Smoothing Chokes with Maximum Time Constant on Non-Standard Cores

E. I. Goldshtein

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

In the practice of designing smoothing chokes (SC), problems are encountered of creating a choke with the maximum possible value of the time constant:

\[ \tau = \frac{L}{R}. \tag{1} \]

But for a specified value of the SC inductance:

\[ \tau_{\max} = \frac{1}{R_{\min}}. \tag{2} \]

Thus, the problem reduces to finding the conditions that ensure the minimum possible resistance of the SC for specified electromagnetic parameters.

In the present work we shall restrict ourselves to an analysis of the geometry of an SC on a non-standard core for shell-type and two-coil rod-type constructions. In [1] expressions are given for the basic dimension “\(a\)” when calculating for a specified voltage drop [case I, expression (10)] and when calculating for a specified overheating [case II, expression (11)].

Solving the indicated expressions jointly, we obtain the relationship of the resistance \(R\) with the specified electromagnetic parameters, design conditions, and geometric dimensions of the SC:

\[ R = \left[ N_R \left( \frac{K_{\mathrm{cool}}^{2.5}}{n_r} \right) \right]^{2/7}, \tag{3} \]

\[ N_R = \left[ \frac{\rho (2 \sigma \tau)^{2.5}} {(L I_0^2)^{1.5} B_0^2 K_c^2 K_0 K_h^{2.5}} \right] \cdot L^{3.5}, \tag{4} \]

\[ K_h = \frac{\rho_r}{\rho}, \tag{5} \]

\[ K_{\mathrm{cool}} = K_{\infty} + \Theta K_{\mathrm{oc}}. \tag{6} \]

The values of the coefficients \(n_r\), \(K_{\infty}\), and \(K_{\mathrm{oc}}\) are given in Table 1.

It should be noted that in [1] the coefficient of effectiveness of heat removal from the core \(\Theta\) is taken equal to zero; therefore there

\[ K_{\mathrm{cool}} = K_{\infty}. \]

The dimensionless geometry parameters \(x\), \(y\), and \(z\) characterize, respectively, the ratio of the tape width (stack thickness), the width, and the height of the window to the base dimension—the core width.

\[ x=\frac{b}{a};\quad y=\frac{c}{a};\quad z=\frac{h}{a} \tag{7} \]

Taking, in the analysis, the induction, filling factors, and other calculated quantities as constant (\(N_R=\mathrm{const}\)), one can find the geometry of the smoothing choke for which its resistance will be minimal. Minimization of the calculated expression

\[ \Phi_R=\left[\frac{K_{\mathrm{ok}.\ell}^{2.5}}{n_r}\right] \tag{8} \]

was carried out on the digital computer of the Tomsk Polytechnic Institute (programmer—engineer N. M. Bylino).

Table 1

The search for the optimal values of the geometry parameters was carried out over the following range of variation:

\[ 0.5\leq x\leq 2.7;\quad 0.5\leq y\leq 2.7;\quad 1\leq z\leq 5.0. \tag{9} \]

The search step was taken to be the same for all geometry parameters:

\[ \Delta x=\Delta y=\Delta z=0.1. \tag{10} \]

The results obtained in studying expression (8) are given in Table 2.

Table 2

It is of interest to compare the specific techno-economic indicators of a smoothing choke, optimized with respect to resistance, with the corresponding

…with the indicators of an SC optimized with respect to other indicators: overall volume \(V_{\mathrm{g}}\), total volume of active materials \(V\), and volume of conductor material \(V_{\phi}\). Table 3 gives the results of such a comparison; data from [1] and [2] were used.

Table 3

The data of Table 3 show that the indicator \(R\) increases most strongly at \(\Theta = 0\), whereas at \(\Theta = 1\) the resistance of an SC to direct current depends comparatively weakly on its geometry. Similarly, when optimizing an SC with respect to the indicator \(R\), other indicators increase, especially the overall volume at \(\Theta = 0\) and the winding volume at \(\Theta = 1\).

List of adopted notation:

\(\tau\) — time constant of the SC, in sec;

\(L\) — inductance of the SC, in H;

\(R\) — resistance of the SC at \(20^\circ\mathrm{C}\), in ohms;

\(a\) — basic core dimension, in cm;

\(N_R\) — design constant;

\(K_{\mathrm{cool}}\) — coefficient of the cooling surface of the SC;

\(n_{\mathrm{g}}\) — geometry coefficient of the SC;

\(\rho\) — electrical resistivity at \(20^\circ\mathrm{C}\);

\(\rho_{\mathrm{g}}\) — the same in the heated state;

\(K_{\mathrm{h}}\) — heating coefficient;

\(\sigma\) — heat-transfer coefficient, W/cm\(^2\);

\(\tau^\circ\) — overheating of the choke, in degrees;

\(I_0\) — direct component of the rectified current, in A;

\(B_0\) — direct component of induction, Wb/cm\(^2\);

$K_c$ — coefficient of filling of the core with steel;
$K_o$ — coefficient of filling of the window with the winding;
$K_{oo}$ — coefficient of the cooling surface of the winding;
$K_{oc}$ — coefficient of the cooling surface of the core;
$\Theta$ — coefficient of efficiency of heat removal from the core;
$x$, $y$, and $z$ — dimensionless geometry parameters;
$b$ — strip width (stack thickness), in cm;
$c$ — window width, in cm;
$h$ — window height, in cm;
$\Phi_R$ — calculated resistance function;
$V_g$ — overall volume of the smoothing choke;
$V$ — total volume of active materials;
$V_o$ — volume of conductor material.

REFERENCES

1. E. I. Goldshtein, “Some questions of designing optimal smoothing chokes,” Proceedings of Higher Educational Institutions, Electromechanics, No. 4, 1964.

2. E. I. Goldshtein. “Universal dependences for selecting optimal geometry parameters of smoothing chokes on non-normalized cores,” Proceedings of TPI, Vol. 149, Tomsk State University Press, 1965.