Bounds for the electrical resistance for non-homogeneous conducting body

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 ﬂ ow 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.


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 P 1 at potential U 1 to a point P 2 at potential U 2 then U 1 -U 2 5 RI, where R is the resistance of the conductor between points P 1 and P 2 , it depends only on the shape and temperature and the material of the conductor. The inverse of electric resistance is the electric conductance G 5 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 vV 1 and vV 2 , which have no common point. The current flows inside the conductor from inner boundary surface vV 1 whose potential is U 1 to the outer boundary surface vV 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][4][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 vV 1 and vV 2 . The electric potential U on the boundary surfaces vV 1 and vV 2 are prescribed, so the following boundary conditions are valid [6][7][8], (1) where r denotes the position vector ( Fig. 1). According to Maxwell's theory [6][7][8] the steady motion of charges is described by the next equations: 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 ¼ σðrÞ is the conductivity of the non-homogenous hollow conductor. In Eq. (2) ∇ is the del operator and the dot between two vectors denotes the scalar product [9]. From the above equations it follows that Introducing a new function u ¼ uðrÞ by the next definition, It is evident that u ¼ uðrÞ satisfies the following boundary value problem, The function u ¼ uðx; yÞ 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 vV 1 and vV 2 maintains the steady flow of the electric current. The amount of charge flowing through surface vV 1 per unit time is I. The determination of I is based on the next equation In Eq. (6), n is the outer normal unit vector of the inner boundary surface vV 1 and dA is the area element of vV 1 . The electrical resistance R and the conductance G of the hollow conductor is defined as [6,8], σðrÞ vu vn dA : Note that if then uðrÞ ¼ u 0 ðrÞ; where u 0 ðrÞ is a unique solution of the following Dirichlet type boundary-value problem In this case 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 numericalanalytical method are used R-functions to solve the boundary value problems of electrodynamics [11][12][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 ¼ FðrÞ which is continuously differentiable in V∪vV satisfies the boundary conditions (13) then the inequality relation (14) is valid Proof. The proof of inequality (14) 3 can be derived by the Cauchy-Schwarz inequality relation (15), A simple computation leads to the result Z 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 FðrÞ ≡ uðrÞ: 3. LOWER BOUND FOR G, UPPER BOUND FOR R Theorem 2. Let q ¼ qðrÞ be a vector field defined in the hollow space domain V∪vV, which satisfies the following equations in this case In lower bound formula (18) equality is reached only if q ≡ λ∇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)
σðrÞn$qdA: Proof. The proof of (24) can be obtained from (18) with under-mentioned qðrÞ

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; f; ϑ is x ¼ rcosfsinϑ; y ¼ rsinfsinϑ; z ¼ rcosϑ: 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 σðr; αÞ ¼ σ 0 exp α r a 1 : Application of Theorem 1 to the function Àa 2 expðαÞ À a 2 E i ð1; ÀαÞα here E i ð1; xÞ is the exponential integral [17,18]. Putting the following function f ðrÞ ¼ f ðrÞ ¼ r −1 in the lower bound formula (24) gives Lengthy, but elementary calculations shows that, the exact value of electrical conductivity which is obtained from the solution of boundary value problem (5) is (29) G L ð2:5Þ ¼ Gð2:5Þ ¼ 1:59105310 6 1 Ω ≦G U ð2:5Þ ¼ 1:981333310 6 1 Ω: (30) Figure 3 shows the upper and the lower bounds of the conductance G as a function of α for −2 ≤ α ≤ 2: In this example a 1 ¼ 0:3 m; a 2 ¼ 0:5 m; σ 0 ¼ 7:69310 6 1=Ωm: Figure 4 shows the plot of function F ¼ FðrÞ, the plot of exact analytical solution u ¼ uðrÞ and the plot of u F ¼ u F ðrÞ which is obtained from Finite Element (FE) approximation in the case of Example 1 for α ¼ 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.   Figure 6 shows the upper and the lower bounds as a function of power index for −2 ≤ n ≤ 2:

CONCLUSIONS
A mathematical model is developed to determine the steadystate 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.