On the Design of Transformer Windings with Minimum Active Resistance
Abstract Generated abstract
This article addresses the optimization of power transformer coil windings to minimize active resistance and additional copper or aluminum losses from eddy currents, with the core geometry and principal winding dimensions treated as fixed. It derives analytical conditions for selecting the number of elementary conductors, the number of turns per coil, and the radial and axial conductor dimensions for disk, continuous coil, and helical windings, rather than relying solely on numerical computation. The resulting formulas give optimal conductor dimensions and layer counts as functions of insulation thickness, conductor material, coil width, and cooling or insulation spacing, with tabulated examples showing the effect on additional losses. The analysis also considers cases where the axial conductor dimension must be chosen for design reasons, estimating the associated small increase in additional losses.
Full Text
On the Design of Transformer Windings with Minimum Active Resistance
I. D. Kutyavin
The article considers a local problem of optimizing the dimensions of coil windings of power transformers, proceeding from the minimum of additional losses in copper (aluminum) due to eddy currents, with the geometry of the core held constant. A fundamental solution of this problem is possible only in a general techno-economic determination of the optimal geometry and parameters of transformers, taking into account additional losses in copper. However, the particular solution of this problem proposed in the article is also of undoubted theoretical and practical interest.
Consideration of this question is given in [1], where the author analytically determines the optimal dimensions of elementary conductors and the number of turns in a coil, proceeding from the maximum filling of the winding area with copper. The article states that “for coil windings it is impossible to determine analytically the minimum active resistance of the winding,” and this problem is solved on an electronic digital computer. Meanwhile, it is possible analytically to determine the conditions for designing coil windings (disk, continuous coil, and helical) with minimum active resistance.
Let us consider the solution of this problem for the case when the following are specified: the axial height of the winding—\(h\), the radial width of the coil—\(b\), and the number of winding turns—\(w\); the axial distance between the copper of adjacent coils—\(\delta\) (consisting of the radial cooling duct and the insulation of the coil on two sides); the thickness of the insulation of an elementary conductor on two sides—\(i\). As the variable quantities to be optimized we shall take the number of elementary conductors in the coil \(m\), the number of turns in the coil \(w_k\), and the radial and axial dimensions of the elementary conductor \(x\) and \(y\) (Fig. 1).
The coefficient of increase of the active resistance of the winding, taking into account active-power losses due to eddy currents [1, 2],
\[ k_r \cong 1+\frac{m^2}{9}\xi^4 =1+\frac{c^2 m^2 x^4 y^2}{9(y+\delta)^2} \quad \text{for } m>2, \tag{1} \]
where
\[ c=\pi f\frac{\mu_0}{\rho}, \tag{2} \]
\(f\) is the number of periods of the current \((2\omega)\), \(\mu_0=0.4\pi\cdot10^{-8}\), and \(\rho\) is the calculated specific resistance of the winding material \((\Omega,\ \text{cm})\).
The number of elementary conductors in a coil (layer) \(m\) is related to the coil width \(b\) and the conductor dimension \(x\) by the following expression:
\[ m=\frac{b}{x+i}. \tag{3} \]
The dc resistance of the copper of the coil (layer) per running centimeter of the mean conductor is
\[ R_0=\frac{\rho}{mxy}=\frac{\rho(x+i)}{bxy}. \tag{4} \]
The active resistance of the copper of the coil (layer) per running centimeter of the mean conductor, taking (1)—(4) into account, is
\[ R=R_0k_r=\frac{\rho}{by}\left[\frac{x+i}{x}+\frac{b^2c^2y^2}{9(y+\delta)^2}\frac{x^3}{(x+i)}\right]. \tag{5} \]
The number of coils in the winding is
\[ n=\frac{h+\delta}{y+\delta}. \tag{6} \]
The number of turns in a coil (layer) is
\[ w_k=\frac{w}{n}=\frac{w(y+\delta)}{h+\delta}. \tag{7} \]
The current in the cross section of the coil (layer) is
\[ I_{\mathrm{н}}w_k=\frac{I_{\mathrm{н}}w(y+\delta)}{h+\delta}=\mathrm{var}. \tag{8} \]
The active-power losses in the copper of the coil per running centimeter of its mean length, taking (5) and (8) into account, are
\[ P=(I_{\mathrm{н}}w_k)^2R= \left(\frac{I_{\mathrm{н}}w}{h+\delta}\right)^2 \frac{\rho}{b} \left[ \frac{(x+i)(y+\delta)^2}{xy} +\frac{b^2c^2}{9}\,y\,\frac{x^3}{x+i} \right]. \tag{9} \]
The partial derivatives of expression (9) with respect to the variables \(x\) and \(y\) are:
\[ \frac{\partial P}{\partial x} = -\frac{i(y+\delta)^2}{x^2y} + \frac{b^2c^2x^2y^2(2x+3i)}{9(x+i)^2} =0, \tag{10} \]
\[ \frac{\partial P}{\partial y} = -\frac{(x+i)(\delta^2-y^2)}{xy^2} + \frac{b^2c^2x^3}{9(x+i)} =0. \tag{11} \]
The simultaneous solution of (10) and (11) gives the following expression for the optimal value of \(y\):
\[ y_0=\delta\frac{1+V}{2+V}<\delta, \tag{12} \]
where
\[ V=\frac{x}{i}. \tag{13} \]
Substituting \(y_0\) from (12) into equation (11), we find:
\[ b.c.i=U_1=\frac{3}{V^2}\sqrt{2V+3}. \tag{14} \]
Figure 1 gives the curve of the dependence \(u(v)\), constructed from expression (14), which can be used to determine \(V\) from the value \(u_1\) found in advance from (14) (through the constants \(b\), \(c\), and \(i\)).
Fig. 1.
In items 4–6 of the table, the optimal values \(x_0, y_0, m_0\) are given as functions of the insulation thickness \(i\) for \(b=10\ \text{cm}\); \(c=0.92\) (for copper); \(\rho=2.14\cdot 10^{-6}\ \Omega\ \text{cm}\); \(\delta=1.0\ \text{cm}\).
Item 7 shows the additional copper losses due to eddy currents \(P_d\), expressed as percentages of the principal losses and calculated from the formula (from expression 1):
\[ P_d\% = 100 \frac{c^2 m_0^2 x_0^4 y^2}{9(y_0+\delta)^2}. \tag{15} \]
Table
| No. | Quantity | 0.01 | 0.02 | 0.03 | 0.05 | 0.07 | 0.10 |
|---|---|---|---|---|---|---|---|
| 1 | \(i\) | 0.01 | 0.02 | 0.03 | 0.05 | 0.07 | 0.10 |
| 2 | \(U_1\) | 0.09 | 0.18 | 0.28 | 0.46 | 0.64 | 0.92 |
| 3 | \(V\) | 13.2 | 8.8 | 6.7 | 4.8 | 4.0 | 3.1 |
| 4 | \(x_0\) | 0.13 | 0.18 | 0.20 | 0.24 | 0.28 | 0.31 |
| 5 | \(y_0\) | 0.94 | 0.91 | 0.89 | 0.85 | 0.83 | 0.80 |
| 6 | \(m_0\) | 70 | 51 | 44 | 34 | 29 | 24 |
| 7 | \(P_d\%\) | 3.3 | 5.3 | 6.3 | 7.9 | 9.7 | 10.2 |
| 8 | \(U_2\) | 54.3 | 27.1 | 18.1 | 10.9 | 7.80 | 5.43 |
| 9 | \(V\) | 12 | 7.2 | 5.5 | 3.9 | 3.3 | 2.7 |
| 10 | \(x_0\) | 0.12 | 0.15 | 0.17 | 0.20 | 0.23 | 0.27 |
| 11 | \(m_0\) | 79 | 61 | 51 | 41 | 33 | 27 |
| 12 | \(P_d\%\) | 4.2 | 6.0 | 7.0 | 8.7 | 10.5 | 11.3 |
However, the optimal dimension \(y_0\) proved to be small (item 5 of the table). If, for design reasons, the value of \(y\) is taken different from \(y_0\), then, to determine \(x_0\) and \(m_0\), one may use equation (10), from which it follows that
\[ \frac{3(y+\delta)}{b.c.i\,y} = U_2 = \frac{V^2\sqrt{2V+3}}{1+V}. \tag{16} \]
Having determined \(U_2\) from the left-hand side of (16), we find \(V\) from the curve in Fig. 2 and then calculate \(x_0\) from (13) and \(m_0\) from (3).
Items 10 and 11 of the table give the values of \(x_0\) and \(m_0\) as functions of \(i\) for \(y=1.5\ \text{cm}\) and for the other constants adopted earlier.
The additional active-power losses in the winding, in percent, calculated from (15), are shown in item 12. When the value of \(y\) is increased from \(y_0\) (item 5) to \(1.5\ \text{cm}\), the additional losses increase by \(0.7\)–\(1\) percent.
Fig. 2.
In conclusion, we note that the optimum \(m_0\) and \(x_0\) depend on the thickness of the insulation and on the material of the winding conductors, as well as on the radial width of the coil.
REFERENCES
- Dietrich W. Auslegung von Transformatorenwicklungen mit kleinstem Wirkwiderstand, “Elektrotechn.”, vol. 6, 1965.
- I. M. Postnikov. Design of Electrical Machines. Kiev, 1960.