In class we've been talking about circuits with AC sources of the form v(t) = V(adsbygoogle = window.adsbygoogle || []).push({}); _{m}cos(ωt+θ) which produces a current i(t) = I_{m}cos(ωt+Φ). They go on to talk about shifting their time reference by re-writing the function for voltage, V_{m}cos(ωt + θ - 90°) = V_{m}sin(ωt+θ) the current I_{m}cos(ωt+Φ-90°) = I_{m}sin(ωt+Φ) for some reason of which I can only speculate. Next multiplication of the forcing function v_{m}(t) by j somehow transforms what was once V_{m}cos(ωt+θ) into jV_{m}sin(ωt + θ). And then use the superposition theorem to produce the total response for the complex forcing function to be v(t) = V_{m}cos(ωt+θ) + jV_{m}sin(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?

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# The Complex Forcing Function

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