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I am studying a pair of AC signals for voltage and current. In a long run, I found a distortion happening as shown in the attached picture. I want to understand what could possibly have happened in the signal from a mathematical point of view.
The base signal is $y = A\cos(100\pi)$
What kind of other signal added to the base signal can produce the final signal as shown in the picture?
Mathematically, we have this relationship:
$$\cos(2\pi f_1 t) + 2\sin^2(\pi f_1 t) = 1$$
I don't suspect the OP is actually observing additive noise (although any deviation from the expected signal can be referred to as a noise). It's also not clear from the plot of there is phase continuity between the two regions (so there may also be a "noise" of a phase jump as well). It appears that either whatever is generating the sinusoidal source has paused momentarily, or whatever is measuring the source has paused and held it's value. We also see a small linearly increasing ramp such as we would see with a constant current through a fixed capacitance.
The above relationship was derived from the following trigonometric identities:
$$\sin^2(x)+\cos^2(x) = 1 \label{1}\tag{1}$$
$$\cos(x)\cos(y) = \frac{1}{2}(\cos(x+y) - \cos(x-y))\label{2}\tag{2}$$
From \ref{2}:
$$\cos^2(x) = \frac{1}{2}(\cos(2x) + \cos(0))$$
$$2\cos^2(x) = \cos(2x)+1$$
$$\cos(2x) = 2\cos^2(x) - 1\label{3}\tag{3}$$
If we add $2\sin^2(x)$ to \ref{3}:
$$\cos(2x) + 2\sin^2(x) = 2\cos^2(x) - 1 + 2\sin^2(x)$$
$$ = 2(\sin^2(x)+\cos^2(x))-1 = 2-1 = 1$$
Thus we have the relationship first introduced:
$$\cos(2x)+2\sin^2(x) = 1$$
