STUDY OF THE MAGNETIC FIELD OF SIMPLE SYNCHRONOUS RELUCTANCE MOTORS

A. N. AIFERT, E. V. KONONENKO

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

In calculations of synchronous reluctance motors (SRM) of simple salient-pole design, simplified formulas or curves borrowed from the theory of synchronous salient-pole machines are usually used to determine the field form factors \(k_d\) and \(k_q\) [1, 2]. Therefore, in recent years a number of works have appeared in which an attempt is made to refine the calculation of the magnetic field in the air gap of an SRM.

In article [3], the field form factors were obtained by modeling the magnetic field on conducting paper, and the influence of pole height on the field in the gap was investigated. In modeling the field, the stator potential was assumed constant, and the coefficients were determined on the basis of the conductance of the air gap, which, naturally, reduced the accuracy of the results obtained.

An analytical calculation method is given in article [4] for a sinusoidal distribution of the magnetomotive force along the surface of the stator bore by solving Laplace’s equation (1) for the vector potential

\[ \frac{\partial^2 A(x,y)}{\partial x^2} + \frac{\partial^2 A(x,y)}{\partial y^2} = 0. \tag{1} \]

However, in doing so a simplifying assumption was made that along the lines \(x=a\) and \(x=\tau-a\) (Fig. 1) in the air gap the normal component of the magnetic-field intensity \((H_y)\) is constant. This substantially affected the accuracy of the results obtained.

In [5] the calculation is performed for a simplified configuration of the air gap, considering the field separately under the pole and in the interpolar space.

In all these works only the field form factors are determined. However, knowledge of these coefficients is insufficient for calculating an SRM, since, in addition to the first harmonic of induction, it is necessary to know the total magnetic flux and the maximum value of the induction in the air gap. On this basis, we proposed a system of coefficients that completely characterize the magnetic field in the air gap of an SRM, and these coefficients were determined on the basis of a simplified field pattern [6] and by modeling the magnetic field on an electronic integrator [7].

All these methods have significant shortcomings: approximate construction of the field pattern, a large amount of work in processing the resulting field curve, and so on. In addition, it should be noted that in all the listed works the coefficients \(k_d\), \(k_q\) were determined only for

first harmonic of the field curve. Therefore, we finally chose the analytical method for solving Laplace’s equation for the scalar magnetic potential (2) in the specified region of magnetic-field distribution under known boundary conditions.

\[ \frac{\partial^2 U(x,y)}{\partial x^2}+\frac{\partial^2 U(x,y)}{\partial y^2}=0. \tag{2} \]

The essence of the method is that the air-gap region of the SRM (Fig. 1) is divided into two rectangular parts, for each of which the solution of equation (2) is found by the method of G. A. Grinberg [8], and then, using the conditions of continuity of the function at the common boundary, the solution of the entire problem is found. Such a method is used in applied mechanics and, as applied to electric machines, was used by L. E. Dombur to study the magnetic field of the armature of an induction machine [9]. Since the method of investigation is a general one, we shall present only certain features and the final results, without dwelling on the intermediate derivations set forth in the literature [8, 9]. The commonly accepted assumptions [6] are made in the work.

Fig. 1. Air-gap region of a simple synchronous motor.

1. Transverse Field

The boundary conditions for the region in Fig. 1 in the case of a transverse field are:

Rectangle I

\[ U_1(x,-\delta)=U_{sm}\sin\frac{s\pi}{\tau}x. \]

\[ U_1(0,y)=U_1(\tau,y)=0, \tag{3} \]

\[ U_1(x,0)=f_1(x)= \begin{cases} 0, & 0<x<a,\\ f_2(x), & a<x<\tau-a,\\ 0, & \tau-a<x<\tau, \end{cases} \]

where \(s\) is the ordinal number of the harmonics of the stator m.m.f.

Rectangle II

\[ U_2(a,y)=U_2(\tau-a,y)=U_2(x,h)=0, \]

\[ U_2(x,0)=f_2(x). \tag{4} \]

The matching conditions at the common boundary

\[ U_1(x,0)=U_2(x,0), \]

\[ \left. \frac{\partial U_1(x,y)}{\partial y} \right|_{y=0} = \left. \frac{\partial U_2(x,y)}{\partial y} \right|_{y=0}, \tag{5} \]

where

\[ a < x < \tau - a. \]

The solution of the first boundary-value problem for equation (2), with boundary conditions (3) and (4), by Greenberg’s method gives the following expressions for the functions in rectangles I and II:

\[ U_1(x,y)= \sum_{n=1}^{\infty} \left( A_n \frac{\operatorname{sh}\frac{n\pi}{\tau}(y+\delta)} {\operatorname{ch}\frac{n\pi}{\tau}\delta} + d_{ns} \frac{\operatorname{ch}\frac{n\pi}{\tau}y} {\operatorname{ch}\frac{n\pi}{\tau}\delta} \right) \sin \frac{n\pi}{\tau}x, \tag{6} \]

\[ U_2(x,y)= \sum_{k=1}^{\infty} A_k \frac{\operatorname{sh}\frac{k\pi}{b}(y-h)} {\operatorname{ch}\frac{k\pi}{b}h} \sin \frac{k\pi}{b}(x-a), \tag{7} \]

where

\[ d_{ns}= \begin{cases} U_{sm}, & \text{for } n=s,\\ 0, & \text{for } n\ne s. \end{cases} \tag{8} \]

From the matching conditions (5) we obtain two equations:

\[ \sum_{n=1}^{\infty} \left( A_n \operatorname{th}\frac{n\pi}{\tau}\delta + \frac{d_{ns}}{\operatorname{ch}\frac{n\pi}{\tau}\delta} \right) \sin \frac{n\pi}{\tau}x = - \sum_{k=1}^{\infty} A_k \operatorname{th}\frac{k\pi}{b}h \cdot \sin \frac{k\pi}{b}(x-a), \tag{9} \]

\[ \sum_{n=1}^{\infty} A_n \frac{n\pi}{\tau}\sin \frac{n\pi}{\tau}x = \sum_{k=1}^{\infty} A_k \frac{k\pi}{b}\sin \frac{k\pi}{b}(x-a), \tag{10} \]

where

\[ a < x < \tau - a. \]

After transformations [9], these equations can be reduced to the form

\[ A_n \operatorname{th}\frac{n\pi}{\tau}\delta + \frac{d_{ns}}{\operatorname{ch}\frac{n\pi}{\tau}\delta} = - \sum_{k=1}^{\infty} A_k \operatorname{th}\frac{k\pi}{b}h \cdot d_{kn}, \tag{11} \]

\[ A_k \frac{k\pi}{b} = \sum_{n=1}^{\infty} A_n \frac{n\pi}{\tau}\cdot \frac{\tau}{b} d_{kn}, \tag{12} \]

where

\[ d_{kn} = \frac{ 2b\sin \frac{\pi}{2}\left(k-n\frac{b}{\tau}\right) \cos \frac{\pi}{2}(k-n) }{ \pi(k\tau-nb) } - \frac{ 2b\sin \frac{\pi}{2}\left(k+n\frac{b}{\tau}\right) \cos \frac{\pi}{2}(k+n) }{ \pi(k\tau+nb) } \quad \text{for } \frac{n\pi}{\tau}\ne \frac{k\pi}{b}, \tag{13} \]

\[ d_{kn} = \frac{b}{\tau}\cos \frac{\pi}{2}(k-n) \quad \text{for } \frac{n\pi}{\tau}= \frac{k\pi}{b}. \]

Substituting equation (12) into (11), we obtain an infinite system of equations for determining the coefficients

\[ A_n=-\sum_{m=1}^{\infty} m A_m \sum_{k=1}^{\infty}\frac{1}{k}\operatorname{th}\frac{k\pi}{b}h\cdot \operatorname{cth}\frac{n\pi}{\tau}\delta \cdot d_{kn}\cdot d_{km} -\frac{d_{ns}}{\operatorname{sh}\frac{n\pi}{\tau}\delta}. \tag{14} \]

The normal component of the magnetic-field intensity on the surface of the stator bore is determined from the equation

\[ H_{yn}=-\left.\frac{\partial U(x,y)}{\partial y}\right|_{y=-\delta} =-\sum_{n=1}^{\infty}\left( -\frac{A_n}{\operatorname{ch}\frac{n\pi}{\tau}\delta} -d_{ns}\operatorname{th}\frac{n\pi}{\tau}\delta \right)\frac{n\pi}{\tau}\sin\frac{n\pi}{\tau}x . \tag{15} \]

Equation (15) gives the final solution of the problem in the form of harmonics (\(n\) is the order of the harmonic) of the intensity; moreover, as the investigations have shown, the solution holds only for odd values of \(n\) and \(k\).

2. Longitudinal field

In the case of a longitudinal field, the conjugation conditions and boundary conditions for rectangle II remain the same, (4) and (5), while the boundary conditions for rectangle I have the form

\[ U_1(x_1-\delta)=U_{sm}\cos\frac{s\pi}{\tau}x, \]

\[ \left.\frac{\partial U_1(x,y)}{\partial x}\right|_{x=0} = \left.\frac{\partial U_1(x,y)}{\partial x}\right|_{x=\tau} =0 \tag{16} \]

\[ U_1(x,0)=f_1(x)= \begin{cases} 0, & 0<x<a,\\ f_2(x), & a<x<\tau-a,\\ 0, & \tau-a<x<\tau. \end{cases} \]

Thus, in this case for rectangle I a second boundary-value problem is solved. The expression for the function in rectangle II remains unchanged, that is, it is given by equation (7), while the solution of equation (2) under the boundary conditions (16) gives the following equation for rectangle I:

\[ U_1(x,y)=\sum_{n=1}^{\infty}\left( A_n\frac{\operatorname{sh}\frac{n\pi}{\tau}(y+\delta)} {\operatorname{ch}\frac{n\pi}{\tau}\delta} +d_{ns}\frac{\operatorname{ch}\frac{n\pi}{\tau}y} {\operatorname{ch}\frac{n\pi}{\tau}\delta} \right)\cos\frac{n\pi}{\tau}x. \tag{17} \]

Carrying out the same transformations as in the case of the transverse field, we obtain an infinite system of equations for determining the coefficients \(A_n\), differing from (14) only in the values of the coefficients \(d_{kn}\), which in this case will have the form:

\[ d_{kn}= \frac{ 2b\sin\frac{\pi}{2}\left(k-n\frac{b}{\tau}\right) \sin\frac{\pi}{2}(k-n) }{ \pi(k\tau-nb) } + \]

\[ +\frac{ 2b\sin\frac{\pi}{2}\left(k+n\frac{b}{\tau}\right) \sin\frac{\pi}{2}(k+n) }{ \pi(k\tau+nb) }, \qquad \text{for } \frac{n\pi}{\tau}\ne\frac{k\pi}{b}, \tag{18} \]

\[ d_{kn}=\frac{b}{\tau}\sin\frac{\pi}{2}(k-n), \qquad \text{for } \frac{n\pi}{\tau}=\frac{k\pi}{b}. \]

The normal component of the magnetic-field intensity on the surface of the stator bore is determined by the equation

\[ H_{yn}=\left.\frac{\partial И_1(x,y)}{\partial y}\right|_{y=-\delta} = -\sum_{n=1}^{\infty} \left( \frac{A_n}{\operatorname{ch}\frac{n\pi}{\tau}\delta} - d_{ns}\operatorname{th}\frac{n\pi}{\tau}\delta \right) \frac{n\pi}{\tau}\cos\frac{n\pi}{\tau}x . \tag{19} \]

As the investigations have shown, the solution holds only for odd values of \(n\) and even values of \(k\).

Calculation of coefficients characterizing the magnetic field in the air gap

In order to calculate the parameters and take into account the influence of the higher field harmonics on the operation of the synchronous reluctance motor, the field-form coefficients are determined for both the fundamental and the higher field harmonics as follows.

The field-form coefficients are

\[ K_{dn}=\frac{B_{adn}}{B_{ad}}=\frac{H_{yn}^{(d)}}{H_{y\max}}, \tag{20} \]

where \(H_{yn}^{(q)}\) is determined by equation (19);

\[ K_{qn}=\frac{B_{aqn}}{B_{aq}}=\frac{H_{yn}^{(q)}}{H_{y\max}}, \tag{21} \]

where \(H_{yn}^{(q)}\) is determined by equation (15).

Putting \(h=0\) in equations (14) and (15), that is, assuming the air gap to be uniform, we find the maximum value of the magnetic-field intensity

\[ H_{y\max}=\frac{S\pi}{\tau}\operatorname{cth}\frac{S\pi}{\tau}\delta . \tag{22} \]

To determine the coefficients \(\alpha_d, \alpha_q, k_{\phi d}, k_{\phi q}\) [6], it is necessary to determine the fluxes of the first harmonics \((\Phi_{d1}, \Phi_{q1})\), the total magnetic fluxes \((\Phi_d, \Phi_q)\), and also the maximum values of the induction \((B_{md}, B_{mq})\), which are determined from the relations [10]:

\[ \Phi_{d1}=\frac{2}{\pi}\tau l B_{ad1} = \frac{2}{\pi}\tau l\mu_0 H_{y1}^{(d)}, \tag{23} \]

\[ \Phi_{q1}=\frac{2}{\pi}\tau l B_{aq1} = \frac{2}{\pi}\tau l\mu_0 H_{y1}^{(q)}, \tag{24} \]

\[ \Phi_d=\Phi_{d1}\pm\Phi_{d3}\pm\ldots\pm\Phi_{dn} = \frac{2}{\pi}\tau l\mu_0 \left( H_{y1}^{(d)} \pm\frac{1}{3}H_{y3}^{(d)} \pm\ldots\pm\frac{1}{n}H_{yn}^{(d)} \right), \tag{25} \]

\[ \Phi_q=\Phi_{q1}\pm\Phi_{q3}\pm\ldots\pm\Phi_{qn} = \frac{2}{\pi}\tau l\mu_0 \left( H_{y1}^{(q)} \pm\frac{1}{3}H_{y3}^{(q)} \pm\ldots\pm\frac{1}{n}H_{yn}^{(q)} \right), \tag{26} \]

where \(l\) is the calculated length of the stator stack.

\[ B_{md}=\mu_0\sum_{n=1,3..}^{\infty} H_{yn}^{(d)} \quad \text{at } x=0, \tag{27} \]

\[ B_{mq}=\mu_0\sum_{n=1,3..}^{\infty} H_{yn}^{(q)} \quad \text{at } x=a. \tag{28} \]

We obtain the following expressions for calculating the coefficients of the magnetic field.

Calculated coefficients of the pole arc:

\[ \alpha_d=\frac{\Phi_d}{\tau l B_{md}} =\frac{2}{\pi}\, \frac{ \left( H_{y1}^{(d)} \pm \frac{1}{3}H_{y3}^{(d)} \pm \ldots \pm \frac{1}{n}H_{yn}^{(d)} \right) }{ \displaystyle \sum_{n=1,3,\ldots}^{\infty} H_{yn}^{(d)} \;(\text{for } x=0) }, \tag{29} \]

\[ \alpha_q=\frac{\Phi_q}{\tau l B_{mq}} =\frac{2}{\pi}\, \frac{ \left( H_{y1}^{(q)} \pm \frac{1}{3}H_{y3}^{(q)} \pm \ldots \pm \frac{1}{n}H_{yn}^{(q)} \right) }{ \displaystyle \sum_{n=1,3,\ldots}^{\infty} H_{yn}^{(q)} \;(\text{for } x=a) }. \tag{30} \]

Coefficients characterizing the first harmonics of the fluxes,

\[ K_{\phi d}=\frac{\Phi_d}{\Phi_{d1}} = \frac{ \left( H_{y1}^{(d)} \pm \frac{1}{3}H_{y3}^{(d)} \pm \ldots \pm \frac{1}{n}H_{yn}^{(d)} \right) }{ H_{y1}^{(d)} }, \tag{31} \]

\[ K_{\phi q}=\frac{\Phi_q}{\Phi_{q1}} = \frac{ \left( H_{y1}^{(q)} \pm \frac{1}{3}H_{y3}^{(q)} \pm \ldots \pm \frac{1}{n}H_{yn}^{(q)} \right) }{ H_{y1}^{(q)} }. \tag{32} \]

Calculations of the above coefficients were carried out on the M-20 electronic digital computer at \(s=1\) for the following parameter values,

\[ \alpha=0.2;\ 0.4;\ 0.5;\ 0.6;\ 0.8, \]

\[ \delta=0.001\,\tau;\ 0.003\,\tau;\ 0.005\,\tau;\ 0.01\,\tau;\ 0.02\,\tau, \]

\[ h=0.02\,\tau;\ 0.04\,\tau;\ 0.07\,\tau;\ 0.1\,\tau;\ 0.2\,\tau;\ 0.3\,\tau;\ 0.4\,\tau;\ 0.5\,\tau. \]

The studies showed that, in solving the infinite systems of equations (14), it is sufficient to limit oneself to the number of equations \(n=15\), since with a further increase in \(n\) the values of the harmonics of the field intensity practically do not change.

As the calculations showed, the field-form coefficients along the longitudinal axis \((k_d)\) for values \(h>0.04\,\tau\) change very little, whereas the analogous coefficients along the transverse axis \((k_q)\) for \(h<0.2\,\tau\) assume different values as \(h\) changes. As can be seen from Fig. 2, the coefficients \(k_q\) change more sharply when \(\alpha\) is decreased and the air gap is increased. Thus, for values \(h<0.2\,\tau\), the coefficients \(k_q\) must be calculated for each value of the pole height. For values \(h>0.2\,\tau\), these coefficients remain practically constant, and in calculations one may use the curves calculated for the value \(h=0.2\,\tau\).

The values of the coefficients \(k_d\), \(k_q\), \(\alpha_d\), \(\alpha_q\), \(k_{\phi d}\), \(k_{\phi q}\) at \(h=0.2\,\tau\) are presented in Figs. 3, 4, and 5 as functions of the pole overlap and for different values of the air gap.

An analysis of the harmonic composition of the magnetic-field intensity in the air gap of an SRM shows that the third harmonic component is most sharply expressed, and for certain parameter ratios the fifth and seventh harmonic components as well. Figure 6 presents the harmonic composition for the case \(\alpha=0.5\), \(\delta=0.01\,\tau\), \(h=0.2\,\tau\). Such a sharply expressed harmonic composition indicates the need to take their influence on the operation of the SRM into account; therefore Figs. 7 and 8 show the dependences of the field-form coefficients \(k_{dn}\), \(k_{qn}\) for the higher harmonics of the field curve on pole overlap at \(\delta=0.01\,\tau\) and \(h=0.2\,\tau\). In determining these coefficients, the calculation was carried out for values of \(\alpha\) at intervals of \(0.05\,\tau\).

Fig. 2. Dependence of the coefficients \(K_q\) on the pole height.

Fig. 3. Dependence of the coefficients \(K_d\), \(K_q\) on the pole overlap at \(h = 0.2\tau\).

Fig. 4. Dependence of the coefficients \(a_d\), \(a_q\) on pole coverage at \(h = 0.2\tau\).

Fig. 5. Dependence of the coefficients \(k_{\Phi d}\), \(k_{\Phi q}\) on pole coverage at \(h = 0.2\tau\).

a. Direct-axis field

b. Quadrature-axis field

Fig. 6. Harmonic composition of the field curve along the direct and quadrature axes at \(\alpha=0.5\);
\(\delta=0.01\tau\); \(h=0.2\tau\).

Fig. 7. Dependence of the coefficients \(k_{d\eta}\) on pole coverage,
at \(h=0.2\tau\); \(\delta=0.01\tau\).

![Graph with vertical axis $k_{qn}$ and horizontal axis $\alpha$; curves labeled 3, 5, 7, 9, 11, 13.]

Fig. 8. Dependence of the coefficients \(k_{qn}\) on the pole overlap
for \(h=0.2\tau;\ \delta=0.01\tau\).

Conclusions

1. The coefficients characterizing the magnetic field in the air gap of simple synchronous reluctance motors have been determined by an exact analytical method of solving Laplace’s equation in the region of magnetic-field distribution.

2. The solution is obtained directly in the form of voltage harmonics, which is a major advantage of the given method.

3. The pole height has a substantial influence on the field along the transverse axis for values \(h<0.2\tau\).

4. The sharply pronounced harmonic composition of the field curve indicates the need to take into account the influence of higher harmonics when calculating the operating characteristics of synchronous reluctance motors.

REFERENCES

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