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Application of conformal mapping methods to solve planar inverse problems in instrumentation and measurement

Dipl.-Ing. Dr. Norbert Eidenberger

Dipl.-Ing. Dr. Norbert Eidenberger

Supervisory Committee:

Univ.-Prof. Dipl.-Ing. Dr. Bernhard Zagar
a.Univ.-Prof. Dipl.-Ing. Dr. Christian Magele

Final exam:

March 07, 2014

This thesis covers the development of a capacitive sensor system which measures the geometry of knife bands in order to determine their sharpness. In this case the sharpness is defined by the blade angle and the blade height under the assumption of an infinitely small cutting edge radius. These parameters uniquely determine the capacitance density along the x-axis of the developed sensor system shown in Fig.1. The computation of the geometry parameters from discrete measured capacitance density values involves the solution of an inverse electrostatic problem. In order to obtain a solution which is valid for a large range of knife band geometries an analytic closed form solution based on a parameter model is required.

The cross section of the knife band illustrates the geometry parameters defining its sharpness.

Figure 1: The cross section of the knife band illustrates the geometry parameters defining its sharpness.

The sensor system is modelled using the conformal mapping method. This method yields the field equation describing the electric field between the knife band and the sensor electrodes in terms of the geometry parameters. The field equation is the basis for a parameter model which uniquely links the capacitance density and the geometry parameters.

In general this parameter model cannot be inverted due to the complex functions involved and the geometry parameters cannot be directly expressed as a function of the capacitance density. Thus, no solution of the inverse problem exists in closed form, but optimization methods can be used to obtain a solution. The selected optimization method uses the parameter model to construct an objective function and computes the unique combination of parameters which corresponds to the discrete measured capacitances. The computed parameters determine the sharpness of the knife band sufficiently well.

The developed method accurately determines the knife band geometry and can be used to determine many similar knife band geometries in good approximation. The described modelling and evaluation methods were successfully used to design and to characterize a sensor prototype. In addition, the knowledge obtained about conformal mapping methods during the sensor development was used to analyse a magnetostatic measurement setup.

Keywords: instrumentation, measurement, capacitive sensing, conformal mapping, inverse problem, mathematical modelling