Calculation of the Characteristics of Capacitor Braking of a Synchronous Reluctance Motor

Yu. M. Achkasov, B. A. Zakharov

(Presented by the scientific-technical seminar of the Scientific Research Institute of AEM)

When a capacitor synchronous reluctance motor (SRM), connected according to the circuit in Fig. 1,a, is disconnected from the inverter transformer TR or from the supply network, owing to the residual magnetization of the steel or to the residual charge of the capacitance in two stator phases, a self-excitation current will flow. The SRM will enter the regime of asymmetric capacitor braking (an asynchronous generator with self-excitation from capacitors).

The mechanical characteristics of capacitor braking of a DRS-150M type SRM, taken from the screen of a PMKh-1 mechanoscope, are shown in Fig. 1,b; braking was carried out as follows: simultaneously with disconnection of the motor, the starting capacitance \(C_{\text{p}}\) was connected in parallel with the working capacitance \(C_{\text{r}}\). In the graphs of Fig. 1,b the following are denoted:

Fig. 1

\[ M^d=\frac{M_{\text{t}}}{M_{\text{p}}} \]
— relative braking torque;

\(M_{\text{t}}\) — torque of capacitor braking of the SRM;

\(M_{\text{p}}\) — starting torque of the SRM under three-phase symmetrical supply from a sinusoidal-voltage network;

\[ \Omega=\frac{n}{n_c} \]
— relative speed;

\(n\) — instantaneous value of the speed, \(\dfrac{\text{rev}}{\text{min}}\);

\(n_c\) — synchronous speed, \(\dfrac{\text{rev}}{\text{min}}\).

Analysis of the braking characteristics (Fig. 1,b) makes it possible to reveal a feature of asymmetrical capacitor braking of an SRM. With a small capacitance value (\(C = 25\ \mu\text{F}\)), the maximum braking torque occurs at a speed below synchronous \((\Omega = 0.92)\), and its magnitude is small, since it is produced mainly due to losses in the stator. The lower critical speed is high \((\Omega = 0.62)\); braking is ineffective.

With an increase in the connected capacitance, owing to the demagnetizing action of the rotor currents, the voltage on the capacitance decreases, while the stator current increases slowly, as can be seen from the experimental graphs in Fig. 2,a, b.

Fig. 2

In Fig. 2,a, b the following are denoted:

\[ I^\circ = \frac{I}{I_{\text{p}}} \]
— relative current;

\(I\) — instantaneous value of the current;

\(I_{\text{p}}\) — starting current in the three-phase symmetrical circuit;

\[ C^\circ = \frac{C_{\text{r}} + C_{\text{p}}}{C} \]
— relative capacitance;

\(C_{\text{r}}\) — operating capacitance;

\(C_{\text{p}}\) — starting capacitance.

At the same time, as the capacitance increases, the self-excitation frequency decreases and the initial slip increases, which causes an increase in the rotor current. All this leads to an increase in the braking torque due to the growth of total losses. At capacitances greater than \(85\ \mu\text{F}\), because of the small resistance of the short-circuited rotor winding of the SRM, the reaction of the rotor currents reduces the voltage to such an extent that the growth of the stator current ceases, and this leads to the fact that the increase in rotor losses is not fully compensated by the decrease in stator losses, and the initial braking torque remains almost unchanged, while its maximum shifts downward. At the same time, the zone of action of capacitor braking narrows. At a capacitance of more than \(200\ \mu\text{F}\), breakdown of self-excitation is observed because of demagnetization of the machine.

Since improved SRMs are made with a lower active rotor resistance in comparison with asynchronous motors (AMs), the excitation time of the SRM is greater than that of the AM, owing to the greater degree of damping of the main flux by the short-circuited rotor winding. In order to simplify the calculation, we adopt the following assumptions:

1. We neglect the pulsating torque of the SRM because of its smallness in comparison with the asynchronous torque.

1. We do not take into account losses in the rotor steel, since the rotor current has a low frequency.

2. Saturation of the steel is taken into account by averaged parameters of the equivalent circuit of the SRM.

3. We do not take into account the influence of the negative-sequence field on the magnitude of the torque, owing to its smallness.

Under the indicated assumptions, the equivalent circuit of the SRM for capacitor braking, Fig. 3, is valid, where the designations are:

Fig. 3

\(x_c\) — capacitive reactance of the capacitor at a frequency of 50 Hz (ohms);

\(r_s, x_s, x_m, x'_R, \dfrac{r'_R}{s}\) — motor parameters at a frequency of 50 Hz (ohms);

\(F=\dfrac{f_1}{50}\) — stator-current frequency referred to the rated frequency.

For capacitances \(C^\partial=3\div5\), the dependence of the braking torque on speed is well described by the expression

\[ M^\partial = M^\partial_c \cdot \sqrt{\frac{\Omega_{\mathrm{в}}-\Omega_{\mathrm{н}}}{1-\Omega_{\mathrm{н}}}}, \tag{1} \]

where

\(\Omega_{\mathrm{в}}\) — rotational speed of the SRM corresponding to the greatest torque, in fractions of synchronous speed;

\(\Omega_{\mathrm{н}}\) — lower critical speed, in fractions of synchronous speed.

The values of the relative speeds are determined from the formulas obtained in [2]

\[ \Omega_{\mathrm{в}}=\frac{1.4}{\sqrt{C^\partial}};\qquad \Omega_{\mathrm{н}}=\frac{0.53}{\sqrt{C^\partial}}. \tag{2} \]

The braking torque at synchronous speed is determined by the formula obtained for symmetrical capacitor braking of an asynchronous motor (three capacitances are connected in delta and connected to the phases of the stator windings) in [1]

\[ M^\partial_c=\frac{m\cdot E_m^2(\Omega-F)}{\omega_c\cdot M_{\mathrm{н}}\cdot r'_R}, \tag{3} \]

here \(E_m\) — motor emf at \(f=50\) Hz (V);

\(\omega_c\) — synchronous speed \(\left(\dfrac{1}{\mathrm{s}}\right)\);

$M_{\mathrm{n}}$ — rated motor torque (N·m);

$m$ — number of phases.

In the present case, only the SRM phases are traversed by the self-excitation current, i.e., $m=2$, and since with such a circuit, according to [3], losses in the machine are twice as large as in the symmetrical case, formula (3) is written as follows:

\[ M_c^{\partial}=\frac{4E_m^2(\Omega-F)}{\omega_c M_{\mathrm{n}} r_R'} . \tag{4} \]

We take account of the stator iron losses by increasing the total losses. We determine the emf from the experimental magnetization curve of the SRM in relative units $I_m=f(E_m)$, from the magnitude of the conductance of the magnetizing circuit:

\[ b_m=\frac{-(x_s-x_c)F^2}{(r_sF)^2+(x_s-x_c)^2} -\frac{x_R'(F-\Omega)^2}{(r_R'F)^2}. \tag{5} \]

We find the actual value of the current frequency in the stator from the formula obtained in [4],

\[ F=\Omega-\omega, \tag{6} \]

where $\omega$ is the relative frequency of the current in the rotor, determined from the expression

\[ \omega=\frac{-c_4+\sqrt{c_4^2-4c_3c_5}}{2c_3}, \tag{7} \]

where the coefficients, expressed through the relative parameters of the equivalent circuit, are:

\[ \begin{gathered} c_3=a_2b_3+a_3b_2+a_5b_5-a_4b_4-a_1b_7;\\ c_4=a_3b_3+a_5b_6+a_4b_7;\qquad a_1=r_s+r_R';\\ a_2=\Omega(r_sx_R'-2r_R'x_s);\quad a_3=(x_c-\Omega^2x_R')\cdot r_R';\\ a_4=\Omega(r_s+2r_R');\quad a_5=\Omega^2\cdot r_R';\\ b_2=2\Omega(x_s+x_R');\quad c_5=-a_5b_7;\\ b_3=x_c-\Omega^2(x_s+x_R');\\ b_5=2\Omega x_s\cdot x_R';\quad b_7=\Omega r_s\cdot r_R';\\ b_6=r_sr_R'+(x_c-\Omega^2x_s)x_R'. \end{gathered} \]

In accordance with the above, we propose the following calculation procedure:

1. At a relative rotor speed $\Omega=1$ and $C^{\partial}=\mathrm{const}$, determine the current frequency in the rotor from (7).
2. Determine the actual frequency in the stator $F$ from (6).
3. From expression (5), find the value $b_m$ and, from the magnetization curve, determine $E_m$.
4. Substituting $\lambda$, $F$, and $E_m$ into (4), find $M_c^{\partial}$.
5. Substituting $M_c^{\partial}$ into (1) and specifying $\Omega_{\mathrm{v}}$, construct the mechanical characteristic.

Thus, the proposed method makes it possible to approximately calculate analytically the characteristics of asymmetrical capacitor braking of an SRM. In comparison with the methods presented in [1]

and [4], this method does not require the use of complex dependencies and iterative approximations for the calculation; moreover, the error of the method does not exceed 10%.

REFERENCES

1. L. V. Bushtyan, L. M. Petrov. Calculation of the characteristics of short-circuited motors under capacitor braking. Electrical Machine Building and Electrical Equipment. Republican interdepartmental thematic scientific-technical collection, issue 7, 1963.

2. Yu. M. Achkasov, B. A. Zakharov. Mechanical characteristics of a capacitor synchronous reluctance motor. Proceedings of the VII Scientific and Technical Conference. Static Converters in Automation and Electric Drive. Tomsk, 1971.

3. T. P. Gubenko, Ya. V. Kostyrko. The ratio of losses in two-phase and three-phase modes in low-power asynchronous electric motors. Izv. vuzov, “Electromechanics,” No. 7, 1971.

4. S. I. Kitsis. Analytical determination of the characteristics of an asynchronous self-excited generator in braking mode. Energetika i transport, No. 5, 1971.