Figure 1: Illustration of the principle of conformal mapp ing. A conformal mapping function maps a reference setup to the sensor setup. The same function also maps the potential distribution from which the field equation can be determined.

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# Design of Planar E-Field Sensors for Measurements of Three Dimensional Objects

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Dipl.-Ing. Norbert Eidenberger

Industrial production processes pose increasingly higher demands on the development of sensors and sensor networks. One of the challenges consists in the measurement of object geometries while the objects are moving through the production line. This thesis studies a sensor system which is intended for quality control in knife band production. In the production process the knife bands move with up to 3 m/s. Nevertheless, the geometric parameters must be measured with a precision of 5 um. The influences of the industrial environment such as dirt, vibration, changing illumination, etc. are especially challenging. A possible solution to overcome these challenges and attain the desired resolution consists in utilizing electric field sensors. Being purely reactive these sensors, in the form of capacitance sensors, theoretically provide infinite resolution only limited by noise in the front end hardware. It remains to be determined whether a sensor system can be developed which suits the requirements.

The first step consists in designing a sensor setup and establishing an appropriate mathematical model. When developing an electric field sensor knowledge about the electric field structure is important. In this case only variations in the cross-section geometry of the knife band influence the electric field. Thus, a two dimensional model sufficiently describes the electric field. Figure 1 illustrates the two dimensional setup model. The corresponding Poisson equation can be solved utilizing conformal mapping methods. These methods map the contours of a simple reference setup to the contours of the complex contours of the setup model. Conformal means that at small scales the lengths and the angles of the mapped contours are preserved. Different methods are utilized for the construction of the appropriate conformal mapping function:

- Schwarz-Christoffel transform (SCT)
- Joukowski transform
- Polar transform

The application of the SCT requires solving the so-called SCT parameter problem. A novel procedure for solving this problem has been developed. The procedure combines a series expansion and an optimization algorithm which compensates the approximation error. This procedure yields equation (1) which describes the electric field in the sensor setup. This field equation permits efficient sensor design and simplifies the sensor signal evaluation.

The design of the sensor setup needs to consider conflicting requirements:

- high resolution (spatial as well as geometrical)
- high signal strength (SNR).

In order to obtain a high spatial resolution small sensing electrodes are required. However, small electrodes produce only small capacitance values (pF-range) and thus small signal changes which results in a low signal-to-noise ratio (SNR). Optimization methods need to be employed to achieve a workable compromise between these requirements. Different sensor setups of increasing complexity are developed, starting with single electrode capacitive sensors up to sensor arrays which can be considered to be electric field camera systems. The evaluation of the sensed capacitances bases on a capacitance-to-digital converter by Analog Devices (AD7746) which has been specially developed for measuring capacitances in the pF-range.

The electric field strength is measured via capacitive sensors. The sensing electrodes are placed in a plane which lies perpendicular with respect to the band direction. The knife band causes a field distribution in this plane. Thus, the evaluation of the measurement data equals to solving an inverse problem where the geometry of the three dimensional object needs to be determined from the two dimensional measurement data. The problem is ill posed because different objects can cause the same field distribution in the two dimensional plane. Only the knowledge of the general object shape permits a meaningful evaluation of the measurement data. For knife bands the field equation has been determined, therefore the inverse problem can be solved. The determination of the different geometric parameters such as blade angle, blade height and air gap length requires at least three measurements at different positions in the measurement plane. Then the evaluation algorithms can utilize the field equation to determine all relevant geometric parameters from the measurements.

Keywords: capacitive sensors, capacitance measurement, conformal mapping, Schwarz-Christoffel transform, inverse problem.