Electrical Machines : Single phase transformer and equivalent circuit, phasor diagram

 

Single Phase Transformer – Detailed Notes (Electrical Engineering) ⚡

A Single Phase Transformer is a static electrical device used to transfer electrical energy between two circuits through electromagnetic induction while changing voltage or current level but keeping the frequency constant.

It is widely used in power distribution systems, electrical machines, and electronic equipment.

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1. Principle of Operation

A single-phase transformer works on the principle of Electromagnetic Induction, discovered by Michael Faraday.

When AC supply is applied to the primary winding, it produces an alternating magnetic flux in the core which links with the secondary winding and induces voltage in it.

EMF Equation of Transformer

E = 4.44 f N \Phi_m

Where:

• E = RMS induced EMF

• f = supply frequency (Hz)

• N = number of turns

• Φₘ = maximum flux in Weber

For windings:

• Primary EMF

$$E_1 = 4.44 f N_1 \Phi_m$$

• Secondary EMF

$$E_2 = 4.44 f N_2 \Phi_m$$

Turns Ratio

$$k = \frac{N_2}{N_1} = \frac{V_2}{V_1}$$

If

• k > 1 → Step-up transformer

• k < 1 → Step-down transformer

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2. Construction of Single Phase Transformer

Main parts:

1️⃣ Magnetic Core

• Made of laminated silicon steel

• Reduces eddy current losses

2️⃣ Primary Winding

• Connected to AC supply

• Produces magnetic flux

3️⃣ Secondary Winding

• Connected to load

• Receives induced EMF

4️⃣ Insulation

• Separates windings and core

5️⃣ Transformer Oil (in large transformers)

• Cooling and insulation

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3. Types of Single Phase Transformers

1. Core Type Transformer

• Windings placed on two limbs

• Easier cooling

2. Shell Type Transformer

• Windings placed on central limb

• Better magnetic path

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4. Ideal Transformer

Assumptions:

• No copper loss

• No iron loss

• No leakage flux

• Infinite permeability

Relations:

$$\frac{V_1}{V_2} = \frac{N_1}{N_2}$$ $$V_1{I}_1=V_2{I}_{\mathrm{2}}$$

Efficiency = 100%

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5. Practical Transformer

In practical transformers losses occur:

1️⃣ Copper Loss

$$I^2R$$

2️⃣ Iron Loss

• Hysteresis loss

• Eddy current loss

3️⃣ Leakage Flux

4️⃣ Magnetizing Current

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6. Equivalent Circuit of Single Phase Transformer

To simplify analysis, the transformer is represented by an equivalent circuit.

Components

Component	Meaning
R₁	Primary winding resistance
X₁	Primary leakage reactance
R₂	Secondary winding resistance
X₂	Secondary leakage reactance
R₀	Core loss resistance
X₀	Magnetizing reactance

1 Equivalent Circuit of a Single-Phase Transformer  

The analysis of a transformer can be carried out by using an equivalent circuit which can be derived by considering the following: -

  The primary and secondary windings have finite resistances considered as lumped parameters. 

 The leakage fluxes are modelled as leakage reactance in the equivalent circuit. 

 The core-loss component of current is modelled using a shunt resistance. 

 The magnetization of the core is modelled using a magnetizing reactance as a shunt branch.  

    

   R1      X1

V1 ───ΩΩΩ───jX───┬──────────────
                  │
                 R0
                  │
                 X0
                  │
                  └────R2'──X2'──Load

Where

$$R_2' = \frac{R_2}{k^2}$$ $$X_2' = \frac{X_2}{k^2}$$

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7. Phasor Diagram of Single Phase Transformer

A Transformer Phasor Diagram shows the phase relationship between voltage, current, and flux.

No Load Condition

Steps:

1. Apply voltage V₁

2. Magnetizing current I₀ flows

3. Flux Φ is produced

4. Induced EMF E₁ and E₂

Relations:

• Flux lags V₁ by 90°

• Induced EMF lags flux by 90°

Thus:

$$E_1 \approx -V_1$$

No-Load Phasor Representation

          V1
           ↑
           |
           |
Flux → ----+----
           |
           |
           ↓
          E1

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On Load Condition

When load is connected:

Secondary current I₂ flows.

Primary current becomes

$$I_1 = I_0 + I_2'$$

Where

$$I_2' = \frac{I_2}{k}$$

Voltage drops occur in

• R₁

• X₁

• R₂

• X₂

. Phasor Diagram of Transformer (No Load)

At no load, the secondary is open and only excitation current flows.

Steps to Draw in Exam

1️⃣ Draw V₁ as reference vector

2️⃣ Draw flux Φ lagging V₁ by 90°

3️⃣ Induced EMF E₁ and E₂ lag flux by 90°

4️⃣ No-load current I₀ lags V₁

Phasor Representation

                V1
                ↑
                │
                │
Φ  ─────────────┼────────────
                │
                │
                ↓
                E1

Relations:

• Flux Φ lags V₁ by 90°

• E₁ and E₂ lag Φ by 90°

• E₁ ≈ −V₁

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 Phasor Diagram on Load

When load is connected:

Secondary current I₂ flows.

Steps to Draw

1️⃣ Draw V₂ reference

2️⃣ Draw I₂ depending on power factor

3️⃣ Add voltage drops​

4️⃣ Obtain E₂

5️⃣ Refer current to primary

$$I_2' = \frac{I_2}{k}$$

6️⃣ Add magnetizing current

$$I_1 = I_0 + I_2'$$

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Phasor Diagram (Lagging Load)

             V1
              ↑
              │
              │
              │
              │
E1  ←─────────┼─────────→ I1
              │
              │
             I0

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Important Transformer Relations

Voltage ratio

$$\frac{V_1}{V_2}=\frac{N_1}{N_2}$$

Current ratio

$$\frac{I_1}{I_2}=\frac{N_2}{N_1}$$

Power

$$V_1I_1 \approx V_2I_2$$

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Transformer Losses

1 Copper Loss

$$P_c=I^{2}R$$

2 Core Loss

• Hysteresis loss
• Eddy current loss

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Condition for Maximum Efficiency

Maximum efficiency occurs when

$$Copper\ Loss = Iron\ Loss$$

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Efficiency of Transformer

Efficiency is the ratio of output power to input power.

$$\eta = \frac{Output}{Input} \times 100$$
$$\eta = \frac{V_2 I_2 cos\phi}{V_2 I_2 cos\phi + Iron Loss + Copper Loss}$$

Maximum efficiency occurs when:

$$Copper\ Loss = Iron\ Loss$$

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Voltage Regulation

Voltage regulation measures change in secondary voltage from no-load to full-load.

$$VR = \frac{V_{NL} - V_{FL}}{V_{FL}} \times 100$$

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Applications of Single Phase Transformer

• Domestic power supply

• Distribution transformers

• Electronic circuits

• UPS systems

• Battery chargers

• Instrument transformers

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✅ Summary

A Single Phase Transformer is an essential electrical machine that works on Electromagnetic Induction to transfer electrical power between circuits with different voltage levels. Its performance is analyzed using equivalent circuits, phasor diagrams, efficiency, and voltage regulation.

Important Viva Questions (Very Common)

Q1: Transformer works on which principle?
➡ Electromagnetic induction

Q2: Frequency change in transformer?
➡ Frequency remains constant

Q3: Why laminated core is used?
➡ To reduce eddy current losses

Q4: Maximum efficiency condition?
➡ Copper loss = Iron loss