## AC generator

October 21st, 2016

Let’s consider the device shown in Fig.1

Fig.1. Schematic diagram of a simple AC generator: A -view profile, B – view from the front

It is a frame that rotates with constant angular velocity ω in a uniform magnetic field created by the two pole pieces. Each end of the frame is connected to one of the rings which planted the axis; the axis and rings isolated from one another. On the surface of rings slide brushes that connect the frame with the outer electrical circuit.

When rotating frame in the magnetic field there appear the alternating magnetic flux which is equal:

$\Phi=B_nS$,      (1)

where $B_n$  is a normal component of magnetic induction, $S$ is the square of the frame.

The normal component of the magnetic induction changes according to the law   (fig.2):

$B_n=B\ cos\alpha$  (2)

Fig.2. The normal component of magnetic induction

The angle $\alpha$ can be determined from the definition of angular velocity $\omega=\frac {\alpha}{t}$, then $\alpha = \omega t$.

Substituting α into the formula (2), we get the formula to change the module through induction vector angular velocity

$B_n=B \cos \omega t$ (3)

When the frame is rotating its plane crosses the alternating magnetic flux:

$\Phi=B_nS=BS\cos \omega t$  (4)

The phenomenon of electromagnetic induction generates in the frame the electromotive force, which changes according to law:

$\epsilon= - \frac {d \Phi}{dt}$   (5)

Substituting in (5) the expression for Φ from (4) and differentiating, we get:

$\epsilon= B S \omega \sin \omega t$  (6)

Replacing $B S \omega$ on $\epsilon_0$, we write the expression (6) in the form:

$\epsilon= \epsilon_0 \sin \omega t$ (7)

When connecting the frame to the electrical circuit with resistance $R$ in the electrical circuit proceeds AC:

$I= \frac { \epsilon_0}{R} \sin \omega t$ (8)

By entering a designation $I_0= \frac { \epsilon_o}{R}$ let’s rewrite formula (8) as:

$I=I_0 \sin \omega t$  (9)

Formula (9) corresponds to the case where the initial position of the frame plane  is perpendicular to the vector  induction. Where   between the plane of the frame and the vector of induction of the initial time there is an angle $\phi$, called the initial phase, the formula of the depending on amperage occasionally takes the form:

$I=I_0 \sin ( \omega t + \phi)$ (10)

(The information has been taken from e-book «Electrotechniсs with Fundamentals of Industry Electronics” (author V.D. Shvets), which has the Recommendation of Ministry Education and Science of Ukraine: protocol № 1/11-8959 from 26.06.2015. Copyright is reserved by certificate Number 55970 from 08.08.2014.)