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István Ecsedi Institute of Applied Mechanics, Faculty of Mechanical Engineering and Informatics, University of Miskolc, Miskolc-Egyetemváros, Hungary

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Ákos József Lengyel Institute of Applied Mechanics, Faculty of Mechanical Engineering and Informatics, University of Miskolc, Miskolc-Egyetemváros, Hungary

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## Abstract

The paper deals with the capacitance of cylindrical two-dimensional capacitor which consists of Cartesian orthotropic dielectric material. The determination of the capacitance of capacitor with orthotropic dielectric material by a suitable coordinate transformation is reduced to the computation of capacitance of an isotropic capacitor. It is proven that the capacitance of a Cartesian orthotropic capacitor can be obtained in terms of an isotropic capacitor whose dielectric constant is the geometric mean of the dielectric constant of the orthotropic capacitor.

## Abstract

The paper deals with the capacitance of cylindrical two-dimensional capacitor which consists of Cartesian orthotropic dielectric material. The determination of the capacitance of capacitor with orthotropic dielectric material by a suitable coordinate transformation is reduced to the computation of capacitance of an isotropic capacitor. It is proven that the capacitance of a Cartesian orthotropic capacitor can be obtained in terms of an isotropic capacitor whose dielectric constant is the geometric mean of the dielectric constant of the orthotropic capacitor.

## 1 Introduction

Combining different computation methods with analytical procedures is a common method for solving electrical problems. Paper  developes a computation method determining eddy currents effect in electrical steel sheets. Ecsedi and Baksa formulate a mathematical model to obtain upper and lower bounds for a three-dimensional hollow non-homogeneous body .

This study includes the determination of a cylindrical two-dimensional capacitor. The considered capacitor consists of homogeneous and orthotropic dielectric materials. The capacitance is the ability of a capacitor to store electric charge per unit voltage across its inner and outer surfaces. The capacitance is a function depending on the geometry of a capacitor and the permittivity of its dielectric material. The solution of the capacitance of capacitor with Cartesian orthotropic dielectric materials is reduced to the problem of capacitor with isotropic dielectric material with a suitable homogeneous linear coordinate transformation. Knowledge of electricity, which is necessary for formulating and solving the set task, can be found in detail in books [3–6]. This type of the presented approach was used in paper  to solve the Saint-Venant's torsion problem of Cartesian orthotropic elastic bars.

## 2 Governing equations of electrostatic field for two-dimensional capacitor

Figure 1 shows a two-dimensional hollow plane domain $A$ whose inner boundary curve is $∂A1$ and outer boundary curve is $∂A2$. The origin of the Cartesian coordinate system $Oxy$ is an inner point of closed curve $∂A1$, and the unit vectors of the $Oxy$ coordinate system are $ex$, $ey$ and $r=xex+yey$ denotes the position vector of an arbitrary point $P$ in $A¯=A∪∂A1∪∂A2$. To give the concept of capacitor for two-dimensional hollow domain shown in Fig. 1 the following boundary value problem is defined
$∇·ε·∇U=0,r∈A,$
$U(r)=U1,r∈∂A1,U(r)=U2,r∈∂A2,U1≠U2.$
In Eqs (1) and (2) $U=U(r)$ is the electric potential, $ε$ is a two-dimensional second order positive definite tensor called the permittivity tensor of the Cartesian orthotropic dielectric material and $∇$ is the two-dimensional Nabla operator
$∇=ex∂∂x+ey∂∂y.$
In Eq. (1) the scalar product is denoted by dot. The representation of permittivity tensor in the coordinate system $Oxy$ is as follows
$ε=εxex∘ex+εyey∘ey,$
where the circle between two vectors denotes their diadic product. The matrix representation of permittivity tensor will be used
$ε=εx00εy.$
In the case of isotropic dielectric material
$ε=ε(ex∘ex+ey∘ey)=ε=ε1,$
where $1$ is the two-dimensional second order unit tensor.
Denote $C$ the capacitance of the two-dimensional capacitor. The unit of $C$ is $F/m$. The electric energy of the capacitor can be obtained as [3–6]
$W=C2(U1−U2)2.$
Formula (7) is reformulated by the use of a new function $u=u(r)$. The connection between $U=U(r)$ and $u=u(r)$ is as follows
$U(r)=(U1−U2)u(r)+U2.$
It is evident that $u=u(r)$ satisfies the following Dirichlet type boundary value problem
$∇·ε(r)·∇U=0,r∈A,$
$u(r)=1,r∈∂A1,u(r)=0,r∈∂A2.$
The specific electric energy can be computed as [3–6].
$w=12E·D=12E·ε·E=12∇U·ε·∇U=12U1−U22∇u·ε·∇u,$
where $E$ is the electric field vector and $D$ is the electric displacement vector. The whole electric energy of the two-dimensional Cartesian orthotropic capacitor can be formulated as
$W=∫AwdA=12U1−U22∫A∇u·ε·∇u dA.$
Comparison of Eq. (7) with Eq. (12) gives an explicit formula for the capacitance $C$
$C=∫A∇u·ε·∇u dA.$

It must be noted that the capacitance of a capacitor of width $h$ is $h$ times that which is given by formula (13), where $h$ is the length of the capacitor.

## 3 Transformation of the governing boundary value problem

The law of the homogeneous linear transformation between the coordinates $(X,Y)$ and $(x,y)$ is defined as
$X=axx,Y=ayy,$
$ax=εy/εx4,ay=εx/εy4.$
Let
$F∼i(X,Y)=0,(X,Y)∈∂A∼i$
be the equation of boundary curve $∂A∼i$ $(i=1,2)$. The plane domain whose inner boundary is curve $∂A∼1$ and its outer boundary curve is $∂A∼2$ in the plane $O∼XY$ is denoted by $A∼$ (Fig. 2).
Let
$Fi(x,y)=0,(i=1,2)$
be the equation of the boundary curve $∂Ai$ $(i=1,2)$ (Fig. 1). It is evident that
$Fi(x,y)=F∼i(axx,ayy),(i=1,2).$
The one-to-one map given by Eq. (14) preserves the area. This fact follows from Eq. (19)
$dA∼=dXdY=|∂(X,Y)∂(x,y)|dxdy=|ax00ay|dxdy=|εy/εx400εx/εy4|dA=dA.$
Let $u=u(x,y)$ be defined as
$u(x,y)=u∼(axx,ayy),$
which is the solution of the boundary value problem formulated by the following equations
$∂2u∼∂X2+∂2u∼∂Y2=0,(X,Y)∈A∼,$
$u∼(X,Y)=1,(X,Y)∈∂A∼1,$
$u∼(X,Y)=0,(X,Y)∈∂A∼2.$
Substituting the following expressions
$u∼=u∼X,Y,ε=ε1$
in Eq. (13) yields
$C(ε)=C∼=ε∫A[(∂u∼∂X)2+(∂u∼∂Y)2]dA∼.$
From Eq. (20) it follows that
$∂u∂x=ax∂u∼∂X,∂2u∂x2=ax2∂2u∼∂X2,$
$∂u∂y=ay∂u∼∂Y,∂2u∂y2=ay2∂2u∼∂Y2.$
A simple computation results that
$εx∂2u∂x2+εy∂2u∂y2=εxax2∂2u∼∂X2+εyay2∂2u∼∂Y2=εxεy(∂2u∼∂X2+∂2u∼∂Y2),(x,y)∈A,(X,Y)∈A∼$

## 4 Determination of the capacitance of anisotropic capacitor

According to Eqs (26) and (27) it follows that
$Cεx,εy=∫A∇u·ε·∇u dA=∫A∼εxax2∂u∼∂X2+εyay2∂u∼∂Y2dA∼=εxεy∫A∂u∼∂X2+∂u∼∂Y2dA∼=Cεxεy.$
Eq. (29) shows that the capacitance of a Cartesian orthotropic capacitor can be expressed in terms of an isotropic capacitor whose dielectric constant is the geometric mean of the dielectric constants of the orthotropic capacitor. Eq. (29) gives a possibility to obtain the capacitance of a Cartesian orthotropic two-dimensional capacitor in terms of an isotropic capacitor whose dielectric constant is
$ε=εxεy.$

## 5 Example

Let the isotropic two-dimensional capacitor be a cylindrical capacitor whose inner boundary circle is $∂A∼1$ and outer boundary circle is $∂A∼2$ and
$X2+Y2−R12=0,(X,Y)∈∂A∼1,$
$X2+Y2−R22=0,(X,Y)∈∂A∼2,$
and $0 (Fig. 3). The capacitance of isotropic hollow circular capacitor [3–7] is
$C(ε)=2πε(lnR2R1)−1.$
In the present problem
$Fi(x,y)=F∼i(axx,ayy)=0,(i=1,2).$
A detailed form of the equation of boundary contour $∂Ai$ is as follows (Fig. 4)
$x2αi2+y2βi2−1=0,(i=1,2),$
$αi=Riεx/εy4,βi=Riεy/εx4.$
The capacitance of anisotropic capacitor with elliptical boundary curves is
$C(εx,εy)=2πεxεy(lnR2R1)−1.$
The solution of the boundary value problem
$εx∂2u∂x2+εy∂2u∂y2=0r∈A,$
$u(r)=1,r∈∂A1,u(r)=0,r∈∂A2,$
is as follows
$u(x,y)=k ln((ax2x2+ay2y2)/R22),$
where $k=(ln(R1R2)2)−1.$
The equation of the boundary curve in polar coordinates
$r=x2+y2,φ=arctan (y/x)$
is
$|OPi¯|=ρi(φ)0≤φ≤2π,=αiβi(αi2sin2φ+βi2cos2φ)−0.5,(i=1,2).$
The function $u=u(x,y)$ can be represented in polar coordinates as
$v(r,φ)=u(r cos φ,r sin φ)=kln[r2R22(ax2cos2φ+ay2sin2φ)].$
Let $Vj=Vj(r)$ be defined as
$Vj(r)=v(r,φj),ρ1(φj)≤r≤ρ2(φj).$

The graphs of $Vj(r)$ for $ρ1(φj)≤r≤ρ2(φj)$ are shown in Figs 5 and 6 for $φj=0$ and $φj=π/2$. The contour lines of the function $u=u(x,y)$ are presented in Fig. 7. The following numerical data were used $εx=8·10−12$ (F m−1), $εy=8.5·10−12$ (F m−1), $R1=0.025$ m, $R2=0.04$ m.

## 6 Conclusions

The considered two-dimensional cylindrical capacitor consists of homogeneous Cartesian orthotropic dielectric material. The capacitance of orthotropic capacitor is expressed in terms of a homogeneous capacitor whose permittivity is the geometrical mean of the principle values of the Cartesian orthotropic capacitor.

## References

• 

J. Pippuri and A. Arkkio, “2D – 1D time–harmonic model for rotating electrical machines,” Pollack Period., vol. 1, no. 3, pp. 7990, 2006.

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• 

I. Ecsedi and A. Baksa, “Bounds for the electrical resistance for non-homogeneous conducting body,” Pollack Period., vol. 18, no. 1, pp. 172176, 2023.

• Export Citation
• 

K. Simonyi, Foundations of Electrical Engineering: Fields – Networks – Waves. Oxford: Pergamon Press, 2016.

• 

E. M. Purcell and D. J. Morin, Electricity and Magnetism. CambridgeCambridge University Press, 2013.

• 

R. E. Thomas, A. J. Rosa, and G. J. Toussaint, The Analysis and Design of Linear Circuits. New York: Wiley, 2023.

• 

A. K. Raychaudhuri, Classical Theory of Elecricity and Magnetism. Berlin: Springer, 2022.

• 

I. Ecsedi and A. Baksa, “A method for the solution of uniform torsion of Cartesian orthotropic bar,” J. Theor. Appl. Mech., vol. 52, pp. 129143, 2022.

• Export Citation
• 

J. Pippuri and A. Arkkio, “2D – 1D time–harmonic model for rotating electrical machines,” Pollack Period., vol. 1, no. 3, pp. 7990, 2006.

• Export Citation
• 

I. Ecsedi and A. Baksa, “Bounds for the electrical resistance for non-homogeneous conducting body,” Pollack Period., vol. 18, no. 1, pp. 172176, 2023.

• Export Citation
• 

K. Simonyi, Foundations of Electrical Engineering: Fields – Networks – Waves. Oxford: Pergamon Press, 2016.

• 

E. M. Purcell and D. J. Morin, Electricity and Magnetism. CambridgeCambridge University Press, 2013.

• 

R. E. Thomas, A. J. Rosa, and G. J. Toussaint, The Analysis and Design of Linear Circuits. New York: Wiley, 2023.

• 

A. K. Raychaudhuri, Classical Theory of Elecricity and Magnetism. Berlin: Springer, 2022.

• 

I. Ecsedi and A. Baksa, “A method for the solution of uniform torsion of Cartesian orthotropic bar,” J. Theor. Appl. Mech., vol. 52, pp. 129143, 2022.

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