Mathematics

# The Laplace Transform and its Application to Circuit Problems

Contents

## ABSTRACT

This paper presents an overview of the Laplace transform along with its application to basic circuit analysis. There is a focus on systems which other analytical methods have difficulty solving. The concept of Laplace Transformation plays a vital role in diverse areas of science and technology such as electric circuit analysis, communication engineering, control engineering, linear system analysis, statistics, optics, quantum physics, etc.

CHAPTER ONE

1.0 Introduction

1.1 Aims and objectives

1.2 Scope of study

1.3 Background of study

1.4 Definition of terms

CHAPTER TWO

2.0 The Laplace transform-

2.1 Unit step function-

2.2 Impulse function

2.3 Periodic function

2.4 Inverse Laplace transform

2.5 Convolution

CHAPTER THREE

3.0 Application to circuit analysis

3.1 Complete circuit analysis

3.2 Transfer function

CHAPTER FOUR

4.0 Applications-

4.1 Problem I

4.2 Problem II

4.3 Problem III

4.4 Problem IV

4.5 Summary

References

## 1.0 INTRODUCTION

The Laplace Transform is an integral transform method which is particularly useful in solving linear ordinary differential equations. It finds very wide applications in various areas of physics, optics, electrical engineering, control engineering, mathematics, signal processing and probability theory.

The Laplace transform-is an important concept from the branch of mathematics called functional analysis. It is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, mechanical systems, control theory and optical devices using algebraic methods. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. The analysis of electrical circuits and solution of linear differential equations is simplified by use of Laplace transform.In actual Physics systems the Laplace transform can be interpreted as a transformation from the time domain, where input and output are functions of time to the frequency in the domain, where input and output are functions of complex angular frequency. The basic process of analyzing a system using Laplace transform involves conversion of the system transfer function or differential equation into s –domain, using s –domain to convert input functions, finding an output function by algebraically combing the input and transfer functions, using partial functions to reduce the output function to simpler components and conversion of output equation back to time domain.

Furthermore, we use the Laplace transform to transform the circuit from the time domain to the frequency domain obtain the solution and apply the inverse Laplace transform to the result to transform it back to the time domain.

## AIMS / OBJECTIVES

The major purpose of this work is to explore the procedure for solving a complete circuit problem by transform technique.

This paper provides the reader with a solid foundation in the fundamentals of Laplace transform and circuit theory, and gain an understanding of some of the very important and basic applications of these fundamentals to electric circuit and solution to related problems.

To apply basic circuit analysis method (transform domain circuit) to obtain desired response function, write the differential equation in the time domain and solve for a desired variable using transform method.

## SCOPE OF STUDY

The aim of this monograph as stated earlier is to supply a treatment of the Laplace transform and its applications to circuit analysis. This project intends to study the behavior of circuit. How does it respond to a given input? How do the interconnected elements and devices in the circuit interact? The study is about creating and solving equations for (1) the inputs, (2) the transmission or intermediate processing function and (3) the output of electrical circuit. It concentrates upon transient analysis and the solution of circuit equations with differential and integral terms using the Laplace transform.

Accordingly, the main body of this thesis is divided into three parts. Part one provides an elementary discussion of the unilateral Laplace transform in addition to derivation of the basic properties of this transform. Part two will consider some properties of the Laplace transform that are very helpful in circuit analysis. A brief discussion of the Heaviside function, the Delta function, Periodic functions and the inverse Laplace transform. Finally, the third part will outline with proper examples how the Laplace transform is applied to circuit analysis. Furthermore, discuss solutions to few problems related to circuit analysis.

## BACKGROUND OF STUDY

The Laplace transform was discovered originally by Leonhard Euler, the eighteenth-century Swiss mathematician but the technique is named in the honor of Pierre-Simon Laplace a French mathematician and astronomer (1749-1827) who used the transform in his work on probability theory and developed the transform as a technique for solving complicated differential equation. Although the Laplace transform is often taught simply as a method of solving electrical circuit, differential equations, its use and influence is much wider than that in the field of electronics and communication. The use of Laplace transform has produced a literature and a tradition that is the foundation of transient analysis. The transform itself did not become popular until Oliver Heaviside a famous electrical engineer began using a variation of it to solve electrical circuit. Most of electrical engineering was invented by 1925, reduced to practice by 1935 and mathematically analyzed and scientifically understood by 1945.

The Laplace transform is denoted byL{f(t) }has it function f(t) with t (t> 0) that transforms it to a function f(s) with a real argument s. In order for any function of time f(t) to be Laplace transformable it must satisfy the following Dirichlet conditions;

f(t)Must be piecewise continuous which means that it must be single valued but can have a finite number of finite isolated discontinuities for t>0.

f(t)Must be of exponential order, which means f(t)must remain less than

∫▒e^(〖-a〗_0 t) Ast approaches ∞ where s is a positive constant and a_0is a real positive number.

If there is any function f(t)that satisfies the Dirichlet conditions then

F(s)=∫_0^∞▒〖f(t) e^(-st) dt〗

Written as L{f(t) } is called the Laplace transform of f(t)Here s can be either a real variable or a complex quantity. In analyzing a system, one usually recounts Time–Invariant, Linear Differential Equations of second or high orders. Generally, it is difficult to obtain solutions of these equations in closed form via the solution methods in ordinary differential equations. One way to circumvent this problem is to apply LAPLACE TRANSFORMATION. Laplace transforms converts a differential equation into an algebraic equation in terms of the transform function of the unknown quantity intended. Transform domain equivalent circuit are developed for representing the voltage current relationship of all circuit components the use of these equivalent circuit permits the application of basic algebraic circuit analysis schemes to be applied directly to complex circuit.

The Laplace transform is significant for a number of reasons.

First, it can be applied to a wider variety of inputs than other methods of analysis.

Second, it provides an easy way to solve circuit problems involving initial conditions, because it allows us to work with algebraic equations instead of differential equations.

Third, the Laplace transform is capable of providing us, in one single operation, the total response of the circuit comprising both the natural and forced response.

## The Laplace Transform

The Laplace transform of a function f(t)defined for all real numbers t≥0 is the function F(s), defined by:

F(s)=L{f(t) }=∫_(0^-)^∞▒e^(-st) (1.0)

Where:

F(s) Indicates the Laplace transform of the function f(t)on condition that;

f(t)=0,

t<0,

s= Complex variable known as Laplace variable given by s=σ+jω

L= Laplacian transform operator

Since the arguments of the exponent in (1.0) must be dimensionless, it follows thats has the dimensions of frequency and units of inverse seconds (s^(-1) ).

In (1.0), the lower limit of 0^-is short notation to means lim┬(ϵ→0)⁡- ϵ and assures the inclusion of the entire Dirac delta function δ(t)at 0 if there is such an impulse in

f (t) at 0.

The Laplace transform is an integral transformation of a function f(t) from the time domain into the complex frequency domain, giving F(s).

We assume in (1.0) that f(t) is ignored for t<0. To ensure that this is the case, a function is often multiplied by the unit step. Thus, f(t) is written as f(t)u(t)or f(t), t≥0

The Laplace transform in (1.0) is known as the one-sided or unilateral Laplace transform. The two-sided or bilateral Laplace transform is given by

F(s)=∫_(-∞)^∞▒〖f(t) e^(-st) dt〗 (1.1)

The one sided Laplace transform in (1.0), being adequate for purpose of this project, is the only type of Laplace transform that will be discussed in this monograph.

## Applications of Laplace Transform

“The Laplace transform has been applied to various problems: to evaluation of payments, to reliability and maintenance strategies, to utility function analysis, to the choice of investments, to assembly line and queuing system problems, to the theory of systems and elements behavior, to the investigation of the dispatching aspect of job/shop scheduling, for assessing econometric models, to study dynamical economic systems”. Grubbstr¨om and Yinzhong (1990)

The Laplace transform technique is applicable in many fields of science and technology such as:

* Control Engineering

* Communication

* Signal Analysis and Design

* System Analysis

* Solving Differential Equations

Properties and Theorem

Given the functions f(t) and g(t)and their respective Laplace transform F(s)and G(s):

f(t)=L^(-1) {F(s) }

g(t)= L^(-1) {G(s) }

The following is a list of properties of unilateral Laplace transform:

### Linearity

L{af(t)+bg(t)}=aF(s)+bG(s)

### Frequency differentiation

L{tf(t) }=〖-F〗^’ (s)

L{t^n f(t) }=〖(-1)〗^n F^((n) ) (s)

### Differentiation

L{F^’ }=sL{f}-f(0^-)

L{f”}=s^2 L{f}-sf^’ (0^- )-f^’ (0^-)

L{f^((n)) }=s^n L{f}-s^(n-1) f〖(0〗^-)-…-f^((n-1)) 〖(0〗^-)

### Frequency integration

L{f(t)/t}=∫_s^∞▒F(σ)dσ

### Integration

L{∫_0^t▒f(r)dr}=L{u(t)*f(t) }=1/s F(s)

### Scaling

L{f(at) }=1/a F(s/a)

### Initial value theorem

f(0^+ )=lim┬(s→∞)⁡sF(s)

### Final value theorem

f(∞)=lim┬(s→0)⁡sF(s)

All poles in left-hand plane, the final value theorem is useful because it gives the long term behavior without having to perform partial fraction decomposition or other difficult algebra. If a function poles are in the right hand plane (e.g. e^tor sin⁡(t)) the behavior of this formulae is undefined.

### Frequency shifting

L{e^at f(t) }=F(s-a)

L^(-1) {F(s-a) }=e^at f(t)

### Time shifting

L{f(t-a)u(t-a) }=e^(-as) F(s)

L^(-1) {e^(-as) F(s) }=f(t-a)u(t-a)

Note: u(t)is the Heaviside step function

### Convolution

L{f(t)*g(t) }=F(s)∙G(s)

Periodic function period T

L{f}=1/(1-e^(-Ts) ) ∫_0^T▒〖e^(-st) f(t)dt〗

### 1.5 Basic Terminology

There are a few key terms that need to be understood at the beginning of this project, before I can continue. This is only a partial list of all terms that will be used throughout this thesis, but these key words are important to know before I begin the main narrative of this text.

### Time domain

The time domain is described by graphs of power, voltage and current that depends upon time. The “Time domain” is simply another way of saying that the circuits change with time, and that the major variable used to describe the system is time. Another name is”Temporal”.

### Frequency domain

The frequency domain are graphs of power, voltage and/or current that depends upon frequency. Variable frequencies in wireless communication can represent changing channels or data on a channel. Another name is the “Fourier domain”. Other domains that an engineer might encounter are the “Laplace domain” (or the “s domain” or”complex frequency domain”), and the “Z domain”. When combined with the time, it is called a “Spectral” or “Waterfall”

### Circuit Response

Circuits generally have inputs and outputs. In fact, it is safe to say that a circuit isn’t useful if it doesn’t have one or the other (usually both). Circuit response is the relationship between the circuit’s input to the circuit’s output. The circuit response may be a measure of either current or voltage.

The final value, when all elements have a constant or periodic behavior, is known as the steady-state value of the circuit. The circuit response at steady state (when things aren’t changing) is also known as the “steady state response”. The steady state solution is called the particular solution.

### Transient Response

A transient response occurs when:

A circuit is turned on or off.

A sensor responds to the physical world changes

### Static electricity is discharged

An old car with old spark plugs (before resistors were put in spark plugs) drives by

Transient means momentary, or a short period of time. Transient means that the energy in a circuit suddenly changes which causes the energy storage elements to react. The circuit’s energy state is forced to change. When a car goes over a bump, it can fly apart, feel like a rock, or cushion the impact in a designed manner. The goal of most circuit design is to plan for transients, whether intended or not.

Transient solutions are determined using a homogeneous solution technique.

### Electric Circuit

An electric circuit is an interconnection of electrical elements. A simple electric circuit consists of three basic components; a battery, a lamp and connecting wires, such a simple circuit can exist by itself. It has several applications such as a torch light, a search light and so forth. Electrical circuit are used in numerous electrical systems to accomplish different task, however the major concern of this work is the analysis of circuit.

### Electric Charge

A charge is an electrical property of the atomic particles of which matter consists, measured in Coulombs.

Q=It (Coulombs) (1.2)

### Electric Current

This is the time rate of change of charges measured in Amperes (A). Mathematically the relationship between current i, charge q and time t is

i= dq/dt (1.3)

Where current is measured in Amperes (A) and

1Ampere = 1 Coulomb/second

The charge transferred between time t_0and t is obtain by integrating both sides of (1.4) we obtain

q=∫_(t_0)^t▒idt (1.4)

### Direct Current (DC)

A direct current is a current that remains constant with time. If the current does not change with time, but remains constant it is called direct current

### Alternating Current (AC)

An alternating current is a current that varies sinusoidally with time. A time-varying current is represented by the symbol i. A common form of time-varying current is the sinusoidal current or alternating current (AC).

### Voltage

As explained briefly on charges, to move the electron in a conductor in a particular direction requires some work or energy transfer. This work is performed by an external electromotive force (E.M.F) typically represented by a battery. This E.M.F is also known as voltage or potential difference.

The voltage V_ab between two point a and b in an electric circuit is the energy (or work) needed to move a unit charge from a to b, mathematically

V_ab=dw/dq (1.5)

Where;W is energy in Joules (J) and q is charge in Coulombs (C). The voltage V_abor simply V is measured in Volts (V) thus,

Voltage (or potential difference) is the energy required to move a unit charge through an element. Measured in Volts (V)

### Ohm’s Law

Ohm’s law deals with the relationship between voltage and current in an ideal conductor, this relationship states that the potential difference (voltage) across an ideal conductor proportional to the current through it.

The constant of proportionality is called resistance R Ohm’s law is given by

V=IR (1.6)

### Resistors

The relationship between resistors, voltage and current is seen in Ohm’s law above. A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. Resistors act to reduce current flow and at the same time act to lower voltage level within the circuit.

## Restricts electron flows

### Capacitors

A capacitor stores electric charge. The capacitance of a capacitor is measured in Farads and this is proportional to the ratio of the charge stored and voltage with the charge stored Q being a relationship of current and time.

C=Q⁄V (1.7)

But,

Q=It Thus

C=It⁄V (1.8)

### Inductors

An inductor also called a coil or reactor is similar to capacitors, mathematically however they are virtually the opposite. An inductor is a passive electronic component that stores energy in the form of a magnetic field.

## References

Abramovitz M. and Stegun (1972), Laplace Transforms, New York Dover.

Alexander K.C. and Sadiku N.O. (2010);Fundamentals of electric circuit, McGraw-Hill Higher Education, Columbus, Ohia.

Dass H.K. (2009), Advanced Engineering mathematics, S.chand and Company ltd, New Delhi.

Dyke P. (2014), An Introduction to Laplace Transforms and Fourier Series, Springer-Verlag, London.

Polking J., Boggess A. and Arnold D. (2006), Differential Equations with Boundary Value Problems, Pearson Prentice Hall, New Jersey.

Siebert McC.W. (1986), circuits, signals and systems, MIT press, Cambridge, Massachusetts.

Stroud K.A. (2011), AdvancedEngineeringMathematics, Palgrave Macmillan, Hampshire.