STARTING CHARACTERISTICS OF A SYNCHRONOUS RELUCTANCE MOTOR AT VARIABLE SUPPLY-NETWORK FREQUENCY

E. V. KONONENKO, G. I. LUKYANOV

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

In the development of systems of frequency-controlled electric drive, synchronous reluctance motors (SRM) are finding increasingly wide application alongside asynchronous motors. SRMs are usually brought up to speed as asynchronous motors. Therefore, study of the influence of changes in frequency and voltage on the starting characteristics is necessary, making it possible to identify rational laws of frequency regulation during starting.

To investigate the starting characteristics of an SRM at variable frequency of the supply network, we take as the initial equations the following equations, written in relative units [1]:

\[ \left. \begin{aligned} U_d &= p\Psi_d + r i_d - \Psi_q(\alpha - p\Theta),\\ U_q &= p\Psi_q + r i_q + \Psi_d(\alpha - p\Theta),\\ \Psi_d &= x_d(p)i_d,\\ \Psi_q &= x_q(p)i_q, \end{aligned} \right\} \tag{1} \]

where

\[ p=\frac{d}{d\tau} \]
is the relative frequency of the network;

\[ \alpha=\frac{f_1}{f_n} \]
is the differentiation operator with respect to synchronous time;

\(\Theta\) is the angle between the applied voltage and the transverse axis of the rotor.

The equation of electromagnetic torque is

\[ M=\Psi_d\cdot i_q-\Psi_q i_d \tag{2} \]

Taking into account that during starting SRMs operate in the asynchronous mode and, according to [1],

\[ \frac{d\Theta}{d\tau}=\alpha\cdot s, \]

equations (1) can be written in the following form:

\[ \left. \begin{aligned} U_d &= p\Psi_d + r i_d - \Psi_q\cdot \alpha(1-s),\\ U_q &= p\Psi_q + r i_q + \Psi_d\cdot \alpha(1-s) \end{aligned} \right\} \tag{3} \]

Usually, in studying the asynchronous operating modes of an SRM, static starting characteristics are considered. To derive the calculation—

... of the formulas, the complex method is used [2]. Since in the system of relative units the frequency of the currents and voltages in the transformed stator windings along the \(d\) and \(q\) axes is equal to \(\alpha s\), then, in passing from the operational equations to complex ones, \(p\) is replaced by \(j\alpha s\). Solving equations (3) with respect to the current complexes, we obtain

\[ \overline{i}_d = \frac{ \overline{U}_d\left[j\alpha s x_q(j\alpha s)+r\right] + \overline{U}_q x_q(j\alpha s)\,\alpha(1-s) }{ \overline{A} } \]

\[ \overline{i}_q = \frac{ \overline{U}_q\left[j\alpha s x_d(j\alpha s)+r\right] - \overline{U}_d x_d(j\alpha s)\,\alpha(1-s) }{ \overline{A} }, \tag{4} \]

where

\[ \overline{A} = r^2 + j\alpha s\,r\left[x_d(j\alpha s)+x_q(j\alpha s)\right] + x_d(j\alpha s)\,x_q(j\alpha s)\left[\alpha^2(1-2s)\right]. \]

\[ x_d(j\alpha s)= \frac{x_d+j\alpha s\,T_d x_d^{\prime\prime}} {1+j\alpha s\,T_d}, \qquad x_q(j\alpha s)= \frac{x_q+j\alpha s\,T_q x_q^{\prime\prime}} {1+j\alpha s\,T_q}. \]

The rotor speed at variable network frequency is equal to

\[ \omega=\omega_n\cdot \alpha(1-s). \tag{5} \]

Then the complexes of the voltage components are

\[ \overline{U}_d=\gamma_u U_n e^{j(\alpha s\tau+\delta_0)} \]

\[ \overline{U}_q=-j\gamma_u U_n e^{j(\alpha s\tau+\delta_0)}, \tag{6} \]

where

\[ \gamma_u=\frac{U}{U_n}, \]

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

Substituting (6) into (4), the expressions for the current complexes may be written in the following form:

\[ \overline{i}_d = \frac{\gamma_u U_n e^{j(\alpha s\tau+\delta_0)}}{\overline{A}} \left\{ r-j\alpha s\,x_q(j\alpha s)\,[\alpha(1-2s)] \right\}, \]

\[ \overline{i}_q = \frac{\gamma_u U_n e^{j(\alpha s\tau+\delta_0)}}{\overline{A}} \left\{ -jr-x_d(j\alpha s)\,[\alpha(1-2s)] \right\}. \tag{7} \]

In investigating steady-state asynchronous modes, it is convenient to operate with space-wave vectors. For the space vector of current the following expression is valid:

\[ \overline{i}_s=i_d+ji_q. \tag{8} \]

The instantaneous values of the currents \(i_d\) and \(i_q\) are determined through the corresponding vectors

\[ i_d=\frac{1}{2}\left(\overline{i}_d+i_d^*\right), \qquad i_q=\frac{1}{2}\left(\overline{i}_q+i_q^*\right). \tag{9} \]

Then, taking (9) into account, we have

\[ \overline{i}_s = \frac{1}{2}\left(\overline{i}_d+j\overline{i}_q\right) + \frac{1}{2}\left(i_d^*+ji_q^*\right) = \overline{i}_{s1}+\overline{i}_{s2}, \tag{10} \]

where

\(i_{s1}\) and \(i_{s2}\) are the space vectors of the currents of the positive and negative sequences, respectively,

\(*\) denotes the conjugate vector.

The space vectors of the current after the transformations, taking (7) into account, are equal to

\[ \overline{i}_{s1} = -\frac{\gamma_u U_n e^{j(as\tau+\hat{\delta}_0)}}{\hat{A}} \cdot \left\{ r-\frac{1}{2}j[z(1-2s)]\,[x_d(jas)+x_q(jas)] \right\}, \]

\[ \overline{i}_{s2} = \frac{\gamma_u U_n e^{-j(as\tau+\hat{\delta}_0)}}{\hat{A}^{*}} \left\{ -\frac{1}{2}j[z(1-2s)]\,[x_d(-jas)-x_q(-jas)] \right\}. \tag{11} \]

If the expressions for the operational resistances are substituted into (11), then equations (11), after transformation, take the form

\[ \overline{i}_{s1} = \frac{a+jb}{c+jd}\cdot \gamma_u U_n e^{j(as\tau+\hat{\delta}_0)}, \]

\[ \overline{i}_{s2} = \frac{a'-jb'}{c-jd}\cdot \gamma_u U_n e^{-j(as\tau+\hat{\delta}_0)}, \tag{12} \]

where

\[ a = r(1-a^2s^2T_dT_q) + \frac{1}{2}zs[z(1-2s)]\cdot[(x_d''+x_q)T_d+(x_q''+x_d)T_q]; \]

\[ b = r\alpha s(T_d+T_q) - \frac{1}{2}[z(1-2s)]\cdot[(x_d+x_q)-a^2s^2(x_d''+x_q'')T_dT_q]; \]

\[ c = r^2(1-a^2s^2T_dT_q) - r\alpha^2s^2[(x_d+x_q'')T_q+(x_q+x_d'')T_d] + \]

\[ \qquad + [z^2(1-2s)]\cdot(x_dx_q-a^2s^2T_dT_q\cdot x_d''x_q''); \]

\[ d = r^2\alpha s(T_d+T_q) + rzs[x_d+x_q-a^2s^2(x_d''+x_q'')T_dT_q] + \]

\[ \qquad + [z^2(1-2s)]\cdot zs(x_d''x_qT_d+x_q''x_dT_q); \]

\[ a' = \frac{zs}{2}[z(1-2s)]\cdot[(x_q-x_d'')T_d+(x_q''-x_d)T_q]; \]

\[ b' = \frac{z(1-2s)}{2}\cdot[x_d-x_q-a^2s^2(x_d''-x_q'')T_dT_q]. \]

From expressions (12) it is possible to determine the rms values of the positive- and negative-sequence currents:

\[ I_1 = \gamma_u U_n \left| \frac{a+jb}{c+jd} \right| = \gamma_u U_n \sqrt{\frac{a^2+b^2}{c^2+d^2}}, \]

\[ I_2 = \gamma_u U_n \left| \frac{a'-jb'}{c-jd} \right| = \gamma_u U_n \sqrt{\frac{(a')^2+(b')^2}{c^2+d^2}}. \tag{13} \]

The rms value of the stator current is equal to

\[ I=\sqrt{I_1^2+I_2^2}. \tag{14} \]

Then the current vectors of the positive and negative sequences may be written as

\[ \overline{i}_{s1}=I_1\cdot e^{-j(as\tau+\hat{\delta}_0-\varphi_1)} \]

\[ \overline{i}_{s2}=I_2\cdot e^{-j(as\tau+\hat{\delta}_0-\varphi_2)}, \tag{15} \]

where

\[ \varphi_1=\operatorname{arctg}\frac{ad-bc}{ac+bd}; \qquad \varphi_2=\operatorname{arctg}\frac{a'd-b'c}{a'c+b'd}. \tag{16} \]

The electromagnetic torque according to (2), after transformations, can be represented as the sum of two components

\[ M=\Psi_d i_q-\Psi_q i_d=\operatorname{Re}\left(j\overline{\Psi}_s\cdot i_s^*\right)=M_c+M_{\text{pul}}, \tag{17} \]

where

\[ M_c \]
is the average (time-constant) torque,

\[ M_{\text{pul}} \]
is the pulsating, time-varying torque.

The average torque can be represented as the sum of the torques of the positive and negative sequences

\[ M_c=M_1+M_2, \tag{18} \]

where

\[ \begin{aligned} M_1&=\operatorname{Re}\left(j\overline{\Psi}_{s1}\cdot i_{s1}^*\right) =\frac{1}{2}\operatorname{Re}\left[(j\overline{\Psi}_d-\overline{\Psi}_q)\cdot i_{s1}^*\right],\\ M_2&=\operatorname{Re}\left(j\overline{\Psi}_{s2}\cdot i_{s2}^*\right) =-\frac{1}{2}\operatorname{Re}\left[(j\Psi_d^*-\Psi_q^*)\cdot i_{s2}^*\right]. \end{aligned} \tag{19} \]

In order to obtain formulas for the torques that are convenient for calculation, it is necessary to eliminate the flux linkages from the last expressions. To this end, solving equations (3) in complex form, we find the flux-linkage complexes

\[ \overline{\Psi}_q=-\frac{j\overline{U}_d}{\alpha_q} -\frac{r\left[j\overline{i}_d+(1-s)\overline{i}_q\right]}{\alpha(1-2s)} \]

\[ \overline{\Psi}_d=-\frac{\overline{U}_d}{\alpha_q} +\frac{r\left[\overline{i}_d(1-s)-j s\overline{i}_q\right]}{\alpha(1-2s)}. \tag{20} \]

Taking (20) into account, we find expressions for the components of the electromagnetic torque:

1) of the positive sequence

\[ M_1=\frac{\gamma_u U_n I_1\cos\varphi_1-rI_1^2}{\alpha}; \tag{21} \]

2) of the negative sequence

\[ M_2=-\frac{rI_2^2}{\alpha(1-2s)}; \tag{22} \]

3) of the pulsating torque

\[ M_{\text{pul}}=\frac{1}{2}\operatorname{Re}\left(\overline{\Psi}_d\cdot \overline{i}_q-\overline{\Psi}_q\cdot \overline{i}_d\right)= \]

\[ =-\frac{\gamma_u U_n I_2}{\alpha}\cos(2\alpha s\tau+2\delta_0-\varphi_2) -I_1I_2\frac{2r(1-s)}{\alpha(1-2s)}\times \]

\[ \times \cos(2\alpha s\tau+2\delta_0-\varphi_1-\varphi_2). \tag{23} \]

The amplitude of the pulsating torque is

\[ M_{\text{pul}}=\frac{I_2}{\alpha}\sqrt{ \gamma_u^2U_n^2 +4I_1^2\left[\frac{r(1-s)}{1-2s}\right]^2 -4\gamma_u U_n\cdot \frac{r(1-s)}{1-2s}\cdot I_1\cos\varphi_1 } \tag{24} \]

The obtained expressions (13), (14), (21), (22), and (24) make it possible to calculate the characteristics of an SRM in the starting mode, operating from a network with variable frequency and voltage amplitude. Figures 1 and 2 present the calculated mechanical characteristics and currents of the SRM.

at different values of the frequency for a motor with a power of 3 kW and a rotational speed of 1000 rpm, characterized by the following parameter values: \(x_d=2.57\); \(x_q=0.49\); \(r=0.066\); \(x_d''=0.228\); \(x_q''=0.205\); \(T_d=43\) electrical sec; \(T_q=11.8\) electrical sec; \(H=161\) electrical sec. (all parameters are in relative units). The parameters were calculated according to the method of [2]. The moment of inertia of the load was taken equal to five times the value of the rotor moment of inertia. The law of frequency variation was assumed to be directly proportional to the voltage. \(M_n=0.42M_\delta\).

As follows from these figures, a change in frequency has a substantial effect on the starting characteristics of the SRM and leads to changes in the starting torque and current, the maximum torque in asynchronous operation, and also to distortion of the form of the starting characteristics.

Fig. 1. Mechanical characteristics of the SRM at different values of the relative frequency \(\alpha\)

Fig. 2. Dependence of the SRM stator current on slip at different values of the relative frequency \(\alpha\)

Figure 3 shows the dependences of the starting current, torque, and maximum torque in asynchronous operation on frequency. The starting torque in the frequency range \(\alpha=0.4\text{–}1.0\) changes only slightly and has a maximum at \(\alpha=0.5\); at \(\alpha=0.2\) the starting torque still remains above the rated value. The maximum torque in asynchronous operation in the indicated frequency range also changes very little. At the same time, the starting current, when the frequency is decreased from 1 to 0.2, decreases almost by a factor of two. Thus, an analysis of the starting characteristics of an SRM at variable supply-network frequency indicates that starting

Fig. 3. Dependences of the starting torque \(M_{\mathrm{p}}\), starting current \(I_{\mathrm{p}}\), and maximum torque in asynchronous operation \(M_{\mathrm{m}}\) on the relative frequency \(\alpha\)

The starting of synchronous reluctance motors operating in systems with frequency control of rotational speed is possible at reduced values of the starting-current multiple. In systems for controlling the rotational speed of AC motors with thyristor frequency converters, reducing the starting-current multiple is of substantial importance, since it makes it possible to reduce the size and cost of thyristor frequency converters by decreasing the current of the semiconductor elements.

Thus, the starting of synchronous reluctance motors operating in systems with frequency control can be carried out at reduced starting-current multiples; for this purpose the start should begin at reduced frequencies, and then the voltage and frequency should be increased at such a rate that the starting current remains within the prescribed limits.

REFERENCES

1. E. V. Kononenko, G. I. Lukyanov. Equations of a synchronous reluctance motor at variable frequency of the supply network. Izv. TPI, vol. 229, 1972.

2. E. V. Kononenko. Synchronous Reluctance Machines. “Energiya,” Moscow, 1970.