Chapter 1 says v(t) is an independant voltage source that maintains the same voltage across its terminals regardless of the current through it and i(t) is an independant current source that maintains the same current output regardless of the voltage across it. Are these the definitions you were thinking of in the first post?
Not exactly. We can consider v(t) to be an AC voltage source across a load; then i(t) would be the resulting AC load current.
But, by using lower case and "(t)" we indicate that both are functions of time, that is,
v(t) = Vm[sin(wt)]
and
i(t) = Im[sin(wt + phi)] where phi is the lead or lag angle.
The evaluation of these functions at any instant of time yields a real number--that is all. A plot of v and i vs wt yields two sine waves displaced by the angle phi. But, the voltages and currents so plotted are REAL numbers--no phase angle.
Clearly, the ratio of v and i is not constant unless phi = 0.
The point is that the idea of impedance as a function of time is nonsense.
