# Equation relating rotational velocity and output electrical power

1. Mar 13, 2012

### Charles123

What is the equation that relates the angular velocity of an alternator rotor with its output electrical power, voltage and frequency (not necessarily just one equation) .
And for a DC generator?
I know that when you calculate the power generated by a turbine you multiply its efficiency of conversion and the theoretical power that you can extract from a flow with that swept area by the turbine. I assume that in that efficiency is the factor of conversion to electrical power. How does that work?
regards

2. Mar 14, 2012

### Bob S

Start with Maxwell's equation in differential form
$$curl\space E = - dB/dt$$
Integrate to get Faraday's Law:
$$\oint E\space d\ell= - \frac{d}{dt}\int_{A}^{}B\cdot n \space dA$$
The induced voltage in an N-turn loop around the perimeter of an area A is equal to minus the time derivative of the perpendicular magnetic field integrated over the area inside the loop. Either the magnetic field or the area can be a function of time. If an N-turn loop of area A(t) = Aosin(ωt) rotates in a dc magnetic field B0 with angular frequency ω, then the rms output voltage is
$$V= \frac{\omega NA_0 B_0}{\surd 2}$$

Bob S

Last edited: Mar 14, 2012
3. Mar 14, 2012

### jim hardy

That's a kinda general question...
like asking what make a centrifugal pump work..

Here's a fairly scholarly article on automobile alternators and where they are going
www.rle.mit.edu/per/JournalPapers/JPtpemay04p618.pdf

and the Wikipedia page really is pretty good
http://en.wikipedia.org/wiki/Alternator

Alternators in cars have widely varying efficiency at various speeds and loads. Take 50-60% as ballpark.
Utility central station alternators can be, as you doubtless know, 99% efficient.

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