Electric Circuits and Networks

15.7 SOME THEOREMS ON LAPLACE TRANSFORMS

The property of linearity of Laplace transforms was already noted and made use of in earlier sections. We look at other interesting properties of Laplace transform in this section.

15.7.1 Time-shifting Theorem

If v(t) = f(t) u(t) has a Laplace transform V(s) then v_d(t) = v(t – t_d) = f(t – t_d) u(t – t_d) has a Laplace transform V_d(s) = V(s)e^–st_d.

The shifting operation implied in this theorem is illustrated in Fig. 15.7-1. Note that there is a difference between f(t – t_d) u(t) and f(t – t_d) u(t) u(t – t_d). Time-shifting theorem for unilateral Laplace transform works properly for f(t – t_d) u(t – t_d) but not for f(t – t_d) u(t).

 

 

Fig. 15.7-1 Illustrating the Time-Shift Operation Envisaged in Shifting ...