In this article, we will derive the emf equation of a single-phase AC generator.

A single-phase AC generator consists of two main parts
namely, a **magnetic field system** and an **armature**. The magnetic field system
produces the required magnetic field in the generator. The armature of the ac
generator is rotated by a prime mover like a hydro-turbine or a steam turbine
with an angular velocity Ï‰
radians per second.

When
the armature rotates under the influence of a fixed magnetic field produced by
the field poles, there is a change in the flux linkage of the armature
conductors. Therefore, an emf is generated in the armature conductors according
to **Faraday’s law of electromagnetic
induction**.

This
generated emf in armature conductors is given by,

Where,

- N is the number of armature conductors.
- Ï•
is the flux linkage per armature conductor at any instant of time
*t*.

Here,
it is important to note that when the plane of the armature is perpendicular to
the axis of the magnetic field lines, the maximum magnetic flux *Ï• _{m}* is linked with the
armature conductors. The linked flux is zero when the plane of the armature is
parallel to the axis of magnetic field lines.

At
any other angle, let's say Î¸ (where Î¸ = Ï‰t), the magnetic flux linked with the
armature conductors is equal to the component of the maximum flux Ï•_{m}
which is perpendicular to the coil, i.e.

On
substituting the value of Ï• in equation (1), we get,

`\e=-(d(NÏ•_m cosÏ‰t))/dt`

`\⇒e=-N (d(Ï•_m cosÏ‰t))/dt`

`\⇒e=NÏ‰Ï•_m sinÏ‰t`

`\∴e=E_m sinÏ‰t…(2)`

Where *E _{m}* is the maximum value of
the induced emf in the armature of the single-phase ac generator.

The
expression in equation (2) is the **emf
equation of a single-phase ac generator**. From equation (2), it is clear that a single-phase ac generator generates **sinusoidal
EMF**.