Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov
Vol. 228
1974

COEFFICIENTS OF THE CALCULATED POLE ARC OF ALTERNATING-PULSATING MACHINES WITH MAGNETICALLY SOFT POLE SHOES

Yu. N. KRONEBERG, A. S. ZHIBINOV, V. B. GOMZYAKOV

(Presented to the scientific and technical seminar of the Department of Electrical Equipment, AEMF)

Among electric machines with combined excitation, a considerable group of machines \([1 \div 6\ \text{et al.}]\) is distinguished in which regulation is carried out by varying the flux of poles of the same name. One of the typical pole systems of such machines consists of alternating bars of permanent magnets, magnetized in one direction, and magnetically soft poles (varieties with claw-shaped poles are also known). The initial excitation is provided by the magnets, while regulation is effected by changing the flux of the magnetically soft poles; in this case the flux of the magnets changes only slightly. To increase the depth of regulation, the magnetically soft poles may be made wider than the magnets. In the general case the excitation field has constant and alternating components (as in pulsating machines) and at the same time is alternating in sign (as in alternating machines).

As studies \([4 \div 6]\) show, alternating-pulsating machines are one of the promising varieties of machines with combined excitation, and prerequisites are being created for their introduction as sources of increased-frequency current.

The noted specificity of the field requires refinement of methods for calculating certain parameters, including the coefficient of the calculated pole arc \(\alpha_i\), which is usually defined as the ratio of the mean induction of the excitation field \(B_{\delta 0}\) over the pole pitch to the maximum induction \(B_{\delta m}\) of this field above the middle of the pole:

\[ \alpha_i=\frac{B_{\delta 0}}{B_{\delta m}} = \frac{1}{\pi B_{\delta m}} \int_{-\vartheta_0}^{+\vartheta_0} B_\delta(\vartheta)\,d\vartheta, \tag{1} \]

where \(B_\delta(\vartheta)\) is the law of variation of the air-gap induction along the armature bore;

\(-\vartheta_0,\ +\vartheta_0\) are the angular coordinates of the points at which the induction passes through zero when the origin of coordinates is located on the pole axis.

In alternating-pulsating machines, unlike alternating machines, where the limits of integration are known in advance \((-\pi/2\) and \(+\pi/2)\), the position of the point at which the induction passes through zero depends on the pole dimensions and on the degree of excitation of the machine. This makes it necessary to compute \(\alpha_i\) separately for poles of different polarity.

Let us consider an alternating-pulsating machine with poles of constant width (Fig. 1), in which the permanent magnets are provided with pole shoes. When the inductions of the narrow \(B_{\delta mN}\) (we shall conventionally take it as northern) and wide \(B_{\delta mS}\) (southern) poles are equal, i.e., when the excitation degree

\[ \beta=\frac{B_{\delta mS}}{B_{\delta mN}}=1, \]

the zero value of the air-gap induction will coincide with the middle of the interpolar gap [7]. When \(\beta>1\), the point at which the induction passes through 0 is displaced toward the narrow pole, and when \(\beta<1\), toward the wide pole (points \(\vartheta'_0\) and \(\vartheta''_0\), Fig. 1,b).

In Fig. 1: “permanent magnet”; “magnetically soft pole.”

Fig. 1. Principal dimensions and coordinates of the pole system for calculating \(\alpha_i\):
a) pole system; b) field in the air gap; c) coordinates of variation of the relative specific magnetic permeance

The distribution of induction over the armature surface by the method of magnetic permeances [8, 9] is expressed through the relative specific magnetic permeances of the northern \(\lambda_{\delta N}\) and southern \(\lambda_{\delta S}\) poles, multiplied respectively by the m.m.f. drop in the air gap between the northern pole and the armature \((U_N-U_{\text{arm}})\) and between the southern pole and the armature \((U_S-U_{\text{arm}})\):

\[ B_{\delta}(\vartheta)=\Lambda_{\delta0}\left[\lambda_{\delta N}(U_N-U_{\mathrm{ya}})+\lambda_{\delta S}(U_S-U_{\mathrm{ya}})\right], \tag{2} \]

where \(\Lambda_{\delta0}\) is the magnetic permeance of the field at the middle of the pole. Taking into account that

\[ U_{\mathrm{ya}}-U_S=\beta(U_N-U_{\mathrm{ya}}), \tag{3} \]

we obtain:

\[ B_{\delta}(\vartheta)=\Lambda_{\delta0}(U_N-U_{\mathrm{ya}})(\lambda_{\delta N}-\beta\lambda_{\delta S}). \tag{4} \]

The maximum value of the induction under the north pole is

\[ B_{\delta mN}=\Lambda_{\delta0}(U_N-U_{\mathrm{ya}}). \tag{5} \]

Substituting (4) and (5) into (1), we obtain the coefficient of the calculated pole arc of the north pole

\[ \alpha_{iN}=\frac{2}{\pi}\int_{0}^{\vartheta_0}(\lambda_{\delta N}-\beta\lambda_{\delta S})\,d\vartheta, \tag{6} \]

where the limit of integration is determined from the solution of the equation

\[ \lambda_{\delta N}-\beta\lambda_{\delta S}=0. \tag{7} \]

The expressions for the relative specific magnetic permeances obtained in [9] for the equipotential surface of the poles can also be applied to the machines under consideration, since the pole shoes are equipotential surfaces at the full magnetic potential of the permanent magnet, and their height has little effect on the distribution of induction on the armature surface [10].

In the gap under the poles, between the lines \(KK_1\) drawn sufficiently far from the pole edges at the north pole and \(LL_1\) at the south pole (Fig. 1), it may be assumed with a high degree of accuracy that the field is uniform [9] and that \(\lambda_{\delta N}\), \(\lambda_{\delta S}\) are determined by the expressions

\[ \lambda_{\delta N}= \begin{cases} 1, & (-\vartheta_K<\vartheta<\vartheta_K),\\[4pt] \dfrac{q^4-t^2}{t(q^4-1)}, & (\vartheta_K<\vartheta<\vartheta_L),\\[8pt] 0, & (\vartheta_L<\vartheta<2\pi-\vartheta_L). \end{cases} \tag{8a} \]

\[ \lambda_{\delta S}= \begin{cases} 0, & (-\vartheta_K<\vartheta<\vartheta_K),\\[4pt] \dfrac{q^2(t^2-1)}{t(q^4-1)}, & (\vartheta_K<\vartheta<\vartheta_L),\\[8pt] 1, & (\vartheta_L<\vartheta<2\pi-\vartheta_L), \end{cases} \tag{8b} \]

where the constant \(q\) is determined by the ratio of the distance between the poles \(a\) to the air-gap magnitude \(\delta\)

\[ q=\frac{a}{2\delta}+\sqrt{\left(\frac{a}{2\delta}\right)^2+1}. \tag{9} \]

The parameter \(t\) is related to the angular coordinate \(\vartheta\), measured from the middle of the north pole, by the expression

\[ \vartheta=\frac{\pi}{2}-(1-\alpha_{\mathrm{sr}})\left[ \frac{\pi}{2} -2\arctan\frac{t}{q} +\frac{\delta}{a}\ln\frac{(t+1)(q^2-t)}{(t-1)(q^2+t)} \right], \tag{10} \]

where \(\vartheta_K\) and \(\vartheta_L\) are the angular coordinates of the lines \(KK_1\) and \(LL_1\);

\(\tau\) and \(\pi\) are the pole pitch in linear and angular units. Solving (7) with allowance for (8a, b), we find the value of the parameter \(t_0\) corresponding to the angular coordinate \(\vartheta_0\),

\[ t_0=q\sqrt{\frac{q^2+\beta}{\beta q^2+1}}. \tag{11} \]

Expression (6) for the narrow (northern) pole takes the form

\[ \alpha_{i1}=\alpha_{iN} = \frac{2}{\pi}\int_{0}^{\vartheta_K} d\vartheta + \frac{2}{\pi}\int_{\vartheta_K}^{\vartheta_0} \frac{q^4-t^2}{t(q^4-1)}\,d\vartheta - \frac{2}{\pi}\beta\int_{\vartheta_K}^{\vartheta_0} \frac{q^2(t^2-1)}{t(q^4-1)}\,d\vartheta . \tag{12} \]

After replacing, in the second and third integrals, the variable of integration \(\vartheta\) by \(t\) and carrying out transformations, we obtain in final form:

\[ \alpha_{i1} = \alpha_1+ \frac{2}{\pi}(1-\alpha_{\mathrm{cp}}) \left\{ \operatorname{arctg}\frac{2\delta}{a} + \frac{\delta}{a} \left[ \ln\left(1+\frac{a^2}{4\delta^2}\right) + \beta\ln\beta - (\beta+1)\ln(\beta+1) \right] \right\}. \tag{13a} \]

By analogous calculations we obtain the expression for the wide pole:

\[ \alpha_{i2} = \alpha_2+ \frac{2}{\pi}(1-\alpha_{\mathrm{cp}}) \left\{ \operatorname{arctg}\frac{2\delta}{a} + \frac{\delta}{a} \left[ \ln\left(1+\frac{a^2}{4\delta^2}\right) + \ln\beta - \frac{\beta+1}{\beta}\ln(\beta+1) \right] \right\}. \tag{13б} \]

In formulas (13a, b) \(\alpha_{\mathrm{cp}}=\dfrac{\alpha_1+\alpha_2}{2}\) is the average value of the pole-arc coefficient.

In particular, for \(\alpha_1=\alpha_2\) and \(\beta=1\), expressions (13a, b) coincide and determine the value of the calculated pole-arc coefficient of an alternating machine

\[ \alpha_i = \alpha+ \frac{2}{\pi}(1-\alpha) \left[ \operatorname{arctg}\frac{2\delta}{a} + \frac{\delta}{a}\ln\left(\frac14+\frac{a^2}{16\delta^2}\right) \right]. \tag{14} \]

Expressions (13a, b) can be transformed:

\[ \alpha_{i1}=\alpha_1+\Delta\alpha_\delta+\frac{\delta}{\tau}\Delta_{1\beta}, \tag{15a} \]

\[ \alpha_{i2}=\alpha_2+\Delta\alpha_\delta+\frac{\delta}{\tau}\Delta_{2\beta}, \tag{15б} \]

where

\[ \Delta\alpha_\delta = \frac{2}{\pi}(1-\alpha_{\mathrm{cp}}) \left\{ \operatorname{arctg} \frac{2\dfrac{\delta}{\tau}}{1-\alpha_{\mathrm{cp}}} + \frac{\dfrac{\delta}{\tau}}{1-\alpha_{\mathrm{cp}}} \ln \left[ \frac14+ \left( \frac{1-\alpha_{\mathrm{cp}}}{4\dfrac{\delta}{\tau}} \right)^2 \right] \right\}, \tag{16} \]

\[ \Delta_{1\beta} = \frac{2}{\pi}\,[2\ln2+\beta\ln\beta-(1+\beta)\ln(1+\beta)]. \tag{17a} \]

\[ \Delta_{2\beta}=\frac{2}{\pi}\left[2\ln 2+\ln \beta-\frac{1+\beta}{\beta}\ln(1+\beta)\right]. \tag{17б} \]

The term \(\Delta\alpha_{\delta}\) takes into account the influence of the magnetic field extending beyond the pole shoe when the inductions under the poles are identical, i.e., \(\beta=1\). Like the second term in formula (14), it is determined by the distance between the poles and by the magnitude of the air gap.

Fig. 2. Dependences \(\Delta\alpha_{\delta}=f(\alpha)\): — calculated by formula (16); -------- calculated according to [11].

Figure 2 shows the dependences of the term \(\Delta\alpha_{\delta}\) on the value \(\alpha_{\mathrm{cp}}\) for the entire practically used range of \(\delta/\tau\). For comparison, the same figure also gives the dependences of the analogous term obtained from the approximate formula [11], derived on the basis of graphical constructions of the magnetic field in the air gap. It is seen from the figure that, for average values of \(\delta/\tau\), the indicated dependences nearly coincide. The maximum difference in the value of \(\alpha_i\), calculated from the formulas given here and from [11], is \(1\%\).

The coefficients \(\Delta_{1\beta}\) and \(\Delta_{2\beta}\) take into account the change in the magnetic field beyond the limits of the pole shoe when \(\beta\) deviates from 1. Figure 3 presents the dependences of these coefficients as \(\beta\) varies from 0 to 2, making it possible to analyze the change in the coefficient of the calculated pole arc when regulating an alternating-pulsating machine. They show that under weak excitation of the machine (or in the absence of excitation), as well as under very strong excitation, the change in \(\alpha_i\)

Fig. 3. Dependences $\Delta_1\beta=f(\beta)$, $\Delta_2\beta=f(\beta)$

can reach a significant value, and it must be taken into account. For example, in a machine with $\alpha_{\mathrm{av}}=0.6$ with a large air gap $(\delta/\tau=0.05)$ and weak excitation $(\beta=0.2)$, the changes in $\alpha_{i1}$ and $\alpha_{i2}$ reach 6%. With insignificant differences between the inductions of the narrow and wide poles $(0.8\leq \beta \leq 1.2)$, the change in $\alpha_i$ does not exceed 1%, and it may be neglected.

In alternating-pulsating machines with poles of variable width [3], the calculation of $\alpha_{i1}$ and $\alpha_{i2}$ may be carried out using formulas (15a, b), (16), (17a, b); in this case the value $\delta/\tau$ must be taken with allowance for the lateral bevel of the poles, analogously to [7].

Thus, the analytical solutions obtained make it possible to compute \(\alpha_i\) for alternating-pulsating machines with magnetically soft pole shoes, and the plotted graphical relationships are suitable for use in practical calculations.

REFERENCES

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