EMF equation of an alternator and solved problems

An alternator or a synchronous generator is a device used to convert mechanical energy to Alternating current electric current. The EMF equation of an alternator is a mathematical expression to calculate the effective value of EMF induced in each phase of an alternator or a synchronous generator.

According to Faraday’s law of electromagnetic induction, the average emf induced per conductor is:

$E_a_v = \frac{d\Phi}{dt}$

Where,

Φ is the flux per pole in weber.

t is the time taken for rotation.

If ‘P’ is the number of poles in the alternator and n is the speed of rotation in radials per minute, the change in flux can be given by

$d \Phi= \Phi \times P$

Also the time take for rotation is given by

$dt = \frac{60}{n}$

Hence,

$E_a_v = \frac{\Phi \times P}{\frac{60}{n}} = \frac{\Phi n P}{60} -- Equation [1]$

The frequency of the induced EMF is given by

$f = \frac{n \times P}{120}$

$n = \frac{120 \times f}{P} -- Equation [2]$

Substituting equation [2] in equation [1]

$E_a_v = 2 \Phif$

Let ‘Z’ be the number of conductors per phase and ‘T’ be the number of turns per phase. Every phase contains two turns of the conductor so Z=2T. The average EMF induced in each phase is given by:

$E_\frac{av}{phase} = E_a_v \times Z = 2 \Phi nfZ = 4\Phi fT -- Equation [3]$

The above equation represents the average value of EMF induced per phase of an alternator. As you know the induced voltage is sinusoidal in nature.

The RMS value of a sinusoidal AC voltage is 1.11 times its average value. Hence effective RMS value of EMF induced per phase can be found by multiplying Equation [3] by 1.11.

$E_{rms} = 1.11 \times 4\Phi fT = 4.44\Phi f T -- Equation [4]$

Considering the pitch factor k_p and the distribution factor k_d also, the equation [4] for the RMS value of the voltage generated in each phase of an AC synchronous generator may be rewritten as follows:

$E_{rms} = 4.44\Phi f T k_p k_d$

The EMF equation of an alternator is given above. The pitch factor k_p and the distribution factor k_d can be calculated from the following formulae:

$k_d = \frac{sin \frac{m \alpha}{2}}{m sin \frac{\alpha}{2}}$

$k_p = cos \frac{\beta}{2}$

Where,

α is the slot angle and m is the number of slots per pole per phase.β is the angular distance between a full pitch and short pitch coils. For a full pitch coil, β = 0, and hence k_p is 1.

Solved example:

Let us find the solution for the following problem using the emf equation of an alternator.

Problem: A 3-phase, 50 Hz alternator is running at 600 rpm has a 2-layer winding, 12 turns/coil, 4 slots/pole/phase, and coil-pitch of 10 slots. Let us find the induced EMF per phase if the flux/pole is 0.035 webers.