Chapter 32: Transformer

Master Navigation & Comprehensive Study Guide

Textbook: A Textbook of Electrical Technology (Vol. II — AC & DC Machines) by B.L. Theraja & A.K. Theraja
Chapter Scope: Pages 1115–1210 (96 PDF pages, complete 100% word-for-word digitization with vector-cropped diagrams)

---

📚 Vault Module Structure

To ensure lightning-fast rendering, seamless Obsidian mobile & desktop navigation, and focused topic study, this comprehensive 96-page chapter has been organized into five modular study units:

Ch-32/
├── Ch-32_Index.md                                 <-- Master Curriculum & Navigation Hub (You are here)
├── Ch-32_01_Construction_and_Principles.md        <-- Part 1: Working Principles, Construction, E.M.F. Equation (pp. 1115–1132)
├── Ch-32_02_Equivalent_Circuit_and_Drop.md        <-- Part 2: Equivalent Circuit, Phasors, Total Drops (pp. 1132–1145)
├── Ch-32_03_Testing_and_Regulation.md             <-- Part 3: OC/SC Tests, Voltage Regulation, Kapp Diagram (pp. 1145–1168)
├── Ch-32_04_Efficiency_and_AutoTransformers.md    <-- Part 4: Losses, Efficiency, All-Day Efficiency, Auto-Transformers (pp. 1168–1193)
└── Ch-32_05_Parallel_Operation_and_Tests.md       <-- Part 5: Parallel Operation, Q&A, Objective Tests (pp. 1193–1210)

---

📑 Detailed Table of Contents & Syllabus Mapping

Part 1: Working Principles, Construction & E.M.F. Equation

• Pages: 1115–1132 (PDF pp. 1–18)
• Sections Covered:
  • 32.1: Working Principle of a Transformer
  • 32.2: Transformer Construction (Core-type, Shell-type, Spiral Core, Laminations)
  • 32.3: Core-type and Shell-type Constructions (Comparison, Windings, Insulation)
  • 32.4: Elementary Theory of an Ideal Transformer
  • 32.5: E.M.F. Equation of a Transformer ($E=4.44fN{\Phi }_m$)
  • 32.6: Voltage Transformation Ratio ($K=\frac{E_2}{E_1}=\frac{N_2}{N_1}$)
  • 32.7: Transformer with Losses but no Magnetic Leakage
  • 32.8: Transformer on No-Load (No-load current $I_0$, $I_w$, $I_{\mu }$, no-load phasor diagram)
  • 32.9: Transformer on Load (MMF balance, secondary load current reflection $I_2^{\prime }$)
  • 32.10: Phasor Diagrams on Load (Unity p.f., Lagging p.f., Leading p.f.)
• Worked Examples: Examples 32.1 to 32.14
• Tutorial Problems: Tutorial Problems 32.1 (8 problems) & 32.2 (10 problems)
• Diagrams Included: 27 inline cropped figures (Ch-32_p01_... to Ch-32_p17_...)

---

Part 2: Equivalent Circuit, Magnetic Leakage & Voltage Drop

• Pages: 1132–1145 (PDF pp. 18–31)
• Sections Covered:
  • 32.11: Transformer with Resistance and Leakage Reactance (Primary & secondary leakage flux)
  • 32.12: Total Equivalent Resistance ($R_{01}=R_1+R_2/K^2$, $R_{02}=R_2+K^2{R}_1$)
  • 32.13: Total Equivalent Leakage Reactance ($X_{01}=X_1+X_2/K^2$, $X_{02}=X_2+K^2{X}_1$)
  • 32.14: Equivalent Impedance ($Z_{01}=\sqrt{R_{01}^2+X_{01}^2}$)
  • 32.15: Exact and Approximate Equivalent Circuits (Shunt branch shifting, cantilever circuits)
  • 32.16: Total Approximate Voltage Drop in a Transformer ($\Delta V=I_2{R}_{02}\cos \varphi \pm I_2{X}_{02}\sin \varphi$)
  • 32.17: Exact Voltage Drop Derivation (Quadratic expansion, exact phasor trigonometry)
  • 32.18: Vector Diagram of Transformer on Load (Complete secondary & primary vector diagrams)
• Worked Examples: Examples 32.15 to 32.26
• Diagrams Included: 23 inline cropped figures (Ch-32_p18_... to Ch-32_p31_...)

---

Part 3: Testing, Voltage Regulation & Sumpner's Test

• Pages: 1145–1168 (PDF pp. 31–54)
• Sections Covered:
  • 32.19: Transformer Tests (Purpose, advantages of indirect testing)
  • 32.20: Open-Circuit (No-Load) Test (Core loss determination, finding $R_0$, $X_0$, $Y_0$, $G_0$, $B_0$)
  • 32.21: Separation of Core Losses (Separating hysteresis loss $Af$ and eddy current loss $B{f}^2$)
  • 32.22: Short-Circuit (Impedance) Test (Determination of full-load Cu loss, $R_{01}$, $X_{01}$, $Z_{01}$)
  • 32.23: Why Transformer Rating is in kVA (Independent of power factor)
  • 32.24: Voltage Regulation of a Transformer (Regulation 'down', regulation 'up', conditions for zero and max regulation)
  • 32.25: Percentage Resistance, Reactance, and Impedance (Per-unit relations, independence of referral side)
  • 32.26: Kapp Regulation Diagram (Graphical circle diagram for terminal voltage and drop)
  • 32.27: Sumpner's or Back-to-Back Test (Heat run test without actual loading, separate measurement of losses)
• Worked Examples: Examples 32.27 to 32.58
• Tutorial Problems: Tutorial Problems 32.3 (11 problems)
• Diagrams Included: 16 inline cropped figures (Ch-32_p31_... to Ch-32_p53_...)

---

Part 4: Losses, Efficiency & Auto-Transformers

• Pages: 1168–1193 (PDF pp. 54–78)
• Sections Covered:
  • 32.28: Losses in a Transformer (Core/Iron losses: hysteresis & eddy currents; Copper losses: $I^{2}R$)
  • 32.29: Commercial Efficiency of a Transformer ($\eta =\frac{\text{Output}}{\text{Output}+\text{Losses}}$)
  • 32.30: Condition for Maximum Efficiency (Proof that $\text{Iron Loss}=\text{Copper Loss}$, ${\text{kVA}}_{{\eta }_{max}}={\text{kVA}}_{FL}\sqrt{W_i/W_{cu}}$)
  • 32.31: Variation of Efficiency with Power Factor (Maximum efficiency at unity power factor)
  • 32.32: All-Day Efficiency (Energy efficiency over 24-hour daily load cycle, distribution transformer design)
  • 32.33: Auto-Transformer (Single-winding transformer, step-up & step-down connections, copper savings: $\text{Saving}=K{W}_0$, inductive vs conductive power transfer)
  • 32.34: Conversion of Two-Winding Transformer into Auto-Transformer (kVA multiplication, connection schemes)
• Worked Examples: Examples 32.59 to 32.95
• Tutorial Problems: Tutorial Problems 32.4 (22 problems) & 32.5 (2 problems)
• Diagrams Included: 14 inline cropped figures (Ch-32_p54_... to Ch-32_p78_...)

---

Part 5: Parallel Operation, Questions & Answers & Objective Tests

• Pages: 1193–1210 (PDF pp. 79–96)
• Sections Covered:
  • 32.35: Parallel Operation of Single-Phase Transformers
    • Essential conditions (polarity, voltage ratio, frequency)
    • Desirable conditions (per-unit impedance, $X/R$ ratio)
    • Case 1: Transformers with Equal Voltage Ratios ($E_A=E_B$, load division formulas)
    • Case 2: Transformers with Unequal Voltage Ratios ($E_A\ne E_B$, circulating cross-current $I_C$, exact formulas)
  • Questions and Answers on Transformers: 10 detailed technical interview & exam Q&As
  • Objective Tests 32: 38 multiple-choice exam questions with complete Answer Key
• Worked Examples: Examples 32.96 to 32.111
• Tutorial Problems: Tutorial Problems 32.6 (7 problems) & 32.7 (2 problems)
• Diagrams Included: 9 inline cropped figures (Ch-32_p79_... to Ch-32_p90_...)

---

⚡ Master Formula Quick Reference

1. E.M.F. & Transformation

• R.M.S. Induced E.M.F.: $$E_1=4.44f{N}_1{\Phi }_m=4.44f{N}_1{B}_{max}A$$ $$E_2=4.44f{N}_2{\Phi }_m=4.44f{N}_2{B}_{max}A$$
• Transformation Ratio: $$K=\frac{E_2}{E_1}=\frac{N_2}{N_1}=\frac{I_1}{I_2}$$

2. Equivalent Circuit Parameters

• Referred to Primary: $$R_{01}=R_1+\frac{R_2}{K^2},X_{01}=X_1+\frac{X_2}{K^2},Z_{01}=\sqrt{R_{01}^2+X_{01}^2}$$
• Referred to Secondary: $$R_{02}=R_2+K^2{R}_1,X_{02}=X_2+K^2{X}_1,Z_{02}=\sqrt{R_{02}^2+X_{02}^2}$$
• Shunt Exciting Branch (from O.C. Test): $$\cos {\varphi }_0=\frac{W_0}{V_1{I}_0},I_w=I_0\cos {\varphi }_0,I_{\mu }=I_0\sin {\varphi }_0$$ $$R_0=\frac{V_1}{I_w},X_0=\frac{V_1}{I_{\mu }},G_0=\frac{W_0}{V_1^2},B_0=\sqrt{Y_0^2-G_0^2}$$

3. Voltage Regulation

• Approximate Expression: $$\%\text{ Regulation}=\frac{I_2(R_{02}\cos \varphi \pm X_{02}\sin \varphi )}{{}_0{V}_2}\times 100\%=v_r\cos \varphi \pm v_x\sin \varphi$$ ( $+$ for lagging p.f., $-$ for leading p.f. )
• Condition for Zero Regulation (Leading p.f. only): $$\tan \varphi =\frac{R_{02}}{X_{02}}⟹\cos \varphi =\frac{X_{02}}{\sqrt{R_{02}^2+X_{02}^2}}$$
• Condition for Maximum Regulation (Lagging p.f.): $$\tan \varphi =\frac{X_{02}}{R_{02}}⟹\cos \varphi =\frac{R_{02}}{\sqrt{R_{02}^2+X_{02}^2}}$$

4. Efficiency & Core Loss Separation

• Commercial Efficiency: $$\eta =\frac{x{S}_{FL}\cos \varphi }{x{S}_{FL}\cos \varphi +W_i+x^2{W}_{cu(FL)}}\times 100\%$$
• Condition for Maximum Efficiency: $$W_i=x^2{W}_{cu(FL)}⟹x=\sqrt{\frac{W_i}{W_{cu(FL)}}}$$ $$\text{kVA at }{\eta }_{max}={\text{kVA}}_{FL}\times \sqrt{\frac{W_i}{W_{cu(FL)}}}$$
• Core Loss Separation: $$\frac{W_i}{f}=A+Bf⟹W_h=Af,W_e=B{f}^2$$
• All-Day Efficiency: $${\eta }_{\text{all-day}}=\frac{\sum (P_{out}\times t)}{\sum (P_{out}\times t)+(W_i\times 24)+\sum (W_{cu}\times t)}\times 100\%$$

5. Auto-Transformers

• Copper Saving: $$\frac{\text{Weight of Cu in Auto-transformer}}{\text{Weight of Cu in 2-winding transformer}}=1-K$$ $$\text{Saving of Copper}=K\times W_0$$
• Power Transfer Breakdown: $$\text{Power transferred conductively}=K\times \text{Input Power}$$ $$\text{Power transferred inductively}=(1-K)\times \text{Input Power}$$
• kVA Rating Expansion (Two-winding to Auto): $${\text{kVA}}_{\text{auto}}={\text{kVA}}_{\text{2-w}}\times \frac{1}{1-K}\text{(Step-down)}$$

6. Parallel Operation

• Equal Voltage Ratios ($E_A=E_B$): I_A = I \left(\frac{Z_B}{Z_A + Z_B}

ight), \quad I_B = I \left(\frac{Z_A}{Z_A + Z_B} ight)$$

S_A = S \left(\frac{Z_B}{Z_A + Z_B} ight), \quad S_B = S \left(\frac{Z_A}{Z_A + Z_B} ight)$$ * **Unequal Voltage Ratios ($E_A \ne E_B$):** $$I_C = \frac{E_A - E_B}{Z_A + Z_B} \quad \text{(Circulating Current on No-Load)}$$ $$I_A = \frac{E_A Z_B + (E_A - E_B) Z_L}{Z_A Z_B + Z_L(Z_A + Z_B)}$$ $$I_B = \frac{E_B Z_A - (E_A - E_B) Z_L}{Z_A Z_B + Z_L(Z_A + Z_B)}$$ ---