Operating Characteristics of Synchronous Reluctance Motors at Variable Frequency
Abstract Generated abstract
This paper analyzes the operating characteristics of synchronous reluctance motors supplied from a variable-frequency network, focusing on how frequency, load torque, and voltage regulation affect current, efficiency, power factor, losses, and load angle. The authors derive calculation formulas in relative units using the angle between the stator magnetomotive force and the rotor direct axis, including expressions for torque, current, active current, losses, efficiency, and a voltage law based on maintaining constant maximum torque. Calculations, with approximate allowance for saturation through parameter dependence on load angle, are compared with experimental data for a 2.2 kW motor and show satisfactory agreement. The results indicate that proportional voltage and frequency control gives nearly constant current at constant torque, while energy performance depends strongly on the load torque law, and regulation to maintain maximum torque can preserve high efficiency over a broad speed range.
Full Text
Proceedings of the Tomsk Polytechnic Institute
awarded the Order of the October Revolution
and the Order of the Red Banner of Labor
named after S. M. Kirov
Volume 284 1974
OPERATING CHARACTERISTICS OF SYNCHRONOUS RELUCTANCE MOTORS AT VARIABLE FREQUENCY
E. V. Kononenko, G. I. Lukiyanov
Synchronous reluctance motors (SRM) are used in electric drives with frequency control of the rotational speed. The operating modes of an SRM when supplied from a variable-frequency network depend on various factors: the parameters of the motor itself, the magnitude and character of the change in load torque, the law of voltage regulation as a function of frequency, etc. To evaluate the properties of the motor, to identify design features, and also for the rational use of designed motors in frequency-control systems, it is necessary to study the influence of the indicated factors on the operating modes and the various characteristics of SRMs.
The basic equations describing the processes in an SRM at variable frequency are given in [1]. In the present work an analysis is carried out of the operating characteristics of an SRM at variable frequency.
In analyzing the influence of frequency on the operating characteristics of an SRM, it is convenient to consider the dependences, on frequency, of the currents, efficiency, power factor, and load angles, determined at a constant or any specified value of the load torque. We shall also call these dependences operating characteristics. It is expedient to express the calculation formulas in terms of the load torque. In this case the formulas turn out to be simpler if one operates with the angle \(\beta\) between the axis of the magnetomotive force of the stator winding and the direct axis of the rotor.
For convenience, we transform the analytical expressions in [1], describing the synchronous operating mode of an SRM, so that they contain one trigonometric function of the angle. Then the stator current at any frequency is determined by the following expression (all quantities are in relative units):
\[ I= \frac{\gamma U_{\mathrm{n}}\sqrt{1+\operatorname{tg}^{2}\beta}} {\sqrt{r^{2}+\alpha^{2}x_{d}^{2}+\operatorname{tg}^{2}\beta\left(r^{2}+\alpha^{2}x_{q}^{2}\right)+2r\alpha\cdot\operatorname{tg}\beta\cdot\left(x_{d}-x_{q}\right)}} , \tag{1} \]
where
\(\gamma\) and \(\alpha\) are the relative values of voltage and frequency.
In the rated mode \(U_{\mathrm{n}}=1.0\), \(\gamma=1.0\), \(\alpha=1.0\), \(I=1.0\). The value of the angle \(\beta_{\mathrm{n}}\), corresponding to the rated load torque at rated frequency, can be found from equation (1):
\[ \tan \beta=\frac{-b_1-\sqrt{b_1^2-4a_1c_1}}{2a_1}, \tag{2} \]
where
\[ a_1=r^2+x_q^2-1,\quad b_1=2r(x_d-x_q),\quad c_1=r^2+x_d^2-1. \]
The electromagnetic torque at any frequency is equal to
\[ M=\frac{(\gamma U_{\mathrm{n}})^2(x_d-x_q)\cdot \tan\beta} {r^2+\alpha^2x_d^2-\tan^2\beta\cdot (r^2+\alpha^2x_q^2)+2r\alpha\cdot \tan\beta\cdot (x_d-x_q)}. \tag{3} \]
or
\[ M=I^2\cdot (x_d-x_q)\cdot \frac{\tan\beta}{1+\tan^2\beta}. \tag{4} \]
The values of the angle \(\beta\) corresponding to any specified value of the load torque can be found from expression (3):
\[ \tan\beta=\frac{-b_2\pm\sqrt{b_2^2-4a_2c_2}}{2a_2}, \tag{5} \]
where
\[ a_2=M(r^2+\alpha^2x_q^2),\quad b_2=(x_d-x_q)\cdot[2M\cdot r\alpha-(\gamma U_{\mathrm{n}})^2], \]
\[ c_2=M(r^2+\alpha^2x_d^2). \]
The established synchronous operating mode of the motor corresponds to the smaller value of the angle \(\beta\) according to expression (5). The stator current is determined from expression (4) in the following form:
\[ I=\sqrt{\frac{M(1+\tan^2\beta)}{(x_d-x_q)\cdot \tan\beta}}. \tag{6} \]
The power factor is found as the ratio of the active component of the current to the total current. The active component of the stator current, expressed through the load torque, can be determined by the following formula:
\[ I_a=\frac{M}{\gamma U_{\mathrm{n}}}\cdot \left[ \alpha+\frac{r(1+\tan^2\beta)}{(x_d-x_q)\tan\beta} \right]. \tag{7} \]
The power consumed by the motor is proportional to the active component of the current and is equal to
\[ P_1=\gamma U_{\mathrm{n}}\cdot I_a. \tag{8} \]
The copper losses in relative units are equal to
\[ p_{\mathrm{m}}=rI^2. \tag{9} \]
According to [2, 3], it may be assumed that the mechanical losses vary in proportion to the rotor angular velocity to the power 1.5, i.e.,
\[ p_{\mathrm{mech}}=p_{\mathrm{mn}}\alpha^{1.5}, \tag{10} \]
where
\[ p_{\mathrm{mn}} \text{ — mechanical losses at } \alpha=1.0 \text{, in relative units.} \]
If the losses in the steel are considered proportional to the square of the induction in the machine steel and to the frequency to the power 1.5, then, when the frequency changes, they can be determined with sufficient accuracy from the expression
\[ p_{c\alpha}=p_{c\mathrm{n}}\cdot\frac{(\gamma U_{\mathrm{n}})^2}{\sqrt{\alpha}}, \tag{11} \]
where
\(p_{c\mathrm{n}}\) are the losses in the steel at the rated frequency, in relative units.
Knowing the total losses, one can also find the efficiency,
\[ \eta=1-\frac{\sum p}{P_1}, \tag{12} \]
where
\(\sum p\) are the total losses in the motor.
In frequency control of the rotational speed, simultaneously with changing the frequency it is also necessary to regulate the voltage according to a definite law. The choice of the voltage-control law depends on many factors and requires knowledge of the specific operating conditions of the motor. Therefore let us consider the operation of the motor under the two most characteristic laws of voltage control: 1) proportional to the frequency and 2) proceeding from constancy of the maximum torque developed by the motor.
According to [1], the maximum electromagnetic torque of the motor at any frequency is equal to
\[ M_m= \frac{(\gamma U_{\mathrm{n}})^2(x_d-x_q)} {2(r^2+\alpha^2 x_d x_q)^2} \cdot \left[ \sqrt{(r^2+\alpha^2 x_d^2)(r^2+\alpha^2 x_q^2)} -\alpha r(x_d-x_q) \right]. \tag{13} \]
The maximum electromagnetic torque at rated frequency \(M_{m\mathrm{n}}\) is found from this same expression by substituting \(\gamma=\alpha=1.0\). Dividing \(M_{m\mathrm{n}}\) by \(M_m\), we find
\[ \gamma= \frac{1}{U_{\mathrm{n}}}\cdot \sqrt{ \frac{ \sqrt{(r^2+\alpha^2 x_d^2)(r^2+\alpha^2 x_q^2)} +r\alpha(x_d-x_q) }{ \sqrt{(r^2+x_d^2)(r^2+x_q^2)} +r(x_d-x_q) } } \cdot \sqrt{\frac{M_m}{M_{m\mathrm{n}}}}. \tag{14} \]
Relations (1)–(14) make it possible to analyze the operating characteristics of an SRM at variable frequency. To improve the accuracy of the calculations, it is desirable to take into account the effect of saturation on the parameters of the SRM, since in the general case, with a change in frequency, the magnetic fluxes along the \(d\) and \(q\) axes do not remain constant. The magnetic fluxes change:
1) with a change in load torque;
2) at constant load torque and with a change in frequency, since in this case the angle \(\beta\) changes;
3) because, in regulating the frequency and voltage, constancy of the ratio of the EMF to the frequency is not maintained.
Taking all the listed factors into account causes considerable computational difficulties. Experience has shown that in the range \(\alpha=0.5\div1.5\), calculations can be carried out with sufficient accuracy if the change in the parameters due to the change in the angle \(\beta\) is taken into account. In this case, for calculating the operating characteristics it is sufficient to find the dependences \(x_d, x_q=f(\tan\beta)\), determined at the rated frequency. Such dependences are given in Fig. 1.
Figure 2a shows the operating characteristics of a motor with a rated power of 2.2 kW (at a frequency of 50 Hz) and a rated rotational speed of 1000 rpm. The voltage varies in proportion to the frequency. From these curves it is evident that, when the frequency is varied over fairly
Fig. 1. Dependences of \(x_d\) and \(x_q\) on \(\operatorname{tg}\beta\)
wide limits and \(M_{\mathrm{st}}=\mathrm{const}\), the stator current changes only slightly. Therefore the copper losses also change little. As the frequency decreases, the iron and mechanical losses, as well as the total losses, decrease. At the same time, the consumed power decreases more rapidly, which determines the nature of the change in efficiency. The power factor increases as the frequency decreases. Figure 2a also gives operating characteristics at the rated load torque, determined experimentally. The calculated data agree sufficiently well with the experimental data.
Fig. 2. Operating characteristics of an SRM as a function of frequency:
a) \(M_{\mathrm{st}}=\mathrm{const}\); b) \(M_{\mathrm{st}}=\alpha\)
If the load torque decreases in proportion to the rotational speed (the voltage is regulated in proportion to the frequency), then the angle \(\beta\) corresponding to this load also decreases (Fig. 2b). In this case the nature of the change in efficiency remains the same as for \(M_{\mathrm{st}}=\mathrm{const}\). As the frequency decreases, the power factor deteriorates. Consequently, in the case where the load torque varies in proportion to the rotational speed and the voltage varies in proportion to the frequency, the energy performance of the motor deteriorates markedly when the rotational speed is regulated downward from the rated value.
Figure 3 shows the characteristics of the SRM under different laws of voltage regulation. The load torque remains constant and equal to the rated value, \(U_{\mathrm{n}}=1.0\). Comparing these characteristics, it may be noted that in the range \(\alpha=0.6\text{--}1.4\) they differ only slightly. When the voltage is regulated on the basis of maintaining constancy of the maximum motor torque and \(\alpha<1.0\), the power factor is lower,
Fig. 3. Comparison of the operating characteristics of an SRM under different laws of regulation
than with a proportional change in voltage, since in the first case the voltage supplied to the motor is higher, as a result of which the reactive component of the current increases. The efficiencies are practically identical.
Thus, when the voltage is regulated as a function of frequency on the basis of maintaining constancy of the maximum torque, the energy performance of the SRM remains sufficiently high over a wide range of rotational-speed regulation.
REFERENCES
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E. V. Kononenko, G. I. Lukyanov. Equations of a synchronous reluctance motor at variable frequency of the supply network. Izv. TPI, vol. 229, 1972.
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A. A. Bulgakov. Frequency Control of Asynchronous Electric Motors. Moscow, Publishing House of the Academy of Sciences of the USSR, 1966.
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M. I. Kraytsberg, V. I. Iog, V. V. Sigodin. Frequency Regulation of Asynchronous Electric Motors. Moscow, Informstandartelektro, 1967.