Instrumentation and Measurement Technology Conference - IMTC 2007 Warsaw, Poland, May 1-3, 2007 Estimation of Flow Parameters for the Needs of the Electromagnetic Measurement in Open Channels Based on a Concept of Inner Product Spaces Jacek Jakubowski', Andrzej Michalski1 2 'Electronics Faculty, Military University of Technology, ul. Kaliskiego 2, 00908 Warsaw, Poland Phone: +48-22-683-9768, Fax: +48-22-683-91-25, Email: jjakubowskigwel.wat.edu.pl 2 Electrical Faculty, Warsaw University of Technology, ul. P1. Politechniki 1, 00601 Warsaw, Poland Phone: +48-22-660-74-27, Fax: +48-22-629-29-62, Email: anmigiem.pw.edu.pl Abstract - Electromagnetic flowmeters are said to be the best solution in many applications mainly because of the possibility of mean velocity measurement in the conditions of varyingfilling of the channel. Despite the relatively simple principals of operation the question of the signal processing method that can handle the flow parameter estimation in the noisy environment is still open. The estimators based on classical signal decomposition with the use of Euclidean inner product have high variance. The paper presents a quantitative assessment of a trial to modify the inner product in such a way that it would be "blind" to the existing disturbances. Keywords - flow measurement in open channels, application of inner product spaces. I. INTRODUCTION The electromagnetic flowmeter is a device for liquid flow measurement with the use of Farady's effect as a principal of operation. According to the effect two different types of voltages can be simultaneously generated in any conductive liquid that moves relative to time-variant magnetic field lines. The resulting combined voltage can be collected with the use of two dipped electrodes. They provide us with the information about basic flow parameters - the velocity and height of filling when properly installed in the channel [1] [2]. In the case of no external disturbances the induced voltage waveform can be described by the equation: u(t)=a.v.iI(t)+b.h *i2(t) =WI il(t) +W2 i2(t) (I a) Parameters a and b are constant for a given device, whereas v and h are liquid velocity and height of filling, respectively. Quantity i,(t) describes the waveform of current responsible for the existing magnetic field and i2(t) is its first derivative. Having acquired a single period of the induced signal u(t) as well as the period of waveform i,(t), one can demand the equation (la) to be fulfilled by all of the signal samples. In this way an over-determined set of linear equations with respect to w, and w2 arises. Its LS solution leads to the decomposition of the flow signal into two components: the current component i,(t) and the derivative of current component i2(t). Their amplitudes represent the liquid velocity and filling respectively while the product of them carries information about the volumetric flow. Despite the relatively simple theoretical specification of the measurands, there are lots of problems in real measurements with various disturbances that affect the resulting voltage signal. The frequency of the signal should not exceed the values of several hertz as higher frequencies considerably increase the component i2(t) leading to the saturation of the ADC converter. The dominant component of the disturbances is also of low frequency nature at the same time and can achieve levels many decades higher than the level of the induced flow signal. The disturbances are mainly brought about by random effects connected with the difference between the electrochemical potentials of the liquid and the sort of material the electrodes are made of. The low frequency component is believed to be the main source of the overall complex uncertainty including the uncertainty of the measuring instrumentation. Thus as a measure of an algorithm's ability to suppress the spread in results caused by the disturbances, the basic statistic - the standard deviation representing the standard uncertainty for the resulting value w, or w, was used. Without changing the generality of the consideration we can put into equation (la) an additional waveform d(t) representing such a disturbance [3]: u(t) = WI * il (t) + W2 * i2 (t)+ d(t). (lb) Taking into account the properties of real disturbances [4], the simplest and efficient method to model d(t) is to use a polynomial approximation. According to (lb) any deviation of the acquired signal from the weighted sum of i,(t) and i2(t) generates non-zero coefficients of an approximating polynomial, which thanks to that takes the role of the disturbing component [4]. The computational method is based again on the set of over-determined equations u = Aw. Its least square solution will be also a solution to the associated normal system ATAw= Au. The column matrix A contains the components achieved by data acquisition, initial signal processing resulting in the estimation of the derivative i2(t) and polynomial approximation. The components constitute a non-orthogonal base for the signal model according to (la) or (lb). The fundamental problem is that the system brings about the necessity of calculating and inverting the matrix ATA as the solution treated as an 1-4244-0589-0/07/$20.00 ©2007 IEEE 1