EngOpt 2008 - International Conference on Engineering Optimization Rio de Janeiro, Brazil, 01 - 05 June 2008. Optimal Mathematical Model of Hall-effect Sensor Gideon Villar Leandro, Pedro Augusto Pereira Borges, Fabiano Salvadori, Mauricio Campos, Cassiano Rech, Robinson Figueiredo de Camargo UNIJUI, Ijuí, RS, Brasil gede@unijui.edu.br , pborges@unijui.edu.br , f.salvadori@unijui.edu.br ), campos@unijui.edu.br , cassiano.rech@unijui.edu.br , robinson.camargo@unijui.edu.br 1. Abstract Hall-effect sensors functioning is based on the tension developed in a conductor when this is submitted simultaneously to an electric current and a magnetic field perpendicular to the direction of the current flow. The tension developed is perpendicular to both variables. The Hall tension is a function of the current density, of the magnetic field strength and of the properties of the conductor (load density and mobility of the carriers). In control systems, hall-effect sensors are used to measure tension, current, displacement, etc. In this work, the variable to be monitored is tension. In the stage of specification of the controller(s), mathematical models are used to simulate the functioning of the plant, so that they meet the desired specifications. Thus, it is necessary that each component of the plant or subsystem possesses optimized representations, so that interferences among components are minimum. This work proposes a discrete model to simulate the interactions of the hall-effect sensor in control system. In order to obtain the input and output data, experiments were carried out using three distinct input signals (square wave, sine wave and PWM), with different frequencies, so as to obtain a general model. The technique used to set the parameters of the model proposed was the Memetic Algorithms. We used the black box model, in which the system content is inferred by processing experimental input and output data statistically. Based on the previous knowledge one has about the sensor model and the experimental input and output data, it is possible to recursively identify the plant. The problem of identification of systems is a matter of optimization, because it estimates the model parameters that minimize the difference between the prediction of the output and the input, at every instant. 2. Keywords: Inverse problems; Hall-effect sensors; optimized representations. 3. Introduction The function of a measure system is to attribute a numeric value to variable. In this context, there are two purposes in the measure system possess: to monitor or to control processes. Differently of monitor systems, the control doesn't involve only a quantification system, but also a system of performance that will modify variables of the controlled system. In control systems, the sensors are component fundamental, because, they are the responsible to obtain the physical variables values that someone want to observe and/or to control. In the system model development for simulation and project of controllers, the effects introduced by the sensors in the signals are neglectful usually. They are just used the information that the sensors make available in their terminals. A form of visualizing those possible effects is to obtain the sensor mathematical model and to include it in the system model. With this, the controllers can be more precisely obtained, mainly controllers whose parameters are constant. A certain sensor is selected to be sensitive for determined variables, converting variations of that variables direct or indirectly. However, in practice, a sensor is not sensitive just to the desirable variables, but also to other variables, whose variations can implicate in alterations in the output variables. In a measure system, there are three types of input signals: the wanted input; the modifier input and the interferences. The wanted input are those that do influence at the output. The modifiers input are signals that do influence at the output indirectly. The interferences are signals that, even in the absence of the wanted input, they produce an output. In way to minimize the influence of the modifier input and of the interferences in the output sign wanted, the goal is to reduce them to a value minimum, using filtering techniques, screening, etc. The Hall effect sensors, and others kind of sensor too, are plenty employees in practical applications for obtaining of currents and electric tensions. In this paper was proposed a discreet model to simulate the interactions of the Hall effect sensor in control systems. The structure of the sensor and the discreet models were analyzed used in the simulation of the sensor. To obtaining the input-output data experiments were done experiments with three signals of different input (square wave, sinusoidal wave and PWM), with different frequencies in the attempt of obtaining a general model. The technique used for the obtaining of the parameters of the proposed model was the Memetic Algorithm. The modelling black box was used. In this modelling the content of the system is inferred by statistic process, using experimental data of input-output. Starting from the theoretical knowledge of the sensor and of the experimental data of input-output, it was possible to identify recursively the plant and to obtain the optimal parameters of the sensor model. 4. Hall effect sensor The Hall effect was discovered in 1879 for Edwin Hall.This effect refers to the potential difference (Hall voltage) on the opposite sides of an electrical conductor through which an electric current is flowing, created by a magnetic field applied perpendicular to the current.