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etotheipi

If a voltage source is sinusoidal, then we can introduce a phasor map and come up with equations like$$V_0 e^{i \omega t} = I(R + i\omega L + \frac{1}{\omega C} i)$$where ##I## would also differ from ##V## by a complex phase.

But if you set ##\omega = 0##, which would appear to correspond to the case where the source voltage is constant, you get nonsense results. If there is a capacitor then you get an infinity, and the effects of any inductor disappears completely (as if it were no longer in the circuit) because ##i\omega L = 0##.

I wondered if anyone could explain what's going on here. Is this approach only valid for a sinusoidal source? Or if not, then how do we interpret these weird results? If we were to solve the above example with ##\omega = 0## in the normal way, we'd be dealing with the constant coefficients differential equation ##\ddot{I} + \frac{R}{L}\dot{I} + \frac{1}{LC}I = 0##.

But if you set ##\omega = 0##, which would appear to correspond to the case where the source voltage is constant, you get nonsense results. If there is a capacitor then you get an infinity, and the effects of any inductor disappears completely (as if it were no longer in the circuit) because ##i\omega L = 0##.

I wondered if anyone could explain what's going on here. Is this approach only valid for a sinusoidal source? Or if not, then how do we interpret these weird results? If we were to solve the above example with ##\omega = 0## in the normal way, we'd be dealing with the constant coefficients differential equation ##\ddot{I} + \frac{R}{L}\dot{I} + \frac{1}{LC}I = 0##.

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