OPERATION OF AN SRRD WITH A SHORT-CIRCUITED WINDING ON THE ROTOR IN SYNCHRONOUS STEADY-STATE MODE

A. A. Buchmov, G. M. Goloshchevsky, E. V. Kononenko

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

The synchronous mode is the principal operating mode of a synchronous reactive reduction motor (SRRD), and therefore considerable attention has been paid to the synchronous mode in published articles [1, 2, 3]. However, these studies are devoted mainly to calculating the operating characteristics of the motor in synchronous mode, constructing vector and circle diagrams, and compiling equivalent circuits. The question of the influence of parameters on the torque of an SRRD remains little studied; meanwhile, the presence of a short-circuited cage in the rotor slots, as a means for reliable starting of the motor, makes the problem of studying this influence still more complex and relevant.

In the domestic literature there are works devoted to formulating the equations of a synchronous reactive reduction motor that take into account the presence of a winding on the rotor [5, 4]. In these articles, equations of the SRRD are derived that are universal in the sense that, for given voltages and load torque, they completely determine the behavior of an SRRD with a short-circuited winding on the rotor and can be used in studying both transient and steady-state operating modes.

The voltage-balance equations for the steady-state synchronous mode can be obtained from the basic equations derived in [4] by setting \(p=0\). In accordance with the assumptions adopted in [4], only the fundamental harmonics of the currents and voltages of the phase windings of the stator are taken into account. Therefore their instantaneous values may be replaced by their effective values.

Taking into account the short-circuited winding on the rotor, the system of voltage-balance equations for the steady-state synchronous mode, that is, for \(k_p \omega_p = 1\), takes the form

\[ \begin{aligned} -U\sin\theta &= -\psi_q + r I_d;\\ U\cos\theta &= \psi_d + r I_q;\\ 0 &= -\psi_{rq}\cdot n + r_r I_{rd};\\ 0 &= \psi_{rd}\cdot n + r_r I_{rq}. \end{aligned} \tag{1} \]

where

\(n=\dfrac{k_p-1}{k_p}\) is the speed of rotation of the magnetic field relative to the rotor, in relative units.

Substituting the values of the flux linkages

\[ \begin{aligned} \psi_d&=x_d\cdot I_d+x_{ad}\cdot I_{rd};\\ \psi_q&=x_q\cdot I_q+x_{aq}\cdot I_{rq};\\ \psi_{rd}&=x_{ad}\cdot I_d+x_{rd}\cdot I_{rd};\\ \psi_{rq}&=x_{aq}\cdot I_q+x_{rq}\cdot I_{rq} \end{aligned} \tag{2} \]

into equations (1) and solving them simultaneously, we obtain the expression for the currents of all motor windings:

\[ \begin{aligned} I_d&=\frac{x_q\cos\theta-r\sin\theta}{x_d x_q+r^2}\,U -\frac{x_{ad}\cdot x_q\cdot I_{rd}-x_{aq}\cdot r\cdot I_{rq}}{x_d x_q+r^2};\\[4pt] I_q&=\frac{x_d\sin\theta+r\cos\theta}{x_d x_q+r^2}\,U -\frac{x_d\cdot x_{aq}\cdot I_{rq}+x_{ad}\cdot r\cdot I_{rd}}{x_d x_q+r^2};\\[4pt] I_{rq}&=\frac{nU}{A^2-BC}\cdot \frac{A x_{aq}(r\cos\theta+x_d\sin\theta)-B x_{ad}(x_q\cos\theta-r\sin\theta)}{x_d x_q+r^2};\\[4pt] I_{rd}&=\frac{nU}{A^2-BC}\cdot \frac{C x_{aq}(r\cos\theta+x_d\sin\theta)-A\cdot x_{ad}(x_q\cos\theta-r\sin\theta)}{x_d x_q+r^2}. \end{aligned} \tag{3} \]

where

\[ A=\frac{n\cdot x_{ad}\cdot x_{aq}\cdot r}{x_d x_q+r^2}+r_r; \]

\[ B=-\frac{x_d\cdot x_{aq}^{\,2}\cdot n}{x_d\cdot x_q+r^2}+x_{rq}\cdot n; \]

\[ C=\frac{x_q\cdot x_{ad}^{\,2}\cdot n}{x_d x_q+r^2}-n\cdot x_{rd}. \]

The expressions for the stator currents \(I_d, I_q\) consist of two components. One component is the currents of an SRRD without a winding on the rotor, while the other component is determined by the motor parameters and the currents of the rotor circuits. The currents of the rotor circuits \(I_{rd}, I_{rq}\) are a complicated function of voltage, load angle, and motor parameters. As the active resistance \(R_r\) increases, the magnitude of the rotor currents decreases, and at \(R_r=\infty\), which means the absence of rotor circuits, both currents are equal to zero.

The expression for the electromagnetic torque

\[ M=(\psi_d\cdot I_q-I_d\cdot\psi_q)-\frac{k_p-1}{k_p}(I_{rd}\psi_{rq}-I_{rq}\psi_{rd}). \tag{4} \]

after substitution of the flux-linkage values (2) takes the form

\[ M=(x_d-x_q)I_d I_q+(x_{ad}-n x_{aq})I_{rd}\cdot I_q +(n x_{ad}-x_{aq})I_d I_{rq}+n(x_{rd}-x_{rq})I_{rd}I_{rq}. \tag{5} \]

Substituting the values of the currents \(I_d, I_q, I_{rd}, I_{rq}\) into (5), after transformations sufficiently cumbersome that they need not be included in this article, the expression for the electromagnetic torque can be represented in the form

\[ M = M_m \sin(2\theta+\psi_1)+M_n . \tag{6} \]

It follows from expression (6) that the electromagnetic torque at synchronous rotor speed can be represented as the sum of two components. The first component of the torque is a variable quantity depending on the load angle \(\theta\), while the second is a constant independent of the load. The amplitude of the variable component of the torque \(M_m\) and the constant component \(M_n\), when a short-circuited winding is present on the rotor, are complicated functions of the voltage and the motor parameters. It is impossible to analyze these torque components in general form, and therefore a numerical method of analysis was adopted as the basis in the study.

Fig. 1. Dependence of SRRD torque on \(\theta\) for \(\dfrac{x_d}{x_q}=1.5\), \(r=0.1\), and various \(r_r\):
\(a — r_r=0.3\); \(b — r_r=0.7\); \(v — r_r=\infty\).

Fig. 2. Dependence of SRRD torque on \(\theta\) for \(r=0.1\); \(r_r=0.5\), and various \(\dfrac{x_d}{x_q}\):
\(a — \dfrac{x_d}{x_q}=1.75\); \(b — \dfrac{x_d}{x_q}=1.5\); \(v — \dfrac{x_d}{x_q}=1.25\).

We shall call the constant component in the expression for the electromagnetic torque (6) the asynchronous component of the torque, since it is analogous to the torque of a normal asynchronous short-circuited motor and owes its origin mainly to the interaction of the stator magnetic field with the currents \(I_{rd}, I_{rq}\) flowing in the equivalent rotor windings. This winding on the rotor has a significant influence on the energy characteristics of the SRRD. If, in a machine without a rotor winding, all the electromagnetic power transmitted through the air gap is expended on producing electromagnetic torque (in accordance with the assumptions, iron losses are not taken into account), then when a rotor winding is present part of the energy is expended on losses in its active resistance.

Calculations carried out for the established synchronous operating mode of an SRRD with a starting winding on the rotor according to equations (3), (4) show that the active resistance of the rotor winding has a su—

...a significant influence on the magnitude of the resultant electromagnetic torque. Figure 1 presents torque curves calculated for three cases: without a rotor winding \((r_r=\infty)\) and with winding resistances \(r_r=0.7\) and \(r_r=0.3\).

The electromagnetic torque at \(r_r=0.7\) varies from the maximum positive value to the maximum negative value in generator mode. And if an increase in the maximum torque in motor mode is expedient for increasing the overload capacity of the machine and improving its utilization, then a decrease in torque in generator mode has an adverse effect on the dynamic property of the motor.

Increasing the asynchronous component of the torque by reducing the active resistance of the rotor winding is possible only up to a certain limit. This is evident from the torque curve at \(r_r=0.3\) (Fig. 1, a). At \(r_r=0.3\), the torque curve has no negative values at the given rotor speed; this means that, at no load, synchronous operation is impossible for an SRRD with such parameters. Starting and synchronous operation in this case are possible only in the presence of a static braking torque.

Figure 2 presents the dependences of the resultant electromagnetic torque on the angle \(\theta\) for various values of the ratio of the synchronous inductive reactances along the \(d\) and \(q\) axes. As for motors without a starting winding, this ratio characterizes the geometry of the toothed zone of the motor. It follows from the curves presented that increasing \(\dfrac{x_d}{x_q}\) makes it possible to raise the maximum torque in motor mode, and also to reduce the magnitude of the torque at negative values of the angle \(\theta\). Larger values of negative torque, in turn, make it possible to use a starting cage with a smaller active resistance and, consequently, with a larger asynchronous torque. In other words, increasing the ratio \(\dfrac{x_d}{x_q}\) in any case has a positive effect on the characteristics of the machine.

Analysis of the torque curves presented, calculated for different parameters (Figs. 1, 2), shows that changing the parameters does not affect the shape of the torque curve: in all cases it is sinusoidal, and only the amplitude of the alternating component of the torque and the magnitude of the constant component change.

For synchronous operation, the amplitude of the alternating component of the torque must always be increased, since it determines the main performance indicators of the machine to a greater extent. The magnitude of the constant component of the resultant torque must be sufficient to accelerate the rotor during starting, but less than the amplitude of the alternating component of the resultant torque, since otherwise the motor will not be able to operate in synchronous mode at no load.

Figure 3 presents the dependences of the amplitude of the alternating component of the resultant torque, calculated for the ratio \(\dfrac{x_d}{x_q}=1.5\), as the active resistance of the rotor winding is varied. As can be seen from the curves, increasing \(r\) from 0.6 to 2.0 is accompanied by an increase in the amplitude from 0.08 to 0.151 at a stator-winding active resistance of 0.1. Physically, this is explained by the damping action of the rotor winding on the stator magnetic field; the degree of damping increases as the rotor active resistance decreases.

Thus, by using the starting short-circuited winding on the rotor, we reduce the magnitude of the variable component of the torque. As the active resistance of the rotor winding decreases, its influence increases.

The magnitude of the constant component of the torque decreases as the active resistance of the rotor winding increases (Fig. 4). Moreover, from the calculated torque curves presented it is evident that the influence of the active resistance of the rotor is manifested much more strongly in the region of small values of \(r_r\), and much more weakly from \(r_r = 1.2\) and above.

Fig. 3. Dependence of the amplitude of the variable component of the torque on \(r_r\) for \(\dfrac{x_d}{x_q} = 1.5\) and for different \(r\):
\(a\)—\(r = 0.1\); \(b\)—\(r = 0.2\); \(c\)—\(r = 0.3\)

The active resistance of the stator winding has a somewhat smaller influence on the variable component of the torque. From Fig. 5 it is evident that increasing the resistance from \(r = 0.1\) to \(r = 0.2\) for the ratio

\[ \frac{x_d}{x_q} = 1.5 \]

reduces the amplitude of the variable component by 20%.

The character of the influence of the ratio of the synchronous inductive reactances along the longitudinal and transverse axes on the amplitude of the variable component of the resultant torque is shown in Fig. 5. Curves \(a\), \(b\), \(c\) were calculated for different values of the resistance of the stator winding as a function of the magnitude of the ratio \(\dfrac{x_d}{x_q}\). From these curves it is evident that, as the ratio \(\dfrac{x_d}{x_q}\) increases, the magnitude of the torque amplitude increases significantly. The ratio \(\dfrac{x_d}{x_q}\) has the opposite effect on the magnitude of the constant component.

Fig. 4. Dependence of the constant component of the torque on \(r_r\) for \(r = 0.1\) and for different \(\dfrac{x_d}{x_q}\):
\(a\)—\(\dfrac{x_d}{x_q} = 1.35\); \(b\)—\(\dfrac{x_d}{x_q} = 1.5\);
\(c\)—\(\dfrac{x_d}{x_q} = 1.75\)

Analysis of the calculated curves presented shows that the amplitude of the variable component of the resultant electromagnetic torque in the established synchronous operating mode decreases with an increase in the active resistance of the stator winding and increases with an increase in the active resistance of the rotor winding and in the ratio $\frac{x_d}{x_q}$.

Physically, this can be explained by the fact that, as the stator resistance increases, the electromagnetic power transmitted through the air gap decreases, and consequently so does the electromagnetic torque of rotation. The increase in the amplitude of the variable component of the torque with increasing $r_r$ is due to the fact that, at small resistances of the rotor winding, the degree of damping of the stator magnetic fields is greater, which causes a decrease in the variable component of the torque.

The magnitude of the constant component depends mainly on the value of the active resistance of the rotor winding. This is explained

Fig. 5. Dependence of the amplitude of the variable component of torque on $\frac{x_d}{x_q}$ at $r_r = 0.5$ and for different $r$; $a$ — $r = 0.1$; $b$ — $r = 0.2$; $c$ — $r = 0.3$

by the fact that, as the resistance decreases, the asynchronous torque increases, which mainly determines the magnitude of the constant component.

From the curves presented (Figs. 2, 3) it is seen that, in the no-load mode, the SRRD with a winding on the rotor operates at negative values of the angle $\theta$. The magnitude of this angle in the no-load mode depends on the ratio of the variable and constant components of the torque. The greater the magnitude of the constant component, the greater the value of the negative angle $\theta$. As the load increases, $\theta$ decreases to 0, and then again increases toward larger positive values. When the load is increased above $\theta_{\mathrm{cr}}$, at which the resultant torque has its maximum value, the motor rotor falls out of synchronism. The value of the torque at $\theta_{\mathrm{cr}}$ characterizes the overload capacity of the SRRD.

REFERENCES

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