16.1 Introduction

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16.1 Introduction

Now that we have introduced the Laplace transform, let us see what we can do with it. Keep in mind that with the Laplace transform we actually have one of the most powerful mathematical tools for analysis, synthesis, and design. Being able to look at circuits and systems in the s-domain can help us to understand ho w our circuits and systems really function. In this chapter we will tak e an in-depth look at ho w easy it is to w ork with circuits in the s-domain. In addition, we will briefly look at physical systems. We are sure you have studied some mechanical systems and may have used the same differential equations to describe them as we use to describe our electric circuits. Actually that is a wonderful thing about the physical universe in which we li ve; the same dif ferential equations can be used to describe any linear circuit, system, or process. The key is the term linear.

A system is a mathematical model of a physical process relating the input to the output.

It is entirely appropriate to consider circuits as systems. Historically, circuits have been discussed as a separate topic from systems, so we will actually talk about circuits and systems in this chapter realizing that circuits are nothing more than a class of electrical systems.

The most important thing to remember is that everything we discussed in the last chapter and in this chapter applies to any linear system. In the last chapter, we saw how we can use Laplace transforms to solv e linear differential equations and integral equations. In this chapter, we introduce the concept of modeling circuits in the s-domain. We can use that principle to help us solv e just about an y kind of linear circuit. W e will take a quick look at how state variables can be used to analyze systems with multiple inputs and multiple outputs. Finally, we examine how the Laplace transform is used in netw ork stability analysis and in netw ork synthesis.

16.2 Circuit Element Models

Having mastered how to obtain the Laplace transform and its inverse, we are now prepared to employ the Laplace transform to analyze circuits. This usually involves three steps.

Steps in Applying the Laplace Transform:

• 1. Transform the circuit from the time domain to the s-domain.
• 2. Solve the circuit using nodal analysis, mesh analysis, source transformation, superposition, or any circuit analysis technique with which we are familiar.
• 3. Take the inverse transform of the solution and thus obtain the solution in the time domain.

Only the first step is new and will be discussed here. As we did in phasor analysis, we transform a circuit in the time domain to the frequenc y or s-domain by Laplace transforming each term in the circuit.

For a resistor, the voltage-current relationship in the time domain is

$$v(t) = Ri(t) \tag{16.1}$$

Taking the Laplace transform, we get

$$V(s) = RI(s) \tag{16.2}$$

For an inductor,

$$v(t) = L \frac{di(t)}{dt}$$

(16.3)

Taking the Laplace transform of both sides gives

$$V(s) = L[sI(s) - i(0^{-})] = sLI(s) - Li(0^{-})$$

\n(16.4)

or

$$I(s) = \frac{1}{sL} V(s) + \frac{i(0^{-})}{s}$$

(16.5)

The s-domain equivalents are shown in Fig. 16.1, where the initial condition is modeled as a voltage or current source.

For a capacitor,

$$i(t) = C \frac{dv(t)}{dt}$$

(16.6)

which transforms into the s-domain as

$$I(s) = C[sV(s) - v(0^{-})] = sCV(s) - Cv(0^{-})$$

\n(16.7)

$$\sum_{i=1}^{n} x_i$$ $$V(s) = \frac{1}{sC} I(s) + \frac{v(0^{-})}{s}$$

(16.8)