On the Stability of Motion of AC Machines with an Asymmetric Rotor
Abstract Generated abstract
This paper examines the stability of motion in AC machines with asymmetric rotors, especially synchronous and synchronous-reluctance motors, focusing on pulling into synchronism and oscillatory behavior. It formulates the rotor motion equation under common assumptions that asynchronous torque is proportional to slip and that the proper torque is sinusoidal, then studies the resulting nonlinear equation qualitatively in the phase plane through its singular points. The analysis indicates conditions under which asynchronous operation or synchronization occurs, and shows that the standard equation permits only damped oscillations, not self-excited oscillations. The paper concludes that the usual simplifying assumptions are insufficient for explaining self-oscillation phenomena in these machines.
Full Text
On the Stability of Motion of AC Machines with an Asymmetric Rotor
E. V. Kononenko
(Presented by the scientific seminar of the Electromechanical Faculty)
One of the most important questions in the study of the operation of AC machines with an asymmetric rotor (synchronous and synchronous-reluctance machines) is the investigation of the stability of their motion. In studying stability one may distinguish two problems: pulling into synchronism and oscillations.
Synchronous and synchronous-reluctance motors usually have an asynchronous method of starting; that is, they have on the rotor a special short-circuited winding and, consequently, can operate as asynchronous motors. Operating as an asynchronous motor, such a motor reaches a speed close to synchronous and, under the action of the electromagnetic torque of rotation, is pulled into synchronism under certain conditions.
Along with pulling into synchronism, oscillations of synchronous and synchronous-reluctance machines are of great practical interest. In addition to damped or periodic oscillations arising under the action of external periodic forces, in practice there are observed undamped or even increasing oscillations of synchronous machines. The causes of such oscillations or swings are due to properties of the machine itself, which it acquires under certain relations of the parameters. These oscillations are called self-excited oscillations, or self-oscillations. This phenomenon itself is often called self-swinging.
The essence of this phenomenon is that a machine, having received a random impulse from outside, subsequently rotates with an oscillating speed. Moreover, these oscillations do not die out, but have a constant or even increasing amplitude. In the latter case the machine falls out of synchronism.
In order to study the stability of operation of a synchronous machine, it is necessary to formulate and investigate the equation of motion of the machine.
The rotational speed of the rotor of a synchronous machine in the general case is described by the differential equation
\[ H \frac{ds}{dt} + M = M_{cm}, \tag{1} \]
where \(H\) is the mechanical time constant,
\(s\) is the slip,
\(M\) is the electromagnetic torque,
\(M_{st}\) is the static or resistance torque.
For synchronous (and synchronous-reluctance) machines this equation is a nonlinear differential equation. It has no regular method of solution; therefore an exact mathematical analysis of this equation presents great difficulties.
When a synchronous machine operates on a network with constant voltage and frequency, the voltage vector rotates with constant (synchronous) speed. Let the angle formed by the transverse axis of the rotor pole with the voltage vector be denoted by \(\Theta\). Then the slip may be determined as
\[ s=-\frac{d\Theta}{dt}, \tag{2} \]
and the equation of motion takes the form
\[ H\frac{d^2\Theta}{dt^2}+M=M_{st}. \tag{3} \]
Usually, in studying equation (3), the following assumptions are made. The electromagnetic torque of a synchronous machine is due to the presence on the rotor of short-circuited windings (we shall call this component the asynchronous torque \(M_a\)), and also to the interaction of the magnetic fields of the stator and rotor (we shall call this component the proper torque \(f(\Theta)\)).
Thus,
\[ M=M_a+f(\Theta). \tag{4} \]
In studying pulling into synchronism and oscillations it is usually assumed that, for small slips \((s=0 \div 0.1)\), the asynchronous torque is directly proportional to the slip \([1,2]\)
\[ M_a=m_a s, \tag{5} \]
where \(m_a\) is the coefficient of proportionality.
The proper torque is taken to be equal to:
a) for a synchronous machine
\[ f(\Theta)=M_m\sin\Theta, \tag{6} \]
b) for a synchronous-reluctance machine
\[ f(\Theta)=M_m\sin 2\Theta. \tag{7} \]
Taking into account the accepted assumptions, the equation of motion (3) may be represented in the form
\[ H\frac{d^2\Theta}{dt^2}+m_a\frac{d\Theta}{dt}+f(\Theta)=M_{st} \tag{8} \]
or
\[ \frac{d^2\Theta}{dt^2}+\left(-B+A\frac{d\Theta}{dt}\right)+f_1(\Theta)=0. \tag{9} \]
The resulting equation is nonlinear, since the restoring force \(f_1(\Theta)\) is a nonlinear function.
At present there are no regular methods for integrating (solving) such equations. Equation (9) can be investigated by constructing a family of integral curves in the phase plane \([3, 4]\).
The time \(t\) does not enter explicitly into equation (9); therefore it can be reduced to a first-order equation.
Taking the velocity \(s=\dfrac{d\Theta}{dt}\) as a new variable, we obtain the system of equations:
\[ \frac{d\Theta}{dt}=s; \]
\[ \frac{ds}{dt}=-f_1(\Theta)-\varphi(s), \tag{10} \]
where
\[ \varphi(s)=-B+As. \]
Substituting the first equation of system (10) into the second, we obtain
\[ \frac{ds}{d\Theta}=\frac{-f_1(\Theta)-\varphi(s)}{s}. \tag{11} \]
The solution of equation (11) can be represented most clearly in the phase plane (in the coordinates \(s,\Theta\)) in the form of a family of integral curves.
The form of the family of integral curves (the phase portrait) can be constructed either by direct integration of equation (11), or by the graphical methods of Lienard or isoclines. Graphical methods are applied when direct integration of equation (11) is impossible.
In the case under consideration, direct integration of equation (11) is impossible; therefore, to construct the phase portrait it is necessary to use a graphical method.
For a qualitative study of the motion of the system described by equation (11), it is necessary to consider the behavior of the system at singular points.
Equation (11) can be written in the form
\[ \frac{ds}{d\Theta} = \frac{-f_1(\Theta)-\varphi(s)}{s} = \frac{P(\Theta,s)}{Q(\Theta,s)}. \tag{12} \]
The coordinates of the singular points (that is, the points corresponding to the equilibrium state of the system) are determined from the equations:
\[ P(\Theta,s)=0;\qquad Q(\Theta,s)=0. \]
The coordinates of the singular points are equal (for the synchronous motor) to:
\[ C'_n \begin{cases} \Theta=\Theta_o+2\pi n;\\ s=0; \end{cases} \]
\[ C''_n \begin{cases} \Theta=-\Theta_o+\pi(1+2n);\\ s=0, \end{cases} \tag{13} \]
where
\[ \Theta_{0}=\arcsin \frac{M_{cm}}{M_{m}}; \]
\[ n=0,1,2\ldots . \]
To study the behavior of the system, let us consider its behavior in the neighborhood of a singular point, assuming that
\[ \Theta=\Theta_{0}+\xi, \]
\[ s=\eta. \]
A function of two variables, on passing to an infinitely close point \((\xi,\eta)\), can be obtained by expansion in a Taylor series:
\[ \begin{aligned} P(\Theta,s)&=P(\Theta_{0},O)+P'_{\Theta}(\Theta_{0},O)\xi+P'_{s}(\Theta_{0},O)\eta+P_{1}(\xi,\eta);\\ Q(\Theta,s)&=Q(\Theta_{0},O)+Q'_{\Theta}(\Theta_{0},O)\xi+Q'_{s}(\Theta_{0},O)\eta+Q_{1}(\xi,\eta), \end{aligned} \tag{14} \]
where \(P'_{\Theta}, P'_{s}, Q'_{\Theta}, Q'_{s}\) are partial derivatives,
\(P(\Theta_{0},O), Q(\Theta_{0},O)\) are the values of the functions at the singular point,
\(P_{1}(\xi,\eta), Q_{1}(\xi,\eta)\) are functions of \(\xi,\eta\) of order higher than one.
Placing the origin of coordinates at the singular point, equation (12) may be represented in the form
\[ \frac{P(\Theta,s)}{Q(\Theta,s)} = \frac{a\xi+b\eta+P_{1}(\xi,\eta)} {c\xi+d\eta+Q_{1}(\xi,\eta)}, \tag{15} \]
where
\[ a=P'_{\Theta}(\Theta_{0},O);\quad b=P'_{s}(\Theta_{0},O);\quad c=Q'_{\Theta}(\Theta_{0},O);\quad d=Q'_{s}(\Theta_{0},O). \]
The investigation of this equation makes it possible to determine the state of the system at the singular points.
a) Singular points \(C'_{n}\).
Investigations show that these points are either a stable node or a stable focus, depending on the relations among the parameters entering equation (15).
b) Singular points \(C''_{n}\).
Investigations show that these points are saddles.
Thus, an investigation of the behavior of a synchronous machine whose motion is described by equation (8) at the singular points shows that, in this case, self-oscillations cannot arise.
A qualitative investigation of equation (8) in the phase plane shows that the following operating regimes of a synchronous (synchronous-reluctance) motor are possible.
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If \(M_{cm}>M_{m}\), then an asynchronous operating regime occurs for any initial conditions. In this case the synchronous motor operates with a definite slip.
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If \(M_{cm}<M_{m}\), then two cases are possible. Under certain initial conditions the synchronous motor is pulled into synchronism and will operate normally. Under some initial conditions an asynchronous regime will occur, i.e., the motor will not be able to pull into synchronism.
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If \(M_{cm}=0\), then for any initial conditions the synchronous motor will pull into synchronism.
- In a synchronous motor whose motion is described by equation (8), only damped oscillations can arise. The phenomenon of self-excitation cannot be explained by means of this equation.
Thus, the investigations carried out show that, using the most widely employed equation of motion for synchronous machines, (8), one can study the stability of motion of these machines only partially. One can investigate questions concerning the pulling of the motor into synchronism and forced oscillations. However, the phenomenon of self-excitation cannot be explained in this case. Consequently, assumptions (5) and (6), adopted in deriving the equation, cannot always be justified, in particular when studying self-oscillations of synchronous and synchronous-reluctance machines.
References
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L. N. Belyustina, “On the stability of the operating regime of a salient-pole synchronous motor,” Izvestiya AN SSSR, No. 10, 1954.
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A. M. Langen, “Determination of the maximum permissible moment of entry into synchronism of a reluctance electric motor,” VEP, No. 12, 1956.
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A. A. Andronov and S. E. Khaikin, Theory of Oscillations, ONTI, 1937.
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J. Stoker, Nonlinear Oscillations in Mechanical and Electrical Systems, Moscow, 1953.