On Resistance and Inductance of Solid Conductors

Abstract

The concepts of resistance and inductance are strewn with misunderstanding and errors that result from inaccurate terminology, mistakes in their definitions, and from the fact that they are quantities that characterize elements of ideal electrical circuits as well as real solid conductors. This is shown unfavourably not only in the respective field of the theory of electrical engineering, but also in the calculation of resistance and inductance. In this paper, a brief theoretical analysis of the state of the art is given, and its results are applied in the calculation of the current density, resistance, and inductance of two coaxial solid tubular conductors.

1. Introduction

The fundamental concepts used in electrical engineering and their definitions are the subject of physics in the part electricity and magnetism [1]. Resistance 𝑅 and inductance 𝐿 are technical terms, and their meaning in electrical engineering should be strictly defined. In the basic circuit theory (CT), it is assumed that the values of 𝑅 and 𝐿 are given. The calculation of these quantities is the task of electromagnetic field theory (EM) [1]. EM and CT undoubtedly have much in common but the physical and mathematical characteristics of the basic quantities differ.

The basic quantities in Maxwell’s theory of electromagnetism are the field vectors 𝐄, 𝐁, and 𝐉 [2]. Electromagnetic phenomena in a circuit are expressed only by means of voltage across the elements and by currents flowing through them. If CT is used, the real electrical device is substituted by several ideal elements. In the following, instead of saying that a device is substituted by a resistor with resistance 𝑅 or by an inductor with inductance 𝐿, it will for brevity be said that a device is substituted by resistance 𝑅 or by inductance 𝐿.

In circuits with lumped parameters, voltages and currents may depend on time 𝑡 and are thus functions of 𝑡 but they are not functions of position. In circuits that satisfy this condition, the phenomena are referred to as quasi-stationary [1, 3] and the respective quantities are referred to as slowly varying [4]. Displacement current is usually neglected when examining these phenomena. In the present paper, quasi-stationary phenomena alone will be examined.

The concepts of resistance and inductance are coupled with misunderstanding and errors that result from inaccurate terminology, disregarding valid norms, from mistakes in their definitions and from the fact that they are quantities that characterize elements of ideal electrical circuits as well as real solid conductors. In the paper, a theoretical analysis of the state of the art is given, and its results are applied in the calculation of the resistance and inductance of two coaxial solid tubular conductors. These conductors are the simplest possible arrangement of conductors, except for the solitary conductor, which was examined already in [5]. The formulae for calculating 𝑅 and 𝐿 as derived by Thomson in [5] are not correct, inclusive of the frequency dependence of resistance; the solitary conductor is not a suitable model for the calculation of 𝐿, as has been proved in [6].

2. Ohm’s Law and Resistance

2.1. Ohm’s Law, Resistance, and Impedance in CT

By Ohm’s law, the voltage between resistor poles is in proportion to the current flowing through the resistor: 𝑈=𝐼𝑅. (1) Using the reciprocal value 1/𝑅=𝐺, Ohm’s law can be written as 𝐼=𝐺𝑈. (2) In the present paper, the symbols 𝑈 or 𝑈(𝑡) and 𝐼 or 𝐼(𝑡) will denote voltage and current, respectively, and not the effective value of these quantities. Underlined symbols will denote complex numbers in the following.

In [7], the resistive two-terminal element is defined as passive two-terminal element characterized by a functional relation between the voltage across the terminals and the electric current in the element. The electric energy absorbed by a resistive two-terminal element is not recoverable as electric energy at the terminals. The resistive two-terminal element is characterized by a quantity denoted by 𝑅. The linear resistive two-terminal element is an ideal resistor (see [7]), and 𝑅 satisfies relation (1). The terms resistance (1) and resistance (2) (both denoted by the symbol 𝑅) and impedance are defined in [7]. In the following text, the denotations resistance-1 and resistance-2 are used to prevent mistaking them for relations (1) and (2); the word resistance refers comprehensively to the two resistances.

Definition 1. Resistance-1 for a resistive two-terminal element or two-terminal circuit with terminals A and B equals the voltage 𝑈_AB between the terminals divided by the electric current 𝐼 in the element or circuit: 𝑈𝑅=^AB(𝑡)𝐼(𝑡),𝑅≥0, (3) where the electric current is taken as positive if its direction is from A to B and negative in the opposite case (see [7]).

Definition 2 (see [7]). Impedance of passive linear two-terminal element or two-terminal circuit under sinusoidal conditions equals the quotient of the phasor 𝑈 representing the voltage between the terminals divided by the phasor 𝐼 representing the electric current in the element or circuit: 𝑍=𝑈𝐼. (4)

Definition 3 (see [7]). Resistance-2 is the real part of an impedance 𝑍: 𝑅=ℜ𝑍. (5)

In the current spoken and written utterances in electrical engineering, the term resistance is used without any further differentiation or specification. This leads not only to vague formulations, but also to erroneous statements. The terms defined in [7] also include conductance-1 and conductance-2, both of which are denoted by the symbol 𝐺. Conductance-1 equals the reciprocal value of resistance-1 while conductance-2 equals the real part of admittance.

2.2. Ohm’s Law in EM

Ohm’s law (1) is a consequence of the general relation [2] 𝐉=𝐉(𝐄,𝐁). (6) The dependence of 𝐉 on 𝐁 is manifested markedly in an ionized gaseous medium (Hall effect). Throughout a remarkably wide range of conditions, in both solids and weakly ionized solutions, relation (6) proves to be linear [8]. The proportionality constant only depends on the material of the conductor but not on its shape: 𝐉=𝛾𝐄. (7) In isotropic conductors, the conductivity 𝛾 is a scalar and is equal to the reciprocal value of resistivity (𝛾=1/𝜚). In anisotropic conductors, 𝛾 is a tensor.

Ohm’s law (7) is remarkable in that it holds very accurately in many conductors and over a wide range of values of 𝐸=|𝐄|. The validity of (7) can thus be assumed in the case that the changes in the field 𝐄 or 𝐉 are negligible over a time interval of ≈10⁻¹⁴ s in metallic conductors [4, 9]. For a periodically changing field 𝐄 with frequency 𝑓, law (7) holds if 𝑓≪10¹⁴Hz.

2.3. Calculation of Resistance-1

It follows from the above that law (7) is proved on the assumption of the fulfilment of the respective conditions. To calculate resistance-1 means to calculate 𝑅 in relation (1) or 𝐺 in relation (2). When deriving relation (7), the field 𝐄 was assumed to be homogeneous in the conductor. By (7), this means that the field 𝐉 is also homogeneous in the conductor. Any conductor placed in a nonconducting medium is a tube of the field 𝐉 and by (7) also of the field 𝐄. Assume that the conductor is of length 𝑙, and its ends are equipotential surfaces. They are areas of constant potential, whose gradient is the field 𝐄. Consider a part of this conductor, which is homogeneous (with 𝛾 constant) and has constant fields 𝐄 and 𝐉, and this part is a current tube of cross-section Δ𝑆 between two equipotential levels, whose distance is Δ𝑙. On these assumptions, the vector Δ𝐒 is parallel to Δ𝐥, 𝐄, and 𝐉. After multiplying (7) by the product Δ𝐥⋅Δ𝐒 and after rewriting, we have Δ𝑈=Δ𝑅Δ𝐼, (8) where Δ𝑈=𝐄⋅Δ𝐥,Δ𝐼=𝐉⋅Δ𝐒,Δ𝑅=𝜚Δ𝑙.Δ𝑆 (9) If the assumptions made for a part of the conductor hold for the whole conductor, then the conductor has a constant cross-section 𝑆. Equations that hold for the conductor are (8) and (9) but with Δ left out. Equation (8) is then law (1).

In the real circuit, the real conductor under consideration is replaced by a resistor which has resistance-1 𝑅. Several resistances-1 𝑅_𝑖, 𝑖=1,2,…,𝑛, 𝑛>1, which satisfy relations (1) and (2), can be substituted by one resistor whose resistance-1 is 𝑅. On the assumption of the validity of both Kirchhoff’s laws, it holds for resistors connected in series or in parallel: ^𝑛_𝑖=1𝑅_𝑖=𝑅 (10) or ^𝑛_𝑖=1𝐺_𝑖=𝐺,𝐺_𝑖=1𝑅_𝑖1,𝐺=𝑅. (11)

When proving law (1) in the first paragraph of this section, it was assumed, among other things, that 𝛾 and 𝐉 are constant in the conductor. In the following we will prove the validity of law (1) also for the case when 𝛾=𝛾(𝑃) and 𝐉=𝐉(𝑃), where 𝑃 denotes a point in the conductor denoted by the symbol 𝑉. We will assume that the cross section of the conductor changes along conductor continuously and that the conductivity 𝛾 in the conductor is a continuous and bounded function. These assumptions do not imply any loss of generality because a real conductor is made up of a finite number of such conductors and thus its resistance-1 can be determined using relations (10) and (11).

The conductor is a tube of the field 𝐉, and its ends are equipotential areas. The magnitudes of the cross sections at the conductor ends are 𝑆_𝑏 and 𝑆_𝑒. We will divide the area 𝑆_𝑏 into 𝑆_𝑖,𝑖=1,2,…,𝑛. The sets 𝑆_𝑖 are simply connected sets (see [10]) and every two of them have at most the boundary points in common. It holds that 𝑆_𝑏=^𝑛_𝑖=1𝑆_𝑖,𝑉=^𝑛_𝑖=1𝑇_𝑖, (12) where 𝑇_𝑖 is the tube of field 𝐉 given by set 𝑆_𝑖. By definition, connected sets 𝑆_𝑖 are open sets. This means that they do not include their boundaries, and, consequently, the right and the left sides of relations (12) differ from each other by the set of measure zero, which is made up of the boundaries of the sets 𝑆_𝑖 and 𝑇_𝑖. In view of the continuity of 𝛾, the current flowing through the boundaries is also zero. Inside every part of 𝑆_𝑖 we choose arbitrarily a point 𝑃_𝑖0. We denote by 𝑙_𝑖 the line of vector 𝐉 which passes through the point 𝑃_𝑖0. We choose an integer 𝑚>0 and denote by 𝑃_𝑖𝑚 the point at which the line 𝑙_𝑖 intersects the area 𝑆_𝑒. Using the points 𝑃_𝑖𝑗,𝑗=1,2,…,𝑚, we divide the part of each line 𝑙_𝑖,𝑖=1,2,…,𝑛, between the points 𝑃_𝑖0 and 𝑃_𝑖𝑚 into 𝑚 arcs of the same length Δ𝑙. Between every two points 𝑃_{𝑖,𝑗−1} and 𝑃_𝑖𝑗, we choose arbitrarily a point 𝑃_𝑖𝑗, that is, a total of 𝑛×𝑚 points. The equipotential area passing through the point 𝑃_𝑖𝑗 intersects the tube 𝑇_𝑖 in the area 𝑆_𝑖𝑗. We denote, respectively, 𝐉_𝑖𝑗 and 𝜚_𝑖𝑗, the current density and resistivity at the point 𝑃_𝑖𝑗. The conductor 𝑉 can be substituted by conductors 𝑉_𝑖𝑗, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, with each conductor 𝑉_𝑖𝑗 having a constant cross-section 𝑆_𝑖𝑗, length Δ𝑙, constant resistivity 𝜚_𝑖𝑗, and constant current density 𝐉_𝑖𝑗. The conductors 𝑉_𝑖𝑗, where 𝑖 is constant and 𝑗=1,2,…,𝑚, are arranged in series and form a partial conductor 𝑉_𝑖. The conductors 𝑉_𝑖,𝑖=1,2,…,𝑛, are connected in parallel. The resistance-1 of the conductor 𝑉_𝑖 depends on 𝑚 and 𝑛 and is denoted by 𝑅_𝑖(𝑚,𝑛). By (9) and (10), 𝑅_𝑖(𝑚,𝑛)=^𝑚_𝑗=1𝜚_𝑖𝑗Δ𝑙𝑆_𝑖𝑗. (13) By (11), the conductance-1 of the conductor composed of all the parts 𝑉_𝑖𝑗 is 𝐺(𝑚,𝑛)=^𝑛_𝑖=11𝑅_𝑖().𝑚,𝑛 (14)

Formulae (13) and (14) express the physical essence of the calculation of resistance-1 or conductance-1 and enable calculating numerically their approximate values. The conductor has been replaced by a series and parallel connection of conductors that are so small that their cross section can be considered constant, the same as their conductivity. The calculation of the exact values of resistance-1 and conductance-1 depends on the conductor shape and on the functions 𝛾(𝑃) and 𝐉(𝑃). The procedure of deriving the formula for the calculation of resistance-1 and conductance-1 in the general case is given in the following.

On the equipotential area 𝑆_𝑏 of the conductor considered we choose an area Δ𝑆_𝑏, which is a simply connected set. All the lines of vector 𝐉 passing through the surface Δ𝑆_𝑏 intersect the surface 𝑆_𝑒 in a set Δ𝑆_𝑒 and form a current tube. The part of this tube between the surfaces Δ𝑆_𝑏 and Δ𝑆_𝑒 is a conductor that will be denoted by Δ𝑇. Inside of Δ𝑆_𝑏 we choose a point 𝑃_𝑏. The line of vector 𝐉 passing through this point will be denoted by 𝑙. This line intersects the surface Δ𝑆_𝑒 at the point 𝑃_𝑒. The length of line 𝑙 between the points 𝑃_𝑏 and 𝑃_𝑒, will be denoted ℓ. A parameter 𝑝 will be introduced on the line 𝑙 such that to every point 𝑃∈𝑙 lying between the points 𝑃_𝑏 and 𝑃_𝑒 the value of the parameter 𝑝(𝑃) is assigned, which is equal to the length of the part 𝑙 between the points 𝑃_𝑏 and 𝑃. Thus, it holds that 𝑝(𝑃_𝑏)=0, 𝑝(𝑃_𝑒)=ℓ. Similarly, every equipotential surface 𝑆⊂Δ𝑇 can be assigned the value of the parameter 𝑝(𝑆), which equals the value of the parameter assigned to the intersection of the surface 𝑆 and the line 𝑙. The equipotential surfaces in Δ𝑇 do not intersect one another and are perpendicular to 𝑙. This can be exploited for choosing the coordinates of an arbitrary point 𝑃∈Δ𝑇. This is because the point 𝑃 can uniquely be assigned the equipotential surface 𝑆, on which the point 𝑃 lies, and the value of parameter 𝑝(𝑆). Two coordinates are sufficient to determine the position of 𝑃 on 𝑆. The two coordinates are not important, provided that the resistance-1 Δ𝑅 of the tube Δ𝑇 is determined on the assumption that 𝛾 and 𝐉 are constant on 𝑆 and 𝛾=𝛾(𝑝), 𝐉=𝐉(𝑝). Such a case was solved in [11] in a formally different way than in this paper. It follows from the above that for given 𝛾(𝑝) and 𝐉(𝑝), the surface 𝑆 only depends on Δ𝑆_𝑏 and 𝑝, 𝑆=𝑆(𝑝). There is a function 𝐴(𝑝) such that for every 𝑆 (more exactly for the magnitude of every surface 𝑆), it holds that 𝑆(𝑝)=Δ𝑆_𝑏𝐴(𝑝), and in agreement with (10), Δ𝑅=^ℓ₀d𝑝𝑆𝑝𝛾𝑝=1Δ𝑆_𝑏^ℓ₀d𝑝𝐴𝑝𝛾𝑝. (15) This formula can be rewritten using the generalized mean value theorem of integral calculus [12] 1Δ𝑅=Δ𝑆_𝑏𝐴𝑝𝛾𝑝^ℓ₀ℓd𝑝=Δ𝑆_𝑏𝐴𝑝𝛾𝑝,𝑝∈[0,ℓ]. (16) Let us denote 𝐴=𝐴(𝑝),𝛾=𝛾(𝑝). Relation (16) enables determining the conductance-1 Δ𝐺 of conductor Δ𝑇: 1Δ𝐺==Δ𝑅Δ𝑆_𝑏𝐴𝛾ℓ. (17) If the surface Δ𝑆_𝑏 contracts towards the point 𝑃_𝑏 such that its diameter (the maximum distance of two points in Δ𝑆_𝑏) converges to zero, then Δ𝑆_𝑏→d𝑆_𝑏. The position of infinitesimal surface d𝑆_𝑏 is given by the point 𝑃_𝑏, that is, by its coordinates. The quantities ℓ,𝛾, and 𝐴, that is, ℓ=ℓ(𝑃_𝑏), 𝛾=𝛾(𝑃_𝑏), and 𝐴=𝐴(𝑃_𝑏), depend on 𝑃_𝑏. For Δ𝑆_𝑏→d𝑆_𝑏, Δ𝐺 converges to d𝐺. It follows from (17) that d𝐺=𝛾𝑃_𝑏𝐴𝑃_𝑏ℓ𝑃_𝑏d𝑆_𝑏. (18) The formula for the calculation of the conductance-1 𝐺 of the initially considered conductor, denoted above by 𝑉, is obtained by integrating the preceding relation: 𝐺=_𝑆𝑏𝛾𝑃_𝑏𝐴𝑃_𝑏ℓ𝑃_𝑏d𝑆_𝑏. (19)

3. Faraday’s Law and Inductance

3.1. Current Filaments

In theoretical models, the complex motion of electric charges constituting current in a conductor is approximated either by the current characterized by the current density vector 𝐉 or by a system of current filaments or by surface current. In the following, the word curve refers to a simple finite piecewise smooth closed curve [13].

The term current filament denotes two quantitatively and qualitatively different objects. The current filament-1 is a curve in which current 𝐼(𝑡) of finite magnitude flows. Current filament-1 is the idealization of a “thin” conductor whose cross section is small, relative to its length. The current density in filament-1 is infinitely large.

The current filament-2 is used in the description of solid conductors. If the cross section of solid conductor is in the plane of coordinates 𝑥𝑦 and the current density component 𝐽_𝑧 at the point (𝑥,𝑦) of the cross section is not identically equal to zero, then the current filament-2 passing through the point (𝑥,𝑦) is formed by the lines of vector 𝐉, which pass through the infinitesimal surface d𝑥 d𝑦. The infinitesimal current passing through this filament-2 is d𝐼(𝑡)=𝐽_𝑧(𝑥,𝑦,𝑡)d𝑥 d𝑦.

3.2. Magnetic Field of Current Filament-1

The magnetic field of filament-1, which is in the shape of a curve 𝐶, can be established using the Biot-Savart law: 𝐁=𝜇𝑥,𝑦,𝑧,𝑡⁰𝐼4𝜋_𝐶d𝐂×𝐫𝑟³, (20) where 𝐫 is the position vector of the point (𝑥,𝑦,𝑧), relative to the point at which the current element 𝐼d𝐂 occurs, 𝐼=𝐼(𝑡), 𝑡∈[0,∞). The curve 𝐶 is rectifiable and can thus be approximated by a broken line with arbitrary precision. The broken line, denoted by 𝐶_lin below, is formed by a sequence of line segments. The magnetic field produced by the current filament-1, which is in the shape of broken line 𝐶_lin, approximates the field produced by a current filament-1 which is in the shape of curve 𝐶. The field of a filament-1 in the shape of line 𝐶_lin is a superposition of fields from the individual line segments that form 𝐶_lin. The formula for the calculation of the field of an arbitrary line segment of the current filament-1, which can be denoted 𝒜, can be derived using (20).

The unit vector colinear with a vector 𝐚 will be denoted by the superscript 1 on the right; that is, 𝐚¹=𝐚/𝑎. The line segment 𝒜 has an initial point (𝑥₀,𝑦₀,𝑧₀) and an end point (𝑥₁,𝑦₁,𝑧₁). Let us further denote 𝐚=𝑥₁−𝑥₀,𝑦₁−𝑦₀,𝑧₁−𝑧₀,𝐫₀=𝑥−𝑥₀,𝑦−𝑦₀,𝑧−𝑧₀,𝐫₁=𝑥−𝑥₁,𝑦−𝑦₁,𝑧−𝑧₁,𝑣=𝐫₀⋅𝐚¹,𝐝=𝐫₀−𝑣𝐚¹. (21) The line segment 𝒜 is formed by the points (𝐶_𝑥,𝐶_𝑦,𝐶_𝑧), 𝐶_𝑥=𝑥₀+𝑝𝑥₁−𝑥₀,𝐶_𝑦=𝑦₀+𝑝𝑦₁−𝑦₀,𝐶_𝑧=𝑧₀+𝑝𝑧₁−𝑧₀, (22) where the parameter 𝑝∈[0,1). The field produced by the line segment 𝒜, through which the current 𝐼 flows from the initial point to the end point, is, by (20), 𝐁_𝒜=𝜇𝑥,𝑦,𝑧,𝑡⁰𝐼()4𝜋𝐚×𝐝¹₀d𝑝𝑝²𝑎²−2𝑝𝑎𝑣+𝑟²₀^3/2 (23) since d𝐂×𝐫=𝐚×𝐝d𝑝. After integrating and rewriting, we obtain 𝐁_𝒜=𝜇𝑥,𝑦,𝑧,𝑡⁰𝐚4𝜋¹×𝐝¹𝐼𝑑𝑣𝑟₀+𝑎−𝑣𝑟₁. (24)

The vector 𝐝 has its end at the point (𝑥,𝑦,𝑧) while its origin is the foot of the perpendicular dropped from the point (𝑥,𝑦,𝑧) onto the line 𝑃, on which the line segment 𝒜 lies. 𝑑 is thus the distance of the point (𝑥,𝑦,𝑧) from the line 𝑃. Therefore, the expression on the right-hand side of relation (24) is not defined if 𝐼≠0 and the point (𝑥,𝑦,𝑧) is on the line 𝑃. For (𝑥,𝑦,𝑧)∈𝑃, the integral in (23) is improper, and therefore 𝐁_𝒜𝑥,𝑦,𝑧,𝑡=lim_𝑛→∞𝐁_𝒜𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝑡, (25) where {(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)},𝑛>1, is an arbitrary sequence of points (𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)∉𝑃, which for 𝑛→∞ converges to (𝑥,𝑦,𝑧). 𝑛 greater than 1 is chosen so that the denotation of the sequence points should not collide with the denotation of the initial and end points of the line segment 𝒜. The quantities 𝐫_𝟎,𝐫_𝟏,𝐝, and 𝑣, defined for the point (𝑥,𝑦,𝑧), will for the sequence points depend on 𝑛, which in the following is expressed by the subscript 𝑛. Obviously it holds that lim_𝑛→∞𝑑_𝑛=0. (26) The vectors 𝐚^𝟏 and 𝐝^𝟏 are perpendicular to each other and therefore their vector product in relation (24) is a unit vector. The magnitude of the field 𝐁_𝒜 is thus determined by the expression to the right of the vector product in relation (24).

Let the point (𝑥,𝑦,𝑧) be an internal point of the segment 𝒜. From the validity of (26), it follows that there exists a subscript 𝑁,𝑁>1, such that for all 𝑛>𝑁, it holds that 0<𝑣_𝑛<𝑎, and hence lim_𝑛→∞𝐼𝑑_𝑛𝑣_𝑛𝑟_0𝑛+𝑎−𝑣_𝑛𝑟_1𝑛=∞. (27) This relation also holds when the point (𝑥,𝑦,𝑧) is the initial or end point of the line segment 𝒜 and all the points (𝑥_𝑛,𝑦_𝑛,𝑧_𝑛) lie in a plane that is perpendicular to 𝒜, and passes through the point (𝑥,𝑦,𝑧). If the point (𝑥,𝑦,𝑧)∈𝑃 lies to the left of the point (𝑥₀,𝑦₀,𝑧₀) (for the point to the right of (𝑥₁,𝑦₁,𝑧₁) the procedure is analogous), then there exists a subscript 𝑁,𝑁>1, such that for all 𝑛>𝑁, it holds that 𝑣_𝑛<0, 𝑣_𝑛=−(𝑟²_0𝑛−𝑑²_𝑛)^1/2,𝑎−𝑣_𝑛=(𝑟²_1𝑛−𝑑²_𝑛)^1/2, 𝐼𝑑_𝑛𝑣_𝑛𝑟_0𝑛+𝑎−𝑣_𝑛𝑟_1𝑛=𝐼𝑑_𝑛−𝑑1−^2𝑛𝑟²_0𝑛+𝑑1−^2𝑛𝑟²_1𝑛. (28) From the validity of (26), it follows that for sufficiently large 𝑛, it holds that 𝑑_𝑛≤𝑟_0𝑛,𝑑_𝑛≤𝑟_1𝑛. The square roots in (28) can thus be substituted by their expansion into the Taylor series [13], in which all but the first two terms are neglected; that is, 𝑑1−^2𝑛𝑟²_0𝑛𝑑≈1−^2𝑛2𝑟²_0𝑛,𝑑1−^2𝑛𝑟²_1𝑛𝑑≈1−^2𝑛2𝑟²_1𝑛, (29) yielding 𝐼2𝑑_𝑛𝑟²_0𝑛−𝑑_𝑛𝑟²_1𝑛. (30) For 𝑑_𝑛→0, the limit of this expression equals zero, so that lim_𝑛→∞𝐵_𝒜𝑥_𝑛,𝑦_𝑛,𝑧_𝑛,𝑡=0 (31) and hence 𝐵_𝒜(𝑥,𝑦,𝑧,𝑡)=0.

The integral in (23), if improper, converges, and its value is finite if the value of the limit in expression (25) does not depend on the choice of the sequence {(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)}. For (𝑥,𝑦,𝑧)=(𝑥₀,𝑦₀,𝑧₀) and for (𝑥,𝑦,𝑧)=(𝑥₁,𝑦₁,𝑧₁), this integral does not converge because the value of the limit in expression (25) depends on the choice of the sequence {(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)} and equals either ∞ or zero or the limit does not exist.

Up to now we have proved that the field 𝐁_𝒜(i)outside of 𝑃 is given by relation (24), (ii)is not defined on the line segment 𝒜, inclusive of its end points, and(iii)is zero on the line 𝑃 outside 𝒜. Since the field of current filament-1, which is in the shape of curve 𝐶, can be approximated with arbitrary precision by the field of current filament-1 in the shape of 𝐶_lin, it is proved that the field 𝐁 is not defined on current filament-1.

3.3. Magnetic Field of Current Defined by Current Density

The current density 𝐉 = 𝐉(𝑥,𝑦,𝑧,𝑡) is defined on the Cartesian product 𝑉×[0,∞), where 𝑉 is a closed set of points (𝑥,𝑦,𝑧) and 𝑡∈[0,∞). We assume that for (𝑥,𝑦,𝑧)∉𝑉, there is 𝐽=0. The set 𝑉 is realized by conductors and insulants and it cannot be ruled out that at some of its points there is 𝐽=0. 𝑉 has the property that with each of its points it also contains the whole line of the vector 𝐉 passing through this point, and thus the flux of vector 𝐉 through the boundary of the set 𝑉 (the conductor surface) is zero. In view of the assumption that 𝑉 is a closed set, this means that the motion of electric charge across the conductor surface in the direction of a tangent to the conductor surface cannot be ruled out; at the same time, this is not a case of surface current (through a current layer). The magnetic field can be established using the Biot-Savart law: 𝐁=𝜇𝑥,𝑦,𝑧,𝑡⁰4𝜋_𝑉𝐉×𝐫𝑟³d𝑉, (32) where 𝐫 is the position vector of the point (𝑥,𝑦,𝑧), relative to the point at which there is the current element 𝐉 d𝑉. No existence of currents outside of 𝑉 is assumed, so that the definition range of density 𝐉 can be extended to the whole space; outside of 𝑉 it holds that 𝐽=0. In reality (from the macroscopic viewpoint), the current density cannot be other than piecewise continuous and therefore it is bounded. According to [10], the field 𝐁 given by relation (32) is thus continuous at all the points of the space. The essence of calculating the integral in (32) consists in substituting the conductors in the set 𝑉 by current filaments-2 and then superposing the elementary fields produced by individual filaments.

3.4. Faraday’s Law and Magnetic Flux

The term inductance is closely related to the induction law. Faraday’s law of electromagnetic induction is expressed by Maxwell’s equation in the integral form ℰ_𝐶=_𝐶𝐄⋅d𝐂=−dΦ,d𝑡 (33) where 𝐶 is a closed curve, the integral of the vector of intensity of field 𝐄 along 𝐶 is the electromotoric force ℰ_𝐶, and Φ is the flux of the vector of magnetic field 𝐁 (magnetic flux) through an arbitrary simple finite piecewise smooth surface 𝑆 [13], in the following only surface, bounded by the curve 𝐶: Φ=_𝑆𝐁⋅d𝐒. (34) It is further assumed that the position and shape of the curve 𝐶 do not depend on 𝑡. Φ does not depend on the shape of the surface 𝑆, which is easy to prove if we express 𝐁 by means of the vector potential and apply Stokes’ theorem [4]. Stokes’ theorem can, however, be applied only if its assumptions are fulfilled. Similarly, relations (33) and (34) make sense only when the respective integrals and derivatives are available. This means, inter alia, that the field 𝐁 cannot be arbitrary. If the field 𝐁 is produced by a current given by the current density, then by Section 3.3, the field 𝐁 is continuous and its magnitude 𝐵 is bounded at all points of the space, and therefore the integral on the right-hand side of relation (34) always exists and is finite. Every curve 𝐶 can thus be uniquely assigned a linked flux [14] Φ=Φ(𝐶). The curve 𝐶 is arbitrary (except for the above assumptions); its position and shape need in no way be linked with the conductors considered, that is, with 𝑉 or with the lines of the vector 𝐉 [15].

It is often argued that magnetic flux is linked with the conductor, which is incorrect or inaccurate at best. In fact, there are infinitely many curves in a real solid conductor, having different linked fluxes. This ambiguity is removed by the assumption of a closed conductor which is a filament-1. It will be proved below that this is a fundamental error because the linked flux of filament-1 has not been defined.

Let then current filament-1 have the shape of a closed curve 𝐶 approximated by a broken line 𝐶_lin. The linked flux Φ(𝐶_lin) is the flux Φ(𝑆) of vector 𝐁 through a surface 𝑆 bounded by a curve 𝐶_lin. We choose an arbitrary line segment 𝒜⊂𝐶_lin. The flux Φ does not depend on 𝑆, so that 𝑆 may comprise a rectangle ℛ, one side of which lies on 𝒜. It holds that Φ(𝑆)=Φ(ℛ)+Φ(𝑆−ℛ). We will show that the value of Φ(ℛ) is infinitely large, which will prove that the flux Φ(𝑆), too, is of infinitely large value and hence not defined.

Without loss of generality, we choose the system of Cartesian coordinates 𝑥𝑦𝑧 such that the line segment 𝒜 of length 𝑎>0 lies on the axis 𝑧; its initial point is the origin of the system of coordinates, and the rectangle ℛ lies in the first quadrant of the plane 𝑥𝑧. The flux Φ(ℛ) is a sum of the fluxes of fields generated by line segments of the curve 𝐶_lin. The flux of the field produced by the line segment 𝒜 will be denoted by Φ_𝒜(ℛ). The field generated by the curve 𝐶_lin can be calculated using relation (24). The field 𝐁 produced by the segment line 𝒜 in the rectangle ℛ has only the 𝑦-component 𝐵_𝒜𝑦 nonzero and is given by relation (24), where 𝐚=(0,0,𝑎), 𝐫₀=(𝑥,0,𝑧), 𝐫₁=(𝑥,0,𝑧−𝑎), 𝑣=𝑧, and 𝐝=(𝑥,0,0). The rectangle ℛ is the Cartesian product of intervals [𝑥_𝑏,𝑥_𝑒]×[0,𝑎], where 0=𝑥_𝑏<𝑥_𝑒, and hence Φ_𝒜(ℛ)=^𝑥𝑒_𝑥𝑏^𝑎₀𝐵_𝒜𝑦(𝑥,0,𝑧,𝑡)d𝑧d𝑥. (35) As assumed, 𝑥_𝑏=0, so that the integral on the right-hand side of (35) is improper because the field 𝐁_𝒜 is not defined on 𝒜. We calculate the integral in (35) for 𝑥_𝑏>0 and via a limit transition for 𝑥_𝑏→0+, we determine Φ_𝒜(ℛ) [13]. For 𝑥_𝑏>0, it holds that Φ_𝒜(ℛ)=𝜇₀𝐼2𝜋𝑎ln𝑎+𝑥²_𝑏+𝑎²𝑥_𝑏+𝑥_𝑏−𝑎²+𝑥²_𝑏−𝑎ln𝑎+𝑥²_𝑒+𝑎²𝑥_𝑒−𝑥_𝑒+𝑎²+𝑥²_𝑒. (36) In view of the assumptions made, the limit of the first term on the right-hand side of (36) for 𝑥_𝑏→0+ is equal to +∞; the limit of the remaining terms has a finite value. This proves that the flux Φ_𝒜(ℛ) is not defined. From this, it follows that the linked flux Φ(𝐶) is not defined either unless 𝐶 is a filament-1.

3.5. On Inductance and Its Definition

According to [7], ideal inductor is defined in CT as a linear inductive two-terminal element. The electric energy absorbed by an inductive two-terminal element is stored in magnetic form and is completely restitutable. In [7], inductance is defined by the following definition.

Definition 4. Inductance for an inductive two-terminal element with terminals A and B is the quotient of the linked flux Ψ between the terminals divided by the electric current 𝐼 in the element Ψ(𝑡)𝐿=𝐼(𝑡), (37) where the sign of the linked flux is determined by taking the voltage, in the time integral defining it, as the difference of the electric potentials at terminals A and B and where the current is taken as positive if its direction is from A to B and negative in the opposite case.

In CT, Ψ(𝑡)=^𝑡_𝑡0𝑈_AB(𝜏)d𝜏, (38) where 𝑡₀ is any instant before the first supply of electric energy [7]. The linked flux determined by relation (38) is useful only in CT because an inductor can in CT be only associated with voltage and current. The ideal inductor is dimensionless and does not produce any concrete field 𝐁. Equality (37) can be rewritten as Ψ(𝑡)=𝐿𝐼(𝑡) and differentiated with respect to 𝑡. If 𝐿 does not depend on 𝑡, then 𝑈_AB(𝑡)(𝑡)=𝐿d𝐼.d𝑡 (39) If 𝐿 depends on 𝑡, the differential inductance 𝐿_d=dΨ/d𝐼 is defined [7], which is constant only over a small interval of current. From the viewpoint of the objectives of the present paper, it is not necessary to deal with differential inductance because it does not bring anything new as regards the methodology: 𝐿_d=𝐿 for an ideal inductor.

Prior to calculating the resistance and inductance, it is necessary to replace the real solid conductor by resistors and inductors. An equivalent schematic diagram of the real solid conductor needs to be constructed. In sinusoidal conditions [7], this means that the conductor is replaced by an appropriate impedance. Conductor inductance then becomes inductor inductance in the equivalent schematic diagram and is given by relation (39).

4. Resistance and Inductance of Two Coaxial Tubular Conductors

4.1. Description of Conductors

Consider a pair of coaxial solid tubular conductors. Conductor cross sections and their position in the system of Cartesian coordinates 𝑥𝑦𝑧 are evident in Figure 1. In addition to the Cartesian system, the system of cylindrical coordinates 𝑟𝜙𝑧 is also used. But because of the symmetry with respect to axis 𝑧, the 𝜙 coordinate will be to no avail. The cross section of the inner or of the outer conductor is determined by the inequalities 𝑟_𝑖0≤𝑟≤𝑟_𝑖𝑛or𝑟_𝑜𝑛≤𝑟≤𝑟_𝑜0, (40) respectively. The quantities relating to the inner or the outer conductors are in the following denoted with the subscript 𝑖 or 𝑜, respectively, while the subscript 𝑐 denotes either 𝑖 or 𝑜. The permeability of conductors and of the insulation between the conductors is assumed to be 𝜇₀. The two conductors have the direction of axis 𝑧 and are connected to an ideal voltage source. An electrical appliance is connected to their other end. The source voltage and current 𝐼 in the conductors depend only on time 𝑡∈[0,∞). Conductor resistivity is 𝜚_𝑐, and the ratio of conductor cross-section magnitudes is denoted by 𝑟𝑞=^2𝑜0−𝑟²_𝑜𝑛𝑟²_𝑖𝑛−𝑟²_𝑖0. (41) In view of the symmetry of the arrangement of conductors, it is assumed that current density vector 𝐉 is parallel to the axis 𝑧 and depends only on 𝑟 and 𝑡. 𝐉=(0,0,𝐽_𝑐), where the function 𝐽_𝑐=𝐽_𝑐(𝑟,𝑡) is defined for 𝑟∈[𝑟_𝑖0,𝑟_𝑜0] and 𝐽_𝑐(𝑟,𝑡)=0,for𝑟∈𝑟_𝑖𝑛,𝑟_𝑜𝑛,𝐽_𝑐(𝑟,𝑡)=𝐽_𝑖(𝑟,𝑡),for𝑟∈𝑟_𝑖0,𝑟_𝑖𝑛,𝐽_𝑐(𝑟,𝑡)=𝐽_𝑜(𝑟,𝑡),for𝑟∈𝑟_𝑜𝑛,𝑟_𝑜0. (42)

Figure 1 

A quarter of the cross section of coaxial conductors and of the insulation between the conductors.

The magnitude 𝐵 of the field vector 𝐁 also depends on only 𝑟 and 𝑡. The fact that the current 𝐼 and vectors 𝐉, 𝐁 do not depend on 𝑧 actually means that an infinitely high propagation velocity of electromagnetic field is assumed. In such a case, the current is considered to be slowly varying [4].

4.2. Partial Loops

Let 𝐷_𝑛 be a division of the interval [𝑟_𝑖0,𝑟_𝑖𝑛], determined by the points 𝑟_𝑖0<𝑟_𝑖1<𝑟_𝑖2<⋯<𝑟_𝑖𝑛. The points 𝑟_𝑖0,𝑟_𝑖1,…,𝑟_𝑖𝑛 will be mapped via mapping 𝑚 onto the points 𝑟_𝑜0>𝑟_𝑜1>⋯>𝑟_𝑜𝑛 in the interval [𝑟_𝑜𝑛,𝑟_𝑜0]; that is, the interval ℑ_𝑖𝑘=(𝑟_{𝑖,𝑘−1},𝑟_𝑖𝑘] will be mapped onto the interval ℑ_𝑜𝑘=[𝑟_𝑜𝑘,𝑟_{𝑜,𝑘−1}). Mapping 𝑚 is a one-to-one mapping of the intervals [𝑟_𝑖0,𝑟_𝑖𝑛] and [𝑟_𝑜𝑛,𝑟_𝑜0]. To determine 𝑚 means to determine the lines of vector 𝐉 in the conductors, according to [16] 𝑚𝑟_𝑖=𝑟_𝑜=𝑟²_𝑜𝑛+𝑞𝑟²_𝑖𝑛−𝑟²_𝑖. (43) For simplicity it is assumed that 𝐷_𝑛 is chosen such that for 𝑘=1,2,…,𝑛, and it holds that 𝑎=𝜋𝑟²_𝑖𝑘−𝑟²_{𝑖,𝑘−1}𝜋𝑟,𝑎=^2𝑖𝑛−𝑟²_𝑖0𝑛. (44) In consequence of relation (43), it holds for 𝑘=1,2,…,𝑛 that 𝑞𝑎=𝜋𝑟²_{𝑜,𝑘−1}−𝑟²_𝑜𝑘. (45)

Via division 𝐷_𝑛, coaxial conductors can be divided into 𝑛 partial conductors, with the cross section of the 𝑘th conductor being an annulus. The annulus of the 𝑘th partial conductor in the inner conductor or in the outer conductor is determined by the radii 𝑟_{𝑖,𝑘−1},𝑟_𝑖𝑘 or 𝑟_𝑜𝑘,𝑟_{𝑜,𝑘−1}. The 𝑘th partial conductor in the inner conductor and the 𝑘th partial conductor in the outer conductor form the 𝑘th partial loop.

Conductor resistivity is constant, and so current density in the coaxial conductors will be continuous with respect to 𝑟 and can therefore be approximated with arbitrary accuracy by a function that is constant by parts. It can therefore be assumed that on the 𝑘th partial interval ℑ_𝑖𝑘, or ℑ_𝑜𝑘, the current density magnitude 𝐽_𝑖𝑘(𝑡), or 𝐽_𝑜𝑘(𝑡), is independent of 𝑟. The current flowing through the 𝑘th partial loop is 𝐼_𝑘(𝑡), so that 𝐼_𝑘(𝑡)=𝑎𝐽_𝑖𝑘(𝑡)=−𝑞𝑎𝐽_𝑜𝑘(𝑡). (46) From relation (46) it follows that 𝐽_𝑜𝑘(𝑡)𝐽=−^𝑖𝑘(𝑡)𝑞. (47)

The cross section of each of the conductors through the plane 𝑧=const. is an equipotential area. In this paper, the segment of the two conductors between the planes 𝑧=𝑧₁ and 𝑧=𝑧₂, where 𝑧₁<𝑧₂, 𝑧₂−𝑧₁=𝑧₂₁, will be considered. Across the two conductors in the plane 𝑧=𝑧₁ or 𝑧=𝑧₂ is a voltage 𝑉(𝑧₁,𝑡) or 𝑉(𝑧₂,𝑡).

A segment of the conductor between the planes 𝑧=𝑧₁ and 𝑧=𝑧₂ can be replaced by 𝑛 lumped parameter circuits. Current in the 𝑘th partial loop depends on the resitance-1 of the 𝑘th partial loop. Resistances-1 are given by the resistivities 𝜚_𝑖,𝜚_𝑜 and the segment length 𝑧₂₁. The voltage induced in the segment of the 𝑘th partial loop can be determined using the procedure described below.

Let us choose the points 𝑐_𝑖1<𝑐_𝑖2<⋯<𝑐_𝑖𝑛 and 𝑐_𝑜𝑘=𝑚(𝑐_𝑖𝑘), 𝑘=1,2,…,𝑛, such that 𝑟_{𝑖,𝑘−1}≤𝑐_𝑖𝑘≤𝑟_𝑖𝑘. Let us denote Φ_𝑘=_𝐴𝑘𝐁⋅d𝐀_𝑘, (48) with 𝐴_𝑘 being a rectangle [𝑐_𝑖𝑘,𝑐_𝑜𝑘]×[𝑧₁,𝑧₂] in the plane 𝑦=0. According to Faraday’s law, a voltage 𝜕Φ_𝑘/𝜕𝑡 is induced around the boundary of the rectangle 𝐴_𝑘. If we take into consideration the fact that the magnetic field 𝐁 produced by the current in the conductors under consideration has on the circle 𝑟=const. the direction of the tangent to this circle, then 𝜕Φ_𝑘𝜕𝑡=𝑧₂₁𝜕𝜕𝑡^𝑛_ℓ=1𝐽_𝑖ℓ(𝑡)𝜙_𝑘ℓ, (49) where 𝜙_𝑘ℓ=^𝑐𝑜𝑘_𝑐𝑖𝑘ℬ_𝑖ℓ(𝑟)−ℬ_𝑜ℓ(𝑟)𝑞d𝑟. (50)ℬ_𝑖ℓ(𝑟) or ℬ_𝑜ℓ(𝑟) is a magnetic field produced by the ℓth partial conductor in the inner or the outer conductor per unit current density. For given 𝑐_𝑐1,𝑐_𝑐2,…,𝑐_𝑐𝑛, the magnetic fluxes considered depend on 𝑡 alone. The function 𝐽_𝑖ℓ(𝑡) on the right-hand side of relation (49) also depends on 𝑡 alone, and therefore the derivative 𝜕/𝜕𝑡 can be replaced by the derivative d/d𝑡 in (49). For a sufficiently large 𝑛 the voltage 𝑈_𝐿𝑘 induced in the segment of the 𝑘th partial loop differs little from dΦ_𝑘/d𝑡.

For the equivalent circuit of the segment of the 𝑘th partial loop, it holds that 𝑈_𝑅𝑖𝑘(𝑡)+𝑈_𝑅𝑜𝑘(𝑡)+𝑈_𝐿𝑘(𝑡)+𝑉𝑧₂,𝑡−𝑉𝑧₁,𝑡=0. (51) This equation, written for the segment of all partial loops, forms after modification a system of 𝑛 ordinary differential equations with constant coefficients with respect to the functions 𝐽_𝑖𝑘, 𝑘=1,2…,𝑛, 𝜚_𝑖+𝜚_𝑜𝑞𝐽_𝑖𝑘(𝑡)+^𝑛_ℓ=1𝜙_𝑘ℓd𝐽_𝑖ℓ(𝑡)(𝑡)d𝑡=𝑈,𝑘=1,2…,𝑛, (52) where 𝑈(𝑡)=[𝑉(𝑧₁,𝑡)−𝑉(𝑧₂,𝑡)]/𝑧₂₁.

In the steady state with a sinusoidal voltage source 𝑈(𝑡)=𝑈sin(𝜔𝑡+𝛼), it is of advantage to solve the system (52) for the complex voltage 𝑈exp(𝑗𝜔𝑡) and for the current densities 𝐽_𝑖𝑘exp(𝑗𝜔𝑡) on the partial intervals ℑ_𝑖𝑘,𝑘=1,2,…,𝑛, where 𝑈=𝑈exp(𝑗𝛼), 𝐽_𝑖𝑘=𝐽̂_𝑖𝑘exp(𝑗𝛽_𝑖𝑘), and thus 𝐽_𝑖𝑘(𝑡)=𝐽̂_𝑖𝑘sin(𝜔𝑡+𝛽_𝑖𝑘). Substituting phasors into (52) yields, after minor modification, a system of linear algebraic equations with complex coefficients for the unknown phasors 𝐽_𝑖𝑘, 𝑘=1,2,…,𝑛, 𝜚_𝑖+𝜚_𝑜𝑞𝐽_𝑖𝑘+𝑗𝜔^𝑛_ℓ=1𝜙_𝑘ℓ𝐽_𝑖ℓ=𝑈,𝑘=1,2…,𝑛. (53) The system (53) was solved using the Gaussian elimination with the selection of the maximum element in the matrix of the system. The maximum value of 𝑛 is given by the size of RAM. The calculations were carried out on a computer with 4 Gbyte RAM, which enables solving up to 10 000 equations. The coefficients in the main diagonal of system (53) include the element 𝜚_𝑖+𝜚_𝑜/𝑞, whose value does not depend on 𝑛 while the value of the other coefficients 𝜙_𝑘ℓ decreases with increasing 𝑛. Thus with increasing 𝑛, the main diagonal of the system matrix becomes dominant, which undoubtedly contributes to the stability of the solution of system (53). As regards the low values of the amplitude of densities 𝐽̂_𝑖ℓ, all that has been given about this problematic in [16, 17] holds.

The resistance-1 𝑅_𝑘 per 1 m of the considered segment of partial loops is for all partial loops identical because the validity of (44) is assumed. For the segment of the 𝑘th loop, it holds that 𝑅_𝑘=𝜚_𝑖+𝜚_𝑜𝑞⁻¹𝑎,𝑘=1,2,…,𝑛. (54)

The solution of (53) for specific cases is given in [16]. But attention should be drawn to an error. In Figures 4 and  6 in [16] incorrect values of the initial phase 𝛽_𝑜𝑘 are given (the denotation used in [16], is 𝛼_𝑜𝑘) in the partial conductors of the outer coaxial conductor. In the figures mentioned, the validity of relation (47) was ignored, so that 180^∘ needs to be added to the values 𝛼_𝑜𝑘.

4.3. Current Density in Conductors

The calculation of current density in a pair of coaxial tubular conductors and the determination of its resistance and inductance will be performed via the solution of equation system (53) in the case that the conductors are supplied from a source of sinusoidal voltage.

All the calculation results given below were obtained for 𝑐_𝑖𝑘=𝑟_{𝑖,𝑘−1}+𝑘−1𝑛𝑟_𝑖𝑘−𝑟_{𝑖,𝑘−1},𝑘=1,2,…,𝑛. (55) The current density resulting from the solution of equation system (53) can be considered as superposition of two current densities. The first density is generated by the voltage source; the second density is the density of induced current. The first current density is determined by the phasors 𝐽^sou_𝑐𝑘 and the second by the phasors 𝐽^ind_𝑐𝑘,𝑐=𝑖,𝑜;𝑘=1,2,…,𝑛. The partial conductors have identically large cross sections and therefore 𝐽^sou_𝑐𝑘 do not depend on 𝑘, 𝐽^sou_𝑖𝑘=𝑈𝜚_𝑖+𝜚_𝑜𝑞⁻¹,𝐽^sou_𝑜𝑘1=−𝑞𝐽^sou_𝑖𝑘. (56) The phasors 𝐽^ind_𝑐𝑘,𝑐=𝑖,𝑜;𝑘=1,2,…,𝑛, are determined by the relation 𝐽^ind_𝑐𝑘=𝐽_𝑐𝑘−𝐽^sou_𝑐𝑘, (57) where 𝐽_𝑐𝑘 is the solution of the equation system (53).

Example 1. The inner conductor is of copper, and the outer conductor is of aluminium; the resistivity values at a temperature of 20^∘C have been taken over from [18]: 𝑟_𝑖0=4mm,𝑟_𝑖𝑛=10mm,𝑟_𝑜𝑛=11mm,𝑞=1.5,𝜚_𝑖=1.712×10⁻⁸𝜚Ω⋅m,_𝑜=2.709×10⁻⁸𝑈(𝑡)Ω⋅m,=sin𝜔𝑡,𝑈=1V⋅m⁻¹,𝜔=2𝜋𝑓. (58) Figures 2 and 3 give (in continuous curves) the modules and arguments, respectively, of the phasors 𝐽_𝑐𝑘 and 𝐽^ind_𝑐𝑘,𝑐=𝑖,𝑜;𝑘=1,2,…,𝑛, in Example 1 for 𝑓=10³ Hz.

Figure 2 

Dependence of phasor module of resultant current density 𝐽𝑐𝑘 and of phasor module of induced current density 𝐽ind𝑐𝑘 on distance from axis of conductors in Example 1 for 𝑓=103Hz, 𝑛=1000. The module values have been normalized with respect to 𝐽̂sou𝑖𝑘=2.842×107A⋅m−2.

Figure 3 

Dependence of phasor argument of resultant current density 𝐽𝑐𝑘 and of phasor argument of induced current density 𝐽ind𝑐𝑘 on distance from axis of conductors in Example 1, for 𝑓=103 Hz, 𝑛=1000.

4.4. Inductance and Resistance of a Pair of Conductors

From the given voltage 𝑈 and the calculated current phasor 𝐼=𝑎^𝑛_𝑘=1𝐽_𝑖𝑘=−𝑞𝑎^𝑛_𝑘=1𝐽_𝑜𝑘, (59) inductance 𝐿 and resistance-2 𝑅_𝑠 of 1 metre of the considered pair of coaxial conductors can be determined. The considered segment of the two conductors can be replaced by the impedance 𝑍=𝑈𝐼,𝑍=ℜ𝑍+𝑗ℑ𝑍=𝑅_𝑠+𝑗𝜔𝐿. (60) Using (60), 𝐿 and 𝑅_𝑠 can be calculated for 𝜔>0. Admittedly, the value 𝐿 for 𝜔=0 has no meaning but it is equal to the limit of the values 𝐿 for 𝜔→0+ calculated using (60). For 𝜔=0, it holds that 𝑅_𝑠=𝑅_𝑘/𝑛. The values 𝑅_𝑠,𝐿 and the current phasor 𝐼=𝐼̂exp(𝑗𝛽_𝑐) in the conductors together with the current density depend on 𝑛; see Tables 1 and 2. The calculation of 𝑅_𝑠 and 𝐿 corresponds with the case 𝑛=1 on the assumption that current density in the conductors is independent of the distance from the axis of conductors. This assumption evidently leads to the not very accurate values of 𝐼̂ and 𝐿.

Table 1 

Dependence of current in inner conductor and impedance replacing 1 m of pair of conductors in Example 1 on 𝑛, for 𝑓=50 Hz.

Table 2 

Dependence of current in inner conductor and impedance replacing 1 m of pair of conductors in Example 1 on 𝑛, for 𝑓=10⁶ Hz.

Table 3 gives the dependence on 𝑓 of some quantities characterizing the pair of conductors in Example 1. The values 𝑅_𝑠 and 𝐿 for 𝑓=10⁻⁴ Hz in Table 3 can be regarded as the resistance-2 and inductance values in the case 𝜔=0, as mentioned previously. The values given in Table 3 for 𝑓>10⁻⁴ Hz were obtained by calculating for 𝑛=1000. The current density amplitude in the inner conductor reaches its maximum 𝐽̂_max for 𝑟=𝑟_𝑖𝑛.

Table 3 

Dependence of some quantities characterizing pair of conductors in Example 1 on 𝑓.

The principle of calculating the resistance-2 and inductance of the segment examined consists in replacing the segment of two conductors by impedance 𝑍, as given previously (see (60)). The examined segment of the two conductors is a parallel connection of 𝑛 partial loops, the equivalent impedances of which are 𝑍_𝑘, 𝑘=1,2,…,𝑛, and therefore it holds that 1𝑍=^𝑛_𝑘=11𝑍_𝑘=𝑎^𝑛_𝑘=1𝐽_𝑖𝑘𝑈=𝐼𝑈. (61) The equivalent impedance 𝑍_𝑘 of the segment of partial loops is illustrated in the Gaussian plane in Figure 4, for 𝑓=10³ Hz in Example 1.

Figure 4 

Equivalent and induced impedances 𝑍𝑘 and 𝑍ind𝑘 of partial loops in Gaussian plane in Example 1, for 𝑓=103 Hz, 𝑛=1000. Impedance parts are expressed in Ω⋅m−1. On horizontal axis, the value of resistance-1 of partial loop is marked by •.

For 𝑛=1, the segment of the two conductors is identical to the segment of its only partial loop and it holds that 𝑅_𝑠=𝑅_𝑘/𝑛. This means that resistance-2 of the segment under consideration does not depend on 𝑓, as confirmed by the data for 𝑛=1 in Tables 1 and 2. In the case of 𝑛=1, ℜ(𝑍) can be regarded as resistance-1 of the segment of both conductors. The situation is somewhat different for 𝑛>1.

The segments of partial loops satisfy equation system (53). These equations can be modified by breaking down the sum 𝜚_𝑖+𝜚_𝑜𝑞𝐽_𝑖𝑘+𝑗𝜔𝜙_𝑘𝑘𝐽_𝑖𝑘+𝑗𝜔^𝑛_{ℓ≠𝑘ℓ=1}𝜙_𝑘ℓ𝐽_𝑖ℓ=𝑈,𝑘=1,2…,𝑛. (62) Using the relation 𝐼_𝑘=𝑎𝐽_𝑖𝑘 yields 𝑅_𝑘𝐼_𝑘+𝑗𝜔𝐿_𝑘𝐼_𝑘+𝑗𝜔^𝑛_{ℓ≠𝑘ℓ=1}𝑀_𝑘ℓ𝐼_ℓ=𝑈,𝑘=1,2…,𝑛, (63) where 𝐿_𝑘=𝜙_𝑘𝑘𝑎,𝑀_𝑘ℓ=𝜙_𝑘ℓ𝑎,forℓ≠𝑘. (64) Dividing (63) by the current 𝐼_𝑘 leads to 𝑅_𝑘+𝑗𝜔𝐿_𝑘+𝑗𝜔𝑀_𝑘=𝑍_𝑘,𝑀_𝑘=^𝑛_{ℓ≠𝑘ℓ=1}𝑀_𝑘ℓ𝐼_ℓ𝐼_𝑘. (65) The numbers 𝑅_𝑘,𝐿_𝑘,𝑀_𝑘ℓ, ℓ=1,2,…𝑘−1, 𝑘+1,…,𝑛, are real numbers. 𝑅_𝑘 is the resistance-1 of 1 metre of the 𝑘th partial loop. 𝐿_𝑘 is the self-inductance of 1 metre of the 𝑘th partial loop and 𝑗𝜔𝐿_𝑘 is a pure imaginary number. 𝑀_𝑘ℓ is the inductance of 1 metre of the 𝑘th partial loop from the ℓth partial loop. For ℓ≠𝑘, 𝜔>0, the number 𝐼_ℓ/𝐼_𝑘≠1 is complex, so that the third term on the left-hand side of the first equation in (65) is a complex number, whose parts are generally nonzero. It holds that ℜ𝑍_𝑘=𝑅_𝑘−𝜔ℑ𝑀_𝑘,ℑ𝑍_𝑘=𝜔𝐿_𝑘+𝜔ℜ𝑀_𝑘. (66) The equaivalent impedance of the segment of the 𝑘th partial loop can also be expressed by the sum 𝑍_𝑘=𝑅_𝑘+𝑍^ind_𝑘, (67) where 𝑍^ind_𝑘=𝑗𝜔𝐿_𝑘+𝑀_𝑘 (68) can be called induced impedance. The induced impedances 𝑍^ind_𝑘 are illustrated in the Gaussian plane in Figure 4 for 𝑓=10³ Hz in Example 1. Since some of the ℜ(𝑍_𝑘) values can be negative (see Figure 4), they cannot represent the resistance-1 of a real conductor.

4.5. Joule Power in Conductors

A current 𝑎𝐽_𝑖𝑘 flows through the 𝑘th partial loop, in which heat is released—Joule’s dissipation power 𝑃_J (in W⋅m⁻¹). Let us write 𝑃_J(𝑘)=12^𝑘_ℓ=1𝜚_𝑖𝑎𝑎𝐽̂_𝑖ℓ²+𝜚_𝑜𝐽̂𝑞𝑎𝑞𝑎_𝑜ℓ²=𝑎2𝜚_𝑖+𝜚_𝑜𝑞^𝑘_ℓ=1𝐽̂²_𝑖ℓ. (69) The value of 𝑃_J(𝑘) calculated from relation (69) is equal to the Joule power in the considered segment of the first 𝑘 partial loops. The Joule power of the segment of both conductors under examination is thus 𝑃_J(𝑛).

Irrespective of the fact that ℜ(𝑍_𝑘) can have a negative value, the function 𝑃_Js(𝑘) can be defined by a relation that is similar to relation (69): 𝑃_Js(𝑘)=12^𝑘_ℓ=1ℜ𝑍_ℓ𝑎𝐽̂_𝑖ℓ²=𝑎2^𝑘_ℓ=1ℜ𝑈𝐽_𝑖ℓ𝐽̂²_𝑖ℓ. (70) The values of the functions 𝑃_J(𝑘) and 𝑃_Js(𝑘) are shown in Figure 5 for the segment of a pair of conductors in Example 1 for 𝑓=10³ Hz. Let us add, for completeness, that not only the quantities 𝑃_J(𝑘) and 𝑃_Js(𝑘), but also ℜ(𝑍_𝑘) and ℑ(𝑍_𝑘) are functions of the natural number 𝑘 and thus acquire discrete values. In Figures 4 and 5, the discrete values are joined by continuous curves. The difference 𝑃_J(𝑛)−𝑃_Js(𝑛) is not non-zero but its magnitude converges to zero with increasing 𝑛. For 𝑛=1000, it holds that 𝑃_J(𝑛)=591.50W·m⁻¹𝑃_Js(𝑛)=591.35W·m⁻¹ in Example 1. Since relations (60) and (61) hold, 𝑅_s=ℜ(𝑍) cannot be the resistance-1 of the considered segment of conductors either, even if it holds that 𝑃_Js(𝑛)≐𝑃_Js(𝑛)=12𝑅_s𝐼²̂. (71)

Figure 5 

Dependence of 𝑃J and 𝑃Js values on 𝑘, for the segment of conductors in Example 1, for 𝑓=103 Hz, 𝑛=1000.

4.6. Dependence of Inductance on 𝑟_𝑖0 and 𝑓

It follows from Table 3 that in a pair of conductors, the inductance, current in the conductors, and maximum magnitude of the current density depend on frequency 𝑓. The magnitude of current density in the inner conductor or outer conductor is an increasing or decreasing function, respectively, of the distance from the axis of both conductors, as follows from Figure 2 and from [16]. It is therefore relevant to deal with the dependence of 𝐿 on 𝑟_𝑖0 and 𝑓.

Example 2. The two coaxial conductors are of copper, having the same cross section (𝑞=1), 𝑟_𝑖𝑛=10mm, and insulation thickness 𝜃=𝑟_𝑜𝑛−𝑟_𝑖𝑛=1mm.
Figure 6 gives the dependence of 𝐿 on 𝑟_𝑖0 and 𝑓 in Example 2 (𝑛=1000). For a given 𝑟_𝑖𝑛, the value 𝐿₀=lim_{𝑟𝑖0→𝑟𝑖𝑛+}𝐿 (72) does not depend on 𝑓; however, 𝐿₀ depends on 𝜃. The following Table gives the values 𝐿₀ in Example 2: 𝜃(mm)00.512510𝐿₀10⁻⁸H⋅m⁻¹00.9761.9063.6478.10913.863. (73)

Figure 6 

Dependence of inductance on radius 𝑟𝑖0 and frequency 𝑓 in Example 2. Isolated points • denote values 𝐿 for 𝑟𝑖0=0mm. If 𝑓=0, then 𝐿=9.226×10−8H⋅m−1 for 𝑟𝑖0=0mm.

5. Comparison of Results

No exact calculation of the impedance of a pair of coaxial tubular conductors has so far been made, leaving aside the so-called wire models [19]. The wire model replaces a solid conductor by cylindrical conductors of constant current density. In such a replacement, the conductor cross section is replaced by disjoint circles, which cannot fully cover the conductor cross section. The conductor cross section has a finite magnitude but when it is covered by circles, a finite part of the cross section still remains noncovered, even with the number of circles growing beyond all bounds. The quotient of the area of circles and the area of the cross section replaced by the circles can be ≈0.9 as maximum.

Calculation methods are given in the following that are based on the assumption that the sought current density in solid conductor is the solution of a second-order differential equation [20]. These methods are an analogy with the calculation of current density in tubular conductor [5, 17]. The boundary conditions necessary for the solution can be determined correctly only in the case that the inner conductor is cylindrical and the outer one is tubular with infinitely large radius 𝑟_𝑜0 [8, 17, 21]; in the other cases, the boundary conditions are estimated [20] or the outer conductor is not considered.

The problematic solved in the present paper is to some extent related to the problematic of calculating induced current densities and earth impedances of overhead and buried conductors [22–24], in the solution of which, for example, the neural network artificial intelligence technique [25] can be employed.

Several formulae are given in the literature for an approximate calculation of the inductance of two coaxial solid conductors. It is true that these formulae can give a very exact value of inductance but only for some values of parameters (conductor dimensions and current frequency). However, the border between suitable and unsuitable parameters can only be determined via comparing approximate and exact inductance values.

The results of calculating 𝐿 using (60) will be compared with the values 𝐿_𝑎, 𝑎=H,T,M,S, for a pair of coaxial copper conductors, 𝑟_𝑖0=0mm, 𝑟_𝑖𝑛=5mm, 𝑞=1, 𝜃=0,1,2,3,4,5mm, 𝑓=50,10³,10⁶Hz. The values of inductance 𝐿(𝑛=1000) and the percent deviations Δ_𝑎𝐿=100^𝑎−𝐿𝐿,𝑎=H,T,M,S, (74) are given in Table 4. The formula for calculating 𝐿_H is given in [26]. For the high frequencies 𝑓≥10⁶ Hz, 𝐿_H=𝜇₀𝑟2𝜋ln^𝑜0𝑟_H,where𝑟_H=𝑟_𝑖𝑛for𝑟_𝑖0𝑟=0_𝑖0for𝑟_𝑖0>0. (75) For the low frequencies (𝑓<10⁶ Hz), it is necessary to add 5 ×10⁻⁸H⋅m⁻¹ on the right-hand side of (75), which according to [27] is the internal inductance of inner conductor. As can be seen from Table 4, formula (75) only gives orientational values of the inductance because when it was being derived, a constant current density in the cross section of conductors was assumed and part of the magnetic flux was neglected so that also the dependence on 𝑓 is vague. Formula (75) is acceptable for the low frequencies in the case that the size of conductor cross sections is small with respect to the size of the cross section of insulation between the conductors.

Table 4 

Comparison of inductances for a pair of coaxial copper conductors, 𝑟_𝑖0=0 mm, 𝑟_𝑖𝑛=5 mm, 𝑞=1.

The formula for calculating 𝐿_T is given in [3] 𝐿_T=𝜇₀𝑟2𝜋ln^𝑜0𝑟_𝑖𝑛. (76) When deriving this formula, it was assumed that conductor thicknesses, that is, 𝑟_𝑖𝑛−𝑟_𝑖0 and 𝑟_𝑜0−𝑟_𝑜𝑛, were negligible with respect to 𝑟_𝑖𝑛 and 𝑟_𝑜0. This means that a current of finite magnitude only flows in an infinitely thin layer near the conductor surface. It is therefore no surprise that for a zero thickness of the insulation between conductors, 𝐿_T=0, and the value of 𝐿_T is not very precise for a low 𝑓. Formula (76) has also been published in [28], with added assumption of 𝑞=1.

In [27], a formula is given for the calculation of inductance 𝐿_M “at sufficiently low frequencies” 𝐿_M=𝐿_T+5×10⁻⁸. (77) It is evident from Table 4 that for higher 𝑓 it would be more appropriate to put 𝐿_M=𝐿_T but a problem arises when determining the boundary between low and high frequency. Even if this boundary was established, the value of 𝐿_M exactly in the neighbourhood of this boundary would not be precise.

In [8], the formula for the calculation of inductance is derived on the assumption that 𝑟_𝑖0=0, 𝑟_o0=∞, and 𝑓>10⁵Hz: 𝐿_S=𝜇₀𝑟2𝜋ln^𝑜𝑛𝑟_𝑖𝑛+𝜚_𝑖2𝜔𝜇₀1𝑟_𝑖𝑛+1𝑟_𝑜𝑛. (78) A comparison of the inductance values in Table 4 reveals that Formula (78) is more accurate for the higher frequencies. The inductance 𝐿_S, calculated using (78), does not depend on 𝑟_i1 and 𝑟_o2, that is, no hindrance exactly at the higher frequencies. The unsuitability of (78) for the low frequencies follows from the fact that lim_𝜔→0+𝐿_S=+∞ (79) but, for example, for 𝜃=2mm, lim_𝜔→0+𝐿=1.325×10⁻⁷H⋅m⁻¹ (80) and 𝐿_H=1.585×10⁻⁷H⋅m⁻¹. The reason is that inductance is defined independently of 𝑓 and must thus make sense also for 𝑓=0.

In the calculation of inductance, using (60), the method for calculating current density is of fundamental importance. One of the advantages of the latter method is that the current density can also be determined for 𝑟_i1>0. This advantage shows up in particular in the calculation of inductance at the lower frequencies, when the magnitude of current density is nonnegligible over the whole cross section of conductors.

6. Conclusion

The present paper is concerned with the definition and calculation of resistance 𝑅 and inductance 𝐿 of solid conductors. Attention is devoted, in particular, to the dependence of 𝐿 and 𝑅 on frequency 𝑓 under sinusoidal conditions. General considerations and conclusions are applied in the calculation of 𝐿 and 𝑅 of two infinitely long coaxial solid tubular conductors (coaxial conductors in the following). (i)A general formula (19) is derived for the calculation of conductance-1 and thus also resistance-1 of solid conductors. (ii)Current filaments-1 and filaments-2 are defined. It is proved that a field 𝐁 produced by filament-1 is not defined on this filament. This knowledge is used to prove that the flux of vector 𝐁 produced by a closed filament-1 which is in the shape of a curve 𝐶 through a surface bounded by the curve 𝐶 is not defined and thus the self-inductance of closed filament-1 cannot be calculated. (iii)When deriving the method for the calculation of current density in coaxial conductors, it is assumed that the current density on the cross section of conductors is piecewise constant. When the conductors are supplied from a source of time-dependent voltage, the current density is the solution of a system of differential equations (52). If the voltage source is sinusoidal, then in the steady state, the current density is the solution of a system of algebraic equations (53) with complex coefficients. Using the calculated current density and formula (60), the 𝐿 and 𝑅 of coaxial conductors can be established. (iv)Irrespective of the currently valid terminology and definitions [7], the still prevailing opinion is expressed by the following assertion: “the magnetic field of alternating current in a solid conductor displaces the current towards the conductor surface (skin effect) in consequence of which the resistance of the conductor increases.” It follows from the analysis made that the current is not pushed but the current from the source is partially compensated by induced current. With increasing 𝑓, the resultant current flowing through the conductors decreases and their resistance-2 increases while the resistance-1 of the conductors does not change. (v)A comparison is made of the exact value of the inductance of coaxial conductors established using formula (60), with the inductance values calculated using the approximate formulae (75)–(78) given in the literature.

Acknowledgments

This paper contains the results of research work funded until the end of 2011 from Research Plan no. MSM0021630516 of the Ministry of Education, Youth and Sports of the Czech Republic, and later from project of specific research program of Brno University of Technology no. FEKT-S-11-19.