On the Calculation of Leakage in Transformers with Windings of Unequal Height
Abstract Generated abstract
The paper addresses calculation of leakage inductance and short-circuit reactance in transformers with concentric windings of unequal height whose height midpoints lie in a common plane. It argues that direct averaging of winding heights is not theoretically justified and develops a generalization of Kapp’s formula by replacing the real transformer with an equivalent calculated transformer, introducing a correction coefficient based on winding heights and Rogowski coefficients. The resulting formulas reduce to the usual Kapp expression for equal winding heights and are presented as having comparable approximate accuracy, about 10 percent. Experimental short-circuit tests on a three-winding transformer gave errors of 5.17 percent, 2.08 percent, and 11.2 percent for the calculated reactances.
Full Text
ON THE CALCULATION OF LEAKAGE IN TRANSFORMERS WITH WINDINGS OF UNEQUAL HEIGHT
Yu. V. Frakman
(Presented by Professor I. D. Kutyavin)
The calculation of the short-circuit resistance of windings of unequal height finds practical application in the design of transformers with arbitrarily arranged windings.
Determination of the corresponding leakage inductance by means of the theory of multi-winding transformers [1] reduces to repeated application of Kapp’s formula. The disadvantages of this method of calculation include the large amount of computation and the lower accuracy of the result than with Kapp’s formula, owing to the subtraction of quantities of the same order in the intermediate calculations.
The averaging, admitted in factory calculations, of the heights—close in magnitude—of windings of unequal height is not theoretically justified.
The present work is devoted to a generalization of Kapp’s formula for the case in which the midpoints of the heights of windings of unequal height lie in one plane normal to the axes of the latter.
In principle, in a magnetically balanced two-winding transformer, on the basis of the law of conservation of energy, the energy of the leakage magnetic field and the leakage inductance can be characterized by the height of any one of the windings of unequal height. To clarify the real possibility of using winding heights in Kapp’s formula, we shall make use of Rogowski’s derivation [1, § 67].
Consider a two-winding transformer whose windings satisfy all the conditions adopted in Rogowski’s derivation, but differ in height (Fig. 1). To determine the vector potential of the magnetic field, the space between the ferromagnetic planes must be divided into three zones. In Rogowski’s derivation these zones are defined as follows. Zone 1 contains all regions of the magnetic field in which the current density is not equal to zero. In this zone the current density does not depend on the coordinate \(x\) and varies only in the direction of the \(y\)-axis. Zones 2 and 3 are located on both sides of zone 1 and extend to infinity. At all points of these zones the current density is equal to zero.
When the windings are arranged according to Fig. 1, the width of zone 1 cannot be taken equal to the mean height of the windings or to the height of the lower winding, because such a choice contradicts the definitions of all the zones. If the width of zone 1 is taken equal to that of the taller winding, then only the condition of independence of the current density from the coordinate \(x\) in this zone is violated. At the same time, fulfillment of this condition is achieved quite simply.
Fig. 1 — Fig. 1.
Since the windings are magnetically balanced, the diagram of the distribution of current density along the \(y\)-axis can be represented in the form of two rectangles equal in area:
\[ i_{\mathrm{B}} b h_1 = - i_{\mathrm{c}} c h_2; \tag{1} \]
where
\(i_{\mathrm{B}}\) and \(i_{\mathrm{c}}\) are the current densities in windings 1 and 2;
\(b\) and \(c\) are the radial thicknesses of windings 1 and 2;
\(h_1\) and \(h_2\) are the heights of windings 1 and 2.
Equality (1) will not be violated if its right-hand side is multiplied and divided by \(h_1\) (in Fig. 1, \(h_1 > h_2\)), and then rewritten in the form
\[ i_{\mathrm{B}} b h_1 = - i_{\mathrm{pc}} c h_1; \tag{2} \]
where
\[ i_{\mathrm{pc}} = i_{\mathrm{c}} h_2 / h_1; \]
Here and below the subscript \(\mathrm{p}\) (calculated) is used to denote quantities that have no place in the real transformer.
Equality (2) replaces consideration of a real transformer with windings of unequal height by consideration of a calculated (fictitious) transformer with windings of equal height, equivalent to the real one by the condition of magnetic balance of the windings. From equality (2) we obtain
\[ i_{\mathrm{pc}} c = - i_{\mathrm{B}} b. \tag{3} \]
As a result of arguments differing from those given in § 67 [1] only by the use of equality (3) instead of equality (23, 9) [1], it may be concluded that, in the case considered, the height of the taller winding should be substituted into the Kapp formula.
Rogowski’s conclusion is based on determining the energy of the leakage magnetic field; consequently, the Kapp formula can be used for calculation only in the case when the energies of the magnetic fields in the calculated and real transformers are identical. At the same time, the calculated current density introduced in equality (2) and the corresponding magnetic-field energy are less than the actual ones. To eliminate the noted contradiction it is necessary to increase the volume of the field
leakage of the calculation transformer, while keeping the volume density of the energy of its magnetic field unchanged. The height of the equal-height windings of the calculation transformer was set higher; therefore, the increase in the volume of the leakage field can be achieved only by changing the radial dimensions of the latter. We shall determine the radial dimensions of the calculation transformer on the basis of the following considerations.
The simplifying assumptions adopted in deriving Kapp’s formula [1; 2] are also valid in the case of windings of unequal height, since all the expressions for calculating their leakage inductance by this formula, obtained with the aid of the theory of multiwinding transformers, are linear.
If the curvature of the windings is neglected and the assumption of rectilinearity of the magnetic lines of force is used, the expression for the volume of the leakage field—\(V\)—per unit mean length of a turn of the windings (in [1], correspondingly, per unit length in the direction of the \(z\)-axis) may be written in the form
\[ V=\tau h/\rho; \]
where \(\tau=a+b+c\) is the radial dimension of the leakage field;
\(h\) is the height of the winding;
\(a\) is the gap between the windings;
\(\rho\) is Rogowski’s coefficient.
In deriving Kapp’s formula [1] it is assumed that the entire volume of the leakage field is occupied by the leakage flux of the primary winding. Since the windings of the transformer are magnetically balanced, the converse reasoning is also legitimate, according to which the entire volume of the leakage field is occupied by the leakage flux of the secondary winding. The volume density of the energy of the leakage field \(W_v\) in these cases will be, respectively, equal to
\[ W_{v1}=W/V_1=W\rho_1/h_1\tau; \]
\[ W_{v2}=W/V_2=W\rho_2/h_2\tau; \]
where \(W\) is the total energy of the leakage field.
In a real transformer there is a certain average volume density of energy, \(W_{v\mathrm{cp}}\); therefore the condition of equality of the energies of the magnetic fields of the calculation and real transformers may be written in the form
\[ W_{v1}\frac{h_1}{\rho_1}\tau_p = W_{v\mathrm{cp}}\frac{h_1}{\rho_1}\tau, \tag{4} \]
whence
\[ \tau_p=\varphi\tau; \tag{5} \]
where
\[ \tau_p \]
is the radial dimension of the leakage field of the calculation transformer;
\[ \varphi=W_{v\mathrm{cp}}/W_{v1} \]
is a coefficient showing how many times greater the radial dimension of the leakage field in the calculation transformer is than in the real one.
A comparison of various methods of averaging the quantities \(W_{v1}\) and \(W_v\) showed that, over a sufficiently large range of winding-height ratios, the expression
\[ W_{v\mathrm{cp}}=\sqrt{0.5\left(W_{v1}^2+W_v^2\right)}. \tag{6} \]
gives optimal results.
Substitution of equality (6) into formula (5) makes it possible to obtain the following expression for the coefficient \(\varphi\):
\[ \varphi=\sqrt{0.5\left[1+\left(h_1\rho_2/h_2\rho_1\right)^2\right]}. \]
The preceding considerations make it possible to write Kapp’s formula for calculating the total leakage inductance of a transformer with windings of unequal height, symmetrical with respect to the \(y\)-axis (Fig. 1), in the form
\[ L_{12s}=\frac{\mu_0 w_1^2 l\varphi^2}{h_{\max}} \left(a+\frac{b+c}{3}\right)\rho_{\mathrm{m}} \tag{7} \]
or
\[ L_{12s}=\frac{2\pi\mu_0 w^2\varphi^2}{h_{\max}} \left(aR_a+\frac{1}{3}bR_b+\frac{1}{3}cR_c\right)\rho_{\mathrm{m}}, \tag{8} \]
where \(\mu_0=4\pi\cdot10^{-7}\ \mathrm{H/m}\) is the magnetic permeability of air (or oil) and of the winding material (copper or aluminum), taken as equal to the magnetic permeability of vacuum;
\(w_1\) is the number of turns in the winding to which the total inductance of the transformer \(L_{12s}\) is referred;
\(l\) is the mean length of a winding turn;
\(h_{\max}\) and \(h_{\min}\) are, respectively, the heights of the taller and shorter transformer windings;
\(R_b\), \(R_c\), and \(R_a\) are the mean radii of the windings and of the gap between them;
\(\rho_{\max}\) and \(\rho_{\min}\) are Rogowski coefficients determined from \(h_{\max}\) and \(h_{\min}\) for the radial dimension of the leakage field equal to \(\tau\);
\[ \varphi=\sqrt{0.5\left[1+\left(h_{\max}\rho_{\min}/h_{\min}\rho_{\max}\right)^2\right]}, \]
\[ \rho_{\mathrm{m}}=1-\frac{\varphi\tau}{\pi h_{\max}} \left(1-e^{-\pi h_{\max}/\varphi\tau}\right) \]
is the Rogowski coefficient for the transformer under calculation.
It is appropriate to note that the square of the coefficient \(\varphi\) in the last two formulas corresponds to the product of the radial dimensions.
Formulas (7) and (8), in relation to Kapp’s formula, are more general and reduce to the latter in the case of windings of equal height, since for \(h_{\max}=h_{\min}\), \(\varphi=1\).
Formulas (7) and (8), like Kapp’s formula, do not take into account the influence of the tank or casing and of the yoke beams of the transformer on the inductance under consideration, and in this sense they are approximate. In deriving formulas (7) and (8), simplifying assumptions were not made; therefore the accuracy of these formulas is the same as that of Kapp’s formula, i.e., about 10%.
To verify the last assertion, a transformer was manufactured in which the midpoints of the heights of the unequal-height concentric windings 1, 2, and 3 lay in one plane normal to the axes of the latter. The transformer dimensions required for the calculations are given in the table, where \(q\) and \(r\) denote the inner and outer radii of the windings.
Table
| Winding index | \(w\) | \(h_{\mathrm{w}}\), cm | \(q\), cm | \(r\), cm | Notes |
|---|---|---|---|---|---|
| 1 | 834 | 18 | 2.95 | 3.3 | All windings are made of PEL-0.8 wire |
| 2 | 553 | 12 | 4.5 | 4.85 | All windings are made of PEL-0.8 wire |
| 3 | 1142 | 24 | 5.75 | 6.1 | All windings are made of PEL-0.8 wire |
Calculation of the short-circuit reactances of this transformer by formula (8) gives:
\[ \omega L_{12S}=8.1\ \text{ohm}; \qquad \omega L_{13S}=11.28\ \text{ohm}; \qquad \omega L_{32S}=19\ \text{ohm}. \]
In the corresponding short-circuit tests, the following were obtained:
\[ x_{k12}=8.54\ \text{ohm}; \qquad x_{k13}=11.05\ \text{ohm}; \qquad x_{k32}=21.4\ \text{ohm}, \]
that is, the relative errors are 5.17%, 2.08%, and 11.2%, respectively.
References
- G. N. Petrov. Transformers. Vol. 1, ONTI, 1934.
- B. Hague. Electromagnetic Calculations. ONTI, 1934.