Comparison of impedance measurements in a DSP using ellipse-fit and seven-parameter sine-fit algorithms

Abstract

In this paper, two DSP implemented algorithms for impedance measurements are compared. Previously published results demonstrate the usefulness of sine-fit algorithms and ellipse-fit algorithms for impedance measurements where two channels are simultaneously acquired with analog to digital converters. The comparison between the two implemented algorithms is done by analyzing the average execution time, memory requirements and experimental standard deviation of the estimated impedance parameters.

For the first time, the seven-parameter sine-fit algorithm is adapted so that it does not need the construction and manipulation of its largest matrix thus requiring less overall memory. This improvement can be used to acquire and process more samples (leading to reduced experimental standard deviations of the estimated parameters) with the same memory size.

Introduction

Traditional impedance measurement techniques are being improved [1] due to the development of highly efficient analog to digital converters, the use of digital signal processing techniques and the rapid development of new, more powerful digital signal processors that can now perform millions of operations per second. Impedance measurements are a relevant topic with many new developments published recently. Some works focus on impedance sensor measurements [2] while others make use of recently published signal processing algorithms to improve impedance estimation [3], [4]. In [5] a high impedance spectroscopy probe is analyzed and in [6] a novel method to estimate the individual parameters of the components from an impedance model was presented.

A simple impedance measurement technique presented in [7] and improved in [8] minimizes the analog front-end circuitry to reduce the influence of its frequency dependence. This technique is based on the volt-ampere method where a reference impedance and an unknown impedance are placed in series and supplied with the same current from a sine generator. The voltage across each impedance is simultaneously acquired with a two-channel analog to digital converter and the impedance’s magnitude and phase are extracted using digital signal processing algorithms.

Sine-fit algorithms are an option to estimate the sine signal parameters from a set of acquired samples. Since the ratio between the sine frequency and the sampling rate is, most of the times, not accurately known, the algorithms must also estimate the sine frequency. This makes the problem a nonlinear regression that is solved using an iterative procedure which is called the four-parameter sine-fit algorithm [9]. For impedance measurements, this algorithm can be applied to each channel separately, however the overall experimental uncertainty can be reduced by forcing the common frequency in what is called the seven-parameter sine-fit algorithm [10]. Nevertheless, the iterative nature of the sine-fit algorithms results in a large number of operations that the processor must execute. Another issue that can affect the performance of sine-fit algorithms implemented in DSP devices is the memory requirements. In fact, the seven-parameter algorithm as proposed in [10] must build a matrix with seven columns and 2N rows, corresponding to the total number of samples acquired in both channels. This large matrix and its manipulation together with the limited memory available in the DSP restricts the number of samples that can be processed. In this paper, the algorithm proposed in [10] is shown to be applicable in a memory restricted system by bypassing the need to build the 2N × 7 matrix. This modification allows the DSP to process more samples to estimate the sinewave parameters with the seven-parameter sine-fit.

Although the frequency must be accurately known for a correct estimation of the sine signals’ amplitudes, phases and dc components by the sine-fit algorithms, in impedance measurements an accurate frequency value is not needed. The time information, and thus the sine signals frequency, can be removed from the problem by making an XY plot of the two waves. The result is an ellipse whose parameters can be estimated by ellipse-fit algorithms such as the one presented in [11] and then improved in [12]. The sine signals’ parameters can then be extracted from the ellipse parameters as proposed in [13]. This algorithm has been recently adapted and optimized for use in DSP based impedance measurements [14]. In the modified ellipse-fit method published in [14] there is no limit on the number of samples that can be used since only nine values must be stored and the samples themselves can be discarded after their contribution to the nine stored values is taken into account. These characteristics make the algorithm a prime candidate for efficient implementation in a DSP based impedance measurement instrument.

In this paper, the performance of the DSP implementation of the ellipse-fit algorithm and the seven-parameter sine-fit algorithm is compared through the evaluation of the memory requirements, speed of execution and experimental uncertainty of both algorithms. The system used for the comparison is based on a commercial floating-point DSP kit, with few external electronics, for baseline assessment and requirements definition. The performance tests were executed at frequency f = 1 kHz, for impedances with magnitudes from 100 Ω to 15 kΩ and phases in the ±90° range.

The paper is divided into four sections including the introduction (Section 1) and the conclusions (Section 4). The description of the two algorithms is presented in Section 2 together with the measurement setup and the algorithms DSP implementation details. The performance comparison is performed in Section 3 where the experimental results are shown and analyzed. The conclusions are presented in Section 4.