Methods for Calculating Reactive Power Consumed by Induction Machines

N. V. Lisetskii

(Submitted by the scientific-methodological seminar of the Electromechanical Faculty)

In a number of calculations connected with the construction of reactive-power curves, the determination of losses in networks, the determination of the power of compensating devices, etc., it is necessary to know the dependence of the reactive power consumed by induction machines on their load. This dependence must be obtained as accurately as possible, since induction machines consume about 60% of the reactive power of the system [1].

The most accurate dependence between active and reactive powers for induction machines can be obtained from circle diagrams. However, this method is associated with cumbersome graphical constructions, and therefore it is used comparatively rarely. Usually, the reactive energy consumed by induction machines is calculated by the following approximate formula [1]:

\[ Q = S_{kva}\left\{ \frac{x_{x.x}}{100}\left[ 0.7\frac{U}{U_{nom}} + 0.3\left(\frac{U}{U_{nom}}\right)^5 \right] + \frac{x_p}{100}\left(\frac{P}{P_n}\right)^2 \right\}, \tag{1} \]

where \(S_{kva}\) is the total rated power of the motor in kVA;
\(x_{x.x}\) is the percentage of no-load current at rated voltage;
\(x_p\) is the percentage of the total reactive leakage resistance of the stator and rotor, referred to the total power.

Calculation by the indicated formula, as will be shown below, gives large errors. In this connection it is advisable to find a more accurate dependence between the active and reactive powers of an induction machine, one that does not require cumbersome constructions or calculations. Such a dependence can be obtained directly from the circle diagram constructed for the equivalent circuit of an induction machine refined according to the method of Academician M. P. Kostenko, with the magnetizing branch brought out to its terminals. The refined equivalent circuit and the circle diagram constructed for it are shown in Fig. 1.

On the basis of the circle diagram of Fig. 1, the position of the ends of the current vectors on the axes \(I''_u\), \(I''_p\) can be expressed by the known equation of a circle

\[ I_u'' + I_p'' = \frac{U_\phi^2}{4x^2}, \tag{2} \]

where \(I_a\) and \(I_p\) are, respectively, the active and reactive components of the phase current;

\(U_\phi\) is the phase voltage at the terminals of the machine;

\(x\) is the total reduced inductive reactance of the phase of the induction machine, with the corrections of Academician M. P. Kostenko [2], determined from the expression

\[ x = x_1\left(1+\frac{x_1}{x_\mu}\right) + x'_2\left(1+\frac{x_1}{x_\mu}\right)^2, \]

where \(x_1\) and \(x'_2\) are, respectively, the leakage inductive reactances of the stator and rotor phases, referred to the stator winding, and \(x_\mu\) is the inductive reactance of the magnetizing circuit.

Fig. 1.

To obtain the dependence \(I_p = f(I_a)\), it is necessary to reduce equation (2) to the axes \(I_p, I_a\) (Fig. 1). For this purpose, we first shift the axis \(I'_a\) into the position \(I''_a\) (Fig. 1). In the axes \(I''_a, I''_p\), equation (2) is written as

\[ I_a^2 + \left(I_p - \frac{U_\phi}{2x}\right)^2 = \frac{U_\phi^2}{2x^2} \]

or

\[ I_a^2 + I_p^2 - 2I_p\frac{U_\phi}{2x} + \frac{U_\phi^2}{4x^2} = \frac{U_\phi^2}{4x^2}. \]

After collecting like terms, we have

\[ I_a^2 + I_p^2 - \frac{U_\phi I_p}{x} = 0. \tag{3} \]

We reduce the obtained equation to the axes \(I'_a, I'_p\) (Fig. 1), for which we rotate the coordinate axes \(I''_a, I''_p\) through the angle \(\alpha\) in the direction opposite to the rotation of the vectors; in this case equation (3) will be written as:

\[ (I_a \cos \alpha - I_p \sin \alpha)^2 + (I_a \sin \alpha + I_p \cos \alpha)^2 - \]

\[ - \frac{U_\phi}{x} (I_a \sin \alpha + I_p \cos \alpha) = 0. \]

Transforming the obtained equation, we have:

\[ I_a^2 \cos^2 \alpha - 2 I_a I_p \sin \alpha \cos \alpha + I_p^2 \sin^2 \alpha + I_a^2 \sin^2 \alpha + \]

\[ + 2 I_a I_p \sin \alpha \cos \alpha + I_p^2 \cos^2 \alpha - \frac{U_\phi}{x} (I_a \sin \alpha + I_p \cos \alpha) = 0, \]

\[ I_a^2 + I_p^2 - \frac{U_\phi}{x} (I_a \sin \alpha + I_p \cos \alpha) = 0. \tag{4} \]

According to the data of Academician M. P. Kostenko [2]

\[ \tg \alpha = \frac{2 I_0 r_1}{U_\phi}, \]

where \(I_0\) is the magnetizing component of the no-load current;
\(r_1\) is the active resistance of the stator phase and
\(U_\phi\) is the phase voltage at the machine terminals.

The angle \(\alpha\) will be very small owing to the fact that the magnitude of the voltage drop in the active resistance of the stator from the magnetizing component of the no-load current \((I_0 r_1)\) is much smaller than the phase voltage \(U_\phi\). On this basis, with accuracy sufficient for practice, it may be assumed that

\[ \tg \alpha = \sin \alpha = \frac{2 I_0 r_1}{U_\phi} \quad \text{and} \quad \cos \alpha = 1. \]

With this assumption, equation (4) is written as

\[ I_a^2 + I_p^2 - \frac{U_\phi}{x} \left( I_a \frac{2 I_0 r_1}{U_\phi} + I_p \right) = 0. \]

Shifting the axis \(I'_a\) to the left by the magnitude of the magnetizing component of the no-load current \(I_0\), and the axis \(I'_p\) downward by the magnitude of the active component of the no-load current \(\dfrac{P'_0}{U_\phi}\) (here \(P'_0\) is the no-load loss, referred to one phase), we pass to the axes \(I_a\) and \(I_p\) and obtain an analytical dependence relating the reactive and active current of the induction machine:

\[ \left( I_a - \frac{P'_0}{U_\phi} \right)^2 + (I_p - I_0) - \frac{U_\phi}{x} \left[ \left( I_a - \frac{P'_0}{U_\phi} \right) \frac{2 I_0 r_1}{U_\phi} + \right. \]

\[ \left. + (I_p - I_0) \right] = 0. \tag{5} \]

Separating the variables and solving the resulting quadratic equation with respect to \((I_p - I_o)\), we have:

\[ (I_p - I_o)^2 - \frac{U_\phi}{x}(I_p - I_o) + \left(I_a - \frac{P'_o}{U_\phi}\right) \left(I_a - \frac{P'_o}{U_\phi} - 2 I_o \frac{r_1}{x}\right) = 0, \]

\[ I_p - I_o = \frac{U_\phi}{2x} \pm \sqrt{ \frac{U_\phi^2}{4x^2} - \left(I_a - \frac{P'_o}{U_\phi}\right) \left(I_a - \frac{P'_o}{U_\phi} - 2 I_o \frac{r_1}{x}\right) }, \]

\[ I_p = I_o + \frac{U_\phi}{2x} \pm \sqrt{ \frac{U_\phi^2}{4x^2} - \left(I_a - \frac{P'_o}{U_\phi}\right) \left(I_a - \frac{P'_o}{U_\phi} - 2 I_o \frac{r_1}{x}\right) }. \tag{6} \]

As is seen from equation (6), each active component of the current corresponds to two values of the reactive component. From the physical point of view this is explained by the vector diagram in Fig. 2.

Fig. 2.

It is seen from Fig. 2 that the larger value of the reactive component of the current pertains to the unstable operating mode of the induction machine; therefore, in the subsequent discussion we take only one value, corresponding to the stable operating mode, namely:

\[ I_p = I_o + \frac{U_\phi}{2x} - \sqrt{ \left(\frac{U_\phi}{2x}\right)^2 - \left(I_a - \frac{P'_o}{U_\phi}\right) \left(I_a - \frac{P'_o}{U_\phi} - 2 I_o \frac{r_1}{x}\right) }. \tag{7} \]

If now the right- and left-hand sides of equation (7) are multiplied by \(3U_\phi\), we obtain

\[ Q = Q_o + \frac{U^2}{2x} - \sqrt{ \left(\frac{U^2}{2x}\right)^2 - (P - P_o) \left(P - P_o - 2Q_o \frac{r_1}{x}\right) }. \tag{8} \]

where

where \(Q\) is the total reactive power consumed by the machine;

\(Q_0\) is the no-load reactive power;

\(U\) is the line voltage applied to the machine;

\(P_0\) is the active-power loss at no load;

\(P\) is the power consumed from the network when operating as a motor and delivered to the network when operating as a generator;

\(r_1\) is the active resistance of the stator phase;

\(x\) is the total inductive leakage reactance of the phase of the induction machine, referred to the stator winding, with the corrections of Academician M. P. Kostenko.

To estimate the accuracy of equations (1) and (8), the reactive powers calculated from the indicated equations were compared with those obtained experimentally. The experiments were carried out both for the case in which the induction machine operated as a motor and for the case in which it operated in the generator mode (at negative slips). An induction motor rated at 2.8 kW for a voltage of 380/220 V was used as the test object.

Modern multi-motor machine tools used in many branches of industry are equipped with motors of approximately this rating. Therefore, the relationship between active and reactive powers obtained for this motor will, to a greater extent than for a larger machine, correspond to the analogous relationship for an entire enterprise.

Fig. 3.

For carrying out the experiment, the circuit shown in Fig. 3 was assembled. The induction machine was started by a motor and loaded by changing the resistance of a rheostat in the armature circuit of a direct-current machine operating as a generator. Then the direct-current machine was connected to a power source and, by changing the resistance in the field-winding circuit, was brought into motor operation. Its rotational speed was increased above synchronous speed, and in this way the induction machine was brought into generator operation and delivered active power to the network, while at the same time consuming the reactive power required for its magnetization.

The active power consumed or delivered by the induction machine, at various modes and loads, was measured by wattmeters—

…included in the stator circuit. The reactive power is calculated by the well-known formula

\[ Q=\sqrt{S^2-P^2}, \]

where \(Q\) is the reactive power consumed from the network;
\(P\) is the active power consumed from the network or delivered to the network;
\(S\) is the apparent power, equal to

\[ S=\sqrt{3}\,UI, \]

where \(U\) is the line voltage applied to the induction machine;
\(I\) is the current, measured by ammeters, in the stator circuit.

The curves shown in Fig. 4 indicate the relationship between active and reactive powers for an induction machine, obtained by different methods. The solid line shows the dependence obtained with the aid of equation (8), the dashed line the dependence obtained with the aid of equation (1), and the circles mark the points obtained experimentally.

Fig. 4.

Considering the curves of Fig. 4, it is easy to see that expression (1) gives increased values of reactive power when the induction machine operates as a motor, and reduced values when it operates as a generator. The greatest error in determining reactive power by means of equation (1) is obtained at the rated load of an induction machine operating in motor mode. For the experiment carried out, this error amounted to 20% relative to the reactive power determined from the experiment.

Equation (8) gives more accurate data in comparison with equation (1), but it does not give the dependence of the no-load reactive power of the machine on the change in the voltage applied to the stator.

This dependence is associated with the magnetization curve; therefore an exact analytical expression for it is impossible. However, the magnitude of the no-load reactive power is 30–40% of the total reactive power consumed by an induction machine [1], and varies, depending on the voltage, within the range of 40–170%; therefore, neglecting its variation may lead to inadmissibly large errors.

For practical calculations, the indicated dependence may be obtained on the basis of equation (1) and the indications given in the literature [1], and may be written by the following equation

\[ Q_0 = Q_{0n}\left[0.7\,\frac{U}{U_n} + 0.3\left(\frac{U}{U_n}\right)^5\right]\frac{U}{U_n}, \tag{9} \]

where \(Q_0\) is the no-load reactive power at a voltage differing from the nominal one;

\(Q_{0n}\) is the no-load reactive power at nominal voltage;

\(U_n\) and \(U\) are the nominal voltage and the voltage applied to the stator of the machine.

To check the accuracy of expression (9), the motor under test was connected to a variable voltage. The motor carried no mechanical load except for bearing friction and air resistance. The voltage at the stator terminals was varied from 70 to 120% of the nominal voltage, which corresponded to cases encountered in practice.

The curves of the dependence of the no-load reactive power of an induction machine on the applied voltage are shown in Fig. 5. The curve drawn with a solid line corresponds to the indicated dependence determined experimentally. The curve drawn with a dashed line shows the analogous dependence constructed with the aid of equation (9).

Fig. 5.

As can be seen from the curves in Fig. 5, the calculated no-load reactive powers, when the voltage is reduced, are obtained somewhat smaller than the actual values, and when the voltage is increased, somewhat larger. However, the discrepancy between the calculated and experimental curves in the operating range of voltage fluctuations is very small.

On the basis of all the foregoing, the following conclusions may be drawn.

1. The calculation of the reactive power consumed by induction machines, as a function of their load, must be carried out as accurately as possible.

2. The most accurate dependence between active and reactive powers for induction machines is given by circle diagrams and by equation (8) derived on their basis.

3. When induction machines are connected to voltages differing from the rated values, or when voltages fluctuate, the change in the no-load reactive power should be taken into account; this can be done using equation (9).

4. Equation (1) may be used only for approximate calculations not connected with capital installations and the operating conditions of critical networks.

REFERENCES

1. Markovich I. M. Power Systems and Their Operating Conditions, Gosenergoizdat, 1952.
2. Piotrovskii L. M. Electrical Machines, Gosenergoizdat, 1949.