EFFECT OF ROTOR ASYMMETRY ON THE STARTING CHARACTERISTICS OF A SYNCHRONOUS RELUCTANCE MOTOR

E. V. Kononenko, T. V. Chesheva

(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)

A synchronous reluctance motor without a starting winding on the rotor and with a laminated core, when operating in the asynchronous mode, is an example of a machine in which only magnetic rotor asymmetry is present. The study of this case is chiefly of theoretical interest not only for investigation of the asynchronous operating mode of a synchronous reluctance motor, but also of other electric machines whose rotor design has pronounced poles without starting windings (reductor, stepper, and other synchronous machines). In the literature this question has already been considered [1 et al.], but it has been covered and developed quite insufficiently.

In calculating and studying the asynchronous mode of a synchronous reluctance motor without a starting winding on the rotor, the system of Park–Gorev differential equations may be used. In doing so, it is necessary to keep in mind that the steady-state values of the operator resistances of the machine will be equal to

\[ X_d(js)=x_d,\qquad X_q(js)=x_q. \tag{1} \]

The system of differential equations for voltage equilibrium in the present case may be written as follows:

\[ u_d=[p x_d+r]i_d-x_q(1-s)i_q, \]

\[ u_q=x_d(1-s)i_d+[p x_q+r]i_q. \tag{2} \]

Solving these equations, we find the values of the current complexes:

\[ \bar{i}_d= \frac{\left[r-j(1-2s)x_q\right]\bar{U}_d} {r^2+jrs(x_d+x_q)+(1-2s)x_d x_q}, \]

\[ \bar{i}_q= \frac{\left[r-j(1-2s)x_d\right]\bar{U}_q} {r^2+jrs(x_d+x_q)+(1-2s)x_d x_q}, \tag{3} \]

where

\[ \bar{u}_d=Ue^{j(st+\delta_0)},\qquad \bar{u}_q=-j\bar{U}_d; \]

\(\delta_0\) is the angle formed by the resultant voltage vector with the rotor \(d\)-axis at the initial instant of time.

In investigations of the steady asynchronous mode it is convenient to operate with space-wave vectors. The space vector of the stator current \(I\) may be represented as the sum of the space vectors of the positive-sequence current \(\bar{I}_1\) and the negative-sequence current \(\bar{I}_2\):

\[ \bar{I}=\bar{I}_1+\bar{I}_2, \]

where

\[ \overline{I}_1= \frac{\left[r-j\frac{1}{2}(1-2s)(x_d+x_q)\right]Ue^{j(st+\delta_0)}} {r^2+jrs(x_d+x_q)+(1-2s)x_dx_q}, \tag{5} \]

\[ \overline{I}_2= \frac{-j\frac{1}{2}(1-2s)(x_d-x_q)Ue^{-j(st+\delta_0)}} {r^2-jrs(x_d+x_q)+(1-2s)x_dx_q}. \tag{6} \]

Let us analyze the law of variation of each of these components. If the variable parameter is taken to be \(k=2s-1\), then the expressions for the currents \(\overline{I}_1\) and \(\overline{I}_2\) can be written in the following form:

\[ \overline{I}_1= \frac{\left[r+j\frac{1}{2}(x_d+x_q)k\right]Ue^{j(st+\delta_0)}} {\left[r^2+j\frac{1}{2}r(x_d+x_q)\right] +\left[j\frac{1}{2}r(x_d+x_q)-x_dx_q\right]k} = \]

\[ = \frac{\overline{A}+\overline{B}k}{\overline{C}+\overline{D}k} \,Ue^{j(st+\delta_0)} \tag{7} \]

\[ \overline{I}_2= \frac{\left[j\frac{1}{2}(x_d-x_q)k\right]Ue^{-j(st+\delta_0)}} {\left[r^2-j\frac{1}{2}r(x_d+x_q)\right] +\left[-j\frac{1}{2}r(x_d+x_q)-x_dx_q\right]k} = \]

\[ = \frac{\overline{E}k}{C^*+D^*k}\cdot Ue^{-j(st+\delta_0)}. \tag{8} \]

Equation (7) is the equation of a circle that does not pass through the origin. Consequently, as the slip varies within the limits \(\pm\infty\), the end of the direct-sequence current vector \(\overline{I}_1\) describes a circle. Equation (7) makes it possible to find the vector of the center of the circle and the radius vector. The vector of the center of the circle described by the end of the current vector \(\overline{I}_1\) is

\[ \overline{N}_1= \left[ \frac{r}{r^2+x_dx_q} - j\frac{x_dx_q+\frac{1}{4}(x_d+x_q)^2} {(x_d+x_q)(x_dx_q+r^2)} \right]Ue^{j(st+\delta_0)}. \tag{9} \]

The radius vector of this circle is

\[ \overline{R}_1= \left[ -j\frac{\frac{1}{4}(x_d-x_q)} {r^2+x_dx_q} \right]Ue^{j(st+\delta_0)}. \tag{10} \]

The modulus of the diameter of the circle is equal to

\[ d_{01}= \frac{\frac{1}{2}(x_d-x_q)U} {r^2+x_dx_q}. \tag{11} \]

Equation (8) is the equation of a circle passing through the origin. The radius vector of this circle is equal to

\[ \overline{R}_2= \frac{ \frac{1}{2}(x_d-x_q)\left[r-j\frac{1}{2}(x_d+x_q)\right]Ue^{-j(st+\delta_0)} } {(x_d+x_q)(r^2+x_dx_q)}. \tag{12} \]

The modulus of the diameter of this circle, described by the end of the negative-sequence current vector \(\overline{I}_2\), is equal to

\[ d_{02}= \frac{(x_d-x_q)\sqrt{r^2+\frac{1}{4}(x_d+x_q)^2}} {(x_d+x_q)(r^2+x_dx_q)}\cdot U . \tag{13} \]

If the magnitude of the active resistance of the stator winding is neglected, i.e., if it is assumed that \(r=0\), then equations (7) and (8) take the form

\[ \overline{I}_1= \frac{1}{2}\, \frac{j(x_d+x_q)Ue^{j(st+\delta_0)}}{x_dx_q}, \tag{14} \]

\[ \overline{I}_2= \frac{1}{2}\, \frac{j(x_d-x_q)Ue^{-j(st+\delta_0)}}{x_dx_q}. \tag{15} \]

The amplitude of the stator current for \(r=0\) will be

\[ I= \frac{\sqrt{x_d^2+x_q^2}}{x_dx_q}\cdot U . \tag{16} \]

From equations (14)–(16) it is evident that, for \(r=0\), the stator current does not depend on the magnitude of slip. From the physical point of view, neglecting the active resistance of the stator winding leads to the following. At \(r=0\) the stator winding becomes a superconductor; therefore the rotor asymmetry has no effect on the operation of the machine, since the negative-sequence flux caused by the rotor asymmetry cannot penetrate into the stator. Consequently, in order to take account of the rotor asymmetry, it is necessary in investigations and calculations to operate with the real values of the active resistances in the stator winding.

Analysis of the expressions obtained for the currents \(\overline{I}_1\) and \(\overline{I}_2\) makes it possible to conclude that the positive- and negative-sequence currents of a synchronous reluctance machine without starting windings on the rotor are determined mainly by the parameters of the synchronous mode, \(x_d\) and \(x_q\). Figure 1,a gives curves of the moduli of the currents \(\overline{I}_1\) and \(\overline{I}_2\) and of the amplitude of the stator current vector \(\overline{I}\) as functions of slip for two values of the active resistance of the stator winding: \(r=0.06\) and \(r=0.2\). In these curves there is a dip at subsynchronous speed. This dip is not observed if the active resistance of the stator winding is neglected. The main influence on the shape of the curve \(I=f(s)\) is exerted by the reverse field caused by the salient-pole character of the machine. This field rotates relative to the stator with a speed \((1-2s)\), variable in magnitude and direction, and is damped by currents of the corresponding frequency arising in it. At slips close to 0.5, the damping action of the stator winding weakens, since as the frequency decreases the influence of active resistance increases. At subsynchronous speed the reverse field becomes stationary with respect to the stator winding, and for this reason its damping influence is weakened. Hence it is clear that the boundaries of the slip region in which the action of the reverse field is manifested most strongly are determined by the magnitude of the active resistance of the stator winding, expanding as it increases (Fig. 1). The amplitude of the resultant reverse field at \(s=0.5\) depends on the degree of magnetic asymmetry, determined by the difference between the synchronous inductive reactances along the \(d\) and \(q\) axes. The curve \(I=f(s)\) is symmetric with respect to \(s=0.5\) because of the absence of electrical asymmetry of the rotor.

Fig. 1. Starting characteristics of a synchronous reluctance motor without starting windings on the rotor, having the following parameters: \(x_d=2.3\); \(x_q=0.45\); \(x''_d=x''_q=0.2\); \(T_d=T_q=0\). Solid lines—calculation for \(r=0.06\), dashed lines—calculation for \(r=0.2\):
a) current starting characteristics; b) torque starting characteristics.

The electromagnetic torque of a synchronous reluctance motor without starting windings is not constant in time, and it can be represented as the sum of the average torque \(M_c\) and the alternating torque \(M_p\), pulsating in time with the frequency of twice the slip. Since there is no current in the rotor core, the occurrence of the torque \(M_c\), equal to the sum of the torques of the positive \(M_1\) and negative \(M_2\) sequences, is explained only by deformation of the magnetic flux due to the salient-pole nature of the rotor.

The analysis carried out makes it possible to establish that in this case the torques of the positive and negative sequences are equal in magnitude and direction and can be written in the following form:

\[ M_1=M_2=-\frac{\frac{1}{4}r(1-2s)(x_d-x_q)^2U^2} {\left[r^2+(1-2s)x_dx_q\right]^2+r^2s^2(x_d+x_q)^2}. \tag{17} \]

Then the average torque will be equal to

\[ M_c=M_1+M_2=2M_1. \]

It follows from the analysis of this equation that in a synchronous reluctance motor without a starting winding on the rotor, when operating in an asynchronous mode \((s=\mathrm{const})\), a torque constant in time arises. It is necessary

it should be emphasized that this is the mean value of the asynchronous torque, whose magnitude does not depend on the position of the rotor in space.

From equation (18) it is evident that, when the slip varies within the limits from 1.0 to 0.5, the torque \(M_c\) is positive, while for \(s<0.5\) it changes sign to the opposite. Consequently, a synchronous reluctance motor without a starting winding on the rotor, when the rotational speed varies from zero to subsynchronous, can operate as a motor; with a further increase in speed it changes over to generator operation. Therefore, independent starting of a synchronous reluctance motor and its synchronization are possible only if there is a starting winding on the rotor. The curves \(M_c=f(s)\) for two values of the active resistance of the stator winding are presented in Fig. 1, b.

An increase in the active resistance of the stator winding amounts mainly to an increase in the starting torque. At the same time, this circumstance has no significant effect on the magnitude of the dip. At \(r=0\) the mean torque is zero. That is, the influence of magnetic asymmetry at \(r=0\) cannot be detected. This important circumstance should be taken into account if the starting process of a salient-pole machine is being investigated in general.

It is of interest to analyze the expression for the amplitude of the pulsating torque and its dependence on slip. The amplitude of the pulsating torque may be represented as follows:

\[ M_{\mathrm{pl}}= \sqrt{ \frac{b_1^2}{c^2+d^2} \left[ U^2 + 4\frac{r^2+b^2}{c^2+d^2}r^2 \left(\frac{1-s}{1-2s}\right)^2 - 4U\frac{rc+bd}{c^2+d^2} \left(\frac{1-s}{1-2s}\right) \right] }, \tag{19} \]

where

\[ b_1=\tfrac{1}{2}(1-2s)(x_d-x_q), \]

\[ b=\tfrac{1}{2}(1-2s)(x_d+x_q), \]

\[ c=r^2+(1-2s)x_d x_q, \]

\[ d=rs(x_d+x_q). \]

By this formula \(M_{\mathrm{pl}}\) can be calculated for all values of slip except \(s=0.5\). In Fig. 1, b, the dependences \(M_{\mathrm{pl}}=f(s)\) are presented for two values of \(r\). From these curves it is evident that the pulsating torque depends little on slip, except in the region near subsynchronous rotational speed. The dip in this region depends mainly on the ratio of the synchronous inductive reactances \(x_d\) and \(x_q\), while an increase in the active resistance of the stator winding is manifested mainly in a widening of the dip zone and a slight decrease of \(M_{\mathrm{pl}}\) at starting.

When investigating the asynchronous mode of a synchronous reluctance motor, it is important to trace the influence not only of the magnetic asymmetry of the rotor on the starting characteristics, but also how the characteristics change if there is a starting winding on the rotor either along the direct axis or along the quadrature axis. In Figs. 2 and 3 the starting characteristics of a synchronous reluctance motor are presented with the following parameters: \(x_d=2.3;\ x_q=0.45;\ x''_d=0.2;\ x''_q=0.2;\ r=0.06\). But in one case

Fig. 2. Starting characteristics of a synchronous reluctance motor with one winding on the rotor, having the following parameters: \(x_d = 2.3\); \(x_q = 0.45\); \(x''_d = 0.2\); \(x''_q = 0.2\); \(T_d = 40\); \(T_q = 0\).
a) current starting characteristics; b) torque starting characteristics

Fig. 3. Starting characteristics of a synchronous reluctance motor with one winding on the rotor, having the following parameters: \(x_d = 2.3\); \(x_q = 0.45\); \(x''_d = x''_q = 0.2\); \(T_d = 0\); \(T_q = 10\).
a) current starting characteristics; b) torque starting characteristics

(Fig. 2) there is no winding along the direct axis, and in the other case (Fig. 3) there is none along the quadrature axis. The calculations of these characteristics were carried out by an exact method on the “Promin” digital computer. From the data presented it is evident that, when there is no starting winding on the rotor along the direct axis \([T_d = 0]\), the synchronous reluctance motor cannot develop a speed greater than subsynchronous (Fig. 3, b). Comparing the curves \(M_c = f(s)\), presented in Fig. 2, b and Fig. 3, b, one can detect a significant difference in the degree of influence of the rotor-winding parameters on the starting characteristics. The difference in the starting characteristics (Fig. 2, b and Fig. 3, b) is explained primarily by the magnitude of the difference in the operational resistances along the \(d\) and \(q\) axes, which depends on the parameters of the rotor windings (Fig. 4).

The influence of the ratio of the established values of the operational resistances along the \(d\) and \(q\) axes on the starting characteristics of a synchronous

of a reluctance motor can be traced by comparing the starting characteristics considered above for three cases:

1) the machine has a starting winding only along the direct axis;

2) the machine has a starting winding only along the quadrature axis;

3) the machine has no starting windings on the rotor.

The starting characteristics for these machines are presented in Figs. 1–3, and Fig. 4 presents the dependences of the moduli of the operational impedances on slip for all three cases. The largest dip in the starting characteristics is observed in the synchronous reluctance motor that has no winding along the direct axis (Fig. 3). The magnitude of the difference between the operational impedances along the \(d\) and \(q\) axes corresponding to this variant is the greatest (Fig. 4):

Fig. 4. Dependences of the moduli of the steady-state values of the operational impedances along the \(d\) and \(q\) axes on slip for a synchronous reluctance motor having the following parameters: \(x_d=2.3\); \(x_q=0.45\); \(x''_d=x''_q=0.2\); \(T_d=40\); \(T_q=10\)

\[ x_d(js)=x_d, \]

\[ X_q(js)=\frac{x_q+jsx''_qT_q}{1+jsT_q}. \tag{20} \]

The smallest dip in the starting characteristics is observed in the case when there is no winding on the rotor along the quadrature axis; this case corresponds to the smallest difference between the operational impedances along the \(d\) and \(q\) axes (Fig. 2).

\[ X_d(js)=\frac{x_d+jsx''_dT_d}{1+jsT_d}, \tag{21} \]

\[ x_q(js)=x_q \]

A synchronous reluctance motor without windings on the rotor is an intermediate case between the first two. The difference between the operational impedances along the \(d\) and \(q\) axes for this case (Fig. 1), which is the difference between the synchronous inductive reactances, determines the magnitude of the dip in the starting characteristics, which is greater than in the machine with a starting winding only along the direct axis, and smaller than in the machine with a starting winding only along the quadrature axis.

Thus, the form of the starting characteristics of the asynchronous mode of a synchronous reluctance motor is determined mainly by the ratio of the established values of the operational resistances along the \(d\) and \(q\) axes. Consequently, this is the principal factor that determines the possibility of operation of a synchronous reluctance motor in the asynchronous mode and the conditions for normal pull-in to synchronism. This important circumstance must be kept in mind when designing such motors.

REFERENCES

1. G. G. Barshchevskii. Electromagnetic torque of a reluctance machine. Elektrichestvo, 1965, No. 3.

2. K. P. Kováčh, Über Asynchronmotoren mit asymmetrischem Läufer, Archiv für Elektrotechnik, 1964, 49 Band, Heft 3.