Pulling into Synchronism of a Synchronous Reluctance Motor

E. V. Kononenko, A. F. Fink

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

The improved synchronous reluctance motors (S.R.M.) currently being developed compare favorably with synchronous machines in their simple design and high reliability. In addition, these motors have sufficiently high energy performance, owing to which they are relatively economical and can be used as a drive wherever a synchronous rotational speed is required.

However, in the theory of these machines there are a number of questions that have not yet found a sufficiently satisfactory solution. One such question is the process by which the rotor is pulled into synchronism. The present work aims to fill, in part, the gap in this area.

Qualitative analysis of the process

Starting of an S.R.M. is carried out by direct connection to the mains. Since the rotor is equipped with a short-circuited winding, the starting process proceeds in the same way as in asynchronous machines. At small slips the rotor of an S.R.M. is pulled into synchronism under the action of the synchronous torque.

The problem posed, concerning the pulling of the S.R.M. rotor into synchronism while neglecting transient electromagnetic processes in the windings, leads to the investigation of the differential equation of rotor motion

\[ T_j \cdot \frac{d^2 \Theta}{dt^2} + M_a + M_s = M_c, \tag{1} \]

where \(T_j\) is the mechanical time constant;
\(\Theta\) is the angle formed by the transverse axis of the rotor pole with the voltage vector;
\(M_a\) is the asynchronous torque;
\(M_s\) is the synchronous torque;
\(M_c\) is the static or load torque.

In studying pulling into synchronism it is usually assumed that, at small slips \((S = 0 \div 0.1)\), the asynchronous torque is directly proportional to the slip [1, 2]

\[ M_a = M_n \frac{S}{S_n} = M_m \frac{S}{S_m}, \tag{2} \]

where \(M_n, M_m\) are the rated and maximum synchronous torques, taken from the rectilinear part of the asynchronous characteristic of the synchronous-reluctance motor (Fig. 1); \(S_n, S_m\) are the slips at which the synchronous-reluctance motor develops the rated and maximum torques in the asynchronous mode.

The synchronous torque is taken to be

\[ M_s = M_m \cdot \sin 2\Theta . \tag{3} \]

Taking (2) and (3) into account, the equation of motion of the rotor can be written in the form

\[ T_j \frac{d^2 \Theta}{dt^2} = M_c - M_m \cdot \frac{S}{S_m} - M_m \sin 2\Theta \tag{4} \]

or

\[ \frac{T_j}{M_m}\cdot \frac{d^2 \Theta}{dt^2} = m - \frac{S}{S_m} - \sin 2\Theta, \tag{5} \]

where

\[ m=\frac{M_c}{M_m}. \]

The equation obtained is nonlinear, since the magnitude of the synchronous torque is a nonlinear function. The investigation of equation (5) may be carried out by constructing a family of integral curves in the phase plane [1—3].

Fig. 1. Starting characteristic of a synchronous-reluctance motor.

Since time does not enter explicitly into equation (5), it can be reduced to an equation of first order. For this purpose we write

\[ S=\frac{1}{\omega_1}\frac{d\Theta}{dt}; \]

\[ \frac{d\Theta}{dt}=\omega_1 S, \tag{6} \]

\[ \frac{d^2\Theta}{dt^2}=\omega_1\frac{dS}{dt}. \tag{7} \]

Multiplying the numerator and denominator of equation (7) by \(d\Theta\), we obtain

\[ \frac{d^2\Theta}{dt^2} = \omega_1 \frac{dS}{d\Theta}\cdot \frac{d\Theta}{dt} = \omega_1^2 \cdot S \frac{dS}{d\Theta}. \tag{8} \]

Multiplying and dividing this equality by \(S_m^2\), we obtain

\[ \frac{d^2\Theta}{dt^2} = (S_m\omega_1)^2 \left(\frac{S}{S_m}\right) \frac{d\left(\dfrac{S}{S_m}\right)}{d\Theta}. \tag{9} \]

Taking (9) into account, the equation of rotor motion (5) can be written in the form

\[ \frac{T_j \cdot S_m^2 \omega_1^2}{M_m} Y\frac{dY}{d\Theta} = m-Y-\sin 2\Theta, \tag{10} \]

where

\[ Y=\frac{S}{S_m}. \]

Equation (10) was obtained for a two-pole machine. If the number of pole pairs is equal to \(P\), then equation (10) takes the form

\[ \frac{T_j S_m^{2}\omega_1^{2}}{P M_m}\cdot Y\frac{dY}{d\Theta}=m-Y-\sin 2\Theta . \tag{11} \]

The expression entering equation (11),

\[ \frac{T_j S_m^{2}\omega_1^{2}}{P M_m}=A, \]

remains unchanged for a given rotor design.

Fig. 2. Phase portrait during the pulling of the motor into synchronism.

The phase space of the system under consideration is the surface of a circular cylinder with its axis parallel to the \(Y\) axis, since the equation does not change when \(\Theta\) is replaced by \(\Theta+\pi\). It is convenient to consider the surface of the circular cylinder unrolled onto a plane. To construct the integral curves, first on the plane \((Y,\Theta)\) we determine the directions of the tangent at each point of this plane, \(dY/d\Theta\), for specified values of \(A\) and \(m\), and then construct the integral curves. The drawing of the integral curves \(Y=f(\Theta)\) may be started from any point of the plane \((Y,\Theta)\), assuming that the angle \(\Theta\) varies periodically from \(\pi/2\) to \(-\pi/2\), and continued until the motor at the end of the process falls into synchronism.

Figure 2 shows the construction of integral curves for the values \(A=2\) and \(m=0.5\). In accordance with equation (11), for the indicated values of \(A\) and \(m\) we obtain

\[ \frac{dY}{d\Theta}=\frac{0.5-Y-\sin 2\Theta}{2Y}. \tag{12} \]

In the case under consideration, the construction begins at

\[ \frac{S}{S_m}=3 \]

and continues to

\[ \frac{S}{S_m}=0. \]

In making the construction it was taken into account that the direction of the curve at any point must correspond to the directions of the tangents. The limiting curve is shown by a dashed line. As follows from Fig. 2, if the curve lies below the limiting curve, the motor reaches synchronism. The curves presented in Fig. 2 illustrate very clearly the course of the process of pulling the rotor into synchronism.

Figure 3 gives the construction of integral curves for the values \(A=6\) and \(m=0.5\).

Fig. 3. Phase portrait for the case in which the motor is not pulled into synchronism.

As follows from Fig. 3, increasing the moment of inertia to three times its value leads to the rotor of the synchronous reluctance motor not being pulled into synchronism, and to its oscillating for a long time along the limiting curve.

The plotting of the curves, as is obvious, is possible only for single, definite values of \(A\) and \(m\).

If one confines oneself to a completely general investigation of the question of for what values of \(A\) and \(m\) the motor does not fall into synchronism, then equation (12) can be solved by the method of successive intervals or with the aid of a continuous-action electronic computer.

Sudden change in the load of the s. r. m.

If the load of a synchronous reluctance motor is changed instantaneously, the motor rotor will rotate with an oscillating speed. This means that periodic changes of the angle \(\Theta\) arise, and the point \(B\), which determines the initial operating mode of the s. r. m., moves along the characteristic \(M=f(\Theta)\) to point \(C\) (Fig. 4). In this case a dynamic torque \(\Delta M\) arises, equal to the difference between the electromagnetic mo-

moment and the resisting torque \(M_{\mathrm{co}}\). Under the action of this torque the rotor will rotate with variable slip \(S\). At the value \(\Theta=\Theta_c\) the electromagnetic torque becomes equal to the resisting torque. However, owing to the inertia of the rotating masses, the rotor continues to move, passing point \(C\) and increasing the angle \(\Theta\). In moving from point \(C\) to point \(D''\), the rotor undergoes acceleration under the action of the electromagnetic torque, which here is greater than the resisting torque. The kinetic energy stored during the acceleration, \(A_{\text{acc}}\), is transformed into potential energy during the motion of the rotor from point \(C\) to \(D''\). At point \(D''\) all the kinetic energy obtained during acceleration proves to have been expended in overcoming the braking forces, and the slip \(S\) is equal to zero. However, the motion cannot cease, since an excess electromagnetic torque \(\Delta M_2\) acts on the rotor. Under the action of this torque the rotor again approaches point \(C\), acquiring slip \(S_c\) and possessing kinetic energy \(A_{\text{br}}\). The magnitude

Fig. 4. Change of operating condition under a sudden change in the load of an SRM.

of this energy is proportional to the area \(ABCA\). At point \(B\) all the kinetic energy again proves to have been expended, \(S=0\), and the process begins anew. Owing to the presence of losses in the motor, the oscillations become smaller in amplitude with each cycle, and the dependence \(S(\Theta)\) is represented in the form of a spiral.

The energy stored by the rotor during the acceleration process is determined as follows:

\[ A_{\text{acc}}=\int_{\Theta_c}^{\Theta_s} \Delta M\,d\Theta \tag{13} \]

and is represented graphically in Fig. 4 as the area \(CD''D'C\). The braking energy is expressed mathematically as the integral

\[ A_{\text{br}}=\int_{\Theta_0}^{\Theta_c} \Delta M\,d\Theta \tag{14} \]

and is represented graphically in the form of the area \(ABCA\). These areas are called the acceleration and braking areas. In general form the equality of the areas, according to [4], is determined by

\[ A_{\text{acc}}=A_{\text{brake}} \quad \text{or} \quad \int \Delta M\, d\Theta = 0. \tag{15} \]

One can imagine a case in which the energy obtained during braking is balanced by the energy obtained during acceleration not at the point \(D''\), but at the point \(D\). This point may be called critical, since with any further increase of the angle \(\Theta\) the rotor will be acted upon not by accelerating, but by braking forces. In this case the point \(D\) is a point of unstable equilibrium, since the slightest change in the angle \(\Theta\) leads to the appearance of a braking or accelerating torque. Either braking of the rotor begins, followed by its return to the stable equilibrium point \(C\), or a progressive increase of the angle and falling out of synchronism.

The trajectory along which the rotor will move is called the critical trajectory. In a first approximation it can be calculated by the phase-plane method, with allowance for the magnitude of the asynchronous torque. Let us determine the maximum slip acquired by the rotor as it moves along the critical trajectory.

Equation (5), taking account of (2.8), after simple transformations takes the following form:

\[ T_j \frac{\omega_1^2}{P} S \frac{dS}{d\Theta} + M_n \frac{S}{S_n} + \gamma M_n \sin 2\Theta = M_c, \tag{16} \]

where

\[ \gamma=\frac{M_m}{M_n} \]

is the overload capacity of the motor.

After dividing equation (16) by \(M_n\), we obtain

\[ \omega_1^2 \frac{T_j}{P M_n}\frac{dS}{d\Theta} + \frac{1}{S_n} S + \gamma \sin 2\Theta = m', \tag{17} \]

where

\[ m'=\frac{M_c}{M_n} \]

is the mechanical torque on the shaft in relative units.

Expressing

\[ K=\frac{\omega_1^2 T_j}{P M_n}, \]

we obtain

\[ KS \frac{dS}{d\Theta} + \frac{1}{S_n} S + \gamma \sin 2\Theta = m'. \tag{18} \]

Dividing all expressions by \(K\) and denoting

\[ \frac{1}{S_n}=N; \qquad \frac{N}{K}=C; \]

\[ \frac{\gamma}{K}=k; \qquad \frac{m'}{K}=L, \]

we obtain

\[ S \cdot \frac{dS}{d\Theta} + CS + k \sin 2\Theta = L. \tag{19} \]

The equation of motion in phase coordinates can be represented in the form

\[ \frac{dS}{d\Theta} = \frac{L-CS-k\sin 2\Theta}{S}. \tag{20} \]

Figure 5 shows the integral curve of equation (20), where \(\Theta_0\) is the initial load angle; \(\Theta_c\) is the angle corresponding to the new value of the load. This angle can be determined for \(S=0\) from equation (20)

\[ \sin 2\Theta_c=\frac{L}{k}, \tag{21} \]

\(\Theta_s\) is the maximum load angle, equal to \(\Theta_s=\dfrac{\pi}{2}-\Theta_c\), for which

\[ \sin 2\Theta_s=\sin(\pi-2\Theta_c)=\frac{L}{k}. \tag{22} \]

Fig. 5.

Taking (22) into account, from equation (20), for \(\Theta=\Theta_s\) and \(S=0\), we obtain an expression for the slope of the tangent to the integral curve at the point \(\Theta_s\)

\[ \tg \alpha=\left[\frac{dS}{d\Theta}\right]_{\Theta_s} =-C+\frac{L-k\sin 2\Theta_s}{S} =-C+\frac{0}{0}. \tag{23} \]

Resolving the indeterminacy in equation (23), we obtain

\[ \frac{L-k\sin 2\Theta_s}{S} = -\frac{2k\cos 2\Theta_s}{\left[\dfrac{dS}{d\Theta}\right]_{\Theta_s}}, \tag{24} \]

and, taking into account that \(\cos 2\Theta_s=-\cos 2\Theta_c\), from equation (23) we obtain

\[ \tg \alpha=-C+\frac{2k\cos 2\Theta_c}{\tg \alpha} \tag{25} \]

or

\[ \tg \alpha_{1,2}=-\frac{C}{2}\pm\sqrt{\frac{C^2}{4}+2k\cos 2\Theta_c}. \tag{26} \]

Since

\[ \cos 2\Theta_c=\sqrt{1-\sin^2 2\Theta_c} =\sqrt{1-\frac{L^2}{k^2}}, \]

then

\[ \alpha_{1,2}=-\frac{C}{2}\pm \sqrt{\frac{C^{2}}{4}+2\sqrt{k^{2}-L^{2}}}. \tag{27} \]

The maximum slip \(S_m\), corresponding to the rotor position \(\Theta_c\), can be determined if one moves from the rotor position \(\Theta_s\) in the direction of the point \(\Theta_c\). For this purpose we use the graph-analytical method of successive intervals \([2, 3]\). Let us divide the segment \(\Theta_s-\Theta_c\) into \(n_c\) parts of length \(l\), as shown in Fig. 5. In this case

\[ \left[\frac{\Delta S}{\Delta \Theta}\right]_n = \frac{S_{n-1}-S_{n+1}}{2l} = \frac{L-CS_n-k\sin 2\Theta_n}{S_n}, \tag{28} \]

whence

\[ S_{n+1}=S_{n-1}-2l\,\frac{L-CS_n-k\sin 2\Theta_n}{S_n}. \tag{29} \]

According to Fig. 5, in order to solve this equation it is necessary to know the value \(S_1\). For small values of \(l\), the slip \(S_1\) can be determined approximately:

\[ S_1=l\cdot \tg \alpha. \tag{30} \]

Conclusions

1. The application of the phase-plane method makes it possible to judge visually the character of the motion of the rotor of a synchronous reluctance motor. Construction of the picture of the processes on the phase surface also makes it possible to obtain a quantitative solution of problems connected with electromechanical transient processes in a synchronous reluctance motor during pulling into synchronism and sudden load changes.

2. The application of the phase-plane method makes it possible, quite simply, to determine the magnitude of the maximum slip during motion of the motor rotor along the critical trajectory.

References

1. K. P. Kovach, I. Rác. Transient Processes in Alternating-Current Machines, GEI, 1963.

2. J. Basta. Ein Beitrag zum nichtlinearen Pendeln von Synchronmaschin. E und M 74 (1957).

3. J. Stoker. Nonlinear Oscillations in Mechanical and Electrical Systems. Foreign Literature Publishing House, 1953.

4. V. A. Venikov. Transient Electromechanical Processes in Electric Systems. “Energiya” Publishing House, 1964.