ANALYSIS OF PROCESSES OF REACTIVE-POWER VARIATION IN AN INDUCTION MOTOR UNDER FREQUENCY CONTROL

E. V. KONONENKO, Yu. G. MESHCHERYAKOV

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Revealing the essence of the processes of variation of the reactive power of an induction machine under frequency control is necessary for developing a correct solution to questions of selecting an economical operating mode and the corresponding laws of frequency control. The control laws currently in use do not provide for obtaining the best value of the power factor while maintaining a high efficiency over a wide control range; consequently, high economy of the motor also cannot be achieved. In order to improve the utilization of the motor, it is necessary to have a sufficiently complete understanding of the features of the distribution of reactive energy in it when the operating mode changes. Such questions have not found noticeable reflection in the theory of frequency control.

In the present work the task is set of investigating the nature of the variation of reactive power in the working range of control, and of determining the operating conditions of the motor with a minimum reserve of reactive power.

The nature of the variation of reactive power is conveniently revealed by investigating the dependence of reactive power on absolute slip. Taking slip as the independent variable facilitates the analytical solution of the problem and preserves the physical meaning of the processes. Slip, in essence, is an economic indicator of the motor operating mode, determining the values of the efficiency and the power factor. In addition, slip is a controlled parameter. Frequency-control systems constructed on the principle of slip control are becoming widespread in practice. Therefore, using slip as the independent variable not only facilitates investigation of the processes in the motor, but also has direct practical meaning.

The investigation is carried out under the assumption that the motor is supplied by a sinusoidal, symmetrical three-phase voltage and has a linear magnetic circuit. For the analysis a T-shaped equivalent circuit with constant parameters is used. The motor parameters are expressed in per-unit values. As base quantities, the nominal values of the voltage at the stator terminals \(U_{1n}\), the stator current \(I_{1n}\), the synchronous angular velocity of rotation of the magnetic field

\[ \omega_{1n}=\frac{2\pi f_{1n}}{p}, \]

as well as the base impedance

\[ z_b=\frac{U_{1n}}{I_{1n}}, \]

are adopted; the base power—

ness \(P_b=m_1 I_{1n}U_{1n}\); base torque
\(M_b=\dfrac{m_1 I_{1n}U_{1n}}{\omega_{1n}}\); base magnetic flux
\(\Phi_b=\dfrac{2\pi U_{1n}}{c_e\omega_{1n}p}\)
(\(p\) is the number of pole pairs; \(m_1\) is the number of phases; \(c_e\) is a constant including the number of turns and the winding coefficient).

The motor parameters in relative units are: \(u=\dfrac{U_1}{U_{1n}}\)—the voltage at the stator terminals; \(i_1=\dfrac{I_1}{I_{1n}}\)—the stator current; \(i_2'=\dfrac{I_2'}{I_{1n}}\)—the rotor current; \(i_0=\dfrac{I_0}{I_{1n}}\)—the magnetizing current; \(\alpha=\dfrac{\omega_1}{\omega_{1n}}=\dfrac{f_1}{f_{1n}}\)—the stator-current frequency; \(\beta=\dfrac{\omega_1-\omega_2}{\omega_{1n}}\)—the absolute slip; \(\omega=\dfrac{\omega_2}{\omega_{1n}}\)—the rotor rotational speed; \(\mu=\dfrac{M}{M_b}\)—the shaft torque; \(\varphi=\dfrac{\Phi}{\Phi_b}\)—the magnetic flux in the air gap; \(q=\dfrac{Q}{P_b}\)—reactive power; \(r_1, x_1, r_2', x_2', r_0, x_0\)—equivalent-circuit parameters referred to \(z_b\).

The absolute slip, rotor rotational speed, and shaft torque are taken as independent variables. The motor parameters are expressed through the independent variables as follows.

The rms value of the rotor current is

\[ i_2'=\sqrt{\frac{\mu\beta}{r_2'}} . \tag{1} \]

The complex stator current is determined through the rotor current and the parameters of the two parallel branches of the equivalent circuit, \(z_0=r_0+jx_0\) and \(z_2'=\dfrac{r_2'}{\beta}+jx_2'\):

\[ \bar{i}_1=\bar{i}_2'\frac{\dot{z}_2'+\dot{z}_0}{z_0}; \tag{2} \]

\[ \left|\frac{\dot{z}_2'+\dot{z}_0}{z_0}\right| = \sqrt{ \frac{ \left(r_0+\dfrac{r_2'}{\beta}\right)^2+(x_0+x_2')^2 }{ r_0^2+x_0^2 } }, \tag{3} \]

The rms value of the stator current after substituting (1) and (3) into (2) is

\[ i_1= \sqrt{ \frac{\mu\beta}{r_2'z_0^2} \left[ \left(r_0+\frac{r_2'}{\beta}\right)^2+x_{02}^2 \right] }, \tag{4} \]

where

\[ z_0^2=r_0^2+x_0^2;\qquad x_{02}=x_0+x_2'. \]

The magnitude of the e.m.f. in the stator winding is

\[ e_1=i_2'z_2'\alpha = \alpha \sqrt{ \frac{\mu\beta}{r_2'} \left[ \left(\frac{r_2'}{\beta}\right)^2+x_2'^2 \right] }. \tag{5} \]

Magnetizing current

\[ i_0=\frac{e_1}{z_0\alpha} = \sqrt{ \frac{\mu\beta}{r_2' z_0^2} \left[ \left(\frac{r_2'}{\beta}\right)^2+x_2'^2 \right] }. \tag{6} \]

Magnetic flux in the air gap

\[ \varphi=\frac{e_1}{\alpha} = \sqrt{ \frac{\mu\beta}{r_2'} \left[ \left(\frac{r_2'}{\beta}\right)^2+x_2'^2 \right] }. \tag{7} \]

The stator-current frequency is \(\alpha=\omega+\beta\).

\[ \tag{8} \]

The reactive power consumed by the motor from the network can be represented as the sum of the powers of the magnetic field in the air gap \(q_0=x_0\alpha i_0^2\) and of the leakage fields of the stator \(q_1=x_1\alpha i_1^2\) and rotor \(q_2=x_2'\alpha i_2'^2\):

\[ q=\alpha\left(x_0 i_0^2+x_1 i_1^2+x_2'i_2'^2\right). \tag{9} \]

After substituting the currents and frequency in (9) by their expressions (1), (4), (6), (8), the reactive-power function takes a form convenient for investigation:

\[ q=\mu(\omega+\beta)\left(c_1\beta+c_2\frac{1}{\beta}+c_3\right). \tag{10} \]

The quantities \(c_1\div c_3\) depend only on the active and inductive resistances of the motor and are constant terms:

\[ c_1=\frac{x_2'}{r_2'} \left[ 1+\frac{x_0x_2'}{z_0^2} +\frac{x_1(r_0^2+x_{02}^2)}{z_0^2x_2'} \right]; \]

\[ c_2=\frac{r_2'(x_0+x_1)}{z_0^2}; \qquad c_3=\frac{2x_1r_0}{z_0^2}. \]

The reactive-power function with respect to slip (10) is investigated for an extremum. From the condition

\[ \frac{\partial q}{\partial \beta}=0 \]

the minimum of the function is found. It corresponds to a cubic equation in \(\beta\):

\[ 2\beta^3+\omega\beta^2-\frac{c_2}{c_1}\omega=0. \tag{11} \]

As a result of solving equation (11), the law of slip regulation is found, ensuring operation of the motor with the smallest reserve of reactive energy. Depending on the magnitude of the rotational speed, two calculation formulas are obtained. In the region of high values, approximately at \(\omega>0.3\), the slip is calculated by the formula:

\[ \beta_q= \frac{\omega}{6} \left\{ 2\cos \left[ \frac{\pi}{3} -\frac{1}{3}\arccos \left( 1-\frac{108c_2}{c_1\omega^2} \right) \right] -1 \right\}. \tag{12} \]

In the range of low rotational speeds the following formula is used:

\[ \beta_q= \sqrt[3]{-\sigma_{1\omega}+\sigma_{2\omega}} + \sqrt[3]{-\sigma_{1\omega}-\sigma_{2\omega}} -\frac{\omega}{6}. \tag{13} \]

where

\[ \sigma_{1\omega}= \left(\frac{\omega}{6}\right)^3 -\frac{c_2\omega}{2c_1}; \]

\[ \sigma_{2\omega}= \sqrt{ \sigma_{1\omega}^2+ \left(\frac{\omega}{6}\right)^6 }. \]

Thus, to ensure operation of the motor with the smallest reserve of reactive energy, it is necessary to regulate the slip as a function of the rotational speed. The nature of the change in load torque does not affect the optimal (in the sense of the minimum of reactive power) law of slip regulation (12), (13) under the assumptions adopted here.

The meaning of the dependence of the optimal slip on speed can be explained by the influence of the optimal value of slip on the magnitude of the stator-current frequency, which is established as a function of the required rotational speed and the magnitude of the optimal slip. When the motor operates at high speed, the influence of slip on the frequency is relatively small, and if it is neglected—that is, if the frequency is taken as the independent variable—then, as follows from formula (10), taking (8) into account, the minimum of reactive power will be observed at one value of slip for any speed. However, such a situation will not correspond to the actual picture of phenomena in the case of deep downward speed regulation from the rated value, when the magnitude of the slip becomes comparable with the magnitude of the speed; this is taken into account by adopting the speed as the independent variable. The dependences of reactive power on slip \(q(\beta)\) for different rotational speeds at rated load torque are shown in Fig. 1. The dependence of slip on rotational speed \(\beta_q(\omega)\), satisfying the conditions for the minimum of reactive power, is shown in Fig. 2. The calculation was made by formulas (10), (12), (13) for an A051-4 motor.

Fig. 1. Dependence of reactive power on slip for different rotational speeds

Fig. 2. Dependence of slip on speed corresponding to the minimum of reactive power

According to Fig. 1, the region of small slips \(\beta < s_n\) corresponds to operation of the motor with an increased reserve of reactive energy. A decrease in slip from the rated value is accompanied by a noticeable increase in reactive power, especially intense in the region of small slips. This is due to the fact that, under the condition of constant torque, a decrease in slip must be compensated by an increase in the power of the magnetic field in the air gap. The latter can be achieved only by increasing the magnetizing current. In this case the magnetic flux assumes values exceeding the rated value. Figure 3 gives the dependences of the magnetic

Fig. 3. Dependences of the magnetic flux
and the squares of the currents on slip

flux \(\varphi(\beta)\) and the squares of the currents \(i_1^2(\beta)\), \(i_2^2(\beta)\), \(i_0^2(\beta)\) on the absolute slip, calculated from formulas (7), (4), (1), (6) for the A051-4 motor with rated torque on the shaft. From Fig. 3 the influence of the character of the change in currents in the working slip zone on the reactive power is clearly seen. In the slip zone from the rated value to the value corresponding to the minimum stator current,

\[ \beta_i = \frac{r_2'}{\sqrt{r_0^2 + x_{02}^2}}, \tag{14} \]

simultaneously with the increase in the power of the magnetic field in the air gap, the power of the leakage fields decreases owing to a reduction in the stator and rotor currents, as a result of which the growth of the total reactive power is somewhat retarded. However, with a further decrease in slip, beginning from the value corresponding to the minimum stator current (14), the increase in reactive power is already caused by an increase not only in the power of the magnetic field in the air gap, but also in the power of the stator leakage field. This slip region is characterized by a sharp increase in the reactive power consumed from the network, which must be taken into account when selecting and evaluating the economy of motor operating regimes with minimum losses and minimum current, which are realized at small slips.

With an increase in slip above the value corresponding to the minimum reactive power, the total value of the reactive power increases more slowly than in the preceding case, owing to the decrease in the power of the magnetic field in the air gap as the power of the leakage fields increases.

In the zone of high rotational speeds, approximately beginning with $\omega = 0.3 \div 0.4$, the minimum of reactive power is attained at slips close to the rated value.

To maintain the minimum reactive power over the entire control range, the slip should be regulated as the speed changes according to Fig. 2. Because of the increased influence of slip on the value of frequency in the region of low rotational speeds, a reduction in speed must be accompanied by an increasing reduction in slip.

The region of high rotational speeds is characterized by a small change in slip. The limiting value of slip satisfying the conditions for the minimum of reactive power as $\omega \to \infty$ can be calculated from the formula obtained from (10) under the assumption $\omega \gg \beta$:

\[ \beta = \sqrt{ \frac{x_0 + x_1} {x_0 x_2^{\prime 2} + x_1 (x_0 + x_2')^2 + x_2' x_0^2} }. \tag{15} \]

The limiting value of slip (15) only slightly exceeds the rated value.

Conclusions

1. In the range of working slips under frequency control, the magnitude of the reactive power consumed by the motor varies within wide limits.

2. At a sufficiently high rotational speed, operation of the motor with small slips of the order of the value corresponding to the minimum current is accompanied by a significant increase in the reactive power consumed.

3. To maintain a high power factor, it is advisable, as the rotational speed increases, to increase the slip to values differing from the rated value by no more than $(15—20\%)$.