Algorytm estymacji składowych impedancji w trójkanałowym układzie akwizycji danych

W artykule przedstawiono algorytm do estymacji składowych impedancji, wyznaczonych na podstawie spróbkowanych wartości napięć związanych z tą impedancją oraz napięcia generatora zasilającego układ pomiarowy. W celu wykorzystania algorytmu dopasowania do elipsy, do zredukowania wymiaru macierzy danych wejściowych, zastosowano analizę głównych składowych. Uzyskane wyniki porównano z algorytmem dopasowania do elipsy dla danych pierwotnych, nie uwzględniających ograniczenia wynikającego z pomiaru napięcia generatora.

The paper presents an algorithm for estimating impedance components that were determined on the basis of sampled voltages associated with this impedance as well as a voltage of the signal generator feeding the measuring system. Since all signal processing circuit elements introduce errors, a set of N points described by equations (5) does not lie on the plane but it creates a cloud in the 3-D space. To determine the parameters of the plane (5b) on which the measurement results should be located, the principal component analysis can be used [10]. In this method the data dimension is reduced by searching for a plane which maximizes the variance of the data collected in X. For sequences of the sampled signal values, the covariance matrix of a sample (6) can be calculated. Then there are determined eigenvalues λ and eigenvectors a_j of the covariance matrix C. A base of the plane which is the best 2-D approximation of the data contained in X is defined by the eigenvectors a₁ and a₂2. For the determined base of the plane, the data can be transformed to the coordinate system defined by the vectors a₁ and a₂ (8). The adjusted coordinate values of points in the coordinate system Oxyz, are obtained after transformation (9). The first two columns of this matrix are the input data for the ellipse-fit algorithm. The results obtained with use of the principal component analysis were compared with those from the ellipse-fit algorithm for raw data, without taking into account constraints of the generator voltage measurement. Properties of the proposed algorithm and particularly the influence of incoherent sampling were examined with the Monte Carlo method. The influence of incoherent sampling on the random characteristics of the relative measurement error of impedance components is shown in Fig. 3 in the form of histograms of error values of the module δ|Z| and the phase angle δφ of the impedance Z.