## Alternator and Synchronous Generator EMF Equation

Table of Contents

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 = 2Tâ€¦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

Î¦ = Flux per pole (Weber)

N = rotor speed (RPM)

Kd= Distribution factor =

Where Distribution factor =Â **K _{dÂ }**=

**K _{c }**or

**K**=

_{P}**Cos Î±/2**

### If inducedÂ EMFÂ 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 **Î¦P** Webers.

** dÎ¦Â **=Â

**Î¦**Â and alsoÂ

*P***= 60/N seconds**

*d*Î¦Âthen induced E.M.F per conductor ( average) =â€¦.. *(i)*

But we know that:

** f = PN / 120** or

**N= 120**

*f*/ PPutting the value ofÂ **N**Â inÂ **EquationÂ (i)**, we get,

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

If there are Z conductors in series per phase,

then synchronous generator average E.M.F per phase = 2 f Î¦ Z Volts = 4 fÎ¦T Volts â€¦.. (Z = 2T)

Also we know that;

#### Form Factor = RMS Value / Average Value

= RMS value = Form Factor x Average Value,

VAV = 1.11 x 4fÎ¦T = 4.44fÎ¦T 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 Î¦TPH

#### V = 4.44 Kf KC KD f Î¦T 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 âˆš3 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

Ï† = flux per pole (Webers)

N = rotor speed (RPM)

As we know flux per pole is â€˜Ï†â€™. Therefore each conductor is cut by a flux of â€˜Ï†Pâ€™. and The time taken by a pole to complete on revolution is â€™60/Nâ€™ seconds.

So the average EMF per conductor becomes

Average EMF per conductor = Ï†P/(60/N) = Ï†NP/60

Where alternator speed, N is given by

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

Where â€˜fâ€™ is the frequency of induced EMF. Therefore the average EMF per conductor becomes

**Average EMF per conductor = Ï†NP/60 = (Ï†P/60) x (120f/P)**

**Average EMF per conductor = 2fÏ† volts**

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

**EMF per phase = 2fÏ† x Z = 2fÏ†Z**

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

**EMF per phase = 2fÏ†Z = 2fÏ†(2T) = 4fÏ†T**

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 4fÏ†T

**VrmsÂ per phase = 4.44 fÏ†T volts**

Now lets introduce â€˜coil span factor kcâ€˜ and â€˜distribution factor kdâ€˜ to get the actual induced EMF per phase.

**Vrms per phase = VphÂ = 4.44 kckdfÏ†T volts**

Or

Vph = 2.22 kckdfÏ†Z 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

Ï† = Flux per pole (Weber)

Z = No. of conductors

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

For star-connected alternator, the line voltage VL is âˆš3 times the phase voltage

**VLÂ = âˆš3 Vph**