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# Unexpected phase shift in results

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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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There is a mistake in the last line: \eqalign{V_{C_1}(t) &= V_1(t) - RI_{C_1}(t) \cr &= \sin(t) - \dfrac{\sin(t) + \cos(t) - \mathrm{e}^{-t}}{2} \cr &= \dfrac{\sin(t) - \cos(t) + \mathrm{e}^{-t}}{2} \cr &= \dfrac{\cos(t-90^\circ) - [-\sin(t-90^\circ)] + \mathrm{e}^{-t}}{2}\cr &= \dfrac{\cos(t-90^\circ) + \sin(t-90^\circ) + \mathrm{e}^{-t}}{2}\cr &= I_{C_1}(t - 90^\circ)}, where the last equality is to be understood as true at the limit $t\to +\infty$, that is, it becomes more and more true as $t$ becomes large, as the "charging" term $e^{-t}$ vanishes.

I answer here to the comments of @Carloc, as this may be of some help for other persons.

You wrote that Vc1=Ic1, this is wrong, volts cannot be amperes.

Indeed, volts are not Ampere. But in physics (and electricity is part of physics), the intermediate computations are very often dimensionless, as you could seen in physics books.

In fact, this method is even used everywhere in theoretical physics to simplify the appearance of the computations: I quote here Wikipedia at Maxwell equations, section "Formulation in Gaussian units convention" (please, read the whole section there):

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of ε0 and μ0 into the units of calculation, by convention. [...] The equations are particularly readable when length and time are measured in compatible units like seconds and lightseconds i.e. in units such that c = 1 unit of length/unit of time. [...]

There are many many other examples, but again, that's not the point here. The point is that writting dimensionless equations is the way problems are often solved and presented everywhere in the Academy.

You wrote cos(t-90°), this is wrong because the argument of trigonometric functions must be radiants, [continued below]

By definition $1^\circ = \pi/180$ rad, so $90^\circ = \pi/2$ rad. Notice that both radians and degrees are not units, but only a convention. Angles are essentially dimensionless in physics.

and t is seconds instead. The same (t-90°) is again wrong because you cannot add seconds and degrees.

Again, you are apparently unaware of the customs in electrical physics and electromagnetism, where the pulsation $\omega$ is normalized to 1 rad/s, and droped from the computations.

These points are mathematical truth, not my opinions.

Isolated from the context of the OP, these points are indeed true. But they turned to be pointless and even a burden in the context of theoretical problems, as they are solved in the academy, especially after the OP has noted the normalized values of the resistor, capacitors and frequencies in his schematic.

I would like also to address the answer of Olin. While there is nothing wrong in what he said (in fact, I'm pretty certain the OP knew the material there very well), this does not really apply here, as the voltage at the terminals of the cap is not sinusoidal. What the OP desired was a correct mathematical derivation, and to understand why his derivation disagreed with the simulation.

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