# The Complex Forcing Function

1. Oct 17, 2015

### icesalmon

In class we've been talking about circuits with AC sources of the form v(t) = Vmcos(ωt+θ) which produces a current i(t) = Imcos(ωt+Φ). They go on to talk about shifting their time reference by re-writing the function for voltage, Vmcos(ωt + θ - 90°) = Vmsin(ωt+θ) the current Imcos(ωt+Φ-90°) = Imsin(ωt+Φ) for some reason of which I can only speculate. Next multiplication of the forcing function vm(t) by j somehow transforms what was once Vmcos(ωt+θ) into jVmsin(ωt + θ). And then use the superposition theorem to produce the total response for the complex forcing function to be v(t) = Vmcos(ωt+θ) + jVmsin(ωt+θ).
My confusion deals with their "shifting of the reference time" and how they transformed the forcing function into a sinusoid. Do they re-write cosines as sines just to use eulers identity later on?

2. Oct 17, 2015

Got it now