Equivalent Circuit of Transformer - GATE EE Electrical Machines Free MCQ

Concept:

In a transformer, the turns ratio is given by

Where V₁ is the primary voltage

V₂ is the secondary voltage

N₁ is the primary turns

N₂ is the secondary turns

I₁ is the primary current

I₂ is the secondary current

The equivalent resistance of the transformer referred to the primary side = R₁ + n²R₂

The equivalent resistance of the transformer referred to the secondary side = R₂ + R₁/n²

Calculation:

Given that, primary voltage (V₁) = 2200 V

Secondary voltage (V₂) = 220 V

Primary resistance (R₁) =100 Ω

Secondary resistance (R₂) = 10 Ω

Turns ratio = 2200/220 = 10

∴ Equivalent resistance of the transformer referred to the primary side

=> R₁ + n²R₂ = [100 + (10)² × 10] = 1100 Ω

∴ Equivalent resistance of the transformer referred to the secondary side

=> R₂ + R₁/n² = [10 + 100/(10)²] = 11 Ω

Concept:

Referred value in Transformer:

In order to simplify the calculation, it is theoretically possible to transfer the voltage, current, and impedance of one winding to the other winding and combined them to a single value for each quantity.

Considered a transformer has turns ration 'a' which is given by,

Where,

I₁ and I₂ are primary and secondary current respectively.

V₁ and V₂ are primary and secondary voltage respectively.

N₁ and N₂ are numbers of turn in primary and secondary respectively.

For an ideal transformer:

Input Power = Output Power

Equivalent secondary Impedance in terms of primary Impedance:

Equivalent primary Impedance in terms of secondary Impedance:

Calculation:

Given,

Z_in is resistive and equal to 2.5 Ω

Angular frequency ω = 5000 rad/s

 

Inductive reactance X_L = jωL = j 5000 × 1 × 10^-3 = j 5 Ω 

Capative reactance X_C = 1 / jωC = 1 / j(5000 × 10 × 10^-6) = - j 20 Ω 

The input impedance of the transformer from the secondary side of the transformer will be

Z' = XL + R/a² = j 5 + 2.5/a²

Circuit diagram will be as follows  

The input impedance of the transformer from the primary side of the transformer will be

It is given that the input impedance is resistive, therefore there won't be any reactance term in the input impedance

As there is no reactance in the input impedance, make reactance equal to zero in the above equation

Now the given resistive term of the input impedance is equal to 2.5 Ω 

Therefore the value of a = 2.0, b = 0.5 

Concept:

Consider a two winding single phase transformer as shown below,

N₁ = primary winding turns

N₂ = secondary winding turns

V₁ = primary winding voltage

V₂ = secondary winding voltage

I₁ = current through the primary winding

I₂ = current through the secondary winding

Transformation ratio: It is defined as the ratio of the secondary voltage to the primary voltage. It is denoted by K.

Transformer equivalent circuit with respect to secondary can be represented a show below

Where R₀₂ = Effective resistance of the transformer as referred to the secondary side of the transformer.

R₀₂ = R₂ + R₁'    ------- (2)

R₁' = Equivalent primary resistance as referred to the secondary winding

R₁' = R₁ × K ²  ------ (3)

Similarly, effective resistance of the transformer as referred to the primary side of the transformer is given as,

R01 = R₁ + R₂' 

R₂' = Equivalent secondary resistance as referred to the primary winding

R₂' = R₂ / K 2

Calculation:

Given data

V₁ = 2000 V, V₂ = 250 V, R₁ = 1.15 Ω, R₂ = 0.0155 Ω

From equation(1)

K= V₂ / V₁

K = 250 / 2000

K = 1 / 8

Effective resistance of the transformer as referred to the secondary of the transformer.

From equations(2) & (3)

R₀₂ = 0.0155 + 1.15 / 8²

R02 = 0.0335 Ω

Power output P = 40 kVA

V₂ I₂ = 40 × 10³

I₂ = 40000 / 250

I₂ = 160 A = I_fl

I_fl is the full load current flowing through the secondary.

We required to find power loss at half full load condition

So, current at half full load is given as,

∴ Total power loss at half full load condition is given as,

P_hfl = I²_hfl × R₀₂

Phfl = 80² × 0.0335

Phfl = 214.2 W

Concept:

Consider a two winding single phase transformer as shown below,

N1 = primary winding turns

N2 = secondary winding turns

V1 = primary winding voltage

V2 = secondary winding voltage

I1 = current through the primary winding

I2 = current through the secondary winding

Transformation ratio: It is defined as the ratio of the secondary voltage to the primary voltage. It is denoted by k.

K=N2N1=V2V1=I1I2" tabindex="0">K=N2N1=V2V1=I1I----- (1)Transformer equivalent circuit with respect to secondary can be represented a show below

Where R02 = Effective resistance referred to the secondary side of the transformer.

R02 = R2 + R1' ........ (2)

R1' = Primary winding resistance as referred to the secondary side.

R1' = R1 × k2 ......... (3)

Similarly, the effective resistance referred to the primary side of the transformer is given as,

R01 = R1 + R2' 

R2' = Secondary winding resistance as referred to the primary side.

R2' = R2 / k2

Calculation:

Given total impedance of the transformer referred to primary R₀₁ = 60 Ω 

Transformation ratio k = V₂ / V₁ = 2000 / 4000 = 0.5

Total impedance of the transformer referred to secondary when primary and secondary windings have negligible resistances is given as

R₀₂ = R₀₁ × k²

= 60 / 4 = 15 Ω

Apply the voltage division rule to find voltage across the load

V_L = 2000 × 20 / (15 + 20)

V_L = 1143 V

Concept:

The equivalent circuit of the transformer is

I₁ = Primary winding current

I₂ ' =Secondary winding current referred to the primary side

I₀ = NO - load current

I_w = Core loss component

I_m = magnetizing component No-load current is almost the same when we apply the load also.

The magnetizing branch circuit is connected in parallel to the supply voltage as the No-load current is constant while we apply load also. Therefore Core loss is also constant as I₀ remains constant

Concept:

Transformers are used for impedance matching applications. This is explained with the help of the following figure:

The load impedance reflected on the primary side is redrawn as shown:

For the load to match with the source resistance:

Calculation:

With R_L = 8 Ω, and R_S = 200 Ω, the transformation ratio will be:

Calculation:

Given:

Primary resistance of Transformer R₁ = 1.5 Ω

The primary reactance of Transformer X₁ = 2j Ω

Secondary resistance of Transformer R₂ = 0.015 Ω

Secondary reactance of Transformer X₂ = 0.02j Ω

Primary Current I₁ = 20 A

Secondary Current I₂ = 200 A

Let Primary Induced Emf and Secondary Induced Emf be E₁ and E₂

Let primary Voltage be V₁

Circuit Diagram Shown:

E₂ = 200 (0.015 + j0.02)

E₂ = 5∠ 53.13

Turns ratio (N) is given by

Now, by Applying KCL in Primary Circuit

Concept:

Consider a two winding single phase transformer as shown below,

N₁ = primary winding turns

N₂ = secondary winding turns

V₁ = primary winding voltage

V₂ = secondary winding voltage

I₁ = current through the primary winding

I₂ = current through the secondary winding

Transformation ratio: It is defined as the ratio of the secondary voltage to the primary voltage. It is denoted by k.

Concept:

Consider a two winding single phase transformer as shown below,

N₁ = primary winding turns

N₂ = secondary winding turns

V₁ = primary winding voltage

V₂ = secondary winding voltage

I₁ = current through the primary winding

I₂ = current through the secondary winding

Transformation ratio: It is defined as the ratio of the secondary voltage to the primary voltage. It is denoted by k.

Concept:

Working of Transformer on No load:

I₁ = Supply current

I₂ = Load current

I'₂ = Load current reflected on the primary

I₀ = No-load current

When the transformer is operating at no-load condition, the load current (I₂) on the secondary side is zero. Since I₂ is zero, its reflected current on the primary side (I'₂) is also zero. 

Supply current I₁ is the sum of no-load current (I₀) and load current reflected on the primary side (I'2). 

Means supply current is no-load current itself

No-Load current has two components:

Active or power component (I_i): The active component of the no-load current is always in phase with supply voltage (V1) and is responsible for iron losses.

Magnetizing component (Im): The magnetizing component of the no-load current is responsible for flux production in the core of the transformer, hence it is out of phase with supply voltage (V1) or in phase with flux (ø).

​Phasor diagram:

• The active component of the no-load current is always in phase with supply voltage (V₁).
• The magnetizing component of the no-load current is responsible for flux production in the core of the transformer, hence it is in phase with flux (ø).
• Induced emf in the primary winding (E₁) lags the flux ø by 90 degrees.
• The applied voltage (V₁) is equal and opposite to the induced emf on the primary side (E₁) as losses on the primary side are negligible because the no-load current is 5% to 8% of the rated current. 

From Turns Ratio:

where, N₁ = No. of turns on the primary side

N₂ = No. of turns on the secondary side

E₁ = EMF induced on the primary side

E₂ = EMF induced on the secondary side

 Therefore, from the above phasor diagram, it is clear that the induced voltage in the primary and secondary induced windings of the transformer are in the same phase.