Bounds for the electrical resistance for non-homogeneous conducting body

István Ecsedi; Attila Baksa

doi:10.1556/606.2022.00621

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Pollack Periodica

Volume/Issue:: Volume 18: Issue 1

Bounds for the electrical resistance for non-homogeneous conducting body

Authors:

István Ecsedi

István Ecsedi Institute of Applied Mechanics, Faculty of Mechanical Engineering and Informatics, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary

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Attila Baksa

Attila Baksa Institute of Applied Mechanics, Faculty of Mechanical Engineering and Informatics, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary

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attila.baksa@uni-miskolc.hu https://orcid.org/0000-0003-2727-2498

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Pages:: 172–176

Online Publication Date:: 21 Oct 2022

Publication Date:: 07 Mar 2023

Article Category:: Research Article

DOI:: https://doi.org/10.1556/606.2022.00621

Keywords:: electrical resistance; hollow body; lower and upper bounds; steady-state

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Abstract

A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape has a three-dimensional hollow body. The equations of the Maxwell's theory of electric current flow in a non-homogeneous isotropic solid conductor body are used to formulate the corresponding electric boundary value problem. The determination of the steady motion of charges is based on the concept of the electrical conductance. The derivation of the upper and lower bound formulae for the electrical conductance is based on Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bound formulae.

Abstract

1 Introduction

Electrical resistance of an electrical conductor is a measure of the difficulty to pass a steady electric current through the conductor. The well-known elementary form of Ohm's law states that when the conductor carries a current I from a point P₁ at potential U₁ to a point P₂ at potential U₂ then U₁–U₂ = RI, where R is the resistance of the conductor between points P₁ and P₂, it depends only on the shape and temperature and the material of the conductor. The inverse of electric resistance is the electric conductance G = 1/R. This paper deals with the electric resistance of a three-dimensional non-homogeneous conductor body. Examination of non-homogeneous structural elements is a very important task. Maróti's study [1] deals with the bending vibration of axially non-homogeneous beams. The buckling problem of axially functionally graded beams is considered in paper [2]. For prescribed frequency and buckling loads Maróti and Elishakoff [2] determined the Young's modulus in axial direction as a function of axial coordinate. The non-homogeneous isotropic hollow conductor is bounded by two closed surfaces $\partial V_{1}$ and $\partial V_{2}$ , which have no common point. The current flows inside the conductor from inner boundary surface $\partial V_{1}$ whose potential is $U_{1}$ to the outer boundary surface $\partial V_{2}$ whose potential is $U_{2}$ , $U_{1} > U_{2}$ . Two-side estimation will be proven for the electrical conductance of non-homogeneous isotropic hollow three-dimensional conductor. The mathematical formalism follows the methods, which were used in papers [3–5]. In paper [3] upper and lower bounds are proven for the electrical resistance of homogeneous isotropic ring like axisymmetric conductor. In paper [4] the capacitance of two-dimensional cylindrical capacitor, which consists of non-homogeneous dielectric materials is studied. Examples illustrate the applications of the derived bounding formulae of capacitance [4]. A mathematical heat transfer model is developed for the steady-state heat transfer problem for homogeneous isotropic body of rotation in [5] and it is used to obtain estimations of thermal heat transfer conductance.

Let us consider the steady motion of charges in the non-homogeneous hollow conductor shown in Fig. 1. The conductor body occupies the space domain V and its boundary surfaces are

\partial V_{1}

and

\partial V_{2}

. The electric potential U on the boundary surfaces

\partial V_{1}

and

\partial V_{2}

are prescribed, so the following boundary conditions are valid [6–8],

U (r) = U_{i} = c o n s t a n t, r \in \partial V_{i} (i = 1,2),

(1)

where

r

denotes the position vector (Fig. 1). According to Maxwell's theory [6–8] the steady motion of charges is described by the next equations:

j = σ E, \nabla \cdot j = 0, E = - \nabla U .

(2)

Fig. 1.

Hollow non-homogeneous conductor body bounded by closed surfaces

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Differential form of Ohm's law formulates that at constant temperature in isotropic conductor the current density vector

j

is proportional to the electric field vector

E .

Here

σ = σ (r)

is the conductivity of the non-homogenous hollow conductor. In Eq. (2)

\nabla

is the del operator and the dot between two vectors denotes the scalar product [9]. From the above equations it follows that

σ (r) Δ U + \nabla σ \cdot \nabla U = 0, r \in V, Δ = \nabla \cdot \nabla .

(3)

Introducing a new function

u = u (r)

by the next definition,

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

(4)

It is evident that

u = u (r)

satisfies the following boundary value problem,

\begin{array}{l} σ (r) Δ u + \nabla σ \cdot \nabla u = 0, r \in V, \\ u = 1, r \in \partial V_{1}, u = 0, r \in \partial V_{2} . \end{array}

(5)

The function

u = u (x, y)

plays crucial role in the expressions of electrical resistance and electrical conductance. An electric current in the conductor is the continuous passage of the current along that conductor. The constant potential difference between the closed surfaces

\partial V_{1}

and

\partial V_{2}

maintains the steady flow of the electric current. The amount of charge flowing through surface

\partial V_{1}

per unit time is I. The determination of I is based on the next equation

\begin{array}{l} I = - \int_{\partial V_{1}} j \cdot n d A = (U_{1} - U_{2}) \int_{\partial V_{1}} σ (r) n \cdot \nabla u d A = \\ (U_{1} - U_{2}) \int_{\partial V_{1}} σ (r) \frac{\partial u}{\partial n} d A . \end{array}

(6)

In Eq. (6), n is the outer normal unit vector of the inner boundary surface

\partial V_{1}

and

d A

is the area element of

\partial V_{1}

. The electrical resistance R and the conductance G of the hollow conductor is defined as [6, 8],

\begin{array}{l} R = \frac{U_{1} - U_{2}}{I} = \frac{1}{\int_{\partial V_{1}} σ (r) \frac{\partial u}{\partial n} d A}, \\ G = \frac{I}{U_{1} - U_{2}} = \int_{\partial V_{1}} σ (r) \frac{\partial u}{\partial n} d A . \end{array}

(7)

From Eq. (5) it follows that

\int_{V} u [σ (r) Δ u + \nabla σ \cdot \nabla u] d V = \int_{\partial V_{1}} u σ (r) n \cdot \nabla u d A - \int_{V} σ (r) {| \nabla u |}^{2} d V = 0,

(8)

G = \int_{V} σ (r) {| \nabla u |}^{2} d V, R = \frac{1}{\int_{V} σ (r) {| \nabla u |}^{2} d V} .

(9)

Note that if

\nabla σ \cdot \nabla u = 0, r \in V,

(10)

then

u (r) = u_{0} (r),

where

u_{0} (r)

is a unique solution of the following Dirichlet type boundary-value problem

Δ u_{0} = 0, r \in V, u_{0} = 1, r \in \partial V_{1}, u_{0} = 0, r \in \partial V_{2} .

(11)

In this case

G = \int_{V} σ (r) {| \nabla u_{0} |}^{2} d V .

(12)

There are several approximation methods to get the solution of the boundary-value problem Eq. (5), most of which use the results of variational calculus for example as Ritz method, finite element method [8, 9]. Other methods are also known and they used, for example finite difference methods, method of weighted residuals, boundary element method [10]. It must be mentioned that, many numerical-analytical method are used R-functions to solve the boundary value problems of electrodynamics [11–13]. The efficiency of the R-Function Method (RFM) to solving the boundary value problems of electrostatics in very complicated domain is illustrated in paper by Kravchenko and Basarab [14]. They considered a boundary-value problem of electrodynamics in the fractal regions of the Sierpiski carpet and the Koch island types [14]. Iványi solved a number of two-dimensional boundary value problems of static and stationary electromagnetisms by variational method connecting of them with the use of R-functions [12, 13, 15, 16]. It is not the aim of this paper is to give a detailed list of different analytical and numerical methods, which are used widespread in electrical engineering calculations.

2 Upper bound for G and lower bound for R

Theorem 1.

If the function $F = F (r)$ which is continuously differentiable in $V \cup \partial V$ satisfies the boundary conditions (13) then the inequality relation (14) is valid

F (r) = 1, r \in \partial V_{1}, F (r) = 0, r \in \partial V_{2},

(13)

G \leq \int_{V} σ (r) {| \nabla F |}^{2} d V .

(14)

Proof.

The proof of inequality (14)₃ can be derived by the Cauchy-Schwarz inequality relation (15),

{(\int_{V} σ (r) \nabla F \cdot \nabla u d V)}^{2} \leq \int_{V} σ (r) {| \nabla F |}^{2} d V \cdot \int_{V} σ (r) {| \nabla u |}^{2} d V .

(15)

A simple computation leads to the result

\int_{V} σ (r) \nabla F \cdot \nabla u d V = \int_{\partial V_{1}} σ (r) n \cdot \nabla u d A = \int_{V} σ (r) {| \nabla u |}^{2} d V .

(16)

The combination of the inequality relation (15) with Eq. (16) and using formula (9) gives (14). A brief discussion shows that the sign of equality in relation (14) is valid only if $F (r) \equiv u (r) .$

3 Lower bound for G, upper bound for R

Theorem 2.

Let $q = q (r)$ be a vector field defined in the hollow space domain $V \cup \partial V$ , which satisfies the following equations

\nabla \cdot q = 0, r \in V, n \cdot q = 0, r \in \partial V_{1},

(17)

in this case

G \geq \frac{{(\int_{\partial V_{1}} σ (r) n \cdot q d A)}^{2}}{\int_{V} σ (r) q^{2} d V}, \int_{V} q^{2} d V \neq 0 .

(18)

In lower bound formula (18) equality is reached only if $q \equiv λ \nabla u,$ where $λ$ is an arbitrary constant which is different from zero.

Proof.

The proof of lower bound formula (18) is based on the Cauchy-Schwarz inequality relation (19)

{(\int_{V} σ (r) p \cdot q d V)}^{2} \leq \int_{V} σ (r) p^{2} d V \int_{V} σ (r) q^{2} d V .

(19)

Let

p = \nabla u

(20)

be in inequality relation (19). A simple calculation yields the result

\int_{V} σ (r) \nabla u \cdot q d V = \int_{\partial V} u σ (r) n \cdot q d A - \int_{V} u \nabla \cdot [σ (r) q] d V = \int_{\partial V_{1}} σ (r) n \cdot q d A .

(21)

Substitution Eq. (21) into Cauchy-Schwarz inequality (19) gives

{(\int_{\partial V_{1}} σ (r) n \cdot q d A)}^{2} \leq \int_{V} σ (r) {| \nabla u |}^{2} d V \int_{V} σ (r) q^{2} d V .

(22)

From inequality relation (22) the proof of lower bound formula, (18) can be obtained immediately.

Theorem 3.

Let $f = f (r)$ be non-identically constant function in $V \cup \partial V$ , which satisfies the Laplace equation in $V$

\nabla \cdot \nabla f = Δ f = 0, r \in V .

(23)

The following lower bound formula is valid for G

G \geq \frac{{(\int_{\partial V_{1}} \frac{\partial f}{\partial n} d A)}^{2}}{\int_{V} \frac{{| \nabla f |}^{2}}{σ (r)} d V} .

(24)

Proof.

The proof of (24) can be obtained from (18) with under-mentioned $q (r)$

q (r) = \frac{\nabla f}{σ (r)}, r \in V \cup \partial V .

(25)

4 Numerical examples

In the numerical examples the spherical coordinate system is used. The connection between the Cartesian coordinates $x, y, z$ and the spherical coordinates $r, ϕ, ϑ$ is $x = r \cos ϕ \sin ϑ, y = r \sin ϕ \sin ϑ, z = r \cos ϑ .$ Developed numerical example relate to axisymmetric electrical problems.

Example 1.

The meridian section of hollow spherical domain is shown in Fig. 2. The specific conductivity is a given function of the radial coordinate

σ (r, α) = σ_{0} \exp (α \frac{r}{a_{1}}) .

Fig. 2.

Meridian section of hollow spherical domain

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Application of Theorem 1 to the function

F (r) = F (r) = 1 - \frac{1 / r - 1 / a_{1}}{1 / a_{2} - 1 / a_{1}}

gives

G (α) \leq G_{U} (α) = - \frac{1}{{(a_{1} - a_{2})}^{2}} {4 π σ_{0} a_{1} a_{2} [\begin{array}{l} a_{1} \exp (\frac{α a_{2}}{a_{1}}) + E_{i} (1, - \frac{α a_{2}}{a_{1}}) α a_{2} \\ - a_{2} \exp (α) - a_{2} E_{i} (1, - α) α \end{array}]},

(26)

here

E_{i} (1, x)

is the exponential integral [17, 18]. Putting the following function

f (r) = f (r) = r^{- 1}

in the lower bound formula (24) gives

G (α) \geq G_{L} (α) = \frac{4 π σ_{1} a_{1} a_{2}}{- a_{1} \exp (- \frac{α a_{2}}{a_{1}}) + E_{i} (1, \frac{α a_{2}}{a_{1}}) α a_{2} + a_{2} \exp (- α) - a_{2} E_{i} (1, α) α} .

(27)

Lengthy, but elementary calculations shows that, the exact value of electrical conductivity which is obtained from the solution of boundary value problem (5) is

G (α) = \frac{4 π σ_{1} a_{1} a_{2}}{- a_{1} \exp (- \frac{α a_{2}}{a_{1}}) + a_{2} \exp (- α) + α a_{2} E_{i} (1, \frac{α a_{2}}{a_{1}}) - α a_{2} E_{i} (1, α)}

(28)

that in this case is

G (α) = G_{L} (α)

. The validity of upper bound (14) for

α = - 2.5

and

α = 2.5

examplifies as follows

G_{L} (- 2.5) = G (- 2.5) = 2.651169 \times 10^{6} \frac{1}{Ω} ≦ G_{U} (- 2.5) = 3.301497 \times 10^{6} 1 / Ω,

(29)

G_{L} (2.5) = G (2.5) = 1.59105 \times 10^{6} \frac{1}{Ω} ≦ G_{U} (2.5) = 1.981333 \times 10^{6} 1 / Ω .

(30)

Figure 3 shows the upper and the lower bounds of the conductance G as a function of α for $- 2 \leq α \leq 2 .$ In this example $a_{1} = 0.3 m, a_{2} = 0.5 m, σ_{0} = 7.69 \times 10^{6} 1 / Ω m .$

Fig. 3.

Upper and lower bounds for the conductance of non-homogeneous spherical conductor as a function of α for $- 1.5 \leq α \leq 1.5$

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Figure 4 shows the plot of function $F = F (r)$ , the plot of exact analytical solution $u = u (r)$ and the plot of $u_{F} = u_{F} (r)$ which is obtained from Finite Element (FE) approximation in the case of Example 1 for $α = 2.5$ . The FE model is developed in ABAQUS with DC3D8 elements (node numbers are $706 356$ ) and for definition of the nonlinear material a special user subroutine usdfld() is applied.

Fig. 4.

Comparison of exact solution, approximate solution and FE approximation

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Example 2.

The meridian section of axisymmetric hollow domain bounded by two spherical surfaces as it is shown in Fig. 5. The following data are used $a_{1} = 0.3 m, a_{2} = 0.5 m, b = 0.025, σ_{1} = 7.69 \times 10^{6} 1 / m Ω, σ (r, n) = σ_{1} {(\frac{r}{a_{1}})}^{n} .$

Fig. 5.

The meridian section of axisymmetric hollow domain bounded by two spherical surfaces with different centre points

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

Let $F (r, ϑ) = \ln \frac{R (ϑ)}{r} {(\ln \frac{R (ϑ)}{a_{1}})}^{- 1}$ be in Theorem 1 and in Theorem 3 $f (r) = r^{- 1} .$

Figure 6 shows the upper and the lower bounds as a function of power index for $- 2 \leq n \leq 2 .$

Fig. 6.

Upper and lower bounds for the conductance as a function of power index n

Citation: Pollack Periodica 18, 1; 10.1556/606.2022.00621

5 Conclusions

A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape is a three-dimensional hollow body. The hollow body considered is bounded by two closed surfaces which have no common points. The derivation of the upper and lower bound formulae for the electrical conductance is based on the two types of Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bounds for the conductance. The derived upper and lower bound formulae of electric conductance can be used to check the results of numerical computations obtained by finite element method, boundary element method and by any other numerical methods.

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András Timár (Faculty of Engineering and Information Technology, University of Pécs, Hungary)
Barry H. V. Topping (Heriot-Watt University, UK, Faculty of Engineering and Information Technology, University of Pécs, Hungary)

POLLACK PERIODICA
Pollack Mihály Faculty of Engineering
Institute: University of Pécs
Address: Boszorkány utca 2. H–7624 Pécs, Hungary
Phone/Fax: (36 72) 503 650

E-mail: peter.ivanyi@mik.pte.hu

or amalia.ivanyi@mik.pte.hu

Indexing and Abstracting Services:

SCOPUS
CABELLS Journalytics

2024
Scopus
CiteScore
CiteScore rank
SNIP
Scimago
SJR index	0.385
SJR Q rank	Q3

2023
Scopus
CiteScore	1.5
CiteScore rank	Q3 (Civil and Structural Engineering)
SNIP	0.849
Scimago
SJR index	0.288
SJR Q rank	Q3

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 2025	Online subsscription: 381 EUR / 420 USD Print + online subscription: 456 EUR / 520 USD
Subscription Information	Online subscribers are entitled access to all back issues published by Akadémiai Kiadó for each title for the duration of the subscription, as well as Online First content for the subscribed content.
Purchase per Title	Individual articles are sold on the displayed price.

Pollack Periodica
Language	English
Size	A4
Year of Foundation	2006
Volumes per Year	1
Issues per Year	3
Founder	Faculty of Engineering and Information Technology, University of Pécs
Founder's Address	H–7624 Pécs, Hungary, Boszorkány utca 2.
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)