Swept area theory: an approach in nonlinear and switched network analysis

Abstract

In this work, according to the concept of 'area' on the v-i plane, a new approach called swept area theory, under both nonlinear continuous and discontinuous conditions, is developed. Novel conservative functions, such as area velocity and closed area over time (CAT), involved in this theory, are proposed. An analysis is carried out, by means of these functions, over nonlinear R, L, and C elements and over the ideal switch. In addition, jump discontinuities are discussed in detail. The CAT is related to the harmonic reactive powers and under sinusoidal steady state becomes proportional to the classical reactive power. A balance rule concerning harmonic reactive powers over nonlinear resistor under continuous conditions is obtained and discussed as a novel interesting result. This aspect impacts on a possible expanded definition of reactive power under distorted conditions. Thanks to the switching power, a novel quantitative relation between hard-switching commutations and CAT is obtained, with both theoretical and applicative relevance. More explanation is presented through a demonstration that shows how an ideal switch and power converters can become sources of reactive power. Issues of principle regarding the ideal switch model with respect to the real one are another important result of this work.So far, only time-invariant nonlinear networks have been dealt with, whereas time-variant networks are usually considered by means of switches. The general problem and solution methods of switched networks are presented in [19][20][21][22][23]. Classical issues in the presence of switching are network solution and inconsistent initial conditions. Network solution (with related problems of uniqueness of solution and stability) is fulfilled by several methods, of which one of the main is the complementarity approach, where switching is basically external constraint to a time-invariant multiport [24]. Inconsistent initial conditions, caused by switching, imply discontinuities on state variables and impulsive behavior on some voltages or currents, as addressed in depth in [25,26]. Nevertheless, as a whole, it appears to lack general principles as well as applications of generalized power in the field of switched networks; a notable exception is the 'connection energy' [27].Another important aspect that appeared previously is the 'area' on the v-i plane. From this idea in [28], a conservative function was proposed, called 'mean generalized content' (MGC), which is balanced and under periodical continuous steady-state conditions is invariant on nonlinear resistors. Therefore, the MGC could be seen as a generalization of the reactive power in distorted conditions [29]. In [30], the MGC was extended to circuits with ideal switches. In that paper, the definition of switching power (SP) was introduced, and a relationship between switching and reactive power was outlined.Following the aforementioned area approach, this paper intends to develop the swept area theory (SAT), which widely uses the concepts of trajectory and area on ...

Retracted: High‐speed FSK modulator using switched‐capacitor resonators

“…Figure (2) shows the topology of the source-free resonator after switching. Initial voltage of the capacitor, V 0 , is calculated using continuity of electric charge as shown in (10): Initial current of the inductor, I 0 , can also be obtained using the continuity of magnetic flux, ϕ, as: 15) or:…”

“…Transient properties of switched circuits have been utilized in a wide range of applications such as time-varying matching networks, reconfigurable networks, tunable filters and phase shifters [1][2][3][4][5][6][7]. Although most switched circuits are traditionally analyzed in a steady-state fashion, there are some reports that present an analytical transient representation for switched circuits [8][9][10][11][12][13][14][15]. The most important consideration in analyzing a switched network is the switching boundaries.…”