EFFECT OF ROTOR MAGNETIC ASYMMETRY ON TRANSIENT PROCESSES IN AC MACHINES

V. S. GRINBERG, E. V. KONONENKO

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

Transient processes in AC electrical machines have a major influence both on the operation of the electrical machines themselves and on the operation of electric drives and power systems in which these machines are used. Therefore this question has been, and remains, an object of close scientific investigation.

AC electrical machines are divided into two principal types: 1) synchronous machines and 2) induction machines. They differ in construction, application, and so on. The main difference between these two types of AC machines from the standpoint of theory is that, in the general case, the rotors of synchronous machines possess magnetic and electrical asymmetry, whereas the rotors of induction machines are symmetrical in both respects [1]. However, in the literature the question of the influence of rotor asymmetry—both magnetic and electrical—on transient processes has not been sufficiently treated. Therefore a detailed study of the influence of rotor asymmetry on the transient processes of AC electrical machines is of definite interest.

The processes occurring in AC machines are very complex and varied. In order to simplify analytical investigations, it is convenient to use generally accepted assumptions that have been verified by many years of experience [2, 3, etc.]:

1) the phase windings of the stator are symmetrical; the rotor is symmetrical with respect to its longitudinal \((d)\) and transverse \((q)\) axes;

2) only the fundamental harmonics of the spatial distribution of the magnetomotive forces and magnetic fields created by the windings are taken into account;

3) the magnetic circuits are unsaturated;

4) iron losses, mechanical losses, and current displacement in the windings are absent;

5) the machine is connected on the stator side to a network of infinitely large power; the network voltage is sinusoidal and symmetrical;

6) zero-sequence currents are equal to zero.

Under the stated assumptions, instead of a real electrical machine, an idealized electrical machine is analyzed. In this connection, one may consider a certain generalized electrical machine, of which a synchronous or induction machine is a special case, and formulate differential equations used for the analysis of both types of machines. However, the difference between synchronous and induction machines, consisting in the presence of rotor symmetry,

be reflected in their analytical study. The asymmetry of the rotor of synchronous machines necessitates writing the original differential equations in a coordinate system fixed with respect to the rotor, since only in this system are periodic coefficients eliminated. To achieve the same goal, the original differential equations of asynchronous machines, i.e., machines with a symmetrical rotor, need only be written in a coordinate system rotating at any speed relative to the rotor. The number of equations of a machine with a symmetrical rotor can be reduced by half if they are written in vector form. The number of equations of a synchronous machine in vector form, because of the asymmetry of its rotor, will be equal to the number of equations with real coefficients. Preference is given to the latter, since they are more convenient for solution than equations in vector form [1].

Real transient processes of AC electrical machines are electromechanical processes, i.e., they are accompanied by a change in the rotational speed over time. In the present work, the connection to the network of an electrical machine rotating at a constant speed is considered, i.e., only electromagnetic transient processes are considered. Symmetrical transient regimes in AC machines at constant rotational speed and constant parameters are described by a system of linear differential equations with constant coefficients. In this case, the voltage-balance equations of the electrical circuits can be considered independently of the equation of rotor motion.

In the general case of rotor asymmetry of an AC machine, the voltage-balance equations of the stator windings in the per-unit system, under the condition that the rotor \(q\)-axis leads the \(d\)-axis by 90 electrical degrees, in axes fixed with respect to the rotor, in motor operation, have the form

\[ \begin{aligned} U_d &= [p x_d(p) + r] i_d - x_q(p)(1-s)i_q,\\ U_q &= x_d(p)(1-s)i_d + [p x_q(p) + r] i_q . \end{aligned} \tag{1} \]

The voltages \(U_d\) and \(U_q\) can be represented in the form

\[ \begin{aligned} U_d &= U\cos(st+\delta_0);\\ U_q &= U\sin(st+\delta_0), \end{aligned} \tag{2} \]

where
\(U\) is the amplitude of the phase voltage applied to the stator winding,
\(\delta_0\) is the angle formed by the resultant voltage vector with the rotor \(d\)-axis at the instant of switching on \((t=0)\).

In the case where the steady-state regime is considered, the time-differentiation operator \(p\) is replaced by \(js\), and the system of equations (1), taking (2) into account, is solved by the complex method [3]. In the study of transient regimes, however, the operational method is widely used, making it possible to obtain the solution of nonhomogeneous differential equations in a single step.

Passing from real quantities to their images, equation (1), taking (2) into account, in operational form under zero initial electromagnetic conditions can be written as

\[ U \cdot \frac{p^2\cos\delta_0 - sp\sin\delta_0}{p^2+s^2} = [p x_d(p)+r] i_d(p) - x_q(p)(1-s)i_q(p), \]

\[ U \cdot \frac{p^2\sin\delta_0 - sp\cos\delta_0}{p^2+s^2} = x_d(p)(1-s)i_d(p) + [p x_q(p)+r] i_q(p). \tag{3} \]

Electromagnetic initial conditions are understood here as the initial conditions for the quantities of currents or flux linkages. In equations (3) the Carson transform is used. In the present work, the influence of only the magnetic asymmetry of the rotor on transient processes is considered. By magnetic asymmetry of the rotor we mean the difference between the magnetic permeances along the \(d\)-axis and along the \(q\)-axis of the rotor. An example of a rotor having only magnetic asymmetry in an AC machine is a synchronous reluctance motor without windings on the rotor [3]. In this case

\[ x_d(p)=x_d \quad \text{and} \quad x_q(p)=x_q . \tag{4} \]

The characteristic equation of system (3), taking (4) into account, has the form

\[ p^2+(\alpha_d+\alpha_q)p+\alpha_d\alpha_q+(1-s)^2=0, \tag{5} \]

where

\[ \alpha_d=\frac{r}{x_d} \quad \text{and} \quad \alpha_q=\frac{r}{x_q}. \]

Solving (3), one can write the images of the currents \(i_d\) and \(i_q\):

\[ i_d(p)=\frac{U}{x_d x_q}\,p\, \frac{(p\cos\delta_0-s\sin\delta_0)(p x_q+r)+x_q(1-s)(p\sin\delta_0+s\cos\delta_0)} {(p^2+s^2)(p-p_1)(p-p_2)}; \]

\[ i_q(p)=\frac{U}{x_d x_q}\,p\, \frac{(p\sin\delta_0+s\cos\delta_0)(p x_d+r)-(p\cos\delta_0-s\sin\delta_0)x_d(1-s)} {(p^2+s^2)(p-p_1)(p-p_2)}, \tag{6} \]

where \(p_1\) and \(p_2\) are the roots of the characteristic equation (5).

The images of the currents \(i_d\) and \(i_q\) in (6) are fractional-rational functions with respect to \(p\). To pass to the originals, it is convenient here to use the expansion theorem, which can be used in the absence of multiple and zero roots [4 et al.]. Therefore we first examine the nature of the roots of the characteristic equation (5). Solving it with respect to \(p\), we find an expression for the roots \(p_1\) and \(p_2\)

\[ p_{1,2}=-\frac{r(x_d+x_q)}{2x_d x_q} \pm \sqrt{ \left[ \frac{r(x_d-x_q)}{2x_d x_q} \right]^2-\omega^2 } =\alpha \pm z, \tag{7} \]

where

\[ \omega=(1-s) \text{ is the rotor rotational speed;} \]

\[ \alpha=-\frac{r(x_d+x_q)}{2x_d x_q}, \tag{7a} \]

\[ z=\sqrt{ \left[ \frac{r(x_d-x_q)}{2x_d x_q} \right]^2-\omega^2 } =\frac{\lambda}{2x_d x_q}. \tag{7b} \]

The first term \(\alpha\) of both roots \(p_1\) and \(p_2\) is determined only by the machine parameters. It is always real and negative. The second term \(z\) is determined by the ratio of parameters and by the absolute value of the speed at which the machine is connected to the network, \(|\omega_{\mathrm{on}}|\). Under the condition

\[ |\omega_{\mathrm{on}}|< \frac{r(x_d-x_q)}{2x_d x_q} \]

the roots \(p_1\) and \(p_2\) will be real and distinct. In this case the dependence of the roots \(p_1\) and \(p_2\) of the characteristic equation (5) is describ-

Fig. 1. Dependence of the roots of the characteristic equation on the switching speed

is given by the equation of a circle of radius

\[ R = \frac{r(x_d - x_q)}{2x_d x_q} \]

in the axes \(\omega_{\mathrm{sw}} - p_{1,2}\) (Fig. 1). Under the condition

\[ |\omega_{\mathrm{sw}}| > \frac{r(x_d - x_q)}{2x_d x_q} \]

the roots \(p_1\) and \(p_2\) will be complex and conjugate. In this case the real part of the roots \(p_1\) and \(p_2\) is always negative and equal to \(\alpha\), while the dependence of the coefficient \(\beta\) for the imaginary part of the roots \(p_1\) and \(p_2\) in the same axes is described by the equation of an equilateral hyperbola with the distance between the vertices equal to

\[ 2R = \frac{r(x_d - x_q)}{x_d x_q}. \]

At the switching speed

\[ |\omega_{\mathrm{sw}}| = \frac{r(x_d - x_q)}{2x_d x_q} \]

the roots will be real, multiple, and negative:

\[ p_1 = p_2 = \alpha. \]

Thus, under the condition

\[ |\omega_{\mathrm{sw}}| \ne \frac{r(x_d - x_q)}{2x_d x_q} \]

one may use the expansion theorem to find the originals \(i_d(t)\) and \(i_q(t)\).

After finding the originals and transformations, taking into account (7), (7a), and (7b), the transient currents \(i_d(t)\) and \(i_q(t)\) can be written in the form:

\[ i_d(t)=U\left\{[A_{d0}\cos(st+\delta_0)+B_{d0}\sin(st+\delta_0)]+\right. \]

\[ \left.+\left[(A_{d1}\cos\delta_0-A_{d2}\sin\delta_0)\frac{\operatorname{sh}(zt)}{\lambda} -(A_{d0}\cos\delta_0+B_{d0}\sin\delta_0)\operatorname{ch}zt\right]e^{at}\right\}, \tag{8} \]

\[ i_q(t)=U\left\{[A_{q0}\sin(st+\delta_0)-B_{q0}\cos(st+\delta_0)]+\right. \]

\[ \left.+\left[(A_{q1}\sin\delta_0+A_{q2}\cos\delta_0)\frac{\operatorname{sh}(zt)}{\lambda} -(A_{q0}\sin\delta_0-B_{q0}\cos\delta_0)\operatorname{ch}zt\right]e^{at}\right\}, \tag{9} \]

where

\[ A_{d0}=\frac{ra-s(1-2s)b\cdot x_q}{a^2+s^2b^2}, \tag{10} \]

\[ B_{d0}=\frac{srb+(1-2s)ax_q}{a^2+s^2b^2}, \tag{11} \]

\[ A_{d1}=\frac{-rb+(sb^2+2xc)x_q}{a^2+s^2b^2}, \tag{12} \]

\[ A_{d2}=\frac{sr(b^2-2x_dx_qa)+(1-2s)b\cdot x_q}{a^2+s^2b^2}, \tag{13} \]

\[ \begin{gathered} a=r^2+x_dx_q(1-2s),\\ b=r(x_d+x_q),\\ c=r^2+x_dx_q(1-s)(1-2s). \end{gathered} \tag{14} \]

The coefficients \(A_{q0}\), \(B_{q0}\), \(A_{q1}\), and \(A_{q2}\) are calculated by formulas (10), (11), (12), and (13), respectively, with \(x_q\) replaced in the latter by \(x_d\). The expression for the transient torque is

\[ M(t)=(x_d-x_q)i_d(t)i_q(t). \tag{15} \]

Substituting (8) and (9) into (15), after some transformations the transient torque can be represented in the form

\[ M(t)=M_c+M_{\text{pul}}+[M_u+M_{uk}+M_u\delta_0+M_k\delta_0]e^{2at}+M_{s\beta}e^{at}, \tag{16} \]

where

\(M_c\) is the constant component of the torque in the steady-state regime \((t=\infty)\);

\(M_{\text{pul}}\) is the torque component in the steady-state regime, pulsating at twice the slip frequency;

\(M_u\) is a free component decaying with damping coefficient \(2\alpha\); its initial value \((t=0)\) does not depend on the angle \(\delta_0\), and the character of its variation in time does not depend on whether the roots of characteristic equation (5) are complex or real;

\(M_{uk}\) is a free component of the torque; in contrast to the component \(M_u\), its initial value depends on the angle \(\delta_0\);

\(M_u\delta_0\) is a free component whose initial value does not depend on the switching angle \(\delta_0\), but whose character of variation in time is determined by the character of the roots; for

\[ |\omega_{\text{sw}}|<\frac{r(x_d-x_q)}{2x_dx_q} \]

\(M_u\delta_0\) is the sum of two aperiodic components decaying with damping coefficients \(2P_1\) and \(2P_2\); for

\[ |\omega_{\text{sw}}|>\frac{r(x_d-x_q)}{2x_dx_q}, \]

\(M_u\delta_0\)

is a periodic component pulsating with frequency \(2\beta\) and decaying with coefficient \(2\alpha\);

\(M_k \delta_0\) is the free component of the torque; its initial value, unlike the component \(M_u \delta_0\), depends on the angle \(\delta_0\); when the nature of the roots changes, it varies in time analogously to \(M_u \delta_0\);

\(M_s \beta\) is the free component of the torque, whose law of variation in time is determined by the nature of the roots. The initial value of \(M_s \beta\) depends on the angle \(\delta_0\). For real roots, \(M_s \beta\) is the sum of two periodic components pulsating with the slip frequency and decaying with damping coefficients \(P_1\) and \(P_2\). For complex roots, \(M_s \beta\) is the sum of two periodic components pulsating with frequencies \((s+\beta)\) and \((s-\beta)\) and decaying with coefficient \(\alpha\).

Thus, the nature of the time variation of the free components of the transient torque corresponding to \(M_u \delta_0\), \(M_k \delta_0\), and \(M_s \beta\) in (16) is determined by the nature of the roots of the characteristic equation (5). The initial values of the free components of the torque corresponding to \(M_{ik}\), \(M_k \delta_0\), and \(M_s \beta\) in (16) are determined by the angle \(\delta_0\). The nature of the time variation and the initial value of the free component corresponding to \(M_i\) in (16) depend neither on the nature of the roots nor on the angle \(\delta_0\).

Conclusions

1. At a constant rotational speed, the degree of magnetic asymmetry of the rotor of an alternating-current electrical machine determines the nature of the time variation of certain components of the transient electromagnetic torque.

2. With an unchanged voltage switching phase, the initial values of part of the free components of the transient torque are determined by the initial position of the magnetically asymmetric rotor relative to the stator.

References

1. I. I. Treshchev. Methods for Investigating Electromagnetic Processes in Alternating-Current Machines. Energiya, 1969.

2. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes of a Synchronous Machine. Gosenergoizdat, 1960.

3. E. V. Kononenko. Synchronous Reluctance Machines. Energiya, 1970.

4. S. G. Ginzburg. Methods for Solving Problems on Transient Processes in Electrical Circuits. Moscow, 1967.