Calculation of the Field of Rectangular Interpoles in DC Machines
Abstract Generated abstract
This paper addresses calculation of the magnetic field of rectangular interpoles in DC machines, motivated by the importance of the commutating electromotive force and the difficulty caused by a nonuniform air gap and significant leakage flux. The authors develop an empirical method based on field modeling with conductive paper, deriving second-order regression expressions for flux-tube parameters under and outside the interpole as functions of machine geometry for the P1 to P6 frame-size series. Verification on a P51 machine using a Hall-emf sensor showed agreement between calculated and measured induction distributions within about 5 percent. The paper also proposes a difference-of-exponentials approximation for representing the interpole field in commutation calculations, with a reported discrepancy of about 3 to 4 percent for the tested case.
Full Text
CALCULATION OF THE FIELD OF RECTANGULAR INTERPOLES IN DC MACHINES
E. G. Chebotkov, A. I. Skorospelkin
(Recommended by the scientific seminar of the departments of electrical machines and general electrical engineering)
One of the most important factors determining the nature of the commutation process is the commutating e.m.f. induced by the field of the interpoles in the short-circuited section. At the same time, calculation of the interpole field is difficult, since the air gap beneath it is nonuniform and, moreover, the leakage field of the pole is of great importance, because the commutation zone is usually chosen to be somewhat wider than the interpole. Hence it is clear how necessary the calculation of interpole fields is.
Below, a method is proposed for calculating the field of rectangular interpoles as applied to machines of the П1—6 frame-size series.
The direction chosen for developing the method is as follows: obtaining an empirical expression for the magnetic reluctance as a function of the geometry of the machine. The problem was solved by modeling the field on conducting paper.
Modeling makes it possible to obtain the field pattern shown in Fig. 1. The flux tubes are constructed in such a way that their lower width is identical and equal to
\[ b_1=\frac{\pi D_a}{360}. \]
Fig. 1
Measurements carried out on field models show that the width of a tube varies according to a trapezoidal law from \(b_m\) to \(b_1\), where \(b_m\) is the upper width of the flux tube. Knowing the parameters of the flux tube—the height \(h\), and the upper and lower widths \(b_m\) and \(b_1\)—one can, from expressions (1, 2) [1], calculate the induction in the lower cross section of the tube.
Thus it is necessary to obtain the dependences of the parameters of the flux tubes on the geometry of the machine.
Since it is necessary to calculate the field both under the interpole and outside it, the problem of obtaining the parameters of the flux tubes as functions of the geometry is, correspondingly, divided into two parts:
1) determination of the parameters of the flux tubes located under the interpole;
2) determination of the parameters of the flux tubes outside the interpole.
Preliminary experiments performed in constructing the field of the interpole made it possible to identify the factors on which the parameters of the flux tubes depend.
The flux tubes located under the interpole have practically the same width along their height, except for the last one, which rests on the edge of the interpole. The upper width of this tube is equal to the width of the following one.
The height of the flux tubes located under the interpole depends on two factors—the magnitude of the gap under the center of the interpole, \(\delta_w\), and the angle \(\varphi\) between the center of the interpole and the center of the flux tube being studied.
The parameters of the flux tubes outside the interpole depend on three factors:
1) the air gap under the edge of the interpole, \(\delta_{\mathrm{kr}}\);
2) the angle between the edges of the interpole and the main pole, \(\varphi\);
3) the angle between the edge of the interpole and the center of the flux tube being studied, \(\gamma\).
On the basis of the indicated conditions, two plans of a second-order rotatable experiment were drawn up for two and three independent variables. The intervals of variation of the independent variables were chosen within the limits of the change in the geometry of machines of the P series from sizes 1 to 6.
The modeling was carried out on high-resistance conductive paper by means of the EGDA 9/60 integrator [2]. The model was made on a scale of \(20:1\) in order to reduce the error due to the nonuniformity of the paper resistance. The dimensions of the flux tubes were determined with a curvimeter having a scale division of \(1\ \mathrm{mm}\).
Processing of the experimental results made it possible to obtain the following dependences for the parameters of the flux tubes:
1) under the interpole
\[ \frac{h_1}{D_a}\cdot 10^3 = 17 + 7{,}26x_1 + 2{,}65x_2 + 0{,}12x_1^2 + 0{,}9x_2^2 + 0{,}0437x_1x_2, \]
where
\[ x_1 = 141{,}5\,\frac{\delta_w}{D_a} - 2{,}12 \]
\[ x_2 = 0{,}315\varphi - 1{,}57; \tag{1} \]
2) outside the interpole
\[ \frac{h_{II}}{D_a}\cdot 10^2 = 6 + 1{,}47x_1 + 0{,}1x_2 + 1{,}1x_3 + 0{,}29x_1^2 + 0{,}02x_2^2 + 0{,}24x_3^2 - \]
\[ {}- 0{,}085x_1x_2 - 0{,}507x_1x_3 + 0{,}1x_2x_3 \tag{2} \]
\[ \frac{b_{\mathrm{II}}}{D_a}\cdot 10^3 =5-0{,}4x_1+0{,}25x_2-0{,}7x_3+0{,}2x_1^2+0{,}27x_1x_2-0{,}24x_1x_3, \tag{3} \]
where
\[ x_1=44{,}9\,\frac{\delta_{\mathrm{cr}}}{D_a}-2{,}24 \]
\[ x_2=0{,}42\psi-4{,}2, \]
\[ x_3=0{,}60\gamma-2{,}02. \]
The expressions obtained (1–3) make it possible to determine the parameters of the flux tubes required for calculating the field of the interpoles in the neutral zone (1–2) [1].
To verify the reliability of dependences (1–3), an experiment was carried out to determine the induction from an interpole in the neutral zone of machine P51 with the following geometric parameters: \(\delta_w=2{,}5\ \mathrm{mm}\), \(b_w=22\ \mathrm{mm}\), \(D_a=162{,}5\ \mathrm{mm}\), at an interpole current of \(10\ \mathrm{A}\).
The induction distribution was recorded using a Hall-emf sensor fastened to an armature tooth. The experimental data are presented in Fig. 2 as points.
Using the dependences obtained above (1–3) and (1–2) [1], the field of the interpole in the neutral zone was calculated for this machine. The calculation results are presented in Fig. 2 as a curve corresponding to an interpole current of \(10\ \mathrm{A}\).
Analysis of the data presented in Fig. 2 shows that the discrepancy between the calculated and experimental data does not exceed 5%.
Fig. 2
For calculation of commutation, the interpole field must be represented in the form of a function \(B=f(x)\), where \(x\) is time or its derivative. The graphical dependence obtained in Fig. 2 is approximated in the form of the difference of the functions \(y_1\) and \(y_2\) as
\[ B=y_1-y_2=A_1\cdot e^{-c_1\cdot x^2}-A_2\cdot e^{-c_2\cdot x^2}, \tag{4} \]
where
\(x\) is the angle in degrees measured from the midpoint of the interpole;
\(A_1, c_1\) are the parameters of the function \(y_1\), determined by the method of least squares for the section of the field curve located outside the interpole;
\(A_2, c_2\) are the parameters of the approximating function \(y_2\), determined by the method of least squares from the values \(y_2 = y_1 - B\).
For the field of the interpole of the P51 machine at \(i_{\mathrm{dp}} = 10\) A, the following approximating expression was obtained:
\[ B = 0.379 e^{-0.0135x^2} - 0.094 e^{-0.065x^2} \tag{5} \]
The discrepancy between the graphical data of Fig. 2 and the induction values from the approximating function (5) does not exceed \(3\text{--}4\%\).
Thus, analytical expressions have been obtained that make it possible to calculate the field of a rectangular interpole in the commutation zone.
An approximating expression for the interpole field has been proposed, which is needed in calculating commutation.
References
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A. I. Skorospeshkin, E. G. Chebotkov. Calculation of the field of the main poles in the commutation zone of series-P DC machines. Proceedings of TPI, vol. 190, 1968.
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P. F. Filchakov, V. I. Panchishin. EGDA Integrators. Kiev, 1961.