Alternating Current Electrical Generator Model

Electric generators turn motion into alternating-current electric power by exploiting electromagnetic induction. A loop that is placed in a magnetic field induces an motional electromagnetic force (emf). A simple alternating current (AC) generator is illustrated here. ABCD is mounted on an axle PQ. The ends of the wire of the loop are connected to 2 brushes contacting two slip rings continuously at position X & Y. Two carbon brushes are made to press lightly against the slip rings.

The motor features a external magnet (called the stator because it’s fixed in place) and an turning coil of wire called an armature ( rotor or coil, because it rotates). The armature induces an emf, because any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil.

The key to producing motional emf is in change in the magnetic flux experienced by the coil loop.

Faraday's law states the induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. The induced electromotive force or emf, ? in any closed circuit is equal to the rate of change of the magnetic flux , ? through the circuit.

|?| = | d(?)/ dt  |

where ? = N.B.A cos ( B&A)

|?|  is the magnitude of the electromotive force (emf) in volts

? is the magnetic flux through the circuit (in webers).

N is the number of turns of wire in the loop
B is the magnetic field

A is area of coil

angle B&A is the angle between vector magnetic field and vector perpendicular to the area

Lenz's law states an induced current is always in such a direction as to oppose the motion or change causing it.
The law provides a physical interpretation of the choice of sign in Faraday's law of induction, indicating that the induced emf and the change in flux have opposite signs. The the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it

? = - d(?)/ dt

For the case of a rotating loop,

? = - d(?)/ dt

From eariler equstion as ? = N.B.A cos ( B&A)

? = - d(N.B.A cos ( B&A))/ dt

the physical setup of Bz and normal vector of area A when t = 0 s, such that angle B&A = ( ? + ?/2 ).
and taking out the constants from the differential equation,

? = - N.B.A d( cos ( ? + ?/2 )/ dt

from mathematical trigometry identity, cos ( ? + ?/2) = -sin ( ?  )

? = - N.B.A d( -sin ( ?  )/ dt

To derive an expression for the induced emf across the slip rings when the coil is spun at a angular frequency, ?, knowing ? = ?.t

? = - N.B.A d( -sin ( ?.t  )/ dt )

? = N.B.A d( sin ( ? )/ dt )    which the equation used by the custom function  getCurrent () = d( sin ( ? )/ dt )
When a closed ciruit in connected to the rotating loop, using Ohm's law

? =  N.B.A.d( sin ( ? )/ dt ) = I.R      which the equation used by the model

When mechanical energy is used to rotate the loop, the armature induced a emf described by the right hand rule. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil.

Use the rotating handle function input field ?(t)= _________ to see what happens when the rotating handle cranks the loop in the Bz magnetic field. The checkbox current flow & electron flow alows different visualization since I = d(Q)/dt and Q= number of charge*e. The Play & Pause button allows freezing the 3D view for visualizing these induced currents, for checking for consistency with the right hand rule.

When the rotating handle is moved by the input field ?(t)= 2*t , induced current runs through ABCD (select the checkbox labels?) in a manner described by
? =  N.B.A.? .cos ( ?.t  ) = I.R.

since ?(t)= 2*t , imply 2 = ?

?/R =  N.B.A.? .cos ( ?.t  )/R = I. which is the modeled equation.

If N = 1, B = 2 x10-6 T, A = 1.5*1.5 m*m, R = 1 ?

?/R =  1.2.1.5*1.5.2x10-6 .cos ( 2.t  )/1 = I.

? = 9x10-6 cos ( 2.t  ) = I