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I've always thought of just these 3 things that a Phasor can describe -- the magnitude and phase of the sine function of these 3, vertically centered at 0 and of the same frequency.

Are there more?

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What can you do with phasors?

A phasor is a compact method of writing the important parts of something that varies sinusodially over time. In a phasor you have the information of magnitude and phase, but you omit the information of the actual frequency. Therefore phasor calculations assume that all phasors vary with the same frequency. This is why you only use one frequency variable (\$\omega\$ or \$f\$) in calculations.

So you can describe anything which varies over time in a sinusodial manner with a phasor, as long as you restrict the system to a constant frequency which is equal for all things described with the phasor.

This is not as restricting as it sounds first hand, because from Fourier transformation we know that we can assemble most waveforms by summing up sinusodial waveforms. This is what makes phasor calculations handy: You can apply a transformation to (nearly) any input signal to know of which sinusodials it is composed. Then calculate with phasor calculations the system behaviour at these frequencies and sum up the results. You get the response/output of your system to this input signal. To get the output signal in time domain, you need to inverse transform it.

A graphical help for this are frequency response plots like the bode plot: You see pretty quickly how the system reacts to different frequencies.

What can you NOT describe with phasors?

Anything, where changing the frequency does more than change magnitude and phase. Particularly, this happens if your systems dimensions come near the wavelength of your time-varying signal (then the magnitude depends on the location within the system and the location depends on wave speed and frequency). It also happens if you have dispersion in your system, i.e. the speed at which your signal travels through space depends on the frequency (then you can not add up the magnitudes of all signal at any location because they do not arrive there at the same time). And it also happens if you have nonlinear elements (diodes, transistors,...) in your system - they change the waveform by their nonlinearity and thereby deviate from the sinusodial prerequisite. You need to solve such systems with differential equations in the time domain - not in the frequency domain where phasors are used.

Something that can be solved with phasors, but not right away, is, when you have multiple sources with different waveforms and/or frequencies in your system. In this case you need to calculate the whole input-output-relation separately for each source and sum up the different frequency responses in the end.

So which things can you describe with a phasor?

Everything which can be described by a sinusiodial waveform (this includes cosine) or a sum of these. It is neither restricted to a specific physical domain (electrics, mechanics, acoustics,...), not even to physical quantities at all. You could also describe stock exchange stuff or image processing with phasors if you find a suitable system representation. Phasors are in general a method to solve differential equations in the frequency domain rather than in the time domain - if the differential equations describing the system meet the discussed prerequisites, you can use phasors to solve the equations.

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