Rayleigh's Theorem: Difference between revisions
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*[[Signals and systems|Signals and Systems]] |
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Rayleigh's Theorem is derived from the equation for Energy |
Rayleigh's Theorem is derived from the equation for Energy |
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*<math> W = \int_{-\infty}^{\infty}p(t)\,dt </math> |
*<math> W = \int_{-\infty}^{\infty}p(t)\,dt </math> |
Latest revision as of 23:36, 12 October 2006
Rayleigh's Theorem is derived from the equation for Energy
If we assume that the circuit is a Voltage applied over a load then
for matters of simplicity we can assume
This leaves us with
This is the same as the dot product so to satisfy the condition for complex numbers it becomes
If we substitute and
Substituting this back into the original equation makes it
The time integral becomes This simplifies the above equation such that
Proving that the energy in the time domain is the same as that in the frequency domain