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 eiωt 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:
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:
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.
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 = A1eiωt, etc.
We can factor out common terms such as common eiωt 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 eiωt 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.
NEXT: True waves
©P. Ceperley, 2007
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