On the Calculation of Leakage Inductance of Transformers with Windings Arbitrarily Arranged on One Leg of the Magnetic Core

Yu. V. Frakman

(Recommended by the scientific seminar of the Department of Electric Power Stations and Electrical Networks and Systems)

From the standpoint of calculating leakage inductance, any relative position of transformer windings may be called arbitrary. Existing methods of calculation are relatively complicated and insufficiently visual. Therefore, the method of calculation set forth below will find practical application in the design of power transformers. In the logical substantiation of the proposed method, modern physical concepts of the leakage process are developed to some extent, which gives the present work a certain theoretical interest.

The reasoning is based on Kapp’s formula, known for its simplicity and clarity.

The possibility of calculating the leakage inductance of the windings mentioned by means of the theory of multiwinding transformers [1], through repeated use of Kapp’s formula, makes it possible to extend to an arbitrary arrangement of windings the simplifying assumptions adopted in deriving the latter formula and, in particular, to regard the magnetic lines of force as straight.

To determine the direction of the magnetic lines of force in the leakage field between the calculated sections of the windings (used in the derivation of Kapp’s formula), let us consider the mechanical interaction of the latter. The resultant repulsive force of the windings [2] is always directed along the straight line joining their centers of inertia. If it is taken into account that the leakage field of windings located on one leg possesses axial symmetry, this regularity may also be extended to the calculated sections of the windings. Then, in accordance with the concept of the transverse thrust of the magnetic lines of force, the latter may be regarded as directed normally to the straight line joining the centers of inertia of the calculated sections of the windings.

Let us characterize the relative arrangement of the windings under consideration by the angles $\gamma$ formed by the normals to the axes of the windings and the straight line joining the centers of inertia of their calculated sections—Fig. 1.

For a concentric arrangement of windings ($\gamma = 0$), as also in the case of an alternating arrangement of symmetrical disk windings ($\gamma = \pi/2$), the heights and radial dimensions of the latter are measured, respectively, in the longitudinal and transverse directions of the magnetic lines of force. We shall assume that this procedure for determining the dimensions of the windings is also valid for any intermediate position $(0 \leq \gamma < \pi/2)$ of...

Fig. 1

placement of the latter. In this case the calculated picture of the leakage field of the actual transformer will not change if the actual transformer is replaced by a calculated (fictitious) one in which the distance between the centers of inertia of the calculated cross sections of the windings is the same as in the actual transformer, while the heights of the windings and the radial dimensions of the leakage field are smaller by a factor of \(\cos \gamma\) than the corresponding dimensions of the actual transformer, measured respectively along and normal to the geometric axis of the windings of the latter—Figs. 1 and 2. In these figures the dimensions of the calculated transformer are marked with the subscript \(n\) (intermediate position of the windings).

In the calculated transformer \(\gamma = 0\); therefore, to determine the leakage inductance of its windings one can directly use Kapp’s formula. However, for practical calculations it is more convenient, using the relation indicated above, to express the dimensions of the calculated transformer in terms of the dimensions of the actual one. In this case Kapp’s formula for calculating the leakage inductance of equal-height transformer windings arranged arbitrarily on one leg can be written in the form

\[ L_{12s} = \frac{\mu_0 w_1^2 l}{h_{\mathrm{ob}}\cos\gamma} \left( a+\frac{b+c}{3} \right)\rho \tag{1} \]

or

\[ L_{12s} = \frac{2\pi\mu_0 w_1^2}{h_{\mathrm{ob}}\cos\gamma} \left( aR_a+\frac{1}{3}bR_b+\frac{1}{3}cR_c \right)\rho, \tag{2} \]

where \(L_{12s}\) — the total leakage inductance of the transformer, referred to winding 1;

\[ \mu_0 = 4\pi \cdot 10^{-7}\,[\mathrm{H/m}] \]

— the magnetic permeability of air or oil and of the winding material (copper or aluminum), taken equal to the magnetic permeability of vacuum;

\(w_1\) — the number of turns in winding 1;

\(b, c\), and \(a\) — the radial dimensions of the windings and of the gap between them;

Fig. 2

\(R_b, R_c\) and \(R_a\) — mean radii of the windings and of the gap between them;

\(h_{\mathrm{w}}\) — winding height;

\(\tau = a + b + c\) — radial dimension of the leakage field;

\[ \rho = 1 - \frac{\tau}{\pi h_{\mathrm{w}}}\left(1 - e^{-\pi h_{\mathrm{w}}/\tau}\right) \]

— Rogowski coefficient;

\(\sigma_{\mathrm{н}}\) and \(\sigma_{\mathrm{o}}\) — radial and axial distances between the centers of inertia of the calculated sections of the windings of the actual transformer;

\[ \gamma = \operatorname{arctg}\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{н}}}. \]

In the case when the windings of the actual transformer differ in height, formulas (1) and (2), according to [3], should be rewritten as follows:

\[ L_{12s} = \frac{\mu_0 w_1^2 l \varphi^2}{h_{\max}\cos\gamma} \left(a + \frac{b+c}{3}\right)\rho_m, \tag{3} \]

\[ L_{12s} = \frac{2\pi\mu_0 w_1^2 \varphi^2}{h_{\max}\cos\gamma} \left(aR_a + \frac{1}{3}bR_b + \frac{1}{3}cR_c\right)\rho_m, \tag{4} \]

In formulas (3) and (4), \(h_{\max}\) and \(h_{\min}\) are, respectively, the heights of the taller and the shorter winding;

\[ \rho_{\max} = 1-\frac{\tau}{\pi h_{\max}} \left(1-e^{-\pi h_{\max}/\tau}\right); \]

\[ \rho_{\min} = 1-\frac{\tau}{\pi h_{\min}} \left(1-e^{-\pi h_{\min}/\tau}\right); \]

\[ \varphi = \sqrt{0.5\left[1+\left(h_{\max}\rho_{\min}/h_{\min}\rho_{\max}\right)^2\right]}; \]

\[ \rho_m = 1-\frac{\varphi\tau}{\pi h_{\max}} \left(1-e^{-\pi h_{\max}/\varphi\tau}\right); \]

\[ \gamma = \operatorname{arctg}\frac{\sigma_{\mathrm{o}}}{\varphi\sigma_{\mathrm{н}}}; \]

the remaining designations coincide with those adopted in formulas (1) and (2).

According to the law of the transition of quantity into quality, the accumulation of quantitative changes leads abruptly to qualitative changes. Examination of formulas (1)--(4) shows that (for a finite distance between windings) the jump occurs when the equality \(\gamma=\pi/2\) is reached and consists in the mutual replacement of certain concepts, at \(\gamma=0\), concerning the height and radial thickness of the windings. Thus, formulas (1)--(4) are valid in the region bounded by the inequality \(\gamma<\pi/2\). It is appropriate to note that formulas (3) and (4), in relation to formulas (1) and (2), are more general and become the latter in the case of windings of equal height.

When the equality \(\gamma=\pi/2\) is observed, the calculation should use the results of the derivation (formula) of Rogovskii for symmetric interleaved disk windings [4].

Table 1

Table 2

| 1--2 | | \multicolumn{5}{c}{Value of angle \(\gamma\)} |
|---|---|---:|---:|---:|---:|---:|
| | | \(0^\circ\) | \(17^\circ 10'\) | \(35^\circ 40'\) | \(54^\circ 40'\) | \(75^\circ\) |
| \(x_{\mathrm{p}}\) | ohm | 11.78 | 12.35 | 14.5 | 20.4 | 45.5 |
| \(x_{\mathrm{k}}\) | ohm | 11.95 | 12.4 | 13.8 | 18.3 | 50.4 |
| \(\Delta x_{\mathrm{p}}\) | % | 1.42 | 0.4 | 5.07 | 11.5 | 9.73 |

Table 3

| 1--3 | | \multicolumn{5}{c}{Value of angle \(\gamma\)} |
|---|---|---:|---:|---:|---:|---:|
| | | \(0^\circ\) | \(16^\circ 20'\) | \(34^\circ 40'\) | \(54^\circ 20'\) | \(76^\circ 20'\) |
| \(x_{\mathrm{p}}\) | ohm | 21 | 21.9 | 25.6 | 36 | 89 |
| \(x_{\mathrm{k}}\) | ohm | 19.4 | 20 | 22.8 | 32.1 | 98.5 |
| \(\Delta x_{\mathrm{p}}\) | % | 8.25 | 9.5 | 12.3 | 12.15 | 9.65 |

To verify the reasoning presented, two transformers were manufactured; the characteristics of their windings are given in Table 1. In this table, \(q\) and \(r\) denote, respectively, the inner and outer radii of the windings, and \(R\) denotes the dc resistance of the latter.

Tables 2 and 3 present and compare (\(\Delta x\%\)) the results of calculation (\(x_{\mathrm{p}}\)) by formula (4) and of the short-circuit experiment (\(x_{\mathrm{k}}\)) for windings 1 and 2, and in Table 3 for windings 1 and 3, for various values of the angle \(\gamma\).

Comparison of the experimental and calculated results shows that the accuracy of the formulas obtained is the same as that of Kapp’s formula, i.e., about 10 percent.

References

1. G. N. Petrov. Transformers, vol. 1, ONTI, 1934.
2. F. Engels. Dialectics of Nature, GIPL, 1952.
3. Yu. V. Frakman. On the calculation of transformer leakage with windings of unequal height. Proceedings of TPI, No. 172, 1967.
4. B. Hague. Electromagnetic Calculations, ONTI, 1934.