Study of the Geometry of Standardized Tape-Wound Magnetic Cores for Smoothing Chokes*

E. I. Goldshtein

(Presented by the scientific seminar of the Department of Electric Power Stations and Networks)

In the author’s works [1, 2], problems of investigation were formulated and a criterion was proposed for assessing the approximation of the geometry of a series (a group of a series) to the conditions of the optimal geometry of a smoothing choke (s. ch.) of arbitrary configuration. However, in [1] the analysis was performed analytically and only for calculation for a specified voltage drop (the first calculation case— I). In [2] the results of an investigation on an electronic digital computer are given, but not for all technical-and-economic indices and only for calculation for a specified overheating (the second calculation case— II).

In the present work we shall investigate the optimal geometry of standardized tape-wound magnetic cores according to all technical-and-economic indices of the s. ch. under different principles of variation of dimensions within a group of a series for calculation cases I and II.

For the investigation we use the approximation criterion in the form [2]

\[ \Delta M=\sum_{t_{\min}}^{t_{\max}} M(t), \tag{1} \]

for \(t=t_{\min},\ t_1,\ t_2,\ldots t_{\max}\).

In expression (1)

\(t\) — the variable parameter of the geometry,

\(M(t)\) — the function being optimized. For analysis according to the indices of weight \((G)\), cost \((C)\), volume of active materials \((V)\), and winding volume \((V_0)\), we use the generalized function [1]:

\[ \Phi_{\mathrm{I}}=n_{\mathrm{g}}^{-0.6}\left(K_{vc}\beta+K_{v0}\right). \tag{2} \]

\[ \Phi_{\mathrm{II}}=\left(n_{\mathrm{g}}K_{\mathrm{dp}}\right)^{-(3/7)}\cdot\left(K_{vc}\beta+K_{v0}\right). \tag{3} \]

In analysis according to the index of overall volume \((V_{\mathrm{g}})\),

\[ V_{\mathrm{gI}}=n_{\mathrm{g}}^{-0.6}K_{vg} \tag{4} \]

\[ V_{\mathrm{gII}}=\left(n_{\mathrm{g}}K_{\mathrm{dp}}\right)^{-(3/7)}\cdot K_{vg}. \tag{5} \]

The coefficients \(n_{\mathrm{g}}, K_{vc}, K_{v0}\), and \(K_{vg}\) are determined only by the design and geometry of the s. ch.; expressions for these coefficients are given in [1].

* The work was carried out under the supervision of Professor Doctor I. D. Kutyavin.

The coefficient of the calculated cooling area \((K_{\mathrm{dr}})\) characterizes the relation between the total cooling surface \((S_{\mathrm{dr}})\) and the basic dimension “\(a\)”:

\[ K_{\mathrm{dr}}=\frac{S_{\mathrm{dr}}}{a^2}. \tag{6} \]

In the general case, in accordance with the considerations set forth in another work by the author [3],

\[ K_{\mathrm{dr}}=K_{\mathrm{ob}}+\Theta K_{\mathrm{cc}}. \tag{7} \]

The calculation expressions for the cooling coefficients of the winding \((K_{\mathrm{ob}})\) and the magnetic core (core) are given in Table 1.

Table 1

In the analysis, the coefficient of effectiveness of heat removal from the core \((\Theta)\) is taken for two limiting cases

\[ \Theta_1=0 \quad \text{and} \quad \Theta_2=1. \tag{8} \]

The reduction coefficient \(\beta\) is determined from the expressions in [3]:

\[ \beta_{c_0}=0;\qquad \beta_v=1;\qquad \beta_G=\frac{\gamma_c K_c}{K_0\gamma_0};\qquad \beta_{\mathrm{p}}=\frac{\gamma_cK_c a_c}{\gamma_0K_0a_0}. \tag{9} \]

To minimize the approximation criterion (1), a special search program was compiled (programmer—M. F. Panikhina). The printout gave the basic geometry parameters (for example, \(x_b\) and \(y_b\), or \(x_b\) and \(z_b\), or \(y_b\) and \(z_b\)) that ensure the minimum of the corresponding criterion for the adopted values of the varied parameter \((z, y\) or \(x)\).

Table 2 gives the principal information on the search program, in particular the values of the varied parameter, the search ranges of the basic parameters, etc.

Table 2

In addition, Table 2 gives the values of the geometry parameters of normalized magnetic cores of the ShL and PL types for low-power transformers.

The main results of the study of the basic geometry are given in Tables 3–6. In Tables 3–5 a comparison is made of the approximation criteria calculated for the optimal basic geometry \((\Delta M_{\mathrm{opt}})\) and for the geometry parameters of the currently valid standards \((\Delta M_{\mathrm{shl}}\) and \(\Delta M_{\mathrm{pl}})\), with the values for the optimal geometry taken as 100%.

Table 3

Table 4

Table 5

Table 6

A certain interest lies in comparing with one another the indices of normalized magnetic cores of the same design while varying different geometric parameters. For such a comparison we use the average approximation criterion (per one standard size):

\[ \Delta M_{\mathrm{av}}=\frac{\Delta M}{n}, \tag{10} \]

where \(n\) is the number of values of the parameter being varied.
Let us note at once that the results of such a comparison cannot be regarded as absolute, since, in turn, the magnitude of the approximation criterion depends on the selected range of variation of the parameter being varied and on the number of standard sizes in this range. Therefore, the results obtained below make it possible to compare quite definite series of magnetic cores with preselected—and hence possibly nonoptimal—limits of variation of the parameter being varied. Nevertheless, such a comparison is also of interest, all the more so because in the literature known to the author this question has hardly been treated. Tables 7 and 8 give the results

Table 7

comparison of different principles for constructing standardized designs. As 100% there were taken the values of the corresponding indices \((M_{\text{opt}})\) of a choke of arbitrary configuration, calculated for optimal geometry parameters (see [3]).

Table 8

Conclusions

1. For rod-type chokes of minimum weight in all calculation cases, and for shell-type chokes of minimum weight designed for a specified overheating, the magnetic cores of optimal low-power transformers (the ShL and PL series) can be successfully used.

2. For chokes with variation in window height, the optimality conditions for copper consumption and overall volume practically coincide \((x_b=2.0;\ y_b=0.5)\) and make it possible, in comparison with the geometry of minimum-weight rod-type transformers and chokes \((x_b=2.0;\ y_b=1.6)\), to reduce the indices by \((100 \div 200)\%\) in copper consumption and by \((20 \div 90)\%\) in overall volume.

3. The index of overall volume of the ShL magnetic-core series is \((20 \div 40)\%\) higher than the corresponding index for the optimal basic geometry of the series with variation in tape width; the use of speci-

…geometry for smoothing chokes with minimum copper consumption ($z_{\mathrm{b}} = 2.0$; $y_{\mathrm{b}} = 0.5$) provides a reduction of the index by $(60 \div 90)\%$ in comparison with the series of ShL magnetic cores.

1. In developing standards for smoothing chokes of minimum cost, the parameters of the optimum geometry given in the tables may be used for the corresponding design case.

2. The indices of standardized magnetic cores, as was to be expected, exceed the corresponding indices for an arbitrary configuration and optimum geometry; this excess is especially large for the winding-volume indices because of the range of variation of $x$ chosen by us ($x \leq 2.0$, rather than $x \leq 2.7$, as for cores of arbitrary geometry).

3. For the cases studied, according to most indices, the principle of variation by strip width has advantages. For smoothing chokes of minimum weight, designed for a specified voltage drop, application of the principle of variation by window width makes it possible to reduce the corresponding index slightly; for smoothing chokes of minimum overall dimensions, variation by window height is preferable.

4. For rod-type smoothing chokes of minimum weight, any of the considered principles of variation may be used; for shell-type smoothing chokes of minimum weight, variations by window width and by strip width are practically equivalent.

REFERENCES

1. E. I. Goldshtein. On the choice of the geometry of smoothing-filter chokes. Izvestiya TPI, vol. 130, 1964.

2. E. I. Goldshtein. On the choice of the geometry of standardized tape cores for chokes and transformers. Standartizatsiya, 1964.

3. E. E. Goldshtein. Universal relationships for selecting the optimum geometry parameters of smoothing chokes on nonstandardized cores (present collection).