Effective resistance and reactance of a conductor of trapezoidal cross-section placed in semiclosed slot

Abstract

The paper is an extension of a previous one and discusses the distribution of current in a conductor which has a cross-section in the form of a symmetrical trapezium and is placed in a slot with a narrow opening. An approximate solution is obtained by assuming that the upper and lower surfaces of the conductor are defined by concentric cylinders. It is found that the current density at any point may be expressed as an infinite series of Bessel functions, each term involving an arbitrary constant which is determinable by reference to the assumed boundary conditions. From this result, the complex impedance per unit length of the conductor is determined and hence also the effective resistance and the internal reactance.List of principal symbols / = total current in conductor, A J -current density, A/m 2 a = conductivity of conductor, i Q -' m " 1 /A = permeability H = magnetic field strength, A/m R d = d.c. resistance per unit length of conductor, D R e = a.c. resistance per unit length of conductor, Q. X e = internal reactance per unit length of conductor, Q. OJ = 2?r x frequency r, 6 = polar co-ordinates R { , R 2 , 0\, 0 2 -slot parameters a =(uaco) l l 2 , m -1 2i X = 0 2 l0 l n -harmonic number y = <xR{ I v = modified Bessel function of first kind of order v K v = modified Bessel function of second kind of order v OQ, b 0 , tf v , b w = parameters defined by eqns. 34-37 «o> v o> w v> v v = parameters defined by eqns. 32, 33, 38 and 39 ber, bei, ker, kei = real and imaginary parts of Bessel functions of a complex argument (defined in Reference 4) v is, in general, nonintegral

Effect of non‐uniform resistivity on eddy‐currents in the rotor bar

“…Because of a quite non‐uniform current density distribution in the bar, especially for slip s = 1, the temperature and the bar resistivity are also non‐uniformly distributed and moreover they are time dependent. There are many papers relating to analytical and numerical method of eddy‐current analysis in a deep bar, but in all of them a constant resistivity has been assumed[1, 2, 3, 4, 5, 6, 7, 8, 9]. It is therefore necessary to search for an efficient method, implementable in a CAD program, for computation of eddy‐current density and temperature distribution with a non‐uniform resistivity taken into account.…”