(New page: === Energy === I will calculate the energy expended by the signal <math>sin(2t)</math> from <math> t = 0 </math> to <math> t = 8\pi </math> - <math>E = \int_{0}^{8\pi} \mid sin(2t) \mid^...) |
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I will calculate the energy expended by the signal <math>sin(2t)</math> from <math> t = 0 </math> to <math> t = 8\pi </math> - | I will calculate the energy expended by the signal <math>sin(2t)</math> from <math> t = 0 </math> to <math> t = 8\pi </math> - | ||
| − | <math>E = \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx </math | + | <math>E = \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx </math> |
Integration shows us that: | Integration shows us that: | ||
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<math> E = 4\pi </math> | <math> E = 4\pi </math> | ||
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| + | ==Power== | ||
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| + | I will now calculate the average power of the same function from 0 to 8<math>\pi</math>. Power is very easy to calculate once you have the Energy. | ||
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| + | <math>P = \frac{1}{8\pi-0}\int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx </math> | ||
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| + | Now for the easy part. Since I already know <math>\int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx = 4\pi</math> all that's left to do is divide by 8<math>\pi</math> which yields | ||
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| + | <math>P = \frac{\pi}{2} | ||
Latest revision as of 17:25, 4 September 2008
Energy
I will calculate the energy expended by the signal $ sin(2t) $ from $ t = 0 $ to $ t = 8\pi $ -
$ E = \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx $
Integration shows us that:
$ \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx = t/2-\frac{\sin(2t)\cos(2t)}4 $ evaluated from 0 to 8$ \pi $.
$ E = 4\pi $
Power
I will now calculate the average power of the same function from 0 to 8$ \pi $. Power is very easy to calculate once you have the Energy.
$ P = \frac{1}{8\pi-0}\int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx $
Now for the easy part. Since I already know $ \int_{0}^{8\pi} \mid sin(2t) \mid^2\, dx = 4\pi $ all that's left to do is divide by 8$ \pi $ which yields
$ P = \frac{\pi}{2} $
