The Laplace Transform and the Frequency Domain

Circuits having a wide variety of inputs and responses are modeled by differential equations. Their responses are described by the solutions to these differential equations. Since differential equations can be mathematically difficult to work with, the Laplace transform can be used to convert them into more easily manipulated algebraic equations. We have previously discussed the Laplace transform in the Math/Physics and Differential Equations sections of this website. Practical applications of the transform have been shown in some of the example problems involving DC RLC circuit analysis. While some of the information in the following pages will be a review of that material, we will focus our efforts here on the use of the Laplace transform in AC circuit and frequency response analysis.

Advantages of using the Laplace transform:

1. Can be applied to a wider variety of inputs than phasor analysis.
2. Works well with circuit problems that have initial conditions.
3. The total response of the circuit (natural and forced responses) can be obtained in a single operation.

Definition of the Laplace transform

The Laplace transform of a function is defined as the following: $$\mathcal{L}[f(t)] = F(s) = \int_{0^-}^{\infty} f(t)e^{-st} dt \qquad,(eqn\;1)$$

Note that "s" is a complex quantity: $$s = A + jB$$ Since the "st" term inside the integral must be dimensionless, "s" must be in units of inverse seconds. $$s^{-1} = \frac{1}{s} = frequency$$ Also, notice that: $$lower\;limit = 0^{-1}$$ This signifies a time just below t=0 and allows us to see a response at t=0 as well as any discontinuity of f(t) at that point in time.

You may notice that standard use of the Laplace transform requires an understanding of Improper Integrals. In order for a function to have a Laplace transform, the integral must converge to a finite value (thus implying the use of limits with improper integrals). Since all of our applications of the Laplace transform to circuit analysis involve functions that converge, we will not deal with limits in our examples.

Furthermore, determining Laplace transforms by evaluating complicated integrals can be extremely time consuming, laborious and error prone. For this reason we will not bother doing so in our example problems. Instead, we will refer to a table of common Laplace transforms such as the one found here

In the next page we will look at a simple problem involving the Laplace transform.

Continue on to Laplace transform example problem...