On the Calculation of Magnetic Fields in Synchronous Reluctance Machines

E. V. Kononenko, A. N. Aifert

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

One of the basic elements in the design and calculation of electrical machines is the calculation of magnetic fields. Knowledge of magnetic fields makes it possible to calculate the magnetic circuit and the inductive reactances of the machine. However, in the theory of synchronous reluctance machines this question has not received a sufficiently satisfactory solution.

In the present article, questions are considered concerning the calculation of the magnetic field in the air gap of a synchronous reluctance machine having an ordinary salient-pole rotor and operating in a steady synchronous regime.

In investigations and calculations of synchronous reluctance machines, the two-reaction theory is used, which necessitates the application of the principle of superposition of magnetic fields. The latter is strictly valid only for unsaturated magnetic circuits. Therefore, when calculating magnetic fields in the air gap, the following assumptions are adopted:

1) the magnetic permeability of steel is infinitely large in comparison with the permeability of air \[1, 2\];

2) when allowing for saturation of the steel, it is assumed that saturation affects only the magnitude of the magnetic flux.

The reduction in the magnitude of the magnetic flux is taken into account in the same way as the slotting of the stator and rotor surfaces—by increasing the air gap by means of a saturation coefficient \[3\];

3) in calculations, only the first harmonic of the magnetomotive force (m.m.f.) produced by the windings is taken into account;

4) the magnetic field is regarded as plane-parallel.

Magnetic fields, depending on the configuration of the air gap, are usually characterized by various coefficients. These coefficients must determine the relation between the maximum induction in the air gap, the total magnetic flux and its first harmonic component, and also take into account the reduction of the magnetic flux due to the saliency of the rotor. The magnetic-field coefficients must be determined both along the direct and along the quadrature axes of the rotor.

The magnetic field in the air gap of a synchronous reluctance machine can be characterized by the following system of coefficients:

1. The calculation coefficients of the pole arc along the direct \((\alpha_d)\) and quadrature \((\alpha_q)\) axes, which determine the relation between the maximum in-

induction in the air gap and the total magnetic flux (Figs. 1 and 2)

\[ \alpha_d=\frac{\Phi_d}{\tau l B_{ad}};\qquad \alpha_q=\frac{\Phi_q}{\tau l B_{qm}}, \tag{1} \]

where \(\tau\) is the pole pitch,
\(l\) is the calculated length of the stator core.

Fig. 1. Pattern of the magnetic-field distribution in the air gap of a synchronous reluctance machine along the longitudinal axis.

Fig. 2. Pattern of the magnetic-field distribution in the air gap of a synchronous reluctance machine along the transverse axis.

1. The first-harmonic fluxes along the axes are determined by the coefficients

\[ \kappa_{\phi d}=\frac{\Phi_d}{\Phi_{d1}};\qquad \kappa_{\phi q}=\frac{\Phi_q}{\Phi_{q1}}. \tag{2} \]

1. Field form factors along the direct \((k_d)\) and quadrature \((k_q)\) axes, determining the reduction in the amplitude of the first harmonic of the field due to the saliency of the rotor

\[ k_d=\frac{B_{ad1}}{B_{ad}};\qquad k_q=\frac{B_{aq1}}{B_{aq}}. \tag{3} \]

The coefficients \(k_d, k_q\) of a synchronous reluctance machine are analogous to the same coefficients of a salient-pole synchronous machine. It is not difficult to establish that, under the adopted assumptions,

\[ k_{\Phi d}= \frac{\alpha_d \tau l B_{ad}} {\dfrac{2}{\pi}\tau l B_{ad1}} = \frac{\pi}{2}\cdot\frac{\alpha_d}{k_d}, \tag{4} \]

\[ k_{\Phi q}= \frac{\alpha_q \tau l B_{qm}} {\dfrac{2}{\pi}\tau l B_{aq1}} = \frac{\pi}{2}\cdot\frac{\alpha_q}{k_q}\sin\frac{\alpha\pi}{2}, \tag{5} \]

where it is assumed that

\[ B_{qm}=B_{aq}\sin\frac{\alpha\pi}{2}. \]

From equations (4), (5) it follows that, in order to calculate the magnetic field, of the six coefficients introduced above it is sufficient to determine only four \((\alpha_d,\alpha_q,k_d,k_q)\).

In order to calculate these coefficients, it is necessary to find the distribution pattern of the magnetic field in the air gap of the machine, which is described by Laplace’s equation. Under certain assumptions, an analytic solution of the field equation for synchronous reluctance machines can be obtained by means of the method of conformal mappings or eigenfunctions. For a rotor design of any complexity, the field pattern can be obtained by solving Laplace’s equation by the finite-difference method, using an automatic digital computer or an electrolytic tank, and also by carrying out experimental investigations of the field by the modeling method.

As a first approximation, in determining the coefficients one may use a simplified field pattern [4, 5]. In this case it is assumed that the lines of force run radially in the air-gap region and along a circular arc (with its center at the pole tip) in the interpolar space.

The induction at any point of the air gap along the direct axis in this case can be determined from the equation

\[ B_d=\frac{\mu_0}{\delta_x}F_{ad}\cos\frac{\pi}{\tau}x, \tag{6} \]

where \(F_{ad}\) is the amplitude of the m.m.f. along the direct axis,
\(\delta_x\) is the length of the lines of force in the air gap.

Under the assumptions made, \(\delta_x\) is taken equal to:

a) within the pole

\[ \delta_x=\delta'=\kappa_\delta\cdot\kappa_\mu\cdot\delta, \tag{7} \]

b) in the interpolar space

\[ \delta_x=\delta'+\varepsilon x_1, \tag{8} \]

where \(\delta\) is the actual value of the air gap;
\(\kappa_\delta\) is the air-gap coefficient;

\(\kappa_\mu\) is the saturation coefficient, equal to the ratio of the m.m.f. of the magnetic circuit to the m.m.f. of the air gap,

\[ x_1=x-\frac{a\tau}{2}; \]

\[ \varepsilon=\operatorname{arctg}\frac{4\delta_n}{(1-\alpha)\tau} \]
is an empirical coefficient.

The total magnetic flux along the direct axis is determined by the integral

\[ \Phi_d=2l\int_0^{\frac{\tau}{2}} B_d\,dx \]

and can be represented as the sum of two components; the first of them \((\Phi_d')\) is equal to the magnetic flux under the pole shoe, and the second \((\Phi_d'')\) to the flux in the interpole space:

\[ \Phi_d=\Phi_d'+\Phi_d'' = \frac{2l\mu_0 F_{ad}}{\kappa_\delta\cdot\kappa_\mu\cdot\delta} \int_0^{\frac{a\tau}{2}}\cos\frac{\pi}{\tau}x\cdot dx + \frac{2l\mu_0 F_{ad}}{\kappa_\delta\cdot\kappa_\mu\cdot\delta} \int_{\frac{a\tau}{2}}^{\frac{\tau}{2}} \frac{\cos\frac{\pi}{\tau}x\cdot dx}{1+\frac{\varepsilon}{\delta'}x_1}. \]

Carrying out term-by-term integration, we obtain

\[ \Phi_d'=\frac{2}{\pi}\tau l B_{ad}\cdot \sin\frac{a\pi}{2}, \tag{9} \]

\[ \Phi_d''=2lB_{ad}\frac{\delta'}{\varepsilon}\cdot A, \tag{10} \]

where

\[ A= \cos\left(\frac{a\pi}{2}-b\right) \left[ \operatorname{Ci}\left(\frac{1-a}{2}\pi+b\right)-\operatorname{Ci}(b) \right] - \sin\left(\frac{a\pi}{2}-b\right) \left[ \operatorname{Si}\left(\frac{1-a}{2}\pi+b\right)-\operatorname{Si}(b) \right]; \]

\[ b=\frac{\pi}{\tau}\cdot\frac{\delta'}{\varepsilon}; \qquad B_{ad}=\frac{\mu_0F_{ad}}{\kappa_\delta\cdot\kappa_\mu\cdot\delta}. \]

Then the calculated coefficient of the pole arc along the direct axis will have the form

\[ \alpha_d=\frac{2}{\pi}\sin\frac{a\pi}{2}+\frac{2\delta'}{\tau\varepsilon}A. \tag{11} \]

Expanding the induction curve (6) in a series, for the amplitude of the first harmonic we obtain

\[ B_{ad1}=\frac{4}{\tau}\mu_0F_{ad} \int_0^{\frac{\tau}{2}} \frac{\cos^2\frac{\pi}{\tau}}{\delta x}\,dx \]

or

\[ B_{ad1}=B_{ad1}'+B_{ad1}'' = \frac{4}{\tau}B_{ad} \int_0^{\frac{a\tau}{2}}\cos^2\frac{\pi}{\tau}x\cdot dx + \frac{4}{\tau}B_{ad} \int_{\frac{a\tau}{2}}^{\frac{\tau}{2}} \frac{\cos^2\frac{\pi}{\tau}x\cdot dx}{1+\frac{\varepsilon}{\delta'}x_1}. \]

Integrating, we obtain

\[ B'_{ad1}=B_{ad}\frac{\alpha\pi+\sin\alpha\pi}{\pi}, \tag{12} \]

\[ B''_{ad1}=B_{ad}\frac{2\delta'}{\tau\varepsilon} \left\{\ln\left[1+\frac{1-\alpha}{2b}\pi\right]+A_1\right\}, \tag{13} \]

where

\[ \begin{aligned} A_1={}&\cos 2\left(\frac{\alpha\pi}{2}-b\right) \left[\operatorname{Ci}2\left(\frac{1-\alpha}{2}\pi+b\right)-\operatorname{Ci}(2b)\right] \\ &-\sin 2\left(\frac{\alpha\pi}{2}-b\right) \left[\operatorname{Si}2\left(\frac{1-\alpha}{2}\pi+b\right)-\operatorname{Si}(2b)\right]. \end{aligned} \]

The field form factor along the longitudinal axis is

\[ k_d=\frac{B_{ad1}}{B_{ad}}= \frac{\alpha\pi+\sin\alpha\pi}{\pi} +\frac{2\delta'}{\tau\varepsilon} \left\{\ln\left[1+\frac{1-\alpha}{2b}\pi\right]+A_1\right\}. \tag{14} \]

The distribution of induction in the air gap along the transverse axis is determined by the equation

\[ B_q=\frac{\mu_0}{\delta_x}F_{aq}\sin\frac{\pi}{\tau}x. \tag{15} \]

The total magnetic flux along the transverse axis is equal to

\[ \Phi_q=2l\int_0^{\frac{\tau}{2}}B_q\,dx \]

or

\[ \Phi_q=\Phi'_q+\Phi''_q =2lB_{aq}\int_0^{\frac{\alpha\tau}{2}}\sin\frac{\pi}{\tau}x\cdot dx +2lB_{aq}\int_{\frac{\alpha\tau}{2}}^{\frac{\tau}{2}} \frac{\sin\frac{\pi}{\tau}x\cdot dx}{1+\frac{\varepsilon}{\delta'}x_1}, \]

where

\[ B_{aq}=\frac{\mu_0F_{aq}}{k_\delta\cdot k_\mu\cdot\delta}. \]

Integrating, we obtain

\[ \Phi'_q=\frac{2}{\pi}\tau lB_{aq}\left(1-\cos\frac{\alpha\pi}{2}\right), \tag{16} \]

\[ \Phi''_q=2lB_{aq}\frac{\delta'}{\varepsilon}A_2, \tag{17} \]

where

\[ \begin{aligned} A_2={}&\cos\left(\frac{\alpha\pi}{2}-b\right) \left[\operatorname{Si}\left(\frac{1-\alpha}{2}\pi+b\right)-\operatorname{Si}(b)\right] \\ &+\sin\left(\frac{\alpha\pi}{2}-b\right) \left[\operatorname{Ci}\left(\frac{1-\alpha}{2}\pi+b\right)-\operatorname{Ci}(b)\right]. \end{aligned} \]

The calculated coefficient of the pole arc along the transverse axis is

\[ \alpha_q= \frac{\Phi_q}{\tau\cdot lB_{aq}\sin\frac{\pi\alpha}{2}} =\frac{2}{\pi}\, \frac{1-\cos\frac{\alpha\pi}{2}}{\sin\frac{\alpha\pi}{2}} +\frac{2\delta'}{\tau\varepsilon}\cdot \frac{A_2}{\sin\frac{\alpha\pi}{2}}. \tag{18} \]

Expanding the induction curve (15) in a series, for the amplitude of the first harmonic we obtain

\[ B_{aq1}=\frac{4}{\pi}\mu_0 F_{aq}\int_0^{\frac{\tau}{2}} \frac{\sin^2 \frac{\pi}{\tau}x}{\delta_x}\,dx \]

or

\[ B_{aq1}=B'_{aq1}+B''_{aq1} =\frac{4}{\tau}B_{aq}\int_0^{\frac{\alpha\tau}{2}}\sin\frac{\pi}{\tau}x\cdot dx +\frac{4}{\tau}B_{aq}\int_{\frac{\alpha\tau}{2}}^{\frac{\tau}{2}} \frac{\sin^2\frac{\pi}{\tau}x\cdot dx}{1+\frac{\varepsilon}{\delta'}x_1}. \]

Carrying out the integration, we obtain

\[ B'_{aq1}=B_{aq}\frac{\alpha\pi-\sin\alpha\pi}{\pi}, \tag{19} \]

\[ B''_{aq1}=B_{aq}\frac{2\delta'}{\tau\varepsilon} \left\{ \ln\left[1+\frac{1-\alpha}{2b}\pi\right]-A_1 \right\}. \tag{20} \]

Then

\[ k_q=\frac{B_{aq1}}{B_{aq}} =\frac{\alpha\pi-\sin\alpha\pi}{\pi} +\frac{2\delta'}{\tau\varepsilon} \left\{ \ln\left[1+\frac{1-\alpha}{2b}\pi\right]-A_1 \right\}. \tag{21} \]

Fig. 3. Curves for determining the coefficients \(\alpha d\) and \(\alpha q\) for
\(\delta_n=0.2\tau\).

The equations given above make it possible to calculate all the magnetic-field coefficients of interest to us for various values of the air gap and of the rotor-pole configuration. As follows from equations (11), (14), (18), (21), each coefficient is represented as the sum of two components: the first component is determined by the magnetic flux under the poles, and the second by the flux in the interpolar space.

Fig. 4. Curves for determining the coefficients \(\kappa_d\) and \(\kappa_q\) at \(\delta_{\mathrm{п}} = 0.2\tau\).

In Fig. 3 the dependences \(\alpha_d\) and \(\alpha_q = f(\alpha_1, \delta')\) are presented for a constant height of the rotor pole. From the data given it follows that in synchronous reluctance machines \(\alpha_d\) is always less than \(\dfrac{2}{\pi}\), while \(\alpha_d > \alpha_q\).

The curves \(\kappa_d\) and \(\kappa_q = f(a_1\delta')\), shown in Fig. 4, give, in comparison with the data of [1], somewhat overestimated values. This is explained by the fact that in synchronous reluctance machines the pole height is comparatively small, whereas in [1] it is assumed to be infinitely large.

REFERENCES

1. M. P. Kostenko, B. E. Konik. Determination of the fundamental and third harmonics of the armature field and the pole field of a salient-pole synchronous machine. Elektrichestvo, No. 3, 1951.

2. T. G. Soroker, I. I. Gorzhevskii. Calculation of magnetic fields in the air gap of a salient-pole synchronous machine. Elektrichestvo, No. 6, 1952.

3. M. P. Kostenko, L. M. Piotrovskii. Electrical Machines. Part II, GEI, 1958.

4. Pohl, R. Theory of pulsating-field machines. Journ. IEE, 93, No. 31, 1946.

5. M. M. Langen. Two-speed synchronous-reluctance electric motor. VEI, No. 11, 1960.