Why a phasor consists of the magnitude of a voltage multiplied

[jeff1evesque]

Does anyone know why a phasor consists of the magnitude of a voltage multiplied with a sinusoid? For instance,

$$V_Scos(\omega t + \theta)$$

I am searching all over the web, and cannot find a general explanation why. I've found some incomplete mathematical proof showing the following:
$$Ae^{j \theta} = V = V_Scos(\omega t + \theta)$$
but this doesn't explain the general concept.


Thanks so much,

JL

 

[Bob S]

If you put 120 volts AC 60 Hz across a 1 milliFarad (1000 uF) capacitor, how do you write the equation for the current in it? If there is both a voltage across and a current through the capacitor, can you explain why the capacitor does not get hot? If the voltage is V(t) = 120 sin(wt), is the current of the form I = I₀ sin(wt)?

 

[es1]

Does anyone know why a phasor consists of the magnitude of a voltage multiplied with a sinusoid?

This kind of thing used to trip me up all the time before I stopped fighting it.

The reason phasors are defined that way is because it is useful to do so. Many hard mathematical operations are greatly simplified by using that definition. Then once people saw the utility of the definition it stuck.

There is nothing intrinsic in the sinusoid functions that says phasors must be defined that way. In fact, other definitions are possible and could even be more useful depending on the problem at hand. But if you want to solve problems involving sinusoidal signals going through reactive components it's going to be tough to beat the mathematical efficiency of the operations using the phasor definition.

Personally, I found searching for the history behind the definition to be very helpful. Google and wikipedia can be very useful for this. When you see, some guys wanted to solve problem X so he did Y, then it seems much less mystical and more approachable (for me anyway).

P.S.
I think your definition is misleading (but not necessarily incorrect due to the word 'sinusoid') as you did not mention phase which is a key concept of phasors. Personal taste I guess.
http://en.wikipedia.org/wiki/Phasor_(sine_waves )

 

[souravr]

Hi!
friend, from the concept of e.m.f. generation i can say you that if a conductor is rotated in a magnetic field, then emf induces in the conductor due to rate of change of flux with respect to time. If the conductor is rotated in that magnetic field for 360 degrees in space, then it's described circular path can be analyzed into a simple harmonic motion (irrespective whether it is a sine or a cosine wave); Hence you will get a magnitude and as well a sinusoid.

 

[chaoseverlasting]

A phasor is basically a tool for us to use when dealing with similar frequencies. It makes it easier for us to analyze alternating signals.

We use the 'sinusoid' because all alternating signals can be expanded as a set of alternating sinusoidal signals of different frequencies (check out the theory of Fourier Transforms).

Again, the eulers form you've put up is an easy way for us to represent sinusoids and can more easily lead to results than if we were dealing with sinusoids only (I mean the complex representation of the sine wave and the complex mathematics involved).