CALCULATION OF THE STARTING CHARACTERISTICS OF SYNCHRONOUS RELUCTANCE MOTORS

E. V. KONONENKO

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

The asynchronous operating mode of synchronous reluctance motors (SRMs) is at present one of the least studied. This has led to the fact that in [1] the parameters of the rotor winding and the starting characteristics of SRMs are calculated by formulas used in the calculation of asynchronous motors with a squirrel-cage rotor.

In [2], in studying the asynchronous operating mode of an SRM, the possibility is proposed of considering separately the phenomena along the longitudinal (d) and transverse (q) axes of the rotor. In this case the SRM is represented by equivalent circuits along the d and q axes, identical to the known equivalent circuits of asynchronous motors. From the equivalent circuits the currents and torque are determined. The analysis we have carried out makes it possible to conclude that the separate consideration of phenomena along the d and q axes is valid only when the stator winding is superconducting \((r=0)\), or when the rotor is made symmetrical \([x_d(js)=x_q(js)]\). In real motors with an asymmetrical rotor the method under consideration is very approximate. The error of the approximate method increases with an increase in the active resistance of the stator winding and in the asymmetry of the rotor. Using this method, it is impossible, for example, to detect a dip in the curve of the electromagnetic torque at half-synchronous speed even in the limiting case of electrical asymmetry, when there is a single-axis winding on the rotor.

The use of simplified methods became possible because exact methods for calculating the starting characteristics of synchronous salient-pole motors are rather complex [3—8]. Since synchronous motors are usually made with a full starting winding, and the active resistances of their stator windings are small, studies have shown that an approximate method for calculating the average asynchronous torque, based on the assumption \(r=0\), gives satisfactory agreement with the exact method [6, 9, 10, etc.]. This contributed substantially to the very wide spread of approximate methods for calculating the starting characteristics of synchronous salient-pole machines, based on the separate consideration of phenomena along the d and q axes, without any restrictions as to the parameters [6, 9—14].

SRMs, in contrast to synchronous motors, are made of small power, and the active resistance of their stator windings is comparatively large. In addition, SRMs are designed with the greatest possible rel—

by the ratio of the synchronous reactances \(x_d/x_q\). The latter leads to the fact that, even if the rotor has a complete starting winding, the steady-state values of the operational reactances \([x_d(js)\) and \(x_q(js)]\) in the asynchronous mode may differ considerably in magnitude, especially at small rotor slips. Therefore, the use of approximate methods for calculating the starting characteristics of a SynRM is not justified and may lead to significant errors. All this makes it necessary to study the asynchronous operating mode of a SynRM in greater detail and to choose a method for calculating starting characteristics that is acceptable for practice.

In determining the currents and torques of a SynRM in the asynchronous mode, it is expedient to proceed from the complete voltage-equilibrium equations. These equations, under the condition that the electromagnetic torque is taken as positive in the motor operating mode, and that the \(q\) axis leads the \(d\) axis by \(90^\circ\), using the per-unit system, can be written in operator form as:

\[ \begin{aligned} u_d&=[p x_d(p)+r]i_d-x_q(p)(1-s)i_q;\\ u_q&=x_d(p)(1-s)i_d+[p x_q(p)+r]i_q. \end{aligned} \tag{1} \]

For studying the asynchronous operating mode of a SynRM, we shall restrict ourselves to considering only the starting characteristics of the steady-state mode, taking into account that this method is considered a reliable and the most widely used tool in the study of starting processes of synchronous and asynchronous motors.

In deriving the calculation formulas it is expedient to use the complex method. For this purpose, in equations (1) the currents and voltages are replaced by complexes, and \(p\) by \(js\).

If a symmetrical system of voltages is applied to the stator winding, and \(u_d=U\cos(t+d_0)\), then the voltage components \(u_d, u_q\) at constant rotor speed are determined by the known equations of linear transformations

\[ u_d=\frac{2}{3}\left[u_a\cos\gamma+u_B\cos(\gamma-120^\circ)+u_c\cos(\gamma+120^\circ)\right] =U\cos(st+\delta_0); \]

\[ u_q=-\frac{2}{3}\left[u_a\sin\gamma+u_c\sin(\gamma-120^\circ)+u_c\sin(\gamma+120^\circ)\right] =U\sin(st+\delta_0). \tag{2} \]

Here \(\gamma=(1-s)t+\gamma_0\) is the angle formed by the \(d\) axis with the axis of stator phase \(a\);

\[ \delta_0=a_0-\gamma_0. \]

Assuming that real quantities are determined as the real parts of complexes, we may write

\[ \overline{u}_d=Ue^{j(st+\delta_0)};\qquad \overline{u}_q=-jUe^{j(st+\delta_0)}. \tag{3} \]

Solving equations (1) under the assumptions made, we find the values of the current complexes:

\[ \overline{i}_d=\frac{1}{A}\left[r-j(1-2s)x_q(js)\right]Ue^{j(st+\delta_0)}; \]

\[ \overline{i}_q=\frac{1}{A}\left[-jr-(1-2s)x_d(js)\right]Ue^{j(st+\delta_0)}, \tag{4} \]

where

\[ \overline{A}=r^2+jrs[x_d(js)+x_q(js)]+(1-2s)x_d(js)x_q(js). \]

In investigations of the steady-state asynchronous mode it is convenient to operate with vectors of spatial waves. Denoting the instantaneous values of quantities relating to winding \(d\) as real, and

those pertaining to the \(q\) winding as imaginary, the stator-current space vector can be represented as follows:

\[ \overline{i}_s=\overline{i}_d+j i_q . \tag{5} \]

Taking into account that

\[ i_d=\operatorname{Re}[\overline{i}_d]=\frac{1}{2}(\overline{i}_d+i_d^*);\quad i_q=\operatorname{Re}[\overline{i}_q]=\frac{1}{2}(\overline{i}_q+i_q^*), \]

equation (5) can be written as

\[ \overline{i}_s=\frac{1}{2}(\overline{i}_d+j\overline{i}_q)+\frac{1}{2}(i_d^*+j i_q^*)=\overline{i}_{s1}+\overline{i}_{s2}, \tag{6} \]

where \(\overline{i}_{s1}\) is the positive-sequence current space vector;
\(\overline{i}_{s2}\) is the negative-sequence current space vector;
and the asterisk \(*\) denotes conjugate complexes.

Assuming that the real multi-circuit rotor windings of the SRM are replaced by two equivalent ones [15],

\[ x_d(js)=\frac{x_d+j s x_d''T_d}{1+j s T_d};\quad x_q(js)=\frac{x_q+j s x_q''T_q}{1+j s T_q}. \tag{7} \]

Then, after the corresponding transformations, equation (6) can be represented in the form¹:

\[ \overline{i}_s=I_1 e^{j(st+\delta_0-\varphi_1)}+I_2 e^{-j(st+\delta_0-\varphi_2)} . \tag{8} \]

The complexes of the flux linkages of the stator windings are determined from the equalities

\[ \overline{\psi}_d=x_d(js)\overline{i}_d;\quad \overline{\psi}_q=x_q(js)\overline{i}_q . \tag{9} \]

By analogy with equation (6), the space vector of the stator-winding flux linkage is equal to

\[ \overline{\psi}_s=\frac{1}{2}(\overline{\psi}_d+j\overline{\psi}_q)+\frac{1}{2}(\psi_d^*+j\psi_q^*)=\overline{\psi}_{s1}+\overline{\psi}_{s2}. \tag{10} \]

1. Power Consumed

The instantaneous value of the power consumed by the SRM when operating in the steady-state asynchronous mode is determined by the equation

\[ P_1=\operatorname{Re}[\overline{u}_s i_s^*]. \tag{11} \]

Substituting into this equation the conjugate current vector (8) and taking into account that the voltages applied to the stator winding are symmetrical and, consequently,

\[ \overline{u}_s=U e^{j(st+\delta_0)}, \]

after transformations we obtain

\[ P_1=U I_1\cos\varphi_1+U I_2\cos(2st+2\delta_0-\varphi_2). \tag{12} \]

It follows from equation (12) that the power consumed by the SRM from the network can be represented as the sum of two components: a constant component, proportional to the active component of current \(I_1\), and a variable component, pulsating in time at twice the slip frequency.

2. Stator Current

Knowing the components of the space vector of the current [8], the instantaneous values of the current in the phases of the stator winding can be determined by the method set forth in [8].

The rms value of the current for any slip values, except \(s=0\) and \(s=1.0\), is determined by the formula

\[ I=\sqrt{I_1^2+I_2^2}. \tag{13} \]

For \(s=1.0\), the current magnitude depends on the position of the rotor in space and varies within the limits from

\[ I_{\max}=I_1+I_2 \text{ to } I_{\min}=I_1-I_2 . \tag{16} \]

3. Electromagnetic Torque

The electromagnetic torque of rotation of an SRD when operating in the steady asynchronous mode can be represented in the form [8]

\[ M=\operatorname{Re}\left[j\overline{\psi}_s i_s^*\right]=M_c+M_{\text{pul}}, \tag{14} \]

where

\[ M_c=\operatorname{Re}\left[j\overline{\psi}_{s1} i_{s1}^*+j\overline{\psi}_{s2} i_{s2}^*\right]=M_1+M_2- \tag{15} \]

the average torque, equal to the sum of the torques of the positive and negative sequences;

\[ M_{\text{pul}}=\operatorname{Re}\left[j\overline{\psi}_{s1} i_{s2}^*+j\overline{\psi}_{s2} i_{s1}^*\right]- \tag{16} \]

the variable component of the torque, pulsating in time at twice the slip frequency.

The average and pulsating torques can be expressed through the current and flux-linkage complexes along the \(d\) and \(q\) axes. Substituting expressions (6) and (10) into equations (15) and (16), after simple transformations we obtain

\[ M_c=\frac{1}{2}\operatorname{Re}\left(\overline{\psi}_d i_q^*-\overline{\psi}_q i_d^*\right) =\frac{1}{2}\operatorname{Re}\left(\psi_d^*\overline{i}_q-\psi_q^*\overline{i}_d\right); \tag{17} \]

\[ M_{\text{pul}}=\frac{1}{2}\operatorname{Re}\left(\overline{\psi}_d\overline{i}_q-\overline{\psi}_q\overline{i}_d\right) =\frac{1}{2}\operatorname{Re}\left(\psi_d^* i_q^*-\psi_q^* i_d^*\right). \tag{18} \]

To reduce the amount of computational work, it is advisable to represent the torques as functions of the currents and parameters of the SRD. For this purpose, the flux linkages must be eliminated from the torque equations. Expressions for the flux linkages \(\psi_d,\psi_q\) are obtained by solving the system of equations (1) and (9). After the corresponding transformations, we find that

\[ \begin{aligned} \overline{\psi}_d&=-j\overline{u}_d-\frac{r}{1-2s}\left[js\overline{i}_d+(1-s)\overline{i}_q\right];\\ \overline{\psi}_q&=-\overline{u}_d+\frac{r}{1-2s}\left[(1-s)\overline{i}_d-js\overline{i}_q\right]. \end{aligned} \tag{19} \]

Then the equations for the torques of the positive and negative sequences can be found as

\[ M_1=\operatorname{Re}\left[j\overline{\psi}_{s1}i_{s1}^*\right] =\frac{1}{2}\operatorname{Re}\left[\left(j\overline{\psi}_d-\overline{\psi}_q\right)i_{s1}^*\right] =UI_1\cos\varphi_1-rI_1^2, \tag{20} \]

\[ M_2=\operatorname{Re}\left[j\overline{\psi}_{s2}i_{s2}^*\right] =\frac{1}{2}\operatorname{Re}\left[\left(j\psi_d^*-\psi_q^*\right)i_{s2}^*\right] =-\frac{r}{1-2s}I_2^2; \tag{21} \]

The sum of the torques \(M_1\) and \(M_2\) gives the familiar equation for the average torque of rotation [3, 4, etc.].

The pulsating torque (18), taking into account that

\[ \bar i_d=\bar i_{s1}+\bar i_{s2}^{*}; \qquad \bar i_q=-j\left(\bar i_{s1}-\bar i_{s2}^{*}\right), \]

will be determined by the equation

\[ M_{\text{pul}}=UI_2\cos(2st+2\delta_0-\varphi_2) -\frac{2r(1-s)}{1-2s}\,I_1I_2\cos(2st+2\delta_0-\varphi_1-\varphi_2). \tag{22} \]

From equation (22) we determine the amplitude of the pulsating torque

\[ M_p=I_2\sqrt{U^2+4R^2I_1^2-4URI_1\cos\varphi_1}, \tag{23} \]

where

\[ R=r\frac{1-s}{1-2s}. \]

The amplitude of the pulsating torque can be calculated, using equation (23), for any values of slip except \(s=0.5\). It is known that at \(s=0.5\) the negative-sequence current in the stator winding is absent (\(I_2=0\)); however, in this case \(M_p\ne0\). From an analysis of the expression for the modulus of the negative-sequence current vector it follows that

\[ I_2=(1-2s)B. \]

Then, at \(s=0.5\), equation (23) takes the form

\[ M_{p\,(s=0.5)}=rI_1B. \tag{24} \]

A comparison of the starting characteristics calculated by the method set forth above and determined experimentally is presented in Fig. 1. The good agreement between the calculated and experimental data makes it possible to conclude that the proposed method for calculating the starting characteristics of the SynRM reflects with sufficient accuracy the processes occurring in them, and can be recommended for practice.

Fig. 1. Comparison of the theoretical and experimentally determined starting characteristics of a SynRM. Experimental points are denoted by \( \times \).

Appendix 1

Procedure for calculating the starting characteristics of a SynRM

\[ \text{1) }\ I_{1a}=\frac{ac+bd}{c^2+d^2}\,U; \qquad \text{2) }\ I_{1p}=\frac{ad-bc}{c^2+d^2}\,U; \]

\[ \text{3) }\ I_1=\sqrt{I_{1a}^{\,2}+I_{1p}^{\,2}}; \]

\[ \text{4) }\ I_{2a}=\frac{a^1c+b^1d}{c^2+d^2}\,U; \qquad \text{5) }\ I_{2p}=\frac{a^1d-b^1c}{c^2+d^2}\,U; \]

\[ \text{6) }\ I_2=\sqrt{I_{2a}^{\,2}+I_{2p}^{\,2}}; \qquad \text{7) }\ I=\sqrt{I_1^{\,2}+I_2^{\,2}}; \]

\[ a=r\left[(1-s^2T_dT_q)+\frac12s(1-2s)\{(x_d+x_q'')T_q+(x_d''+x_q)T_d\}\right]; \]

\[ b=rs(T_d+T_q)-\frac12(1-2s)\left[x_d+x_q-s^2(x_d''+x_q'')T_dT_q\right]; \]

\[ \begin{gathered} c=r^{2}(1-s^{2}T_dT_q)-rs^{2}\bigl[(x_d+x_q'')T_q+(x_d''+x_q)T_d\bigr]+\\ \qquad +(1-2s)(x_dx_q-s^{2}x_d''x_q''T_dT_q);\\[6pt] d=r^{2}s(T_d+T_q)+rs\bigl[x_d+x_q-s^{2}(x_d''+x_q'')T_dT_q\bigr]+\\ \qquad +s(1-2s)(x_d''x_qT_d+x_dx_q''T_q);\\[6pt] a'=\frac{1}{2}(1-2s)s\bigl[(x_q-x_d'')T_d-(x_d-x_q'')T_q\bigr];\\[6pt] b'=\frac{1}{2}(1-2s)\bigl[x_d-x_q-s^{2}(x_d''-x_q'')T_dT_q\bigr]; \end{gathered} \]

\[ 8)\quad M_1=UI_{1a}-rI_1^2;\qquad 9)\quad M_2=-\frac{r}{1-2s}I_2^2 \]

\[ 10)\quad M_c=M_1+M_2; \]

\[ 11)\quad M_n=I_2\sqrt{ U^2+4I_1^2r^2\left(\frac{1-s}{1-2s}\right)^2 -4UI_{1a}r\frac{1-s}{1-2s} }. \]

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1. The calculation formulas for determining the amplitudes of the positive- and negative-sequence currents \(I_1\) and \(I_2\) are given in Appendix 1. ↩