A SIMPLIFIED METHOD FOR CALCULATING ELECTROMECHANICAL TRANSIENT CONDITIONS OF AN INDUCTION MOTOR

M. P. TABINSKY

(Presented by the scientific seminar of the Electromechanical Faculty)

The operation of AC electric drives intended for service with a high frequency of starting, braking, and regulating switchings, as well as the development of pulse speed control, has revealed the great influence of electromagnetic transient processes on the dynamics, reliability, stability, and accuracy of operation of an induction motor [1].

Thanks to the works of outstanding Soviet scientists—V. A. Venikov, A. V. Ivanov-Smolensky, E. Ya. Kazovsky, V. A. Shubenko, A. A. Yanko-Trinitsky, and others—as well as to the works of foreign scholars—K. P. Kovacs, I. Racz, V. Shuisky, and others—a coherent theory has been created which, with the aid of differential equations, describes AC machines in transient conditions.

An analytical solution of the differential equations describing transient processes in an induction motor is impossible in most cases. The problem is solved by laborious numerical methods or with the aid of computing machines.

Studies of the transient processes of induction motors carried out at the Department of Electric Drive and Industrial Automation of the Tomsk Polytechnic Institute have shown that, on the basis of the theory of physical modeling, it is possible to create a simple method for calculating transient conditions of an induction motor that has satisfactory accuracy.

For a symmetrical induction machine (that is, with a symmetrical supply voltage and with a symmetrical stator and rotor), there exist four basic similarity criteria [2]:

\[ \frac{r_1}{x_\mu} = \mathrm{idem} \tag{1} \]

\[ \frac{r_2'}{x_\mu} = \mathrm{idem} \tag{2} \]

\[ \frac{x_1}{x_\mu} = \mathrm{idem} \tag{3} \]

\[ \frac{x_2'}{x_\mu} = \mathrm{idem}, \tag{4} \]

where \(r_1;\ r_2';\ x_1;\ x_2';\ x_\mu\) are the resistances of the equivalent circuit of an induction motor.

In addition, it is necessary to satisfy an additional similarity condition in order to preserve the time scale of the mechanical motion:

\[ \frac{J\omega_c S_k}{M_k}=\mathrm{idem}, \tag{5} \]

where \(J\) is the moment of inertia of the rotating masses;

\(\omega_c\) is the synchronous speed of the motor;

\(M_k, S_k\) are the critical torque and slip of the motor.

Since the values of the inductive and active resistances are absent from reference literature, it is necessary to express the similarity criteria of induction motors through catalog data.

Usually, for induction motors it is assumed that:

\[ r_1 \cong r_2', \tag{3} \]

\[ x_1 \cong x_2'. \tag{4} \]

The value of the resistance \(r_2'\), for a known rated slip \(s_n\), can be calculated from the formula [5]:

\[ r_2' = s_n \frac{U_{\mathrm{phase}}}{I_{n\,\mathrm{phase}}},\ \Omega \tag{6} \]

The resistance \(x_\mu\) is found from the formula:

\[ x_\mu=\frac{U_{\mathrm{phase}}}{I_m},\ \Omega \tag{7} \]

The no-load current \(I_m\) is determined by the formula [6]:

\[ I_m=I_n\left[\sin\varphi_n-\cos\varphi_n\left(k_m-\sqrt{k_m^2-1}\right)\right], \tag{8} \]

where

\[ k_m=\frac{M_{\max}}{M_n}. \]

Taking into account formulas (6), (7), and (8), the similarity criteria (1) and (2) can be expressed through catalog data by the formula:

\[ \frac{r_1}{x_\mu}\cong\frac{r_2'}{x_\mu} = s_n\left[\sin\varphi_n-\cos\varphi_n\left(k_m-\sqrt{k_m^2-1}\right)\right] = \mathrm{idem} \tag{9} \]

To express the 3rd and 4th similarity criteria through catalog data, we shall use the formula for critical slip:

\[ s_k=\pm\frac{r_2'}{\sqrt{r_1^2+\left(x_1+x_2'\right)^2}} \]

Taking into account that \(x_1 \cong x_2'\), formula (9) can be transformed into the form

\[ s_k=\pm \frac{\dfrac{r_2'}{x_\mu}} {\sqrt{\dfrac{r_1^2}{x_\mu^2}+\dfrac{(2x_1)^2}{x_\mu^2}}}. \tag{10} \]

It follows from formula (10) that if \(S_k=\mathrm{idem}\), then, when the conditions

\[ \frac{r_2'}{x_\mu}=\mathrm{idem} \quad\text{and}\quad \frac{r_1}{x_\mu}=\mathrm{idem}, \]

are satisfied, likewise

\[ \frac{x_1}{x_\mu}=\frac{x_2'}{x_\mu}=\mathrm{idem}. \]

Consequently, a necessary condition for similarity is also

\[ s_k = \mathrm{idem}. \]

The critical slip, from the motor nameplate data, is found by the formulas [6]:

\[ s_k=\frac{s_n+\sqrt{s_n\cdot A}}{1+\sqrt{s_n\cdot A}}, \tag{11} \]

where

\[ A= \frac{\dfrac{M_{\max}}{M_n}-1} {\dfrac{M_{\max}}{M_n}-1} \tag{12} \]

The foregoing makes it possible to conclude that the similarity criteria (1)–(4) are satisfied if

\[ \cos\varphi_n=\mathrm{idem} \tag{13} \]

\[ s_n=\mathrm{idem} \tag{14} \]

\[ \frac{M_{\max}}{M_n}=\mathrm{idem} \tag{15} \]

\[ \frac{M_{\mathrm{start}}}{M_n}=\mathrm{idem} \tag{16} \]

In other words, the transient regimes of induction motors whose mechanical characteristics in relative units coincide, for which \(\cos\varphi_n\) is the same, and for which the condition

\[ \frac{J\omega_c s_k}{M_k}=\mathrm{idem} \]

is observed, are similar in time.

Many investigators note [8; 9] that the duration of the transient process of an induction motor, calculated without taking electromagnetic transients into account, agrees quite accurately with the actual duration of the transient process (the error is no more than 10–20%). This is explained by the fact that the electromagnetic transients produce an alternating-sign torque, whose mean value over the time

SMCh — static mechanical characteristic.
PMCh — transient mechanical characteristic.
PPMCh — pulsating transient mechanical characteristic.

Fig. 1.

the transient process is almost equal to zero. In [5] a mathematical proof is given that the mean value of the pulsating torque is equal to zero for particular cases.

If one analyzes the transient mechanical characteristic, which represents the dependence \(M_{\mathrm{dv}} = f(n)\) in the transient regime (starting, braking, etc.), then one may conclude that it can be decomposed into two characteristics: the static mechanical characteristic, representing the dependence of the motor torque on speed in the steady-state regime, and the pulsating transient mechanical characteristic, which represents the dependence of the pulsating motor torque on speed (Fig. 1).

Such a separation of the transient mechanical characteristic was first carried out by V. A. Glebov [9].

The considerations set out above formed the basis of a simplified method for calculating electromechanical transient regimes of an induction motor.

Fig. 2.

For the calculation it is necessary to have a series of pulsating transient mechanical characteristics for the transient regime, obtained for different values of the moment of inertia referred to the motor shaft, with the static torque equal to zero \((M_{\mathrm{st}}=0)\). Such a series of pulsating transient mechanical characteristics, obtained for the starting regime of the A41-4 induction motor, is shown in Fig. 2. On each characteristic the duration of the transient process for which it was obtained is indicated.

It should be noted that, for a referred moment of inertia equal to three moments of inertia of the motor rotor, the transient mechanical characteristic hardly differs from the static mechanical characteristic; therefore ten pulsating transient mechanical characteristics obtained with the moment of inertia increasing in an arithmetic progression are quite sufficient:

\[ J = J_{\rho} + (n - 1)\,0.3J_{\rho}, \]

where

\[ J_{\rho} \text{ is the moment of inertia of the motor rotor;} \]

\[ n \text{ is the ordinal number of the characteristic.} \]

The indicated series of pulsating transient mechanical characteristics can be used for calculating electromechanical starting regimes of induction motors for which

\[ \frac{M_{\mathrm{start}}}{M_{\mathrm{n}}}, \quad \frac{M_{\max}}{M_{\mathrm{n}}}, \quad \cos \varphi_{\mathrm{n}} \]

and \(s_{\mathrm{n}}\) differ by no more than \(20 \div 30\%\) from the corresponding values for the A41-4 motor.

The calculation procedure is as follows:

1) The mechanical transient process is calculated from the static mechanical characteristic by graphical or grapho-analytical integration of the equation of motion. The dependence \(n=f(t)\) and the duration of the transient process are obtained.

2) From the duration of the transient process, the nearest pulsating transient mechanical characteristic is selected from the series in Fig. 2.

3) Having recalculated the pulsating transient mechanical characteristic into named units, it is combined with the static mechanical characteristic of the motor whose transient process is being calculated, and the transient mechanical characteristic is obtained.

4) The transient process is recalculated using the transient mechanical characteristic.

As experimental verification has shown, the qualitative agreement between experiment and calculation is complete; the quantitative discrepancy does not exceed 20%.

Example:

The A51-6 motor has the following technical data:

\[ P_{\mathrm{n}} = 2.8 \text{ kW}; \quad I_{\mathrm{n}} = 6.6 \text{ A}; \quad n_{\mathrm{n}} = 950 \text{ rpm}; \quad \eta_{\mathrm{n}} = 84.5\%; \]

\[ \cos \varphi_{\mathrm{n}} = 0.80; \quad \frac{I_{\mathrm{start}}}{I_{\mathrm{n}}} = 5.5; \quad \frac{M_{\mathrm{start}}}{M_{\mathrm{n}}} = 1.3; \quad \frac{M_{\max}}{M_{\mathrm{n}}} = 1.8; \]

\[ GD^{2} = 0.17 \ \text{kg m}^{2}. \]

The static mechanical characteristic of this motor, calculated from catalog data, as well as the result of calculating the mechanical transient process by the grapho-analytical me-

...are shown in Fig. 3. The duration of the transient process is 0.13 sec. This time is closely matched by pulsating transient mechanical characteristic No. 3 in Fig. 2.

Fig. 3.

After converting the pulsating transient mechanical characteristic into named units and adding it to the static mechanical characteristic of the A51-6 motor, the transient mechanical characteristic shown in Fig. 4a by the bold line is obtained. For comparison, Fig. 4b gives the transient mechanical characteristic of the same motor obtained from experimental data [10].

PMCh — transient mechanical characteristic.
SMCh — static mechanical characteristic.
PPMCh — pulsating transient mechanical characteristic.

Fig. 4a

The computational error lies within the range of 15–20%, despite the rather considerable discrepancy between the static mechanical characteristics of motors A41-4 and A51-6.

Fig. 4b.

REFERENCES

1. V. A. Shubenko. Problems of dynamics, power engineering, and reliability of automated induction electric drives. Summary report on published works submitted for the degree of Doctor of Technical Sciences. Sverdlovsk, 1963.

2. V. A. Venikov, L. A. Zhukov. Transient processes in electrical systems. GEI, 1958.

3. M. P. Kostenko, L. M. Piotrovskii. Electrical Machines. GEI, 1958.

4. R. Richter. Electrical Machines, Vol. IV, Induction Machines. ONTI, 1939.

5. K. P. Kovacs, I. Racz. Transient Processes in Alternating-Current Machines. GEI, 1963.

6. M. S. Tuganov. Determination of the parameters of squirrel-cage induction motors from catalog data. VEhP, No. 7, 1962.

7. G. P. Khalizev. Electric Drive and Fundamentals of Control. GEI, 1962.

8. L. P. Petrov. Dynamic characteristics of squirrel-cage induction motors. VEhP, No. 5, 1959.

9. V. A. Glebov. Experimental investigation of starting processes of induction motors. Elektrichestvo, No. 12, 1956.

10. V. A. Shubenko. Problems of the dynamics of automated electric drives. Elektrichestvo, No. 1, 1960.