Analog Circuits

Analog circuits are essential in everyday life and in electrical engineering. Even though the digital world is increasing and digital computation is attractive to people today, nature does work in an analog way. Even if the internet and the telecommunication cores are digital, the input and output interfaces are analog. Moreover, the transmission systems start with analog information to be transmitted by antennas. The power electrical and electronic systems are based on analog devices, even though their control is based on digital systems. Sensors and actuators are founded on an analog signal. Therefore, focusing on analog circuits is timely.

Moreover, at the roots of modeling and designing analog circuits is mathematics. For this reason, many papers on analog circuits have been published on the MDPI journal Mathematics in recent years. Based on the previous considerations, the idea of proposing an Editorial to focus the activity of the journal in this area arose. Sixteen papers were selected. In particular, attention has been devoted to remarking on the more appealing topics of analog circuits, such as that of strange attractors and that of using analog computation paradigms for qualitative solutions to differential equations. Moreover, the use of artificial intelligence (AI) methods to design and analyze complex analog circuits has been considered as a further important topic in the area.

In summary, the selected papers have been classified into four areas:

• Chaos and analog computation [1,2,3,4,5,6,7,8,9];
• Algebraic approaches to the analysis of electrical networks [10];
• AI tools for the design of analog devices [11,12,13,14];
• Industrial applications [15,16].

Below, the essential concepts, including some details of the specified papers, will be presented.

Petrzela in [1] gives a complete overview of the main concepts related to chaotic electronic circuits. After a detailed report of the various literature in the field, about 200 papers, the author studies in detail a chaotic electronic device. The device has been studied both numerically and by using experiments showing the complex behavior of the system. Several chaotic attractors are reported in this paper providing the reader with a complete understanding of the main principles of applied chaos. The article indeed represents concepts and experiments that will introduce the readers of Mathematics to the topic of nonlinear analog chaotic devices. The topic of analog computation also emerges from the proposed discussion.

The dynamics of the Clapp Oscillator is discussed by Petrzela in [2], who reports the mathematical model. The paper focuses on transistor-based devices, and a detailed numerical analysis is reported in the first part of the paper, including the characterization of the shape of the obtained attractors. The second part of the paper focalizes the electronic realization of the dynamical model of the system by using techniques based on operational amplifiers. This is convenient for the system characterization, having reduced implementation problems and allowing it to give accurate measurements that are generally difficult to achieve with the transistor implementation. The behavior of this electronic circuit is hyperchaotic. The paper is therefore a further contribution to the class of hyperchaotic devices.

The contribution [3] by Ding et al. covers many of the classic topics in the area of dynamical systems. The authors focus on the study of a fractional-order, chaotic, three-echelon supply chain system. They therefore started the evaluation of such type of system from a mathematical point of view, and their efforts are devoted to the numerical simulations. Moreover, the analog computation idea arises, and they present an analog implementation of the previously studied dynamics in terms of analog electronic circuit. The results are very interesting and strongly confirm that analog device-based circuits can be an alternative to dynamical simulation and, in some cases, more flexible, allowing many experiments in order to qualitatively evaluate the original system.

In [4], Petrzela and Rujzlh present a complete study of the chaotic oscillations in an electronic circuit based on classical discrete components, such as the Darlington configuration. The authors propose mathematical models and a rich gallery of attractors derived both by using numerical simulations and with experimental prototypes. The paper confirms the actual wide interest in discovering strange attractors in classical electronic configurations working in nonlinear conditions.

Synchronization is one of the main aspects to consider when coupling chaotic and hyperchaotic circuits. Alattas et al. [5] discuss in the paper a sliding mode controller in order to synchronize more circuits. The study has been approached from an accurate analytical discussion and experimentally, proving in the real devices the suitability of the proposed approach. Moreover, an analysis of the Lyapunov exponents has been performed to establish the hyperchaotic behavior of the considered devices; indeed, a good synchronization level is obtained, emphasizing the suitability of the approach, not only in the case of chaos but with circuit with higher-order chaos.

The analog implementation of a chaotic circuit is presented in [6]: Almatroud et al. address their study to the generation of pseudorandom numbers. The focus of the study is the secure encryption of medical images.

Fractional Nonlinear Order Systems exhibit complex dynamical behavior. Moreover, the implementation of real Fractional Order System can be suitably performed by using their approximation with high-order analog circuits. In this area, the contribution [7] by Rahman et al. includes new chaotic circuits, their implementations, and the classical use in secure information communication.

The contribution [8] briefly shows the research by Solano et al., addressing the solutions to differential equations using principles of electrical networks. The study in our opinion addresses research to define solvers of differential equations based on analog devices. This reinforces the idea that analog computation until now has been innovative for nonlinear differential equation solvers.

The paper [9] presents an absolutely new circuit based on the implementation of quaternion-based devices by using analog electronic circuits. The authors present an innovative Lorenz-like hyperchaotic circuit, with impressive strange attractors. The devices concern secure analog communications, and therefore, the synchronization topic is also approached in the contribution.

The Geometric Algebra (GA) is proposed in [10] by Montoya et al. as an useful tool to analyze power networks. Today the Time-Frequency analysis of multiphase electrical circuits arises as a fundamental topic due to the introduction of more energy sources to power networks. This leads to the introduction of high-frequency components in the power flow. Therefore, the paper does appear timely, and the introduction of innovative methods in the area of electrical circuits could lead to innovative ideas in the field.

The paper [11] explores the possibility of using artificial intelligence (AI) techniques in order to discover faults in the CAD design of analog devices. The study by Dieste-Velasco gives a contribution in the area of the CAD design of electronic analog circuits and therefore the integration of AI tools with of classical electronic devices’ CAD design modules.

The contribution [12] emphasizes a further example of how AI techniques, such as fuzzy logic, can be usefully used in the design of operational amplifiers. The optimized Sugeno paradigm is adopted. The study opens a further opportunity: to link fuzzy-logic-based tools with standard CAD tools applied to analog device projects.

The basic elements for the design of nonlinear analog circuits are presented in [13]. The efforts of the Rojec et al. are oriented towards the formalization of a new paradigm based on topological considerations in order to describe the circuit and to propose an optimization algorithm based on evolutionary concepts. Several case studies are shown, and a Phyton-based code is used. The results open new frontiers in the design of analog circuits with nonlinear characteristics.

Power electronics include a wide range of analog devices. The design of power electronic circuits is an important challenge today. Therefore, even if the mathematical models of more devices are well known, and the behaviors of this class of systems can be well defined, Rodríguez and co-authors propose in [14] an innovative idea to establish, inside the system, an algorithm of choice in order to obtain the best switching operation. The aim has been achieved by using a particular evolution-based tool, L-Shade. The paper therefore presents a further example of how AI can improve the performances of analog circuits.

The realization of any electronic circuit requires, in the final production, a printed circuit board (PCB). In [15], Guo et al. remark on the importance of this aspect. Today, the PCBs must be tested for accuracy due also to the semiconductors’ company customers. Therefore, the evaluation of PCB boards is a fundamental step. The authors clearly present the problem and give details about the finite-element procedure that allows the thermal analysis of the board to establish its reliability evaluation and sensitivity analysis. A case study is also shown.

The area of analog electronics covers all the conditioning circuits for sensing devices. In [16], Petkovšek et al. present in their contribution an analog circuit for thermistors-based measurements of temperature, which are used, when coupled to microcontroller devices, for conditioning the sensor. The paper is an example of how digital equipment, analog computations and sensors can be integrated together to give accurate low-cost measurement systems.

In

Table 1. Correlation matrix of the selected key topics.

	Fractional Order	Design Methods	Artificial Intelligence	Chaotic Oscillations	Hyperchaotic Oscillations	Analog Computation	Mathematical Methods	Networks	Synchronization	Circuits for Sensing
[1]		X		X		X	X		X
[2]				X	X				X
[3]	X			X		X			X
[4]				X			X		X
[5]				X	X				X
[6]				X		X			X
[7]	X			X					X
[8]				X		X	X		X
[10]							X	X	X
[11]		X	X					X
[12]			X
[13]		X	X
[15]										X
[16]		X								X
[14]		X	X				X
[9]				X			X		X

, the correlations among the various papers and key topics are presented in order to give the readers a compact and complete understanding of the related subjects.

Analog circuits are conceptually and practically always relevant. Moreover, the mathematical aspects, which are strongly related to the dynamics of the circuit, are often topics needing deeper investigations. The aim of this Editorial is to highlight the role of mathematics in the research on analog circuits and to further stimulate mathematical scientists in this direction.