Single-Phase Transformer GATE Exam Quick Notes

Basic Concepts

Single-Phase Transformer - Fundamentals

Definition & Principle

  • Static electrical machine transforming AC voltage levels

  • Works on principle of mutual induction

  • Energy transfer through magnetic coupling

EMF Equation

\[E = 4.44 \times f \times N \times \Phi_m\]

  • E = RMS induced EMF (V)

  • f = Frequency (Hz), N = Number of turns

  • \(\Phi_m\) = Maximum flux (Wb)

Transformation Ratios

\[\frac{N_1}{N_2} = \frac{E_1}{E_2} = \frac{V_1}{V_2} = a \quad \text{and} \quad \frac{I_1}{I_2} = \frac{1}{a}\]

Ideal vs Real Transformer

Ideal Transformer

  • 100% efficiency, no losses

  • Infinite permeability, zero resistance

  • No leakage flux, perfect coupling

Real Transformer

  • Core losses: Hysteresis + Eddy current

  • Copper losses: \(I^2R\) losses in windings

  • Leakage flux causing voltage drops

  • Magnetizing current required

Referred Values (Secondary to Primary)

\[V_2' = aV_2, \quad I_2' = \frac{I_2}{a}, \quad R_2' = a^2R_2, \quad X_2' = a^2X_2\]

Equivalent Circuit

Equivalent Circuit Parameters

Exact Equivalent Circuit

  • \(R_1, X_1\): Primary resistance and reactance

  • \(R_2', X_2'\): Secondary parameters referred to primary

  • \(R_0, X_0\): Core loss resistance and magnetizing reactance

  • \(I_0\): No-load current = \(I_c + I_m\)

Approximate Equivalent Circuit

\[\begin{aligned} R_{eq} &= R_1 + R_2' \\ X_{eq} &= X_1 + X_2' \\ Z_{eq} &= \sqrt{R_{eq}^2 + X_{eq}^2} \end{aligned}\]

Validity of Approximation

  • \(I_0 \ll I_1\) (typically 2-5% of rated current)

  • Shunt branch moved to supply side

Testing

Open Circuit Test

Procedure

  • LV side: Rated voltage applied

  • HV side: Open circuit

  • Measurements: \(V_0, I_0, W_0\)

Calculations

\[\begin{aligned} \cos\phi_0 &= \frac{W_0}{V_0 I_0} \\ R_0 &= \frac{V_0^2}{W_0} \\ X_0 &= \frac{V_0}{I_0 \sin\phi_0} \\ I_c &= I_0 \cos\phi_0, \quad I_m = I_0 \sin\phi_0 \end{aligned}\]

Purpose

  • Determines core loss parameters

  • Core losses = \(W_0\) (constant)

Short Circuit Test

Procedure

  • HV side: Reduced voltage (5-12% of rated)

  • LV side: Short-circuited

  • Measurements: \(V_{sc}, I_{sc}, W_{sc}\) at rated current

Calculations

\[\begin{aligned} R_{eq} &= \frac{W_{sc}}{I_{sc}^2} \\ Z_{eq} &= \frac{V_{sc}}{I_{sc}} \\ X_{eq} &= \sqrt{Z_{eq}^2 - R_{eq}^2} \\ \cos\phi_{sc} &= \frac{R_{eq}}{Z_{eq}} \end{aligned}\]

Purpose

  • Determines equivalent circuit parameters

  • Copper losses at rated current = \(W_{sc}\)

Performance Analysis

Voltage Regulation

Definition

Change in secondary voltage from no-load to full-load as percentage of no-load voltage

Formula

\[\text{Regulation} = \frac{V_{20} - V_{2FL}}{V_{20}} \times 100\%\]

Approximate Formula

\[\text{Regulation} \approx \frac{I_2(R_{eq}\cos\phi \pm X_{eq}\sin\phi)}{V_2} \times 100\%\]

Sign Convention

  • + for lagging power factor (inductive load)

  • - for leading power factor (capacitive load)

Efficiency

Basic Formula

\[\eta = \frac{P_{out}}{P_{out} + P_{losses}} = \frac{P_{out}}{P_{out} + P_{core} + P_{copper}}\]

Losses

  • Core losses: \(P_{core} = W_0\) (constant)

  • Copper losses: \(P_{copper} = x^2 W_{sc}\) (where x = fraction of full load)

Efficiency at Any Load

\[\eta = \frac{x \cdot S_{rated} \cdot \cos\phi}{x \cdot S_{rated} \cdot \cos\phi + W_0 + x^2 W_{sc}}\]

Maximum Efficiency

Condition for Maximum Efficiency

Maximum efficiency occurs when:

\[\text{Copper losses} = \text{Core losses}\]
\[x^2 W_{sc} = W_0\]

Loading for Maximum Efficiency

\[x_{max\eta} = \sqrt{\frac{W_0}{W_{sc}}}\]

Maximum Efficiency Value

\[\eta_{max} = \frac{x \cdot S_{rated} \cdot \cos\phi}{x \cdot S_{rated} \cdot \cos\phi + 2W_0}\]

Key Point

Maximum efficiency is independent of power factor but the loading depends on it

Parallel Operation

Parallel Operation

Conditions for Parallel Operation

  1. Same voltage ratios (within ±0.5%)

  2. Same percentage impedances (within ±7.5%)

  3. Same impedance angles (X/R ratios)

  4. Same polarity and phase sequence

Load Sharing

\[\frac{I_1}{I_2} = \frac{Z_2}{Z_1} = \frac{kVA_2}{kVA_1}\]

Load Current Distribution

\[I_1 = I_{total} \times \frac{Z_2}{Z_1 + Z_2}, \quad I_2 = I_{total} \times \frac{Z_1}{Z_1 + Z_2}\]

Circulating Current

\[I_{circ} = \frac{\Delta V}{Z_1 + Z_2}\]

Per Unit System

Per Unit Analysis

Base Quantities

\[\begin{aligned} S_{base} &= \text{Transformer rating (VA)} \\ V_{base} &= \text{Rated voltage} \\ I_{base} &= \frac{S_{base}}{V_{base}} \\ Z_{base} &= \frac{V_{base}^2}{S_{base}} \end{aligned}\]

Per Unit Values

\[\begin{aligned} Z_{pu} &= \frac{Z_{actual}}{Z_{base}} = \frac{V_{sc}}{V_{rated}} \\ R_{pu} &= \frac{P_{sc}}{S_{rated}} \\ X_{pu} &= \sqrt{Z_{pu}^2 - R_{pu}^2} \end{aligned}\]

Advantages

  • Eliminates voltage level complications

  • Simplifies parallel operation analysis

GATE Problem Types

Common GATE Problem Types

Standard Problems

  1. Given OC and SC test data \(\to\) Find efficiency and regulation

  2. Parallel transformer load sharing calculations

  3. EMF equation and turn ratio problems

  4. Maximum efficiency condition problems

  5. Voltage regulation at different power factors

  6. Per unit impedance calculations

Problem Solving Steps

  1. Identify given data and required parameters

  2. Apply appropriate test formulas (OC/SC)

  3. Calculate equivalent circuit parameters

  4. Apply performance equations

  5. Check units and reasonableness

Common GATE Mistakes

Calculation Errors

  • Sign convention in regulation formula (\(\pm\))

  • Referring secondary quantities to primary

  • Confusion between rated and test voltages

  • Wrong base values in per unit calculations

Conceptual Errors

  • Confusing core losses with copper losses

  • Wrong assumptions about ideal vs real transformer

  • Misunderstanding parallel operation conditions

  • Incorrect interpretation of test data

Quick Reference

Key Formulas Summary

EMF and Ratios

\[\begin{aligned} E &= 4.44 f N \Phi_m \\ \frac{N_1}{N_2} &= \frac{V_1}{V_2} = a, \quad \frac{I_1}{I_2} = \frac{1}{a} \end{aligned}\]

Regulation and Efficiency

\[\begin{aligned} \text{Regulation} &= \frac{I_2(R_{eq}\cos\phi \pm X_{eq}\sin\phi)}{V_2} \times 100\% \\ \eta &= \frac{P_{out}}{P_{out} + W_0 + x^2 W_{sc}} \\ x_{max\eta} &= \sqrt{\frac{W_0}{W_{sc}}} \end{aligned}\]

Parallel Operation

\[\begin{aligned} \frac{I_1}{I_2} &= \frac{Z_2}{Z_1}, \quad I_{circ} = \frac{\Delta V}{Z_1 + Z_2} \end{aligned}\]

Typical Values for GATE

Standard Values

  • No-load current: 2-5% of rated current

  • Core losses: 0.2-1% of rating

  • Copper losses: 0.5-2% of rating

  • Efficiency: 95-99%

  • Regulation: 2-6% at full load

Per Unit Values

  • \(R_{pu}\): 0.005-0.02 (small transformers)

  • \(X_{pu}\): 0.02-0.10 (distribution transformers)

  • \(Z_{pu}\): 0.04-0.12 (typical range)

Final Tips for GATE

Preparation Strategy

  • Master the basic equivalent circuit

  • Practice OC and SC test problems extensively

  • Understand phasor diagrams for different loads

  • Focus on regulation and efficiency calculations

  • Remember sign conventions clearly

Key Relationships

  • \(P_{core} = W_0\) (constant from OC test)

  • \(P_{copper} = x^2 W_{sc}\) (variable with load)

  • Maximum efficiency when \(P_{core} = P_{copper}\)

  • Regulation depends on load power factor