the electromagnetic rotating torque by the method of successive intervals, we shall use the machine with parameters [3]:

\[ x_1=x_2=3{,}07,\qquad x_m=2{,}99,\qquad r_1=r_2=0{,}01, \]

where, in relative units,

\(x_1=L_1\) — stator inductance,
\(x_2=L_2\) — rotor inductance,
\(x_m=M\) — mutual inductance,
\(r_1\) and \(r_2\) — active resistances of the stator and rotor windings.

Let us consider an acceleration regime corresponding to acceleration of the motor to nominal speed in 12 periods, i.e., we take the acceleration

\[ 2a=\frac{1}{12\cdot 2\pi}=0{,}01327, \]

then the angular velocity

\[ \omega=2at=0{,}01327\,t. \]

It should be noted that, when calculating by the method of successive intervals, the entire course of the calculation will remain exactly the same if the acceleration is not constant but is, for example, a specified (graphically or analytically) function of time.

The electromagnetic torque of an induction machine is determined by the equation:

\[ M_e=\frac{3}{2}x_m(i_\beta i_a-i_\alpha i_b), \tag{1} \]

where \(i_a\) and \(i_b\) are the currents along the direct and quadrature axes of the stator of a three-phase induction machine transformed into a two-phase one; \(i_\alpha\) and \(i_\beta\) are, analogously, the rotor currents referred to the stator frequency.

To determine the stator and rotor currents, we shall use the calculation formulas [9]:

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

\[ \Delta i_b=\frac{-AK_2+BK_1-CK_4-DK_3}{K_9}, \tag{2} \]

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

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

Substituting the specified parameter values and the chosen time interval

\[ \Delta t=0.523\left(\frac{1}{12}\text{ period or }\frac{1}{600}\text{ sec.}\right) \]

into the expression for the coefficients [9], we obtain:

\[ x=\frac{r_1}{2}+\frac{L_1}{\Delta t}=5.875;\qquad z=\frac{r_2}{2}+\frac{L_2}{\Delta t}=5.875, \]

\[ y=\frac{M}{\Delta t}=5.717;\qquad u=\frac{M\omega}{2}=0.0194t; \]

\[ v=\frac{L_2\omega}{2}=0.02037t, \]

\[ K_1=z(xz-y^2)+v(xv-yu)=10.76+0.0001273t^2, \]

\[ K_2=y(uz-yv)=0.000567t, \]

\[ K_3=y(xz-y^2)=10.47, \]

\[ K_4=y(xv-yu)=0.03573t, \]

\[ K_5=y(xz-y^2)+u(xv-yu)=10.47+0.000124t^2, \]

\[ K_6=x(uz-yv)=0.000583t, \]

\[ K_7=x(xz-y^2)=10.76, \]

\[ K_8=x(xv-yu)=0.0367t, \tag{3} \]

\[ K_9=(xz-y^2)^2+(xv-yu)^2=3.36+0.000039t^2, \]

\[ A=u_{am}-r_1 i'_a=u_{am}-0.01i'_a, \]

\[ B=u_{bm}-r_1 i'_b=u_{bm}-0.01i'_b, \]

\[ C=-r_2 i'_\alpha-L_2\omega i'_\beta-M\omega i'_b =-0.01i'_\alpha-\omega(3.07i'_\beta+2.99i'_b), \]

\[ D=-r_2 i'_\beta+L_2\omega i'_\alpha+M\omega i'_a =-0.01i'_\beta+\omega(3.07i'_\alpha+2.99i'_a). \]

To carry out the calculation, we compile Table 1. In the first column of the table we enter the initial conditions. With a sinusoidal applied voltage, assuming that at the instant of switching on the voltage in phase \(a\) passes through zero, we fill in the first two rows through to the end of the table. For the chosen time interval one may likewise fill in the following 9 rows through to the end of the table. These operations, as follows from Table 1, are not laborious.

Table 1

etc.

To speed up the calculation, the coefficients $K_2$ and $K_6$ may be neglected, in view of their insignificance in the initial part of the calculation, when the rotational speed is small, and they may be taken into account when they begin to affect the magnitude of the current increment.

To obtain the second column, we substitute the initial conditions into the expressions for $A$, $B$, $C$, and $D$.

$$ A=u_{am}-r_1 i'_a=u_{am}; $$

$$ B=u_{вm}-r_1 i'_в=u_{вm}; $$

$$ C=0,\quad D=0, $$

where $u_{am}$ and $u_{вm}$ are the average values of the voltage over the time interval $\Delta t$.

As was indicated in [9], these average values may be taken, first, as the arithmetic mean between the voltages at the beginning of the interval

$$ u'_a=U\sin(\omega_1 t'+\alpha_0)=\sin\alpha_0=0, $$

$$ u'_в=U\sin\left(\omega_1 t'+\alpha_0+\frac{\pi}{2}\right) =\sin\left(\alpha_0+\frac{\pi}{2}\right)=1 $$

and the voltages at the end of the interval

$$ u''_a=U\sin(\omega_1 t''+\alpha_0) =\sin\left(\frac{\pi}{6}+\alpha_0\right)=0{,}5, $$

$$ u''_в=U\sin\left(\omega_1 t''+\alpha_0+\frac{\pi}{2}\right) =\sin\left(\frac{2}{3}\pi+\alpha_0\right)=0{,}866, $$

i.e.

$$ u_{am1}=\frac{u'_a+u''_a}{2}=\frac{0+0{,}5}{2}=0{,}25, $$

$$ u_{вm1}=\frac{u'_в+u''_в}{2}=\frac{1+0{,}866}{2}=0{,}933, $$

where $U=1$ is the rated voltage,

$\omega_1=1$ is the angular frequency,

$\alpha_0=0$ is the initial phase,

$t'=0$ is the time at the beginning of the interval,

$t''=\Delta t=\dfrac{\pi}{6}$ is the time at the end of the interval;

secondly, as the actual mean value corresponding to the time instant

\[ t_m = t + \frac{\Delta t}{2}, \]

\[ u_{am2} = U \sin (\omega_1 t_m + \alpha_0) = \sin \frac{\pi}{12} = 0.259, \]

\[ u_{bm2} = U \sin \left(\omega_1 t_m + \alpha_0 + \frac{\pi}{2}\right) = \sin \frac{7}{12}\pi = 0.966. \]

Calculation practice shows that a more correct value for \(u_{am}\) will be the mean between the arithmetic mean and the actual mean, i.e.,

\[ u_{am} = \frac{u_{am1} + u_{am2}}{2} = \frac{0.25 + 0.259}{2} = 0.255, \]

\[ u_{bm} = \frac{u_{bm1} + u_{bm2}}{2} = \frac{0.933 + 0.966}{2} = 0.949. \]

Such a calculation is in no way reflected in the duration of the calculation, since, for a sinusoidal variation of the voltage, at

\[ \Delta t = \frac{1}{12} \]

of a period it is necessary to calculate only three mean values \(u_{am}\) for the calculation of the whole process.

Having determined \(A\), \(B\), \(C\), and \(D\), we find the current increments from equation (2) and the currents from equations (4):

\[ \begin{aligned} i_a &= i'_a + \Delta i_a, \qquad & i_\alpha &= i'_\alpha + \Delta i_\alpha, \\ i_b &= i'_b + \Delta i_b, \qquad & i_\beta &= i'_\beta + \Delta i_\beta, \end{aligned} \tag{4} \]

where \(i'_a\), \(i'_b\), \(i'_\alpha\), and \(i'_\beta\) are the values of the currents at the beginning of the interval.

Using the calculated current values, by means of equation (1), we determine the electromagnetic rotating torque.

We then proceed to the calculation of the next, third column, for which the calculated data of the second column will serve as the initial conditions, and so on.

As a drawback it should be noted that, in the calculation, the accuracy that can be guaranteed by a slide rule often proves insufficient; therefore, in order to avoid errors and reduce the calculation error, the computations should be carried out with the aid of an arithmometer.

Calculation of the electromagnetic rotating torque of an asynchronous machine operating in the asynchronous braking regime at negative synchronous speed

The numerical determination of the electromagnetic rotating torque when an asynchronous machine is switched on was carried out with the aid of an integraph [10] for a number of parameter ratios during rotation of the machine with negative synchronous speed. The same calculations were performed by E. Ya. Kazovskii [4] on the basis of formulas obtained by him with the aid of complex operational equations.

In article [4] a comparison is given of the calculated curve of the transient electromagnetic rotating torque of an asynchronous machine operating in ...

in the asynchronous braking mode, with the curve obtained by means of the integraph.

To carry out the calculation of the process under consideration by the method of successive intervals, it is necessary to know the basic machine parameters \(r_1, r_2, L_1, L_2\), and \(M\).

To determine them we shall use the data of the machine from one of the examples [4]

\[ a_s=a_r=0.04036;\quad a'_s=a'_r=0.34;\quad \sigma=0.118. \]

Using the known relations

\[ T_s=\frac{1}{a_s}=\frac{x_1}{r_1},\quad T'_s=\frac{1}{a'_s}=\frac{x'_1}{r_1}\quad \text{and}\quad \sigma=1-\frac{x_m^2}{x_1x_2}, \]

where \(T_s\) is the time constant of the stator winding with the rotor winding open,

\(T'_s\) is the time constant of the stator winding with the rotor winding closed, one can determine:

\[ x_m=0.94x_1,\quad x_1=x_2,\quad r_1=r_2. \]

Since the data given in the example under consideration are insufficient for finding numerical values of the parameters (the active resistances are not given), it will be necessary to confine ourselves only to a qualitative comparison of the calculation results.

Let us take: \(x_1=L_1=2\), then \(x_m=M=1.88\), \(r_1=r_2=0.0807\).

For negative synchronous speed \(\omega=-1\).

As in the preceding example, we choose the time interval

\[ \Delta t=0.523\left(\frac{1}{12}\ \text{period}\right). \]

Using expressions (23) and (25) [9], we compute the auxiliary quantities and calculation coefficients:

\[ x=\frac{r_1}{2}+\frac{L_1}{\Delta t}=3.864,\quad y=\frac{M}{\Delta t}=3.595, \]

\[ z=\frac{r_2}{2}+\frac{L_2}{\Delta t}=3.864,\quad u=\frac{M\omega}{2}=-0.94, \]

\[ v=\frac{L_2\omega}{2}=-1. \]

\[ K_1=z(xz-y^2)+v(xv-yu)=8.268,\quad K_5=y(xz-y^2)+u(xv-yu)=7.696, \]

\[ K_2=y(uz-yv)=-0.137,\quad K_6=x(uz-yv)=-0.147, \]

\[ K_3=y(xz-y^2)=7.240,\quad K_7=x(xz-y^2)=7.783, \]

\[ K_4=y(xv-yu)=-1.743,\quad K_8=x(xv-yu)=-1.874, \]

\[ K_9=(xz-y^2)^2+(xv-yu)^2=4.291, \]

\[ A=u_{am}-0.0807\,i'_{\alpha},\quad C=-0.0807\,i'_{\alpha}+2i'_{\beta}+1.88\,i'_{g}, \]

\[ B=u_{am}-0.0807\,i'_{\beta},\quad D=-0.0807\,i'_{\beta}-2i'_{\alpha}-1.88\,i'_{a}. \]

Substituting the values of the coefficients into equations (2), we obtain the calculation formulas:

\[ \Delta i_a = 1.927 A - 0.0319 B - 1.687 C - 0.406 D, \]

\[ \Delta i_b = 0.0319 A + 1.927 B + 0.406 C - 1.687 D, \]

\[ \Delta i_\alpha = -(1.794 A - 0.0343 B - 1.814 C - 0.437 D), \]

\[ \Delta i_\beta = -(0.0343 A + 1.794 B + 0.437 C - 1.814 D). \]

To carry out the calculation, we compile Table 2.

Table 2

etc.

We specify the initial conditions at \(t = 0\): \(i'_a = 0\), \(i'_b = 0\), \(i'_\alpha = 0\), \(i'_\beta = 0\), \(u'_a = 0\), and \(u'_b = 1\), and enter them in the first column of Table 2.

The course of the calculation was discussed in detail in the preceding example.

If braking occurs at a variable speed and the nature of the speed variation is known, then, in order to calculate the electromagnetic torque, it is necessary to compile a table similar to Table 1,

in which the coefficients depending on the change in speed are to be entered, and then the calculation is to be carried out in the same order as shown in the examples considered.

Conclusions

1. The calculation of the electromagnetic torque in asynchronous machines at constant and variable rotational speed by the method of successive intervals reduces to simple arithmetic and algebraic operations, with a considerably smaller number of computational operations than in other analytical calculation methods. The accuracy of the calculation depends both on the assumptions adopted and on the size of the interval chosen. For practical calculations one may take the time interval \(\Delta t = \frac{1}{12}\) of a period, and only in rare cases, when greater accuracy is required, should the interval be reduced to \(\Delta t = \frac{1}{18}\) of a period.

2. In calculation by the method of successive intervals it is necessary to operate only with the basic parameters of the machine: the active resistances of the stator and rotor windings \(r_1\) and \(r_2\), the inductances of the windings \(L_1\) and \(L_2\), and the mutual inductance of the stator and rotor windings \(M\). Taking account of the active resistance of the stator winding does not introduce any noticeable complications into the calculation equations.

Literature

1. Kantor R. M. Analysis of transient processes in asynchronous machines. Bulletin of VEI, No. 2, 1935.

2. Kas’yanov V. T. Calculation of starting, braking, reversing, and speed- and load-regulation processes of DC motors. VEI, No. 10, 1944.

3. Kazovskii E. Ya. Generalized consideration of transient regimes in asynchronous and synchronous machines. Elektrosila, Nos. 2–3, 1945.

4. Kazovskii E. Ya. Transient processes in asynchronous machines during switching-on and short circuits. Electricity, No. 6, 1947.

5. Kazovskii E. Ya. Transient processes in asynchronous machines during switching-on and load application. VEI, No. 2, 1949.

6. Zaretskii N. S. Study of a three-phase asynchronous motor in transient regimes. Proceedings of LETI, 1947.

7. Shubenko V. A., Pinchuk I. S. Graphical method for calculating transient processes in an asynchronous motor. Electricity, No. 2, 1950.

8. Yanko-Trinitskii A. A. Equations of transient electromagnetic processes of an asynchronous motor and their solution. Electricity, No. 3, 1951.

9. Sinailov G. A. Calculation of the processes of switching on asynchronous machines by the method of successive intervals. Proceedings of TPI, Vol. 72, 1952.

10. Gilfillan E. S., Kaplan E. L. Transient Torques in Squirrel-Cage Induction motors, With Special Reference to Plugging. Tr. AIEE, Vol. 60, p. 1200, 1941.