What Are Phasors and How Do They Work in Wave Analysis?

[tony873004]

My textbook's explanation doesn't do it for me, so I have to ask here.

According to the textbook: "A phasor is an abstract vector whose y-component represents the value of a wave disturbance. The vector's magnitude equals the amplitude of the wave, and the angle it makes with the y-axis, measured counterclockwies, is the wave phase."

What I don't get is that the amplitude of the wave is fixed. So according to this definition, the magnitude (length) of the phasor should change as the angle changes, and its length should be 0 when the angle with the y-axis is 0. But the animation on this page shows a constant length for the phasor:

http://www.physics.udel.edu/~watson/phys208/phasor-slow.html

 

[LowlyPion]

My textbook's explanation doesn't do it for me, so I have to ask here.

According to the textbook: "A phasor is an abstract vector whose y-component represents the value of a wave disturbance. The vector's magnitude equals the amplitude of the wave, and the angle it makes with the y-axis, measured counterclockwies, is the wave phase."

What I don't get is that the amplitude of the wave is fixed. So according to this definition, the magnitude (length) of the phasor should change as the angle changes, and its length should be 0 when the angle with the y-axis is 0. But the animation on this page shows a constant length for the phasor:

http://www.physics.udel.edu/~watson/phys208/phasor-slow.html

Maybe this site will offer you more insight?
http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html

 

[cepheid]

A phasor is a vector in the complex plane. It is a compact way of representing the phase information in an oscillatory function (a wave). Recall that a complex number is, in some sense, just two real numbers. One of them is the magnitude, and the other is the phase.

1. The first real number, the magnitude, is the length of the vector, and as, your textbook said, represents the amplitude of the signal. You have stated in your original post that the amplitude of a (sinusoidal) wave is fixed (which is true). Since the magnitude of the phasor represents the amplitude of the wave, why are you surprised, then, that it is fixed?

2. The second real number making up the complex number is its phase. The phase is the angle the vector makes with the positive real axis. The phase of a sinusoidal signal increases with time (theta = omega*t), until it reaches 2pi (i.e. 0) and starts all over again. This is exactly represented by the phase of the complex number as the phasor rotates in the complex plane.

Magnitude and phase are not the only two real numbers we could use to represent this complex number. Another choice would be the real part and the imaginary part (i.e. the x and y components of this vector). Notice that as the phasor rotates, these x and y components DO oscillate. So it is either the real part, or the imaginary part (your choice) of the phasor that represents the instantaneous value of the oscillatory function. THAT is what should be changing length with time.

This complex representation has huge mathematical advantages. For one thing, phasors (i.e. complex exponentials) are much easier to manipulate than trigonometric functions. Not only that, but finding the magnitude and phase of the superposition of a whole bunch of different sinusoids at a given instant (tedious if you use sines and cosines) is reduced to doing a vector sum of a bunch of phasors in the complex plane...the magnitude and phase of the resultant phasor is the amplitude and phase of the superposition of all the waves.

 

[tony873004]

Thanks for the link, LP. It's got helpful animations!

And thanks for the explanation, cepheid. That's more in-depth than the textbook's description!