Leszek Dziczkowski

E-Cored Coil With a Circular Air Gap Inside the Core Column Used in Eddy Current Testing

The problem of an axially symmetric E-cored coil with a circular air gap inside the core column located above a two-layered conductive half-space is presented. The truncated region eigenfunction expansion method is used to derive expressions describing the magnetic vector potential of the filamentary coil. The final expressions for the impedance of the rectangular cross-sectional coil are obtained, and calculations for various frequency values are carried out. The results are compared with those from the COMSOL package, which shows a very good agreement.

Elimination of Coil Liftoff From Eddy Current Measurements of Conductivity

The construction of eddy current conductivity meters is based on developing a method to effectively eliminate the distance effects between the surface of the coil and the workpiece under test in conductivity measurements. As a result of numerous experiments, it was observed that each actual measuring coil can be replaced by the one that is characterized by only two parameters. Following this procedure makes it possible to use only one universal mathematic formula for each of the coils. Appropriate calculations are provided to certify the theory. This paper describes the method of determining the equivalent parameters of the coil, a scaling method for a conductivity meter based on the elimination of liftoff on conductivity results, and a proposition for an effective method to measure the conductivity of rough elements.

Determination of the Penetration Depth of Eddy Currents in Defectoscopic Tests

In the defectoscopic tests by means of the eddy currents method only a certain superficial layer of the tested element is inspected. The reason of this phenomenon is connected with a very important feature of the eddy currents. The induced eddy currents generate its own magnetic field which obstructs penetration for the primary magnetic field. It is crucial to know the penetration depth of eddy currents. It allows planning successfully the diagnosis process. There are two cases worth mentioning: when the eddy current method is treated as the additional method complementary to the ultrasound method (because it does not detect superficial defects) and when the eddy current method is used as the main method for the thin elements diagnosis. The most frequently used evaluation method of eddy currents penetration depth is connected with determination of the e-folding decrease of electric current. The definition is convenient to use because it is simplified by using in the mathematical formula (allowing determination of the depth) frequency of eddy current and conductivity of the diagnosed elements. However the simplifications are not sufficient in practice. When we change the frequency of eddy currents during the survey or the probe then the depth of penetration is also changed, then we can measure the depth of the defects. While measuring the conductivity of a proper material element it is obligatory to prepare an adequate size of the sample that is free of defects. Knowing the value of penetration depth is then very helpful. On the other hand, when we have a sample of a specified size and we want to measure its conductivity then the knowledge of the depth of penetration of eddy currents helps us to select the proper frequency. In the paper there is described a proposal of a different definition of the penetration depth of eddy current, much more useful and accurate according to the authors. To obtain much more precise results, the new eddy current method was proposed. This method takes into account not only the parameters of the diagnosed sample and the eddy current frequency but the characteristic of the measuring device as well. The above mentioned method is based on the universal mathematical model of impact of conductive thin foil on the measuring coil impedance change. The procedure of calculations is easy to carry out online.

I-cored Coil Probe Located Above a Conductive Plate with a Surface Hole

Abstract This work presents an axially symmetric mathematical model of an I-cored coil placed over a two-layered conductive material with a cylindrical surface hole. The problem was divided into regions for which the magnetic vector potential of a filamentary coil was established applying the truncated region eigenfunction expansion method. Then the final formula was developed to calculate impedance changes for a cylindrical coil with reference to both the air and to a material with no hole. The influence of a surface flaw in the conductive material on the components of coil impedance was examined. Calculations were made in Matlab for a hole with various radii and the results thereof were verified with the finite element method in COMSOL Multiphysics package. Very good consistency was achieved in all cases.

An Analytical Model of an I-Cored Coil With a Circular Air Gap

Probes used in eddy current testing contain ferrite cores of various shapes. In this paper, an axially symmetric mathematical model of an I-cored coil with an air gap was shown. The coil was placed above a two-layered conductive half-space and the solution domain was truncated radially. Using the truncated region eigenfunction expansion method, expressions describing the magnetic vector potential and the impedance of such a coil were derived. The calculations were carried out in MATLAB, both for an air-cored coil and the I-cored one containing an air gap. The results were verified using the COMSOL Multiphysics package in which the finite-element method is applied and the values show a very good agreement. Moreover, impedance components for the I-cored coil with and without an air gap were compared. The difference between the two types was significant, particularly for low-frequency values.

Enhancement of conductometer functions with the measurements of surface roughness

The paper describes an innovative calibration method that makes it possible to efficiently measure the conductivity of workpieces with rough surfaces, where standard specimens with the already known properties are used for calibration. The calibration process leads to the determination of equivalent parameters of a probe. The measurements results are processed on line with the use of a simple mathematical model for the considered phenomenon. Finally, the determined conductivity value only slightly depends on the surface condition of the examined workpiece. During the conductivity determination there was calculated a parameter–namely, the eddy-current factor of surface roughness–that compensates the impact of the surface roughness. The parameter can be treated as a specific roughness factor of the eddy-currents. The factor has been determined for several samples with already known mechanical values obtained with the help of Ra and Rz methods.

Mathematical Model of the Damped Variable Cross Section Beam in Transportation

The paper concerns the problem of vibrations of beamlike system with variable cross section. The beam is treated as the movable system in transportation. The considered problem focuses on modelling and dynamic analysis of geometrically nonlinear beam systems in rotational motion within the context of damping. The major scientific purpose of the paper is to elaborate the mathematical model of such a system. Additionally, the main motion impact on the local vibrations due to the mathematical sense is determined. Moreover, it is necessary to remember the interactions between damping forces of the above mentioned mechanisms and the transportation effect. The main motion of the system is treated as transportation, whereas the vibrations of the system are treated as relative motion. There are two types of systems considered: simple vibrating longitudinally and simple vibrating transversally in the plane transportation. The most interesting elements of the analysis determine the dynamic state of the system and present the mutual coupling of vibration amplitudes, natural frequency, and transportation velocity. Analysis of systems moving with low velocities or vibrating only locally treats the systems as already known models in literature. There are many scientific articles where the forms of vibrations of these systems have been described. Due to the obtained results it will be possible to confront mathematical models with the known stationary and non-stationary systems. As regards complex and simple systems running at high speed, the resonance phenomenon can be noticed, and depending on the amplitude and frequency of vibrations, we consider the following cases: when the amplitude reaches theoretical infinity leading in practice to permanent damage of the mechanism or when the amplitude of vibration reaches a certain speed which can cause the decrease of durability of the whole system. The adequate practical usage of the above mentioned researches is justified by its wide range of applications. In the majority of technical cases, further analysis of the systems is considered to be far too much simplified when we ignore the elements of flexibility, damping, or the nonlinear geometry of the beam. All the mentioned influences are presented in the derived mathematical model in form of equations of motion.