On the Choice of the Optimal Gap in Chokes of Smoothing Filters
Abstract Generated abstract
This paper addresses the selection of the optimal air gap in smoothing filter chokes, where simplified engineering formulas that neglect flux bulging and leakage can produce appreciable errors at higher ampere turns. It develops a calculation procedure using a fictitious gap coefficient that combines bulging and leakage effects, derives graphical dependences for the relative gap and coefficient, and illustrates the method for a U-shaped magnetic circuit using permeance data from prior experimental curves. Comparison with experimental results for several choke configurations suggests that the proposed dependences give errors within about 10 percent over the stated relative-gap range, and a revised coefficient is recommended for E-310 electrical steel.
Full Text
On the Choice of the Optimal Gap in Chokes of Smoothing Filters*)
Postgraduate student E. I. Goldstein
For modern grades of electrical steels, in particular for E-310, the total length of the optimal gap $\delta_{\mathrm{opt}}$ can be determined from the formula in [1, p. 107]:
\[ \delta_{\mathrm{opt}} = 1.6 \cdot 10^{-4} I_0 W \frac{Q_z}{Q_c}, \qquad [\mathrm{cm}], \tag{1} \]
where
$I_0$ — the dc component of the rectified current;
$W$ — the number of turns of the choke winding;
$Q_z$ — the “widened” cross-sectional area of the gap;
$Q_c$ — the cross-sectional area of the core steel.
In the practice of engineering calculations, because of the difficulty of determining the value $Q_z$, “bulging” and leakage of the magnetic flux are neglected [1, formulas (4, 5)]:
\[ \delta_{\mathrm{opt}} = 1.6 \cdot 10^{-4} I_0 W \qquad [\mathrm{cm}] \tag{2} \]
But the use of formula (2) leads to appreciable errors, especially when $I_0 W > 1000$ ampere-turns; therefore, in most cases the value of the optimal gap has to be refined experimentally. The objective of the present work is to obtain graphical dependences suitable for choosing the optimal gap with allowance for both “bulging” and leakage of the magnetic flux.
To relate the “widened” and the true ($Q_0$) cross sections of the gap we use expression (3):
\[ Q_z = Q_0 k_\phi , \tag{3} \]
where the coefficient of the fictitious gap $K_\phi$ relates the real gap $\delta$ to the gap $\delta_\phi$ of a fictitious choke having the same geometrical dimensions and a conductance of the main gap $G_0$ equal to the total conductance $G_p$ of the magnetic system of the choke under consideration, i.e.,
\[ K_\phi = \frac{\delta}{\delta_\phi} = \frac{G_p}{G_0}. \tag{4} \]
*) The work was carried out under the supervision of Doctor of Technical Sciences, Professor I. D. Kutyavin.
The relation of \(Q_0\) to \(Q_C\) is determined by expression (5):
\[ Q_C = Q_0 K_C , \tag{5} \]
where \(K_C\) is the coefficient of filling of the core with steel. From (1), taking into account (3) and (5), we obtain:
\[ \delta_{\mathrm{opt}} = 1.6 \cdot 10^{-4} I_0 W \frac{K_\Phi}{K_C}. \tag{6} \]
Let us pass to the relative air gap \(\delta'\) and the specific ampere-turns \(aw_0\):
\[ \delta' = \frac{\delta}{l_C}, \tag{7} \]
\[ aw_0 = \frac{I_0 W}{l_C}, \tag{8} \]
where \(l_C\) is the length of the line of force in the steel.
From (6), (7), and (8) we obtain:
\[ \frac{\delta'_{\mathrm{opt}}}{K_\Phi} = 1.6 \cdot 10^{-4} aw_0 \frac{1}{K_C}. \tag{9} \]
To determine the optimal gap from expression (9), it is necessary to have the dependence \(\delta'/K_\Phi = f(\delta')\).
In [2] a method is given for obtaining the dependence of the bulging coefficient \(K_{\mathrm{v}}\) on the relative gap. In the same way, dependences may be obtained for the leakage coefficient \(K_P\) and the fictitious-gap coefficient \(K_\Phi\), taking into account that:
\[ K_\Phi = K_{\mathrm{v}} + K_P - 0.5. \tag{10} \]
Figures 1 and 2 show a U-shaped circuit and its equivalent circuit. Table 1 gives the principal geometry parameters, as well as the coordinates of the bulging and leakage fields used in determining the individual permeances.
Table 1.
| Main dimensions, mm | Main dimensions, mm | Main dimensions, mm | Main dimensions, mm | Geometry parameters | Geometry parameters | Geometry parameters | Field coordinates | Field coordinates | Field coordinates | Field coordinates |
|---|---|---|---|---|---|---|---|---|---|---|
| \(a\) | \(b\) | \(c\) | \(h\) | \(x\) | \(y\) | \(z\) | \(y'_{\mathrm{v}}\) | \(\theta''_{\mathrm{v}}\) | \(\theta_a\) | \(\theta_{ra}\) |
| 20 | 24 | 30 | 80 | 1.2 | 1.5 | 4 | \(0.5c\) | \(a'\) | \(a\) | \(a\) |
The bulging coefficient is determined by expression (11):
\[ K_{\mathrm{v}} = 0.5 \left( 1 + g_{\mathrm{pr}} \frac{\delta}{ax} \right), \tag{11} \]
\(0.5\) is the coefficient accounting for the presence of two gaps;
\(g_{\mathrm{pr}}\) is the reduced permeance.
\[ g_{\mathrm{pr}} = g_a + x \left(g'_{\mathrm{v}} + 0.5 g''_{\mathrm{v}}\right). \tag{12} \]
The specific permeance \(g_a\) is determined as the sum of the specific permeances from face “\(a\)” \((g_{ra})\) and from the “ribs” of the end surface \((g_{rta})\).
\[ g_a = g_{ra} + g_{rta}. \tag{13} \]
Fig. 1.
Fig. 2.
Visible labels in the figure: \(G_{\mathrm{rpa}}\), \(\delta_p\), \(G_p\), \(G'_p\), \(G_0 K_\delta\), \(\varepsilon\).
Similarly:
\[ g'_{\mathrm{v}} = g'_{\mathrm{gv}} + g'_{\mathrm{rtv}}, \tag{14} \]
\[ g''_{\mathrm{v}} = g''_{\mathrm{gv}} + g''_{\mathrm{rtv}}. \tag{15} \]
The specific permeances from the face and the “edges” are determined from experimental curves obtained by B. K. Bulem [3, Figs. 6, 8, 9]. The use of the indicated curves makes it possible to take into account the effect on the magnitude of the specific permeance of the pole dimension in the direction perpendicular to the face under consideration (dimension \(\delta\)). To determine the leakage coefficient it is necessary to calculate the total leakage permeance \(G_{\mathrm{r}}\):
\[ G_{\mathrm{r}} = G_{\mathrm{t}} = 2G_{\mathrm{rga}} + 2G_{\mathrm{rpa}} \tag{16} \]
The permeance between faces “v”
\[ G_{\mathrm{t}} = \mu_0 \frac{b}{c}\, l'_{\mathrm{r}}, \tag{17} \]
where the calculated length of the leakage field \(l'_{\mathrm{r}}\) is determined from the corresponding coordinate of the bulging field
\[ l'_{\mathrm{r}} = h - \theta'_{\mathrm{v}}. \tag{18} \]
The permeance between faces “a,” \(G_{\mathrm{rga}}\), is determined, in accordance with the recommendations, from the curves of Kremp and Calderwood [3, Fig. 13]:
\[ G_{\mathrm{rga}} = 0.5\mu_0 g_{\mathrm{rga}} l''_{\mathrm{r}}, \tag{19} \]
where the calculated length of this leakage field is:
\[ l''_{\mathrm{r}} = h - \theta_{\mathrm{a}}. \tag{20} \]
The permeance between the “edges” of the faces “a” is determined analogously:
\[ G_{\mathrm{rpa}} = 0.5\mu_0 g_{\mathrm{rpa}} l''_{\mathrm{r}}. \tag{21} \]
Here the specific permeance, when using the curve of Kremp and Calderwood, is:
\[ g_{\mathrm{rpa}} = 0.1. \tag{22} \]
From expressions (16)–(22), taking into account the relation of the geometric dimensions to the parameters of the core geometry, we obtain:
\[ K_{\mathrm{r}} = 0.5 + 0.333 \frac{\delta}{a} \left[ \frac{1}{y}\left(z - \frac{\theta'_{\mathrm{v}}}{a}\right) + \frac{1}{x}\left(z - \frac{\theta_{\mathrm{a}}}{a}\right) \left(g_{\mathrm{rga}} + 0.1\right) \right] \tag{23} \]
It should be noted that expression (23) was obtained for the leakage coefficient by the number of flux linkages, i.e., for an alternating-current magnetic system, with the following relation between the reduced \((G'_{\mathrm{r}})\) and geometrical \((G_{\mathrm{r}})\) leakage permeances [4]
\[ G'_{\mathrm{r}} = 0.333G_{\mathrm{r}}. \tag{24} \]
Table 2 gives the main stages of the calculations necessary to obtain the dependence
\[ \frac{\delta'}{K_{\Phi}} = f(\delta'), \]
constructed in Fig. 3. On
In this same figure the dependence \(K_{\phi}=f(\delta')\) is given, and the results of its experimental verification, carried out for two values of the relative gap, are shown.
Fig. 3.
A comparison of the calculated and experimental data for the optimal air gap is made in Table 3; moreover, it should be noted that the curves in Fig. 3 are given for one gap, and therefore the calculated \(\delta'\) is for one gap of the U-shaped system.
Table 2.
| \(\dfrac{\Omega}{\Delta}=\dfrac{a}{\delta}\) | \(2,5\) | \(3\) | \(4\) | \(6\) | \(10\) | \(15.\) | Note |
|---|---|---|---|---|---|---|---|
| \(\dfrac{\gamma'_v}{\Delta}\) | \(1,88\) | \(2,25\) | \(3\) | \(4,5\) | \(7,5\) | \(11,3\) | \(\Delta=\delta\) |
| \(\dfrac{\theta'_v}{\Delta}\) | \(0,7\) | \(0,8\) | \(1,2\) | \(2,4\) | \(4,7\) | \(7,7\) | \(\Omega=a\) |
| \(g'_{gv}\) | \(0,74\) | \(0,68\) | \(0,86\) | \(1,18\) | \(1,5\) | \(1,71\) |
Table 2
| Quantity | 2,5 | 3 | 4 | 6 | 10 | 15 | Note |
|---|---|---|---|---|---|---|---|
| $\dfrac{\Omega}{\Delta}=\dfrac{a}{\delta}$ | 2,5 | 3 | 4 | 6 | 10 | 15 | |
| $g'_{\mathrm{rtv}}$ | 0,35 | 0,34 | 0,33 | 0,31 | 0,29 | 0,28 | |
| $g'_{\mathrm{v}}$ | 1,09 | 1,02 | 1,19 | 1,49 | 1,79 | 1,99 | |
| $\dfrac{\Omega}{\Delta}$ | 2,5 | 3 | 4 | 6 | 10 | 15 | $\Omega=a$ |
| $\dfrac{a}{\delta}$ | 1,25 | 1,5 | 2 | 3 | 5 | 7,5 | $\Delta=0,5\delta$ |
| $\dfrac{\theta''_{\mathrm{v}}}{\Delta}$ | 2,5 | 3 | 4 | 6 | 10 | 15 | |
| $g''_{\mathrm{gv}}$ | 1,6 | 1,66 | 1,75 | 1,93 | 2,16 | 2,33 | |
| $g''_{\mathrm{rtv}}$ | 0,35 | 0,34 | 0,33 | 0,31 | 0,29 | 0,28 | |
| $g''_{\mathrm{v}}$ | 1,95 | 2,0 | 2,08 | 2,24 | 2,45 | 2,61 | |
| $\dfrac{\Omega}{\Delta}$ | 2,5 | 3 | 4 | 6 | 10 | 15 | $\Omega=ax$ |
| $\dfrac{a}{\delta}$ | 1,04 | 1,25 | 1,67 | 2,5 | 4,18 | 6,25 | $\Delta=0,5\delta$ |
| $\dfrac{\theta_a}{\Delta}$ | 2,08 | 2,5 | 3,34 | 5,0 | 8,36 | 12,5 | $2x=2,4$ |
| $g_{\mathrm{ga}}$ | 1,46 | 1,5 | 1,6 | 1,75 | 1,98 | 2,13 | |
| $g_{\mathrm{rta}}$ | 0,35 | 0,34 | 0,33 | 0,31 | 0,29 | 0,28 | |
| $g_a$ | 1,81 | 1,84 | 1,93 | 2,06 | 2,27 | 2,41 | |
| $\dfrac{a}{\delta}$ | 2,5 | 3 | 4 | 6 | 10 | 15 | |
| $g_a$ | 2,06 | 2,14 | 2,26 | 2,4 | 2,63 | 2,80 | |
| $x\left(g'_{\mathrm{v}}+0,5g''_{\mathrm{v}}\right)$ | 2,62 | 2,56 | 2,84 | 3,31 | 3,81 | 4,21 | |
| $g_{\mathrm{pr}}$ | 4,68 | 4,70 | 5,10 | 5,71 | 6,44 | 7,01 | |
| $K_{\mathrm{v}}$ | 1,28 | 1,15 | 1,03 | 0,89 | 0,77 | 0,69 | |
| $\dfrac{\theta'_{\mathrm{v}}}{\delta}$ | 0,7 | 0,8 | 1,2 | 2,4 | 4,7 | 7,7 | |
| $\dfrac{\theta'_{\mathrm{v}}}{a}$ | 0,28 | 0,27 | 0,3 | 0,4 | 0,47 | 0,51 | |
| $\dfrac{1}{y}\left(z-\dfrac{\theta'_{\mathrm{v}}}{a}\right)$ | 2,48 | 2,49 | 2,47 | 2,4 | 2,36 | 2,33 | |
| $\dfrac{1}{x}\left(z-\dfrac{\theta'_a}{a}\right)\left(g_{\mathrm{rga}}+0,1\right)$ | 2,7 | 2,7 | 2,7 | 2,7 | 2,7 | 2,7 | $\dfrac{\theta^{[[unclear]]}_{\mathrm{ga}}}{\Delta}=\dfrac{a^{[[unclear]]}}{0,75a};$ |
| $K_{\mathrm{r}}$ | 1,69 | 1,58 | 1,43 | 1,28 | 1,17 | 1,11 | $g_{\mathrm{rga}}=0,98;$ |
| $K_{\mathrm{f}}$ | 1,97 | 1,73 | 1,46 | 1,17 | 0,94 | 0,8 | $l_C=14,14a$ |
| $\dfrac{\delta'_0}{\delta'_0}$ | 2,82 | 2,35 | 1,77 | 1,18 | 0,71 | 0,47 | $\delta'=\dfrac{100}{(a/\delta)\cdot14,14}$ |
| $\dfrac{\delta'}{K_{\mathrm{f}}}\,\%$ | 1,43 | 1,36 | 1,21 | 1,0 | 0,75 | 0,59 |
In determining the optimal gap, expression (25) was used:
\[ \frac{\delta'_{\mathrm{opt}}}{K_{\Phi}}=\frac{K_{\delta}}{K_C}\, a w_0 , \tag{25} \]
where \(K_{\delta}\) from expression (9) is equal to \(1.6\cdot 10^{-4}\).
Numerous experiments carried out by the author on П- and Ш-shaped cores showed that the best agreement between calculated and experimental data is obtained with the coefficient \(K_{\delta}\) equal to \(1.56\cdot 10^{-4}\), which gives, for \(K_C=0.9\) (for steel Э-310):
\[ \frac{K_{\delta}}{K_C}=1.4\cdot 10^{-4} \tag{26} \]
The data of Table 3 confirm the considerations stated.
Table 3.
| Experiment results \(I_0/a\) | Experiment results \(W\) | Experiment results \(\delta_{\mathrm{opt}},\ \mathrm{mm}\) | Calculation results by expression \(a w_0/a,\ \mathrm{cm}\) | Calculation results by expression, for \(\frac{K_{\delta}}{K_C}=1.78\cdot 10^{-4}\): \(\delta'/K_{\Phi}\) | Calculation results by expression, for \(\frac{K_{\delta}}{K_C}=1.78\cdot 10^{-4}\): \(\delta'\%\) | Calculation results by expression, for \(\frac{K_{\delta}}{K_C}=1.78\cdot 10^{-4}\): \(\delta_{\mathrm{opt}},\ \mathrm{mm}\) | Calculation results by expression, for \(\frac{K_{\delta}}{K_C}=1.4\cdot 10^{-4}\): \(\delta'/K_{\Phi}\) | Calculation results by expression, for \(\frac{K_{\delta}}{K_C}=1.4\cdot 10^{-4}\): \(\delta'\%\) | Calculation results by expression, for \(\frac{K_{\delta}}{K_C}=1.4\cdot 10^{-4}\): \(\delta'_{\mathrm{opt}},\ \mathrm{mm}\) | By expression (2): \(\delta_{\mathrm{opt}}\) sum., \(\mathrm{mm}\) | Note |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 800 | 0.9 | 28.3 | 0.005 | 0.37 | 1.05 | 0.004 | 0.3 | 0.85 | 1.28 | \(l_C=28.3\ \mathrm{cm}\); \(K_C=0.9\) |
| 2 | 800 | 2.0 | 56.6 | 0.010 | 1.2 | 3.4 | 0.008 | 0.77 | 2.2 | 2.56 | \(l_C=28.3\ \mathrm{cm}\); \(K_C=0.9\) |
| 1.6 | 800 | 1.5 | 45.2 | 0.008 | 0.77 | 2.2 | 0.0064 | 0.51 | 1.44 | 2.05 | \(l_C=28.3\ \mathrm{cm}\); \(K_C=0.9\) |
| 2 | 1200 | 5.0 | 85 | — | — | — | 0.012 | 1.72 | 4.86 | 3.84 | \(l_C=28.3\ \mathrm{cm}\); \(K_C=0.9\) |
| 1.8 | 1600 | 8.3 | 102 | — | — | — | 0.0142 | 2.70 | 7.56 | 4.6 | \(l_C=28.3\ \mathrm{cm}\); \(K_C=0.9\) |
CONCLUSIONS
-
The proposed method makes it possible to obtain dependences convenient for engineering calculations: \(K_{\Phi}=f(\delta')\) and \(\frac{\delta'}{K_{\Phi}}=f(\delta')\) at \(x=\mathrm{const}\).
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The errors when using the dependences obtained do not exceed 10% in the range \(0.4 \leq \delta'\% \leq 3.0\).
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It appears advisable to obtain the indicated dependences for standardized cores (\(z=\mathrm{const},\ y=\mathrm{const}\)) for several values of \(x\), covering the possible range of variation of this parameter.
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To determine the optimal gap for steel Э-310, expression (27) should be used:
\[ \frac{\delta'}{K_{\Phi}}=1.4\cdot 10^{-3}\, a w_0 \tag{27} \]
REFERENCES
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Bamdas A. M., Savinovskii Yu. A., Filter Chokes of Radio Equipment, Sovetskoe Radio Publishing House, 1962.
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Goldshtein E. I., “On accounting for field bulging in the calculation of transformers and chokes with air gaps,” Collection of Works of the Tomsk Branch of VNIIEM, 1963.
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Boole B. K., “Calculation of the magnetic permeances of air gaps for round and rectangular poles,” VZEI, 1961.
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Stupel F. A., Electromechanical Relays. Kharkov University Press, 1956.