Methodology for Modeling Capacitor Motors with Three-Phase Stator Windings

Yu. M. Achkasov, B. A. Zakharov

(Presented by the scientific and technical seminar of the Research Institute of AEM)

The calculation of steady-state operating conditions of capacitor motors is the subject of [1]. At present, the dynamic operating conditions of capacitor motors have been studied less thoroughly, which is apparently explained by the difficulty of analytically solving the high-order nonlinear differential equations with variable coefficients by which these motors are described in transient processes when electromagnetic phenomena are taken into account. However, this problem is quite solvable with the use of computer technology.

The application of analog computers for studying electromagnetic transient processes in a two-phase asynchronous motor is considered in [2].

In frequency-control systems, three-phase asynchronous and synchronous motors with symmetrical stator windings, connected in star or delta with phase-shifting capacitances and supplied from single-phase frequency converters, are used. The study of the influence of asymmetry and nonsinusoidality of the supply voltage on transient and steady-state conditions in such frequency-control systems is of great practical interest.

The method of superposition of two circular fields rotating in opposite directions, used for modeling asynchronous machines under asymmetrical conditions, makes it possible to investigate transient processes only at a constant, preassigned voltage asymmetry factor [3]. This method is not applicable to the study of transient processes in three-phase electrical machines when the windings are connected in star or delta and operate in the mode of single-phase capacitor machines, since the asymmetry factor is variable in the transient regime. Let us consider a modeling methodology that makes it possible to take into account the change in the asymmetry factor, using as an example a capacitor synchronous reluctance motor connected according to the circuit in Fig. 1.

Transient processes in synchronous machines under asymmetrical conditions are studied by means of the complete system of differential equations with periodic coefficients on digital computers (DCs). However, the use of DCs requires a large amount of qualified work, and the calculation of asymmetrical conditions is associated with a significant expenditure of machine time [4].

Fig. 1. Connection diagram of a three-phase synchronous reluctance motor to a single-phase voltage.

Analog computers, possessing a high speed of solution, visual clarity of the results obtained, and simplicity of programming and tuning, have not yet found application for studying asymmetrical operating conditions of synchronous machines.

Electromagnetic transients in a three-phase synchronous reluctance motor (SRM) are described by a system of equations [5], which, under frequency control, will have the form:

\[ \begin{aligned} p\Psi_d &= U_d - Ri_d + \alpha(1-s)\Psi_q;\\ p\Psi_q &= U_q - Ri_q - \alpha(1-s)\Psi_d;\\ p\Psi_{rd} &= -\frac{1}{T_d}\cdot\Psi_{rd}+\frac{L_{ad}}{T_d}\cdot i_d;\\ p\Psi_{rq} &= -\frac{1}{T_q}\cdot\Psi_{rq}+\frac{L_{aq}}{T_q}\cdot i_q;\\ i_d &= \frac{1}{L_d^{\prime\prime}}\cdot\Psi_d-\frac{L_{rd}}{L_d L_{rd}}\cdot\Psi_{rd};\\ i_q &= \frac{1}{L_q}\cdot\Psi_q-\frac{L_{aq}}{L_q^{\prime\prime}\cdot L_{rq}}\cdot\Psi_{rd};\\ RS &= \frac{1}{\alpha H}\left(M_{em}-\Psi_d\cdot i_q+\Psi_q\cdot i_d\right);\\ S &= \frac{d\theta}{d\tau};\qquad p_{\delta}=\frac{d}{d\tau};\qquad \frac{d\gamma}{d\tau}=\omega_r=\alpha(1-s). \end{aligned} \tag{1} \]

Here \(\alpha=\dfrac{f}{f_n}\) is the relative frequency, and \(\omega_r\) is the rotor rotational speed. The remaining designations are as in [5]. The structural diagram of the model of the system of equations (1) for \(\alpha=1\) is given in [5].

Having the values of the currents along the \(d\) and \(q\) axes, we determine the phase currents by means of the formulas of direct linear transformations [6],

\[ \begin{aligned} i_a &= i_d\cos\gamma-i_q\sin\gamma;\\ i_b &= \left(-\frac{1}{2}i_d+\frac{\sqrt{3}}{2}i_q\right)\cdot\cos\gamma +\left(\frac{1}{2}i_q+\frac{\sqrt{3}}{2}i_d\right)\sin\gamma;\\ i_c &= \left(-\frac{1}{2}i_d-\frac{\sqrt{3}}{2}i_q\right)\cdot\cos\gamma +\left(\frac{1}{2}i_q-\frac{\sqrt{3}}{2}i_d\right)\sin\gamma. \end{aligned} \tag{2} \]

From the value of the phase current in the capacitor phase, we find the voltage across the capacitor

\[ U_k=\frac{1}{c}\int i_B\cdot d\tau . \tag{3} \]

From the equations of electrical equilibrium for the stator windings of the capacitor synchronous reactive motor (Fig. 1), we determine the magnitudes of the instantaneous values of the phase voltages

\[ \begin{aligned} -U_a &= U_{\text{п}} - U_c;\\ -U_c &= U_a + U_{в};\\ -U_{в} &= U_{к} - U_c. \end{aligned} \tag{4} \]

Using the formulas of the inverse linear transformations, we obtain the voltages along the \(d\) and \(q\) axes [6]:

\[ U_d = \left[ \frac{2}{3} U_a - \frac{1}{3}(U_b + U_c) \right]\cdot \cos \gamma + \frac{1}{\sqrt{3}}(U_b - U_c)\sin \gamma; \tag{5} \]

\[ U_q = -\left[ \frac{2}{3} U_a - \frac{1}{3}(U_b + U_c) \right]\cdot \sin \gamma + \frac{1}{\sqrt{3}}(U_b - U_c)\cos \gamma, \]

The voltages \(U_d, U_q\) obtained in this way reflect the influence of the capacitance on transient processes in the capacitor electric motor.

Using the model of a three-phase SRM [5], according to equations (2, 3, 4, 5), a block diagram was compiled for setting up a model of a capacitor synchronous reactive electric motor with three-phase stator windings (Fig. 2). The necessary periodic functions \(\sin \gamma, \cos \gamma\) are obtained using the well-known “mathematical pendulum” circuit. The supply voltage \(U_{\text{п}}\) of sinusoidal form is produced in the “electronic pendulum” circuit and can easily be transformed into a rectangular one by means of relays available, for example, in the analog computing machine MN14.

Fig. 2. Block diagram for setting up the model of a single-phase capacitor synchronous reactive motor.

When studying a closed-loop frequency-control system, it is necessary to add feedback equations to the existing equations and to introduce the corresponding elements into the model.

Conclusions

The proposed methodology for modeling capacitor electric motors with three-phase stator windings makes it possible to:

1. Apply it to asynchronous and synchronous machines.
2. Study the dynamics of starting and braking modes, transition from one speed to another, etc.
3. Study the influence of the starting capacitance on the magnitude of the impact torque and current, the starting time, and speed overshoot.
4. Study the influence of the working capacitance on the static and dynamic stability of a synchronous motor, as well as the process of pulling into synchronism and the accuracy of speed maintenance.
5. Study transient processes in a capacitor machine while taking into account the full harmonic composition of the supply voltage.

Literature

1. A. I. Adamenko, V. I. Kislenko. Conversion of Single-Phase Current into Polyphase Current. Kiev, “Tekhnika,” 1971.

2. I. P. Kopylov, F. A. Mamedov, A. I. Kozhemyakin. Influence of starting capacitance on electromechanical transients of an asynchronous motor. Izv. vuzov, “Elektromekhanika,” 1971, No. 8.

3. I. P. Kopylov, F. A. Mamedov, V. Ya. Bespalov. Mathematical Modeling of Asynchronous Machines. Moscow, “Energiya,” 1969.

4. G. A. Sipailov, A. V. Loos, E. I. Sobko. Mathematical modeling of electromechanical transients of a single-phase synchronous generator. “Elektrotekhnika,” 1972, No. 10.

5. E. V. Kononenko. Synchronous Reactive Machines. Moscow, “Energiya,” 1970.

6. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes of Synchronous Machines. Moscow, Gosenergoizdat, 1960.