GATE EE

Single-Phase Transformer GATE Exam Quick Notes

Lecture Notes

SEC 01

Basic Concepts

SEC 02

Single-Phase Transformer - Fundamentals

1Single-Phase Transformer - Fundamentals
1Definition & Principle

  • Static electrical machine transforming AC voltage levels

  • Works on principle of mutual induction

  • Energy transfer through magnetic coupling

1EMF 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)

1Transformation Ratios
\[\frac{N_1}{N_2} = \frac{E_1}{E_2} = \frac{V_1}{V_2} = a \quad \text{and} \quad \frac{I_1}{I_2} = \frac{1}{a}\]
SEC 03

Ideal vs Real Transformer

1Ideal vs Real Transformer
1Ideal Transformer

  • 100% efficiency, no losses

  • Infinite permeability, zero resistance

  • No leakage flux, perfect coupling

1Real Transformer

  • Core losses: Hysteresis + Eddy current

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

  • Leakage flux causing voltage drops

  • Magnetizing current required

1Referred Values (Secondary to Primary)
\[V_2' = aV_2, \quad I_2' = \frac{I_2}{a}, \quad R_2' = a^2R_2, \quad X_2' = a^2X_2\]
SEC 04

Equivalent Circuit

SEC 05

Equivalent Circuit Parameters

1Equivalent Circuit Parameters
1Exact 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 = \(I_c + I_m\)

1Approximate Equivalent Circuit
\[\begin{aligned} R_{eq} &= R_1 + R_2' \\ X_{eq} &= X_1 + X_2' \\ Z_{eq} &= \sqrt{R_{eq}^2 + X_{eq}^2} \end{aligned}\]
1Validity of Approximation

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

  • Shunt branch moved to supply side

SEC 06

Testing

SEC 07

Open Circuit Test

1Open Circuit Test
1Procedure

  • LV side: Rated voltage applied

  • HV side: Open circuit

  • Measurements: \(V_0, I_0, W_0\)

1Calculations
\[\begin{aligned} \cos\phi_0 &= \frac{W_0}{V_0 I_0} \\ R_0 &= \frac{V_0^2}{W_0} \\ X_0 &= \frac{V_0}{I_0 \sin\phi_0} \\ I_c &= I_0 \cos\phi_0, \quad I_m = I_0 \sin\phi_0 \end{aligned}\]
1Purpose

  • Determines core loss parameters

  • Core losses = \(W_0\) (constant)

SEC 08

Short Circuit Test

1Short Circuit Test
1Procedure

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

  • LV side: Short-circuited

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

1Calculations
\[\begin{aligned} R_{eq} &= \frac{W_{sc}}{I_{sc}^2} \\ Z_{eq} &= \frac{V_{sc}}{I_{sc}} \\ X_{eq} &= \sqrt{Z_{eq}^2 - R_{eq}^2} \\ \cos\phi_{sc} &= \frac{R_{eq}}{Z_{eq}} \end{aligned}\]
1Purpose

  • Determines equivalent circuit parameters

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

SEC 09

Performance Analysis

SEC 10

Voltage Regulation

1Voltage Regulation
1Definition

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

1Formula
\[\text{Regulation} = \frac{V_{20} - V_{2FL}}{V_{20}} \times 100\%\]
1Approximate Formula
\[\text{Regulation} \approx \frac{I_2(R_{eq}\cos\phi \pm X_{eq}\sin\phi)}{V_2} \times 100\%\]
1Sign Convention

  • + for lagging power factor (inductive load)

  • - for leading power factor (capacitive load)

SEC 11

Efficiency

1Efficiency
1Basic Formula
\[\eta = \frac{P_{out}}{P_{out} + P_{losses}} = \frac{P_{out}}{P_{out} + P_{core} + P_{copper}}\]
1Losses

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

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

1Efficiency at Any Load
\[\eta = \frac{x \cdot S_{rated} \cdot \cos\phi}{x \cdot S_{rated} \cdot \cos\phi + W_0 + x^2 W_{sc}}\]
SEC 12

Maximum Efficiency

1Maximum Efficiency
1Condition for Maximum Efficiency

Maximum efficiency occurs when:

\[\text{Copper losses} = \text{Core losses}\]
\[x^2 W_{sc} = W_0\]
1Loading for Maximum Efficiency
\[x_{max\eta} = \sqrt{\frac{W_0}{W_{sc}}}\]
1Maximum Efficiency Value
\[\eta_{max} = \frac{x \cdot S_{rated} \cdot \cos\phi}{x \cdot S_{rated} \cdot \cos\phi + 2W_0}\]
1Key Point

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

SEC 13

Parallel Operation

SEC 14

Parallel Operation

1Parallel Operation
1Conditions 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

1Load Sharing
\[\frac{I_1}{I_2} = \frac{Z_2}{Z_1} = \frac{kVA_2}{kVA_1}\]
1Load Current Distribution
\[I_1 = I_{total} \times \frac{Z_2}{Z_1 + Z_2}, \quad I_2 = I_{total} \times \frac{Z_1}{Z_1 + Z_2}\]
1Circulating Current
\[I_{circ} = \frac{\Delta V}{Z_1 + Z_2}\]
SEC 15

Per Unit System

SEC 16

Per Unit Analysis

1Per Unit Analysis
1Base Quantities
\[\begin{aligned} S_{base} &= \text{Transformer rating (VA)} \\ V_{base} &= \text{Rated voltage} \\ I_{base} &= \frac{S_{base}}{V_{base}} \\ Z_{base} &= \frac{V_{base}^2}{S_{base}} \end{aligned}\]
1Per Unit Values
\[\begin{aligned} Z_{pu} &= \frac{Z_{actual}}{Z_{base}} = \frac{V_{sc}}{V_{rated}} \\ R_{pu} &= \frac{P_{sc}}{S_{rated}} \\ X_{pu} &= \sqrt{Z_{pu}^2 - R_{pu}^2} \end{aligned}\]
1Advantages

  • Eliminates voltage level complications

  • Simplifies parallel operation analysis

SEC 17

GATE Problem Types

SEC 18

Common GATE Problem Types

1Common GATE Problem Types
1Standard 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

1Problem 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

SEC 19

Common GATE Mistakes

1Common GATE Mistakes
1Calculation 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

1Conceptual Errors

  • Confusing core losses with copper losses

  • Wrong assumptions about ideal vs real transformer

  • Misunderstanding parallel operation conditions

  • Incorrect interpretation of test data

SEC 20

Quick Reference

SEC 21

Key Formulas Summary

1Key Formulas Summary
1EMF and Ratios
\[\begin{aligned} E &= 4.44 f N \Phi_m \\ \frac{N_1}{N_2} &= \frac{V_1}{V_2} = a, \quad \frac{I_1}{I_2} = \frac{1}{a} \end{aligned}\]
1Regulation and Efficiency
\[\begin{aligned} \text{Regulation} &= \frac{I_2(R_{eq}\cos\phi \pm X_{eq}\sin\phi)}{V_2} \times 100\% \\ \eta &= \frac{P_{out}}{P_{out} + W_0 + x^2 W_{sc}} \\ x_{max\eta} &= \sqrt{\frac{W_0}{W_{sc}}} \end{aligned}\]
1Parallel Operation
\[\begin{aligned} \frac{I_1}{I_2} &= \frac{Z_2}{Z_1}, \quad I_{circ} = \frac{\Delta V}{Z_1 + Z_2} \end{aligned}\]
SEC 22

Typical Values for GATE

1Typical Values for GATE
1Standard 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

1Per 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)

SEC 23

Final Tips for GATE

1Final Tips for GATE
1Preparation 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

1Key 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