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## About this book

This book integrates analytical and digital solutions through Alternative Transients Program (ATP) software, recognized for its use all over the world in academia and in the electric power industry, utilizing a didactic approach appropriate for graduate students and industry professionals alike.

This book presents an approach to solving singular-function differential equations representing the transient and steady-state dynamics of a circuit in a structured manner, and without the need for physical reasoning to set initial conditions to zero plus (0+). It also provides, for each problem presented, the exact analytical solution as well as the corresponding digital solution through a computer program based on the Electromagnetics Transients Program (EMTP).

Of interest to undergraduate and graduate students, as well as industry practitioners, this book fills the gap between classic works in the field of electrical circuits and more advanced works in the field of transients in electrical power systems, facilitating a full understanding of digital and analytical modeling and solution of transients in basic circuits.

## Table of Contents

### Chapter 1. Introduction to Fundamental Concepts in Electric Circuit Analysis

Abstract
This opening chapter has two main objectives. The first of them, which has a strictly revisionary character, is to approach fundamental concepts of physics, as well as laws, equations and formulas of electricity and electromagnetism. The second is to anticipate, through an example, the real raison d’être of this book, which is the solution, both analytical and digital, of transient phenomena in linear, invariant and concentrated circuits. This chapter presents then the main elements of an electrical circuit with concentrated parameters that do not vary with the frequency, as well as the dynamic response of basic circuits when disturbances occur, such as energizing or de-energizing switches, enabling a physical understanding of the phenomena and the need for mathematical modelling and computer simulations. However, it is assumed that the reader already has fundamental knowledge of analysis of electrical circuits, laws, theorems and methods of solution, as well as of solving ordinary differential equations. Thus, this chapter aims to create the motivations for the study of mathematical and computational techniques and solutions presented in the following chapters.
José Carlos Goulart de Siqueira, Benedito Donizeti Bonatto

### Chapter 2. Singular Functions

Abstract
This chapter introduces the singular functions, also called generalized functions or distributions, that constitute a special class of functions, because they do not fit perfectly into the conventional mathematical concept of function. Here they will be designated by $${U}_{n}(t)$$, $$n=0, \pm 1, \pm 2, \pm ...,$$ in such a way that for each value of $$n$$ a specific singular function will correspond. They can be considered as a family of functions that are generated from the unit impulse function, $${U}_{0}(t)$$, by successive derivations or integrations. However, for simplicity and also for strictly didactic reasons, the study of them will be done starting from the unit step function, $${U}_{-1}(t)$$. An important reason to study them is that the response of any circuit linear to an arbitrary entry can be determined, indirectly, as long as its response to a unit step or unit impulse is known. This is done through the integral convolution call. Thus, the response of a linear circuit to a unitary impulse, denoted by $$h(t)$$, or to a unitary step, called $$\it r(t)$$, can be used as a characteristic property of such a circuit.
José Carlos Goulart de Siqueira, Benedito Donizeti Bonatto

### Chapter 3. Differential Equations

Abstract
When a linear system, with concentrated parameters and invariant over time, is excited with an input signal $$x\left( t \right)$$, the classic mathematical model for determining the output (or response) $$y\left( t \right)$$ is an n order ordinary linear differential equation with constant coefficients. In case the system is an electrical circuit, the order n is generally at most equal to the sum of the number of capacitances and inductances of the circuit, computing these numbers, however, only after capacitances and inductances in series and / or parallel have been replaced by their respective equivalents. This chapter presents a fundamental review about the theory and solution of ordinary differential equations. Moreover, it presents a systematic method to solve ordinary differential equations in which $$x\left( t \right)$$ is a singular and / or causal function. This method allows to determine the initial conditions at $$t = 0_{ + }$$, from the respective initial conditions at $$t = 0_{ - }$$ and the differential equation itself, without any physical reasoning about the given circuit. Therefore, a good understanding of this chapter is an indispensable condition for the good follow-up of the following chapters.
José Carlos Goulart de Siqueira, Benedito Donizeti Bonatto

### Chapter 4. Digital Solution of Transients in Basic Electrical Circuits

Abstract
This chapter presents the fundamental algorithm of the digital computational solution of electromagnetic transients in electrical circuits used in programs based on the EMTP—Electromagnetics Transients Program. Since the nodal method is used in the fundamental EMTP algorithm, therefore, the discretization of circuit elements to concentrated parameters (inductances and capacitances) is implemented through the trapezoidal integration method, resulting in Norton Equivalent Digital Circuits for these elements. It is emphasized that such digital models depend on the appropriate choice of the integration step (∆t) so that the results of the computer simulation present the acceptable precision for the phenomena under study. The simplicity for computational implementation of the trapezoidal integration method when compared to other numerical integration methods, its precision and stability justified its use in EMTP and programs based on it. Because of the trapezoidal integration method’s own discretization algorithm, under certain conditions numerical oscillations may arise in the computational solution. Therefore, it is essential to understand its origin to evaluate the different methods proposed for its solution in different computer programs based on EMTP.
José Carlos Goulart de Siqueira, Benedito Donizeti Bonatto

### Chapter 5. Transients in First Order Circuits

Abstract
This chapter presents transients in first order circuits. Extensive number of examples are provided with their analytical and digital solution using the software ATP—Alternative Transients Program, using the graphical interface ATPDraw. For didactic reasons, the study is organized as a large collection of various case studies, where the fundamentals of circuit analysis, mathematical time domain solution of ordinary differential equations with singular forcing functions, and physical principles are presented in details. For example, “if all voltages and currents in a circuit remain finite, then the voltage at the terminals of a capacitance and the current through an inductance cannot be varied instantly”. Therefore, a continuous knowledge is acquired along this book, integrating fundamental electrical engineering concepts in transient circuit analysis and illustrating potential applications.
José Carlos Goulart de Siqueira, Benedito Donizeti Bonatto

### Chapter 6. Transients in Circuits of Any Order

Abstract
In this chapter, circuits with any number of capacitances and inductances will be covered. This means that the differential equations can now be of any order. In general, the order of the differential equation for a variable of interest in a given circuit is at most equal to the sum of the number of capacitances and inductances it contains. Logically, before calculating this sum, it is necessary to reduce the capacitances that constitute series, parallel or series-parallel connections to their equivalent capacitances. The same goes for inductances that form series, parallel or series-parallel connections, which must be replaced by their equivalent inductances. Initially, circuits powered by independent sources, starting at $$t = 0$$, containing capacitors and inductors without energy initially stored at $$t = {0_{ - }}$$, that is, circuits initially de-energized, will be addressed. Then those with energy initially stored at $$t = {0_{ - }}$$ will be studied, with or without external sources. The latter will be solved, analytically, with the use of the Thévenin and Norton equivalents of capacitances and inductances initially energized. Typical examples of this second situation, that is, circuits containing capacitive elements with initial voltage at $$t = {0_{ - }}$$ and inductors with initial current at $$t = {0_{ - }}$$, the transients caused by switching. Whenever the examples are not exclusively literal, the respective numerical solutions will also be presented using the ATPDraw program.
José Carlos Goulart de Siqueira, Benedito Donizeti Bonatto

### Chapter 7. Switching Transients Using Injection of Sources

Abstract
This chapter presents another methodology for the study of transient phenomena caused by the opening or closing of one or more switches, in circuits that are or are not operating in a steady-state, before switching. It is a technique applicable, preferably, to linear circuits and invariant over time, and whose principle is to simulate the closing of a switch by injecting a voltage source into the circuit; or the simulation of opening a switch by injecting a current source. In essentially linear circuits, the procedure is based on the substitution theorem, also called the compensation theorem, and on the superposition theorem. In the development of the method it is assumed that the independent sources present in the circuit are sinusoidal, making it clear, at first, that it applies equally to other types of independent sources. In contrast to the preceding chapters, the various examples presented here are solved using both the classic method of ordinary differential equations and the operational method of Laplace transforms.
José Carlos Goulart de Siqueira, Benedito Donizeti Bonatto

### Backmatter

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