DETERMINATION OF THE PARAMETERS OF THE EQUIVALENT CIRCUIT OF AN INDUCTION MOTOR FROM CATALOG DATA

M. P. Tabinskii

(Presented by the scientific seminar of the Electromechanical Faculty)

For analysis of the operation of an induction motor, the so-called Г-shaped, or refined, equivalent circuit with the magnetizing branch brought out to the terminals of the primary circuit is usually used (Fig. 1).

Fig. 1

For determining the parameters of the equivalent circuit with an accuracy acceptable for engineering calculations (maximum relative error no greater than 10–15%), in addition to the technical data of the motors:

\[ P_{\mathrm{n}},\ U_{\mathrm{n}},\ I_{\mathrm{n}},\ \cos\varphi_{\mathrm{n}},\ \eta_{\mathrm{n}},\ \frac{I_{\mathrm{start}}}{I_{\mathrm{n}}},\ \frac{M_{\mathrm{start}}}{M_{\mathrm{n}}},\ \frac{M_{\max}}{M_{\mathrm{n}}}, \]

which are given in reference literature, it is also necessary to know the magnetizing current \(I_{\mu}\) (the reactive component of the no-load current), as well as the current \(I'_{2\mathrm{n}}\) referred to the primary circuit. Since the active component of the no-load current is very small in comparison with \(I_{\mu}\), one may take \(I_{\mu} \simeq I_{0}\).

The value of the current \(I_0\) for motors of the A, AO, and AK series is found from the diagram in Fig. 2, where along the abscissa axis, on a logarithmic scale, \(P_n\) is plotted, and along the ordinate axis the ratio \(\dfrac{I_0}{I_n}\). The current \(I_0\) of induction motors of other series, for which the equivalent circuit of Fig. 1 is valid, can be calculated by the formula:

\[ I_0 = \frac{Q_0}{3U_{\mathrm{ph}}}\cdot 1000 \ [\mathrm{A}]. \tag{1} \]

1—AK-series motors, 750 and 1000 rpm;
2—A- and AO-series motors, 750 rpm;
3—A- and AO-series motors, 1000 rpm;
4—A- and AO-series motors, 1500 rpm;
5—AK-series motors, 1500 rpm;
6—A-series motors, 3000 rpm;
7—AO-series motors, 300 rpm.

Fig. 2

where \(Q_0\) is the reactive power at no-load of the motor \([\mathrm{kvar}]\), which is found by means of the empirical formula [1]:

\[ Q_0 \simeq \frac{P_n}{\eta_n}\cdot m, \tag{2} \]

where \(m\) is a calculation coefficient depending on the rated power factor \([\cos \varphi_n]\), determined from Fig. 3.

After determining the current \(I_0\), one can find the current \(I'_2\) from the vector diagram of the motor voltages and currents at rated operation (Fig. 4). From Fig. 4 it is seen that the current \(I_0\) lags by approximately \(90^\circ - \varphi_n\) behind the current \(I_1 = I_n\).

Therefore the current \(I'_{2n}\) can be found from triangle AOB graphically or by the formula

\[ I'_{2n} = \frac{I_n \cos \varphi_n} {\sin \left\{ \arctg \left[ \frac{I_n - I_0}{I_n + I_0} \ctg \frac{90^\circ - \varphi_n}{2} \right] + 0.5(90^\circ + \varphi_n) \right\}} \tag{3} \]

After determining \(I_0\) and \(I'_{2n}\), the data are sufficient for finding the parameters of the equivalent circuit.

Figure 3

1) Determination of the resistance \(r'_2\)

The resistance \(r'_2\) is found from the formula:

\[ r'_2 = s_n \frac{U_{\mathrm{ph}}}{I_{n\,\mathrm{ph}}} \tag{4} \]

Formula (4) follows from representing the speed characteristic of an induction motor on the rectilinear segment in relative units:

\[ \nu = 1 - \rho'_2 i'_2 ; \]

where

\[ \nu = \frac{n}{n_c}; \qquad \rho'_2 = \frac{r'_2}{R_n}; \qquad R_n = \frac{U_{\mathrm{ph}}}{I_{n\,\mathrm{ph}}}; \qquad i'_2 = \frac{I'_2}{I'_{2n}} . \]

For \(I'_{2n}\) and \(n_n\), equation (5) takes the form

\[ \frac{n_n}{n_c}=1-\frac{r'_2}{R_n} \]

Fig. 4

Whence formula (4) follows.

The resistance \(r'_2\) can also be determined from the mechanical-power losses, which according to [2] are expressed by the formula

\[ 3I_{2n}^{\prime\,2} r'_2 = M_n \cdot \omega_c \cdot s_n \quad [\mathrm{W}] \tag{6} \]

Here

\[ M_n = 9560 \frac{P_n}{n_n} \quad [\mathrm{N\,m}] \tag{7} \]

2. Determination of the resistance \(r_1\)

Substituting the coordinates of the starting and rated operating conditions of the motor into the formula for calculating the current \(I_2'\),

\[ I_2'=\frac{U}{\sqrt{\left(r_1+\frac{r_2'}{s}\right)^2+\left(x_1+x_2'\right)^2}} \tag{8} \]

we can obtain a formula for finding the resistance \(r_1\):

\[ r_1= \frac{ \left(\frac{U_n}{I_{2n}'}\right)^2 -r_2'^2\left(\frac{1}{s_n^2}-1\right) -\left(\frac{U_n\cdot K}{I_{\text{start}}}\right)^2 }{ 2r_2'\left(\frac{1}{s_n}-1\right) }. \tag{9} \]

Here the coefficient \(K=0.96\div 0.98\) takes into account the current \(I_0\). With lower accuracy one may take \(r_1\approx r_2'/2\).

3) Determination of \(x_1\) and \(x_2'\)

To find \(x_k=x_1+x_2'\), the short-circuit current is used:

\[ z_k=\sqrt{(r_1+r_2')^2+(x_1+x_2')^2} =\frac{U_{\text{phase}}}{I_{\text{start}}\cdot K} \tag{10} \]

The coefficient \(K\) takes into account the no-load current.

\[ K=0.96\div 0.98 \]

Hence we obtain:

\[ x_k=x_1+x_2'= \sqrt{\left(\frac{U_{\text{phase}}}{I_{\text{start}}\cdot K}\right)^2-(r_1+r_2')^2} \quad [\Omega] \tag{11} \]

Separating \(x_k\) into \(x_1\) and \(x_2'\) does not appear possible. With sufficient accuracy for practical calculations, one may take \(x_2'\approx x_1\) [4].

4) Determination of \(x_\mu\)

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

\[ x_\mu\approx \frac{U_{\text{phase}}}{I_0}\quad [\Omega] \tag{12} \]

Example

Motor A 42/4 has the following technical data:

\[ P_n=2.8\ \text{kW},\quad U_n=380\ \text{V};\quad I_n=6.1\ \text{A};\quad n_n=1420\ \text{rpm}; \]

\[ \eta_n=0.835;\quad \cos\varphi_n=0.84;\quad \frac{I_{\text{start}}}{I_n}=5.5;\quad \frac{M_{\text{start}}}{M_n}=1.9; \]

\[ \frac{M_{\max}}{M_n}=2.0;\quad \text{phase connection—star.} \]

The resistance parameters of the equivalent circuit, calculated by the above method, are as follows:

\[ r'_2 = 1.93\ \Omega \qquad x_1 = x'_2 = 2.6\ \Omega \]

\[ r_1 = 1.6\ \Omega \qquad x_\mu = 75\ \Omega \]

The same parameters, determined experimentally, are as follows:

\[ r'_2 = 1.96\ \Omega \qquad x_1 = x'_2 = 2.8\ \Omega \]

\[ r_1 = 1.81\ \Omega \qquad x_\mu = 74\ \Omega . \]

References

1. L. V. Litvak. Construction of the operating characteristics of induction motors from catalog data. Industrial Power Engineering, 1959, No. 4.

2. M. P. Kostenko and L. M. Piotrovskii. Electrical Machines. GEI, 1958.