Transformer Phasor Diagram

TRANSFORMER IS THE MOST USED ELECTRICAL EQUIPMENT IN INDUSTRY. LET US SEE A DIAGRAMMATIC REPRESENTATION OF ITS WORKING PRINCIPLE, CALLED "PHASOR DIAGRAM" ALSO KNOWN AS "VECTOR DIAGRAM"

Before starting the "Phasor  Diagram" we should know how electrical model of a transformer looks. It is called Equivalent circuit of a transformer.

Say the Primary winding has T1 turns & Secondary winding has T2 turns, then,

(E1/E2)=(T1/T2)

EQUIVALENT CIRCUIT OF TRANSFORMER/
https://electricaltechnologyrishi.blogspot.com

Now let us see what are the parameters used:

ZL=Load Impedance (in generally of inductive nature)

I2= Secondary Current or Load Current, Since the load is inductive current shall be lagging in nature.

V2=Secondary Terminal Load Voltage. (We shall consider this as our reference, hence its position has been selected as 0 degree)

R2=Resistance of the secondary winding

X2=Inductance of the secondary winding

E2=Induced voltage at secondary winding

E2=V2+I2(R2+jX2) (Vector addition), In case I2=0 i.e. trafo is on no load this E2 will appear at transformer terminal as no load voltage

E1=Induced voltage at primary winding

E1=(T1/T2) X E2

Φ m=Maximum value of linked flux (Flux responsible for induced voltage i.e. E2 & E1)

I2'=Current at Primary winding due to secondary current=(T2/T1) X I2

R0 & X0=equivalent Loss component & magnetizing inductance component.

Ie=Loss component of the No load current, responsible for core loss

Im=Magnetizing current, responsible for flux generation

I0=No-Load current

I0=Ie+Im(Vector addition)

R1=Resistance of the Primary winding

X1 Inductance of the Primary winding

I1=Primary Current

I1=I0+I2'(Vector addition)

V1=Primary applied terminal voltage

V1=E1'+I1(R1+jX1)(Vector addition)

Now Let us start Construction of the transformer phase diagram.It will be easier to understand the phasor diagram and transformer's working principle it secondary terminal Load voltage i.e. V2 is taken as reference phasor. Lets See.

Transformer Phasor Diagram_Step 1:

V2 as reference phasor

Fig 1: Transformer phasor diagram
V2 as reference/
https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 2:

I2 is lagging the load voltage i.e. V2

Fig 2: Transformer phasor diagram
I2 is lagging V2/
https://electricaltechnologyrishi.blogspot.com

Fig 3: Transformer phasor diagram_ Angle between V2 and I2 is load power factor angle/
https:electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 3: 

Voltage drop due to resistace of the secondary winding shall be parallel with Load current, as resistance doesn't cause any phase angle displacement. Since we have started our phasor diagram from the terminal load voltage so we shall add this voltage drop with the terminal voltage V2.
I2R2 ,parallel with I2 added with V2.

Fig 4: Transformer phasor diagram_Resistive voltage drop is in parallel with I2/
https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 4: 

Add I2X2 drop with I2R2 drop. As inductive voltage leads current by 90 degree hence I2X2 drop shall be 90 degree leading from I2 i.e. I2R2 phasor.

Fig 5: Transformer phasor diagram_Inductive voltage drop is in 90 degree leading with I2/
https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 5: 

Sum of V2, I2R2, I2X2 shall form induced voltage at secondary winding i.e. E2. As mentioned this is a vector sum which can be seen from the below picture. If the transformer is unloaded then this voltage will appear at the transformer terminals. Hence it is No lad voltage also. which is higher than the Terminal load Voltage.

A transformer is rated with its secondary no load voltage. for example for supplying 415 V load voltage you will have to select secondary no load voltage higher than 415 V. n practice this shall be 433 V. So a 11/0.433 kV transformer means it will step down the voltage from 11 kV to 0.433 kV at no load. Depending upon the load secondary load voltage shall be determined. This transformer shall never be specified as 11/0.415 kV. for getting 6.6 kV output transformer secondary rating shall be 6.9 kV.

Fig 6: Transformer phasor diagram_Computation of Secondary induced voltage or no load terminal voltage/
https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 6: 

Determination of maximum flux. Since the voltage induced in transformer is due to the change of flux hence it shall lag the flux by 90 degree. Lets see how.
The in the core is sinusoidal.

Φ =Φ m Sin wt

E2=dΦ /dt=d/dt(Φ m Sin wt)=Φ m Cos wt=Φ m Sin(wt-90)

this derivation indicates that induced voltage lags behind the flux by 90 degree. hence we can say the flux leads the induced voltage by 90 degree.

So we will add the flux phasor 90 ahead of E2.

Fig 7: Transformer phasor diagram_flux leads induced voltage by 90 degree/

https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 7: 

Induced voltage at primary winding shall also lag the flux by 90 degree. However its magnitude shall be E1=(T1/T2)*E2

Fig 8: Transformer phasor diagram_Primary induced voltage lags the flux by 90 degree/

https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 8: 

The flux is generated by magnetizing current. So both of them shall be in phase. Im is in phase with the flux.

Fig 9: Transformer phasor diagram_Magnetizing current shall be in phase with flux./

https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 9: 

No load loss component.
Magnetizing current is an inductive current and no load loss component is resistive current. So loss component will lead magnetizing current. As inductive current lags behind the resistive current by 90 degree, when both resistance and inductance purely resistive and inductive respectively.

Fig 10: Transformer phasor diagram_No load loss component current shall be 90 degree leading of the flux/

https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 10:

No load current.
Vector Sum of Im & Ie shall give I0 i.e. the no load current. From the equivalent circuit also you can see that this current will always flow through the transformer even if the secondary is open circuited.

Fig 11: Transformer phasor diagram_No load current/

https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 11:

Computation of Primary Current or I1
From the equivalent circuit it is clear that primary current is vector sum of No load current (I0) and reflection of secondary current to the primary(I2')
I2'=(T2/T1)*I2
And it shall be exactly 180 degree opposite to I2

Fig 12a: Transformer phasor diagram_I2 reflected to primary/

https://electricaltechnologyrishi.blogspot.com

Vector sum of I2' & I0 shall give I1 i.e. Primary current.

Fig 12b.: Transformer phasor diagram_Primary current I1/

https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 12:

Computation of Input Voltage & Primary induced voltage

As mentioned earlier Primary induced voltage E1=(T1/T2)*E2.
Now reflect it to primary side.

Fig 13a.: Transformer phasor diagram_Primary induced voltage E1 reflected to primary/

https://electricaltechnologyrishi.blogspot.com

Voltage drop due to primary winding resistance shall be parallel with I1. As Primaty input voltage is applied first, and induced voltage E1 is result of applied input voltage hence V1 shall be obtained after adding all voltage drops with primary induced voltage.

Fig 13b.: Transformer phasor diagram_I1R1 drop added with E1'/

https://electricaltechnologyrishi.blogspot.com

I1X1 drop shall be 90 degree ahead of I1R1.

Fig 13c.: Transformer phasor diagram_I1X1 drop added with I1R1/

https://electricaltechnologyrishi.blogspot.com

V1 shall be vector sum of E1', I1R1 & I1X1.

Fig 13c.: Transformer phasor diagram_V1/

https://electricaltechnologyrishi.blogspot.com

Transformer Phasor Diagram_Step 13:

Source power factor
Angle between V1 & I1 is source power factor angle. From the phasor diagram we can see that Source power factor angle is higher than load power factor angle. hence source power factor shall be less than load power factor.

Fig 14.: Transformer phasor diagram_Source power factor angle/

https://electricaltechnologyrishi.blogspot.com

COMPLETE TRANSFORMER PHASOR DIAGRAM

Fig 14.: Transformer phasor diagram_COMPLETE/

https://electricaltechnologyrishi.blogspot.com

So, by these we can draw the phasor diagram of a transformer. Understanding of this is very important for understanding the transformer working principle.

Hope you have enjoyed this article. Please share your opinion in the comment box. And for all of our other posts please visit the page "Site Map"

Thank you.......