Measurements of the thickness and electrical conductivity of nonmagnetic plates by an eddy-current method
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MEASUREMENTS OF THE THICKNESS AND ELECTRICAL CONDUCTIVITY OF NONMAGNETIC PLATES BY AN EDDY-CURRENT METHOD
V. A. Sandovskii
UDC 620.179.14
Investigations are carried out which show that, when using an adequate mathematical model, by solving the appropriate electrodynamic equations, including the results of experimental measurements by the eddy-current method, one can separately determine the thickness and electrical conductivity of nonmagnetic materials and samples in the form of plates, tapes, and foils. Keywords: electrical conductivity, thickness, foil, eddy-current converter, mathematical model.
Eddy-current converters [1] are widely used in technology, in production and under laboratory conditions to measure and monitor articles and materials. If the samples being monitored are fairly thin, the signal, measured by the eddy-current converter, depends not only on the properties of the material of the sample but also on its thickness. We have formulated the problems of measuring the electrical conductivity of plates irrespective of their thickness, and, conversely, of measuring the thickness of plates irrespective of the electrical conductivity of the material. For this purpose, we investigated the possibility of using calculations with the appropriate mathematical model, together with experimental measurements using a contact-type eddy-current transformer converter [1]. Description of the Mathematical Model. For the purpose of calculations, the eddy-current converter is considered in the form of a mathematical model, consisting of two concentric loops, situated above a magnetic plate of thickness d (Fig. 1). The external loop with radius R is the current loop and the inner loop, with radius r, is the measurement loop. The distances to each of the loops from the surface of the plate are equal to h1 and h2, respectively. The signal, introduced by the plate into the measuring loop, can be calculated from the formulas [2] ∞
∫
Z2 = jωμ0 πr J1(δx ) J1( x ) F ( x ) exp ( −αx ) dx ; 0
F ( x ) = ( x 2 − q 2 )(1 − Y ) [( x + q)2 − ( x − q)2 Y ]; q = x 2 + jβ2 ;
Y = exp ( −2ζβ);
ω = 2 πf ;
δ = r / R;
ζ = d / R;
β = R ωμ0σ ;
α = ( h1 + h2 ) / R,
where μ0 is the magnetic constant, J1 are Bessel functions of the first kind and first order, σ is the electrical conductivity of the plate material, and ƒ is the frequency of the current flowing in the outer loop.
Institute of Metal Physics, Russian Academy of Sciences, Ekaterinburg, Russia; e-mail: [email protected]. Translated from Izmeritel’naya Tekhnika, No. 10, pp. 55–60, October, 2012. Original article submitted July 13, 2012. 0543-1972/13/5510-1201 ©2013 Springer Science+Business Media New York
1201
Fig. 1. Superposed eddy-current converter above a plate.
The signal in the measuring loop, when there is no conducting plate, can be calculated as [3] Z1 =
ωμ0 πr π δk
[(2 − k 2 ) K − 2 E ],
where K and E are complete elliptic integrals of the parameter k = 2 δ [(1 + δ )2 + α12 ];
α1 = h1 − h2
R.
In general, the signal of the measuring loop for a converter pl
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