Phasor Diagram

Consider a transformer supplying the load as shown in the Fig. 1.

Fig. 1

       The various transformer parameters are,

                        R₁ = Primary winding resistance

                        X₁ = Primary leakage reactance

                        R₂ = Secondary winding resistance

                        X₂ = Secondary leakage reactance

                         Z_L = Load impedance

                         I₁= Primary current

                          I₂ = Secondary current = I_L = Load current 

now                    Ī₁ = Ī_o + Ī₂'

where                 I_o = No load current 

                           I₂'= Load component of current decided by the load

                               = K I₂ where K is transformer component

       The primary voltage V₁ has now three components,

1. -E₁, the induced e.m.f. which opposes V₁
2. I₁ R₁, the drop across the resistance, in phase with I₁
3. I₁ X₁, the drop across the reactance, leading I₁ by 90^o

       The secondary induced e.m.f. has also three components,
 1. V₂, the terminal voltage across the load
2. I₂ R₂, the drop across the resistance, in phase with I₂
3. I₂ X₂, the drop across the reactance, leading I₂ by 90^o

       The phasor diagram for the transformer on load depends on the nature of the load power factor. Let us consider the various cases of the load power factor.

1.1 Unity power factor load, cosΦ₂ = 1

       As load power factor is unity, the voltage V₂ and I₂ are in phase. Steps to draw the phasor diagram are,
1. Consider flux Φ as reference
2. E₁ lags Φ by 90^o. Reverse E₁ to get -E₁.
3. E₁ and E₂ are inphase
4. Assume V₂ in a particular direction
5. I₂ is in phase with V₂.
6. Add I₂ R₂ and I₂ X₂ to to get E₂.
7. Reverse I₂ to get I₂'.
8. Add I_o and I₂' to get I₁.
9. Add I₁ R₁ and to -E₁ to get V₁.

       Angle between V₁ and I₁ is Φ₁ and cosΦ₁ is primary power factor. Remember that I₁X₁ leads I₁ direction by 90^o and I₂ X₂ leads I₂ by 90^o as current through inductance lags voltage across inductance by 90^o. The phasor diagram is shown in the Fig.2

Fig. 2  Phasor diagram for unity power factor load

Lagging Power Factor Load, cos Φ_2:

       As load power factor is lagging cosΦ₂, the current I₂ lags V₂ by angle Φ₂. So only changes in drawing the phasor diagram is to draw I₂ lagging V₂ by Φ₂ in step 5 discussed earlier. Accordingly direction of I₂ R₂, I₂ X₂, I₂', I₁, I₁ R₁ and I₁X₁ will change. Remember that whatever may be the power factor of load, I₂X₂ leads I₂ by 90^o and I₁X₁ leads I₁ by 90^o.

       The complete phasor diagram is shown in the Fig. 3.

Fig.  3  Phasor diagram for lagging power factor

Loading Power Factor Load, cos Φ_2:

       As load power factor is leading, the current I₂ leads V₂ by angle Φ₂. So change is to draw I₂ leading I₂  by angle Φ₂. All other steps remain same as before. The complete phasor diagram is shown in the Fig. 4

Fig. 4  Phasor diagram for leading power factor