What is the EMF Equation of Single-Phase AC Generator?

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 ω 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,

`\e=-(d(Nϕ))/dt…(1)`

Where,

• N is the number of armature conductors.
• ϕ 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 ϕ_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 θ (where θ = ωt), the magnetic flux linked with the armature conductors is equal to the component of the maximum flux ϕ_m which is perpendicular to the coil, i.e.

`\ϕ=ϕ_m  cos⁡θ=ϕ_m  cos⁡ωt`

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

`\e=-(d(Nϕ_m  cos⁡ωt))/dt`

`\⇒e=-N (d(ϕ_m  cos⁡ωt))/dt`

`\⇒e=Nωϕ_m  sin⁡ωt`

`\∴e=E_m  sin⁡ωt…(2)`

Where E_m is the maximum value of the induced emf in the armature of the single-phase ac generator.

`\E_m=Nωϕ_m…(3)`

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.