EQUATIONS OF A SYNCHRONOUS RELUCTANCE MOTOR AT VARIABLE SUPPLY NETWORK FREQUENCY

E. V. Kononenko, G. I. Lukyanov

(Presented by the scientific seminar of the Department of Electrical Machines and by the Electrical Machines Division of the Scientific Research Institute of AEM)

Under conditions of the rapid development of automation of production processes, contactless AC electric drives with speed control by changing the frequency of the supply network are becoming widespread. At present, in general-purpose electric-drive systems with frequency control of rotational speed, the asynchronous motor has become most widely used.

Many technological processes, in particular in the production of synthetic and chemical fibers, impose high requirements on the accuracy of synchronous rotation of a large number of individual mechanisms (drives), independently of the load of each over the entire control range, since deviations in speed have a substantial effect on product quality [1]. Similar requirements arise in multi-motor electric drives of production lines of dyeing and finishing equipment, in the automation of dosing units in the chemical, food, pharmaceutical, and other branches of industry. Under these conditions, the use of synchronous motors is preferable. Among low-power synchronous motors, synchronous reluctance motors (SRM) have the greatest prospects and broadest application [2, 3, 4].

In [2, 3, 4], some questions associated with the operation of an SRM at variable supply-network frequency are discussed, but only mechanical transient processes are considered. However, a complete and rigorous study of SRM operation can be carried out only with account taken of both mechanical and electromagnetic transient processes. This becomes possible if, in the analysis, one uses a system of equations describing the electromechanical transient processes of the SRM.

It is known that the voltage-balance equations describing the operation of an SRM can be written in the form:

\[ \left. \begin{aligned} U_d &= \frac{d\psi_d}{dt} + r i_d - \psi_q \cdot \omega;\\ U_q &= \frac{d\psi_q}{dt} + r i_q + \psi_d \cdot \omega;\\ 0 &= \frac{d\psi_{rd}}{dt} + r_{rd}\cdot i_{rd};\\ 0 &= \frac{d\psi_{rq}}{dt} + r_{rq} + i_{rq}, \end{aligned} \right\} \tag{1} \]

where

\[ \left. \begin{aligned} \psi_d &= L_d \cdot i_d + L_{ad}\cdot i_{rd};\\ \psi_q &= L_q \cdot i_q + L_{aq}\cdot i_{rq};\\ \psi_{rd} &= L_{ad}\cdot i_d + L_{rd}\cdot i_{rd};\\ \psi_{rq} &= L_{aq}\cdot i_q + L_{rq}\cdot i_{rq}, \end{aligned} \right\} \tag{2} \]

\(\omega\) is the angular speed of rotation of the rotor; the time \(t\) is expressed in seconds.

For convenience in analysis, the equations of the synchronous reluctance motor are written in a system of relative units. The following generally accepted quantities are taken as base quantities:

1) the amplitude rated values of current and voltage at the rated frequency \((f_n)\);

2) the rated apparent power

\[ P_b=\frac{3}{2}\,U_b\cdot i_b; \]

3) the torque

\[ M_b=\frac{P_b\cdot p}{9.81\cdot 2\pi f_n}; \]

4) the rated angular frequency (rotational speed)

\[ \omega_b=\omega_n=2\pi f_n. \]

In order to write the equations of the synchronous reluctance motor in relative units, equations (1) must be divided by the base voltage
\(U_b=\psi_b\cdot\omega_b=L_b\cdot i_b\cdot\omega_b=i_b\cdot z_b\), and equations (2) by the base flux linkage \(\psi_b=L_b\cdot i_b\). If the quantities expressed in relative units are denoted by the same symbols as in physical units, then the form of equations (2) remains unchanged, while equations (1) take the form:

\[ \left. \begin{aligned} U_d &= \frac{d\psi_d}{d\tau}+r i_d-\psi_q\cdot\frac{\omega}{\omega_n};\\ U_q &= \frac{d\psi_q}{d\tau}+r i_q+\psi_d\cdot\frac{\omega}{\omega_n}; \end{aligned} \right\} \tag{3} \]

\[ \left. \begin{aligned} 0 &= \frac{d\psi_{rd}}{d\tau}+r_{rd}\cdot i_{rd};\\ 0 &= \frac{d\psi_{rq}}{d\tau}+r_{rq}\cdot i_{rq}; \end{aligned} \right\} \tag{4} \]

where \(\tau=\omega_n\cdot t\) is synchronous time, electrical seconds.

When studying operating modes of a synchronous reluctance motor in cases where there is no need to determine the currents in the rotor circuits, the currents \(i_{rd}\) and \(i_{rq}\) should be excluded from the equations under consideration. This can be done by solving (2) and (4). After transformations we obtain:

\[ \left. \begin{aligned} \psi_d &= \left( L_d-\frac{pL_{ad}^{2}}{pL_{rd}+r_{rd}} \right)i_d = \left( x_d-\frac{p x_{ad}^{2}}{p x_{rd}+r_{rd}} \right)i_d = x_d(p)\cdot i_d,\\ \psi_q &= \left( L_q-\frac{pL_{aq}^{2}}{pL_{rq}+r_{rq}} \right)i_q = \left( x_q-\frac{p x_{aq}^{2}}{p x_{rq}+r_{rq}} \right)i_q = x_q(p)\cdot i_q. \end{aligned} \right\} \tag{5} \]

Here the notation \(p=\dfrac{d}{d\tau}\) has been introduced, and it is taken into account that in the system of relative units the inductances \(L\) are equal to the corresponding induct—

reactances, determined at the base frequency of the supply network, since

\[ L=\frac{L}{L_{\mathrm{b}}}=\frac{L\cdot \omega_{\mathrm{b}}}{L_{\mathrm{b}}\cdot \omega_{\mathrm{b}}}=\frac{x}{z_{\mathrm{b}}}=x. \tag{6} \]

The electromagnetic torque is determined by the well-known equation

\[ M=\psi_d i_q-\psi_q i_d. \tag{7} \]

In the study of electromechanical transient processes, together with the voltage equilibrium equations it is necessary to consider the rotor equation of motion

\[ M=M_{\mathrm{st}}+\frac{J}{p}\cdot \frac{d\omega}{dt}. \tag{8} \]

In order to express this equation in relative units, it must be divided by the base torque. Then

\[ M=M_{\mathrm{st}}+H\cdot p\frac{\omega}{\omega_{\mathrm{n}}}, \tag{9} \]

where \(H=J\cdot \dfrac{2\pi f_{\mathrm{n}}\cdot \omega_{\mathrm{n}}^{2}}{P_{\mathrm{b}}\cdot p^{2}}\) is the mechanical constant,

\[ J=\frac{GD^{2}}{4} \]

is the moment of inertia, \([\mathrm{kg\,m^{2}}]\).

To study the operation of a synchronous reluctance motor (SRM) at a variable frequency of the supply network, the ratios

\[ \alpha=\frac{f}{f_{\mathrm{n}}}\quad \text{and}\quad \gamma=\frac{U}{U_{\mathrm{n}}}. \tag{10} \]

are introduced.

In the case when the rotor speed must be expressed through slip, we use the relation

\[ s=\frac{\omega_1-\omega}{\omega_1}=1-\frac{\omega}{\alpha\cdot \omega_{\mathrm{n}}}, \]

whence

\[ \omega=\alpha\omega_{\mathrm{n}}(1-s). \tag{11} \]

Assuming that the rate of change of the frequency of the supply network is significantly less than the rate of the transient mechanical processes, i.e., taking that at any instant the condition \(\alpha=\mathrm{const}\) holds, we may write

\[ \frac{d\omega}{d\tau}=-\alpha\omega_{\mathrm{n}}\frac{ds}{d\tau}. \tag{12} \]

The angle between the vector of the voltage applied to the stator winding and the transverse axis of the rotor is denoted by \(\Theta\). Then

\[ \Theta=\int_{0}^{t}(\omega_1-\omega)\,dt=\int_{0}^{\tau}as\,d\tau; \]

\[ \frac{d\Theta}{d\tau}=as;\quad \frac{d^{2}\Theta}{d\tau^{2}}=a\cdot \frac{ds}{d\tau}; \tag{13} \]

\[ \frac{\omega}{\omega_{\mathrm{n}}}=a-\frac{d\Theta}{d\tau}. \tag{14} \]

In this case the equation of motion of the rotor will take the form

\[ M=M_{\mathrm{st}}-Hp^{2}\Theta, \tag{15} \]

and the voltage-balance equations

\[ \begin{aligned} U_d&=p\psi_d+ri_d-\psi_q(a-p\Theta);\\ U_q&=p\psi_q+ri_q+\psi_d(a-p\Theta);\\ \psi_d&=x_d(p)i_d;\\ \psi_q&=x_q(p)i_q. \end{aligned} \tag{16} \]

In equations (15) and (16), all inductive reactances of the synchronous reluctance motor are determined for the base (nominal) frequency of the supply network. The change in frequency is taken into account only by the coefficient \(\alpha\). The voltage components along the \(d\) and \(q\) axes are determined from the known expressions

\[ \begin{aligned} U_d&=-\gamma U_{\mathrm{n}}\sin\Theta;\\ U_q&=+\gamma U_{\mathrm{n}}\cos\Theta. \end{aligned} \tag{17} \]

The obtained system of equations (15) and (16) describes not only transient but also steady-state operating modes of synchronous reluctance motors supplied from a network with variable frequency. Setting \(p=0\), from (16) we obtain the voltage-balance equations for the steady-state operating mode

\[ \begin{aligned} U_d&=rI_d-\alpha x_q\cdot I_q;\\ U_q&=rI_q+\alpha x_d\cdot I_d, \end{aligned} \tag{18} \]

where \(U_d, U_q, I_d, I_q\) are the rms values of the voltages and currents.

Solving (17) and (18) jointly with respect to the currents, we find

\[ I=\sqrt{I_d^2+I_q^2} = \frac{U_{\mathrm{n}}\gamma}{2(\alpha^2x_dx_q+r^2)} \cdot \sqrt{[2r+\alpha(x_d-x_q)\sin2\Theta]^2+ \alpha^2[x_d+x_q-(x_d-x_q)\cos2\Theta]^2}. \tag{19} \]

In a number of cases it is expedient to operate with the angle \(\beta\) between the axis of the magnetomotive force and the direct axis of the rotor.

The expression for the current as a function of the angle \(\beta\) can be obtained from (19), if the relation between the angles \(\Theta\) and \(\beta\) is used

\[ \tan\Theta= \frac{\alpha x_q\sin\beta-r\cos\beta} {\alpha x_d\cos\beta+r\sin\beta}. \tag{20} \]

Then

\[ I= \frac{2U_{\mathrm{n}}\gamma} {\sqrt{[2r+\alpha(x_d-x_q)\sin2\beta]^2+ \alpha^2[(x_d+x_q)+(x_d-x_q)\cos2\beta]^2}}. \tag{21} \]

Expressions (19) and (21) are also valid in the case when direct current flows through the stator winding, i.e., when \(\alpha=0\). In this case (19) and (21) are equal to:

\[ I=\frac{U_{\mathrm{n}}\gamma}{r}. \tag{22} \]

The electromagnetic torque for the steady-state operating mode after transformations is determined as follows:

\[ M= \frac{U_{\mathrm{n}}^{2}\gamma^{2}(x_d-x_q)} {2(\alpha^2x_dx_q+r^2)^2} \cdot \bigl[(\alpha^2x_dx_q-r^2)\sin2\Theta+\alpha r(x_d-x_q)\cos2\Theta -\alpha r(x_d-x_q)\bigr] \tag{23} \]

or

$$ M=-\frac{2U_{\mathrm n}^{2}\gamma^{2}(x_d-x_q)\sin 2\beta}{[2r+\alpha(x_d-x_q)\sin 2\beta]^2+\alpha^2[(x_d+x_q)+(x_d-x_q)\cos 2\beta]^2}. \tag{24} $$

For \(\alpha=0\), expressions (23) and (24) have the following form:

$$ M=-\frac{U_{\mathrm n}^{2}\gamma^{2}}{2r^2}(x_d-x_q)\sin 2\Theta \tag{25} $$

and

$$ M=-\frac{U_{\mathrm n}^{2}\gamma^{2}}{2r^2}(x_d-x_q)\sin 2\beta. \tag{26} $$

The maximum electromagnetic torque is equal to

$$ M_m=\frac{U_{\mathrm n}^{2}\gamma^{2}(x_d-x_q)}{2(\alpha^2x_dx_q+r^2)^2}\cdot \left[\pm\sqrt{(\alpha^2x_d^2+r^2)(\alpha^2x_q^2+r^2)}-\alpha r(x_d-x_q)\right]. \tag{27} $$

To determine the power factor, we shall use the equivalent circuit [5], which is shown in Fig. 1,

Fig. 1. Equivalent circuit of a SRM at variable supply-network frequency

where

$$ x_u=\frac{x_d-x_q}{2};\qquad x_c=\frac{x_d+x_q}{2}. $$

From this equivalent circuit, the complex total impedance of the stator winding is determined as

$$ z=r+\alpha x_u\sin 2\beta+j\alpha(x_c+x_u\cos 2\beta)=R+j\alpha x, \tag{28} $$

where \(R=r+\alpha x_u\sin 2\beta\) is the equivalent active resistance of the stator winding,

\(x=\alpha(x_c+x_u\cos 2\beta)\) is the equivalent inductive reactance.

Then the power factor can be determined as follows:

$$ \cos\varphi=\frac{R}{z}=\frac{R}{\sqrt{R^2+\alpha^2x^2}}= $$

$$ =\frac{2r+\alpha(x_d-x_q)\sin 2\beta} {\sqrt{[2r+\alpha(x_d-x_q)\sin 2\beta]^2+\alpha^2[(x_d+x_q)+(x_d-x_q)\cos 2\beta]^2}}. \tag{29} $$

The electromagnetic power of the SRM at variable supply-network frequency is

$$ P_{\mathrm{em}}=M\cdot\alpha. \tag{30} $$

The consumed power is

$$ P_1=\frac{2U_{\mathrm n}^{2}\gamma^{2}[2r+\alpha(x_d-x_q)\sin 2\beta]} {[2r+\alpha(x_d-x_q)\sin 2\beta]^2+\alpha^2[(x_d+x_q)+(x_d-x_q)\cos 2\beta]^2}. \tag{31} $$

or

\[ P_1=\frac{U_{\mathrm{н}}^{2}\gamma^{2}}{2\left(\alpha^{2}x_d x_q+r^{2}\right)} \left[2r+\alpha(x_d-x_q)\sin 2\Theta\right]. \tag{32} \]

The efficiency is

\[ \eta=\frac{P_{\mathrm{em}}}{P_1} =\frac{\alpha(x_d-x_q)\sin 2\beta} {2r+\alpha(x_d-x_q)\sin 2\beta} \tag{33} \]

or

\[ \eta= \frac{\alpha(x_d-x_q)} {2r+\alpha(x_d-x_q)\sin^{2}\Theta} \cdot \left[ (\alpha^{2}x_d x_q-r^{2})\sin 2\Theta +\alpha r(x_d+x_q)\cos 2\Theta -\alpha r(x_d-x_q) \right]. \tag{34} \]

The expressions obtained do not take into account iron losses or mechanical losses.

Analogous expressions for currents, torques, powers, power factor, efficiency, etc., for steady-state operation were obtained in [3] as functions of the tangent of the angles \(\beta\) and \(\Theta\), and of the ratio of the parameters \(x_d\) and \(x_q\). Consequently, these expressions are a special case of the more general equations (16), which describe both transient and steady-state operating conditions of an SRM at variable frequency of the supply network.

Thus, the equations presented make it possible to investigate a wide range of questions connected with the operation of an SRM at variable frequency, knowledge of which is important both in design and in operation of an SRM.

REFERENCES

1. J. Franke, A. Schönung. Steuerung statischer Umformer zum Speisen der Antriebe von Chemiefaser—Spinnmaschinen, ETZ—B, 1968, H. 21.

2. D. P. Petelin. Some questions of the statics of frequency control of a synchronous motor. Proceedings of the Conference on Automated Electric Drive with Alternating Current, Publishing House of the USSR Academy of Sciences, 1958.

3. E. M. Berlin, B. A. Egorov, V. D. Kulik, I. S. Skosyrev. Frequency-control systems with synchronous-reluctance motors. “Energiya,” 1968.

4. V. V. Ralle. Some questions of operation of a synchronous-reluctance motor at variable frequency. In: Investigation of Electromagnetic Processes of Electromechanical Systems. “Naukova dumka,” 1965.

5. E. V. Konsnenko, B. P. Garganeev, A. I. Aifert. Circle diagrams of synchronous-reluctance motors. “Elektromekhanika,” 1967, No. 2.