W artykule przedstawiono ocenę niepewności aparaturowej pomiaru modułu i kąta fazowego impedancji za pomocą algorytmu dopasowania do sinusoidy oraz dwóch wersji algorytmu dopasowania do elipsy - klasycznej i zmodyfikowanej o algorytm analizy składowych głównych. Metodą Monte Carlo wyznaczono budżety niepewności dla badanych algorytmów oraz macierze niepewności. Wykazano, że podstawowym źródłem niepewności aparaturowej jest błąd wzmocnienia - dla modułu impedancji i szum wejściowy - dla kąta fazowego.
EN
An estimation of the instrument uncertainty of magnitude and phase angle measurement of impedance by using sine fitting algorithm, and the two versions of ellipse-fitting algorithm – the classical and the modified by algorithm of principal component analysis (PCA) is presented in the paper. In the sine fitting algorithm, based on LMS method, the values of orthogonal components of voltage and current (3) are used to calculate the impedance components |Z| and , from Eq. (4). In the classical ellipse fitting algorithm, based on the determined value of the parameter vector a (Eq. (6)), the impedance components are calculated from Eq. (7). In the modified ellipse fitting algorithm, the measuring system is supplemented by an additional acquisition channel of the generator signal. The classical ellipse fitting algorithm is then preceded by a fitting to the plane algorithm, using the method of principal components analysis [7]. Histograms in Figs. 1 and 2 show relative measurement errors impedance components obtained by the Monte Carlo method. Uncertainty budgets were determined for the tested algorithms as well as the uncertainty matrices. In the Tabs. 2 and 3 are shown the contributions to the standard uncertainty of the various uncertainty sources. It has been shown that the basic source of the uncertainty is the gain error - for the magnitude of impedance, and the input noise - for the phase angle. Components values of the combined standard uncertainty of impedance values estimation and the shape of probability distribution depend on the form of a processing algorithm.
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.
EN
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 aj 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.
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