Learning Goal:

To use Laplace transform concepts to transform any circuit from the time domain to the s$-$domain.

We develop a general technique for analyzing the complete response of any linear circuit. $$Performing analysis in the time$-$domain, even for simple circuits containing inductors and capacitors, results in differential equations whose solutions are difficult to calculate. As often happens in engineering problems, we can transform the problem from the time domain to a different mathematical "space", in this case the s$-$domain based on the Laplace transform. In the s$-$domain, Ohm$$s law can be used to relate the s$-$domain voltage and current for resistors, inductors, and capacitors to the complex impedance of these elements. $$KVL and KCL also hold in the s$-$domain, so all of our circuit analysis tools can be used to construct a set of equations in the s$-$domain for the circuit. Since these are algebraic, not differential, equations, they are relatively easy to solve in the s$-$domain. The s$-$domain solutions can be inverse Laplace$-$transformed to find the complete response of both voltages and currents in the time domain.We begin by examining the relationship between the Laplace transform of the voltage and the Laplace transform of the current for resistors, inductors, and capacitors. We will take into account the possibility that the inductors and capacitors may have initial stored energy. $$What we discover will lead us to a general technique for transforming any circuit from the time domain to the s$-$domain, where the describing equations are algebraic and thus relatively easy to solve. $$