16.7 Summary

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16.7 Summary

• 1. The Laplace transform can be used to analyze a circuit. We convert each element from the time domain to the s-domain, solve the problem using an y circuit analysis technique, and con vert the result to the time domain using the inverse transform.
• 2. In the s-domain, the circuit elements are replaced with the initial condition at t = 0 as follo ws. (Please note, v oltage models are

given below, but the corresponding current models w ork equally well.):

Resistor:

$$v_R = Ri \rightarrow V_R = RI$$

\nInductor: $v_L = L\frac{di}{dt} \rightarrow V_L = sLI - Li(0^-)$
\nCapacitor: $v_C = \int i \, dt \rightarrow V_C = \frac{1}{sC} - \frac{v(0^-)}{s}$

• 3. Using the Laplace transform to analyze a circuit results in a com plete (both transient and steady state) response because the initial conditions are incorporated in the transformation process.
• 4. The transfer function H(s) of a network is the Laplace transform of the impulse response h(t).
• 5. In the s-domain, the transfer function H(s) relates the output response Y(s) and an input excitation X(s); that is, H(s) = Y(s)∕X(s).
• 6. The state v ariable model is a useful tool for analyzing comple x systems with several inputs and outputs. State variable analysis is a powerful technique that is most popularly used in circuit theory and control. The state of a system is the smallest set of quanti ties (known as state v ariables) that we must kno w to determine its future response to an y given input. The state equation in state variable form is

$$\dot{\mathbf{x}} = \mathbf{A}x + \mathbf{B}z$$

while the output equation is

$$\mathbf{y} = \mathbf{C}x + \mathbf{D}z$$

• 7. For an electric circuit, we first select capacitor voltages and inductor current as state v ariables. We then apply KCL and KVL to obtain the state equations.
• 8. Two other areas of applications of the Laplace transform co vered in this chapter are circuit stability and synthesis. A circuit is stable when all the poles of its transfer function lie in the left half of the s plane. Network synthesis is the process of obtaining an appropriate network to represent a given transfer function for which analysis in the s-domain is well suited.