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In the following video (not necessary to watch)...

https://www.youtube.com/watch?v=K_vWYEjgVRg&list=PLdnqjKaksr8pXF2SpDyyD7ouAVlz96_Ra&index=23

...the creator takes the projection of the phasor onto the $y$ axis to be the instantaneous value of the signal. I've seen a couple other lecture-notes online where the writers do the same thing.

However, I always thought that since the $y$ axis was the imaginary axis, and the $x$ axis was the real-axis, we should consider the projection of the phasor onto the $x$ axis as the instantaneous value of the signal.

Indeed, in the top answer to this question...

What exactly is a phasor?

...the author considers the projection of the phasor onto the $x$ axis.

Is there a convention for which axis represents the actual signal?

Thanks!

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There is really very little standardization of this kind of thing. Various textbooks have various conventions about all kinds of things related to phasors and the complex number representation of sine waves. Some say that phasors rotate, others consider them as static objects. Some say that phasors are complex numbers, others avoid mentioning complex numbers (with the implication that students would later drop the training wheels of the phasor representation and switch to complex numbers, which is what engineers and scientists actually use). And finally, various people use various phase and sign conventions, which are related to the phase conventions used in Fourier analysis.

To avoid confusion and complicated math, I think it's helpful to think of phasors as lying in the complex plane (so we can switch easily between the two languages), and to think of the conventions as conventions for mapping sinusoidal functions to points in this plane. Ignoring a possible real, positive scaling factor (which some people would take to be 1 and others $1/\sqrt{2\pi}$), there are three conventions that I've encountered:

(A) $\sin\rightarrow i$, $\cos\rightarrow 1$

(B) $\sin\rightarrow -i$, $\cos\rightarrow 1$

(C) $\sin\rightarrow 1$, $\cos\rightarrow i$

Any of these conventions can be related to any of the others by a 90-degree rotation and/or flip of the complex plane across one axis.

The advantage of B and C is that they are consistent with the conventions used by scientists and engineers in defining complex impedances. The advantage of C over B is that there is one less negative sign to remember. The video is using C. I teach the system using C myself. The main disadvantage of C would be that it's not consistent with the ways that people usually define the Fourier transform.

Some similar info on math.SE, describing A and B: https://math.stackexchange.com/a/2306802/13618

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A phasor is a rotating vector. If, at time t it makes angle $(\omega t + \phi)$ with the x axis, its projection (for a phasor of unit magnitude) on to x axis is, by definition, $\cos(\omega t + \phi)$, and its projection on to y axis is $\sin(\omega t + \phi)$. Which of these we choose to represent a sinusoidal displacement, voltage or whatever is pretty arbitrary. Whether either of them is a sine or a cosine depends only on the time we choose to call $t=0$, or, equivalently, on our choice of the phase constant, $\phi$.

Phasors come into their own when we wish to add sinusoids of the same frequency but not necessarily the same phase or amplitude. The relative orientations of the phasors stay constant, allowing easy vector addition. We can then take the x or y projection of the resultant vector, if we want the value of the resultant sinusoid at time t.

"However, I always thought that since the 𝑦 axis was the imaginary axis, and the 𝑥 axis was the real-axis [...]"

Phasors, as usually understood, do not require the complex plane. The $y$ and $x$ projections that I talked about earlier are both real numbers. Using complex numbers to represent oscillating quantities does very much the same job as using phasors, and is even more convenient, but it is, strictly speaking, a different technique. With complex numbers the analogue to choosing x or y axes is choosing to represent quantities by the real or imaginary parts of complex numbers. It is usual to choose the real parts.

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