ETEP z Measurements for the Characterization of Quasi-Periodic Waveforms L. Peretto, R. Sasdelli Abstract zyxwvutsrq Quasi-periodic signals feature discrete-spectrum waveforms with lines located at frequencies expressed by non- rational numbers. In this paper, a new digital method for the spectral analysis zyxw of quasi-periodic bi-tone signals is proposed. The results of some simulations showing its good pe~ormance are reported and discussed. 1 Introduction zyxwvuts Nonperiodic voltages and/or currents can be ob- served in electrical circuits with time-varying loads in steady-state operation; hence, considering their wave- forms as periodic at industrial frequency is a mislead- ing assumption which can turn into large measurement errors. To better discuss these situations, let us model a nonlinear power system by means of the multi-port block in Fig. 1 having m+q ports. The m input ports are zyxwv I I 4 h s20 I I OL2 Nonlinear system s, Fig. zyxwvutsrq 1. Schematic diagram of a multi-port nonlinear system; SI, ..., S , : ac sources; L1. .... L,: loads. supplied by the ac sources S,, . .., S, and the q output ports are connected to the loads L1, ..., L,. The m sources can be synchronous with a given frequency or not. In the first case, the spectral components of the generic output signal x(t) are located at harmonically related frequencies; but, in the presence of a time- variance of the system parameters, x(t) can feature a continuous spectrum. If the input sources are not synchronous, x(t) is a discrete-spectrum signal whose lines could be located at noncommensurable frequen- cies even in the lack of a time-variance of the system parameters. In several contributions that can be found in the literature, see e.g. [ 1-71, the term quasi-periodic is used to denote this special case of almost-periodic function in the sense of Bohr [8]. Though the same term was also used in papers [9-121 with reference to originally periodic waveforms that become nonperiodic due to a time-variance of the system parameters, only the former case will be referred to as quasi-periodic in this paper. 2 Theoretical background Instruments based on digital signal processing (DSP) techniques can be designed to measure, also in quasi-periodic conditions, the parameters (rms values, powers, quality indices) usually determined to character- ize the operation of the electrical circuits under periodic conditions. To do this, they must implement parameter definitions generalized to quasi-periodic conditions, see e.g. [6,13]. Information on the behavior of an electrical cir- cuit can be derived from the spectral analysis of the voltage and current waveforms, no matter they are pe- riodic or not. The characterization of periodic signals requires finite observation intervals, equal at least to one period, whereas the implementation of definitions based on integration intervals tending to infinity is involved in the case of measurements on electrical circuits with nonperiodic discrete-spectrum signals. The techniques for the analysis of the latter kind of signals can be grouped into two main categories: win- dowing and statistical techniques. A statistical ap- proach for the analysis of quasi-periodic signals over finite observation intervals will be described in the following Section 3. 2.1 Definitions Let us assume that the ac generators supplying the m input ports of the generic nonlinear multi-port system in Fig. 1 feature different fundamental angular frequencies, denoted by w1 ,..., o+ ,..., zy w,. If no element of the set { wj) is a multiple of the others, the m gener- ators can be referred to as tone sources; the output signals of the considered multi-port system can be named multi-tone signals, see e.g. [5]. Any multi-tone signal x(t), representing a voltage or a current at a generic output port in Fig. 1, can be expressed by means of this general form: ETEP Vol. 12, No. 1, January/February 2002 I1