ABOUT THE BOOK

LRC circuits which are composed of inductors, resistors and capacitors form the basics of the electric circuits used in the industry. This book is a comprehensive study designed to help science and engineering students understand the mathematical framework used to analyze the LRC circuits encountered in many different science and engineering courses. Main mathematical tools including differential equation solutions, complex algebra and phasor diagram method are all presented in a concise, yet apprehensible, manner. Rarely touched in the literature and occasionally discussed in the lectures, the equivalency between the mechanical systems and electric circuits topic is also presented in a separate chapter with explanatory design examples.

ABOUT THE AUTHOR

The author, Arif E. ERKOCA holds a PhD degree in Physics. He taught electricity and magnetism, electric circuits, mechanics and optics labs during his Master’s studies at the Middle East Technical University in Ankara, Turkey (2002 - 2004) and at the University of Arizona in Tucson, AZ, USA (2004 - 2010). He is currently acting as the business development director at Electronics Valley Inc. (consultancy and training services for aerospace professionals) and at Bupat Global LLC (value added electronics distributor company in the aerospace industry). Mr. ERKOCA has also been privately tutoring hundreds of engineering and science students since 2010, and he is the founder of ASHLAR STEM ACADEMY, a web portal designed to educate young generations in the fields of STEM.

AUTHOR'S NOTES:LRC Circuits, which are composed of inductors, resistors and capacitors, is a subject of many science and engineering courses, where only part of the picture is shared with the students. In some courses, the students learn about phasor method to solve for the impedance and in some others they learn how to exploit complex algebra to do the same. The equivalency between the electric circuits and the mechanical systems is even rarely pronounced. Not to mention, it is almost not possible to find any detailed study which presents all these aspects as a complete review.

A basic series LRC circuit is described by a second order differential equation. In this book, I present the generic solution to this equation, and the use of complex algebra and phasor method as the mathematical tools required to analyze more complex circuits. Furthermore, I also highlight the appearance of the resonance phenomenon in LRC circuits. I show that LC, LR and RC circuits are all types of LRC circuits in the limits R->0, C-> ∞ and L->0, respectively.

It is quite fascinating to see the resemblance between the electric circuits and the mechanical systems. In this book, I also present how the differential equations which govern both systems turn out to be identical. Thus, for a given mechanical system which is composed of masses, springs and rough surfaces, an equivalent electric circuit with inductors, capacitors and resistors can be constructed.

In real life, like everything else, the electric components are also far from being ideal. As an example, I contrast the electric characteristics of a real capacitor with those of an ideal one in the final chapter of this book.

I hope many will benefit from this comprehensive review of basic circuits.

Arif E. ERKOCA

CHAPTERS

Chapter 1: LRC CIRCUIT IN SERIES …………………………… 1

Chapter 2: LRC CIRCUIT WITH A DC SOURCE ……………….. 13

Chapter 3: LOSSLESS CIRCUIT () ……………………… 23

Chapter 4: LRC CIRCUIT IN STEADY STATE () ……….. 31

Chapter 5: LRC CIRCUIT / COMPLETE SOLUTION ……………. 45

Chapter 6: RC CIRCUITS WITH DC & AC SOURCES …………. 57

Chapter 7: LR CIRCUITS WITH DC & AC SOURCES ………….. 67

Chapter 8: EQUIVALENCY (MECHANICS ELECTRICITY) ….. 75

Chapter 9: FINAL PROJECT …………………………………….. 89

(A real capacitor powered by a square wave signal)

APPENDIX: REAL CAPACITOR .………….………………………. 107

REFERENCES …………………………………………………….. 115

In Chapter 1, a series LRC circuit is introduced and the differential equation for the charge on the capacitor is constructed via the use of the laws of Physics. This equation is solved for an LRC circuit powered by an AC source with a sinusoidal wave function.

In Chapter 2, the solutions to the observed quantities in a series LRC circuit as in presented in Chapter 1 are obtained when AC source is replaced with a DC source.

In Chapter 3, the LC circuit (lossless circuit, i.e R=0) is introduced together with the resonance phenomenon, which seems crucial to understand before moving on to the study of the LRC circuits in steady state.

In Chapter 4, the focus is on the steady state (t->∞) behavior of circuits. The phasor diagrams and complex algebra in analyzing the steady state behavior of LRC circuits are also introduced in this chapter.

In Chapter 5, the complete solution for the observed quantities in an LRC circuit is presented and the numerical results are presented and discussed in detail.

In Chapter 6, the characteristics of RC circuits are determined from the results of LRC circuits after taking the limit L->0.

In Chapter 7, the characteristics of LR circuits are determined from the results of LRC circuits after taking the limit C->∞.

In Chapter 8, the equivalency between mechanical systems and electrical circuits is investigated. For given mechanical systems, the electric equivalent circuits are constructed.

In Chapter 9, a real capacitor is studied. The imperfections of a real capacitor such as its internal resistance and inductance are shown to be affecting its electrical characteristics.

# LRC CIRCUITS - A Comprehensive Review

October 6, 2020