Issues in Selecting the Operating Mode of an Asynchronous Motor with Variable Frequency

E. V. Kononenko, Yu. G. Meshcheryakov

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

The possibilities of frequency control make it possible to ensure operation of an asynchronous motor in the most economical mode over the entire speed-control range. In the general case, the economy of the operating mode is determined by the capital and current costs corresponding to it. However, if the capital costs are considered independent of the motor operating mode, then its economy will be determined by current costs, which include primarily the costs due to energy losses in the motor and the cost of reactive-energy compensation. In this case, for economical use of the motor, the optimal values of efficiency and power factor should be maintained over the entire control range. The general conditions corresponding to such a mode can be found without resorting to techno-economic calculations. This is achieved on the basis of a joint analysis of the laws of frequency control that provide, separately, the least values of reactive power and losses.

The investigation is carried out for an asynchronous motor with a linear magnetic circuit, under the assumptions and notation corresponding to [1].

The power losses in the motor consist of electrical losses in the stator and rotor windings, iron losses, and mechanical losses. The sum of the listed losses is expressed by the formula:

\[ \sum p = r_1 i_1^2 + r_2' i_2'^2 + (k_{\mathrm{г}} + k_{\mathrm{в}}\alpha)\varphi^2 \alpha + k_{\mathrm{м}}\omega^{3/2}. \tag{1} \]

Here \(i_1\), \(i_2'\), \(\varphi\), \(\alpha\), \(\omega\) correspond to the notation in [1], \(k_{\mathrm{в}}\) and \(k_{\mathrm{г}}\) are the loss coefficients for eddy currents and hysteresis, and \(k_{\mathrm{м}}\) is the coefficient of mechanical losses.

After replacing in (1) the currents, magnetic flux, and frequency by their values [1], the power losses are brought to a form convenient for investigation:

\[ \sum p = \left( \gamma_1 \xi^2 + \gamma_2 \beta + \gamma_3 \frac{1}{\beta} + \gamma_4 \right)\mu, \tag{2} \]

where

\[ \gamma_1 = \frac{x_2'^2}{r_2'}(k_{\mathrm{г}} + 2k_{\mathrm{в}}\omega); \qquad \gamma_2 = 1 + \frac{x_0^2 r_1}{r_2' z_0^2} + \frac{x_2'^2}{r_2'}\omega (k_{\mathrm{г}} + k_{\mathrm{в}}\omega); \]

\[ \gamma_3 = \frac{r_2' r_1}{z_0^2} + r_2'\omega (k_{\mathrm{г}} + k_{\mathrm{в}}\omega); \qquad \gamma_4 = \frac{2r_0 r_1}{z_0^2} + r_2'(k_{\mathrm{г}} + 2k_{\mathrm{в}}\omega) + \frac{\omega^{3/2}}{\mu}. \]

The terms $\nu_1 \div \nu_4$ are functions of the rotational speed.

Function (2) is investigated for an extremum. From the condition

\[ \frac{\partial \sum p}{\partial \beta}=0 \]

the minimum of the function is found. It corresponds to the equation:

\[ 2\nu_1\beta^3+\nu_2\beta^2-\nu_3=0 \tag{3} \]

The law of slip control that ensures operation of the motor with minimum losses is calculated by the following formula, obtained as a result of solving equation (3):

\[ \beta_p=-\frac{\nu_2}{6\nu_1} \left[ 2\cos\left(\frac{\pi-\varphi}{3}\right)-1 \right], \tag{4} \]

where

\[ \varphi=\arccos\left(\frac{1}{4\nu_1}-\frac{\nu_3\nu_1^2}{2\nu_2^3}\right). \]

The law of slip control corresponding to operation of the motor with minimum reactive power is expressed, in accordance with [1], by dependences of the form:

\[ \left. \begin{aligned} \beta_q &= \frac{\omega}{6} \left\{ 2\cos\left[ \frac{\pi}{3} -\frac{1}{3}\arccos\left(1-\frac{108c_2}{c_1\omega^2}\right) \right]-1 \right\};\\ \beta_q &= \sqrt[3]{-\sigma_{1\omega}+\sigma_{2\omega}} + \sqrt[3]{-\sigma_{1\omega}-\sigma_{2\omega}} -\frac{\omega}{6}, \end{aligned} \right\} \tag{5} \]

where

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

The graphs of the dependences $\beta_p(\omega)$ and $\beta_q(\omega)$, calculated by formulas (4) and (5) for the A051-4 motor as the speed varies from nominal to zero, are shown in Fig. 1.

Both control laws $\beta_p(\omega)$ and $\beta_q(\omega)$ have a common regularity: slip increases with increasing speed. However, the intensity of the increase in slip is different: in the minimum-loss mode, slip increases almost in proportion to speed, whereas, in order to maintain the minimum of reactive power, it is necessary in the region of low speeds to increase slip very sharply, and, as the motor is further accelerated, to change it only slightly.

The absolute value of the slips under minimum losses $\beta_p$ is much smaller than the corresponding values that ensure the minimum of reactive power $\beta_q$. As a result, between the values $\beta_p$ and $\beta_q$ in the operating speed range there exists a broad slip zone, the upper boundary of which is formed by the slips corresponding to the minimum of reactive power, and the lower boundary by the slips satisfying the conditions for minimum losses. As the slip changes within the limits from the lower boundary to the upper one, the power factor increases to its greatest value and the efficiency decreases from its maximum. Outside the indicated zone, a change in slip in either direction is accompanied by a deterioration both of the efficiency and of the power factor. Consequently, from the standpoint of economy, the optimal slip values should lie within the zone $\beta_p<\beta<\beta_q$.

Knowledge of the boundaries of such a zone has practical significance. In order to facilitate the implementation of control laws, one should strive for their maximum simplification. At the same time, it is important to know within what limits a deviation of the slip from the calculated value may be allowed,

without noticeably degrading the motor’s energy characteristics. The adoption of control laws most convenient for regulating the drive, with a certain departure from the conditions of the motor’s most economical operation, ultimately contributes to improving the economy of the entire drive by reducing the costs of the control system.

For calculating the limiting slip values $\beta_p$ and $\beta_q$, formulas (4) and (5) are used. Formula (4) can be replaced by a simpler one, analogous to the formula for optimal slip at minimum losses, obtained in [2] under the condition that frequency is an independent variable. Investigation of equation (2) for low speeds confirms that, for the minimum-loss mode, the assumption that slip has no effect on frequency over the entire control range is admissible, since the change in losses when the frequency changes is restrained by the electrical losses in the stator and rotor windings, which do not depend on frequency. On this basis, for calculating the slips corresponding to the minimum losses, and for subsequent investigations, it is sufficient to use the formula obtained under the assumption $\omega \gg \beta$ as a result of investigating function (2) for a minimum:

\[ \beta = \sqrt{\frac{a_3 + r_2'^{\,\omega}(k_{\mathrm{g}} + k_{\mathrm{v}}^{\omega})}{1 + a_1 + a_2^{\omega}(k_{\mathrm{g}} + k_{\mathrm{v}}^{\omega})}}, \tag{6} \]

where

\[ a_1=\frac{(x_0+x_2')^2 r_1}{r_2' x_0^2}; \qquad a_2=\frac{x_2'^{\,2}}{r_2'}; \qquad a_3=\frac{r_2' r_1}{x_0^2}. \]

From the simultaneous solution of equations (5) and (6), the rotational speed is found at which the operating conditions of the motor with minimum losses and minimum reactive power coincide, i.e., the motor operates with maximum values of both efficiency and power factor. In the operating range of rotational speeds, such a mode corresponds to a value of the speed close to zero at the slip

\[ \beta_{\mathrm{e}0}=\sqrt{\frac{a_3}{1+a_1}}. \tag{7} \]

For all other slips, the economical mode will be characterized by a definite (optimal) ratio of efficiency and power factor. As the speed increases, the reactive power, owing to its proportionality to the frequency, increases rapidly, and, accordingly, the influence of the power factor on the economy of the motor increases. The optimal value of slip ensuring economical operation of the motor shifts from the minimum of losses toward the minimum of reactive power. At nominal rotational speed, the optimal ratio between efficiency and power factor in general-purpose motors is usually observed at the nominal slip or at a somewhat smaller value of slip (by 15–20%).

In accordance with the described motor operating mode, one may choose, on the assumption that the optimal dependence of slip on speed $\beta_{\mathrm{e}}(\omega)$ includes two slip values, one of which corresponds to the minimum-loss mode at a speed equal to zero, while the other corresponds to the nominal mode at the nominal rotational speed. The simplest dependence satisfying this condition is the equation of a straight line:

\[ \beta_{\mathrm{e}}=\beta_{\mathrm{e}0}+(\beta_{\mathrm{eн}}-\beta_{\mathrm{e}0})\omega, \tag{8} \]

where $\beta_{\mathrm{e}0}$ is determined by formula (7), $\beta_{\mathrm{eн}}=(0.8 \div 0.9)s_{\mathrm{н}}$.

Expression (8) is an approximate law of economical frequency control. From the standpoint of ease of implementation, it is suffi

it is quite sufficient to confine oneself to this linear dependence. The error due to the assumption of linearity will have practically no significant value.

Changing the slip in accordance with (8) presupposes its reduction as the speed decreases. In control according to the known laws of frequency regulation (apart from the minimum-loss control considered here), the slip either does not change as a function of speed (modes with rated slip, rated flux, minimum current), or increases as it decreases (the mode

\[ \frac{u_1}{u_{1\mathrm{n}}}=\frac{f_1}{f_{1\mathrm{n}}}\sqrt{\frac{M}{M_{\mathrm{n}}}}). \]

As a result, as the speed decreases, the motor is used ever less economically.

Fig. 1. Slip dependences on speed corresponding to minimum losses (\(\beta_p\)), minimum reactive power (\(\beta_c\)), and the mode

\[ \frac{u_1}{u_{1\mathrm{n}}}=\frac{f_1}{f_{1\mathrm{n}}}\sqrt{\frac{M}{M_{\mathrm{n}}}} \]

(\(\beta_k\)).

Figure 1 shows a graph of the variation of slip as a function of speed under control according to the basic law of frequency regulation

\[ \frac{u_1}{u_{1\mathrm{n}}}=\frac{f_1}{f_{1\mathrm{n}}}\sqrt{\frac{M}{M_{\mathrm{n}}}}. \]

The slip was calculated in accordance with [2] for an AO51-4 motor with rated shaft torque, using the formula:

\[ \beta_k= \frac{r_2'}{(b^2+c^2\omega^2)} \left[ \left( \frac{m_1u_{1\mathrm{n}}^2\omega^2}{2\omega_{1\mathrm{n}}M_{\mathrm{n}}} -r_1\omega \right) - \sqrt{ \left( \frac{m_1u_{1\mathrm{n}}^2\omega^2}{2\omega_{1\mathrm{n}}M_{\mathrm{n}}} -r_1\omega \right)^2 - (b^2+c^2\omega^2)(d^2+e^2\omega^2) } \right], \tag{9} \]

where

\[ b=r_1\left(1+\frac{x_2'}{x_0}\right); \qquad c=x_1+x_2'+\frac{x_1x_2'}{x_0}; \]

\[ d=-\frac{r_1}{x_0}; \qquad e=1+\frac{x_1}{x_0}. \]

It is clear from Fig. 1 that the operating region of the motor with high values of efficiency and power factor under control according to law (9) is limited to the upper part of the control range. As the speed decreases, the slip increases, which contradicts not only the principles of economical operation but also, as is known, is accompani-

is caused by a reduction in overload capacity and by excessive heat release in the windings. Obviously, the mode closest to the economical one is maintained in the upper part of the speed range when control is carried out according to the law $\beta = s_{\mathrm{n}} = \mathrm{const}$, and in the lower part when control is carried out according to the minimum of losses (6).

Conclusions

1. The energy indicators of the motor operating mode are determined by the magnitude of its operating slip.

2. The optimal slip values lie in the zone between the values corresponding to the minimum of reactive power and the minimum of losses; moreover, they are closer to the latter in the region of low speeds.

3. To maintain an economical mode when regulating the speed downward from the rated speed, the absolute slip should be reduced. This is especially important in the case of motor operation at a low rotational speed.

References

1. E. V. Kononenko, Yu. G. Meshcheryakov. Analysis of the processes of change in the reactive power of an asynchronous motor under frequency regulation. In the present collection.

2. A. A. Bulgakov. Frequency Control of Asynchronous Electric Motors. Nauka Publishing House, 1955.