Analytical-Numerical Approach to the Skin and Proximity Effect in Lines

energies

Article Analytical-Numerical Approach to the Skin and Proximity Eﬀect in Lines with Round Parallel Wires

Paweł Jabło ´nski* , Dariusz Kusiak and Tomasz Szczegielniak Department of Automation, Electrical Engineering and Optoelectronics, Faculty of Electrical Engineering, Czestochowa University of Technology, Armii Krajowej 17, 42-200 Czestochowa, Poland; [email protected] (D.K.); [email protected] (T.S.) * Correspondence: [email protected]; Tel.: +48-34-325-0306

Received: 14 November 2020; Accepted: 18 December 2020; Published: 19 December 2020

Abstract: Power and communication lines with round wires are often used in electrical engineering. The skin and proximity effects affect the current density distribution and increase resistances and energy losses. Many approaches were proposed to calculate the effects and related quantities. One of the simplest approximate closed solutions neglects the dimensions of neighboring wires. In this paper, a solution to this problem is proposed based on the method of successive reactions. In this context, the solution with substitutive filaments is considered as the first approximation of the true solution. Several typical arrangements of wires in single-phase communication lines or three-phase bus ducts are considered to detect the limits of applicability of the first approximation. The error of the first approximation grows with wire radius to skin depth ratio and wire radius to wire spacing ratio. When the wire radius to skin depth ratio is up to 1, and the gap between the wires is above the wire radius, the error is at a level of 1%. However, lowering the distance and/or skin depth leads to a much larger error in the first approximation.

Keywords: multi-wire bus duct; skin effect; proximity effect; cylindrical conductors; current filament; current density; successive reactions

1. Introduction Power and communication lines with wires of circular cross-section (often called round wires) are very often used in electrical engineering. Two wires are used in single-phase cables or in communication technology. Three-phase cables with and without neutral wire are often arranged in flat or trefoil geometry. When an alternating current passes in a wire, it generates an alternating magnetic field, which induces eddy currents in the wire so that the current is pushed towards the conductor surface (skin effect) [1]. The current also generates eddy currents in neighboring conductors, so the current density in them is also disturbed (proximity effect). Therefore, the current density is considerably disturbed compared to the uniform density occurring during the direct current passage. As a result, the effective cross-section area drops and the resistances of the bus duct elements enlarge, which lowers bus duct ampacity and power transfer efficiency. Hence, the knowledge of current distribution in power and communication lines is important to properly assess or optimize their properties. Despite the fact that the phenomenon is well known, and much research has been done in this field, the topic is still vivid, and new methods are still being developed, e.g., [2]. When analyzing known approaches, it seems they can be categorized into differential, integral and circuit-based. In the first approach, the subject is reduced to solving the Helmholtz equation in the conductors and the Laplace equation in the non-conducting regions. However, except for coaxial conductors, the equations are hard to solve. For example, when considering a twin line with round wires, one would like to use the method of separation of variables in bipolar coordinates. Unfortunately, the variables in

Energies 2020, 13, 6716; doi:10.3390/en13246716 www.mdpi.com/journal/energies Energies 2020, 13, 6716 2 of 21 the Helmholtz equation do not separate in that coordinate system. The situation is even worse for systems with multiple wires. Therefore, the differential approach requires either some simplification or numerical support. One of the first analytical solutions was found by Carson [3], who considered wave propagation along two parallel round wires. The solution was expressed as infinite series with coefficients, which have to be found via successive approximations. Later Arnold [4] used this result as well as some others to obtain formulae for the proximity factor defined as AC resistance to DC resistance for round parallel conductors in single and three-phase cases. There were many approaches with semi-analytical solutions, where fields are expressed as infinite series, yet the coefficients in the series have to be determined numerically by solving a system of linear algebraic equations. For example, Hussain and Bringer [5] obtained E and H fields for a system of parallel conductors, Ahmed et al. [6] used the method of fundamental solutions to express E and H fields as a linear combination of such solutions, and Machado et al. [7] used infinite series together with multipole expansion in the dielectric. Nevertheless, the final calculations always require the determination of coefficients in the series, which is mostly done via series truncation and solving a linear system of equations (but successive approximations are possible, too). Therefore, the most general approach is just using a purely numerical approach, like the finite element method (FEM) (e.g., [8–11]) or boundary element method (BEM) (e.g., [12]). Furthermore, combined models were used, like FEM + BEM [13] or Bubnov–Galerkin + variable separation method [14,15]. A pure numerical approach is quite general but does not allow us to obtain closed formulae, which makes it difficult to formulate general conclusions. The second approach uses an integral formulation. One of the earliest solutions was found by Manneback [16], who solved an integral equation for eddy currents in a round conductor due to nearby current filament. An interesting approach useful for high frequencies, when currents are constrained to a thin surface layer, was considered by Smith [17]. Dlabaˇcand Filipović[18] solved the integral equation for two round parallel wires by expressing the solution in the form of power series; of course, this required series truncation and solving a linear system of algebraic equations. The most general approach leads to the Fredholm integral equation of the second kind. Again, it can be solved numerically, e.g., via the collocation method, like in [19]. Finally, the circuit-based approach uses the idea of splitting the conductors into a certain set of parallel subconductors. Then each subconductor is assigned DC resistance and inductance based on assumed current density, which is usually considered constant (although unknown) if the subconductor’s cross-section area is small enough. Then equations known from circuit theory are used to formulate a system of algebraic equations [20]. In a way, a similar approach was proposed by Coufal [2,21], but with voltage excitation, which is more realistic. Nevertheless, all the approaches require some kind of numerical support. The Manneback solution was later used to model a set of round wires by treating the neighboring wires as filaments, e.g., [22,23]. This is one of the simplest approximate approaches. However, the main drawback is not taking into account the finite dimensions of neighboring wires. The method of including the dimensions into the solution was presented independently in [24,25]. In this work, the method is generalized for lines with multiple wires. In this approach, the solution with substitutive current filaments is regarded as the first approximation of the true solution. Several typical lines with round wires are considered as examples to check the performance of the method as well as the limits of applicability of the first approximation.

2. Methodology

2.1. Problem Description Let us consider a multi-wire bus duct with round conductors (Figure1). The bus duct consists of K long wires placed parallel to each other in a nonconductive and non-magnetic environment. The length is assumed much larger than the distances between the wires. All the wires are non-magnetic, have constant conductivity σk and invariable circular cross-section of radius Rk, where k = 1, 2, ... , K. Energies 2020, 13, 6716 3 of 21 Energies 2020, 13, x FOR PEER REVIEW 3 of 21 whereIt is assumed 푘 = 1,2, that… , 퐾. the It is wires assumed carry that sinusoidal the wires carry currents sinusoidal with angular currents frequency with angularω and frequency complex 휔 aroot-mean-squarend complex root- (RMS)mean- valuesquare (RMS). The frequencyvalue ℐ푘. isThe assumed frequency low is enough assumed to neglect low enough the displacement to neglect Ik thecurrents displacement (electromagnetic currents wave(electromagnetic length is much wave larger length than is circuitmuch larger size). than circuit size).

Figure 1. MultiMulti-wire-wire bus duct with round parallel wires.wires.

The above assumptions allow us to use 2D calculations in the cross-section of the configuration. The above assumptions allow us to use 2D calculations in the cross-section of the configuration. Due to the default assumption of linearity, the current density in wire i can be expressed as follows: Due to the default assumption of linearity, the current density in wire 푖 can be expressed as follows: 퐾 퐾 XK XK ss pp J 퐽== ∑W푊ik ℐ ==W푊ii ℐ + ∑ W푊ikℐ ==퐽J ++퐽J (1(1)) i 푖 푖푘Ii푖 푖푖I푖i 푖푘I푘k 푖i 푖 i k=푘=1 1 푘k=11,,푘k≠,푖i where 푊 are functions of size, wire position, frequency, and material properties, but independent where Wik푖푘are functions of size, wire position, frequency, and material properties, but independent of ofcurrents currents in thein the wires. wires. It is It convenient is convenient to separate to separate the part the dependentpart dependent on current on current in wire ini .wire This 푖 part. This is partrelated is related to the so-called to the so skin-called eff ectskin (superscript effect (superscript “s”). The “s”). remaining The remaining part is attributed part is attributed to eddy currentsto eddy currentsinduced ininduced wire i due in wire to current 푖 due passage to current in the passage neighboring in thewires, neighboring whichis wires, called which the proximity is called eff theect proximity effect (superscript “p”). Finding closed exact analytical forms for 푊 is possible only in (superscript “p”). Finding closed exact analytical forms for Wik is possible only푖푘 in special cases with specialcoaxial cases symmetry. with coaxial Another symmetry. geometry Another requires geometry either a requires kind of numericaleither a kind approach of numerical or introducing approach oradditional introducing simplifications. additional simplifications.

2.2. Approximate Analyt Analyticalical Solution In the casecase ofof aa system system with with multi-wire multi-wire round round conductors, conductors, the the approximate approximate analytical analytical solution solution can canbe found, be found, among among others, others, in [22]. in The [22 approach]. The approach is based onis based introducing on introducing simplifications simplifications and treating and the treatingneighboring the neighboring wires as infinitely wires thinas infinitely filaments. thin This fila leadsments. to This a fully leads analytical to a fully solution analytical to a substitutive solution to problema substitutive and is oftenproblem suffi cientand inis practice.often sufficient This approximate in practice. solution This willapproximate be temporarily solution marked will with be temporarilya tilde. It is convenientmarked with to usea tilde. polar It coordinates is convenient(r, θ to) associated use polar withcoordinates wire i—see (푟, 휃 Figure) associated2. If (r kwith, θk) wireare coordinates 푖—see Figure of the 2. If axis (푟푘 of, 휃 wire푘) arek thencoordinates it follows of that:the axis of wire 푘 then it follows that:

1 Γ푖푅푖 퐼I0(Γ푖푟r) 푊̃ (푟, 휃) = 1 ΓiRi 0 Γi (2) Weii푖푖(r, θ) = 2 ( ) (2) 휋푅2푖 22 I퐼1(Γ푖푅R푖 ) πRi 1 i i

1 푟 푟푘 ! 푊̃푖푘(푟, 휃) = 1 Λ ( r , 휃 − 휃푘; rk , Γ푖푅푖) (3) We (r, θ) = 휋푅2 Λ 푅 , θ θ ; 푅 , Γ R (3) ik 푖2 R푖 − k R푖 i i πRi i i where 퐼푛 is the modified Bessel function of the first kind of order 푛, and: where In is the modiﬁed Bessel function of the ﬁrst kind of order n, and: 1 + j Γ = √j휔휇 휎 = (4) 푖 p 0 푖 1훿+ j Γi = jωµ0σi = 푖 (4) δi is the propagation constant, in which 훿푖 = √2⁄(휔휇0휎푖) is the skin depth for wire 푖 and j is the p imaginaryis the propagation unit. Function constant, Λ inequals which: δi = 2/(ωµ0σi) is the skin depth for wire i and j is the imaginary unit. Function Λ equals: ∞ 퐼 (휌훾) X −푛 푛 Λ(휌, 휑; 휉, 훾) = −훾 ∑∞ 휉 n In(ργ) cos 푛휑 (5) Λ(ρ, ϕ; ξ, γ) = γ ξ− 퐼푛−1(훾) cos nϕ (5) − 푛=1 In 1(γ) n=1 − and is interpreted as eddy current density induced at point of polar coordinates (휌, 휑) in a unit radius round wire aligned with 푧 axis and having propagation constant 훾 due to a parallel current filament with the current π crossing point (휉, 0)—see Figure 3.

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and is interpreted as eddy current density induced at point of polar coordinates (ρ, ϕ) in a unit radius round wire aligned with z axis and having propagation constant γ due to a parallel current ﬁlament Energies 2020, 13, x FOR PEER REVIEW 4 of 21 Energies 20with20, 13 the, x FOR current PEERπ crossing REVIEW point (ξ, 0)—see Figure3. 4 of 21

Figure 2. Approximate approach to the proximity effect in multi-wire bus duct with round wires. Figure 2.Figure Approximate 2. Approximate approach approach to the to the proximity proximity effect eﬀect in in multi-wire multi-wire bus ductbus duct withround with wires.round wires.

FigureFigure 3. 3.InterpretationInterpretation ofof function functionΛ. Λ. Figure 3. Interpretation of function Λ. Such an approach neglects the finite dimensions of the cross-section of the adjacent conductors. Such an approach neglects the finite dimensions of the cross-section of the adjacent conductors. SuchThe an question approach is, what neglects error causes the finite such neglect.dimensions To estimate of the this, cross a comparison-section of with th thee adjacent exact solution conductors. The questionis needed. is, However,what error there causes is no such such method. neglect. One way To is estimate to use a solution this, obtaineda comparison via other methodswith the exact The question is, what error causes such neglect. To estimate this, a comparison with the exact solution (e.g.,is needed. finite elements) However, assumed there as is more no accurate.such method. Another One approach way isis toto construct use a solution a more precise obtained via solution is needed. However, there is no such method. One way is to use a solution obtained via other methodsapproximation (e.g., finite of the originalelements) problem assu solution.med as Bothmore methods accurate. were Another used in the approach research. is to construct a other methods (e.g., finite elements) assumed as more accurate. Another approach is to construct a more precise approximation of the original problem solution. Both methods were used in the more precise2.3. Improving approximation the Solution of the original problem solution. Both methods were used in the research. research. The true solution differs from that given by Equations (1)–(5) because the parts of current in a neighboring wire which are closer to the considered wire interacts stronger than it is expressed by 2.3. Improving the Solution 2.3. ImprovingEquation the (3). Solution In these terms, the above solution can be regarded as a first approximation of the true solution. A more accurate approximation can be obtained via the method of successive reactions. The true solution differs from that given by Equations (1)–(5) because the parts of current in a TheThe true eddy solution currents differs induced from in a that wire given due to theby neighboringEquations (1) substitutive–(5) because filaments the areparts now of a current new in a neighboring wire which are closer to the considered wire interacts stronger than it is expressed by neighboringsource wire of the which magnetic are field, closer which to the affects considered the neighboring wire wires interacts themselves stronger and induces than it additional is expressed by Equationeddy (3). currents.In theseThe terms, additional the above currents solution are another can sourcebe regarded of a magnetic as a fieldfirst andapproximation induce new eddy of the true Equation (3). In these terms, the above solution can be regarded as a first approximation of the true solution.currents A more and accurate so forth. a Ofpproximation course, this image can isbe just obtained an auxiliary via construction the method of ourof successive minds. In fact, reactions. solution. A more accurate approximation can be obtained via the method of successive reactions. The eddyonly currents the total induced of the reactions in a wire can bedue observed. to the neighboring However, by calculatingsubstitutive the successivefilamentsreactions, are now a new The eddywe currents approach induced the true solution. in a wire due to the neighboring substitutive filaments are now a new source of the magnetic field, which affects the neighboring wires themselves and induces additional source of the magnetic field, which affects the neighboring wires themselves and induces additional eddy currents. The additional currents are another source of a magnetic field and induce new eddy eddy currents. The additional currents are another source of a magnetic field and induce new eddy currents and so forth. Of course, this image is just an auxiliary construction of our minds. In fact, currents and so forth. Of course, this image is just an auxiliary construction of our minds. In fact, only the total of the reactions can be observed. However, by calculating the successive reactions, we only the total of the reactions can be observed. However, by calculating the successive reactions, we approach the true solution. approach the true solution. To formalize this idea, let us introduce superscripts (푚) and [푚] to denote 푚-th reaction and To formalize this idea, let us introduce superscripts (푚) and [푚] to denote 푚-th reaction and 푚-th approximation of the solution, respectively. Then we can write: 푚-th approximation of the solution, respectively. Then we can write: 푀 푀 푀 푀 [푀] (푚) (0) (1) (푚) ̃ [푀] 퐽푖[푀] = ∑ 퐽푖(푚) = ⏟퐽푖(0 ) + 퐽푖( 1) + ∑ 퐽푖(푚) = 퐽̃푖 + Δ퐽푖[푀] 퐽푖 = ∑ 퐽푖 = ⏟퐽푖 + 퐽푖 + ∑ 퐽푖 = 퐽푖 + Δ퐽푖 (6) 푚=0 퐽̃ 푚⏟= 2 (6) 푚=0 푖 푚⏟= 2 퐽̃푖 [푀] Δ퐽푖[푀] Δ퐽푖 where where (0)( ) ̃ ( ) 퐽푖(0) 푟, 휃 = ℐ푖푊̃푖푖 푟, 휃 (7) 퐽푖 (푟, 휃) = ℐ푖푊푖푖(푟, 휃) (7)

Energies 2020, 13, 6716 5 of 21

To formalize this idea, let us introduce superscripts (m) and [m] to denote m-th reaction and m-th approximation of the solution, respectively. Then we can write:

M M [M] X (m) ( ) ( ) X (m) [M] J = J = J 0 + J 1 + J = eJ + ∆J (6) i i i i i i i m=0 | {z } m=2 eJ | {z } i [M] ∆Ji

where (0) J (r, θ) = We (r, θ) (7) i Ii ii K (1) X J (r, θ) = We (r, θ) (8) i Ik ik k=1,k,i

[M] Function eJ represents the approximate solution presented in Section 2.2, whereas ∆J is the i i correction due to the ﬁnite size of the wires after considering M higher-order reactions. Reaction m 2 in wire i can be calculated by solving the equation: ≥ X 2 (m) 2 (m) (m 1) A (X) Γ A (X) = µ J − (X) (9) ∇ i − i i − j j,i and then calculating: (m) (m) J = jωσ A (10) i − i i The point is that Equation (9) is not easy to solve. If it were, the whole procedure would not be necessary because the true solution could be obtained in one-step. The proposition is to represent (m 1) − J j at point X as follows:

(m 1) (m 1) J − (X) = J − (Y)δ(X Y)dY (11) j x j − Sj where δ(X Y) is a 2D Dirac delta function placed at point Y. Physically, this is equivalent to think of − (m 1) − ( ) wire j as an infinite set of infinitesimal filaments with currents J j Y dY. Solution for a round wire and one filament is, however, analogical to that which is given by Equation (3). Hence, by the method of superposition reaction m in current density can be expressed as follows (Figure4): ! (m) 1 X (m 1) r rY J (r, θ) = J − (Y)Λ , θ θ ; , Γ R dY (12) i 2 x j Y i i πR Ri − Ri i j,i Sj

2.4. Numerical Implementation Integral (12) is hard to evaluate analytically; therefore, some numerical support will be required. In its simplest form, wire, j can be divided into Q “small” fragments of areas Sj,q and then:

! Q ! (m 1) r rY X (m 1) r rj,q − ( ) − x J j Y Λ , θ θY; , ΓiRi dY J j Yj,q Sj,qΛ , θ θj,q; , ΓiRi (13) Ri − Ri ≈ Ri − Ri Sj j=1

(m 1) (m 1) − = − where Yj,q is the formal center of the fragment Sj,q. Let us denote J j,q J j Yj,q and introduce designation: ! Sj,q rp rj,q Λ = Λ , θp θ ; , Γ R (14) i,j;p,q 2 R − j,q R i i πRi i i Energies 2020, 13, x FOR PEER REVIEW 5 of 21

퐾 (1) ̃ 퐽푖 (푟, 휃) = ∑ ℐ푘푊푖푘(푟, 휃) (8) 푘=1,푘≠푖 ̃ [푀] Function 퐽푖 represents the approximate solution presented in Section 2.2, whereas Δ퐽푖 is the correction due to the finite size of the wires after considering 푀 higher-order reactions. Reaction 푚 ≥ 2 in wire 푖 can be calculated by solving the equation:

2 (푚) 2 (푚) (푚−1)( ) ∇ 퐴푖 (푋) − Γ푖 퐴푖 (푋) = −휇 ∑ 퐽푗 푋 (9) 푗≠푖 and then calculating:

(푚) (푚) Energies 2020, 13, 6716 퐽푖 = −j휔휎푖퐴푖 6 of 21 (10) The point is that Equation (9) is not easy to solve. If it were, the whole procedure would not be Then a numerical version of Equation (12) can be rewritten as follows: necessary because the true solution could be obtained in one-step. The proposition is to represent (푚−1) (m) X X (m 1) 퐽 at point 푋 as follows: = − 푗 Ji,p J j,q Λi,j;p,q (15) j,i q 퐽(푚−1)(푋) = ∬ 퐽(푚−1)(푌)훿(푋 − 푌)푑푌 푗 푗 (m) (m) (11) It is convenient to arrange the discrete current푆푗 densities Ji,p in vectors Ji as well as introduce matrices Λij with elements Λi,j;p,q. Then it follows that: where 훿(푋 − 푌) is a 2D Dirac delta function placed at point 푌. Physically, this is equivalent to think (m) X (m 1) (푚−1) = − = of wire 푗 as an infinite set of infinitesimalJi filamentsΛi,jJj , withm currents2, 3, 4, ... 퐽푗 (푌)푑푌. Solution (16)for a round j i wire and one filament is, however, analogical, to that which is given by Equation (3). Hence, by the Energiesmethod 2020, 1of3, superpositionx FOR(0) PEER(1) REVIEW reaction 푚 in current density can be expressed as follows (Figure 4): 6 of 21 with Ji and Ji evaluated from Equations (7) and (8). Such an approach is equivalent to representing each wire as a set of the finite number of current filaments distributed throughout the wire cross-sections. (푚) 1 (푚−1) 푟 푟푌 Then a Hence,numerical wire j, version when퐽푖 ( acting푟, 휃 of) = onEq theuation2 neighboring∑ ∬ (12)퐽푗 can wires, (be푌 can) Λrewritten( be modeled, 휃 − 휃 as푌; as follows a, setΓ푖푅 of푖)Q:푑푌 current filaments (12) 휋푅푖 푆 (m 1) 푅푖 푅푖 placed at points Y and carrying currents푗≠푖 J푗 − S (Figure5). j,q (푚) j,q j,q (푚−1) 퐽푖,푝 = ∑ ∑ 퐽푗,푞 Λ푖,푗;푝,푞 (15) 푗≠푖 푞 (푚) (푚) It is convenient to arrange the discrete current densities 퐽푖,푝 in vectors 퐉푖 as well as introduce matrices 횲푖푗 with elements Λ푖,푗;푝,푞. Then it follows that:

(푚) (푚−1) 퐉푖 = ∑ 횲푖,푗퐉푗 , 푚 = 2,3,4, … (16) 푗≠푖 (0) (1) with 퐉푖 and 퐉푖 evaluated from Equations (7) and (8). Such an approach is equivalent to representing each wire as a set of the finite number of current filaments distributed throughout the wire cross-sections. Hence, wire 푗, when acting on the neighboring wires, can be modeled as a set of (푚−1) 푄 current filaments placed at points 푌푗,푞 and carrying currents 퐽푗,푞 푆푗,푞 (Figure 5). FigureFigure 4. 4.InterpretationInterpretation ofof Equation Equation (12). (12).

2.4. Numerical Implementation Integral (12) is hard to evaluate analytically; therefore, some numerical support will be

required. In its simplest form, wire, 푗 can be divided into 푄 “small” fragments of areas 푆푗,푞 and then: 푄 ( ) 푟 푟푌 ( ) 푟 푟푗,푞 ∬ 퐽 푚−1 (푌)Λ ( , 휃 − 휃 ; , Γ 푅 ) 푑푌 ≈ ∑ 퐽 푚−1 (푌 )푆 Λ ( , 휃 − 휃 ; , Γ 푅 ) (13) 푗 푅 푌 푅 푖 푖 푗 푗,푞 푗,푞 푅 푗,푞 푅 푖 푖 푆푗 푖 푖 푖 푖 Figure 5. AnFigure exemplary 5. An exemplary representation representation of of3 3wires wires푗 = withwith1 sets sets of currentof current filaments filaments (dots); the (dots); areas the areas associated with the wire fragments represented by the filaments are approximately equal.(푚−1) (푚−1) whereassociated 푌푗,푞 withis the the formal wire fragments center of represented the fragment by the 푆 푗filaments,푞 . Let us are denote approximately 퐽푗,푞 = equal.퐽푗 (푌푗,푞) and + introduce designationHaving found: reaction m, we can use it to construct reaction m 1 and so forth. Let us observe Havingthat found this procedure reaction is semi-analytical—it 푚, we can use gives it an to analytical construct solution reaction but requires 푚 a+ bit1 ofand a numerical so forth. Let us approach. However, in contrast to many푆푗, numerical푞 푟푝 methods, it푟푗 does,푞 not require solving a system of observe that this procedure is semiΛ푖,푗-;analytical푝,푞 = Λ—(it gives, 휃푝 − 휃an푗,푞 ;analytical, Γ푖푅푖) solution but requires a( 14bit) of a algebraic linear equations but only performing2 a summation. Moreover, matrices Λi,j are calculated 휋푅푖 푅푖 푅푖 numerical approach.only once, and However, then they in can contrast be used to to find many corrections numerical of any methods, order. The successive it does not corrections require solving a

are weaker and weaker so that the calculations can be ﬁnished when correction m is weak enough. system of algebraic linear equations but only performing a summation. Moreover, matrices 횲푖,푗 are The algorithm can be summarized as follows: calculated only once, and then they can be used to find corrections of any order. The successive corrections are weaker and weaker so that the calculations can be finished when correction 푚 is weak enough. The algorithm can be summarized as follows:

1. Select a set of filaments representing the wires and calculate matrices 횲푖,푗; (1) 2. For assumed currents ℐ푖 in all wires, calculate vectors 퐉푖 as the first approximation of the proximity effect; (푚) 3. Calculate vectors 퐉푖 For 푚 = 2,3, … to obtain presumed accuracy or to consider the presumed number of corrections 푀; [푀] 푀 (푚) 4. Use vectors 횫퐉푗 = ∑푚=2 퐉푗 to represent the correction of current density in wire 푖 as follows: 1 푟 푟 [푀]( ) [푀] 푗,푞 Δ퐽푖 푟, 휃 = 2 ∑ 퐽푗,푞 푆푗,푞Λ ( , 휃 − 휃푗,푞; , Γ푖푅푖) (17) 휋푅 푅푖 푅푖 푖 푗≠푖,푞 5. Repeat steps 2–4 for other currents in wires if necessary.

2.5. Power Losses and Resistance The power losses in wire 푖 per unit of length can be calculated using the standard integral over wire cross-sections as follows: ∗ 퐽푖 퐽푖 푃 = ∬ 푑푆 (18) 푖 휎 푆푖 푖 In the simplest approach, this integral can be calculated using the decomposition of wire cross-section into sectors related to the filaments. A more sophisticated but also more accurate is

Energies 2020, 13, 6716 7 of 21

1. Select a set of filaments representing the wires and calculate matrices Λi,j; ( ) 2. For assumed currents in all wires, calculate vectors J 1 as the first approximation of the Ii i proximity effect; (m) = 3. Calculate vectors Ji For m 2, 3, ... to obtain presumed accuracy or to consider the presumed number of corrections M; M [M] = P (m) 4. Use vectors ∆Jj Jj to represent the correction of current density in wire i as follows: m=2 ! [M] 1 X [M] r rj,q ∆J (r, θ) = J S Λ , θ θ ; , Γ R (17) i 2 j,q j,q j,q i i πR Ri − Ri i j,i,q

5. Repeat steps 2–4 for other currents in wires if necessary.

2.5. Power Losses and Resistance The power losses in wire i per unit of length can be calculated using the standard integral over wire cross-sections as follows: J J i i∗ Pi = x dS (18) σi Si In the simplest approach, this integral can be calculated using the decomposition of wire cross-section into sectors related to the ﬁlaments. A more sophisticated but also more accurate is using Equation (6) in the above formula and perform analytical integration. Brief information on this is presented in AppendixA. Then power can be used to ﬁnd resistance, e.g., in a twin line with opposing 2 currents (P1 + P2)/I .

3. Results and Discussion

3.1. Analyzed Conﬁgurations Several typical arrangements of current bus ducts with round wires are considered below. They are as follows:

twin single-phase line (Figure6a); • three-phase line in ﬂat arrangement (Figure6b,c); • three-phase line in trefoil arrangement (Figure6d,e); • three-phase line in a square arrangement (Figure6f). • In the case of the single-phase line, two versions are analyzed: opposing and same currents of equal RMS value. In the case of three-phase lines, versions of equal RMS currents and positive ( = exp( j120 ), = exp( j120 )), negative ( = exp(+j120 ), IL2 IL1 − ◦ IL3 IL2 − ◦ IL2 IL1 ◦ = exp(+j120 )) and zero sequences ( = = ) are analyzed. In the case when the IL3 IL2 ◦ IL3 IL2 IL1 total of currents is not zero, it is assumed the current returns via additional wire placed far from the considered ones. In each case, it is assumed that the wires are identical, and their radius equals R. The value of Γ is established as (1 + j)/δ, where δ is the skin depth. The spacing between the closest wires in each arrangement is equal to s (the distance between wire axes is d = s + 2R). Analysis of Equation (12) indicates that current density depends on R/δ and d/R ratios. Energies 2020, 13, x FOR PEER REVIEW 7 of 21 using Equation (6) in the above formula and perform analytical integration. Brief information on this is presented in Appendix A. Then power can be used to find resistance, e.g., in a twin line with 2 opposing currents (푃1 + 푃2)/퐼 .

3. Results and Discussion

3.1. Analyzed Configurations Several typical arrangements of current bus ducts with round wires are considered below. They are as follows:  twin single-phase line (Figure 6a);  three-phase line in flat arrangement (Figure 6b,c);  three-phase line in trefoil arrangement (Figure 6d,e);  threeEnergies-phase2020 ,line13, 6716 in a square arrangement (Figure 6f). 8 of 21

Figure 6. Analyzed current lines: (a)—twin single-phase line; (b)—ﬂat three-phase line without N Figure 6.wire; Analyzed (c)—ﬂat three-phasecurrent lines: line with(a)— Ntwin wire; single (d)—trefoil-phase three-phase line; (b line)—flat without three N- wire;phase (e )—trefoilline without N three-phase line with N wire; (f)—three-phase line with N wire arranged in a square. wire; (c)—flat three-phase line with N wire; (d)—trefoil three-phase line without N wire; (e)—trefoil three3.2.-phase Comparison line with with N Literature wire; (f)—three-phase line with N wire arranged in a square. As a test example, a twin line with opposing currents was selected, and the results were compared In thewith case those of obtainedthe single in- [2phase], Problem line, 2. two The versions parameters are of analyzed: the line were opposing as follows: and wire same radius currents of equal RMSR = value.10 mm, In distance the case between of three wire-phase axes d =lines,40 mm, versi wiresons of of hypothetical equal RMS conductivity currents and (σAl =positive36.9/c (ℐ퐿2 = MS/m), where c is a constant, frequency f = 60 Hz. ℐ퐿1 exp(−j120°), ℐ퐿3 = ℐ퐿2 exp(−j120°)), negative (ℐ퐿2 = ℐ퐿1 exp(+j120°), ℐ퐿3 = ℐ퐿2 exp(+j120°)) and Figure7a shows the current density RMS value on the left wire surface for c = 1, which corresponds zero sequences (ℐ = ℐ = ℐ ) are analyzed. In the case when the total of currents is not zero, it is to R/δ = 0.935.퐿3 The퐿2 values퐿1 are expressed in relation to DC current density when the total voltage drop assumedon the the current line equals returns 1 V/m. via The additional reference values, wire whichplaced were far obtainedfrom the by considered analyzing [2 ],ones. were marked In eachwithEnergies case dots. 2020, Figureit, 1 3is, x assumedFOR7b PEER shows REVIEW that the current the wires density are RMS identical value on, and the diametertheir radius of the equals left wire 푅 for. The8c of= 21value1 , of Γ 0.05, 0.02 and 0.01 as in work [2]. The values correspond to R/δ ratios equal to 0.935, 4.18, 6.61 and is establishedleft wire as for(1 +푐 =j)1⁄,훿 0.05,, where 0.02 and훿 is0.01 the as skin in work dep [2th.]. The The values spacing correspond between to 푅 the/훿 ratiosclosest equal wires to in each 9.35, respectively. The current density was expressed in absolute units of A/mm2. In both cases, the full arrangement0.935, is 4.18, equal 6.61 to and 푠 (the9.35, distance respectively. between The current wire densityaxes is was푑 = expressed푠 + 2푅). inAnalysis absolute ofunits Equation of (12) agreement was obtained. indicates A/mmthat current2. In both density cases, the depends full agreement on 푅 was⁄훿 obtained.and 푑⁄ 푅 ratios.

3.2. Comparison with Literature As a test example, a twin line with opposing currents was selected, and the results were compared with those obtained in [2], Problem 2. The parameters of the line were as follows: wire radius 푅 = 10 mm, distance between wire axes 푑 = 40 mm, wires of hypothetical conductivity (휎Al = 36.9/c MS/m), where 푐 is a constant, frequency 푓 = 60 Hz. Figure 7a shows the current density RMS value on the left wire surface for 푐 = 1, which corresponds to 푅/훿 = 0.935. The values are expressed in relation to DC current density when the total voltage drop on the line equals 1 V/m. The reference values, which were obtained by analyzing

[2], were marked with dots.( aFigure) 7b shows the current density RMS(b )value on the diameter of the

FigureFigure 7. 7.Current Current densitydensity valuesvalues (lines)(lines) in twin line compared with reference solution solution (dots) (dots) from from workwork [ 2[2],], problem problem 2: 2:( (aa)) currentcurrent densitydensity relatedrelated toto DC current density at the the same same voltage voltage drop drop on on the the 푅 푑 푓 푅⁄훿 lineline for for aluminum aluminum wires wires (R (= 10= 10 mm, mm,d = 40 = mm, 40 mm,f = 60 Hz,= 60R Hz,/δ = 0.935); = 0.935); (b) absolute (b) absolute values values of current of current density at voltage drop equal to 1 V/m for several values of 푐 = 휎Al⁄휎. density at voltage drop equal to 1 V/m for several values of c = σAl/σ. Table 1 shows the resistance per length of the unit for selected values of 푐. It was calculated via Equation (18) and compared with that given in work [2], Table 1, as well as with that obtained via finite element method (FEMM software, version 4.2, 64-bit 21 April 2019 by David Meeker, MA, USA, http://www.femm.info/wiki/HomePage) with very fine mesh. When the skin depth is comparable with wire radius, the results agree very well. For smaller skin depths, some discrepancies arise. This is probably the result of too rough discretization (each wire was divided into 100 filaments).

Table 1. Resistance values of twin line compared to reference ones for several values of 푐 = 휎Al⁄휎 for 푓 = 60 Hz, 푅 = 10 mm, 푑 = 40 mm.

Resistance per Unit of Length [μΩ/m] 풄 푹⁄휹 [2] FEM This Work 1 0.935 177.1 177 177 0.5 1.32 94.63 94.7 94.6 0.2 2.09 48.72 48.8 48.5 0.1 2.96 33.11 33.2 32.9 0.01 9.35 9.630 9.72 9.52 0.001 29.6 2.849 3.06 2.92

3.3. Current Density Distribution Figures 8 and 9 show current density distribution on wire surfaces for configurations 6a and 6c, respectively, for 푅⁄훿 = 3 and 푑⁄푅 = 2.5 with selected variants of currents. The solid lines represent the improved solution with higher-order reactions up to 푀 = 6. Each wire was divided into 100 filaments to ensure high accuracy. The solutions were compared with the results obtained by means of FEM. The FEM solution is represented by dots in Figures 7 and 8. It follows that the improved semi-analytical solution agrees very well with FEM results. The dashed lines represent the first approximation. The differences between the first approximation and the improved solution are noticeable. Further analysis reveals that the differences grow with 푅⁄훿 ratio and drops with 푑⁄푅 ratio.

Energies 2020, 13, 6716 9 of 21

Table1 shows the resistance per length of the unit for selected values of c. It was calculated via Equation (18) and compared with that given in work [2], Table1, as well as with that obtained via finite element method (FEMM software, version 4.2, 64-bit 21 April 2019 by David Meeker, MA, USA, http://www.femm.info/wiki/HomePage) with very fine mesh. When the skin depth is comparable with wire radius, the results agree very well. For smaller skin depths, some discrepancies arise. This is probably the result of too rough discretization (each wire was divided into 100 filaments).

Table 1. Resistance values of twin line compared to reference ones for several values of c = σAl/σ for f = 60 Hz, R = 10 mm, d = 40 mm.

Resistance per Unit of Length [µΩ/m] c R/δ [2] FEM This Work 1 0.935 177.1 177 177 0.5 1.32 94.63 94.7 94.6 0.2 2.09 48.72 48.8 48.5 0.1 2.96 33.11 33.2 32.9 0.01 9.35 9.630 9.72 9.52 0.001 29.6 2.849 3.06 2.92

3.3. Current Density Distribution Figures8 and9 show current density distribution on wire surfaces for configurations 6a and 6c, respectively, for R/δ = 3 and d/R = 2.5 with selected variants of currents. The solid lines represent the improved solution with higher-order reactions up to M = 6. Each wire was divided into 100 filaments to ensure high accuracy. The solutions were compared with the results obtained by means of FEM. The FEM solution is represented by dots in Figures7 and8. It follows that the improved semi-analytical solution agrees very well with FEM results. The dashed lines represent the first approximation. The differences between the first approximation and the improved solution are noticeable. Further analysis reveals that the differences grow with R/δ ratio and drops with d/R ratio. Figure 10 shows the plots of current density correction given by Equation (17) in wire cross-sections for selected configurations and supply variants. The values are expressed in percentage of uniform current density. It follows that the largest differences appear on wire surfaces. Analysis of analogous plotsEnergies for 20 the20, remaining13, x FOR PEER configurations REVIEW (not attached) leads to similar conclusions. 9 of 21

(a)

Figure 8. Cont.

(b)

Figure 8. Current density distribution on wire surface for 푅⁄훿 = 3 and 푑⁄푅 = 2.5 for twin line; values are expressed in a multiplicity of uniform current density in phase wire: (a) opposing currents; (b) same currents.

(a)

Energies 2020, 13, x FOR PEER REVIEW 9 of 21

Energies 2020, 13, 6716 10 of 21

(a)

(b)

FigureFigure 8. 8. CurrentCurrent density density distribution distribution on on wire wire surface surface for forR /푅δ⁄훿==33and andd /푑R⁄푅==2.52.5 forfor twin twin line; line; valuesvalues are are expressed expressed in a in multiplicity a multiplicity of uniform of uniform current current density density in phase in wire: phase (a) opposingwire: (a) currents; opposing (bcur) samerents; currents. (b) same currents.

3.4. The Effect of Wire Discretization The improved method requires using filaments(b) to represent the neighboring wires. The more filamentsFigure 8. used, Current the higher density accuracy distribution obtained, on but wire also surface the more for time푅⁄훿 = required3 and in푑⁄ computations.푅 = 2.5 for twin In this line; section,values the are e ff expressedect of filament in a numbermultiplicity is tested. of uniform The wires current are divided density into in n phasesectors wire: of similar (a) opposing areas. Figure 11 shows the tested discretization variants. currents; (b) same currents.

(a)

Figure 9. Cont. Energies 2020,, 13,, xx FORFOR PEERPEER REVIEWREVIEW 10 of 21

Energies 2020, 13, 6716 11 of 21 Energies 2020, 13, x FOR PEER REVIEW 10 of 21

(b)

Figure 9. Current density distribution on wire surface for 푅⁄훿 = 3 and 푑⁄푅 = 2..5 for the flat three-phase line with N wire; values are expressed in a multiplicity of uniform current density in phase wire: (a) positive current sequence; (b) zero sequence of currents (return via N wire).

Figure 10 shows the plots of current density(b) correction given by Equation (17) in wire cross-sections for selected configurations and supply variants. The values are expressed in percentage of uniform current density. It follows that the largest differences푅⁄훿 = 3 appear푑⁄푅 on= wire2.5 surfaces. FigurepercentageFigure 9. 9.Current Currentof uniform density density current distribution distribution density. on It wire follows on surface wire that surfacefor theR/ largestδ = for3 and differencesd/R = 2.5and appearfor the on ﬂat wire three-phase surfaces.for the flat Analysis of analogous plots for the remaining configurations (not attached) leads to similar threeAnalysisline-phase with of Nline analogous wire; with values N plotswire; are expressedvalues for the are remaining in expressed a multiplicity configurations in a of multiplicity uniform (not current of attached) uniform density leads incurrent phase to wire:densitysimilar in conclusions. phaseconclusions.(a) positivewire: (a current) positive sequence; current (b sequence;) zero sequence (b) zero of currents sequence (return of currents via N wire). (return via N wire).

Figure 10 shows the plots of current density correction given by Equation (17) in wire cross-sections for selected configurations and supply variants. The values are expressed in percentage of uniform current density. It follows that the largest differences appear on wire surfaces. Analysis of analogous plots for the remaining configurations (not attached) leads to similar conclusions.

(a)

(b(a) )

(c)

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Energies 2020, 13, 6716 12 of 21 Energies 2020, 13, x FOR PEER REVIEW 11 of 21

(d)

Figure 10. The values of correction given by Equation (17) for 푅⁄훿 = 3 and 푑⁄푅 = 2.5; values expressed in percentage of uniform current density in phase wire; (a) twin line with opposing currents; (b) twin line with similar currents; (c) flat three-phase line with N wire with a positive current sequence; (d) flat three-phase line with N wire with zero sequences of currents and return via

N wire. (d)

3.4.Figure The Effect 10. ofThe Wire values Discretization of correction given by Equation (17) for 푅⁄훿 = 3 and 푑⁄푅 = 2.5; values Figure 10. The values of correction given by Equation (17) for R/δ = 3 and d/R = 2.5; values expressed expressed in percentage of uniform current density in phase wire; (a) twin line with opposing in percentageThe improved of uniform method current requires density using in phase filaments wire; (ato) twinrepresent line with the opposingneighboring currents; wires. (b )The twin more currents; (b) twin line with similar currents; (c) flat three-phase line with N wire with a positive filamentsline with similar used, currents;the higher (c) accuracy ﬂat three-phase obtained, line withbut also N wire the with more a positive time required current sequence;in computations. (d) ﬂat In current sequence; (d) flat three-phase line with N wire with zero sequences of currents and return via thisthree-phase section, linethe witheffect N of wire filament with zero number sequences is tested. of currents The wires and return are divided via N wire. into 푛 sectors of similar N wire. areas. Figure 11 shows the tested discretization variants. 3.4. The Effect of Wire Discretization The improved method requires using filaments to represent the neighboring wires. The more filaments used, the higher accuracy obtained, but also the more time required in computations. In this section, the effect of filament number is tested. The wires are divided into 푛 sectors of similar areas. Figure 11 shows the tested discretization variants. (a) (b) (c) (d) (e) (f)

Figure 11. Considered variants of wire discretization. (a) 푛 = 1; (b) 푛 = 4; (c) 푛 = 16; (d) 푛 = 36; (e) 푛 Figure 11. Considered variants of wire discretization. (a) n = 1; (b) n = 4; (c) n = 16; (d) n = 36; (e) n = 64; = 64; (f) 푛 = 100. (f) n = 100. Figure 12 shows the relative error in current density modulus on the wire surface in a twin line Figure 12 shows the relative error in current density modulus on the wire surface in a twin line with opposing and similar currents. The left part of each plot corresponds to opposing currents, with opposing(a) and similar(b) currents. The(c) left part of each(d) plot corresponds(e) to opposing(f) currents, whereas the right part is for similar currents. The four plots correspond to weak and strong skin whereas the right part is for similar currents. The four plots correspond to weak and strong skin effects effectFigures as 11.well Considered as closely variants or loosely of wire placed discretization. wires. It follows(a) 푛 = 1 ;that (b) 푛when = 4; ( cthe) 푛 skin= 16 ;effect (d) 푛 is= 36 weak; (e) 푛(훿 ≥ 푅) as well as closely or loosely placed wires. It follows that when the skin effect is weak (δ R) and the and= 64the; ( fwires) 푛 = 100are. not too close (푑 ≥ 3푅), the error is below 1% even for the first approximation≥ wires are not too close (d 3R), the error is below 1% even for the first approximation (Figure 12a). (Figure 12a). However,≥ for closely placed wires (푑 = 2.2푅) to keep the error at a level of 1%, it is However, for closely placed wires (d = 2.2R) to keep the error at a level of 1%, it is necessary to take necessaryFigure 12to takeshows at theleast relative four filaments. error in currentIn case densityof stronger modulus skin effect on the (훿 wire≤ 푅⁄ surface3 ) and innot a tootwin closely line at least four filaments. In case of stronger skin effect (δ R/3) and not too closely placed wires, withplaced opposing wires, theand number similar ofcurrents. filaments The should left part be around of each ≤16. plot When corresponds the wires areto opposing placed closely, currents, even thewhereas numbermore filaments the of filamentsright arepart shouldrequired. is for similar be It around is currents.probably 16. When Thepossible thefour wires plotsto propose arecorr placedespond a more closely, to weakefficient even and morediscretization. strong filaments skin areeffect required.Hows ever,as wellIt the is as probablyresults closely indicateor possible loosely that placed to in propose low wires.-frequency a It more follows e fficases cientthat with when discretization. not the too skin close effect However, wires is weak, it theis ( 훿usually results≥ 푅) indicateandenough the that wires to in use low-frequencyare thenot firsttoo closeapproximation, cases (푑 ≥ with3푅), not thewhereas tooerror close isthis below wires, approximation 1% it iseven usually for is the enoughnot first enough approximation to use for thehigher first approximation,(Figurefrequency 12a). cases. whereasHowever, As thisthe for approximationfrequency closely placed rises, iswires the not skin enough(푑 = depth2.2 for푅) decreases higherto keep frequency the and error the cases. ateddy a level Ascurrents theof 1% frequency ,due it is to rises,necessaryneighboring the skin to depth take wires at decreases least concentrate four and filaments. the mainly eddy In near currents case theof stronger duewire to surface. neighboring skin effect Therefore, (훿 wires≤ 푅 ⁄eddy concentrate3 ) and current not too mainly density closely near is theplaced wireexpected surface. wires larger, the Therefore, nearnumber the eddyofwire filaments surface, current should whereas density be is veryaround expected small 16. in largerWhen deeper nearthe layers. wires the wire Inare such placed surface, cases closely, whereas, it would even very be smallmoremore in deeperfilaments efficient layers. toare use required. In a suchdenser cases, meIt issh it probablyof would filaments be possible more near ewire ffitocient proposesurfaces to use witha amore denser fewer efficient meshfilaments ofdiscretization. filaments near the wire near wireHowaxis. surfacesever, Of course, the with results it fewer should indicate filaments be kept that in near mindin low the that- wirefrequency this axis. model Ofcases is course, valid with for itnot frequencies should too close be keptat wires which in, mindit displacement is usually that this modelenoughcurrents is valid to c anuse for be frequencies theneglected first approximation,, as at stated which in displacement Section whereas 2.1. currentsthis approximation can be neglected, is not as enough stated infor Section higher 2.1 . frequency cases. As the frequency rises, the skin depth decreases and the eddy currents due to 3.5.neighboring The Effect of wires Skin Depthconcentrate mainly near the wire surface. Therefore, eddy current density is expected larger near the wire surface, whereas very small in deeper layers. In such cases, it would be moreTo assessefficient the to euseffect a denser of skin me depth,sh of filaments all the considered near wire surfaces cases were with analyzed fewer filaments for constant near the distance wire d =axis.3R Ofand course,R/δ itratio should varying be kept in in range mind 0.1thatto this 4. model The correctionis valid for frequencies given by Equation at which displacement (17) related to uniformcurrents current can be density neglected was, as calculated stated in Section at some 2 characteristic.1. points of all the configurations at selected variants of supply currents. The results are plotted in Figures 13 and 14. It follows that for δ > R the correction modulus rises approximately proportionally to R/δ ratio. The differences depend not only on skin depth to radius ratio but also on phases of currents in the wires. In the case of three-phase lines, the phase sequence is important. When the current distortion is high, there are significant harmonics with high R/δ ratio and various phase sequence. In such a case, the first approximation may introduce significant errors.

Energies 2020, 13, 6716 13 of 21

Energies 2020, 13, x FOR PEER REVIEW 12 of 21

(a)

(b)

(c)

(d)

Figure 12. RelativeRelative errors errors in incurrent current density density modulus modulus on wire on wire surface surface in a twin in a twinline for line various for various discretization variants for opposing (angles 0–180°) and similar currents (angles 180–360°); (a) weak discretization variants for opposing (angles 0–180◦) and similar currents (angles 180–360◦); (a) weak skin effect effect and and far far wires; wires; (b (b) )weak weak skin skin effect effect and and clos closee wires; wires; (c) (strongc) strong skin skin effect eff andect andfar wires; far wires; (d) (strongd) strong skin skin effect eff ectand and close close wires. wires.

3.5. The Effect of Skin Depth

Energies 2020, 13, x FOR PEER REVIEW 13 of 21

To assess the effect of skin depth, all the considered cases were analyzed for constant distance 푑 = 3푅 and 푅⁄훿 ratio varying in range 0.1 to 4. The correction given by Equation (17) related to uniform current density was calculated at some characteristic points of all the configurations at selected variants of supply currents. The results are plotted in Figures 13 and 14. It follows that for 훿 > 푅 the correction modulus rises approximately proportionally to 푅⁄훿 ratio. The differences depend not only on skin depth to radius ratio but also on phases of currents in the wires. In the case of three-phase lines, the phase sequence is important. When the current distortion is high, there are significantEnergies 2020 , 13harmonics, 6716 with high 푅⁄훿 ratio and various phase sequence. In such a case, the14 first of 21 approximation may introduce significant errors.

(a) (b)

(e) (f)

Figure 13. Modulus of correction given by. Equation (17) at specific points in configurations of twin Figure 13. Modulus of correction given by. Equation (17) at specific points in configurations of twin line and flat three-phase for selected variants of supply currents at the spacing between the wires line and flat three-phase for selected variants of supply currents at the spacing between the wires equal equal to wire radius (푑⁄푅 = 3) vs. 푅⁄훿 ratio (which is proportional to a square root of frequency); to wire radius (d/R = 3) vs. R/δ ratio (which is proportional to a square root of frequency); values values of current density relative to uniform current in phase wire. (a) twin line with opposing of current density relative to uniform current in phase wire. (a) twin line with opposing currents; (b) currents; (b) twin line with the same current; (c) flat three-phase line with 3 wires and positive twin line with the same current; (c) flat three-phase line with 3 wires and positive sequence of currents; sequence of currents; (d) flat three-phase line with 3 wires and zero sequence of currents; (e) flat (d) flat three-phase line with 3 wires and zero sequence of currents; (e) flat three-phase line with 4 wires three-phase line with 4 wires and positive sequence of currents; (f) flat three-phase line with 4 wires and positive sequence of currents; (f) flat three-phase line with 4 wires and zero sequence of currents. and zero sequence of currents.

Energies 2020, 13, 6716 15 of 21

Energies 2020, 13, x FOR PEER REVIEW 14 of 21

(a) (b)

(e) (f)

Figure 14. Modulus of correction given by. Equation (17) at specific points in non-flat three-phase Figure 14. Modulus of correction given by. Equation (17) at specific points in non-flat three-phase configurations for selected variants of supply currents at the spacing between the wires equal to wire configurations for selected variants of supply currents at the spacing between the wires equal to wire radius (푑⁄푅 = 3) vs. 푅⁄훿 ratio (which is proportional to the square root of frequency); values of radius (d/R = 3) vs. R/δ ratio (which is proportional to the square root of frequency); values of current current density relative to uniform current in phase wire. (a) trefoil three-phase line with 3 wires and density relative to uniform current in phase wire. (a) trefoil three-phase line with 3 wires and positive positive sequence of currents; (b) trefoil three-phase line with 3 wires and zero sequence of currents; sequence of currents; (b) trefoil three-phase line with 3 wires and zero sequence of currents; (c) trefoil (c) trefoil three-phase line with 4 wires and positive sequence of currents; (d) trefoil three-phase line three-phase line with 4 wires and positive sequence of currents; (d) trefoil three-phase line with 4 wires with 4 wires and zero sequence of currents; (e) three-phase line arranged in a square with positive and zero sequence of currents; (e) three-phase line arranged in a square with positive sequence of sequence of currents; (f) three-phase line arranged in a square with zero sequence of currents. currents; (f) three-phase line arranged in a square with zero sequence of currents.

3.6.3.6. The EEffectffect of Spacing betweenbetween WiresWires TheThe spacingspacing between between the the wires wires also also greatly greatly aff ectsaffects the currentthe current distribution. distribution. Figures Figures 15 and 15 16 and show 16 theshow modulus the modulus of current of current density density correction correction given by given Equation by Equation (17) versus (17) distance versus distance between between the wires the at weakwiresskin at weak effect skin (δ = effectR). It ( follows훿 = 푅). thatIt follows the correction that the modulus correction is approximatelymodulus is approximately inversely proportional inversely toproportional the distance to between the distance axes between of the wires. axes Forof the weak wires. skin For eff ectweak and skin distance effect betweenand distance the wires between not the wires not smaller than wire radius, the correction seems negligible. However, as shown in Figures 13 and 14, the correction grows with the skin depth, so for higher harmonics, the correction may be significant.

Energies 2020, 13, 6716 16 of 21 smaller than wire radius, the correction seems negligible. However, as shown in Figures 13 and 14, the correction grows with the skin depth, so for higher harmonics, the correction may be signiﬁcant. Energies 2020, 13, x FOR PEER REVIEW 15 of 21

(a) (b)

(e) (f)

Figure 15. Modulus of correction given by Equation (17) at specific points in twin line and flat Figure 15. Modulus of correction given by Equation (17) at specific points in twin line and flat three-phase lines for selected variants of supply currents for 푅⁄훿 = 1 vs. 푑⁄푅 ratio; values of three-phase lines for selected variants of supply currents for R/δ = 1 vs. d/R ratio; values of current current density relative to uniform current in phase wire. (a) twin line with opposing currents; (b) density relative to uniform current in phase wire. (a) twin line with opposing currents; (b) twin line twin line with the same current; (c) flat three-phase line with 3 wires and positive sequence of with the same current; (c) flat three-phase line with 3 wires and positive sequence of currents; (d) flat currents; (d) flat three-phase line with 3 wires and zero sequence of currents; (e) flat three-phase line three-phase line with 3 wires and zero sequence of currents; (e) flat three-phase line with 4 wires and with 4 wires and positive sequence of currents; (f) flat three-phase line with 4 wires and zero positive sequence of currents; (f) flat three-phase line with 4 wires and zero sequence of currents. sequence of currents.

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(a) (b)

(e) (f)

Figure 16. Modulus of correction given by. Equation (17) at specific points in non-flat three-phase Figure 16. Modulus of correction given by. Equation (17) at speciﬁc points in non-ﬂat three-phase lines lines for selected variants of supply currents for 푅⁄훿 = 1 vs. 푑⁄푅 ratio; values of current density for selected variants of supply currents for R/δ = 1 vs. d/R ratio; values of current density relative relative to uniform current in phase wire. (a) trefoil three-phase line with 3 wires and positive to uniform current in phase wire. (a) trefoil three-phase line with 3 wires and positive sequence of sequence of currents; (b) trefoil three-phase line with 3 wires and zero sequence of currents; (c) trefoil currents; (b) trefoil three-phase line with 3 wires and zero sequence of currents; (c) trefoil three-phase three-phase line with 4 wires and positive sequence of currents; (d) trefoil three-phase line with 4 line with 4 wires and positive sequence of currents; (d) trefoil three-phase line with 4 wires and zero wires and zero sequence of currents; (e) three-phase line arranged in a square with positive sequence sequence of currents; (e) three-phase line arranged in a square with positive sequence of currents; of currents; (f) three-phase line arranged in a square with zero sequence of currents. (f) three-phase line arranged in a square with zero sequence of currents.

3.7. Power Losses The resultsresults in in Sections Sections 3.5 and3.5 and3.6 indicate 3.6 indicate that using that theusing approximate the approximate analytical anal solutionytical presented solution inpresented Section in2.2 Section leads to 2.2 errors leads into currenterrors in density, current which density, are which the bigger, are the the bigger closer, the the closer wires the and wires the smallerand the thesmaller skin depth.the skin Current depth. densityCurrent is density directly is related directly to related power lossesto power via Equationlosses via (18). Equation Therefore, (18). powerTherefore losses, power were losses calculated were both calculated with higher-order both with higher reactions-order taken reactions into account taken (intoP) and account without (푃) them and aswithout resulting them from as resulting the first reactionfrom the ( Pfirst[1]), re andaction the ( di푃ff[1erence]), and the was difference then calculated. was then Figure calculated. 17 shows Figure the di17ff erenceshows presentedthe difference as δP presented% = P Pas[1] 훿푃/P%[1]= 100%(푃 − 푃.[ Power1])⁄푃[1 was] × 100% used. insteadPower ofwas resistance used instead because of − × resistance because the latter leads to ambiguity for three-phase bus ducts, which would require additional explanations. In cases such as twin lines with opposing currents, the error in resistance

Energies 2020, 13, 6716 18 of 21 the latter leads to ambiguity for three-phase bus ducts, which would require additional explanations. In cases such as twin lines with opposing currents, the error in resistance will be the same as δP%, because power losses are proportional to resistance. The calculations were performed for 4 sets of (R/δ, d/R) ratios: (1, 2), (1, 4), (4, 2), (4, 4). In all the cases, typical variants of currents in the bus duct were taken into account. Figure 17 shows cases for which δP% values were at least 1%. As expected, the most signiﬁcant errors occur for the highest tested value of R/δ and closely placed wires, i.e., for parameters (4, 2). In such cases, the errors are nearly 30% in some variants, but typically up to around 15%. The errors for a twin line are relatively small compared to those for three-phase lines. Of course, in practice, the wires are not placed so tightly. Therefore errors are then smaller. For d/R = 4 the errors are usually below 1%. Energies 2020, 13, x FOR PEER REVIEW 18 of 21

Figure 17. FigurePercentage 17. Percentage error δ Perror% for 훿푃 selected% for selected variants variants of supplyof supply in in the the considered considered configurations conﬁgurations (a–f as 푅 푑 (a–f as in Figure 6); numbers in parenthesesR showd and ratios and the preceding number is the in Figure6); numbers in parentheses show δ and R ratios훿 푅 and the preceding number is the number of the wire, wherenumber 4 of is the attributed wire, where to 4 N is attributed wire. to N wire.

4. Conclusions

Energies 2020, 13, 6716 19 of 21

4. Conclusions A semi-analytical method for a set of parallel round wires was proposed in the paper. It can be regarded as an extension of the analytical approach in which the neighboring wires are treated as current filaments, which is called here the first approximation. The first approximation leads to the fully analytical formula with neglecting the dimensions of the neighboring wires. The proposed method also leads to analytical formula, but the dimensions of the neighboring wires are taken into account. This, however, usually requires some numerical computations. In contrast to many numerical methods, it does not require solving an algebraic system of equations. In the further stage of research, the method was used for several typical configurations of single- and three-phase lines. The results were compared with those obtained by means of finite elements, and a full agreement was obtained. The results were also compared with those from the first approximation as well as to results found in the literature. The first approximation offers accuracy in current density about 1% when the skin depth is not less than the wire radius and when the gap between the wires is not less than the wire radius. The error is the bigger, the closer the wires are and the smaller the skin depth. The analysis revealed that if the skin depth is smaller than the wire radius, the error increases almost proportionally to the skin depth. This may introduce errors in power losses and resistances of bus duct elements, especially for a higher frequency, and in the case of three-phase systems—for currents with considerable levels of higher harmonics. Like all methods, also the considered one has its pros and cons. It offers a semi-analytical solution, which sometimes can be useful. There is no necessity to solve an algebraic system of equations because the successive reactions quickly tend to zero. However, the numerical effort in this method is quite noticeable because function Λ must be evaluated many times. When the skin depth is very small compared to the wire radius, the required discretization may be too large to effectively perform calculations with reasonable accuracy and in an acceptable time. Therefore, it seems that the most suitable application is for skin depths above R/5 and closely placed wires (as for distant wires, the first approximation offers quite an acceptable accuracy).

Author Contributions: Conceptualization, P.J.; methodology, P.J.; software, P.J.; validation, T.S. and D.K.; formal analysis, T.S.; writing—original draft preparation, P.J.; writing—review and editing, T.S. and D.K.; visualization, P.J. and D.K.; supervision, P.J. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A Using Equation (6) in Equation (18) leads to the following expression for active power losses in wire i: [M] [M] [M] eJ eJ∗ eJ ∆J ∗ ∆J ∆J ∗ [M] i i < i i i i P = dS + 2 dS + dS. (A1) i x σ x σ x σ Si S S The first term corresponds to the first approximation. The two other are corrections due to higher-order reactions. After putting Equations (7), (8) and (17) into the above formula, the integrals can be evaluated analytically because cosine in Equation (5) makes the mixed terms in the series products integrate to zero. The final formula for the power can be expressed as follows:    XK XK X X X  1   Pi =  ∗Ci,k,l + 2 ∗ Di,k,w + ∗ Di,w,v, (A2) πR2σ  IkIl < IkIw IwIv  i i k,l=1 k=1,k,i w w v Energies 2020, 13, 6716 20 of 21 where  1 ( ) = =  2 i0 ΓiRi for k l i,  Ci,j,k =  Θ(ξkiξli, θki θli, ΓiRi) for k , i and l , i, (A3)  −  0 otherwise, D = Θ(ξ ξ , θ θ , Γ R ). (A4) i,k,w ki wi ki − wi i i In the above formulas, w and v stand for any of additional filaments representing the wires except wire i. Parameter ξ = X X /R is the relative distance of filament k from the axis of wire i, and θ ki | k − i| i ki is the angular coordinate of filament k in the local coordinate system connected with wire i. Functions i n and Θ are defined as follows: ! γIn(γ) in(γ) = , (A5) < In 1(γ) − X ∞ n Θ(ζ, θ, γ) = ζ− in(γ) cos nθ. (A6) n=1

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