The above voltage equation is to be realised using phasor diagrams for various load power factor conditions. For drawing the phasor diagram consider all per phase values and remember following steps.

Steps to draw the phasor diagram :

1. Choose current as a reference phasor.

2. Now if load power factor is cosΦ it indicates that angle between V

_{ph}and I_{a }is Φ as V_{ph}is the voltage available to the load. So show the phasor V

_{ph}in such a way that angle between V_{ph}and I_{a }is Φ. For lagging 'Φ', I_{a }should lag V_{ph}and for leading 'Φ', I_{a }should lead V_{ph}. For unity power factor load Φ is zero, so V_{ph}and I_{a }are in phase.3. Now the drop I

_{a }R_{a}is a resistive drop and hence will always be in phase with I_{a}. So phasor I_{a }R_{a}direction will be always same as I_{a}, i.e. parallel to I_{a}. But as it is to be added to V_{ph}, I_{a }R_{a}phasor must be drawn from the tip of the V_{ph}phasor drawn.4. The drop I

_{a }X_{s}is drop across purely inductive reactance. In pure inductance, current lags voltage by 90^{o}. So 'I_{a }X_{s}' phasor direction will be always such that I_{a}will lag I_{a }X_{s}phasor by 90^{o}. But this phasor is to be drawn from the tip of the I_{a }R_{a}phasor to complete phasor addition of V_{ph}, I_{a }R_{a}and I_{a }X_{s}.5. Joining the starting point to the terminating point, we get the phasor E

_{ph}. Whatever may be the load power factor, I

_{a }R_{a}is a resistive drop, will be in phase with I_{a }while I_{a }X_{s}is purely inductive drop and hence will be perpendicular to I_{a }in such a way that I_{a }will lag I_{a }X_{s }by 90^{o}. This is shown in the Fig. 1.Fig. 1 |

By using the above steps, the phasor diagrams for various load power factor conditions can be drawn.

__1.1 Lagging Power Factor Load__

The power factor of the load is cosΦ lagging so I

_{a }lags V_{ph}by angle Φ. By using steps discussed above, phasor diagram can be drawn as shown in the Fig. 2.Fig. 2 Phasor diagram for leading p.f. load |

To derive the relationship between E

_{ph}and V_{ph}, the perpendicular are drawn on the current phasor from points A and B. These intersect current phasor at points D and E respectively.**.**(E

^{.}._{ph})

^{2}= (OD + DE)

^{2}+ (BE - BC)

^{2}

**.**(E

^{.}._{ph})

^{2}= (V

_{ph }cosΦ + I

_{a }R

_{a})

^{2}+ (V

_{ph}sinΦ - I

_{a }X

_{s})

^{2}

It can be observed that the sign of the I

_{a }X_{s }is negative as against its positive sign for lagging p.f. load. This is because X_{s }consists of X_{ar }i.e. armature reaction reactance. Armature reaction is demagnetising for lagging while magnetising for leading power factor loads. So sign of I_{a }X_{s }is opposite for lagging and leading p.f. conditions.__1.3 Unity Poer Fcator Load__

The power factor of the load is unity i.e. cosΦ = 1. So Φ = 0, which means V

_{ph }is in phase with I_{a}. So phasor diagram can be drawn as shown in the Fig. 3.Fig. 3 Phasor diagram for unity p.f. load |

Consider ΔOBC, for which we can write,

(OC)

^{2}= (OB)^{2}+ (BC)^{2}**.**(E

^{.}._{ph})

^{2}= (OA + AB)

^{2}+ (BC)

^{2}

**.**(E

^{.}._{ph})

^{2}= (V

_{ph }+ I

_{a }R

_{a})

^{2}+ (I

_{a }Xs)

^{2}

As cosΦ = 1, so sinΦ = 0 hence does not appear in the equation.

**Note**: The phasor diagrams can be drawn by considering voltage V

_{ph }as a reference phasor. But to derive the relationship, current phasor selected as a reference makes the derivation much more simplified. Hence current is selected as a reference phasor.

It is clear from the phasor diagram that V

_{ph }is less than E_{ph}for lagging and unity p.f. conditions due to demagnetising and cross magnetising effects of armature reaction. While V_{ph }is more than E_{ph}for leading p.f. condition due to the magnetising effect of armature reaction. Thus in general for any power factor condition,

(

**E**_{ph})^{2}= ( V_{ph }cos + I_{a }R_{a})^{2}+ (V_{ph }sin I_{a }X_{s})^{2}**+ sign for lagging p.f. loads**

**- sign for leading p.f. loads**

and V

_{ph }= per phase rated terminal voltage I

_{a }= per phase full load armature current

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