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What is the EMF equation of an alternator?

The equation used to determine the voltage generated in the armature of an alternator is called the EMF equation of the alternator. Before knowing the EMF equation of a synchronous motor or generator or alternator let us know its basic terms.

An alternator or AC generator (also known as a synchronous generator or dynamo) is a device that converts mechanical energy into electrical energy.

From the principle of generator action (Generator action), it is known that if a conductor cuts 10^8 Maxwell flux per second, then 1-volt voltage is generated in that conductor.

Full pitch turn of an alternator or generator

As shown in Figure 3.1, a full pitch turn around the North Pole cuts the entire flux and produces 10^8 Maxwell fluxes.

Now assume that the coil or turn moves from the first position (Fig. 3.1) to the second position (Fig. 3.2) in one second, so the amount of flux through the coil will be zero. i.e. the net flux through the coil is zero because 50% flux from the North pole, and 50% flux from the South pole neutralize.

Therefore, the average produced voltage or EMF in the turn of the coil from the first position to the second position will be 1 volt.

Now if the flux changes at a greater rate per coil turn and if more coils are used instead of one, the value of the average emf produced increases. That is, the average value of the emf or voltage produced will increase in proportion to the rate of change of flux and the increase in turns of the coil.

In this text, we will prove the EMF equation of alternator in two ways. We detailed in our previous posts What is alternator?, How does it work? I have discussed this. You can read it once if you want.

EMF Equation Of An Alternator or Synchronous Generator

At the beginning we defined alternator. That is, the equation with the help of which the voltage generated in the armature of the alternator is determined, is called the EMF equation of the alternator.

The voltage produced by the alternator depends on the following factors.

1. Frequency of Alternator

2. Flux per phase of alternator

3. Pitch factor of armature winding

4. Number of turns per phase

5. Distribution factor of armature winding.

EMF is generated depending on the total magnetic flux cut in DC generator.

we think,

E_{av} = average generated voltage.
N = number of turns of coil.
\phi_m = flux per pole in Maxwell.
t = time, (seconds)

The time taken for flux to reach zero to maximum means. That is, part of a complete cycle.

The value of emf or voltage produced per phase of an alternator can be expressed with the help of the following equation-

E_{av}=\ N\ \times\frac{\phi_m}{t}\times10^{-8} Volt ………….. (i)

EMF Equation Of An Alternator or Synchronous Generator

As shown in the figure above, the coil travels a distance of \frac{1}{4} in one cycle of voltage in \frac{1}{4f}  seconds.

(since \frac{1}{f} is one cycle per second)

Now substituting   t=\frac{1}{4f}   seconds in equation (1) we get,

E_{av}=\ N\ \times\frac{\phi_m}{t}\times10^{-8} Volt

E_{av}=\ N\ \times\frac{\phi_m}{\frac{1}{4f}}\times10^{-8} Volt

E_{av}=\ 4N\phi_m\times10^{-8} Volt ……….. (ii)

Again we know,

Form factor = \frac{E_{eff}}{E_{ave}}\ = 1.11

So, E_{eff}\ =\ E_{ave}\ \times1.11\ \left(E_{eff}\ =\ effective\ voltage\right)

E_{eff}\ =\ 4Nf\phi_m\times10^{-8}\ \times1.11

E_{eff}\ =\ 4.44Nf\phi_m\times10^{-8}\ …….. (iii)

This is the emf equation per phase of alternator, which is applicable only for full pitch and concentrated winding.

If Fractional pitch and Distributed winding are done instead of Full pitch and Concentrated, then equation (iii) has to be multiplied by Pitch factor ( K_p ) and Distribution factor ( K_d) to get equation per phase of alternator.

The resulting actual EMF equation of an alternator will be,

E\ =\ 4.4Nf\phi_m\times10^{-8}\ \times K_p\times K_d   Volt

EMF Equation of Alternator or Generator – Alternative Method

Think that,

Z = number of conductors

Z = 2T; Here T is the number of coils or turns per phase

P = number of poles

F = frequency of generated emf, (Hz)

Φ = flux per pole, Weber

K_d = distribution factor =  \frac{\sin m\frac{\beta}{2}}{m\ \sin\frac{\beta}{2}}  

K_p   = pitch factor = \cos\frac{\alpha}{2}

K_p  = form factor = 1.11

N = Rotor R.P.M

Flux reduction per conductor per revolution = Φ×P weber

dΦ = Φp

And dt = \frac{60}{N} Seconds

Average EMF produced per conductor

\begin{array}{l}\frac{d\phi}{dt}=\frac{\phi PN}{60}\\ \ \ \ \ \ \ =\ \frac{\phi P}{60}\times\frac{120f}{P}\\ \ \ \ \ \ \ =\ 2\phi f\ \ volt\end{array} 

Hence the average produced emf per phase in the number of conductors

= 2ΦfZ Volt

= 4ΦfT Volt

R.M.S value of EMF produced in each phase,

= 1.11 × 4ΦfT Volt

= 4.44ΦfT Volt ‍

So considering distribution factor and pitch factor,

Actual EMF equation of alternator will be;

= 4.44 ΦfT × K_p ×K_d Volt

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