An integration method of deriving the alternating current resistance and inductance of conductors . ^-4 ——^ ? (38) Letting—2==-^o ^ ^rl^W^_rMX^ R = ^- ^ , ,„., ^ (40) In like manner ^=-L» + - TTw^^ (41) 2 5WV * 4 These equations are identical with (22) and (23) showing thatthe result using complex power series is the same as with realpower series. io6 Scientific Papers of the Bureau of Standards IV. ALTERNATING-CURRENT RESISTANCE ANDANCE OF A RETURN CIRCUIT INDUCT- If a return circuit consists of two parallel cylindrical conductors,whose length is great compared to the diameters of the
An integration method of deriving the alternating current resistance and inductance of conductors . ^-4 ——^ ? (38) Letting—2==-^o ^ ^rl^W^_rMX^ R = ^- ^ , ,„., ^ (40) In like manner ^=-L» + - TTw^^ (41) 2 5WV * 4 These equations are identical with (22) and (23) showing thatthe result using complex power series is the same as with realpower series. io6 Scientific Papers of the Bureau of Standards IV. ALTERNATING-CURRENT RESISTANCE ANDANCE OF A RETURN CIRCUIT INDUCT- If a return circuit consists of two parallel cylindrical conductors,whose length is great compared to the diameters of the wires andto the distance between them, the method of integration can beapplied to the determination of the alternating current resistanceand inductance of the circuit.^ If the wires have the samediameter, the current distribution in one wire is symmetricalabout the line joining the centers of the two wires. If the equationof current distribution is given in polar coordinates, it will beidentical for the two wires if the angles are measured from the linejoining the centers of the Fig. 2.—The cross section of a return circuit, showing coordinates Let P with coordinates p, ^ be a point in one conductor at whichthe current density is to be determined, P^ another point in thesame conductor, and P2 ^ point in the return conductor whosecenter is at a distance s from the center of the first U\ U[, and U^ designate the instantaneous current densityand Z7, U^, and U2 the maximum current density at P, Pi, and Pj,respectively. Also let m ^ represent the mutual inductance betweenthe two filaments at P and Pj, whose distance apart is d^, and m^that between filaments at P and P2, whose distance apart is d^. Equation (3) for this case becomes E = Ucl + jjfn,^ dS, - fjm,^ dS, (42) where dS^ and dS2 are elements of area at Pi and mutual inductance between two long filaments • is m =2/ (log 2/ —i) / log d ^Nicholson has published a formula covering this
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