On the Calculation of the Electromagnetic Field of Slot Leakage in Shock Generators
Abstract Generated abstract
The paper examines slot leakage fields in pulse generators with very shallow stator slots, a geometry used to reduce inductive reactance but for which leakage through the slot bottom may become significant. It formulates a plane parallel magnetic field model for a salient pole stator interpole space, solves Laplace and Poisson equations for the vector potential in three slot regions using Fourier series and boundary conditions, and derives expressions for induction along the slot bottom and walls. Calculations and a supporting measurement indicate that bottom leakage is negligible for slot height greater than about half the slot width, but becomes substantial for smaller ratios, reaching reported values of 7.5 percent to 18 percent of the transverse slot leakage in representative cases.
Full Text
ON THE CALCULATION OF THE ELECTROMAGNETIC FIELD OF SLOT LEAKAGE IN PULSE GENERATORS
V. Z. KHORKOVA
(Presented by the scientific seminar of the Department of Electrical Machines and the Electrical Machines Division of the Scientific Research Institute of Electromechanical Engineering)
In order to reduce the inductive reactance of the slot part of the stator winding in pulse generators, efforts are made to make the height of the stator slots as small as possible. The height of the stator slots of high-breaking-capacity generators is 2–2.5 times smaller than the height of the stator slots of turbogenerators. The height of the stator slots of shock-power generators is still smaller. The ratio between the height and the width of the stator slot in the pulse generators of Kapitsa–Kostenko and of the Elektrosila plant (TO-12-2) is approximately equal to two [1, 2]. Foreign pulse generators also have this ratio of slot height to width [4]. At present there is a tendency toward an even greater reduction of the slots. The ratio between the height and the width of the slot in an experimental pulse generator of the firm CEM (France) is equal to 0.5 [3].
Many authors associate a further reduction in the inductive reactance of the slot part of the stator winding with the partial or complete removal of the stator winding into the air gap [5, 6]. One of the possible structural designs of generators with partial removal of the stator winding into the air gap is a salient-pole stator design with a shallow interpole space. Such an interpole space may be represented as a slot whose height is considerably less than its width. It is known that in the interpole space, along with the leakage flux between the poles, there is a leakage flux to the yoke. It is obvious that, with a sharp decrease in the height of the stator slots of pulse generators, the flux through the bottom of the slot will play a certain role, and it is of interest to estimate its share in the total slot-leakage flux. Let us consider the leakage fields in the interpole space of the salient-pole stator of a pulse generator in which a single-phase concentric winding is placed. We regard the interpole space as a slot lying in the plane of the axes \(x, y\), Fig. 1.
Since the length of the active part of the machine is large in comparison with the dimensions of the slot, the magnetic field may be considered plane-parallel. It is expedient to divide the slot space into three regions, denoted by the numerals I, II, III, Fig. 1. Regions I and III are free of current; the vector potentials of these fields satisfy Laplace’s equation:
\[ \nabla^2 A=\frac{\partial^2 A}{\partial x^2}+\frac{\partial^2 A}{\partial y^2}=0. \tag{1} \]
In region II there are two current-carrying bars, and the vector potential of this region satisfies Poisson’s equation:
\[ \nabla^{2}A=\frac{\partial^{2}A}{\partial x^{2}}+\frac{\partial^{2}A}{\partial y^{2}}=-\mu\delta, \tag{2} \]
where \(\delta\) is the current density at the point under consideration. The distribution of current density in space can be expressed in the form of a Fourier series:
\[ \delta=a_{0}+\sum_{n}a_{n}\cos nk y, \tag{3} \]
Fig. 1. Stator slot
where
\[ a_{0}=2\delta_{0}\frac{b}{b_{n}}; \]
\[ \delta_{0}=\frac{i}{a\cdot b}; \]
\[ k=\frac{2\pi}{ab}; \]
\[ a_{n}=\frac{2\delta}{\pi n}\{\sin nk b_{1}+(-1)^{n}\sin[nk(b_{3}-b_{2})]\}, \]
\(n\) is any integer,
\(i\) is the bar current.
It is obvious that the vector potential is a periodic function of \(y\), with the same period as \(\delta\). The general expressions for the vector potential by regions have the form:
\[ A_{\mathrm{I}}=\sum_{n}B_{n}\left(e^{nkx}+e^{-nkx}\right)\cos nk y; \tag{4} \]
\[ A_{\mathrm{II}}=-2\pi a_{0}(x-s_{1})^{2} +\sum_{n}\left(D_{n}e^{nkx}+E_{n}e^{-nkx}+\frac{4\pi a_{n}}{n^{2}k^{2}}\right)\cos nk y; \tag{5} \]
\[ A_{\mathrm{III}}=-2\pi a_{0}(s_{2}-s_{1})[2x-(s_{1}-s_{2})] +\sum_{n}\left(G_{n}e^{nkx}+F_{n}e^{-nkx}\right)\cos nk y. \tag{6} \]
To determine the unknown coefficients \(B_n, D_n, E_n, G_n, F_n\), we proceed from the condition that at the boundaries between regions II and III the tangential components of the field intensity and the normal components of the magnetic induction are continuous, and on the surface \(S_3\), passing at the level of the slot height, the normal component of the induction is equal to zero. We obtain five equations, from which it is not difficult to find the desired coefficients
\[ B_n=\frac{2\pi a_n}{n^2\kappa^2}\cdot \frac{(e^{-n\kappa s_1}-e^{-n\kappa s_2})e^{n\kappa s_3}+(e^{n\kappa s_1}-e^{n\kappa s_2})e^{-n\kappa s_3}} {e^{n\kappa s_3}+e^{-n\kappa s_3}}; \]
\[ D_n=\frac{2\pi a_n}{n^2\kappa^2}\cdot \frac{-e^{-n\kappa s_2}e^{n\kappa s_3}+(e^{n\kappa s_1}-e^{-n\kappa s_1}-e^{n\kappa s_2})e^{-n\kappa s_3}} {e^{n\kappa s_3}+e^{-n\kappa s_3}}; \]
\[ E_n=-\frac{2\pi a_n}{n^2\kappa^2}\cdot \frac{(e^{n\kappa s_1}-e^{-n\kappa s_1}+e^{-n\kappa s_2})e^{n\kappa s_3}+e^{n\kappa s_2}e^{-n\kappa s_3}} {e^{n\kappa s_3}+e^{-n\kappa s_3}}; \]
\[ G_n=\frac{2\pi a_n}{n^2\kappa^2}\cdot \frac{\left[(e^{n\kappa s_1}-e^{-n\kappa s_1})-(e^{n\kappa s_2}-e^{-n\kappa s_2})\right]e^{-n\kappa s_3}} {e^{n\kappa s_3}+e^{-n\kappa s_3}}; \]
\[ F_n=\frac{2\pi a_n}{n^2\kappa^2}\cdot \frac{\left[(e^{n\kappa s_1}-e^{-n\kappa s_1})-(e^{n\kappa s_2}-e^{-n\kappa s_2})\right]e^{n\kappa s_3}} {e^{n\kappa s_3}+e^{-n\kappa s_3}}. \]
Omitting the intermediate transformations, the expressions for the vector potential by regions can be written in final form as:
\[ A_{\mathrm{I}}=\sum_n \frac{2\pi a_n}{n^2\kappa^2} (e^{-n\kappa s_1}-e^{-n\kappa s_2})(e^{n\kappa x}+e^{-n\kappa x})\cos n\kappa y; \tag{7} \]
\[ A_{\mathrm{II}}=-2\pi a_0(x-s_1)^2-\sum_n \frac{2\pi a_n}{n^2\kappa^2} \left[e^{-n\kappa s_2}e^{n\kappa x}+(e^{n\kappa s_1}-e^{-n\kappa s_1}+e^{-n\kappa s_2})e^{-n\kappa x}-2\right]\cos n\kappa y; \tag{8} \]
\[ A_{\mathrm{III}}=-2\pi a_0(s_2-s_1)\,[2x-(s_1+s_2)]+\sum_n \frac{2\pi a_n}{n^2\kappa^2}\times \]
\[ \times\left[(e^{n\kappa s_1}-e^{-n\kappa s_1})-(e^{n\kappa s_2}-e^{-n\kappa s_2})\right] \left[e^{-2n\kappa s_3}e^{n\kappa x}-e^{-n\kappa x}\right]\cos n\kappa y. \tag{9} \]
Let us determine the distribution of induction along the bottom of the slot and over the height of the slot along the iron surface. The normal component of the induction to the bottom of the slot (in the section \(x=0\)) is equal to:
\[ B_{x\mathrm{I}}=-\frac{\partial A_{\mathrm{I}}}{\partial y} =\mu_0\,\frac{2i}{\delta_{\mathrm{p}}}\cdot \frac{\delta_{\mathrm{p}}^2}{4\pi^2ab} \sum_n \frac{\sin n\kappa\delta_1+(-1)^n\sin[n\kappa(\delta_3-\delta_2)]}{n^2} \times \]
\[ \times (e^{-n\kappa s_1}-e^{-n\kappa s_2})\sin n\kappa y. \tag{10} \]
The curve of the induction distribution over the slot width for one of the considered variants
\[ \left(h_{\mathrm{p}}=\frac{1}{2}\delta_{\mathrm{p}};\quad \delta_1=\delta_3-\delta_2;\quad \Delta_{\mathrm{iz}}=0.1\,\delta_{\mathrm{p}};\quad s_1=s_3-s_2=\Delta_{\mathrm{iz}}\right) \]
is presented in Fig. 2.
The calculations show that the maximum value of the induction \(B_x\) depends on the relation between the height and width of the slot and on the thickness of the rod insulation.
From Fig. 2 it is obvious that, in order to determine the leakage flux through the bottom of the slot, the limits of integration cannot be taken as
\[ y=0\div \frac{\delta_{\mathrm{p}}}{2}, \]
as was done in [7], and, consequently, the conclusion that, when the distances between the coil and the center line of two adjacent poles, on the one hand, and the pole surface, on the other hand, are equal, the leakage flux
Fig. 2. Distribution of induction along the slot bottom,
\(h_p = 0.5 b_p\)
(through the yoke) is zero regardless of the coil length, is fundamentally incorrect. Figure 3 presents the values of \(B_{x\max}\) as fractions of the maximum value of the induction over the slot height for various \(\Delta_{iz}\) and
Fig. 3. Dependence of the maximum value of induction at the slot bottom on the ratio
\[
\frac{h_p}{b_p}:
\]
1. \(\Delta_{iz}=0.1\,b_p,\) 2. \(\Delta_{iz}=0.05\,b_p\)
\(k_1\). For slots with \(k=\dfrac{h_p}{b_p}>1\), the leakage flux through the slot bottom is indeed negligibly small; however, for \(k \ll 1\) it increases sharply. For
\(\kappa = 0.5\), the leakage flux through the bottom of the slot (for \(\Delta_{\mathrm{ins}} = 0.1 b_{\mathrm{p}}\)) is \(7.5\%\), and for \(\kappa = 0.3\) it is \(18\%\) of the leakage flux across the slot.
The normal component of the induction at the slot walls (in the section \(y = \dfrac{b_{\mathrm{p}}}{2}\)) is equal, by regions, to:
1st region \(0 \leqslant x \leqslant s_1\)
\[ B_{y\mathrm{I}} = -\frac{\partial A_1}{\partial x} = \mu_0 \frac{2i}{b_{\mathrm{p}}}\, \frac{\theta_{\mathrm{p}}^2}{4\pi^2 a b} \sum_n \frac{\sin n\kappa\theta_1 + (-1)^n \sin [n\kappa(\theta_3-\theta_2)]}{n^2} \times \]
\[ \times \left(e^{-n\kappa s_1} - e^{-n\kappa s_2}\right)\operatorname{sh} n\kappa x; \tag{11} \]
2nd region \(s_1 \leqslant x \leqslant s_2\)
\[ B_{y\mathrm{II}} = \mu_0 \frac{2i}{b_{\mathrm{p}}} \left\{ \frac{x-s_1}{a} + \frac{\theta_{\mathrm{p}}^2}{4\pi^2 a b} \sum_n \frac{\sin n\kappa\theta_1 + (-1)^n \sin [n\kappa(\theta_3-\theta_2)]}{n^2} \times \right. \]
\[ \left. \times \left[ e^{-n\kappa x}\operatorname{sh} n\kappa s_1 - e^{-n\kappa s_2}\operatorname{sh} n\kappa x \right] \right\}; \tag{12} \]
3rd region \(s_2 \leqslant x \leqslant s_3\)
\[ B_{y\mathrm{III}} = \mu_0 \frac{2i}{b_{\mathrm{p}}} \left\{ 1 - \frac{\theta_{\mathrm{p}}^2}{2\pi^2 a b} \sum_n \frac{\sin \kappa\theta_1 + (-1)^n \sin [n\kappa(\theta_3-\theta_2)]}{n^2} \times \right. \]
\[ \left. \times \left(\operatorname{sh} n\kappa s_1-\operatorname{sh} n\kappa s_2\right) \left(e^{-2n\kappa s_3} e^{n\kappa x} + e^{-n\kappa x}\right) \right\}. \tag{13} \]
The curve of the distribution of induction over the slot height for the variant \(\left(h_{\mathrm{p}}=\dfrac{1}{2}b_{\mathrm{p}};\ \Delta_{\mathrm{ins}}=0.1b_{\mathrm{p}}\right)\) is presented in Fig. 4.
The form of the curve and the values of the induction in the planes \(S_1, S_2, S_3\) differ somewhat from the values that would occur in a calculation by the method of total currents (dashed curve). The induction values calculated in different slot sections for slots of different geometry are given in Table 1.
Fig. 4. Distribution of induction over the slot height: \(h_{\mathrm{p}}=0.5b_{\mathrm{p}}\)
Calculations show that deviations of the induction value from unity increase as the slot height decreases and the insulation thickness increases. However, for \(h_{\mathrm{p}} > 0.5b_{\mathrm{p}}\), and also in calculations in the first approximation, these deviations may be neglected.
In most cases the flux through the bottom of the slot may also be neglected—the relative magnitude of the maximum value of the induction \(B_x\) for \(h_{\mathrm{p}} > 0.5b_{\mathrm{p}}\) does not exceed \(5\%\). For \(h_{\mathrm{p}} < 0.5b_{\mathrm{p}}\), neglecting the leakage flux through the bottom of the slot introduces substantial errors. To confirm the theoretical propositions concerning the existence of leakage fluxes
the following experiment was conducted through the bottom of the slot. Measuring frames were placed on the bottom and walls of the slot, with dimensions
\[ \frac{h_{\mathrm{p}}}{b_{\mathrm{p}}}=\frac{8}{78}; \]
No. 1 along the height of the slot, of width \(h_{\mathrm{p}}\),
No. 2 along the width of the slot, of width \(b_{\mathrm{p}}\),
No. 3 —"— \(\dfrac{1}{2} b_{\mathrm{p}}\),
No. 4 —"— \(\dfrac{1}{4} b_{\mathrm{p}}\).
The emf induced by the leakage flux in the frames was, respectively, 115 mV, 7.5 mV, 9.6 mV, 22 mV.
Table 1
Values of induction in various sections of the slot
| No. | \(\dfrac{\Delta_{\mathrm{kz}}}{b_{\mathrm{p}}}\) | \(\dfrac{h_{\mathrm{p}}}{b_{\mathrm{p}}}\) | \(B_{y\mathrm{I}}\) | \(B_{y\mathrm{II}}\) | \(B_{y\mathrm{III}}\) | \(B_x\) |
|---|---|---|---|---|---|---|
| 1 | 0.05 | 5 | 0.0013 | 1.0 | 1.004 | 0.0012 |
| 2 | 0.05 | 2 | 0.0034 | 0.998 | 1.010 | 0.0028 |
| 3 | 0.05 | 1 | 0.0073 | 0.990 | 1.022 | 0.0064 |
| 4 | 0.05 | 0.5 | 0.0165 | 0.977 | 1.049 | 0.0145 |
| 5 | 0.05 | 0.2 | 0.048 | 0.951 | 1.157 | 0.0415 |
| 6 | 0.1 | 5 | 0.0036 | 0.999 | 1.005 | 0.0023 |
| 7 | 0.1 | 2 | 0.0093 | 0.989 | 1.012 | 0.0062 |
| 8 | 0.1 | 1 | 0.023 | 0.975 | 1.023 | 0.0140 |
| 9 | 0.1 | 0.5 | 0.062 | 0.945 | 1.073 | 0.0364 |
| 10 | 0.1 | 0.3 | 0.180 | 0.850 | 1.160 | 0.0530 |
Note: As unity there was taken the value of the induction in section \(S_2\), calculated by the method of total currents:
\[ B_{y\mathrm{II}}=\mu_0 \frac{2i}{b_{\mathrm{p}}}. \]
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