Comments on Unexpected phase shift in results
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Unexpected phase shift in results
I find the current flowing through the capacitor
$$\begin{align} I_{C_1}(t)&=\dfrac{\mathrm{d}}{\mathrm{d}t}\left[V_1(t)-I_{C_1}(t)R_1\right] \\ {}&= \dfrac{\mathrm{d}}{\mathrm{d}t}\left[\sin(t)-I_{C_1}(t)\right] \end{align}$$
and by solving this differential equation we get
$$I_{C_1}(t) = \dfrac{\sin(t)+\cos(t)-\mathrm{e}^{-t}}{2}$$
To find the voltage of the capacitor we use Ohm's law:
$$V_{C_1}(t) = V_1(t)-I_{C_1}(t)R_1 = \sin(t)-\cos(t)+\mathrm{e}^{-\frac{t}{2}}$$
But when I plot them on Desmos I get a phase shift of 90 degrees between voltage of the capacitor and current through the capacitor which doesnt make sense it should be 45 degrees what am I doing wrong?
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I get a phase shift of 90 degrees between voltage of the capacitor and current through the capacitor which doesn't make sense it should be 45 degrees
You don't need a whole circuit to see that the phase shift should be 90°. You can see that from a capacitor in isolation.
The current thru a capacitor is proportional to the derivative of the voltage across it. If the voltage on a cap is a sine, then the current is a cosine, which has 90° leading phase relative to the voltage. Since this is what a capacitor inherently does, it doesn't matter what the rest of the circuit is trying to do. The above will always be true (for an ideal capacitor).
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