UDC 621.3.011.33.001.24

ON THE QUESTION OF THE LEAKAGE INDUCTANCE OF A CONDUCTOR LOCATED IN A RECTANGULAR SLOT

V. A. Lukutin

(Presented by the scientific seminar of the Department of TOE)

Considerable attention is devoted in the domestic and foreign literature [2] to the determination of the leakage inductance of rectilinear conductors located in slots of rectangular cross section with walls made of ferromagnetic materials. This is explained by the fact that such a problem arises in the investigation of leakage inductance in rotating electrical machines.

It is known that the calculation of inductances, including leakage inductances, reduces either to determining the magnetic flux linked with a conductor or a group of conductors, or to calculating the magnetic field. We shall use the second approach and shall find the vector magnetic-potential function in the region of the slot. To simplify the problem, we shall adopt the assumptions usual in such cases: we shall take the magnetic permeability of the ferromagnetic medium of the slot walls to be equal to infinity and shall not take into account the bulging of the magnetic lines of force at its open edge.

Fig. 1

The figure shows the cross section of the slot and its dimensions, as well as the cross section of a rectangular conductor with a constant current creating the magnetic field. Since it is assumed that at the edge of the slot a line of force coincides with the \(x\)-axis, one may construct a mirror image of the region with respect to this axis with a current of the opposite direction (shown by the dashed line). After such a construction, a closed rectangular region bounded by a ferromagnetic medium is obtained. To describe the magnetic field in the two-dimensional region, we shall seek the vector-potential function \(A(x,y)\), having only one projection, coinciding with the direction of the current in the conductor. The magnetic field will satisfy Poisson’s equation

\[ \frac{\partial^2 A}{\partial x^2} + \frac{\partial^2 A}{\partial y^2} = -\mu \delta (x,y) \tag{1} \]

under the known boundary conditions:

\[ B_x=\frac{\partial A}{\partial y}=0,\quad \text{for } y=\pm n, \]

\[ B_y=-\frac{\partial A}{\partial x}=0,\quad \text{for } x=\pm m,\ y=0. \tag{2} \]

The boundary-value problem under consideration (the Neumann problem for Poisson’s equation) will be solved if it is possible to integrate equation (1). However, another possibility may also be used. From the theory of the calculus of variations it is known that integration of Poisson’s equation is equivalent to finding the extremum of the functional

\[ J=\int_{-m}^{m}\int_{-n}^{n} \left[ \left(\frac{\partial A}{\partial x}\right)^2+ \left(\frac{\partial A}{\partial y}\right)^2 -2\mu\delta A \right]\,dxdy . \tag{3} \]

We shall solve this latter problem by the Ritz method. Taking into account the symmetry with respect to the coordinate axes, we choose the minimizing sequence in the form of three terms

\[ \begin{aligned} A(x,y)={}&D_1(3n^2y-y^3)+D_2y^3(n^2-y^2)^2(m^2-x^2)^2+{}\\ &+D_3(3n^2-y^3)(m^2-x^2)^2 =D_1\varphi_1+D_2\varphi_2+D_3\varphi_3 . \end{aligned} \tag{4} \]

The selected solution (4) satisfies the boundary conditions (2) and contains the unknown coefficients \(D_\kappa\). To compute them we set up the Ritz system of equations:

\[ D_1[\varphi_1,\varphi_1]+D_2[\varphi_1,\varphi_2]+D_3[\varphi_1,\varphi_3]=(\mu\delta,\varphi_1); \]

\[ D_1[\varphi_2,\varphi_1]+D_2[\varphi_2,\varphi_2]+D_3[\varphi_2,\varphi_3]=(\mu\delta,\varphi_2); \tag{5} \]

\[ D_1[\varphi_3,\varphi_1]+D_2[\varphi_3,\varphi_2]+D_3[\varphi_3,\varphi_3]=(\mu\delta,\varphi_3). \]

Here the following notation is used:

\[ (\mu\delta,\varphi_\kappa)= \int_{-m}^{m}\int_{-n}^{n}\mu\delta\varphi_\kappa\,dxdy, \]

\[ [\varphi_\kappa,\varphi_p]= \int_{-m}^{m}\int_{-n}^{n} \left[ \frac{\partial\varphi_\kappa}{\partial x}\cdot \frac{\partial\varphi_p}{\partial x} + \frac{\partial\varphi_\kappa}{\partial y} \frac{\partial\varphi_p}{\partial y} \right]\,dxdy . \tag{6} \]

The magnetic field in the region under study is produced by a constant current; therefore the current density in the conductor can be determined from the formula

\[ \delta=\frac{I}{2a(d-c)}. \]

After carrying out the operations (6) and solving the system of equations (5) in general form, we obtain formulas for calculating the coefficients. The intermediate computations are quite cumbersome; therefore we omit them and give here only the final results:

\[ D_1= \frac{\mu I10^{-3}}{m^3n^7z(d-c)} \left[ 13{,}35m^2n^2(1285m^4+1622m^2n^2+74{,}4n^4)\Pi_1+ \right. \]

\[ \left. +51{,}1\Pi_2-5{,}81n^2(257m^2+14n^2)\Pi_3 \right]; \]

\[ \begin{aligned} D_2 &= \frac{\mu\,10^{-3}}{m^7 n^{13} z (d-c)} \bigl[4610 m^4 n^6 \Pi_1 + 738(21m^2+85n^2)\Pi_2 \\ &\qquad\qquad\qquad\qquad -89.7 n^4(33m^2+140n^2)\Pi_3\bigr];\\[4pt] D_3 &= \frac{\mu\,10^{-3}}{m^7 n^9 z (d-c)} \bigl[-87.2 m^4 n^4(257m^2+14n^2)\Pi_1 -14.9(33m^2+140n^2)\Pi_2 \\ &\qquad\qquad\qquad\qquad +1.53 n^4(1897m^2+360n^2)\Pi_3\bigr]; \tag{7}\\[4pt] z &= 197.5m^4 + 810m^2n^2 + 38n^4;\\ \Pi_1 &= 6n^2(d^2-c^2) - (d^4-c^4);\\ \Pi_2 &= [15m^4 - 10m^2a^2 + 3a^2]\,[6n^4(d^4-c^4)-8n^2(d^6-c^6)+3(d^8-c^8)];\\ \Pi_3 &= [15m^4 - 10m^2a^2 + 3a^4]\Pi_1. \end{aligned} \]

Substituting the coefficients determined from (7) into the selected solution (4), we obtain an analytical expression for the vector-potential function in the region of the slot.

Let us now proceed directly to the determination of the leakage inductance of the conductor in the slot. This inductance is determined by the magnetic flux crossing the slot from one ferromagnetic wall to the other and closing within the conductor itself. It is determined from the vector-potential function found, by the known formula [2]

\[ L=\frac{1}{2a(d-c)I}\int_{-a}^{a}\int_{c}^{d}(A-A_0)\,dx\,dy. \tag{8} \]

Under the integral sign, \(A_0\) denotes the vector potential defining the initial line of force. Here it is convenient to take as the initial line the one that spans the slot at its edge, i.e., coincides with the \(x\)-axis. For this line \(A_0=0\).

Substituting into (8) the function (4) determined above, we obtain

\[ \begin{aligned} L &= \frac{\mu_0 10^{-3}}{4(d-c)^2 z} \biggl[ \frac{13.35}{mn^5}(1285m^4+1622m^2n^2+74.4n^4)\Pi_1^2 \\ &\quad +\frac{8.2}{m^7n^{13}}(21m^2+85n^2)\Pi_2^2 +\frac{1.02}{m^7n^5}(189.7m^2+36n^2)\Pi_3^2 \\ &\quad +\frac{102}{m^3n^7}\Pi_1\Pi_2 -\frac{11.6}{m^3n^5}(257m^2+14n^2)\Pi_1\Pi_3 \\ &\quad -\frac{1.98}{m^7n^9}(33m^2+140n^2)\Pi_2\Pi_3 \biggr], \qquad \frac{\mathrm{H}}{\mathrm{m}}. \tag{9} \end{aligned} \]

In the particular case when the slot is completely filled by a conductor made of nonferromagnetic material, formula (9) is considerably simplified. Substituting in (9) \(m=a\), \(d=n\), \(c=0\), we obtain

\[ L= \frac{\mu_0 n^2(131m^4+524m^2n^2+24.9n^4)} {4m^2(197.5m^4+180m^2n^2+38n^4)}, \qquad \frac{\mathrm{H}}{\mathrm{m}}. \tag{10} \]

Conclusions

1. An approximate analytical expression has been obtained for the vector-potential function (4, 7). It is convenient not only because it makes it possible to determine easily the magnetic induction at any point of the slot region,

energy of the magnetic field in this region and the mechanical forces acting on a current-carrying conductor, but also makes it possible to investigate the influence of the geometrical dimensions of the slot and of the conductor on all the quantities listed.

1. The analytical approximate formulas for the leakage inductance of a conductor in a slot (9, 10) are very simple in form, take into account all dimensions of the conductor region, and make it possible to estimate the dependence of the per-unit-length leakage inductance on these dimensions.

References

1. V. I. Krylov, L. V. Kantorovich. Approximate Methods of Analysis. Moscow, Gostekhizdat, 1952.

2. K. Binns, P. Lawrenson. Analysis and Calculation of Electric and Magnetic Fields. Moscow, “Energiya,” 1970.

3. P. L. Kalantarov, L. A. Tseitlin. Calculation of Inductances. Moscow, “Energiya,” 1970.