Phasor (physics)

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A phasor is a vector drawn to represent a wave, such that the vector sum of several phasors can be used to determine the intensity and phase of the several waves after interference. Phasors are used directly in optics, radio engineering and acoustics. The constant length of the phasor gives the amplitude and the angle it makes with the x-axis gives the phase angle. Because the mathematics of waves frequently carries over to electronics, phasors can be used there in rudimentary circuit analysis of AC circuits. Finally, phasors can be used to describe the motion of an oscillator, with its various properties including projections onto the x and y axes having different physical meanings.

1 Interference
2 Simple harmonic oscillator
3 References
4 External links

[edit] Interference

Example of vector addition

Phasors are most commonly used to visually solve problems of the type "several waves of similar frequency but different phases and amplitudes are interfered at a point, what is the resulting intensity?" To solve this problem, draw one phasor for each of the waves, and then simply perform vector addition on them. The length of the resulting vector is the amplitude of the resulting wave, and its length can be squared to find the intensity. Note that, while the sum of several sine waves is not necessarily another sine wave, the sum of several sine waves of the same frequency is, allowing the resultant phase to be read as the angle of the resultant phasor.

An interesting, related use of phasors arises in the inverted question, "what phase difference do I need between three identical waves for perfect cancellation?" In this case, simply imagine taking three vectors of equal length and placing them head to tail such that the last head matches up with the first tail. Clearly, the shape which satisfies these conditions is an equilateral triangle. Since the angle between each phasor to the next is 120 degrees, or one third of a wavelength $λ / 3$ , so the phase difference between each wave must also be 120. The problem is solved for four phasors with a square and so forth.

Three waves in perfect destructive interference

In the example of three waves, the phase difference between the first and the last wave was 240 degrees, while for two waves destructive interference happens at 180 degrees. In the limit of many waves, the phasors must form a circle for destructive interference, so that the first phasor is nearly parallel with the last. This means that for many sources, destructive interference happens when the first and last wave differ by 360 degrees, a full wavelength $λ$ . This is why in single slit diffraction, the minima occurs when light from the far edge travels a full wavelength further than the light from the near edge.

[edit] Simple harmonic oscillator

A phasor can be used to model the behavior of a particle in simple harmonic motion, in which case, the y value of the phasor corresponds to the particle's current displacement, and one should imagine the phasor rotating around the origin as the object oscillates. The maximum displacement is given by the phasor's length. If we are provided with the period of rotation (i.e oscillation), and we call the length of the phasor A, then the end of the phasor travels a distance $2π A$ in time T, so the end of the phasor travels with velocity $2π A / T$ , which is the maximum speed of the oscillator. Since the maximum velocity occurs when there is no displacement (and so zero phase) so the phasor is completely in the x direction, we can see that the current velocity is given by $2π x / T$ , where x is the current x value.