Ensuring Constant Overload Capacity of a Variable-Frequency Capacitor Synchronous Reluctance Motor
Abstract Generated abstract
The paper addresses voltage and frequency control of a single-phase capacitor synchronous reluctance motor supplied through a phase-shifting capacitor, with the aim of maintaining constant overload capability over a variable-frequency operating range. It develops analytical dependences for the required supply voltage as a function of frequency, load, and capacitance, using symmetrical components and pull-out torque relations while accounting for reverse-sequence effects and iron losses. The authors show that constant overload capacity can be maintained when voltage, frequency, and phase-shifting capacitance are coordinated, particularly by selecting capacitance to produce a circular rotating field at the relevant load condition.
Full Text
ENSURING CONSTANT OVERLOAD CAPABILITY OF A VARIABLE-FREQUENCY CAPACITOR SYNCHRONOUS RELUCTANCE MOTOR
Yu. M. ACHKASOV, B. A. ZAKHAROV
(Presented by the Scientific and Technical Seminar of the Research Institute of AEM)
An important question in the theory of variable-frequency control of an alternating-current motor is the determination of the dependences of the supply-voltage magnitude on frequency in order to obtain stable operation and to ensure regimes that are best from the standpoint of economy within a certain control range. Economical operation of the motor is achieved by implementing frequency-control laws that ensure a minimum of losses in the motor. Stable operation of the motor is ensured if the value of the overload capability is changed in the appropriate manner by regulating the supply-voltage magnitude as a function of frequency [1].
In [2], a frequency-control law was obtained for a single-phase capacitor synchronous reluctance motor (OKSRD), the stator windings of which are connected in a star with one phase-shifting capacitor. Regulation of the voltage according to this law ensures an unchanged overload capability of the motor when operating with a constant phase-shifting capacitance. In this case the control range is limited by the permissible distortion coefficient of the supply voltage.
All other conditions being equal, the rotors of symmetrical machines, which have a circular rotating field in the air gap, rotate most uniformly. With an elliptical field, the resultant magnetic-induction vector rotates in space with variable speed, causing nonuniform rotation of the rotor during one revolution. Therefore, in systems for precise speed maintenance it is expedient to choose the phase-shifting capacitance from the condition of obtaining symmetry of the magnetizing forces at each frequency [3], which, at constant load torque, will ensure a circular rotating field in the machine air gap. In this case the law of voltage regulation as a function of frequency can be simplified. The electromagnetic torque of the OKSRD is determined by the difference of the torques of the positive and negative sequences [2]
\[ M_{\mathrm{e}} = C_{1} \left[ \frac{ E_{1}^{2}\cdot \dfrac{x_{\theta}^{\prime}}{\operatorname{tg}\theta}\cdot a }{ \left(\dfrac{a x_{\theta}^{\prime}}{\operatorname{tg}\theta}\right)^{2} + a^{2}(x_{\theta}^{\prime})^{2} } - \frac{ E_{2}^{2}\cdot r_{R}^{\prime} }{ \left(\dfrac{r_{R}^{\prime}}{2}\right)^{2} + a^{2}(x_{R}^{\prime})^{2} } \right]. \tag{1} \]
where
\[ C_1=\frac{0.0162\cdot m\cdot p}{f_{\mathrm{n}}\cdot x}. \]
The maximum electromagnetic torque will occur at a load angle \(\Theta=45^\circ\). At rated load, the reverse-sequence e.m.f. is absent.
At the greatest load, \(E_2\) has a definite value, which is determined by the method of symmetrical components. Then \(E_2\) is found through the e.m.f. asymmetry coefficient \((K_E)\)
\[ E_2=K_E E_1 \tag{2} \]
Denoting
\[ k_R=\frac{r_R'}{\left(\dfrac{r_R'}{2}\right)^2+(x_R')^2\cdot a^2}, \tag{3} \]
after substituting (2), (3) into (1), we obtain
\[ M_{\mathrm{e}}=C_1\left[\frac{1}{2\cdot x_0'\cdot a}-k_R\cdot k_E^2\right]\cdot E_1^2. \tag{4} \]
Taking into account that the direct-sequence e.m.f. is equal to
\[ E_1=(z_1-z_s)\cdot\left[\frac{(a-a^2)z_2+a x_c}{3z_1z_2+x_c(z_1+z_2)}\cdot\frac{U}{a-a^2}\right], \tag{5} \]
we substitute (5) into (4) and, solving with respect to the voltage, obtain
\[ U=\sqrt{\frac{M_{\mathrm{em}}}{C_1(1-2k_R\cdot k_E^2\cdot x_0'\cdot a)\cdot A^2}}, \tag{6} \]
where
\[ A=\frac{(z_1-z_s)\cdot[(a-a^2)z_2+a x_c]}{(a-a^2)\cdot[3z_1z_2+x_c(z_1+z_2)]}; \]
\[
M_{\mathrm{em}}=\lambda M_{\mathrm{n}}
\]
is the maximum torque;
\[
\lambda
\]
is the required overload capability;
\[
M_{\mathrm{n}}
\]
is the rated torque.
If the drive operates under a load varying from rated to maximum, it is more rational to select the phase-shifting capacitors from the condition of obtaining a circular rotating field at maximum load at each frequency. The dependence of voltage on frequency for this mode can be obtained from the formula for the maximum electromagnetic torque of a three-phase synchronous reluctance motor [4]:
\[ M_{\mathrm{em}}=\frac{U^2(x_d-x_q)}{2(x_d\cdot x_q+r^2)^2}\cdot[z^2-r(x_d-x_q)]; \tag{7} \]
where
\[ z^2=\sqrt{(r^2+x_d^2)\cdot(r^2+x_q^2)}; \]
\(r;\ x_d;\ x_q\) are the relative values of the active and synchronous inductive resistances along the \(d\), \(q\) axes,
\(U\) is the relative phase voltage.
To derive this dependence, we proceed as follows. We take into account the iron losses as a function of frequency by substituting into (7) the equivalent active resistance
\[ r_{\mathrm{e}}=r+r_c, \]
where
\[ r_c=r_{c\mathrm{n}}(a)^{1.5}; \]
\(r_{\mathrm{sn}}\) is the active resistance accounting for iron losses at the nominal frequency.
Introduce the relative-frequency parameter \((\alpha)\). Then the expression for the pull-out torque at any frequency will have the form
\[ M_{\mathrm{em}} = \frac{U^{2}\alpha (x_d - x_q)} {2\left(r_{\mathrm{e}}^{2}+\alpha^{2}x_d x_q\right)^{2}} \cdot \left[z_{\alpha}^{2}-r_{\mathrm{e}}(x_d-x_q)\right]; \tag{8} \]
where
\[ z_{\alpha}^{2} = \sqrt{\left(r_{\mathrm{e}}^{2}+\alpha^{2}x_d^{2}\right) \left(r_{\mathrm{e}}^{2}+\alpha^{2}x_q^{2}\right)} . \]
Solving (8) with respect to the voltage, we obtain
\[ U = \sqrt{ \frac{ 2M_{\mathrm{em}}\left(r_{\mathrm{e}}^{2}+\alpha^{2}x_d x_q^{2}\right) }{ \alpha(x_d-x_q)\cdot \left[z_{\alpha}^{2}-r_{\mathrm{e}}\alpha(x_d-x_q)\right] } }. \tag{9} \]
The load angle at which the motor develops the pull-out torque is found from the formula
\[ \theta = \operatorname{arctg} \frac{\alpha^{3}x_d\cdot x_q-r_{\mathrm{e}}^{2}} {2r_{\mathrm{e}}(x_d+x_q)} . \tag{10} \]
Regulation of the voltages of the capacitor synchronous reluctance motor according to expressions (6), (9) will ensure a constant overload capability for the connection circuit under consideration of a three-phase synchronous reluctance motor to a single-phase voltage.
Implementation of these dependencies requires coordinated regulation of three quantities: voltage, frequency, and phase-shifting capacitance as the load on the shaft changes. This is achieved most simply provided that the control system makes it possible to change the magnitude of the voltage and its frequency at the terminals of the capacitor synchronous reluctance motor independently of one another, while the phase-shifting capacitance is regulated as a function of frequency.
References
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E. M. Berlin, B. A. Egorova, V. D. Kulik, I. S. Skosyrev. Frequency-Control Systems for Synchronous Reluctance Motors. Leningrad, Energiya, 1968.
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A. I. Zaitsev, Yu. M. Achkasov, B. A. Zakharov. The law of frequency control of a single-phase capacitor synchronous reluctance motor. Proceedings of Higher Educational Institutions of the USSR, Electromechanics, 1973, No. 1.
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Yu. M. Achkasov, B. A. Zakharov. Conditions for obtaining symmetry of the magnetizing forces of a three-phase synchronous reluctance motor operating in single-phase capacitor mode. Reports of the 7th Scientific and Technical Conference “Static Converters in Automation and Electric Drives.” Tomsk, TSU Publishing House, 1971.
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E. V. Kononenko. Synchronous Reluctance Machines. Moscow, Energiya, 1970.