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14.1: Introduction to Laplace Transforms

If $$y(x)$$ is a function of $$x$$, where $$x$$ lies in the range $$0$$ to $$\infty$$, then the function $$\bar{y}(p)$$ defined by

$\bar{y}(p) = \int_0^{\infty} e^{-px} y(x) \,dx \label{14.1.1}$

is called the Laplace transform of $$y(x)$$. However, in this chapter, where we shall be applying Laplace transforms to electrical circuits, $$y$$ will most often be a voltage or current that is varying with time rather than with "x". Thus I shall use $$t$$ as our variable rather than $$x$$, and I shall use $$s$$ rather than $$p$$ (although it will be noted that, as yet, I have given no particular physical meaning to either $$p$$ or to $$s$$.) Thus I shall define the Laplace transform with the notation

$\bar{y}(s)=\int_0^{\infty}e^{-st}y(t)dt,$

it being understood that t lies in the range $$0$$ to $$\infty$$.

For short, I could write this as

$\bar{y}(s)=\textbf{L}y(t).$

When we first learned differential calculus, we soon learned that there were just a few functions whose derivatives it was worth committing to memory. Thus we learned the derivatives of $$x^n, \sin x, \ e^x$$ and a very few more. We found that we could readily find the derivatives of more complicated functions by means of a few simple rules, such as how to differentiate a product of two functions, or a function of a function, and so on. Likewise, we have to know only a very few basic Laplace transforms; there are a few simple rules that will enable us to calculate more complicated ones.

After we had learned differential calculus, we came across integral calculus. This was the inverse process from differentiation. We had to ask: What function would we have had to differentiate in order to arrive at this function? It was as though we were given the answer to a problem, and had to deduce what the question was. It will be a similar situation with Laplace transforms. We shall often be given a function $$\bar{y}(s)$$ and we shall want to know: what function $$y(t)$$ is this the Laplace transform of? In other words, we shall need to know the inverse Laplace transform:

$y(t)= \textbf{L}^{-1} \bar{y}(s) \label{14.1.4}$

We shall find that facility in calculating Laplace transforms and their inverses leads to very quick ways of solving some types of differential equations – in particular the types of differential equations that arise in electrical theory. We can use Laplace transforms to see the relations between varying current and voltages in circuits containing resistance, capacitance and inductance. However, these methods are quick and convenient only if we are in constant daily practice in dealing with Laplace transforms with easy familiarity. Few of us, unfortunately, have the luxury of calculating Laplace transforms and their inverses on a daily basis, and they lose many of their advantages if we have to refresh our memories and regain our skills every time we may want to use them. It may therefore be asked: Since we already know perfectly well how to do AC calculations using complex numbers, is there any point in learning what just amounts to another way of doing the same thing? There is an answer to that. The theory of AC circuits that we developed in Chapter 13 using complex numbers to find the relations between current and voltages dealt primarily with steady state conditions, in which voltages and current were varying sinusoidally. It did not deal with the transient effects that might happen in the first few moments after we switch on an electrical circuit, or situations where the time variations are not sinusoidal. The Laplace transform approach will deal equally well with steady state, sinusoidal, non-sinusoidal and transient situations.