STUDY OF A THREE-PHASE SYNCHRONOUS RELUCTANCE MOTOR OPERATING IN SINGLE-PHASE CAPACITOR MODE

Yu. M. ACHKASOV, B. A. ZAKHAROV

(Presented by the scientific and technical seminar of the Research Institute of AEM at TPI)

Of the entire variety of circuits for connecting AC motors with symmetrical three-phase stator windings to a single-phase network, attention is deserved by the circuit with star-connected stator windings and a series capacitance (Fig. 1). As applied to an asynchronous motor, this circuit makes it possible, in single-phase connection, to obtain the full power of the three-phase mode when using one capacitance as the phase-shifting element, with the power factor of the original three-phase motor equal to 0.5 [1].

Fig. 1. Winding connection circuit

In an unbalanced mode, asymmetrical currents flow through the motor windings, i.e., an asymmetrical supply mode of a symmetrical three-phase motor takes place. This makes it possible to study this circuit by the method of symmetrical components.

With a sinusoidal supply voltage in the steady-state operating mode, the equations [3] are valid for the circuit of Fig. 1.

For the currents of the positive \((I_1)\) and negative \((I_2)\) sequences:

\[ \left. \begin{aligned} \dot I_1 &= \frac{(a-a^2)Z_2+aZ_c}{3Z_1Z_2+Z_c(Z_1+Z_2)} \cdot \frac{U}{a-a^2}, \\ \dot I_2 &= \frac{(a-a^2)Z_1+a^2Z_c}{3Z_1Z_2+Z_c(Z_1+Z_2)} \cdot \frac{U}{a-a^2} \end{aligned} \right\} \tag{1} \]

For the phase currents:

\[ \dot i_A = \frac{U\,(Z_1+Z_2+Z_c)} {3Z_1Z_2+Z_c(Z_1+Z_2)} \]

\[ I_{\mathrm{B}}=\frac{U(a^{2}Z_{c}+aZ_{1})}{3Z_{1}Z_{2}+Z_{c}(Z_{1}+Z_{2})} \tag{2} \]

\[ I=\frac{U(aZ_{2}+a^{2}Z_{1}-Z_{c})}{3Z_{1}Z_{2}+Z_{c}(aZ_{1}+Z_{2})}. \]

For the phase voltages:

\[ \dot U_{\mathrm{A}}= \frac{(a-a^{2})Z_{1}Z_{2}+Z_{c}(aZ_{1}-a^{2}Z_{2})} {3Z_{1}Z_{2}+Z_{c}(Z_{1}+Z_{2})} \cdot \frac{U}{a-a^{2}} \]

\[ \dot U_{\mathrm{B}}= \frac{(a-a)Z_{1}Z_{2}+Z_{c}(Z_{1}-Z_{2})} {3Z_{1}Z_{1}+Z_{c}(Z_{1}+Z_{2})} \cdot \frac{U}{a-a^{2}} \tag{3} \]

\[ \dot U_{\mathrm{C}}= \frac{(a^{2}-a)Z_{1}Z_{2}+Z_{c}(a^{2}Z_{1}-aZ_{2})} {3Z_{1}Z_{2}+Z_{c}(Z_{1}+Z_{2})} \cdot \frac{U}{a-a^{2}}. \]

In equations (1–3):

\[ U \]
— line voltage;

\[ a=e^{-j\frac{4\pi}{3}} \]

\[ a=e^{-j\frac{2\pi}{3}} \]
— phase factors;

\[ Z_{c}=-jx_{c} \]
— capacitive reactance. We neglect the value of the active resistance of the capacitor.

\(Z_{1}; Z_{2}\) — total impedances to currents of the positive and negative sequences.

We determine the total impedances \(Z_{1}\) and \(Z_{2}\) from the equivalent circuits of a single-phase synchronous reluctance motor for the positive and negative sequences of the synchronous operating mode (Fig. 3), where

\(r_{s}; x_{s}\) — active and reactive resistance of a stator winding phase;

\(x_{ad}\) — armature-reaction reactance along the direct axis;

Fig. 3. a) motor equivalent circuit for the positive sequence; b) motor equivalent circuit for the negative sequence.

\(r_c\)—active resistance taking into account losses in the steel;

\(\Theta\)—angle of displacement of the rotor pole axis relative to the direct-field wave;

\[ x'_{\Theta};\quad r'_{\Theta}=\frac{x'_{\Theta}}{\operatorname{tg}\Theta} \]

—reactive and active resistances, dependent on the angle \(\Theta\), referred to the stator;

\[ x'_{\Theta}=\frac{x_{ad}\cdot x_{ac}}{x_{ad}+x_{aq}}; \]

\(x'_R;\ \dfrac{r'_R}{2}\)—average inductive and active leakage resistances of the rotor short-circuited cage, referred to the stator;

\[ x'_R=\frac{x_{Rd}+x_{Rq}}{2} \tag{4} \]

\[ r'_R=\frac{r_{Rd}+r_{Rq}}{2}, \tag{5} \]

where

\(r'_d, r'_{Rg}, x_{Rd}, x_{Rg}\)—active and reactive leakage resistances of the rotor short-circuited cage along the longitudinal and transverse axes;

\(x_m\)—inductive resistance of the magnetizing circuit;

\[ x_m=\frac{x_{ad}\cdot x_{aq}}{x_{ad}+x_{aq}}, \tag{6} \]

\(x_{ad}\)—reactive resistance of armature reaction along the transverse axis.

These equivalent circuits are a variety of the universal circuit [2], and make it possible to apply methods for calculating capacitor asynchronous motors to capacitor synchronous-reluctance motors.

It is clear from the equivalent circuits that the magnitude \(Z_1\) depends on \(\Theta\), while \(Z_2\) is a constant quantity for an unsaturated machine. By specifying the value of the displacement angle \(\Theta\) from 0 to 45 degrees, we obtain the dependence \(Z_1=f(\Theta)\), and then, using formulas 1, 2, 3, determine the currents and voltages for each value of \(\Theta\).

We determine the magnitude \(Z_c\) from the condition of obtaining, in the circuit of Fig. 1, a circular rotating field, which is possible with symmetry of voltages and currents in the motor phases.

The vector diagram for the symmetrical mode is shown in Fig. 2. It can be seen from the diagram that the voltage across the capacitance \(U_k=U\) and forms an angle of \(90^\circ\) with the current vector of the capacitor phase \(I_B\). The coefficient

Fig. 2. Vector diagram for the balanced mode

the power of each phase is equal to 0.5 and corresponds to an angle of \(60^\circ\) between the phase voltage and current; the line power factor is equal to 0.866 and corresponds to an angle of \(30^\circ\) between the line current \(I_A\) and the supply voltage \(U\). The power of the capacitor bank at rated load is

\[ P_k = U_k \cdot I_B = U I_n , \tag{7} \]

where

\(I_n\) is the rated line current in single-phase connection, which we find from the formula for the consumed active power in symmetrical operation:

\[ P_1 = U \cdot I_n \cos \varphi_l , \tag{8} \]

\(\cos \varphi_l\) is the line power factor.

\[ I_n = \frac{P_1}{U \cos \varphi_l}. \tag{9} \]

Substituting (9) into (7), we obtain

\[ P_k = \frac{P_1}{\cos \varphi_l} = \frac{2}{\sqrt{3}} P_1 , \tag{10} \]

then the required value of the working capacitance is

\[ c = \frac{2 \cdot P_1}{\sqrt{3} \cdot U^2 \cdot 2 \pi f}. \tag{11} \]

Having the values of the currents and voltages in complex form, it is not difficult to determine the power factors of each phase and the line power factor for the given parameters \(Z_1, Z_2, Z_c\).

Having determined, from (1), the currents of both sequences, one can calculate, for each value of the angle \(\Theta\), the rotating torques defined by the algebraic difference of the torques of the positive \((M_1)\) and negative \((M_2)\) components

\[ M = M_1 - M_2 . \]

\[ M = k \frac{mp}{f} \left[ \frac{E_1^2 \cdot \dfrac{x'_\Theta}{\operatorname{tg}\Theta}} {\left(\dfrac{x'_\Theta}{\operatorname{tg}\Theta}\right)^2 + x_\Theta^{\prime 2}} - \frac{E_2^2 \cdot \dfrac{r'_2}{2}} {\left(\dfrac{r'_R}{2}\right)^2 + x_R^{\prime 2}} \right], \tag{12} \]

where

\(k\) is a coefficient depending on the connection diagram of the stator windings and the capacitor;

\(m\) is the number of phases;

\(p\) is the number of pole pairs;

\(f\) is the supply frequency;

\[ \left. \begin{aligned} E_1 &= \left| Z_1 - Z_s \right| \cdot I_1 \\ E_2 &= \left| Z_2 - Z_s \right| \cdot I_2 \end{aligned} \right\} \tag{13} \]

\(E_1, E_2\) are the absolute values of the emf of the positive \((E_1)\) and negative \((E_2)\) sequences.

From the equivalent circuit we find:

\[ Z_s = r_s + j x_s \tag{14} \]

— the total leakage impedance of the stator phase.

\[ Z_1 - Z_s = \frac{ \left(-\frac{x'_{\Theta}}{\operatorname{tg}\Theta}+j x'_{\Theta}\right)(r_c+jx_{ad}) }{ \left(\frac{x'_{\Theta}}{\operatorname{tg}\Theta}+r_e\right)+j(x'_{\Theta}+x_{ad}) } \tag{15} \]

\[ Z_2 - Z_s = \frac{ \left(\frac{r'_R}{2}+j x'_R\right)\cdot j x_m }{ \frac{r'_R}{2}+j(x_m+x'_R) }. \tag{16} \]

Knowing the dependence \(M=f(\Theta)\), we determine the degree of utilization of the motor at the displacement angle corresponding to the rated load

\[ \mu = -\frac{M}{M_n}\cdot 100\ (\%), \tag{17} \]

where

\(M_n\) is the rated torque of the motor in three-phase operation.

Having determined the torque on the motor shaft

\[ M_v=M-M_0, \tag{18} \]

where \(M_0\) is the loss torque, it is easy to find the power delivered by the motor

\[ P_2 = 9.81\cdot 10^{-2} M_v\cdot \omega, \tag{19} \]

where

\(\omega\) is the synchronous angular velocity; the efficiency at any value of the angle \(\Theta\):

\[ \eta=\frac{P_2}{P_1}\cdot 100\ (\%) \tag{20} \]

\(P_1\) is determined from expression (8).

Thus it is possible to calculate analytically the operating characteristics of a three-phase synchronous reluctance motor operating with single-phase connection.

REFERENCES

1. Academy of Sciences of the Ukrainian SSR. Institute of Electrodynamics. A. I. Adamenko. Methods for Investigating Asymmetrical Asynchronous Machines. Naukova Dumka Publishing House, Kiev, 1969.

2. O. B. Pevzner. “A Method for Investigating Single-Phase and Capacitor Synchronous Reluctance Electric Motors by Means of a Universal Equivalent Circuit.” Contactless Electrical Machines, Part I. Synchronous Machines. Moscow, 1966.

3. G. B. Merkin. Capacitor Electric Motors for Industry and Transport. Energiya, Moscow, 1966.