A complex representation of oscillations - complex phasors
We'll start off by repeating Euler's formula and our cosine formula for a wave:
We see that the real part of eix has a cosine in it just like the oscillating cosine equation above. To connect the two concepts, we use the fact that the real part of the right side of Euler's formula is the cos x. Thus, we can write the second equation as:
where the "Re" in the equation stands for "real part of". Some authors use the symbol ℜ for this instead of Re. The real part of a complex number is the term or terms that do not contain an i. Often the phase shift ϕ is incorporated into the constant A as:
What about the eiωt term? A couple of pages ago, we discussed ei ϕ and found that it represented a vector in the complex plane that started at the origin and had a length of one (or 1). It made a polar angle of ϕ with the real (or x) axis.
Now we are replacing the ϕ with ωt. The time factor, t, constantly increases making the product iωt also to constantly increase with time. The iωt is the polar angle in eiωt, meaning that the polar angle increases with time. Thus the complex point representing eiωt on the complex plane will be moving around in a circle of radius 1. As above, we usually add a vector from the origin to the point, just to help with the visualization.
We show this in the animation to the right. When ωt is larger than 360 degrees (6.28 radians) then the vector starts a new trip around the circle. After all an angle of 360 degrees is equivalent to 0 degrees in most situations.
Below we show another animation of an oscillating parameter expressed as the complex equation
Multiplication of an oscillation by a complex constant
There are many physical phenomena and electrical circuit elements that change the amplitude and phase of an oscillation or wave by a fixed amount. Without using complex notation, there is no convenient way to mathematically express this. However with the use of the complex notation, this operation is merely multiplication by a complex constant. To further understand this, consider the following multiplication of an oscillation by the complex constant B:
Here we see that the resulting product has an amplitude equal to the product of amplitudes, and the phase of the constant adds to the phase of the oscillation by the constant amount ϕ. We can compare this with a similar observation above which concerned multiplication of two complex constants. Here one of the factors has the eiωt term which makes the resulting phase of the oscillation continue to increase with time. After the multiplication, the phase of the rotating vector is shifted by the fixed angle ϕ. In the animation below, we see this shift and the effect of the oscillations, which of course are just the real part of the rotating complex vectors.
A sampling of applications of multiplying an oscillation or wave by a complex constantIn each, the final signal, oscillation, or wave can be calculated by multiplying the initial signal by the appropriate complex constant.
- Microphone signal passing through a stereo amplifier.
- Telephone signal passing through 3 miles of wire.
- Fiber optic signal passing through in-line amplifier.
- Sound passing through the bone structure of the inner ear.
- AC power passing through transformers in residential transformers.
- Sound impinging on a microphone and converted to an electrical signal.
- Radar wave impinging on an airplane and returning towards ground antenna.
- Ultrasound signal impinging on fetus and returning towards detector.
- Cell phone wave from tower traveling a half mile to user's phone.
- Starlight passing through interstellar dust.
- Underwater song of a whale passing through 6 miles of ocean water.
- Ultrasonic clicks from a bat impinging on a moth and reflecting towards the bat.
- Oscillations in exterior temperature working their way into the interior of an unheated structure.
- Oscillation of the suspension system of a truck causing oscillations in the structure of a bridge supporting the truck.
- Oscillations in ocean levels caused by tides working their way 60 miles up a river estuary.
- Oscillations in AC power voltage driving AC current in a reactive load, such as motors or fluorescent lights.
NEXT: More on complex phasors. LAST: More on complex numbers.
©P. Ceperley, 2007.