Phasors

A graphical method that helps in the understanding waves and oscillations, and also helps with calculations, such as wave addition, is called "phasor diagram". Sadly enough, this has nothing to do with Star Trek or the "phasor" weapons used in science fiction movies, although phasor diagrams would undoubtably used in their design when that time should come, since phasor diagrams play a central role in the of understanding of lasers.

If you have trouble understanding this diagram, the animation below may help.

The oscillations and vibrations really only involve motion along one direction or with regard to one parmameter, however in this method we artificially add a second dimension just for the sake of understanding. We imagine a rigid rotor or vector moving around in circles around the origin, as illustrated in the diagram to the right. It is constructed so that the projection or shadow of the tip of the vector moves back and forth exactly like the oscillation we are studying. The angle that the vector makes will the x-axis is made equal to ωt + ϕ. Because of the t, or time, factor that increases as time increases, the angle will constantly increase, making the vector rotate at a constant velocity. The projection that the vector makes on the horizontal or x-axis will be

x = r cos (ωt + ϕ),

so that this process does give us the cosine wavefunction in the end. The angle ϕ is the angle the vector make with the x-axis initially, i.e. at time t = 0.

Phasor diagrams are similar to an analog clock face. In such a clock face we use the rotating hands to keep track of time. Time, however, has nothing rotating or circular about it. To the average man or woman who is not an astronomer, time is a one dimensional progression, more like a time line. Of course time also has a repeating nature to it, like a time line that keeps replaying. But, for mechanical convenience, we use a circular clock face to keep track of time, because repeating motions are easiest to construct with rotating circular devices. Similarly, for computational ease we will use these circular phasors to track oscillations and waves, because these diagrams make computational manipulations of oscillations and waves easier.

To better understand this, study the animation below.

Animated phasor showing shadow (or projection) oscillating back and forth, simulating an oscillation. The angle θ = ωt + ϕ is shown by the angle numbers around the edge (read with the red vector as a pointer.) The position of the tip of the shadow on the ruler is equal to x = r cos (ωt + ϕ). Mouse over the image to see the action and mouse off to stop it. You need a flash player installed on your computer to see this animation.

Presented with a real oscillation to simulate, such as AC power or the pressure oscillations in a musical note, the phasor method instructs us to imagine a rotating vector, whose projection (or shadow) is the observed oscillation. The oscillation shown here has an amplitude of 1, i.e. r = 1. Of course, we can change the vector length for any amplitude. It is perhaps more common to make the projection (or shadow) on the x-axis as is shown in the previous diagram, instead of below the circle as is shown in this animation. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

Another phasor animation:

Animated phasor relating oscillations of a physical object, of our beautiful bungy jumper in this case, to a phasor diagram and a graph of her oscillations (taken at her "center of mass".) For these purposes, the phasor diagram has been rotated 90 degrees and projected horizontally. Also, we assume that the elastic rope is always under some tension, in order to insure cosine dependence.

Mouse over the image to see the action. When the animation is finished, mouse off and on again to replay it. Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

Adding two waves in a phasor diagram

Ok, a phasor diagram is suppose to make it easier to add waves. But, exactly how is that done?

Someone who is very familiar with vector addition may have recognized the equations we laboriously derived for adding cosine waves above. I'll repeat them here:

and

Notice that all the terms are of the form A₁cosϕ or A₁sinϕ . These are just the x and y components (or projections) of the vector A₁ plotted on the phasor diagram to the right. Note that the bold symbol A₃ stands for the whole vector, while the italized A₃ stands only for its amplitude. That is to say:

and

making our equations become:

.

These are just the equations for adding two vectors: we add their x components together to find the x component of the sum, call it x₃, and add their two y components to find the y component of the sum, called y₃. Then the magnitude and tangent of the angle of this the sum vector should be (as we have):

Thus if we add two phasor vectors together, we will get the phasor vector for the correct wavefunction according to the complicated cosine addition formulas.

Below we have an animation for the process of adding two oscillations using phasor diagrams. We use the oscillations of two people on the ends of bungy cords, however the oscillations could also be the oscillations of electrical voltages in AC power or the oscillations of the air pressure as a sound wave passes through. Read the caption below the animation for details on it.

An animated phasor diagram showing phasor addition of two oscillations, that of the woman and that of the man suspended on bungy cords. Like before, the animation shows the relation between the phasor diagram, the actual oscillation, and a graph of the oscillations.

Click on "together" to see the action. When the animation is finished, click on this or other buttons to see this or related animations. Move the mouse off the animation to suspend it and back on to restart it.

The "together" animation shows the pair oscillating in unison or "in phase" with a phase difference of 0 degrees. The gray sum vector represents the sum of the two oscillations and for zero phase difference has an amplitude equal to the sum of the amplitudes of the two people. Zero phase difference results in the largest possible sum oscillations.

One use of summing the two oscillations is to find the motion of the center of mass of the man and woman. The center of mass motion is simply the sum of the motions divided by two. It is shown as a green cross and moves up and down with the motion.

The "in opposition" button delays the release of the woman to put the pair oscillating with 180 degree phase shift between them, or in opposition with each other. When the man is down, the woman is up, etc. This phase results in a sum amplitude equal to the difference between the two amplitudes. In this case the sum is the smallest possible for all possible phases. Note that the center of mass hardly oscillates at all. If the oscillations of the two people were of exactly the same amplitude, then the sum, as well, as the center of mass oscillations would have zero amplitude in this case of 180 degrees phase difference. That is to say that with 180 degree phase difference the two oscillations would cancel in a summing process or would destructively interfere.

You should compare the relative directions of the phasor arrows in the two cases (after the woman has been released. In the first case, with 0 degrees phase shift, the arrows are together, cause the maximum possible sum. In the second case, the arrows are spaced 180 degrees apart, causing the minimum possible sum.

The "random phase" button delays the woman's release a variable amount so that a different phase shift will occur each time this button is clicked. The resulting sum oscillation is the vector sum of the man's and woman's phasors. The summing parallelogram is seen in light gray in the phasor diagram to the left. The sum oscillations have an amplitude some where between the amplitudes of the two extremes cases, that of 0 and that of 180 degree phase difference. In general there will be a phase difference between the sum and either person's oscillations.

The "difference" button shows the difference between the man's and woman's oscillations. One application for a difference is to calculate the distance between the two people as a function of time. We illustrate this length by the length of the blue ribbon stretched between their hands. You can see that its length varies in an oscillatory fashion. This button also adds a random phase shift between the man's and woman's oscillations.

Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

All the above "buttons" show cases in which the frequency of the two oscillations are the same. However the phasor method also works for adding or subtracting oscillations of differing frequencies. We show this next:

An animated phasor diagram showing phasor addition of two oscillations when the frequencies of the two oscillations are different. This allows one of the rotating phasor vectors to rotate faster than the other, continuously changing the phase angle difference between the two. At times, the two phasors will add up in the maximum way, demonstrating constructive interference. At other times, they will subtract, demonstrating destructive interference. The result is that the amplitude of the sum varies with time in a repetitive way. This phenomena is called "beating" and is commonly encountered in the addition of two sound waves of only slightly different frequency or tone. The resulting sound "warbles", i.e. rapidly varies in amplitude. It turns out that the frequency of the amplitude variations, i.e. of the beating, is equal to the difference between the two initial frequencies.

Click on either button to see the action. Mousing over the other button can be used to speed up or slow down the action without restarting it. Mouse off the animation to suspend it and mouse on to continue it. Click on a button to restart it. Clicking will restart the action whether or not it has finished. Each time the action is restarted, a new frequency difference will be used, changing the rate of beating. In these animations, we neglect damping, which will normally reduce the oscillations as time goes by. We will discuss damping in future lessons.

We needed to compress the horizontal axis to show many more oscillations than in the previous animation in order to be able to see the beating phenomena. Press − in the +/− button to hid the cartoon features of this animation and + to restore them. This animation can be downloaded free at George Mason University Archival Repository. Please read the fair use policy for this work.

Options:

• If you are having trouble with your flash player, you can see
  • a still image by clicking here
  • or a animated gif file (2.5MB) of part of the action by clicking here.
    
  
  
• To hear examples of acoustical beating, click on the captions below the following flute players.

Why do phasor diagrams work?

We have already answered this above, at least in part, but it is an important point that could stand repeating and further refinement. Our answer:

• Most simple oscillations, ones having a "linear" restoring force, can be mathematically represented with cosine (or sine) functions, for example: y(t) = A cos(ωt + ϕ).
• This is a result of the type differential equations that these oscillating systems are constrained to follow and has nothing to do with circular motion.
• It just so happens that a cosine function is also, independently of oscillating phenomena, the projection of a vector rotating at constant angular speed around the origin.
• Thus, if we should want, we could choose to have the projection of a rotating vector represent an oscillation. This would have the handy feature that the argument of cosine function in the oscillation which has units of radians, now can be represented as an actual physical angle.
• If we have two vectors rotating around the origin, then their sum vector will also rotate around the origin. This sum vector will have an x-component equal to the sum of the x-components of the two original vectors. Thus the projection of the sum vector is the sum of the two oscillations.
• Whereas there was no simple illustration for adding two cosine functions, with the phasor method there is a simple graphical method for adding two rotating vectors.
• So, given oscillations to add and subtract, we construct the required rotating vectors, called a phasor diagram, and graphically add and calculate the resulting sum and/or difference oscillation.

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NEXT: More on Complex Numbers.

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© P. Ceperley, 2007.

Good references on phasors.

Note that searching under "phasors" will produce links to complex phasors, which we will cover in a later posting. Below are some of the better links to the idea of simple phasors.

• Wikipedia.
• Nice animation from S. Schneider of ITU.
• The complete list of S. Schnieder's physics animations.
• Nice clean animation of phasors and phasor addition.

The algebra of complex phasors

Complex addition of two oscillations.

We can add two oscillations together as

   .

With this representation it turns out that we can factor out the real operator as well as the time dependent part, the e^iωt term:

    .

At this point, many people consider the two terms in the last part of the equation as complex amplitudes. That is, they define:

.

where the tilda, ~, above an amplitude means it is complex, i.e. has an amplitude and phase shift, or alternately has real and imaginary parts.

If we plot the two complex amplitude in the complex plane (as we discussed above), we can use standard vector methods to add the x component of one amplitude to the x component of the other amplitude and the same for the y components, as illustrated here.

We would express the result of our work as:

    ,

where

.

Most engineers and scientist find it bothersome to carry around the Re operator, preferring to keep it in their head and off the page, so they will write:

,

where

In fact, the tildes are also usually not kept, although this author personally likes them and will often use them to emphasize that a number is complex.

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If we want, we can get equations similar to what we got in the cosine method. To add two complex vectors (or numbers), we add their x components and separately add their y components:

.

The magnitude of the complex vector A is given by:

and the angle or argument is:

.

These are the same results as can be achieved with the cosine method with a lot less effort and a lot more understanding. It is more common just to add the complex vectors with values in them and not use equations like that shown just above.

The algebra of oscillations and waves

So to recap, we can express the addition of two waves as:

v₃ = v₁ + v₂ ,

where v₁, v₂, and v₃ are all complex phasors. We can substitute details of each of them as needed, such as substituting

v₁ = A₁e^iωt, etc.

We can factor out common terms such as common e^iωt terms, common phase shifts, common amplitude multipliers, etc. In fact we can do most of the algebraic steps we normally do in simple algebra.

While the above work is more complicated than a simple addition, it is much, much less work than working with cosines and sines as we did above. It takes the best part of the simple phasor method of graphically adding oscillations and adds a powerful notation to it. Thus, a person now has both a graphical method and an algebraic method to work out the details of oscillating phenomena and technology. It is simple and transparent enough to allow a person to really understand wave behavior both geometrically and mathematically.

One of the most powerful aspects of this notation is the complex phasor's exponential form. This particular mathematical form is particularly easy to be factored. This allows the time dependent part to be factored out and collected, before doing the addition. While this may seem like mathematical trickery, be advised, that such trickery is ever-present in certain areas of math, e.g. in integration, a major component of calculus, and trickery there is what allows many problems to be solved.

In fact, the factoring out of the time dependence is so ever present in these calculations that many electrical engineering text books above the introductory level, do not even bother to write the e^iωt factor in the equations, assuming that it is always there and will always be factored out.

It turns out that other common mathematical operations are also easier to carry out in the complex representation, most notably differentiation and integration, adding even more reasons to use the notation over the cosine notation.

With the complex notation, the effect of many electrical components can simply be expressed as (a) a complex factor to multiply by or(b) two or more complex amplitudes added together. While the cosine notation is occasionally used for some applications where the complex notation cannot be used, in most physics and electrical engineering calculations involving waves and oscillations, the complex notation dominates.

In summary, use of complex phasors provides us with a compact notation for manipulating oscillations and waves algebraically. The sum of two waves is just expressed as a simple sum. A product of a complex phasor and a complex constant just changes the amplitude and phase. We can now write fairly complicated equations for waves and solve them for the unknown factor as we do for other engineering topics. There are, however, limitations to this method.

Limitations of the complex method.

We might point out, without special precautions, the complex method does not produce correct results in cases where the wavefunction needs to be taken to a higher power than one or when two wavefunctions are multiplied together. Note that earlier we multiplied a wave (or oscillation) by a complex constant, NOT by another waveform or oscillation. At the same time, in the most common of such higher power cases, in the case where a wavefunction is simply multiplied by itself or by another wavefunction, there is a nice special complex formula for the time averaged result. For example, the formula for time averaged power delivered by AC voltage and current is:

,

where i in this case is the complex current amplitude and the asterisks denote the "complex conjugate" found by inverting the sign of the angle or argument (or alternately by inverting the sign of the imaginary part of the complex number). On the other hand, in the general non-linear case, the complex notation does not give correct results and the cosine representation needs to be used, unless some other special precautions are taken. In spite of this limitation, complex phasor notation dominates all mathematics concerning oscillations and waves in today's technology.

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NEXT: True waves

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©P. Ceperley, 2007

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Good references on WAVES

Good general references on resonators, waves, and fields

Wikipedia on waves. Zona Land on waves, lots of animations. The physics classroom on waves. Thinkquest on waves. Dr. Dan Russell, Kettering University Applied Physics, nice animations. All sorts of demonstration setups and animations from the Univ. of California at Berkeley. And many more sites.... search on waves and traveling waves. Scroll down farther for a list of the various related topics covered in postings on this blog.

Think Quest -- Wave Express technorati.com The Physics Classroom HyperPhysics intutor.com Wikepedia