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.

EMF equation of an alternator

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 kp and the distribution factor kd 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 kp and the distribution factor kd 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 kp 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.

Solution

Given data:

The number of poles can be calculated as follow:

    \[    p = \frac{120f}{n} = 4 \]

The total number of slots

    \[    S = 4 \times 3 \times 10 = 120 \]

Slot angle

    \[    \alpha = \frac{180 \times 10}{120} = 15^{\circ}   \]

    \[    k_d = \frac{sin \frac{4 \times 15^{\circ} }{2}}{4 \times sin \frac{15^{\circ} }{2}}=0.958 \]

Slot angle

    \[    Pole-pitch= \frac{120}{10} = 12 slots  \]

    \[    \beta= (12 - 10) x 15^{\circ}  = 30^{\circ}   \]

    \[    k_p = cos \frac{30}{2} = 0.966 \]

The emf equation of an alternator is

    \[    E_{rms} = 4.44\Phi f T k_p k_d = 4.44 \times 0.035 \times 50 \times 480 \times 0.958 \times 0.966 \]

    \[    E_{rms} =  3451 V \]

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