Study of the Rotor Motion Equation of a Synchronous Reluctance Motor Using the Phase Plane Method
Abstract Generated abstract
This paper examines the nonlinear rotor motion equation of a synchronous reluctance motor during asynchronous operation, synchronous operation, and pull-in to synchronism. Using the phase plane method, the authors reduce the electromechanical model to a second-order rotor equation and analyze the conservative special case with zero damping, constructing phase portraits for different values of the load torque. The study identifies stable and unstable equilibrium points, separatrices, closed trajectories corresponding to rotor oscillations, and trajectories associated with loss of synchronism or asynchronous operation. It concludes that phase-plane analysis provides a visual and, in simple cases, quantitative means for assessing electromechanical transients in synchronous reluctance motors.
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STUDY OF THE EQUATION OF MOTION OF A SYNCHRONOUS RELUCTANCE MOTOR ROTOR BY THE PHASE PLANE METHOD
E. V. Kononenko, A. F. Fink
(Submitted by the scientific seminar of the Electromechanical Faculty)
In the course of operation of a synchronous reluctance motor (SRM), its operation may be classified into the following regimes: asynchronous, synchronous, and the regime of pulling into synchronism.
In the asynchronous regime the motor operates during starting and when it falls out of synchronism under overloads, while in the synchronous regime it operates under normal service conditions. The regime of pulling into synchronism is an intermediate one, when the motor passes from the asynchronous regime to the synchronous one.
In studying the operation of an SRM, the method of rotating magnetic fields is taken as the basis.
The transient electromechanical processes of an SRM are described by the well-known system of equations:
\[ \frac{d\psi_d}{dt}+r\cdot i_d=U_d+\omega\cdot\psi_q, \]
\[ \frac{d\psi_q}{dt}+r\cdot i_q=U_q-\omega\psi_d, \]
\[ \frac{d\psi_{dr}}{dt}+r_{dr}\cdot i_{dr}=0, \tag{1} \]
\[ \frac{d\psi_{qr}}{dt}+r_{qr}\cdot i_{qr}=0, \]
where quantities without additional subscripts refer to the stator windings, and those with the subscript \(r\) to the rotor windings along the \(d\) and \(q\) axes, respectively,
\[ T_j\cdot\frac{d^2\Theta}{dt^2}+M=M_c, \tag{2} \]
where \(T_j\) is the mechanical time constant, \(\Theta\) is the angle formed by the transverse axis of the rotor poles with the voltage vector, \(M=\psi_d\cdot i_q-\psi_q\cdot i_d\) is the electromagnetic torque.
\(M_c\) is the torque of the static resisting forces. Assuming that the electromagnetic transient processes proceed considerably faster—
than the mechanical ones, the equation of motion of the rotor can be represented in the form [1, 2]
\[ \frac{d^{2}\Theta}{dt^{2}}+K\cdot\frac{d\Theta}{dt}+\sin 2\Theta=M_c, \tag{3} \]
where \(K\) is the proportionality coefficient. This equation is nonlinear, since the magnitude of the synchronous torque is a nonlinear function. It has no regular method of solution; therefore an exact mathematical analysis of this equation presents great difficulties. Equation (3) can be studied by constructing a family of integral curves on the phase plane [1—5]. Equation (3) is equivalent to the system
\[ \frac{dS}{dt}=M_c-\sin 2\Theta-K\cdot S;\quad \frac{d\Theta}{dt}=S. \tag{4} \]
The phase space of the system under consideration is the surface of a circular cylinder with an axis parallel to the \(S\)-axis, since the equation is not changed when \(\Theta\) is replaced by \(\Theta+\frac{\pi}{2}\).
The phase trajectories of the system are determined by the differential equation
\[ \frac{dS}{dt}=\frac{M_c-\sin 2\Theta-K\cdot S}{S}=f(\Theta,S). \tag{5} \]
The qualitative picture on the phase surface is determined by the equilibrium states, separatrices, and closed trajectories (limit cycles).
In this article the problem is posed of studying the special case of equation (5), when \(K=0\) (a conservative system). Then equation (5) takes the form:
\[ \frac{dS}{d\Theta}=\frac{M_c-\sin 2\Theta}{S}. \tag{6} \]
Its general integral will be
\[ \frac{S^{2}}{2}=M_c\Theta+\frac{\cos 2\Theta}{2}+C' \tag{7} \]
or
\[ S=\pm\sqrt{2M_c\Theta+\cos 2\Theta+C} =\pm\sqrt{Y(\Theta)+C}, \tag{8} \]
where \(C'\) is the constant of integration, and
\[ C=2C' \]
the coordinates of the equilibrium state are determined by the equalities
\[ M_c-\sin 2\Theta=0,\quad S=0. \tag{9} \]
Let us note that the function \(\frac{S^{2}}{2}\), taken with the opposite sign, represents the potential energy of the system.
\[ U(\Theta)=-\int_{0}^{\Theta}(M_c-\sin 2\Theta)\cdot d\Theta, \tag{10} \]
and the constant of integration \(C'\) in expression (7) is the constant of the total energy of the system, depending on the initial conditions.
From (10), (11) it follows that singular points can occur for values of \(\Theta\) satisfying the equation
\[ U'(\Theta)=0. \]
For a point of stable equilibrium \((\Theta_c)\), it must be that
\[ U''(\Theta_c)=2\cos 2\Theta_c>0. \tag{11} \]
For a point of unstable equilibrium it must be that
\[ U''(\Theta_s)=2\cos 2\Theta_s<0. \tag{12} \]
The construction of integral curves in the \((\Theta,S)\) plane reduces to the following.
For various values of \(C\), one determines \(S=\pm\sqrt{Y(\Theta)+C}\). The results of the calculations are presented in the form of a family of curves in the \((\Theta,S)\) plane. The nature of the curves depends on the magnitude of \(M_c\).
Fig. 1. Phase portrait for the case \(k=0,\ M_c=0\).
a) Figure 1 shows the phase portrait for the case \(M_c=0\). For \(C=-1.0\) we obtain one point of stable equilibrium, \(\Theta_c\)—a center with coordinates
\[ \Theta=0,\quad S=0. \]
For \(C=0.5\) we obtain two points of unstable equilibrium,
\[ \Theta_s=\pm\frac{\pi}{2}. \]
The phase trajectories intersecting at these points are called separatrices. For \(-1<C<0.5\) we obtain a number of closed integral curves nested one inside another. For \(C>0.5\) the curves do not intersect the abscissa axis. For \(C<-1\) the integral curves have no real roots. All integral curves inside the separatrices are closed and do not envelop the cylinder. They correspond to oscillations of the SRM rotor.
The curves outside the separatrices, \(C>0.5\), are also closed, but they correspond to another type of periodic motion—the asynchronous operating mode of the SRM with slip, constant in direction but variable in magnitude.
b) Figure 2 shows the phase portrait for the case
\[ 0<M_c<M_m. \]
It follows from the curves presented that the curves inside the separatrix are closed, while outside it they are open, since when \(\Theta\) is increased by \(\dfrac{\pi}{2}\) the slip \(S\) increases in absolute value with each revolution.
Oscillations about the position of stable equilibrium \(\Theta_c\) in this case are possible only for sufficiently small deviations. If the initial deviation is large, the SRM rotor will pass through the position of unstable equilibrium \(\Theta_s\) and fall out of synchronism.
c) For \(M = M_m\) the phase portrait is shown in Fig. 3. In this case we have one point of unstable equilibrium. There are no closed integral curves enclosing the cylinder.
d) For \(M > M_m\) there are no singular points and no closed curves not enclosing the cylinder.
Fig. 2. Phase portrait for the case \(k=0\), \(0<M_c<M_m\).
The idea of the nature of the course of the processes can be extended if, following [3], the processes occurring in time are determined from the phase trajectories.
Let, at the instant of time \(t_n\), the representative point have coordinates \(\Theta_n\) and \(S_n\), and at the instant \(t_{n+1}\) reach a point with coordinates \(\Theta_{n+1}\) and \(S_{n+1}\). We introduce the notation
\[ \Delta \Theta = \Theta - \Theta_n;\quad \Delta_n \Theta = \Theta_{n+1} - \Theta_n; \]
\[ \Delta t = t - t_n;\quad \Delta_n t = t_{n+1} - t_n. \tag{13} \]
The system of equations (4) can be solved in quadratures
\[ \frac{d(\Delta \Theta)}{dt} = S;\quad \frac{dS}{dt} = f(\Theta_n + \Delta \Theta) = f(\Delta \Theta), \tag{14} \]
\[ S^2 - S_n^2 = 2 \int_{0}^{\Delta \Theta} f(\Delta \Theta)\, d(\Delta \Theta) = F(\Delta \Theta) - F(0), \tag{15} \]
\[ \Delta_n t = \int_{0}^{\Delta_n \Theta} \frac{d(\Delta \Theta)} {\sqrt{F(\Delta \Theta)-F(0)+S_n^2}} = \int_{0}^{\Delta_n \Theta} \varphi(\Delta \Theta)\cdot d(\Delta \Theta). \tag{16} \]
The calculation of (16) can be carried out by expanding \(\varphi(\Delta \Theta)\) in a series and subsequently integrating this series.
Fig. 3. Phase portrait for the case \(k=0\), \(M_c=M_m\)
Conclusions
- The application of the phase-plane method makes it possible to judge visually the nature of the motion of the rotor of a synchronous reluctance motor. Constructing the picture on the phase surface in the simplest cases also makes it possible to obtain a quantitative solution of problems associated with electromechanical transient processes in a synchronous reluctance motor.
REFERENCES
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