Lapalce transform is a generalization of the Fourier transform in the sense that it allows “complex frequency” whereas Fourier analysis can on ly handle “real frequency”. Like Fourier transform, Lapalce transform allows us to analyze a “linear circuit” problem, no matter how complicated the circuit is, in the frequency domain in stead of in he time domain.
Mathematically, it produces the benefit of converting a set of differential equations into a corresponding set of algebraic equations, which are much easier to solve. Physically, it produces more insight of the circuit and allows us to know the bandwidth, phase, and transfer characteristics important for circuit analysis and design.
Most importantly, Laplace transform lifts the limit of Fourier analysis to allow us to find both the steady-state and “transient” responses of a linear circuit. Using Fourier transform, one can only deal with he steady state behavior (i.e. circuit response under indefinite sinusoidal excitation).
Using Laplace transform, one can find the response under any types of excitation (e.g. switching on and off at any given time(s), sinusoidal, impulse, square wave excitations, etc.
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