INFLUENCE OF PARAMETERS ON THE STARTING CHARACTERISTICS OF SYNCHRONOUS RELUCTANCE MOTORS

E. V. Kononenko

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

When designing synchronous reluctance motors (SRMs) that satisfy specified starting conditions, it is necessary to know the influence of various parameters on motor operation. In the present article, in studying the influence of parameters on the starting characteristics, the calculations were carried out according to the method set forth in [1]. By starting characteristics are meant the dependences of the average torque \(M_c\) and the stator current on slip. The influence of parameters and rotational speed on the amplitude of the pulsating torque was also investigated. In order to reduce the time required, all calculations were performed on the “Promin” digital computer.

The investigations carried out showed that the form of the starting characteristics of an SRM is determined mainly by the magnitude and ratio of the steady-state values of the operational resistances along the \(d\)- and \(q\)-axes \([x_d(js)\) and \(x_q(js)]\). The presence of magnetic asymmetry leads to the fact that the curve \(M_c\), unlike in asynchronous motors, passes through zero at \(s > 0\), while at synchronous rotational speed \(M_{c0} \ne 0\) (Fig. 1).

Fig. 1. Mechanical characteristics of an SRM having the following parameters: \(x_d = 2.3\); \(x_q = 0.45\); \(x_d'' = 0.2\); \(x_q'' = 0.2\); \(T_d = 40\); \(T_q = 10\). The dotted curve was calculated by an approximate method.

Analyzing the equation for the average torque of an SRM, one can find that at synchronous speed

\[ M_{c0}=-\frac{1}{2}\,\frac{r(x_d-x_q)^2}{(r^2+x_dx_q)^2}\,U^2. \]

It follows from this equation that \(M_{c0}\) is negative, and its magnitude depends on the degree of magnetic asymmetry, determined by the difference between the synchronous inductive reactances along the \(d\)- and \(q\)-axes, and on the active resistance of the stator winding \(r\). In machines with a nonsalient-pole rotor \((x_d=x_q)\), \(M_{c0}=0\), regardless of the value of \(r\).

The considered properties of the starting characteristics have a substantial influence on the starting of an SRM, especially at the final stage—during pull-in to synchronism. The use of approximate methods for calculating starting characteristics, based on the separate consideration of phenomena along the \(d\)- and \(q\)-axes, does not make it possible to reveal these features. For comparison, Fig. 1 gives the mechanical characteristics of an SRM calculated by exact [1] and approximate methods. The formulas for calculating the starting characteristics by the approximate method are given in the appendix. Comparing the torque curves \(M_c\), we see that neglecting the active resistance of the stator winding in the example considered leads to inadmissibly large errors at all values of slip. The approximate method, at slips \(s=0.3 \div 1.0\), gives results that agree satisfactorily with the exact method, despite the fact that \(r\) is relatively large. This is explained by the fact that, for the adopted parameter values, the operator reactances of the SRM along the \(d\)- and \(q\)-axes at \(s \ge 0.3\) are practically equal to one another, as is seen from Fig. 2. In the case when \(x_d(js)=x_q(js)\), the approximate method becomes exact.

Fig. 2. Dependences of the moduli of the operator reactances of an SRM on slip. Curve 1—for the direct axis; curve 2—for the quadrature axis.

At slip values less than 0.3, the difference in the operator reactances along the \(d\)- and \(q\)-axes increases, which leads to a divergence between the results of calculating the torque \(M_c\) by the exact and approximate methods. The difference in the results of calculating \(M_c\) also depends on the magnitude of the active resistance of the stator winding. Figure 3 gives the dependences of the torque \(M_c\) on \(r\), calculated for two values of slip (0.02 and 0.05). Comparing the \(M_c\) curves, we see that the use of the approximate method for calculating the starting characteristics of an SRM in the region of small slips leads to significant errors. A comparison of the rms values of the stator current showed that the approxi-

Fig. 3. Dependences of the torque \(M_c\) on the active resistance of the stator winding, calculated by exact (solid lines) and approximate (dashed lines) methods.

The approximate method in the case under consideration gives results that practically coincide with the exact method over the entire range of variation of slip.

The curves of variation of the amplitude of the pulsating torque \(M_p\) and of the stator current for an SRM having the same parameters as in the preceding cases, as functions of slip and of the active resistance of the stator winding, are shown in Fig. 4. From the data presented it is seen that the amplitude of the pulsating torque at synchronous rotational speed practically does not depend on the magnitude of the active resistance of the stator winding.

Fig. 4. Curves of variation of the amplitude of the pulsating torque (solid lines) and of the stator current (dashed lines) with slip.

The form of the mechanical characteristics depends on the parameters of the rotor winding. To confirm this, Fig. 5 presents a family of characteristics calculated for an SRM having the parameters: \(x_d=2.3\); \(x_q=0.45\); \(x''_d=0.2\); \(x'_q=0.2\); \(r=0.06\); \(T_q=0.25T_d\). From the data presented it is seen that, in the absence of starting windings on the rotor \((T_d=0)\), the SRM cannot develop a speed greater than half-synchronous. With an increase in the time constants of the rotor winding, the starting characteristics of the motor improve, since the dip in the torque curve at a speed of ...

Fig. 5. Influence of the parameters of the rotor winding on the mechanical characteristics of an SRM.

the more the speed approaches the semisynchronous value, the smaller it becomes. For \(T_d \ge 30\), no dip is found in the mechanical characteristic. This is explained by the fact that, as the time constants of the rotor winding increase, the difference between the operational resistances along the \(d\) and \(q\) axes in the asynchronous mode decreases.

Fig. 6. Mechanical characteristics of SRMs having the following parameters
\(x_d = 2.3;\ x_q = 0.45;\ x_d'' = -0.2;\ x_q'' = -0.2;\)
\(r = 0.06;\) curve 1 — \(T_d = 0,\ T_q = 10;\)
curve 2 — \(T_d = 40,\ T_q = 0.\)

Studies have also shown that windings along the longitudinal and transverse axes of the rotor have different effects on the starting characteristics. Figure 6 gives the mechanical characteristics of SRMs having on the rotor windings only along the longitudinal or only along the transverse axes. From the curves it is seen that, when there is on the rotor a winding only along the transverse axis (curve 1), the torque \(M_c\) is negative at slips lying within the range \(0.23 \div 0.5\). If, however, the rotor has a winding only along the longitudinal axis (curve 2), in the semisynchronous-speed zone the mechanical characteristic has a dip, but the torque \(M_c\) remains positive. This is explained by the fact that in the first case the difference of the operational resistances along the \(d\) and \(q\) axes is significant for all values of slip, whereas for the second case this difference, for \(s > 0.1\), is considerably smaller, as follows from Fig. 2.

Studies also show that the active resistance of the stator winding has a considerably smaller effect on the magnitude of the dip in the torque curve than do the time constants of the rotor winding, as follows from a comparison of Figs. 7 and 6.

Fig. 7. Mechanical characteristics of SRMs having parameters: \(x_d=2.3\);
\(x_q=0.45\); \(x_d''=0.2\); \(x_q''=0.2\); \(T_q=0\).
a) \(r=0.06\); \(T_d=10;\ 40;\ 100\),
b) \(T_d=40\); \(r=0.06;\ 0.2;\ 0.3\).

The leakage resistance of the rotor windings and the magnitude of the air gap have the same effect on the starting characteristics of an SRM as in an induction motor.

Conclusions

The investigations carried out have shown that the form of the starting characteristics of an SRM is determined mainly by the magnetic and electrical asymmetry of the rotor. The dip in the average-torque curve is determined by the difference between the operational resistances \(x_d(js)\) and \(x_q(js)\), and its magnitude depends on the active resistance of the stator winding.

The approximate method for calculating the starting characteristics gives satisfactory results only in cases where the difference between the operational resistances along the \(d\)- and \(q\)-axes is small. For \(s \leq 0.1\), the torque must be calculated by the exact method, regardless of the magnitude of the electrical asymmetry of the rotor.

Appendix

Calculation of the starting characteristics of an SRM by the approximate method

In calculating the starting characteristics by the approximate method, the possibility is allowed of considering separately the phenomena along the longitudinal ...

… and transverse axes of the rotor of the SynRM. In this case the expressions for the current phasors along the \(d\) and \(q\) axes can be determined from equivalent circuits analogous to the known equivalent circuits of induction motors, or analytically:

\[ \overline{i}_d=\frac{1}{r+jx_d(js)}\,Ue^{j(st+\delta_0)}; \]

\[ \overline{i}_q=\frac{1}{r+jx_q(js)}\,Ue^{j(st+\delta_0)}. \tag{1} \]

The steady-state values of the operational resistances of the SynRM along the \(d\) and \(q\) axes can be represented in the form [2]

\[ jx_d(js)=j\frac{x_d+jsx_d''T_d}{1+jsT_d}=R_{ds}+jX_{ds}, \]

\[ jx_q(js)=j\frac{x_q+jsx_q''T_q}{1+jsT_q}=R_{qs}+jX_{qs}, \tag{2} \]

where

\[ R_{ds}=\frac{(x_d-x_d'')sT_d}{1+s^2T_d^2}; \qquad X_{ds}=\frac{x_d+s^2x_d''T_d^2}{1+s^2T_d^2}; \]

\[ R_{qs}=\frac{(x_q-x_q'')sT_q}{1+s^2T_q^2}; \qquad X_{qs}=\frac{x_q+s^2x_q''T_q^2}{1+s^2T_q^2}; \]

Taking (2) into account, equations (1) can be transformed:

\[ \overline{i}_d=I_de^{j(st+\delta_0-\varphi_d)}; \qquad \overline{i}_q=I_qe^{j(st+\delta_0-\varphi_q)}, \tag{3} \]

where

\[ I_d=\frac{1}{\sqrt{(r+R_{ds})^2+X_{ds}^2}}; \qquad I_q=\frac{1}{\sqrt{(r+R_{qs})^2+X_{qs}^2}}. \]

It is not difficult to prove that the rms value of the current in the stator winding in this case is equal to

\[ I=\sqrt{\frac{1}{2}\left(I_d^2+I_q^2\right)}. \tag{4} \]

The flux-linkage phasors along the \(d\) and \(q\) axes are determined by the equations

\[ \overline{\psi}_d=x_d(js)\,\overline{i}_d; \]

\[ \overline{\psi}_q=x_q(js)\,\overline{i}_q. \]

The average torque, under the accepted assumptions, is determined as

\[ M_c=\frac{1}{2}\left(M_d+M_q\right) =\frac{1}{2}\left(I_d^2R_{ds}-I_q^2R_{qs}\right), \tag{5} \]

where

\[ M_d=\operatorname{Re}[j\overline{\psi}_d i_d^*] =\operatorname{Re}[jx_d(js)\,\overline{i}_d i_d^*] =I_d^2\operatorname{Re}[jx_d(js)]; \]

\[ M_q=\operatorname{Re}[j\overline{\psi}_q i_q^*] =\operatorname{Re}[jx_q(js)\,\overline{i}_q i_q^*] =I_q^2\operatorname{Re}[jx_q(js)]. \]

REFERENCES

1. E. V. Kononenko. Calculation of the starting characteristics of synchronous reluctance motors. In the present collection.

2. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes in a Synchronous Machine, GEI, 1960.