Integral equation approach for proximity effect in a line constituted by two solid conductors of circular cross section

Abstract

Preliminary notesThis paper presents an analysis of the proximity effect in a line consisting of two identical solid conductors of circular cross section. Integral equation for the current distribution in one of the conductors is solved approximately by assuming the current density in the form of a finite double polynomial series and by applying the point matching procedure. The current distribution obtained by the presented method has been checked by implementing the FEMM 4.0 software package. Also, the a.c. to d.c. resistance ratio is calculated and the results are compared to the available ones, and a good agreement is exhibited. Keywords: a.c. to d.c. resistance ratio; circular cross section conductors; current distribution; proximity effect Metoda integralnih jednadžbi za analizu efekta blizine u dvožičnom vodu s kružnim vodičimaPrethodno priopćenje U ovom radu predstavljena je analiza efekta blizine u vodu sastavljenom od dva identična vodiča kružnog poprečnog presjeka. Integralna jednadžba za raspodjelu gustoće struje u jednom od vodiča riješena je približno uzimajući da je gustoća struje u obliku konačnog dvojnog stupanjskog reda s nepoznatim koeficijentima koji su određeni metodom podešavanja u točkama. Raspodjela struje dobivena prikazanom metodom provjerena je uporabom softverskog paketa FEMM 4.0. Također, izračunat je omjer otpornosti pri izmjeničnoj i istosmjernoj struji, a rezultat je pokazao dobro poklapanje s raspoloživim rezultatima.Ključne riječi: omjer otpornosti pri izmjeničnoj i istosmjernoj struji; provodnici kružnog poprečnog presjeka; raspodjela struje; efekt blizine

Analytical–Numerical Solution for the Skin and Proximity Effects in Two Parallel Round Conductors

“…Using approximate analytical expressions-e.g., various approximate formulas for alternating current resistance can be found in [11][12][13] and approximations for inductances are given in [14]; integral equation with current density approximated with finite power series was proposed in [15,16]; multipole expansion with finite number of terms was used in [17].…”

Transient Current Density in a Pair of Long Parallel Conductors

“…As mentioned above, the value of L lit is not accurate. If the accurate value L = 8.404 × 10 −8 H•m −1 , determined as in [2], were used in (16), the difference between the transient currents would be even greater than in Figure 9.…”

“…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][17][18][19][20][21][22][23][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].…”

“…According to this formula, L lit = 10 −7 H•m −1 . Figure 9 illustrates the voltage U(t), the transient current I (16) (t) calculated using formula (16) and the transient current I(t) calculated by the method proposed in the present article for t ∈ [0, 4T], I(0) = 0. As mentioned above, the value of L lit is not accurate.…”

The Proximity Effect in Twin Line with Round Conductors Placed in Conductive Medium

“…The approach was later expanded in [4,5] and used to model power and signal lines with neighboring wires substituted with current filaments [6][7][8]. A similar method was later presented in works [9,10]. Jabłoński [11] considered a general solution for round wire affected by an arbitrary time-harmonic magnetic field.…”