Transient Current Density in a Pair of Long Parallel Conductors

applied sciences Article Transient Current Density in a Pair of Long Parallel Conductors Oldˇrich Coufal   Citation: Coufal, O. Transient Current Density in a Pair of Long Parallel Conductors. Appl. Sci. 2021,11, 6920. https://doi.org/10.3390/app11156920 Academic Editors: Giovanni Petrone and Fabio La Foresta Received: 10 June 2021 Accepted: 23 July 2021 Published: 27 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Electrical Power Engineering, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 12, 61600 Brno, Czech Republic; [email protected] Abstract: Two infinitely long parallel conductors of arbitrary cross section connected to a voltage source form a loop. If the source voltage depends on time, then due to induction there is no constant current density in the loop conductors. It is only recently that a method has been published for accurately calculating current density in a group of long parallel conductors. The method has thus far been applied to the calculation of steady-state current density in a loop connected to a sinusoidal voltage source. In the present article, the method is used for an accurate calculation of transient current using transient current density. The transient current is analysed when connecting and short-circuiting the sources of sinusoidal, constant and sawtooth voltages. For circular cross section conductors, the dependences of maximum current density, maximum current and the time of achieving steady state on the source frequency, the distance of the conductors and their resistivity when connecting the source of sinusoidal voltage are examined. Keywords: transmission and distribution lines; induction; current density; mathematical modelling; ordinary differential equations 1. Introduction The article deals with the calculation of transient current density and transient current in a long loop formed by two long parallel solid conductors of arbitrary cross section and a voltage source. The source voltage is assumed to be dependent on time t . Due to induction, the current density in the conductors is a function of t too. It is only recently that a method has been published for accurately calculating current density in a group of long parallel conductors [ 1 ]. The method has thus far been applied to the calculation of steady-state current density in a loop connected to a sinusoidal voltage source [2]. The original contributions of the present article are as follows: • adjustment of the method of current density calculation proposed in [ 1 ] to a form suitable for numerical calculation of transient current density; • presentation of results of the calculation of transient current density and transient current; • analysis of transient current when connecting on and short-circuiting a voltage source; • results of the calculation of the dependences of maximum current density, maximum current and the time of achieving steady state on the distance of circular cross section conductors, the frequency of sinusoidal voltage source and the resistivity of conductors. In the existing methods for the calculation of transient current in a pair of conductors of circular cross section, the current is not determined by transient current density. Either there is a constant current density in the conductor cross section [ 3 ] or the loop is replaced by a circuit with lumped parameters. The circuit is a series connection of a resistor with the resistance R and an inductor with the inductance L . The quantities R and L are specified prior to the calculation of transient current, which is problematic [ 2 ]. Using the proposed method, it is therefore possible to solve problems that have been solved in a steady state (e.g., [4–7]). In the calculation of current density, a quasi-stationary process is assumed and the displacement current and leakage current between conductors are neglected. The conducAppl. Sci. 2021,11, 6920. https://doi.org/10.3390/app11156920 https://www.mdpi.com/journal/applsci Appl. Sci. 2021,11, 6920 2 of 12 tors do not move. The permeability of the conductors and their surroundings equals the vacuum permeability µ0 . In the chosen coordinate system xyz , the conductors are parallel to the axis zand their cross sections do not depend on z. Figure 1gives an example of the cross sections of the conductors A1 and A2 in the plane xy , the cross sections are denoted by the same symbols as the conductors. The resistivity of the conductors is a function of x and y , $=$(x , y) . The voltage source the conductors are connected to produces a potential V(z , t) between the conductors, while the cross section of each conductor is an equipotential area. When calculating the current density, the voltage U(t) = V(z1 , t)−V(z2 , t)/(z2−z1) , where z2>z1 , is assumed to be specified. The current I(t) in the loop is the flux of current density vector J through the cross section of one of the loop conductors. Jz(x , y) = J(x , y) is the only non-zero component of J. 0 8 16 24 0 8 16 56 66 y(mm) A1 A2 ✲ ✻ x(mm) Figure 1. The cross sections of the conductors A1and A2. 2. Calculation of Current Density and Current in Conductors The essence of the method for calculating current density as published in [ 1 ] consists in the conductors being replaced by the partial conductors Ai , ∀i . The symbol ∀ means that the quantity after it takes the values 1, 2, . . . , N successively. The cross section of each partial conductor Ai is also denoted by the symbol Ai . Each cross section A` , `= 1, 2, can be replaced with arbitrary precision by the union a(A`) of identical disjunct rectangles that have been formed using the net of parallel lines x=kx∆x,y=ky∆y, where kx , ky are integers and ∆x and ∆y are the chosen lengths of the rectangle sides. The rectangles of the net, which are subsets in a(A`) , `= 1, 2, are the cross sections of the partial conductors Ai , ∀i . Ai⊂a(A`) if (Xi , Yi)∈A` , where (Xi , Yi) is the centre of the rectangle Ai . The partial conductors Ai , which form a(A1) and a(A2) , must be numbered such that the graph with N vertices (Xi , Yi) and N− 1 edges is a path graph [ 1 , 8 ]. It is assumed that, in the partial conductor Ai , ∀i , there is a constant resistivity $i=$(Xi , Yi) and that the current density Ji(t)does not depend on xand y. According to Coufal [ 1 ], the current densities Ji(t) , ∀i , in conductors A1 and A2 are a solution to the system of N−1 ordinary differential equations $iJi(t)−$i+1Ji+1(t) + µ0 4π N ∑ k=1 φik ˙ Jk(t) = δiU(t),i=1, 2, . . . , N−1, (1) and one algebraic equation N ∑ k=1 Jk(t) = 0. (2) The differentiation of a function with respect to time t is denoted by a dot over the symbol of the function. The coefficients φik do not depend on t , and their calculation is described in the Appendix of [ 9 ]. Jk(t)(z2−z1)µ0φik/( 4 π) is the contribution of the k th partial conductor to the magnetic flux through the segment (between the planes z=z1 and Appl. Sci. 2021,11, 6920 3 of 12 z=z2 ) of the loop formed by partial conductors Ai and Ai+1 . The value of the coefficient δion the right-hand side of Equation (1) is given by the relation δi=   1 if (Ai⊂a(A1)) ∧(Ai+1⊂a(A2)), −1 if (Ai⊂a(A2)) ∧(Ai+1⊂a(A1)), 0 otherwise. (3) Prior to the numerical solution to the system formed by Equations (1) and (2), Equation (2) is substituted by the equation N ∑ k=1 ˙ Jk(t) = 0. (4) The system of Equations (1) and (4) can after rearrangement be replaced by a single equation φ˙ J(t) = 107D+107U(t)δ, (5) where the matrix φ is of the type (N , N) , on whose first N− 1 rows are the fluxes φik , ∀k , i= 1, 2, . . . , N− 1, while on the N th row are ones (coefficients on the left-hand side of Equation (4) ); 10 7= 4 π/µ0 ; the matrices ˙ J ,Dand δ are of the type (N , 1 ) . ˙ J= [ ˙ Ji(t)] ; D = [Di] , Di=$i+1Ji+1−$iJi for i= 1, 2, . . . , N− 1, DN= 0. δ= [δi] , δi is defined by relation (3) for i= 1, 2, . . . , N− 1, δN= 0. Multiplying Equation (5) by the inverse matrix φ−1from the left yields the equation ˙ J=107MJ+107U(t)M, where MJ=φ−1D,M=φ−1δ. (6) The matrices M J and Mare of the type (N , 1 ) . The rows of the matrix M J are linear combinations of the current densities Ji(t) . The elements of matrix Mare numbers. Equation (6) is a linear inhomogeneous system of ordinary differential equations with constant coefficients. According to Rektorys and Vitasek [ 10 ], there is just one solution J= [Ji(t)] to Equation (6) in the interval t∈[0, ∞)that satisfies the initial conditions J(0) = J0,J0= [Ji(0)]. (7) The results given below for the solution to Equation (6) were obtained using the Runge–Kutta method [10–12]. Equation (6) can be written in the form [˙ Ji(t)] = [Fi(t,Jk(t),∀k)]. If Ji(t),∀i, is the current density in the ith partial conductor at time t, then Ji(t+h) = Ji(t) + 1 6(k1i+2k2i+2k3i+k4i) is the current density in the i th partial conductor at time t+h , where h is the integration step, k1i=hFi(t,Jk(t),∀k), k2i=hFi(t+h/2, Jk(t) + k1k/2, ∀k), k3i=hFi(t+h/2, Jk(t) + k2k/2, ∀k), k4i=hFi(t+h,Jk(t) + k3k,∀k). All the values of transient current I(t) given below were calculated using the transient current density in the conductor A1 I(t) = ∆x∆y∑ i Ji(t), where Ai⊂a(A1). The current in the conductor A2is −I(t). Appl. Sci. 2021,11, 6920 4 of 12 3. Results and Discussion 3.1. Example 1 The cross sections of the conductors forming the long loop are illustrated in Figure 1. The temperature of the conductors is 300 K, both conductors are of copper and their resistivity $= 1.725 × 10 −8Ω· m [ 13 ]. At time t= 0, the conductors are connected to the sinusoidal voltage source U(t) = ˆ Usin(ωt+α) , ˆ U= 1 V · m −1 , the frequency f=ω/( 2 π) and the period T=1/ f. This example is for f= 1 kHz, the same as in [ 2 ], Variant 4; however, in Figure 6 in [ 2 ], the results of the calculation of current density in conductors are only given for the steady state, i.e., for t→∞ . Figure 2illustrates the dependences of current density on x and t for a constant yin the conductors A1and A2. −3 −2 −1 0 1 2 3 4 0 5 10 15 60 65 J(MA·m−2) y= 8 mm y= 12 mm 16 56 1 2 = o 2 33 4 4 1 22 3 44 ❄ x(mm) Figure 2. Example 1, α= 0, f= 1 kHz: The dependences of current density on x and t for a constant y in the conductors A1 and A2 . The curve marked by the character o represents the current density at time t=o×T/ 4; J= 0 for t= 0. The axis x at the point marked by the arrow ends in the value x= 16 mm and continues with the value x= 56 mm. Between these values is the gap between the conductors, and the current density is zero. The calculation of transient current density provides a large amount of data that can be displayed in a similar way to Figure 2. For example, the maximum value of |Ji| is equal to 6.095 MA · m −2 in the partial conductor that is down on the left in the conductor A2 , in the interval t∈[0, T]at time t=3.3 ×10−4s, for α=0, f=1 kHz. The current densities Ji(t) , ∀i , are non-periodic oscillating functions that with increasing t converge to periodic functions. For t→∞ , it can be assumed that for the sinusoidal voltage source ˆ Usin(ωt+α) and the cosinusoidal voltage source ˆ Ucos(ωt+α) the current density in the ith partial conductor is Ji(t) = ˆ Jisin(ωt+εi),∀i(8) and Ji(t) = ˆ Jicos(ωt+εi),∀i(9) respectively. Substituting the assumed solutions to (8) and (9) into (1) and (4) yields the first and second equation systems, respectively. The two systems form a system of 2 N equations for N unknown amplitudes ˆ Ji and N unknown initial phases εi . Probably the simplest way of finding a solution to the two systems is to multiply the first system by the imaginary unit j and add the second system of equations. This after adjustment yields a system of N equations for the complex current densities ˆ Jiexp( j εi)exp( j ωt) , ∀i . Solving this system gives current density phasors in the partial conductors, as stated in [1]. With increasing t, the transient current converges to the steady sinusoidal current Appl. Sci. 2021,11, 6920 5 of 12 Isin(t) = ˆ Isin(ωt+β). Figure 3illustrates the current I(t)in three time intervals. −300 −200 −100 0 100 200 300 400 0 1 2][8 910][28 29 30 I(A) ˆ I −ˆ I ❄ ❄ t(T) Figure 3. Example 1, α= 0, f= 1 kHz: The dependence of current on t in three time intervals: [ 0, 2 T] , [8T, 10T]and [28T, 30T]. Theoretically speaking, the steady state is achieved after an infinitely long time. In practice, the steady state is reached at time t≥tsin , where tsin is determined using the chosen value of the deviation ∆I.tsin is such a least time that for t≥tsin it holds 1− |Ilm(t)|/ˆ I≤∆I, (10) where Ilm(t) is the value of the current at the local maximum or minimum. In Example 1 ( α= 0, f= 1 kHz), it holds tsin = 29.49 T for ∆I= 0.0032. For ∆I= 0.003 in Example 1, it holds tsin . = 0.03 − 0.04 s for f∈[ 50, 10 6] Hz. The time tsin depends substantially on the initial phase α of the voltage source at time t= 0. In Figure 4, the function I(t) is illustrated for several values α in Example 1 for f= 1 MHz. It can be seen in Figure 4that the steady state is achieved at the instant t= 0 for αsin = 90 ◦ . The value αsin depends on f . For example, for f=50 Hz, αsin =60◦. −0.4 −0.2 0 0.2 0.4 0 0.5 1 1.5 2 I(A) ˆ I −ˆ I α= 0 45◦ 90◦ 135◦ 180◦ t(T) Figure 4. Example 1, f= 1 MHz: The dependence of the current I(t) on the initial voltage phase α for t∈[0, 2T]. Appl. Sci. 2021,11, 6920 6 of 12 3.2. Example 2 The conductors are the same as in Example 1, but, in contrast to Example 1, they are connected to the constant voltage source U(t) = ˆ U , ˆ U= 1 V · m −1 at time t= 0. The voltage source is being short-circuited at time tsho. Figure 5illustrates the function I(t) when the voltage source is being connected and later short-circuited. The graph of the function I(t) has two parts. The first is for t∈[ 0, tsho] increasing while the second part is decreasing in the interval t∈(tsho,∞). 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 I(kA) Cu Cu Al Al t(0.01 s) Figure 5. Example 2: The dependence of transient current on time when the source of constant current is connected at time t= 0 and is later short-circuited at time tsho = 0.005, 0.01, 0.05 s. The curves (solid lines) denoted by Cu correspond to copper conductors. The curves (thin lines) denoted by Al correspond to aluminium conductors at a temperature of 300 K, $= 2.733 × 10 −8Ω· m [ 13 ]; tsho =0.01, 0.03 s. If Figure 5is redrawn in semilogarithmic coordinates, then the second part of the graph of the function I(t) is a straight line. This means that ln I(t) is a linear function of t and that the slope |∆ln I(t)|/∆t is constant. It follows from the results in Figure 5that with increasing conductor resistivity the time tsin decreases and the slope of the drop in current increases after the short-circuiting of the voltage source. It would undoubtedly be interesting to investigate the disconnection of the source. However, solving such a problem is beyond the scope of the article. Source disconnection is realised using a switch, in which an electric arc is excited between the switch contacts. To calculate transient current density and transient current when disconnecting the source, it is necessary to replace the source by a physical and mathematical model of the concrete switch. 3.3. Example 3 The cross section of the two conductors is a circle with the radius r= 8 mm. The cross sections A1 and A2 have their centres at points ( 0, 0 ) and (d , 0 ) , respectively, d≥ 2 r . Both conductors are of copper and they are connected to a sinusoidal voltage source at time t=0, the same as in Example 1. The solution to Example 3 allows extending and complementing the results of solving Example 1. With the sinusoidal source connected, the main effect of the transient state is an increase in the maximum value of the current density Jmax =max ∀i,t∈[0,T)|Ji(t)| and in the maximum value of the current Imax =max t∈[0,T)|I(t)| Appl. Sci. 2021,11, 6920 7 of 12 with respect to the steady state. The maximum values Imax and Jmax are obtained during the first period T. The amplitude of the steady current density attains the maximum ˆ Jmax =max ∀i ˆ Ji at the point (r , 0 ) . The point at which the transient current density attains the value Jmax is on the surface of the conductor A1 (and at the symmetrical point on the surface of the conductor A2 ) and its position depends on d and f . The instant at which Jmax is attained differs from the instant at which Imax is attained (see Figure 6). 0.01 0.1 1 5 −8−4048 J(MA·m−2) 1 2 3 x(mm) Figure 6. Example 3, d= 100 mm, f= 1 kHz, α= 0: The dependence of current density on x for y= 0 in the conductor A1 . (1) The amplitude of current density in steady state; (2) t= 0.49 ms, the value of the current Iis maximum; (3) t=0.37 ms, maximum current density. Table 1gives the values Jmax/ˆ Jmax and Imax/ˆ I in dependences on f , d and $ . The resistivity of Al conductors is $= 2.733 × 10 −8Ω· m [ 13 ]. Alloy conductors are of an alloy of 75 mass% Cu + 25 mass% Al, $=1.76 ×10−7Ω·m [13]. Table 1. Example 3, α=0: The dependences of MJ=Jmax/ˆ Jmax and MI=Imax/ˆ Ion f,dand $. f(Hz) d= 16 mm 32 mm 0.1 m 1 m 1 m 1 m Cu Cu Cu Cu Al Alloy 50 MJ1.001 1.082 1.232 1.441 1.291 1.002 103MJ1.005 1.250 1.358 1.421 1.439 1.599 105MJ1.666 1.802 2.057 2.236 1.499 1.425 50 MI1.033 1.106 1.248 1.453 1.294 1.002 103MI1.537 1.801 1.890 1.939 1.920 1.659 105MI1.818 1.979 1.990 1.995 1.993 1.983 The transient current I(t) can be considered steady for t≥tsin . The instant tsin is determined by Formula (10) and depends on the chosen value ∆I , the distance d of the axes of conductor cross sections, the voltage source frequency f= 1 /T and the conductor resistivity. Table 2gives the values of tsin for several values of f , ∆I and d . The conductors in Example 3 are of copper and the respective data are given in Table 2, with the exception of the last column, where the data given hold for aluminium conductors ( $= 2.733 × 10 −8Ω· m [ 13 ]). With increasing conductor resistivity, the time tsin decreases. This statement is supported by comparing the values in the penultimate and last columns of Table 2. Appl. Sci. 2021,11, 6920 8 of 12 Table 2. Example 3, α= 0: The dependences of time tsin (expressed by the number of whole periods Tand rounded up) on f,∆Iand d. f(Hz) ∆I(−)16 mm 32 mm 0.1 m 1 m 1 m, Al 50 0.02 2 2 2 3 2 1030.02 9 16 26 47 30 1050.02 525 1425 2478 4574 2887 50 0.01 2 2 3 4 3 1030.01 11 18 31 55 35 1050.01 695 1695 3000 5394 3405 50 0.005 2 2 3 4 3 1030.005 13 21 35 64 40 1050.005 864 1965 3380 6215 3923 3.4. Example 4 The conductors are the same as in Example 3. At time t= 0, they are connected to a source of sawtooth voltage (see Figure 7). The voltage is periodic, in the first period U(t) = ˆ U(1−2t/T),ˆ U=1V ·m−1,t∈[0, T). (11) The solution to Example 4, obtained by solving Equation (6), is given in Figure 7. In the interval t∈[ 0, 2 T) , the current I(t) is transient, and for t≥ 2 T the current can be considered steady. −4 −2 0 2 4 0 1 2 3 −2 −1 0 1 2 I(kA) U(V·m−1) U I t(T) Figure 7. Example 4, d= 16 mm, f= 60 Hz, ˆ U= 1 V · m −1 : The dependences of the current I in the conductor A1and the voltage Uon time t. In the theory of circuits with lumped parameters, the non-sinusoidal voltage (thus also (11)) is in the calculation of steady current I(t) approximated by the first n terms of the Fourier series [ 14 ]. For a source voltage that is equal to the k th harmonic, k= 1, 2, . . . , n , the current Ik(t) is then calculated. The principle of superposition is used to obtain the current I(t) = n ∑ k=1 Ik(t). An advantage of such a calculation is that it allows the use of phasors. The Fourier series can also be used in the calculation of steady current density, and thus also the steady current, in two parallel long conductors connected to a source of non-sinusoidal voltage via the method published in [ 1 ]. When assessing the suitability of such a procedure, it is Appl. Sci. 2021,11, 6920 9 of 12 necessary to take into account that the Fourier expansion is an infinite series while the calculation can only be performed with a finite number n of series terms. The finite number of terms only approximates the given voltage course, which also manifests itself in the resultant current. Instead of n repetitions of the calculation with individual harmonics, a single calculation can be done of transient current density and current with the given voltage. The transient current at time t≥tsin can be considered steady. Figure 8gives the sawtooth voltage in Example 4 and its approximation by n terms of the Fourier series [ 14 ] U(t) = 2ˆ U π ∞ ∑ k=1 sin(kωt) k(12) and the transient current calculated with the source voltage approximated by n terms of the expansion (12) (an accurate transient current is in Figure 7). −4 −2 0 2 4 0 1 2 −2 −1 0 1 2 I(kA) U(V·m−1) U I t(T) Figure 8. Example 4, d= 16 mm, f= 60 Hz, ˆ U= 1 V · m −1 : Approximation of the voltage U(t) by n terms of the series (12) for n= 5 (solid line) and n= 20 (thin line). The current I in the conductor A1 when approximating the voltage by ten terms of the series (12). 3.5. Discussion J. C. Maxwell proposed a method for the calculation of current density in a solitary conductor [ 15 ]. Hundreds of works have since been published (more recently, e.g., [1,16–24] ) that deal with the calculation of current density in one or two conductors, and exceptionally in more conductors. The calculation of current density in one conductor is accurate, but it can only be applied to the solitary conductor connected to the ideal current source, which, however, cannot be implemented [ 1 ]. The published methods, with the exception of those in [1,3,24], are approximate, with the current density error unknown. The self-inductance of two conductors cannot be calculated without accurate current density [ 2 ] and thus approximate formulae continue to be used for the calculation of self-inductance of two coaxial conductors and two circular cross section conductors [ 2 ]. According to these formulae, inductance does not depend on conductor resistivity and voltage source frequency, in contrast to the inductance calculated using accurate current density. The subject of the present article is transient current in two solid long parallel conductors (conductors in the following) connected to the voltage source. A definite contribution of the article is that current is determined using transient current density. In the article, a numerical method is proposed and used for solving a system of differential equations for the calculation of transient current density. The present method of calculating transient current, not only in the conductors considered in the present article, consists in calculating transient current in the circuit C with lumped elements, which is a series connection of