Determination of the capacitance of capacitor with orthotropic dielectric material

István Ecsedi; Ákos József Lengyel

doi:10.1556/606.2023.00828

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Pollack Periodica

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Determination of the capacitance of capacitor with orthotropic dielectric material

Authors:

István Ecsedi

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

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

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akos.lengyel@uni-miskolc.hu https://orcid.org/0000-0002-0885-388X

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Online Publication Date:: 13 Oct 2023

Publication Date:: 13 Oct 2023

Article Category:: Research Article

DOI:: https://doi.org/10.1556/606.2023.00828

Keywords (English):: capacitance; two-dimensional capacitor; orthotropic dielectric material

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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

1 Introduction

Combining different computation methods with analytical procedures is a common method for solving electrical problems. Paper [1] 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 [2].

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 [7] 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

\partial A_{1}

and outer boundary curve is

\partial A_{2}

. The origin of the Cartesian coordinate system

O x y

is an inner point of closed curve

\partial A_{1}

, and the unit vectors of the

O x y

coordinate system are

e_{x}

e_{y}

and

{r = x e}_{x} + y e_{y}

denotes the position vector of an arbitrary point

P

\bar{A} = A \cup \partial A_{1} \cup \partial A_{2}

. To give the concept of capacitor for two-dimensional hollow domain shown in Fig. 1 the following boundary value problem is defined

\begin{array}{c} \nabla \cdot (ε \cdot \nabla U) = 0, & r \in A \end{array},

(1)

\begin{array}{c} \begin{array}{c} U (r) = U_{1}, & r \in \partial A_{1}, \\ U (r) = U_{2}, & r \in \partial A_{2}, \end{array} & U_{1} \neq U_{2} \end{array} .

(2)

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

\nabla

is the two-dimensional Nabla operator

\nabla = e_{x} \frac{\partial}{\partial x} + e_{y} \frac{\partial}{\partial y} .

(3)

In Eq. (1) the scalar product is denoted by dot. The representation of permittivity tensor in the coordinate system

O x y

is as follows

ε = ε_{x} e_{x} \circ e_{x} + ε_{y} e_{y} \circ e_{y},

(4)

where the circle between two vectors denotes their diadic product. The matrix representation of permittivity tensor will be used

ε = [\begin{array}{c} ε_{x} & 0 \\ 0 & ε_{y} \end{array}] .

(5)

In the case of isotropic dielectric material

ε = ε (e_{x} \circ e_{x} + e_{y} \circ e_{y}) = ε [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] = ε 1,

(6)

where

1

is the two-dimensional second order unit tensor.

Denote

C

the capacitance of the two-dimensional capacitor. The unit of

C

(F / m)

. The electric energy of the capacitor can be obtained as [3–6]

W = \frac{C}{2} {(U_{1} - U_{2})}^{2} .

(7)

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) = (U_{1} - U_{2}) u (r) + U_{2} .

(8)

It is evident that

u = u (r)

satisfies the following Dirichlet type boundary value problem

\begin{array}{c} \nabla \cdot (ε (r) \cdot \nabla U) = 0, & r \in A \end{array},

(9)

\begin{array}{c} u (r) = 1, & r \in \partial A_{1}, \\ u (r) = 0, & r \in \partial A_{2} . \end{array}

(10)

The specific electric energy can be computed as [3–6].

w = \frac{1}{2} E \cdot D = \frac{1}{2} E \cdot ε \cdot E = \frac{1}{2} \nabla U \cdot ε \cdot \nabla U = \frac{1}{2} {(U_{1} - U_{2})}^{2} \nabla u \cdot ε \cdot \nabla u,

(11)

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 = \int_{A} w d A = \frac{1}{2} {(U_{1} - U_{2})}^{2} \int_{A} \nabla u \cdot ε \cdot \nabla u d A .

(12)

Comparison of Eq. (7) with Eq. (12) gives an explicit formula for the capacitance

C

C = \int_{A} \nabla u \cdot ε \cdot \nabla u d A .

(13)

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

\begin{array}{c} X = a_{x} x, & Y = a_{y} y \end{array},

(14)

\begin{array}{c} a_{x} = \sqrt[4]{ε_{y} / ε_{x}}, & a_{y} = \sqrt[4]{ε_{x} / ε_{y}} \end{array} .

(15)

Let

\begin{array}{c} {\tilde{F}}_{i} (X, Y) = 0, & (X, Y) \in \partial {\tilde{A}}_{i} \end{array}

(16)

be the equation of boundary curve

\partial {\tilde{A}}_{i}

(i = 1,2)

. The plane domain whose inner boundary is curve

\partial {\tilde{A}}_{1}

and its outer boundary curve is

\partial {\tilde{A}}_{2}

in the plane

\tilde{O} X Y

is denoted by

\tilde{A}

(Fig. 2).

Let

\begin{array}{c} F_{i} (x, y) = 0, & (i = 1,2) \end{array}

(17)

be the equation of the boundary curve

\partial A_{i}

(i = 1,2)

(Fig. 1). It is evident that

\begin{array}{c} F_{i} (x, y) = {\tilde{F}}_{i} (a_{x} x, a_{y} y), & (i = 1,2) \end{array} .

(18)

The one-to-one map given by Eq. (14) preserves the area. This fact follows from Eq. (19)

\begin{array}{c} d \tilde{A} = d X d Y = | \frac{\partial (X, Y)}{\partial (x, y)} | d x d y = | \begin{array}{c} a_{x} & 0 \\ 0 & a_{y} \end{array} | d x d y \\ = | \begin{array}{c} \sqrt[4]{ε_{y} / ε_{x}} & 0 \\ 0 & \sqrt[4]{ε_{x} / ε_{y}} \end{array} | d A = d A . \end{array}

(19)

Let

u = u (x, y)

be defined as

u (x, y) = \tilde{u} (a_{x} x, a_{y} y),

(20)

which is the solution of the boundary value problem formulated by the following equations

\begin{array}{c} \frac{\partial^{2} \tilde{u}}{\partial X^{2}} + \frac{\partial^{2} \tilde{u}}{\partial Y^{2}} = 0, & (X, Y) \in \tilde{A} \end{array},

(21)

\begin{array}{c} \tilde{u} (X, Y) = 1, & (X, Y) \in {\partial \tilde{A}}_{1} \end{array},

(22)

\begin{array}{c} \tilde{u} (X, Y) = 0, & (X, Y) \in {\partial \tilde{A}}_{2} \end{array} .

(23)

Substituting the following expressions

\begin{array}{c} \tilde{u} = \tilde{u} (X, Y), & ε = ε 1 \end{array}

(24)

in Eq. (13) yields

C (ε) = \tilde{C} = ε \int_{A} [{(\frac{\partial \tilde{u}}{\partial X})}^{2} + {(\frac{\partial \tilde{u}}{\partial Y})}^{2}] d \tilde{A} .

(25)

From Eq. (20) it follows that

\begin{array}{c} \frac{\partial u}{\partial x} = a_{x} \frac{\partial \tilde{u}}{\partial X}, & \frac{\partial^{2} u}{\partial x^{2}} = a_{x}^{2} \frac{\partial^{2} \tilde{u}}{\partial X^{2}}, \end{array}

(26)

\begin{array}{c} \frac{\partial u}{\partial y} = a_{y} \frac{\partial \tilde{u}}{\partial Y}, & \frac{\partial^{2} u}{\partial y^{2}} = a_{y}^{2} \frac{\partial^{2} \tilde{u}}{\partial Y^{2}} \end{array} .

(27)

A simple computation results that

\begin{array}{c} \begin{array}{c} ε_{x} \frac{\partial^{2} u}{\partial x^{2}} + ε_{y} \frac{\partial^{2} u}{\partial y^{2}} = ε_{x} a_{x}^{2} \frac{\partial^{2} \tilde{u}}{\partial X^{2}} + ε_{y} a_{y}^{2} \frac{\partial^{2} \tilde{u}}{\partial Y^{2}} \\ = \sqrt{ε_{x} ε_{y}} (\frac{\partial^{2} \tilde{u}}{\partial X^{2}} + \frac{\partial^{2} \tilde{u}}{\partial Y^{2}}), \end{array} & \begin{array}{c} (x, y) \in A, & (X, Y) \in \tilde{A} \end{array} \end{array}

(28)

4 Determination of the capacitance of anisotropic capacitor

According to Eqs (26) and (27) it follows that

\begin{array}{c} C (ε_{x}, ε_{y}) = \int_{A} \nabla u \cdot ε \cdot \nabla u d A = \int_{\tilde{A}} [ε_{x} a_{x}^{2} {(\frac{\partial \tilde{u}}{\partial X})}^{2} + ε_{y} a_{y}^{2} {(\frac{\partial \tilde{u}}{\partial Y})}^{2}] d \tilde{A} \\ = \sqrt{ε_{x} ε_{y}} \int_{A} [{(\frac{\partial \tilde{u}}{\partial X})}^{2} + {(\frac{\partial \tilde{u}}{\partial Y})}^{2}] d \tilde{A} = C (\sqrt{ε_{x} ε_{y}}) . \end{array}

(29)

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 = \sqrt{ε_{x} ε_{y}} .

(30)

5 Example

Let the isotropic two-dimensional capacitor be a cylindrical capacitor whose inner boundary circle is

\partial {\tilde{A}}_{1}

and outer boundary circle is

\partial {\tilde{A}}_{2}

and

\begin{array}{c} X^{2} + Y^{2} - R_{1}^{2} = 0, & (X, Y) \in \partial {\tilde{A}}_{1} \end{array},

(31)

\begin{array}{c} X^{2} + Y^{2} - R_{2}^{2} = 0, & (X, Y) \in \partial {\tilde{A}}_{2} \end{array},

(32)

and

0 < R_{1} < R_{2}

(Fig. 3). The capacitance of isotropic hollow circular capacitor [3–7] is

C (ε) = 2 π ε {(\ln \frac{R_{2}}{R_{1}})}^{- 1} .

(33)

In the present problem

\begin{array}{c} F_{i} (x, y) = {\tilde{F}}_{i} (a_{x} x, a_{y} y) = 0, & (i = 1,2) \end{array} .

(34)

A detailed form of the equation of boundary contour

\partial A_{i}

is as follows (Fig. 4)

\begin{array}{c} \frac{x^{2}}{α_{i}^{2}} + \frac{y^{2}}{β_{i}^{2}} - 1 = 0, & (i = 1,2) \end{array},

(35)

\begin{array}{c} α_{i} = R_{i} \sqrt[4]{ε_{x} / ε_{y}}, & β_{i} = R_{i} \sqrt[4]{ε_{y} / ε_{x}} \end{array} .

(36)

The capacitance of anisotropic capacitor with elliptical boundary curves is

C (ε_{x}, ε_{y}) = 2 π \sqrt{ε_{x} ε_{y}} {(\ln \frac{R_{2}}{R_{1}})}^{- 1} .

(37)

The solution of the boundary value problem

\begin{array}{c} ε_{x} \frac{\partial^{2} u}{\partial x^{2}} + ε_{y} \frac{\partial^{2} u}{\partial y^{2}} = 0 & r \in A \end{array},

(38)

\begin{array}{c} u (r) = 1, & r \in \partial A_{1}, \\ u (r) = 0, & r \in \partial A_{2}, \end{array}

(39)

is as follows

u (x, y) = k \ln ((a_{x}^{2} x^{2} + a_{y}^{2} y^{2}) / R_{2}^{2}),

(40)

where

k = {(\ln {(\frac{R_{1}}{R_{2}})}^{2})}^{- 1} .

The equation of the boundary curve in polar coordinates

\begin{array}{c} r = \sqrt{x^{2} + y^{2}}, & φ = \arctan (y / x) \end{array}

(41)

\begin{array}{c} | \bar{O P_{i}} | = ρ_{i} (φ) & 0 \leq φ \leq 2 π, \\ = α_{i} β_{i} {(α_{i}^{2} \sin^{2} φ + β_{i}^{2} \cos^{2} φ)}^{- 0.5}, & (i = 1,2) . \end{array}

(42)

The function

u = u (x, y)

can be represented in polar coordinates as

v (r, φ) = u (r \cos φ, r \sin φ) = k \ln [\frac{r^{2}}{R_{2}^{2}} (a_{x}^{2} \cos^{2} φ + a_{y}^{2} \sin^{2} φ)] .

(43)

Let

V_{j} = V_{j} (r)

be defined as

\begin{array}{c} V_{j} (r) = v (r, φ_{j}), & ρ_{1} (φ_{j}) \leq r \leq ρ_{2} (φ_{j}) \end{array} .

(44)

The graphs of $V_{j} (r)$ for $ρ_{1} (φ_{j}) \leq r \leq ρ_{2} (φ_{j})$ are shown in Figs 5 and 6 for $φ_{j} = 0$ and $φ_{j} = π / 2$ . The contour lines of the function $u = u (x, y)$ are presented in Fig. 7. The following numerical data were used $ε_{x} = 8 \cdot 10^{- 12}$ (F m⁻¹⁾, $ε_{y} = 8.5 \cdot 10^{- 12}$ (F m⁻¹⁾, $R_{1} = 0.025$ m, $R_{2} = 0.04$ 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

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[2]↑
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I. Ecsedi and A. Baksa, “A method for the solution of uniform torsion of Cartesian orthotropic bar,” J. Theor. Appl. Mech., vol. 52, pp. 129–143, 2022.
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Scopus Cite Score Rank	Civil and Structural Engineering 206/310 (Q3) Computer Science Applications 445/636 (Q3) General Materials Science 295/460 (Q3) Modeling and Simulation 212/274 (Q4) Software 304/373 (Q4)
Scopus SNIP	0,933
Scopus Cites	290
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Pollack Periodica
Publication Model	Hybrid
Submission Fee	none
Article Processing Charge	900 EUR/article
Printed Color Illustrations	40 EUR (or 10 000 HUF) + VAT / piece
Regional discounts on country of the funding agency	World Bank Lower-middle-income economies: 50% World Bank Low-income economies: 100%
Further Discounts	Editorial Board / Advisory Board members: 50% Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription fee 2023	Online subsscription: 336 EUR / 411 USD Print + online subscription: 405 EUR / 492 USD
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Pollack Periodica
Language	English
Size	A4
Year of Foundation	2006
Volumes per Year	1
Issues per Year	3
Founder	Akadémiai Kiadó
Founder's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Publisher	Akadémiai Kiadó
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	1788-1994 (Print)
ISSN	1788-3911 (Online)

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Abstract

Abstract

1 Introduction

2 Governing equations of electrostatic field for two-dimensional capacitor

3 Transformation of the governing boundary value problem

4 Determination of the capacitance of anisotropic capacitor

5 Example

6 Conclusions

References

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