(New page: == Energy and Power == === Energy === The following is the energy expended by the signal <math> sin(2t) </math> from <math> t = 0 </math> to <math> t = 4\pi </math>: <math> E = \int_{0}...) |
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<math> P = \frac{1}{4\pi - 0} \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx </math><br> | <math> P = \frac{1}{4\pi - 0} \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx </math><br> | ||
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| + | Since we already know that the integral equals <math> 2\pi </math>, dividing that by <math> 4\pi </math> will yield the average power. | ||
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| + | <math> P = \frac{2\pi}{4\pi} = \frac{1}{2} </math> | ||
Latest revision as of 10:00, 4 September 2008
Energy and Power
Energy
The following is the energy expended by the signal $ sin(2t) $ from $ t = 0 $ to $ t = 4\pi $:
$ E = \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx $
Integration tables will tell us that:
$ \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx = \mid \frac{t}{2} - \frac{sin(2*2*t)}{4*2} \mid $ evaluated at 0 and subtracted from the value at 4$ \pi $.
$ E = 2\pi $
Power
The following is the average power expended by the signal $ sin(2t) $ from $ t = 0 $ to $ t = 4\pi $:
$ P = \frac{1}{4\pi - 0} \int_{0}^{4\pi} \mid sin(2t) \mid^2\, dx $
Since we already know that the integral equals $ 2\pi $, dividing that by $ 4\pi $ will yield the average power.
$ P = \frac{2\pi}{4\pi} = \frac{1}{2} $
