Alternator and Synchronous Generator EMF Equation

An alternator or synchronous generator (also known as AC generator or dynamo) is a device which converts mechanical energy into electrical energy. In today post, we will show the EMF equation of alternator and AC generators using two methods.

Due to the rotating field magnets, the magnetising current is supplied by a DC shunt generator through two slip rings (recent alternators use electronic starting systems in place of slip rings and commutators). Always keep in mind that most alternators have a stationary armature and a rotating magnetic field.

According to Faraday’s law of electromagnetic induction, which states that if a conductor or coil links with any changing flux, there must be an induced EMF in it, when the rotor rotates, the stator conductors, which are static in the case of an alternator, are cut by magnetic flux and produce induced EMF.

This induced EMF can be calculated using the alternator’s EMF equation, which is as follows:

EMF Equation of Alternator

Lets,

P = No. of poles

Z = No. of conductors or Coil sides in series/phase i.e. Z = 2T…Where T is the number of coils or turns per phase (Note that one turn or coil has two ends or sides)

f = frequency of induced EMF in Hz

Φ = Flux per pole (Weber)

N = rotor speed (RPM)

Kd= Distribution factor =

Where Distribution factor = K_d =

K_c or K_P = Cos α/2

If induced EMF is assumed sinusoidal then,

K_f  = Form factor = 1.11

In one revolution of the rotor i.e. in 60/N seconds, each conductor is cut by a flux of ΦP Webers.

dΦ = ΦP and also dΦ = 60/N seconds

then induced E.M.F per conductor ( average) =….. (i)

But we know that:

f = PN / 120 or N= 120f / P

Putting the value of N in Equation (i), we get,

Average value of EMF per conductor =∴ (N= 120f/P)

If there are Z conductors in series per phase,

 then synchronous generator average E.M.F per phase = 2 f Φ Z Volts = 4 fΦT Volts ….. (Z = 2T)

Also we know that;

Form Factor = RMS Value / Average Value

= RMS value = Form Factor x Average Value,

VAV = 1.11 x 4fΦT = 4.44fΦT Volts.

(Note that is exactly the same equation as the EMF equation of the transformer)

And the actual available voltage of generator per phase

VPH = 4.44 KC KD f ΦTPH

 V = 4.44 Kf KC KD f ΦT Volts.

Where:

V = Actual generated Voltage per phase

KC = Coil Span Factor or Pitch Factor

KD = Distribution Factor

Kf = Form Factor

f = frequency

T = Number of coils or number of turns per phase

Note: If alternator or AC generator is star connected as usually the case, then the Line Voltage is √3 times the phase voltage as derived from the above equation.

Alternatively, let show the EMF equation of AC generator as follow.

EMF Equation of Synchronous Generator

The induced EMF in an alternator is calculated using this equation. Let’s use the formula below to derive the alternator’s EMF equation.

Assume

P = No. Of poles

φ = flux per pole (Webers)

N = rotor speed (RPM)

As we know flux per pole is ‘φ’. Therefore each conductor is cut by a flux of ‘φP’. and The time taken by a pole to complete on revolution is ’60/N’ seconds.

So the average EMF per conductor becomes

Average EMF per conductor = φP/(60/N) = φNP/60

Where alternator speed, N is given by

f = PN/120 or N = 120f/P

Where ‘f’ is the frequency of induced EMF. Therefore the average EMF per conductor becomes

Average EMF per conductor = φNP/60 = (φP/60) x (120f/P)

Average EMF per conductor = 2fφ volts

Let Z = No. of conductors per phase, then the emf pretty phase becomes

EMF per phase = 2fφ x Z = 2fφZ

Let T = No. of turns (two conductors per turn), therefore Z = 2T and the equation becomes

EMF per phase = 2fφZ = 2fφ(2T) = 4fφT

Assume the induced EMF is sinusoidal, then its form factor

Form factor, kf = 1.11

Form factor = RMS value/ Average value

RMS value = form factor x Average value

RMS value per phase = 1.11 x 4fφT

Vrms per phase = 4.44 fφT volts

Now lets introduce ‘coil span factor kc‘ and ‘distribution factor kd‘ to get the actual induced EMF per phase.

Vrms per phase = Vph = 4.44 kckdfφT volts

Or

Vph = 2.22 kckdfφZ volts

This is the EMF equation of the alternator. Where

Vph = Actual induced EMF per phase

Kc = Coil span factor

Kd = Distribution factor

f = Frequency

φ = Flux per pole (Weber)

Z = No. of conductors

T = No. of turns (Z=2T)

For star-connected alternator, the line voltage VL is √3 times the phase voltage

VL = √3 Vph