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?

AC signal distortion


1 Answer 1


Mathematically, we have this relationship:

$$\cos(2\pi f_1 t) + 2\sin^2(\pi f_1 t) = 1$$

Sum of cosine and squared sine

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:

$$\cos(2x)+2\sin^2(x) = 1$$

  • $\begingroup$ Many thanks Dan! I am not sure it's noise or something else. I was analyzing an eletricity meter's random measure error which was reported earlier by testers. Then I supply a meter with a standard high-precision AC source and receiving the meter's real-time wafeform streaming data. Just inside the data, I found this distortion. I think the phase continuity was hold because if there was no the distortion and let the left part waves continue to grow, it will exactly match the right part waves in the picture. $\endgroup$
    – Woody Wu
    Aug 13, 2023 at 12:55
  • 1
    $\begingroup$ It could be in your meter (measurement) as a skip in the reading. If this happens repeatably, you could rule that out with an independent concurrentt measurement (such as with an oscilloscope) $\endgroup$ Aug 13, 2023 at 12:57
  • $\begingroup$ From your math, did it mean, the inserted signal could be something like 2*sin^2(100*pi)? I cannot say this is a high frequency noise, right? Because that would be a greater coefficent beforee the pi, not just the same sinusoildal sqared. So, how could you can this 2*sin^2(100*pi)? In your experience, is this possible a noise? If yes, what kind of noise? Thanks! $\endgroup$
    – Woody Wu
    Aug 13, 2023 at 12:58
  • $\begingroup$ If we normalize your waveform to be $\cos(100\pi)$ then I am saying you can add a synchronous "noise" to it that is $2\sin^2(50\pi)$ which would then cause the output to sum to 1. In my experience the source or the measurement would be held constant for some reason and this would not be any independent external "noise" or anything I would refer to as "noise", but to answer the question as asked, I provided the math that would cause this. Did I answer to your satisfaction? $\endgroup$ Aug 13, 2023 at 13:06
  • $\begingroup$ Sure Dan. Your answer helped a lot! I think it's really caused by 'skip readings' as you guessed. Not like normal sense noises. Thank you very much! $\endgroup$
    – Woody Wu
    Aug 13, 2023 at 13:16

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