Universal Relationships for Selecting Optimal Geometry Parameters of Smoothing Chokes on Nonstandard Cores
Abstract Generated abstract
This paper reports computed design relationships for choosing optimal geometric parameters of open-type smoothing chokes using strip, nonstandardized cores. Building on earlier derivations, it minimizes technical and economic objective functions for several design cases, including specified voltage drop and specified overheating, using the Gauss Seidel method over defined ranges of three geometry parameters. The results are presented as tables for shielded, single-coil core, and two-coil core chokes, with variation in a reduction coefficient representing material weights, costs, and fill factors, and in the effective contribution of core cooling surface. These tabulated relationships are intended to provide a practical engineering procedure for selecting choke geometry that minimizes conductor use, active-material volume, weight, or cost under different assumptions.
Full Text
PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV
Vol. 149 1966
UNIVERSAL RELATIONS FOR SELECTING THE OPTIMAL GEOMETRIC PARAMETERS OF SMOOTHING CHOKES ON NON-NORMALIZED CORES*
E. I. GOLDSHTEIN
(Submitted by the scientific seminar of the Department of Electric Power Stations and Networks)
This paper presents the final results of a study of the geometry of smoothing chokes on strip non-normalized cores. The formulation of the problem, the choice of the research method, and the derivation of the calculation expressions are given in other works by the author [1, 2].
Unlike the works cited above, minimization of the calculation expressions for the principal technical-and-economic indicators of the smoothing choke was carried out on the TPI electronic digital computer by the Gauss–Seidel method (programmer—M. F. Panikhina), which made it possible to carry out a sufficiently large number of studies with a comparatively small expenditure of machine time (on the order of 20–40 seconds per point).
The search for the parameters of the optimal geometry \(x\), \(y\), and \(z\) was carried out in the following ranges of their variation:
\[ 0.5 \leq x \leq 2.7;\quad 0.5 \leq y \leq 2.7;\quad 1 \leq z \leq 5.0. \tag{1} \]
The search step was taken to be the same for all geometry parameters:
\[ \Delta x = \Delta y = \Delta z = 0.1, \tag{2} \]
In the study, the following cases of calculation and design of smoothing chokes were considered:
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The first calculation case (calculation for a specified voltage drop).
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The second calculation case (calculation for a specified overheating). Additional condition—the cooling surface of only the coil (winding) is taken into account
\[ S_{\mathrm{pr}} = S_{\mathrm{ob}}. \tag{3} \]
- The second calculation case (calculation for a specified overheating), but the entire cooling surface of the choke is taken into account
\[ S_{\mathrm{pr}} = S_{\mathrm{ob}} + S_{\mathrm{c}}, \tag{4} \]
where \(S_{\mathrm{c}}\) is the cooling surface of the core.
- The second calculation case (calculation for a specified overheating), but only part of the cooling surface of the core is taken into account
\[ S_{\mathrm{pr}} = S_{\mathrm{ob}} + \Theta S_{\mathrm{c}}, \tag{5} \]
* The work was performed under the supervision of Professor, Doctor I. D. Kutyavin.
where \(\Theta\) is the coefficient of heat-transfer efficiency from the core,
\[ 0 \leqslant \Theta \leqslant 1. \tag{6} \]
In [3] \(\Theta=0.6\); for expression (3) \(\Theta=0\); and for expression (4) \(\Theta=1\). Therefore the analysis was carried out for a series of values of \(\Theta\)
\[ \Theta=0;\ 0.4;\ 0.6;\ 0.8;\ 1.0. \tag{7} \]
To make it possible to use the results obtained in optimizing various technical-economic indices over a wide range of changes in the design conditions (the steel fill factors of the core \(K_c\) and the copper fill factor of the winding window \(K_o\); the costs and specific weights of the active materials \(\alpha_c,\ \alpha_o,\ \gamma_c,\ \gamma_o\)), we introduce the concept of the reduction coefficient \(\beta\).
Table 1
| Optimized index | Shielded reactor \(x_{\mathrm{opt}}\) | Shielded reactor \(y_{\mathrm{opt}}\) | Shielded reactor \(z_{\mathrm{opt}}\) | Core reactor, single-coil \(x_{\mathrm{opt}}\) | Core reactor, single-coil \(y_{\mathrm{opt}}\) | Core reactor, single-coil \(z_{\mathrm{opt}}\) | Core reactor, two-coil \(x_{\mathrm{opt}}\) | Core reactor, two-coil \(y_{\mathrm{opt}}\) | Core reactor, two-coil \(z_{\mathrm{opt}}\) | Notes | Notes |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Overall volume of reactors | 2,7 | 0,5 | 1,5 | 2,7 | 0,5 | 3,0 | 2,6 | 0,6 | 3,0 | I | |
| Overall volume of reactors | 1,8 | 0,5 | 3,8 | 2,7 | 0,5 | 5,0 | 1,6 | 0,5 | 5,0 | \(\Theta=0\) | |
| Overall volume of reactors | 2,7 | 0,5 | 1,9 | 2,7 | 0,5 | 5,0 | 1,5 | 0,5 | 5,0 | II | \(\Theta=0,4\) |
| Overall volume of reactors | 2,7 | 0,5 | 1,6 | 2,7 | 0,5 | 5,0 | 1,5 | 0,5 | 5,0 | \(\Theta=0,6\) | |
| Overall volume of reactors | 2,7 | 0,5 | 1,5 | 2,6 | 0,5 | 4,4 | 1,9 | 0,5 | 5,0 | \(\Theta=0,8\) | |
| Overall volume of reactors | 2,7 | 0,5 | 1,4 | 2,7 | 0,5 | 4,0 | 2,0 | 0,5 | 4,3 | \(\Theta=1,0\) |
Table 2
| \(\beta\) | Shielded reactor \(x_{\mathrm{opt}}\) | Shielded reactor \(y_{\mathrm{opt}}\) | Shielded reactor \(z_{\mathrm{opt}}\) | Core reactor, single-coil \(x_{\mathrm{opt}}\) | Core reactor, single-coil \(y_{\mathrm{opt}}\) | Core reactor, single-coil \(z_{\mathrm{opt}}\) | Core reactor, two-coil \(x_{\mathrm{opt}}\) | Core reactor, two-coil \(y_{\mathrm{opt}}\) | Core reactor, two-coil \(z_{\mathrm{opt}}\) | Notes |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | First design case |
| 0,2 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,6 | 0,5 | 1,0 | First design case |
| 0,5 | 2,7 | 0,5 | 1,0 | 2,6 | 0,5 | 1,2 | 2,0 | 0,5 | 1,2 | First design case |
| 0,86 | 2,3 | 0,5 | 1,0 | 2,3 | 0,5 | 1,5 | 2,1 | 0,7 | 1,4 | First design case |
| 1,0 | 2,2 | 0,5 | 1,0 | 2,5 | 0,6 | 1,6 | 2,0 | 0,7 | 1,5 | First design case |
| 1,5 | 2,2 | 0,6 | 1,2 | 2,5 | 0,7 | 1,7 | 2,2 | 0,9 | 1,8 | First design case |
| 2,0 | 2,0 | 0,6 | 1,3 | 2,6 | 0,8 | 2,0 | 2,2 | 1,0 | 2,0 | First design case |
| 2,3 | 2,2 | 0,7 | 1,4 | 2,5 | 0,8 | 2,1 | 2,3 | 1,0 | 2,2 | First design case |
| 2,6 | 2,2 | 1,0 | 1,5 | 2,6 | 0,9 | 2,2 | 2,2 | 1,1 | 2,2 | First design case |
| 3,0 | 2,3 | 0,8 | 1,6 | 2,5 | 0,9 | 2,3 | 2,1 | 1,1 | 2,4 | First design case |
| 4,0 | 2,1 | 0,8 | 1,7 | 2,6 | 1,0 | 2,6 | 2,2 | 1,3 | 2,7 | First design case |
| 5,0 | 2,2 | 0,9 | 1,8 | 2,6 | 1,1 | 2,8 | 2,2 | 1,4 | 2,9 | First design case |
Table 3
For the Brownian process in the second computational case
| $\beta$ | $\Theta=0,\ x_{\mathrm{opt}}$ | $\Theta=0,\ y_{\mathrm{opt}}$ | $\Theta=0,\ z_{\mathrm{opt}}$ | $\Theta=0.4,\ x_{\mathrm{opt}}$ | $\Theta=0.4,\ y_{\mathrm{opt}}$ | $\Theta=0.4,\ z_{\mathrm{opt}}$ | $\Theta=0.6,\ x_{\mathrm{opt}}$ | $\Theta=0.6,\ y_{\mathrm{opt}}$ | $\Theta=0.6,\ z_{\mathrm{opt}}$ | $\Theta=0.8,\ x_{\mathrm{opt}}$ | $\Theta=0.8,\ y_{\mathrm{opt}}$ | $\Theta=0.8,\ z_{\mathrm{opt}}$ | $\Theta=1.0,\ x_{\mathrm{opt}}$ | $\Theta=1.0,\ y_{\mathrm{opt}}$ | $\Theta=1.0,\ z_{\mathrm{opt}}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 |
| 0.2 | 1.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 |
| 0.5 | 1.2 | 0.5 | 1.0 | 2.2 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 |
| 0.86 | 1.2 | 0.7 | 1.0 | 1.5 | 0.5 | 1.0 | 1.8 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.3 | 0.5 | 1.0 |
| 1.0 | 1.1 | 0.7 | 1.0 | 1.6 | 0.6 | 1.0 | 1.6 | 0.5 | 1.0 | 1.8 | 0.5 | 1.0 | 2.0 | 0.5 | 1.0 |
| 1.5 | 1.2 | 0.9 | 1.0 | 1.6 | 0.7 | 1.0 | 1.6 | 0.7 | 1.0 | 1.9 | 0.7 | 1.0 | 1.8 | 0.6 | 1.0 |
| 2.0 | 1.2 | 1.0 | 1.3 | 1.3 | 0.8 | 1.1 | 1.5 | 0.8 | 1.1 | 1.8 | 0.8 | 1.1 | 2.0 | 0.8 | 1.1 |
| 2.3 | 1.1 | 1.0 | 1.4 | 1.7 | 0.9 | 1.2 | 1.6 | 0.9 | 1.2 | 1.6 | 0.8 | 1.2 | 1.8 | 0.8 | 1.2 |
| 2.6 | 1.2 | 1.1 | 1.5 | 1.3 | 0.9 | 1.4 | 1.5 | 0.9 | 1.3 | 1.7 | 0.9 | 1.3 | 1.9 | 0.9 | 1.2 |
| 3.0 | 1.1 | 1.2 | 1.8 | 1.3 | 1.0 | 1.4 | 1.5 | 0.9 | 1.5 | 1.8 | 1.0 | 1.4 | 2.0 | 1.0 | 1.4 |
| 4.0 | 1.1 | 1.2 | 2.0 | 1.3 | 1.1 | 1.8 | 1.4 | 1.1 | 1.6 | 1.6 | 1.1 | 1.6 | 1.8 | 1.1 | 1.6 |
| 5.0 | 1.1 | 1.3 | 2.3 | 1.2 | 1.2 | 1.9 | 1.4 | 1.2 | 1.9 | 1.5 | 1.2 | 1.8 | 1.7 | 1.2 | 1.8 |
Table 4
For a rod-type single-coil choke in the second calculation case
| $\beta$ | $\Theta=0$ $x_{\mathrm{opt}}$ | $\Theta=0$ $y_{\mathrm{opt}}$ | $\Theta=0$ $z_{\mathrm{opt}}$ | $\Theta=0{,}4$ $x_{\mathrm{opt}}$ | $\Theta=0{,}4$ $y_{\mathrm{opt}}$ | $\Theta=0{,}4$ $z_{\mathrm{opt}}$ | $\Theta=-0{,}6$ $x_{\mathrm{opt}}$ | $\Theta=-0{,}6$ $y_{\mathrm{opt}}$ | $\Theta=-0{,}6$ $z_{\mathrm{opt}}$ | $\Theta=-0{,}8$ $x_{\mathrm{opt}}$ | $\Theta=-0{,}8$ $y_{\mathrm{opt}}$ | $\Theta=-0{,}8$ $z_{\mathrm{opt}}$ | $\Theta=-1{,}0$ $x_{\mathrm{opt}}$ | $\Theta=-1{,}0$ $y_{\mathrm{opt}}$ | $\Theta=-1{,}0$ $z_{\mathrm{opt}}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 |
| 0,2 | 2,6 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 |
| 0,5 | 1,9 | 0,5 | 2,0 | 2,2 | 0,5 | 1,0 | 2,4 | 0,5 | 1,0 | 2,6 | 0,5 | 1,0 | 2,7 | 0,5 | 1,0 |
| 0,86 | 1,9 | 0,7 | 2,1 | 2,0 | 0,6 | 1,4 | 1,8 | 0,5 | 1,4 | 1,8 | 0,5 | 1,3 | 1,9 | 0,5 | 1,2 |
| 1,0 | 2,0 | 0,8 | 2,1 | 1,8 | 0,6 | 1,6 | 1,9 | 0,6 | 1,4 | 2,0 | 0,6 | 1,3 | 1,7 | 0,5 | 1,3 |
| 1,5 | 2,2 | 1,1 | 2,1 | 1,9 | 0,8 | 1,9 | 2,1 | 0,8 | 1,7 | 1,9 | 0,7 | 1,6 | 2,0 | 0,7 | 1,5 |
| 2,0 | 2,3 | 1,3 | 2,4 | 2,1 | 1,0 | 2,1 | 2,3 | 1,0 | 1,9 | 2,1 | 0,9 | 1,8 | 2,3 | 0,9 | 1,8 |
| 2,3 | 2,2 | 1,3 | 2,7 | 2,0 | 1,1 | 2,2 | 2,3 | 1,1 | 2,0 | 2,2 | 1,0 | 2,0 | 2,3 | 1,0 | 1,9 |
| 2,6 | 2,2 | 1,4 | 2,8 | 2,2 | 1,2 | 2,3 | 2,1 | 1,1 | 2,2 | 2,3 | 1,1 | 2,1 | 2,1 | 1,0 | 2,0 |
| 3,0 | 2,2 | 1,5 | 3,0 | 2,3 | 1,3 | 2,6 | 2,2 | 1,2 | 2,5 | 2,4 | 1,2 | 2,3 | 2,2 | 1,1 | 2,2 |
| 4,0 | 2,0 | 1,5 | 3,8 | 2,3 | 1,5 | 3,0 | 2,2 | 1,4 | 2,8 | 2,4 | 1,4 | 2,6 | 2,3 | 1,3 | 2,5 |
| 5,0 | 2,0 | 1,7 | 4,0 | 2,0 | 1,5 | 3,4 | 2,1 | 1,5 | 3,2 | 2,2 | 1,5 | 3,0 | 2,3 | 1,5 | 2,8 |
Table 5
For a core-type two-coil choke in the second calculation case
| $\beta$ | $\Theta=0$, $x_{\mathrm{opt}}$ | $\Theta=0$, $y_{\mathrm{opt}}$ | $\Theta=0$, $z_{\mathrm{opt}}$ | $\Theta=0.4$, $x_{\mathrm{opt}}$ | $\Theta=0.4$, $y_{\mathrm{opt}}$ | $\Theta=0.4$, $z_{\mathrm{opt}}$ | $\Theta=0.6$, $x_{\mathrm{opt}}$ | $\Theta=0.6$, $y_{\mathrm{opt}}$ | $\Theta=0.6$, $z_{\mathrm{opt}}$ | $\Theta=0.8$, $x_{\mathrm{opt}}$ | $\Theta=0.8$, $y_{\mathrm{opt}}$ | $\Theta=0.8$, $z_{\mathrm{opt}}$ | $\Theta=1$, $x_{\mathrm{opt}}$ | $\Theta=1$, $y_{\mathrm{opt}}$ | $\Theta=1$, $z_{\mathrm{opt}}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 |
| 0.2 | 2.2 | 0.5 | 1.2 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 | 2.7 | 0.5 | 1.0 |
| 0.5 | 1.7 | 0.5 | 3.0 | 1.6 | 0.5 | 1.2 | 1.6 | 0.5 | 1.0 | 1.7 | 0.5 | 1.0 | 1.8 | 0.5 | 1.0 |
| 0.86 | 1.3 | 0.5 | 4.2 | 1.4 | 0.6 | 1.9 | 1.6 | 0.7 | 1.3 | 1.5 | 0.6 | 1.3 | 1.4 | 0.6 | 1.1 |
| 1.0 | 1.4 | 0.6 | 4.2 | 1.5 | 0.7 | 1.9 | 1.5 | 0.7 | 1.5 | 1.5 | 0.7 | 1.3 | 1.6 | 0.7 | 1.2 |
| 1.5 | 1.2 | 0.7 | 4.1 | 2.6 | 1.1 | 2.4 | 1.5 | 0.9 | 2.0 | 1.5 | 0.9 | 1.7 | 1.5 | 0.9 | 1.5 |
| 2.0 | 1.3 | 0.9 | 5.0 | 1.4 | 1.0 | 2.9 | 1.6 | 1.1 | 2.3 | 1.6 | 1.1 | 2.0 | 1.4 | 1.0 | 1.8 |
| 2.3 | 1.3 | 1.0 | 5.0 | 1.3 | 1.0 | 3.3 | 1.4 | 1.1 | 2.7 | 1.5 | 1.1 | 2.2 | 1.5 | 1.1 | 2.0 |
| 2.6 | 1.3 | 1.1 | 5.0 | 1.5 | 1.2 | 3.4 | 1.5 | 1.2 | 2.8 | 2.0 | 1.3 | 2.5 | 1.5 | 1.2 | 2.1 |
| 3.0 | 1.3 | 1.2 | 5.0 | 1.5 | 1.3 | 3.7 | 1.5 | 1.3 | 3.1 | 1.5 | 1.3 | 2.6 | 1.5 | 1.3 | 2.3 |
| 4.0 | 1.4 | 1.5 | 5.0 | 1.3 | 1.4 | 4.3 | 1.3 | 1.4 | 3.6 | 1.4 | 1.4 | 3.2 | 1.5 | 1.5 | 2.8 |
| 5.0 | 1.3 | 1.6 | 5.0 | 1.3 | 1.6 | 4.8 | 1.4 | 1.6 | 4.3 | 1.3 | 1.5 | 3.7 | 1.3 | 1.5 | 3.3 |
In the analysis for a minimum expenditure of conductor material,
\[ \beta = 0. \tag{8} \]
In the analysis for a minimum total volume of active materials,
\[ \beta = 1. \tag{9} \]
In the analysis for minimum weight,
\[ \beta = \frac{\gamma_c K_c}{\gamma_0 K_0}. \tag{10} \]
In the analysis for minimum cost,
\[ \beta = \frac{\gamma_c K_c \alpha_c}{\gamma_0 K_0 \alpha_0}. \tag{11} \]
Tables 1–5 present the results obtained in the investigation. The method for selecting the optimal geometry from the data of Tables 1–5 is extremely simple and can be widely used in engineering practice for designing open-type smoothing chokes on non-standardized cores.
REFERENCES
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E. I. Goldshtein. Some problems in the design of optimal smoothing chokes, IVUZ, Electromechanics, No. 4, 1964.
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E. I. Goldshtein. On the choice of the geometry of chokes for smoothing filters. Izvestiya TPI, vol. 130, 1964.
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RTM of radio electronics. Filter chokes of the rod type with laminated detachable cores, Standard calculation NO.475.500 PKB, 1962.