Magnetic flux as affected by eddy currents in electromagnets

Abstract

A method has been developed for determining the current time constant τ at which eddy currents become detrimental to electromagnet operation. Plots of flux density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</tex> versus time, and current <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</tex> versus time were made for various values of τ on electromagnets having a constant air gap and a fixed value of maximum current I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> in the coil. Such an electromagnet can be approximated by a linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RL</tex> circuit as long as eddy current effects are negligible. τ was the time required for the current to reach 0.63 I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> . This corresponds to the time constant in a linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RL</tex> circuit. Values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dB/dt</tex> were measured at τ and the relationship <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dB/dt = C/\tau^{m}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> equals 1 and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> is a constant, was found to be applicable for large values of τ, as expected in a linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RL</tex> circuit. The τ value τ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> at which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> began to decrease from 1, determined the onset of eddy current effects since the circuit is no longer linear. Experimental results were compared to a theoretical solution for a ramp field applied to sheets of thickness δ. Calculations were performed of the time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Deltat</tex> by which the flux density lags the applied field. Reasonably good correlation was obtained between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Deltat</tex> and τ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> . Measurements were made on various soft magnetic materials in various thicknesses. The results provide a guide in selecting material and lamination thickness for a given application.