Maximum Induced Voltage of an AC Generator

In summary, the maximum induced voltage in an AC generator occurs when the loop is parallel to the magnetic field, regardless of the initial angle the loop makes with the field. This is due to the properties of the sine function and its derivatives. The angle of the loop can be defined either as the angle of the tangential or the normal to the loop with the field lines.
  • #1
LePainguin
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I would like to ask a question about the induced voltage of an AC generator. So, according to a graph I found on Google, the maximum induced voltage is reached when the loop is parallel to the loop. (The graph shown below.)
generatorgraph.png

Then I was wondering, what if the loop is initially (when it is at 0 degrees) parallel to the magnetic field instead of it being perpendicular like in the graph. Would it reach the maximum induced voltage when the loop is perpendicular now, or would it still be when it's parallel?
 
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  • #2
LePainguin said:
Then I was wondering, what if the loop is initially (when it is at 0 degrees) parallel to the magnetic field instead of it being perpendicular like in the graph. Would it reach the maximum induced voltage when the loop is perpendicular now, or would it still be when it's parallel?

Why would it? All you're doing is a relabelling of coordinates ##\theta'= \theta -\pi/2##, and you still should expect the same physical results - that the greatest induced EMF occurs when the loop is horizontal.
 
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This is due to a property of the ##\sin (\omega t+\phi)## function regarding its first and second derivative.

Indeed the flux through the loop is $$\Phi=A_0\sin\theta(t)$$ where ##\theta(t)=\omega t +\phi## the angle the loop makes with the field at time t.
The induced voltage is then $$\mathcal{E}=-\frac{d\Phi}{dt}=-\omega A_0\cos\theta(t)$$
The maximum or minimum of the voltage is when $$\frac{d\mathcal{E}}{dt}=0\Rightarrow\omega^2 A_0 \sin\theta(t)=0\Rightarrow \theta(t)=k\pi, k=0,1,2,...$$

So the maximum or minimum is when the angle the loop makes with the field is ##k\pi## which in other words means when the loop is parallel with the field. And this result is independent of the initial angle ##\phi=\theta(0)## that the loop makes with the field at t=0.

PS: My angle ##\theta(t)## is defined as the angle the tangential to the loop makes with the field lines. Your schematic seems to define the angle ##\theta'(t)## as the angle of the normal to the loop makes with the field lines. Hence the two angles relate by the equation $$\theta'(t)=\theta(t)+\frac{\pi}{2}$$.
 
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What is the maximum induced voltage of an AC generator?

The maximum induced voltage of an AC generator is determined by the design and operation of the generator. It is typically calculated by multiplying the maximum magnetic field strength by the number of turns in the generator's coils.

How is the maximum induced voltage affected by the speed of the generator?

The maximum induced voltage is directly proportional to the speed of the generator. This means that increasing the speed of the generator will result in a higher maximum induced voltage, while decreasing the speed will result in a lower maximum induced voltage.

What factors can affect the maximum induced voltage of an AC generator?

The maximum induced voltage of an AC generator can be affected by various factors, such as the strength of the magnetic field, the number of turns in the generator's coils, the speed of the generator, and the type of material used in the coils.

How does the maximum induced voltage of an AC generator compare to that of a DC generator?

The maximum induced voltage of an AC generator is typically higher than that of a DC generator. This is because AC generators use electromagnetic induction to produce a varying output voltage, while DC generators use a commutator to produce a constant output voltage.

Can the maximum induced voltage of an AC generator be controlled?

Yes, the maximum induced voltage of an AC generator can be controlled by adjusting the speed of the generator, the strength of the magnetic field, and the number of turns in the coils. This allows for a more precise control of the output voltage of the generator.

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