V. Siva Kumar G. Kelekanjeri

A closed-form solution for the computation of geometric correction factors for four-point resistivity measurements on cylindrical specimens

The present work deals with the evaluation of geometric correction factors for dc four-point probe measurements on finite cylindrical specimens for a random placement of the four probes on a circular face of the specimen. A closed-form (CF) analytical expression is obtained for the potential difference between the voltage probes via the solution of Laplaces equation using the variable-separation technique. The potential difference thus obtained is subsequently used to compute the geometric correction factor. The applicability of the solution is illustrated for two specific cases using a collinear probe-array, namely probes arranged along a radius and normal to a radius. Correction factors obtained from the CF solution are validated in each case by a finite-element simulation solution obtained using COMSOL Multiphysics 3.2. Additionally, the correction factors from the two methods are compared to analytical approximation factors, evaluated using expressions that exist in the literature. A range defined by a lower and an upper analytical approximation factor is computed for each specific case. Correction factors from the CF and COMSOL solutions show an excellent match with each other and lie within the analytical approximation bounds. The application of the CF solution is demonstrated for experimental measurements on copper specimens of different cylindrical geometries.

Electric field distribution within a metallic cylindrical specimen for the case of an ideal two-probe impedance measurement

A closed form analytical solution for the electric field distribution inside a metallic cylindrical disk specimen has been derived for the problem of a two-probe impedance measurement. A two-probe impedance measurement can be treated as current injection and extraction by means of source and sink electrodes that are placed on opposite sides of a specimen. The analytical formulation is based on Maxwell’s equations for conductors and the derivation has been conducted on the premise of continuum considerations within the specimen. The derived field expressions for axial [Ez(r,z)] and radial [Er(r,z)] fields are expressed in terms of Bessel series. As an extension to this problem, a semi-infinite solution is also given for the case of an infinitely long cylinder. The analytical solutions thus derived have been verified by computer simulations using a commercially available finite element package. The electric field distributions inside the specimen obtained via analytical and finite element solutions are in e...

Computation of the Complex Impedance of a Cylindrical Conductor in an Ideal Two-Probe Configuration

Impedance spectroscopy is an alternating current technique that can be used to probe materials or devices at length scales ranging from the atomic to macroscopic dimensions. Since its first application to solid state materials (Bauerle 1969), it has been used to characterize the electrical response of ionic electrolytes, ferroelectrics, intrinsic conducting polymers, ceramic and polymer matrix composites and biomaterials to name a few (Gerhardt 2005). The technique is based on probing the sample using an ac signal over a wide range of frequencies and studying the polarization phenomena associated with the electrical response. It is ideally suited for studying specimens where there is good electrical contrast at interfaces, either because of different constituent materials or because of space charge formation or dissipation. Impedance spectroscopy has recently been applied to nickel-base superalloys (Zou, Makram et al. 2002; Kelekanjeria and Gerhardt 2006). These metallic alloys contain heterogeneities ranging in size from nanometers to micrometers and the measured impedance response shows interesting dependencies. As a first approximation, for the computations presented in this chapter, the heterogeneous material medium is regarded as a continuum with a uniform conductivity on a macro scale. This treatment is justified because the size of the microstructural heterogeneities is extremely small in relation to the measurement contact area. The problem dealt with here pertains to the specific case wherein circular electrodes are placed on opposite sides of a cylindrical specimen. There are a few cases in the literature, where closed-form solutions in the frequency domain are available for problems similar to the current one. For example, Ney (Costache and Ney 1988; Ney 1991) derived a closed-form solution for the electric field distribution in a solid non-perfectly conducting flat ground plane as a result of electromagnetic interference. The derivation accounted for constriction effect as a result of confinement of current lines near the contact points and skin-effect due to finite conductivity of the ground plane. Bowler (Bowlera 2004) presented closed-form analytical expressions for the electric field distribution in a conducting half-space region due to alternating current injected at the surface. The analytical formulation was conducted in terms of a single, transverse magnetic potential in cylindrical coordinates and the solution was obtained by the use of the Hankel transform. In another publication, Bowler presented closed-form analytical expressions for the electric