On the Selection of the Geometry of Chokes for Smoothing Filters*)

Postgraduate Student E. I. Goldshtein

BASIC OBJECTIVES OF THE WORK

1. An optimal geometry is understood to mean such a combination of the principal dimensions of a product for which one of its indicators—weight, volume, or cost—is minimal. Questions of the optimal geometry of low-power transformers and smoothing chokes have been considered in [1, 2, 3].

In [2] a complete analysis of the geometry of shell-type chokes is carried out for all cases of calculation.

In [3] the results are given of an investigation of both shell-type and core-type chokes, but only for an unspecified winding resistance.

In all the works cited, a graphical-analytical method of investigation is used, which makes it possible to see the behavior of the function under study near extremal values, but makes it difficult to carry out the analysis in general form; moreover, a change in any of the initial conditions requires a new repetition of most of the cumbersome calculations. It seems expedient to develop a sufficiently universal program and to carry out an investigation of the optimal geometry of chokes on an electronic digital computing machine (EDCM).

1. In recent years, much attention has been devoted to the standardization of low-power transformers, magnetic amplifiers, and smoothing chokes [1, 2, 3, 4, 5]. However, in the literature known to the author, the question of the criterion by which one can evaluate the selected geometry of a series of tape cores from the point of view of its maximum approximation to the conditions of optimal geometry obtained for a magnetic core of arbitrary configuration [1, 2, 3] is insufficiently covered. The absence of a sufficiently rigorous criterion makes it difficult, in particular, to assess the advisability of using for smoothing chokes the standards developed for low-power transformers [3].

And, finally, by minimizing the above-mentioned criterion, it becomes possible to select, with sufficient rigor, the geometry for a normalized series of cores.

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

We shall confine ourselves to considering only chokes with laminated cores whose geometry is not subject to the requirements of “waste-free” stamping, with copper winding wire, for two basic calculation cases:

a) without taking the thermal regime into account (case 1);
b) taking the thermal regime into account, but with unspecified winding resistance (case 2).

1. OPTIMAL GEOMETRY FOR A MAGNETIC CORE OF ARBITRARY CONFIGURATION

The geometry of a choke is completely characterized by the dimensionless parameters \(x\), \(y\), and \(z\):

\[ x=\frac{b}{a}; \qquad y=\frac{c}{a}; \qquad z=\frac{h}{a}. \tag{1} \]

(The conventional designations are given at the end of the article.)
In the analysis, one of the following indices of a choke of any construction will be of interest:

a) the total volume of the core and winding;

\[ V=a^3\left(k_{VC}+k_{VO}\right); \tag{2} \]

b) the net weight of the active materials;

\[ G=a^3\left(K_{VC}\gamma_C K_C+K_{VO}\gamma_O K_O\right); \tag{3} \]

c) the cost of the active materials;

\[ C=a^3\left(K_{VC}\gamma_C K_C c_C+K_{VO}\gamma_O K_O c_O\right); \tag{4} \]

d) the overall volume of the choke;

\[ V_g=2a^3K_{Vg}. \tag{5} \]

Expressions for the coefficients \(K_{VC}\), \(K_{VO}\), and \(K_{Vg}\) are given in Table 1. For convenience of the subsequent analysis, we introduce a generalized function \((\Phi)\) of volume, weight, and cost:

\[ \Phi=a^3\left(K_{VC}\beta+K_{VO}\right), \tag{6} \]

where the reduction coefficient \(\beta\) characterizes the type of analysis:

\[ \beta_V=1,\qquad \beta_G=\frac{\gamma_C K_C}{\gamma_O K_O},\qquad \beta_C=\frac{\gamma_C K_C c_C}{\gamma_O K_O c_O}. \tag{7} \]

The base dimension “\(a\)” is determined from formulas analogous to expressions (5-1) and (5-4) from [2]:

a) for case I:

\[ a^3=\left(\frac{N_I}{n_g}\right)^{0.6}; \qquad N_I=\frac{L^2 I_O^2\rho}{R B_O^2 K_C^2 K_C}. \tag{8} \]

b) for case II:

\[ a^3=\left(\frac{N_{II}}{n_g K_{\mathrm{cool}}}\right)^{0.429}; \qquad N_{II}=\frac{\left(L I_O^2\right)^2\rho_g}{2B_O^2K_C^2K_O\sigma\tau}. \tag{9} \]

The values of the coefficients \(n_g\) and \(K_{\mathrm{cool}}\) are given in Table 1.

We shall restrict ourselves to considering only chokes with laminated cores, whose geometry is not associated with the requirements of “waste-free” stamping, with copper winding wire, for two principal calculation cases:

a) without taking the thermal regime into account (case 1);

b) with the thermal regime taken into account, but with an unspecified winding resistance (case 2).

1. OPTIMAL GEOMETRY FOR A MAGNETIC CORE OF ARBITRARY CONFIGURATION

The geometry of the choke is fully characterized by the dimensionless parameters \(x\), \(y\), and \(z\):

\[ x=\frac{b}{a}; \qquad y=\frac{c}{a}; \qquad z=\frac{h}{a}. \tag{1} \]

(The symbols used are given at the end of the article.)

In the analysis, one of the following indices of a choke of any construction will be of interest:

a) the total volume of the core and winding;

\[ V=a^3\left(k_{VC}+k_{VO}\right); \tag{2} \]

b) the natural weight of the active materials;

\[ G=a^3\left(K_{VC}\gamma_C K_C+K_{VO}\gamma_O K_O\right); \tag{3} \]

c) the cost of the active materials;

\[ C=a^3\left(K_{VC}\gamma_C K_C c_C+K_{VO}\gamma_O K_O c_O\right); \tag{4} \]

d) the overall volume of the choke;

\[ V_G=2a^3K_{VG}. \tag{5} \]

Expressions for the coefficients \(K_{VC}\), \(K_{VO}\), and \(K_{VG}\) are given in Table 1. For convenience in the subsequent analysis, let us introduce a generalized function \((\Phi)\) of volume, weight, and cost:

\[ \Phi=a^3\left(K_{VC}\beta+K_{VO}\right), \tag{6} \]

where the reduction coefficient \(\beta\) characterizes the type of analysis:

\[ \beta_V=1,\qquad \beta_G=\frac{\gamma_C K_C}{\gamma_O K_O};\qquad \beta_C=\frac{\gamma_C K_C c_C}{\gamma_O K_O c_O}. \tag{7} \]

The basic dimension “\(a\)” is determined from formulas analogous to expressions (5-1) and (5-4) from [2]:

a) for case I:

\[ a^3=\left(\frac{N_I}{n_G}\right)^{0.6}; \qquad N_I=\frac{L^2 I_O^2 \rho}{R B_O^2 K_C^2 K_C}. \tag{8} \]

b) for case II:

\[ a^3=\left(\frac{N_{II}}{n_G K_{\mathrm{cool}}}\right)^{0.429}; \qquad N_{II}=\frac{\left(L I_O^2\right)^2\rho_G}{2B_O^2K_C^2K_O\sigma\tau}. \tag{9} \]

The values of the coefficients \(n_G\) and \(K_{\mathrm{cool}}\) are given in Table 1.

Table 1.

It is necessary to note that, as in [2], we take into account only the cooling surface of the winding itself; disregarding the cooling of the choke due to heat removal through the core will lead to an error toward underutilization of the choke with respect to heating, whereas taking into account the cooling surface of the core according to [3] may lead to impermissible overheating, since in actuality the thermal contact between the core and the winding is not ideal, and the heat-transfer coefficients from the coil and the core are not equal (cf. [1], p. 151, and [3], p. 114).

Taking the quantities $N_1$ and $N_{\mathrm{II}}$ as constants, we arrive at the problem of determining the conditions for the minimum of the expressions for the specific indicators:

a) for case I:

\[ \Phi'=(n_{\mathrm{G}})^{-0.6}(K_{\mathrm{VC}}\beta+K_{\mathrm{VO}}), \tag{10} \]

\[ V'_{\mathrm{G}}=(n_{\mathrm{G}})^{-0.6}K_{\mathrm{VG}}. \tag{11} \]

b) for case II:

\[ \Phi'=(n_{\mathrm{G}}K_{\mathrm{cool}})^{-0.429}(K_{\mathrm{VC}}\beta+K_{\mathrm{VO}}), \tag{12} \]

\[ V'_{\mathrm{G}}=(n_{\mathrm{G}}K_{\mathrm{cool}})^{-0.429}K_{\mathrm{VG}}. \tag{13} \]

The minimization of expressions (10)—(13) was carried out on the electronic digital computer of the Computing Center of the Siberian Branch of the USSR Academy of Sciences by engineer A. M. Sherin, with the participation of the author of the article. The investigation was carried out in the range of acceptable design values of $x$, $y$, and $z$:

\[ 0.5<x<2.6;\quad 0.5<y<2.6;\quad 1<z<5. \tag{14} \]

In the solution the following initial data were used:

\[ \gamma_{\mathrm{C}}=7.65\ \mathrm{g/cm^3};\quad \gamma_{\mathrm{O}}=8.8\ \mathrm{g/cm^3};\quad K_{\mathrm{C}}=0.9; \]

for a two-coil core-type choke

\[ K_{\mathrm{O}}=0.3;\quad \frac{\alpha_{\mathrm{C}}}{\alpha_{\mathrm{O}}}=0.33;\quad \beta_{\mathrm{C}}=0.86;\quad \beta_{\mathrm{G}}=2.6. \]

for single-coil chokes

\[ K_{\mathrm{O}}=0.34;\quad \frac{\alpha_{\mathrm{C}}}{\alpha_{\mathrm{O}}}=0.374;\quad \beta_{\mathrm{C}}=0.86;\quad \beta_{\mathrm{G}}=2.3. \]

Table 2 gives the principal results of the analysis. There, for comparison, data from [2] and [3] are also given. It must be borne in mind that in [3] the cooling area of the core is taken into account, and $\beta_{\mathrm{C}}=0.32$ (see [3], p. 110).

2. SELECTION OF THE APPROXIMATION CRITERION

In solving the second of the problems posed above, we shall regard as known:

1) the analytical expression of the optimized function of weight, volume, cost, etc.;

\[ \Phi_1=f_1(x_1,y,z,\beta). \tag{14} \]

2) the minimum value of the optimized function for a non-normalized magnetic core of arbitrary configuration under a certain analysis

\[ \Phi_{1\min}=\varphi(\beta). \tag{15} \]

Table 2.

3) the limits of variation of the varied dimension in the series under consideration—the strip thickness or the window height; thus, for cores with variation in window height, the limits may be chosen on design grounds:

\[ z_1 \leq z \leq z_2 . \tag{16} \]

If function (15) is regarded as the function being approximated, and function (14) as the approximating function, then, using the theory of approximation of functions [6], the approximation criterion \(\Delta \Phi'\) may be written in the following form:

\[ \Delta \Phi_1=\int_{z_1}^{z_2} p(z)\,[\Phi_1-\Phi_{1\min}]^S\,dz, \tag{17} \]

where \(S\) is the exponent of the power approximation;
\(p(z)\) is a nonnegative weight function characterizing the probability of using each of the values of \(z\) in the interval \([z_1, z_2]\).

In our case, with sufficient accuracy, a first-degree approximation can be used, since for the particular type of analysis the condition is always satisfied

\[ [\Phi_1-\Phi_{1\min}] \ge 0 . \tag{18} \]

Assuming, for simplicity, that all values of \(z\) are equally probable, we obtain, taking (18) into account:

\[ \Delta \Phi_1=\int_{z_1}^{z_2} [\Phi_1-\Phi_{1\min}]\,dz . \tag{19} \]

The expression (19) obtained above makes it possible to estimate the degree of approximation of the selected geometry to the optimum according to the “integral” criterion \(\Delta \Phi_1\). With an ideal approximation \(\Delta \Phi_1=0\); in practice, \(\Delta \Phi_1>0\) almost always.

By minimizing the integral criterion, one can find \(x'_{\mathrm{opt}}\) and \(y'_{\mathrm{opt}}\), for which \(\Delta \Phi'_1\) is minimal. For a given \(\beta\):

\[ \Phi_{1\min}=const;\qquad \int_{z}^{z_2} \Phi_{1\min}\,dz=const\ne f(x,y). \tag{20} \]

Therefore, to determine the optimum (with respect to a specified index) geometry of normalized cores, it is sufficient to minimize expression (21)

\[ \Delta \Phi'_1=\int_{z_1}^{z_2} \Phi_1\,dz . \tag{21} \]

When determining the optimum geometry from expression (21), in most cases it is also necessary to take into account certain additional conditions due to the specified range of variation of the electrical parameters of the product \((L, I_0, R)\), the temperature regime, etc. Analysis of such conditions makes it possible to establish an additional relation between the sought parameters, for example,

\[ x=\varphi(y)\quad \text{for}\quad z=const . \tag{22} \]

Solving (21) and (22) jointly, we finally obtain:

\[ \Delta \Phi'_1=\int_{z_1}^{z_2} \Phi'_1\,dz , \tag{23} \]

where \(\Phi'_1=f'_1(y,z)\) for \(\beta=const\).

3. OPTIMUM GEOMETRY OF NORMALIZED CHOKES

The additional relation according to expression (22) can be obtained from the condition of full utilization of the choke with respect to the thermal regime at maximum electromagnetic loading

\[ a=\sqrt[5]{\frac{N_1}{n_{\mathrm{G}}}}=\sqrt[2]{\frac{N_{\mathrm{III}}}{K_{\mathrm{cool}}}}, \tag{24} \]

within the limits \(0.5 < y < 1.1\) are well approximated by expressions (28) (shown by the dashed line)

\[ x_{\mathrm{BR}}=1.43y;\qquad x_{1\,\mathrm{coil}}=1.1y;\qquad x_{2\,\mathrm{coils}}=0.6+y. \tag{28} \]

Minimizing \(\Delta \Phi'_1\) according to (21) for \(z_1=1\) and \(z_2=5\), taking into account (10) and (28), the corresponding equations were obtained, whose solutions are given in Table 3. For comparison, the same table gives data obtained for the norm already mentioned above at \(x=2\) and \(y=1.6\).

Table 3.

MAIN CONCLUSIONS

1. For the calculation cases considered, the conditions for optimum geometry are unambiguous and do not depend on the specified choke parameters; furthermore, as a rule, the conditions of minimum weight, overall volume, and cost do not coincide.

2. The results obtained show the fundamental possibility of a rigorous selection of the geometry of standardized cores on the basis of the considerations indicated above.

3. Analysis of the results of the study confirms the conclusion from [2, 5] concerning the possibility of using, for smoothing chokes, minimum-weight strip magnetic cores developed for low-power transformers. It may be assumed that, for smoothing chokes of minimum cost, it is advisable to develop a separate standard.

4. The proposed method can be used in selecting the geometry of other standardized products as well, including transformers, magnetic amplifiers, and alternating-current chokes.

LIST OF ADOPTED NOTATION

\(a\) — basic dimension of the core;
\(b\) — core thickness;
\(c\) — width of the core window;
\(h\) — height of the core window;
\(K_{VC}\) — steel-volume coefficient;
\(K_{VO}\) — winding-volume coefficient;
\(\gamma_C\) — specific weight of steel;
\(\gamma_O\) — specific weight of copper;
\(K_C\) — coefficient of filling the core with steel;
\(K_O\) — coefficient of filling the window with winding;
\(a_C\) — unit cost of steel;
\(a_O\) — unit cost of copper wire;

$K_{V\Gamma}$ — coefficient of the overall volume of the choke;

$n_\Gamma$ — coefficient of the choke geometry;

$K_{\text{cool}}$ — coefficient of the cooling surface of the choke;

$L$ — inductance of the choke in H;

$R$ — resistance of the choke at 20°C in ohms;

$I_0$ — dc component of the rectified current in A;

$B_0$ — dc component of induction, Wb/cm²;

$\rho$ — specific electrical resistance at 20°C;

$\rho_\Gamma$ — specific electrical resistance when heated;

$\sigma$ — heat-transfer coefficient, W/cm² °C;

$\tau$ — temperature rise of the choke in degrees.

References

1. Balian R. Kh., Low-Power Transformers. Sudpromgiz, 1961.

2. Bamdas A. M. and Savinovskii Yu. A., Chokes of Filters for Radio Equipment. Published by Soviet Radio, 1962.

3. Belopolskii I. I., Pikalova L. G., Calculation of Low-Power Transformers and Chokes. GEI, 1963.

4. Rozenblat M. A., Sedykh O. A., Principles for constructing series of toroidal cores for magnetic amplifiers. Standardization, No. 6, 1958.

5. Karetnikova E. I., Paths toward unification of transformers and chokes of the Dzerzhinskii MNTP, collection No. 5, Moscow, 1958.

6. Goncharov V. L., Theory of Interpolation and Approximation of Functions. Fizmatgiz, 1954.