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1

Ellipse-fitting algorithm implementation in the impedance measurement system based on DAQ card with FPGA

Lentka G., Targan M.

Przegląd Elektrotechniczny

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2016

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R. 92, nr 11

124--127

EN

The paper presents an implementation of the ellipse-fitting algorithm in the impedance measurement system based on DAQ card equipped with FPGA chip. The method implementation was tested by simulation means as well as experimentally in the designed and presented measurement system. Finally, the limit values of sampling parameters which assures satisfying accuracy were given.

PL

W artykule omówiono implementację algorytmu ellipse-fitting w systemie pomiarowym impedancji na bazie karty DAQ z układem FPGA. Implementacja metody została przebadana zarówno symulacyjnie jak również eksperymentalnie w zaprezentowanym systemie. Przedstawiono minimalne wartości parametrów próbkowania dla zapewnienia akceptowalnej dokładności.

2

Algorytm estymacji składowych impedancji w trójkanałowym układzie akwizycji danych

Augustyn J.

Pomiary Automatyka Kontrola

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2013

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R. 59, nr 1

1--5

PL

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.

3

Niepewność pomiaru impedancji metodą dopasowania do elipsy

Augustyn J.

Pomiary Automatyka Kontrola

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2011

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R. 57, nr 2

180-183

PL

W artykule przedstawiono porównawczą ocenę niepewności pomiaru impedancji, wyznaczonej za pomocą algorytmu dopasowania do elipsy oraz algorytmu DFT z oknem Hanninga. Wykorzystując metodę Monte Carlo, przeanalizowano wpływ niekoherentnego próbkowania na rozkład prawdopodobieństwa błędu składowych impedancji w układzie współrzędnych biegunowych.

EN

In this paper there is presented comparative evaluation of the result uncertainty of impedance component measurement with use of the ellipse-fitting algorithm and DFT algorithm with Hanning's window under the non-coherent sampling conditions. Impedance compo-nents in both cases are determined on the grounds of pairs of signal samples collected simultaneously, in accordance with the model described by equation (1). After elimination of time, it can be presented as conic curve equation (2). Under asynchronous sampling conditions, the dependence between sampling period Ts and unknown signal period T can be described as (5), in which is a window desinchronisation factor. Then, in order to decrease the influence of the spectral leakage effect, time window w(n) should be used and the values of complex spectrum components should be determined from equation (7), while the unknown impedance components from equation (4). The ellipse-fitting algorithm determines the values of ellipse equation coefficients (8) with use of the least squares method, calculating the eigenvector a corresponding to the least positive eigenvalue . On the basis of the known values of vector a elements, the impedance component values are calculated from equation (12). Characteristics of the compared algorithms have been examined with use of the Monte Carlo method, analysing the influence of non-coherent sampling on the probability distribution of the impedance component error, for two impedances Z1 and Z2 with different values of phase angle. The results of this experiment in the form of bivariate histograms of the resultant relative measurement error of impedance components ?|Z| and ? are presented in Figs. 2-4. The influence of the desinchronisation factor value on random characteristics of the impedance relative measurement error in the form of empirical distribution curves are shown in Figs. 5 and 6.