Calculation of the Field of the Main Poles in the Commutation Zone of P-Series DC Machines
Abstract Generated abstract
The paper addresses calculation of the leakage field of main poles in the commutation zone of P-series DC machines, a factor important for commutation stability especially under deep speed regulation and magnetic saturation. It proposes an empirical method based on conductive-paper field modeling and a complete factorial experiment, deriving regression expressions for flux-tube height and upper width as functions of machine geometry and tube position. These expressions are combined with magnetic-reluctance formulas to compute induction in the neutral zone for both unsaturated and saturated magnetic systems when pole-tip potential distribution is known. Verification on a P51 machine using Hall sensors showed agreement within about 5 percent, and the authors recommend widening the interpole shoe to the commutation-zone width when necessary to reduce the main-pole field.
Full Text
CALCULATION OF THE MAIN POLE FIELD IN THE COMMUTATION ZONE OF P-SERIES DC MACHINES
A. I. Skorospelkin, E. G. Chebotkov
(Recommended by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
One of the factors affecting the commutation stability of DC machines is the leakage field of the main poles in the commutation zone. This factor has a particularly large influence on the commutation of machines with deep speed regulation, since such machines also operate under conditions of strong saturation. As a result, the need to take the main pole field into account when calculating commutation becomes evident.
For calculating magnetic fields in electrical machines, the following methods are used:
1) the mesh method,
2) the method of conformal mappings,
3) Greenberg’s method and its modifications.
The principal common drawback of these methods is their inapplicability to saturated magnetic systems. Each of the methods indicated also has other shortcomings.
The mesh method is complex and cumbersome. To obtain the field pattern in the commutation zone by this method, it is necessary to determine the potential of several thousand points.
The method of conformal mappings, when applied to DC machines, requires preliminary modeling.
Greenberg’s method and its modifications are applicable only for a rectangular magnetic system and a uniform air gap.
Below, another method is proposed for calculating the field in the commutation zone, applicable to P-series machines of frame sizes 1–6.
The direction chosen in developing the method is as follows: obtaining an empirical expression for the magnetic resistance as a function of the geometry of the machine. The problem was solved by modeling the field on conductive paper.
Modeling makes it possible to obtain the field pattern shown in Fig. 1. The flux tubes are constructed so that their lower width is identical and equal to
\[ b_1 = \frac{\pi D_a}{360}. \]
Measurements carried out on field models show that the width
Fig. 1
of the flux tubes changes according to a trapezoidal law from \(b_m\) to \(b_1\). Such a flux tube, in developed form, is shown in Fig. 2, where:
\[ \begin{aligned} b_m&\text{ — upper width of the flux tube,}\\ b_1&\text{ — lower width,}\\ h&\text{ — height,}\\ l_a&\text{ — length of the tube, equal to the armature length.} \end{aligned} \]
If the parameters of the flux tubes \(h\) and \(b_m\) were known, then the induction in the lower section of the tube could be calculated by the formula
\[ B=\frac{\Phi}{S_1}=\frac{F_{ns}}{R_\mu\cdot S_1}, \tag{1} \]
where \(R_\mu\) is the magnetic reluctance of the flux tube,
\[ R_\mu=\frac{1}{\mu_0\cdot l_a}\cdot \frac{h}{b_1-b_m}\cdot \ln \frac{b_1}{b_m}; \tag{2} \]
\(F_{ns}\) is the magnetizing force per pole,
\(S_1=b_1l_a\) is the lower section of the flux tube.
Consequently, it is necessary to obtain dependences as functions of the following parameters:
\[ h,\ b_m=\langle f(\delta_{kr\,g.p.};\ \frac{b_0}{2};\ \delta_{gn};\ b_{gn};\ N), \tag{3} \]
where
\[ \begin{aligned} \delta_{kr\,g.p.}&\text{ — air gap under the edge of the main pole,}\\ \delta_{gn}&\text{ — air gap under the interpole,}\\ b_0&\text{ — interpolar spacing,}\\ b_{gn}&\text{ — width of the interpole,}\\ N&\text{ — number of the flux tube when counted from the pole shoe of the main pole,}\\ \Phi&\text{ — flux of the flux tube.} \end{aligned} \]
The constructed field patterns and the changes in their flux tubes made it possible to assume a linear dependence of the output functions on the independent variables that determine the geometry of the machine. At the same time, the output functions vary with the flux-tube number
Fig. 2
along the broken line shown in Figs. 3a and 3b. Branch I corresponds to flux tubes from the end of the shoe of the main pole to the beginning of the interpole; branch II, from the beginning of the interpole to its middle. Therefore four output functions are considered:
\[ h_I = f\left(\delta_{kr\,gp}; \frac{b_0}{2}; \delta_{gn}; b_{gn}; N\right) \]
\[ b_{mI} = f\left(\delta_{kr\,gp}; \frac{b_0}{2}; \delta_{gn}; b_{gn}; N\right) \]
\[ h_{II} = f\left(\delta_{kr\,gp}; \frac{b_0}{2}; \delta_{gn}; b_{gn}; N\right) \tag{4} \]
\[ b_{mII} = f\left(\delta_{kr\,gp}; \frac{b_0}{2}; \delta_{ga}; b_{gn}; N\right). \]
On the basis of the indicated conditions, a plan was drawn up for a complete factorial experiment for four independent variables, varied at two levels according to [1], as presented in Table 1. The intervals of variation of the independent variables were chosen within the limits of the change in the geometry of machines of series II, from frame size 1 to frame size 6, and are presented in Table 2.
The modeling was carried out on high-resistance conducting paper with the aid of an EGDA 9/60 integrator [2]. The model was made at a scale of 12:1 in order to reduce the error due to nonuniformity of the paper resistance. The dimensions of the flux tubes were determined with a curvimeter having a scale division of 1 mm.
The results of the experiments are brought together in the corresponding columns of Table 1. Their processing made it possible to obtain the regression coefficients \(b_i\) for the output functions \(\overline{y}\).
Consideration of the obtained coefficients made it possible to neglect some of the interactions in order to simplify the final expres-
Table 1
| No. of experiment | \(x_0\) | \(x_1\) | \(x_2\) | \(x_3\) | \(x_4\) | \(x_1x_2\) | \(x_1x_3\) | \(x_1x_4\) | \(x_2x_3\) | \(x_2x_4\) | \(x_3x_4\) | \(x_1x_2x_3\) | \(x_1x_2x_4\) | \(x_1x_3x_4\) | \(x_2x_3x_4\) | \(x_1x_2x_3x_4\) | \(\overline{b}_1(y''_{\mathrm{II}})\) | \(\overline{b}_1(y'_{\mathrm{I}})\) | \(\overline{b}_1(y_{\mathrm{II}})\) | \(\overline{b}_1(y_{\mathrm{I}})\) | \(-y_{\mathrm{II}}10^2(h_1)\) | \(-y_{\mathrm{II}}10^2(h_{11})\) | \(-y'_{\mathrm{I}}10^3(b_{\mathrm{II}})\) | \(-y''_{\mathrm{II}}10^3(b_{\mathrm{III}})\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | 0.208 | 2.7 | 14.7 | 6.8 | 1.1 | 7.3 | 4.9 | 0.2 |
| 2 | + | − | + | + | + | − | − | − | + | + | + | − | − | − | + | − | −0.006 | −0.45 | −1.26 | 0.212 | 3.0 | 7.5 | 2.8 | 0.35 |
| 3 | + | + | − | + | + | − | + | + | − | − | + | − | − | + | − | − | −0.02 | 0.087 | 3.29 | 2.43 | 1.2 | 18.0 | 4.6 | 0.2 |
| 4 | + | − | − | + | + | + | − | − | − | − | + | + | + | − | − | + | 0.075 | 0.125 | 0.137 | −0.025 | 3.2 | 16.0 | 3.8 | 0.2 |
| 5 | + | + | + | − | + | + | − | + | − | + | − | − | + | − | − | − | −0.167 | −1.52 | 2.3 | 4.74 | 1.0 | 8.4 | 5.3 | 0.7 |
| 6 | + | − | + | − | + | − | + | − | − | + | − | + | − | + | − | + | −0.0005 | 0.06 | −0.312 | −0.288 | 2.8 | 7.5 | 3.2 | 0.5 |
| 7 | + | + | − | − | + | − | − | + | + | − | − | + | − | − | + | + | −0.026 | 0 | −0.137 | −0.08 | 1.2 | 18.2 | 4.9 | 0.45 |
| 8 | + | − | − | − | + | + | + | − | + | − | − | − | + | + | + | − | 0.006 | 0.25 | −0.075 | −0.725 | 3.0 | 16.4 | 4.3 | 0.4 |
| 9 | + | + | + | + | − | + | + | − | + | − | − | + | − | − | − | − | −0.008 | 0.037 | −0.087 | −0.05 | 6.4 | 15.0 | 1.3 | 0.01 |
| 10 | + | − | + | + | − | − | − | + | + | − | − | − | + | + | − | + | 0.041 | −0.15 | −1.45 | 2.34 | 6.9 | 15.0 | 1.1 | 0.02 |
| 11 | + | + | − | + | − | − | + | − | − | + | − | − | + | − | + | + | −0.06 | −0.075 | −0.07 | 0.037 | 17.4 | 20.0 | 1.3 | 0.04 |
| 12 | + | − | − | + | − | + | − | + | − | + | − | + | − | + | + | − | 0.019 | −0.062 | 0.23 | 0.012 | 15.4 | 17.8 | 0.8 | 0.04 |
| 13 | + | + | + | − | − | + | − | − | − | − | + | − | − | + | + | + | −0.0006 | −0.288 | 0.07 | −0.25 | 7.0 | 16.3 | 1.0 | 0.03 |
| 14 | + | − | + | − | − | − | + | + | − | − | + | + | + | − | + | − | 0.023 | −0.02 | −0.02 | −0.05 | 6.8 | 14.4 | 1.3 | 0.025 |
| 15 | + | + | − | − | − | − | − | − | + | + | + | + | + | + | − | − | 0.016 | −0.037 | −0.02 | −0.06 | 17.4 | 19.6 | 1.9 | 0.085 |
| 16 | + | − | − | − | − | + | + | + | + | + | + | − | − | − | − | + | −0.018 | −0.087 | 0.075 | 0.025 | 15.0 | 18.0 | 0.7 | 0.08 |
Table 2
| \(X_1\) | \(X_2\) | \(X_3\) | \(X_{4\mathrm{I}}\) | \(X_{4\mathrm{II}}\) | |
|---|---|---|---|---|---|
| Main level \((x_i=0)\) | \(1.846\cdot 10^{-2}\) | \(9.5\) | \(1.231\cdot 10^{-2}\) | \(\left(1+\dfrac{b_0-b_{\mathrm{gn}}\cdot 180}{\pi D_a}\right)^{1/2}\) | \(\left(1+\dfrac{2b_0-b_{\mathrm{gn}}\cdot 180}{\pi D_a}\right)^{1/2}\) |
| Variation interval | \(1.231\cdot 10^{-2}\) | \(3.5\) | \(0.615\cdot 10^{-2}\) | \(\left(\dfrac{b_0-b_{\mathrm{gn}}\cdot 180}{\pi D_a}-1\right)^{1/2}\) | \(\left(\dfrac{b_{\mathrm{gn}}\cdot 180}{\pi D_a}-1\right)^{1/2}\) |
| Upper level \((x_i=\pm 1)\) | \(3.08\cdot 10^{-2}\) | \(13\) | \(1.846\cdot 10^{-2}\) | \(\dfrac{b_0-b_{\mathrm{gn}}\cdot 180}{\pi D_a}\) | \(\dfrac{b_0}{\pi D_a}\cdot 180\) |
| Lower level \((x_i=-1)\) | \(0.615\cdot 10^{-2}\) | \(6\) | \(0.616\cdot 10^{-2}\) | \(1\) | \(\dfrac{b_0-b_{\mathrm{gn}}\cdot 180+1}{\pi D_a}\) |
Thus, the following dependences were obtained for the parameters of the flux tubes:
\[
\frac{h_{\mathrm{I}}}{D_a}\cdot 10^2
= 6.8 + 0.212x_1 + 2.43x_2 + 4.74x_4 + 0.288x_1x_2 - 0.725x_1x_4
\]
\[
{}+ 2.34x_2x_4 - 0.25x_1x_2x_4,
\tag{5}
\]
\[
\frac{b_{m\mathrm{I}}}{D_a}\cdot 10^3
= 2.7 - 0.45x_1 + 0.125x_3 - 1.52x_4 + 0.25x_1x_4 - 0.15x_2x_4
\]
\[
{}- 0.288x_1x_2x_4,
\tag{6}
\]
\[
\frac{h_{\mathrm{II}}}{D_a}\cdot 10^2
= 14.7 - 1.26x_1 + 3.29x_2 + 0.137x_3 + 2.3x_4 - 0.312x_1x_2
\]
\[
{}- 1.45x_2x_4 + 0.238x_1x_2x_3,
\tag{7}
\]
\[
\frac{b_{m\mathrm{II}}}{D_a}\cdot 10^3
= 0.208 - 0.212x_2 + 0.076x_3 - 0.167x_4 - 0.026x_1x_3
\]
\[
{}+ 0.04x_2x_4 - 0.06x_3x_4 + 0.019x_1x_2x_3 + 0.024x_1x_3x_4 + 0.016x_2x_3x_4
\]
\[
{}- 0.08x_1x_2x_3x_4,
\tag{8}
\]
where
\[ x_1 = \frac{ \dfrac{\delta_{\mathrm{kp\,gn}}}{D_a} - 1.848\cdot 10^{-2} }{ 1.232\cdot 10^{-2} }, \]
\[ x_2 = \frac{ \dfrac{b_0-b_{\mathrm{gn}}}{\pi D_a}\,180 - 9.5 }{ 3.5 }, \]
\[ x_3 = \frac{ \dfrac{\delta_{\mathrm{gn}}}{\pi D_a} - 1.23\cdot 10^{-2} }{ 0.615\cdot 10^{-2} }, \]
\[ x_{4\mathrm{I}} = \frac{ N - \left(1+\dfrac{b_0-b_{\mathrm{gn}}}{\pi D_a}\,180\right)\dfrac{1}{2} }{ \left(\dfrac{b_{\mathrm{gn}}}{\pi D_a}\,180 - 1\right)\dfrac{1}{2} }, \]
\[ x_{4\mathrm{II}} = \frac{ N - \left(\dfrac{2b_0-b_{\mathrm{gn}}}{\pi D_a}\,180 + 1\right)\dfrac{1}{2} }{ \left(\dfrac{b_{\mathrm{gn}}}{\pi D_a}\,180 - 1\right)\dfrac{1}{2} }. \]
At the same time, the experiments carried out revealed certain features of the flux tubes under the auxiliary pole and of the tube located directly next to it.
The flux tube located directly next to the auxiliary pole changes its cross section from \(b_m\) to \(b_1\) over the segment \(0.85h\), and over the remaining height from \(0.5b_1\) to \(b_1\).
The flux tubes located under the auxiliary pole practically retain their upper width over the segment \(h_{N-3}\), equal to the height of the flux tube that precedes the given one by three numbers. Over the remaining segment of the height \(h_N - h_{N-3}\), it varies according to a trapezoidal law from \(b_m\) to \(b_1\).
Fig. 3
Thus, the expressions obtained, (5–8), make it possible to determine the parameters of the flux tubes required for calculating the field of the main poles in the neutral zone (1–2).
In setting up the experiments, a check was made of the influence of saturation of the armature tooth zone and of the pole-shoe tip on the field pattern. Saturation of the tooth zone under the main pole does not have a noticeable effect on the field pattern.
Saturation of the main-pole shoe—a decrease in the potential of its tip relative to the pole potential by as much as 50 percent—leads to a change in the field pattern and in the parameters of the flux tubes \(h\) and \(b_m\) by no more than 10 percent. Consequently, in this case as well, relations (1–8) are suitable for calculating the field, provided that the potential distribution over the pole tip is known. It is possible to obtain the potential distribution over the pole tip and to calculate the saturated magnetic system using expressions (1–8) by applying Benedikt’s nomographic method [3].
To verify the reliability of relations (1–8), an experiment was carried out to determine the induction in the pole gap of the P51 machine with the following geometric parameters: \(\delta_{\mathrm{kr}\, n} = 2.5\) mm, \(\delta_{gn} = 2.5\) mm, \(b_{gn} = 22\) mm, \(D_a = 162.5\) mm.
The induction distribution was recorded using Hall emf sensors mounted on an armature tooth. The experimental data are shown in Fig. 4 as points.
Using relations (1–8), the field in the neutral zone was calculated for this machine. The calculation results are presented in Fig. 4 in the form
Fig. 4
curves, where curve I corresponds to an excitation current of 0.2 A, and II to an excitation current of 0.4 A.
Analysis of the data presented in Fig. 4 shows that the maximum deviation between the calculated and experimental data does not exceed 5 percent.
Thus, analytical expressions have been obtained that make it possible to calculate the field of the main poles in the commutation zone for an unsaturated and saturated magnetic system.
The results of the calculation and the experiments make it possible to conclude that the field of the main poles under the interpoles is small. However, the width of the commutation zone is usually chosen somewhat greater than the width of the interpole shoe and, consequently, the field of the main poles in the commutation zone may be significant. Therefore, when \(b_k > b_{gn}\), a sufficient means of reducing the field of the main poles in the commutation zone that can be recommended is increasing the width of the interpole shoe \(b_{gn}\) to the width of the commutation zone \(b_k\).
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V. V. Nalimov, I. A. Chernova. Statistical Methods for Planning Extreme Experiments. “Nauka,” Moscow, 1965.
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P. F. Filchakov, V. I. Panchishin. EGDA Integrators. Kiev, 1961.
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O. V. Benedikt. Nomographic Method for Calculating Complex Highly Saturated Magnetic Circuits of Electrical Machines. GEI, 1953.