Selection of Paper Sheet Thickness for Paper-Oil Power Capacitors
Abstract Generated abstract
This paper addresses the selection of paper sheet thickness in paper-oil power capacitors, where dielectric strength, dimensions, durability, and material cost are interdependent. It derives analytical expressions for the volumes and costs of paper, foil, and impregnating compound as functions of operating voltage, capacitance, paper thickness, number of sheets, pressing coefficient, and section geometry. By differentiating the resulting cost relation, the study obtains an equation for the paper thickness that minimizes material expenditure and notes that section dimensions have only a limited influence on this optimum. Calculations indicate that, for the considered power capacitor design and materials, the most economical paper thickness lies approximately in the range of 10 to 12 microns.
Full Text
Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Volume 204 1971
Selection of Paper Sheet Thickness for Paper-Oil Power Capacitors
V. S. Dmitrevskii
(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)
Paper-oil insulation is widely used in power capacitors.
The choice of the operating field strength and the thickness of the paper sheet determines the dimensions and cost of the capacitor. In order to obtain the most advantageous dimensions, minimum cost, and greatest durability of the capacitor, several variants are calculated, from which the economically most expedient one is selected. Naturally, such a calculation requires a greater expenditure of time.
In connection with the above, the task has been posed of finding the relation between the thickness of the paper sheet, the permissible operating field strength, the parameters of the capacitor, and its cost.
The cost of a capacitor with paper-oil insulation is found as
\[ C_k = C_b + C_f + C_p + C_{\mathrm{corp}} + C_{\mathrm{ins}} + C_{\mathrm{lead}}, \tag{1} \]
where \(C_b, C_f, C_p, C_{\mathrm{corp}}, C_{\mathrm{ins}}, C_{\mathrm{lead}}\) are, respectively, the cost of the paper, foil, impregnation, casing, insulation from the casing, and bushing insulator.
It should be noted that the last three components are small and practically do not depend (or depend little) on the thickness of the paper sheet. Therefore, in what follows we shall approximately assume that the cost of the capacitor is approximately equal to
\[ C_k = C_b + C_{\mathrm{imp}} + C_f. \tag{1a} \]
To determine the cost of the individual components, it is necessary to find their volume.
The initial data for the calculation are the operating voltage \(U_k\) and the capacitance of the capacitor \(C_k\).
The permissible operating field strength in the capacitor is found [1] as
\[ E_p = 3.6 \left( \frac{n\delta}{k_p} \right)^{-0.58} \tag{2} \]
where
\[ \begin{aligned} \delta &\text{ is the thickness of the paper sheet;}\\ n &\text{ is the number of sheets between the plates;}\\ k_p &\text{ is the pressing coefficient, } E_p \text{ in kV/mm.} \end{aligned} \]
The number of series sections in the capacitor
\[ m=\frac{U_{\mathrm{k}}}{U_{\mathrm{s}}}= \frac{U_{\mathrm{k}}}{\dfrac{n\delta}{\kappa_{\mathrm{p}}}E_{\mathrm{p}}} = \frac{U_{\mathrm{k}}}{3.6\left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0.42}} \tag{3} \]
In this expression, \(U_{\mathrm{k}}\) is in kV.
The capacitance of a roll-type section is determined as [2]
\[ C_{\mathrm{s}}=1.77\cdot 10^{-8}\, \frac{\varepsilon b\Delta} {2\dfrac{n\delta}{\kappa_{\mathrm{p}}} \left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}+d_{\mathrm{cp}}\right)} \left(l-\frac{\pi}{4}\Delta\right), \tag{4} \]
where
\[ \begin{aligned} C_{\mathrm{s}}&\text{ is the capacitance of a section, in }\mu\mathrm{F};\\ \varepsilon&\text{ is the relative dielectric permittivity of impregnated paper;}\\ b&\text{ is the foil width, in mm;}\\ \Delta&\text{ is the section thickness, in mm;}\\ d_{\mathrm{cp}}&\text{ is the foil thickness, in mm;}\\ l&\text{ is the section length, in mm.} \end{aligned} \]
The number of parallel sections is found from
\[ p=m\frac{C_{\mathrm{k}}}{C_{\mathrm{s}}} = \frac{ U_{\mathrm{k}} C_{\mathrm{k}} \left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0.58} \left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}+d_{\mathrm{cp}}\right) } { 3.2\cdot 10^{-8}\varepsilon b\Delta \left(l-\dfrac{\pi}{4}\Delta\right) }. \tag{5} \]
The volume of paper in the capacitor is determined by
\[ V_{\mathrm{b}}=pmL(b+2b_{\mathrm{z}})n\delta = \frac{ U_{\mathrm{k}}^{2}C_{\mathrm{k}}\,n\delta \left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0.16} } {23\cdot 10^{-8}\varepsilon} \cdot \frac{b+2b_{\mathrm{z}}}{b}, \tag{6} \]
where \(b_{\mathrm{z}}\) is the margin width;
\(L\) is the length of the plate in the capacitor section, which is found as
\[ L= \frac{\Delta} {2\left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}+d_{\mathrm{cp}}\right)} \left(l-\frac{\pi}{4}\Delta\right). \tag{7} \]
The volume of foil in the capacitor is
\[ V_{\mathrm{f}}=pmLbd_{\mathrm{cp}} = \frac{ U_{\mathrm{k}}^{2}C_{\mathrm{k}}d_{\mathrm{cp}} \left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0.16} } {23\cdot 10^{-8}\varepsilon}. \tag{8} \]
Let us arrange the capacitor sections so that the height of the stack is \(m\Delta\), the width is \(l\), and the length is \(p(b+2b_{\mathrm{z}})\). Naturally, another arrangement of the sections in the capacitor may also be adopted. The impregnating compound is consumed for impregnating the paper of the sections and for impregnating the insulation of the sections from the case:
\[ \begin{aligned} V_{\mathrm{pr}}&= 2\kappa_{2}lm\Delta\Delta_{1} +2\kappa_{2}pm(b+2b_{\mathrm{z}})\Delta\Delta_{1} +2\kappa_{2}p(b+2b_{\mathrm{z}})l\Delta_{1} +\kappa_{1}V_{\mathrm{b}} =\\ &= \frac{\kappa_{2}l\Delta\Delta_{1}U_{\mathrm{k}}} {1.8\left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0.42}} + \frac{ \kappa_{2}U_{\mathrm{k}}^{2}C_{\mathrm{k}}\Delta_{1} \left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0.16} \left(\dfrac{n\delta}{\kappa_{\mathrm{p}}}+d_{\mathrm{cp}}\right) } { 5.7\cdot 10^{-8}\varepsilon \left(l-\dfrac{\pi}{4}\Delta\right) } \frac{b+2b_{\mathrm{z}}}{b} + \end{aligned} \tag{9} \]
\[ + \frac{\kappa_2 U_{\mathrm{k}} C_{\mathrm{k}} l \Delta_1 \left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,58} \left(\frac{n\delta}{\kappa_{\mathrm{p}}}+d_{\mathrm{cp}}\right)} {1,6 \cdot 10^{-8}\varepsilon \Delta \left(l-\frac{\pi}{4}\Delta\right)} \frac{b+2b_3}{b}+ \]
\[ + \frac{\kappa_1 U_{\mathrm{k}}^2 C_{\mathrm{k}} n\delta \left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,16}} {2,3 \cdot 10^{-8}\varepsilon} \frac{b+2b_3}{b}. \]
In this expression:
\(\Delta_1\) is the thickness of the insulation from the case;
\(\kappa_1; \kappa_2\) are the impregnation coefficients of the section and of the insulation from the case, respectively.
All linear dimensions in formulas (6), (8), (9) are substituted in mm, and the volumes have dimensions of \(\mathrm{mm}^3\).
Multiplying the volumes of the corresponding materials by their density and by the cost per unit weight, one can obtain the price of the capacitor. It is necessary to take into account that the cost of paper depends on the thickness of the paper sheet. For KON-1 paper, which is used for manufacturing power capacitors, the dependence of cost on the thickness of the paper sheet can be expressed as
\[ S_{\mathrm{b}} = 1,32 + 62,3 e^{-\frac{1,7\delta}{0,004}} \left[\frac{\mathrm{rub}}{\mathrm{kg}}\right]. \tag{10} \]
Finding the cost of the paper, foil, and impregnating compound and substituting them into expression (1a), we finally obtain:
\[ C_{\mathrm{k}} = V_{\mathrm{b}}\gamma_{\mathrm{b}}S_{\mathrm{b}} + V_{\mathrm{f}}\gamma_{\mathrm{f}}S_{\mathrm{f}} + V_{\mathrm{pr}}\gamma_{\mathrm{pr}}S_{\mathrm{pr}} = \]
\[ = \frac{U_{\mathrm{k}}^2 C_{\mathrm{k}} n\delta \left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,16}} {2,3 \cdot 10^{-8}\varepsilon} \frac{b+2b_3}{b}\, \gamma_{\mathrm{b}} \left(1,32+62,3 e^{-\frac{1,7\delta}{0,004}}\right)10^{-6}+ \]
\[ + \frac{U_{\mathrm{k}}^2 C_{\mathrm{k}} d_{\mathrm{cp}} \left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,16}} {2,3 \cdot 10^{-8}\varepsilon} \,\gamma_{\mathrm{f}}S_{\mathrm{f}}10^{-6} + \frac{\kappa_2 l\Delta\Delta_1 U_{\mathrm{k}}} {1,8\left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,42}} \,\gamma_{\mathrm{pr}}S_{\mathrm{pr}}10^{-6}+ \]
\[ + \frac{\kappa_2 U_{\mathrm{k}} C_{\mathrm{k}}\Delta_1 l \left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,58} \left(\frac{n\delta}{\kappa_{\mathrm{p}}}+d_{\mathrm{cp}}\right)} {1,6 \cdot 10^{-8}\varepsilon\Delta\left(l-\frac{\pi}{4}\Delta\right)} \frac{b+2b_3}{b}\, \gamma_{\mathrm{pr}}S_{\mathrm{pr}}10^{-6}+ \]
\[ + \frac{\kappa_2 U_{\mathrm{k}}^2 C_{\mathrm{k}}\Delta_1 \left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,16} \left(\frac{n\delta}{\kappa_{\mathrm{p}}}+d_{\mathrm{cp}}\right)} {5,7 \cdot 10^{-8}\varepsilon\Delta\left(l-\frac{\pi}{4}\Delta\right)} \frac{b+2b_3}{b}\, \gamma_{\mathrm{pr}}S_{\mathrm{pr}}10^{-6}+ \]
\[ + \frac{\kappa_1 U_{\mathrm{k}}^2 C_{\mathrm{k}} n\delta \left(\frac{n\delta}{\kappa_{\mathrm{p}}}\right)^{0,16}} {2,3 \cdot 10^{-8}\varepsilon} \frac{b+2b_3}{b}\, \gamma_{\mathrm{pr}}S_{\mathrm{pr}}10^{-6}. \tag{11} \]
Analysis of equation (11) shows that, as the thickness of the paper sheet increases, the cost of the capacitor varies along a curve
with a minimum. To find the thickness of the paper sheet corresponding to the minimum cost of the capacitor, we take the derivative
\[ \frac{dC_k}{d\delta} \]
and set it equal to zero. After transformations, neglecting small terms, we obtain:
\[ \begin{aligned} &1.53\,k_p\gamma_b +0.16\,k_p\frac{a_f\gamma_f S_f}{n\delta}\frac{b}{b+2b_z} +4.68\,\frac{k_2\Delta_1 S_{pr}\gamma_{pr}}{l-\frac{\pi}{4}\Delta} +{}\\ &\quad +22.7\,\frac{k_2 l\Delta_1\gamma_{pr}S_{pr}} {U_k\Delta\left(l-\frac{\pi}{4}\Delta\right)} \left(\frac{n\delta}{k_p}\right)^{0.42} +1.6\,k_1k_p\gamma_{pr}S_{pr} =\\ &\quad =(26500\delta-72.3)\gamma_b k_p e^{-\frac{1.7\delta}{0.004}} \end{aligned} \tag{12} \]
The dimensions of the section, as follows from equation (12), have little effect on the choice of the most advantageous thickness of the paper sheet. This makes it possible to select the section dimensions on the basis of the best heat removal and of obtaining the most compact design.
Equation (12) can be solved by specifying the section dimensions \(l\), \(b\), \(\Delta\), and the thickness of insulation from the case \(\Delta_1\). It is expedient to solve equation (12) graphically, assigning various values of \(\delta\). It should be noted that the most advantageous value of \(n\) is taken within the range 4–10.
The calculations performed showed that the most advantageous value of the paper thickness for power capacitors lies within the range 10–12 microns.
By differentiating equation (11) with respect to \(l\) or \(\Delta\), one can find the value of \(l\) and \(\Delta\) corresponding to the minimum cost of the capacitor.
Setting
\[ \frac{dC_k}{d\Delta}=0, \]
we obtain a formula for determining \(\Delta\) corresponding to the minimum cost of the capacitor:
\[ \Delta = 2.13\,\frac{l\left(\dfrac{n\delta}{k_p}\right)^{0.21}}{\sqrt{U_k}} . \tag{13} \]
However, the value \(\Delta\) is limited by technological requirements and does not take into account the cost of manufacturing the capacitor. Therefore formula (13) can be used only for approximate calculations.
In this work, the relationship between the cost of a capacitor and its dimensions has been found. An equation has been obtained that makes it possible to find the optimal thickness of the paper sheet on the basis of the minimum expenditure on materials.
References
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M. A. Greisukh, G. S. Kuchinskii, D. A. Kaplan, G. T. Messerman. Paper-oil insulation in high-voltage structures. Gosenergoizdat, 1963.
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V. T. Renne, Yu. V. Bagalei, I. D. Fridberg. Calculation and design of capacitors. Tekhnika Publishing House, Kiev, 1966.