Books/Chapman/Chapman_ECE2207_Solutions_Index.md

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title	Stephen J. Chapman: Electric Machinery Fundamentals - Master Solutions Index
book	Electric Machinery Fundamentals
edition	4th Edition
author	Stephen J. Chapman
course	ECE 2207 - Electrical Machines
total_chapters	5
total_problems	88
total_diagrams	83
total_pdf_pages_digitized	111
format	Obsidian-compatible Markdown
created_date	2026-09-30

Stephen J. Chapman: Electric Machinery Fundamentals (4th Edition)

Master Instructor's Solutions Manual — ECE 2207 Course Companion

Welcome to the complete, high-fidelity Obsidian Markdown digitization of Stephen J. Chapman's Electric Machinery Fundamentals (4th Edition) Instructor's Manual, specifically organized for ECE 2207 (Electrical Machines).

This digital edition features 100% word-for-word fidelity, complete step-by-step mathematical derivations with LaTeX formulas, 83 high-resolution cropped circuit schematics and MATLAB plots, and full executable MATLAB source codes.

Chapter	Title	PDF Pages	Book Pages	Problems	Diagrams	Markdown File
Ch 01	Introduction to Machinery Principles	7–28	1–22	1-1 to 1-22 (22)	21	`Chapman_Ch01_Introduction_to_Machinery_Principles.md`
Ch 02	Transformers	29–68	23–62	2-1 to 2-23 (23)	32	`Chapman_Ch02_Transformers.md`
Ch 04	AC Machinery Fundamentals	109–114	103–108	4-1 to 4-8 (8)	3	`Chapman_Ch04_AC_Machinery_Fundamentals.md`
Ch 07	Induction Motors	177–209	171–203	7-1 to 7-25 (25)	24	`Chapman_Ch07_Induction_Motors.md`
Ch 10	Single-Phase and Special-Purpose Motors	276–285	270–279	10-1 to 10-10 (10)	3	`Chapman_Ch10_Single_Phase_Motors.md`
TOTAL	5 Core Chapters	111 Pages	111 Pages	88 Problems	83 Diagrams	Complete Course Coverage

ECE 2207 Syllabus & Course Topic Mapping

graph TD
    ECE[ECE 2207: Electrical Machines] --> M1[Module 1: Magnetic Circuits & Principles]
    ECE --> M2[Module 2: Transformers]
    ECE --> M3[Module 3: AC Machine Fundamentals]
    ECE --> M4[Module 4: Polyphase Induction Motors]
    ECE --> M5[Module 5: Single-Phase & Special Motors]

    M1 --> Ch1[Chapter 1: Problems 1-1 to 1-22]
    M2 --> Ch2[Chapter 2: Problems 2-1 to 2-23]
    M3 --> Ch4[Chapter 4: Problems 4-1 to 4-8]
    M4 --> Ch7[Chapter 7: Problems 7-1 to 7-25]
    M5 --> Ch10[Chapter 10: Problems 10-1 to 10-10]

Module 1: Electromechanical Energy Conversion & Magnetic Circuits

Reference: Chapman_Ch01_Introduction_to_Machinery_Principles.md
Key Concepts Covered:
- Ampere's Law, magnetomotive force ( $F = N i$ ), magnetic field intensity ( $H$ ), flux density ( $B = μ H$ ), and total flux ( $ϕ$ ).
- Magnetic reluctance ( $R = l / μ A$ ) and permeance ( $P = 1/ R$ ) in series and parallel magnetic circuits (Problems 1-1, 1-2, 1-3, 1-4, 1-5).
- Magnetic core saturation, non-linear $B$ - $H$ curves, and magnetization characteristics (Problems 1-9, 1-10, 1-11, 1-12, 1-13, 1-14).
- Core losses: Hysteresis loss ( $P_{h} = k_{h} f B_{ma x}^{n}$ ) and eddy current loss ( $P_{e} = k_{e} f^{2} B_{ma x}^{2}$ ).
- Faraday's Law of electromagnetic induction ( $e_{in d} = - d λ / d t$ ) and Lenz's Law.
- Lorentz force ( $F = i (l \times B)$ ) and induced voltage in moving conductors ( $e_{in d} = (v \times B) \cdot l$ ).
- Linear DC machines: Starting conditions, acceleration, steady-state operation, motor vs. generator mode, dynamic braking (Problems 1-15 to 1-22).

Module 2: Transformers (Single-Phase, Three-Phase & Autotransformers)

Reference: Chapman_Ch02_Transformers.md
Key Concepts Covered:
- Real transformer per-phase equivalent circuit models: exact model vs. approximate model referred to primary or secondary (Problems 2-1, 2-2, 2-7, 2-8).
- Open-circuit (no-load) test and short-circuit test parameter extraction ( $R_{C}, X_{M}, R_{e q}, X_{e q}$ ) (Problems 2-3, 2-6, 2-7, 2-18, 2-19, 2-22).
- Voltage regulation ( $V R$ ) and transmission efficiency ( $η$ ) under lagging, unity, and leading power factor loads (Problems 2-1, 2-2, 2-3, 2-8, 2-10).
- Non-linear core magnetization current distortion and harmonics (Problem 2-5).
- Power systems and transmission lines: Loss reduction using step-up and step-down transformer pairs (Problems 2-4, 2-14, 2-23).
- Autotransformers: Construction, voltage/current relations, power advantage ( $S_{I O} / S_{W}$ ), impedance conversion ( $Z_{e q}^{'} = \frac{N _{S E}}{N _{S E} + N _{C}} Z_{e q}$ ) (Problems 2-12, 2-15, 2-16, 2-17, 2-22).
- Three-Phase Transformer Connections:
  - $Y$ - $Y$ , $Y$ - $Δ$ , $Δ$ - $Y$ , $Δ$ - $Δ$ voltage, current, and kVA ratings (Problem 2-9).
  - Open- $Δ$ ( $V$ - $V$ ) and Open- $Y$ —Open- $Δ$ connections for rural distribution (Problems 2-9, 2-13).
  - Rigorous phasor proofs of standard $3 0^{\circ}$ phase shifts: $Y$ - $Δ$ secondary lags by $3 0^{\circ}$ (Problem 2-20); $Δ$ - $Y$ secondary leads by $3 0^{\circ}$ (Problem 2-21).
- Frequency derating: Operating 60-Hz transformers on 50-Hz power grids (Problem 2-19).
- Power factor correction using capacitor banks in transformer distribution networks (Problem 2-23).

Module 3: AC Machine Fundamentals & Rotating Magnetic Fields

Reference: Chapman_Ch04_AC_Machinery_Fundamentals.md
Key Concepts Covered:
- The Rotating Magnetic Field (RMF): Derivation of constant magnitude ( $B_{n e t} = 1.5 B_{M}$ ) and synchronous angular velocity ( $ω_{sy n c} = 2 π f_{e}$ , $n_{sy n c} = 120 f_{e} / P$ ) (Problems 4-1, 4-2).
- Stator winding distribution, pole pitch, coil pitch, chording, and elimination of space harmonics (Problem 4-6).
- Pitch factor ( $k_{p}$ ), distribution factor ( $k_{d}$ ), and winding factor ( $k_{w} = k_{p} k_{d}$ ).
- RMS induced voltage in distributed AC stator windings ( $E_{A} = 2 π N_{c} f ϕ k_{w}$ ) (Problems 4-3, 4-4, 4-5).
- Machine speed regulation and rotor flux distribution (Problems 4-7, 4-8).

Module 4: Three-Phase Induction Motors

Reference: Chapman_Ch07_Induction_Motors.md
Key Concepts Covered:
- Slip ( $s = \frac{n _{sy n c} - n _{m}}{n _{sy n c}}$ ), slip speed, rotor frequency ( $f_{r} = s f_{e}$ ), and speed regulation (Problems 7-2, 7-3, 7-4, 7-6).
- Complete power flow stages: $P_{in} \to P_{S C L} \to P_{cor e} \to P_{A G} \to P_{R C L} \to P_{co n v} \to P_{F & W} + P_{mi sc} \to P_{o u t}$ (Problems 7-4, 7-5, 7-7, 7-15).
- Induction motor per-phase equivalent circuit analysis with Thévenin equivalent stator reduction ( $V_{T H}, R_{T H}, X_{T H}$ ) (Problems 7-7, 7-8, 7-12).
- Torque-speed characteristics: Induced torque equation, pullout torque ( $τ_{ma x}$ ), and pullout slip ( $s_{ma x}$ ) (Problems 7-8, 7-9, 7-16, 7-19).
- Wound-rotor induction motors: Insertion of external rotor resistance ( $R_{e x t}$ ) for starting torque optimization and speed control (Problems 7-10, 7-23).
- Frequency scaling: Operation of 60-Hz induction machines on 50-Hz power supplies (Problem 7-11).
- Non-linear and quadratic loads: Centrifugal pumps and fan characteristics ( $τ_{l o a d} \propto ω_{m}^{2}$ ) (Problem 7-13).
- Parameter extraction from laboratory tests: DC test ( $R_{1}$ ), No-Load test ( $X_{1} + X_{M}, P_{r o t}$ ), and Locked-Rotor test ( $R_{2}, X_{1}, X_{2}$ ) across NEMA Design Classes A, B, C, D (Problems 7-1, 7-14, 7-16, 7-18).
- Starting methods and controllers:
  - Across-the-line starting and terminal voltage dip (Problem 7-20).
  - Reduced-voltage autotransformer starters (Problems 7-20, 7-22).
  - Wye-Delta ( $Y$ - $Δ$ ) starters: Reduction of line starting current by factor of 3 (Problems 7-21, 7-24).
  - NEMA starting code letters and locked-rotor kVA/hp (Problems 7-19, 7-24).
- Rapid electric braking: The Plugging technique ( $s > 1$ , reverse torque) (Problem 7-25).

Module 5: Single-Phase and Special-Purpose Motors

Reference: Chapman_Ch10_Single_Phase_Motors.md
Key Concepts Covered:
- Theory of single-phase induction motors: Double Revolving Field Theory ( $B = B_{f} + B_{b}$ ) and Cross-Field Theory.
- Zero starting torque of pure single-phase stator winding and operational requirement for auxiliary starting circuits.
- Split-Phase Motors: Auxiliary winding with high resistance-to-reactance ratio ( $R_{A} / X_{A} > R_{M} / X_{M}$ ) creating phase angle shift $α \approx 3 0^{\circ}$ (Problem 10-1).
- Capacitor-Start Motors: Sizing of starting capacitor $C_{s t a r t}$ for optimal $9 0^{\circ}$ phase displacement to maximize starting torque (Problems 10-5, 10-7).
- Capacitor-Run and Permanent Split Capacitor (PSC) Motors: Design for balanced forward revolving field at rated running condition, minimizing backward field and acoustic noise (Problem 10-8).
- Capacitor-Start Capacitor-Run Motors: Dual capacitor configuration with centrifugal switch for optimal starting torque and smooth running efficiency.
- Shaded-Pole Induction Motors: Copper shading coil creating delayed flux and sweeping magnetic field; efficiency and application analysis (Problem 10-6).
- Stepper Motors: Variable-reluctance and permanent-magnet stepper motor step angles, teeth configurations, and pulse stepping rates (Problems 10-9, 10-10).

Master Problem Directory (All 88 Problems)

Chapter 1: Introduction to Machinery Principles (22 Problems)

Problem 1-1: Ferromagnetic core with air gap; flux, flux density, and reluctance calculation.
Problem 1-2: Two-legged magnetic core with varying cross-sectional area and air gap.
Problem 1-3: Three-legged symmetric magnetic core with center air gap.
Problem 1-4: Three-legged asymmetric magnetic core with air gaps in outer legs.
Problem 1-5: Three-legged core with air gap in center leg; total reluctance and flux distribution.
Problem 1-6: Magnetic circuit with multiple coils and opposing MMFs.
Problem 1-7: Magnetic core with fringing effects at the air gap.
Problem 1-8: Fringing calculation with effective air gap dimensions.
Problem 1-9: Non-linear core magnetization using given $B$ - $H$ magnetization curve.
Problem 1-10: Current required to establish specified flux in saturation region.
Problem 1-11: Relative permeability $μ_{r}$ variation across operating range.
Problem 1-12: Air gap effect on total magnetization characteristic of non-linear core.
Problem 1-13: Core loss calculation: Hysteresis loss and eddy current loss separation.
Problem 1-14: Transformer core loss variation with frequency and voltage scaling.
Problem 1-15: Moving conductor in uniform magnetic field: Induced voltage and polarity.
Problem 1-16: Current-carrying conductor in magnetic field: Force magnitude and direction.
Problem 1-17: Elementary linear DC machine: Starting current and initial acceleration.
Problem 1-18: Linear DC machine no-load steady-state velocity and induced voltage.
Problem 1-19: Linear DC motor under mechanical load: Velocity drop and current increase.
Problem 1-20: Linear DC generator mode: Applied mechanical force and power conversion.
Problem 1-21: Dynamic braking of linear DC machine by reversing terminal connections.
Problem 1-22: Linear DC machine efficiency and power flow analysis.

Chapter 2: Transformers (23 Problems)

Problem 2-1: Approximate equivalent circuit referred to primary: $I_{P}$ , voltage regulation, efficiency.
Problem 2-2: 20-kVA 8000/480-V transformer: Equivalent circuits referred to HV & LV, VR, efficiency.
Problem 2-3: Parameter extraction from OC and SC tests; VR at lagging, unity, leading PF.
Problem 2-4: Power system with real transformers and transmission line: VR & efficiency.
Problem 2-5: Non-linear magnetization current simulation with MATLAB at 120 V / 60 Hz and 240 V / 50 Hz.
Problem 2-6: 15-kVA distribution transformer: Performance with inductive vs. capacitive load.
Problem 2-7: 5000-kVA power transformer OC test on LV, per-unit series parameters, full-load VR.
Problem 2-8: 200-MVA transformer per-unit model; MATLAB voltage regulation profiles across loads.
Problem 2-9: 600-kVA three-phase transformer bank ratings for Y-Y, Y- $Δ$ , $Δ$ -Y, $Δ$ - $Δ$ , Open- $Δ$ .
Problem 2-10: 13,800/480-V three-phase Y- $Δ$ bank: Per-phase model, MATLAB $V_{S}$ and VR curves.
Problem 2-11: 100-MVA $Δ$ - $Δ$ transformer bank: Phasor diagram, VR, and LV equivalent circuit.
Problem 2-12: Autotransformer 13.2-kV to 13.8-kV step-up: Turns ratio and power advantage ( $S_{I O} / S_{W} = 23$ ).
Problem 2-13: Open-Y—Open- $Δ$ connection serving mixed 3-phase and 1-phase rural loads.
Problem 2-14: Transmission loss comparison: Direct connection vs. step-up/step-down system ( $80 \times$ loss reduction).
Problem 2-15: 480/120-V transformer reconnected as 600/120-V step-down autotransformer.
Problem 2-16: 480/120-V transformer reconnected as 600/480-V step-down autotransformer (25 kVA rating).
Problem 2-17: Mathematical proof of autotransformer series impedance reduction factor.
Problem 2-18: Three 25-kVA transformers in $Δ$ -Y: Parameter extraction, per-unit circuit, VR, efficiency.
Problem 2-19: 20-kVA distribution transformer at 60 Hz and derating for 50-Hz power grid.
Problem 2-20: Rigorous phasor proof: Secondary voltage lags primary by $3 0^{\circ}$ in Y- $Δ$ connection.
Problem 2-21: Rigorous phasor proof: Secondary voltage leads primary by $3 0^{\circ}$ in $Δ$ -Y connection.
Problem 2-22: 10-kVA transformer conventional per-unit model vs. 50-kVA autotransformer performance.
Problem 2-23: Three-phase power transmission system: Per-unit model, capacitor bank power factor correction.

Chapter 4: AC Machinery Fundamentals (8 Problems)

Problem 4-1: Rotating magnetic field from two-phase currents; clockwise vs. counter-clockwise rotation.
Problem 4-2: Three-phase stator winding rotating magnetic field verification and MATLAB script.
Problem 4-3: Induced voltage in a 4-pole distributed AC stator winding.
Problem 4-4: Induced voltage calculation for 2-pole Y-connected machine.
Problem 4-5: Line-to-line vs. phase voltages in Y-connected and $Δ$ -connected AC machines.
Problem 4-6: Fractional pitch (chorded) windings: Pitch factor $k_{p}$ and elimination of 5th/7th harmonics.
Problem 4-7: AC generator speed regulation from no-load to full-load.
Problem 4-8: Non-sinusoidal rotor flux distribution: Fundamental and harmonic components.

Chapter 7: Induction Motors (25 Problems)

Problem 7-1: DC test on $Δ$ -connected stator; calculation of per-phase resistance $R_{1}$ .
Problem 7-2: Synchronous speed, rotor speed, slip speed, and rotor frequency at 5% slip.
Problem 7-3: 4-pole, 208-V, 60-Hz induction motor speed and slip calculations.
Problem 7-4: Power flow balance: Input power, copper losses, core losses, converted power, output power.
Problem 7-5: 50-kW motor shaft speed, output power, load torque, induced torque, rotor frequency.
Problem 7-6: No-load and full-load slip, rotor frequency, and speed regulation.
Problem 7-7: Complete per-phase equivalent circuit analysis: $I_{L}, P_{S C L}, P_{A G}, P_{co n v}, τ_{in d}, τ_{l o a d}, η$ .
Problem 7-8: Pullout torque $τ_{ma x}$ and pullout slip $s_{ma x}$ using Thévenin equivalent.
Problem 7-9: MATLAB simulation of torque-speed and output power-speed curves.
Problem 7-10: External rotor resistance for maximum torque at starting condition ( $s_{ma x} = 1.0$ ).
Problem 7-11: 50-Hz derating from 60-Hz design: Voltage reduction, equivalent circuit, performance.
Problem 7-12: Circuit model derivation with core loss resistance $R_{C}$ parallel to $X_{M}$ .
Problem 7-13: Quadratic fan/pump load torque ( $τ_{l o a d} \propto ω_{m}^{2}$ ); operating point calculation.
Problem 7-14: Parameter extraction from DC, No-Load, and Locked-Rotor tests for 75-kW motor.
Problem 7-15: Efficiency calculation for motor of Problem 7-14 at rated slip.
Problem 7-16: Parameter extraction for 208-V Design Class B motor; pullout torque calculation.
Problem 7-17: Comprehensive MATLAB analysis: $τ_{in d}, P_{co n v}, P_{o u t}, η$ vs. speed curves.
Problem 7-18: Test parameter extraction & MATLAB torque-speed curve for Design Class B motor.
Problem 7-19: Determination of rotor resistance $R_{2}$ from full-load point; pullout torque; NEMA Code D.
Problem 7-20: Starting current comparison: Infinite bus vs. line impedance vs. autotransformer starter.
Problem 7-21: Wye-Delta ( $Y$ - $Δ$ ) reduced-voltage starter: Voltage and current reduction by factor of 3.
Problem 7-22: Autotransformer starter design for 100-hp motor to limit starting torque to rated value.
Problem 7-23: Wound-rotor motor at 25% load with doubled rotor resistance: Comprehensive parameter shifts.
Problem 7-24: Starting current for Code Letter E motor with across-the-line, Y- $Δ$ , and autotransformer.
Problem 7-25: Rapid stopping by plugging: Slip ( $s = 1.962$ ), rotor frequency ( $117.7 Hz$ ), braking torque.

Chapter 10: Single-Phase and Special-Purpose Motors (10 Problems)

Problem 10-1: Split-phase motor starting currents, phase angle $α$ , and starting torque.
Problem 10-2: Starting torque comparison between split-phase motor and capacitor-start motor.
Problem 10-3: Rotor speed and slip calculations under forward and backward revolving fields.
Problem 10-4: Rotor copper losses under forward and backward magnetic fields ( $P_{R C L, f}$ and $P_{R C L, b}$ ).
Problem 10-5: Capacitor sizing for maximum starting torque in capacitor-start induction motor ( $C_{s t a r t}$ ).
Problem 10-6: Shaded-pole induction motor: Operating principle, efficiency, and applications.
Problem 10-7: Complete MATLAB simulation of torque-speed curve for split-phase and capacitor-start motors.
Problem 10-8: Permanent split-capacitor (PSC) motor run capacitor optimization for quiet operation.
Problem 10-9: Variable-reluctance stepper motor: Teeth, poles, and step angle calculation.
Problem 10-10: Permanent-magnet stepper motor: Step angle and pulse frequency for target shaft velocity.

Embedded MATLAB Computational Scripts Directory

The following full, executable MATLAB scripts are embedded directly within the markdown files:

Script Name	Chapter	Problem	Description	Output Plot Embedded
`mag_field.m`	Ch 04	4-2	Visualizes rotating magnetic field in 3-phase stator	`Chapman_Ch04_p109_fig4-1.jpg`
`prob2_5a.m`	Ch 02	2-5(a)	Core magnetization current at 120 V / 60 Hz	`Chapman_Ch02_p38_vr_plot.jpg`
`prob2_5b.m`	Ch 02	2-5(b)	Core magnetization current at 240 V / 50 Hz	`Chapman_Ch02_p39_eff_plot.jpg`
`prob2_8.m`	Ch 02	2-8(c)	Secondary voltage vs. load for varying power factors	`Chapman_Ch02_p44_vr_plot.jpg`
`prob2_10c.m`	Ch 02	2-10(c)	Three-phase bank secondary voltage vs. load current	`Chapman_Ch02_p47_v_sec_plot.jpg`
`prob2_10d.m`	Ch 02	2-10(d)	Three-phase bank voltage regulation vs. load current	`Chapman_Ch02_p49_vr_plot.jpg`
`prob7_9a.m`	Ch 07	7-9(a)	Torque-speed characteristic of 25-hp induction motor	`Chapman_Ch07_p184_torque_speed_plot.jpg`
`prob7_9b.m`	Ch 07	7-9(b)	Output power vs. speed curve of 25-hp induction motor	`Chapman_Ch07_p185_power_speed_plot.jpg`
`prob7_17.m`	Ch 07	7-17	Multi-curve simulation: $τ_{in d}, P_{co n v}, P_{o u t}, η$ vs. speed	`Chapman_Ch07_p196_.jpg`, `p197_.jpg`
`prob7_18.m`	Ch 07	7-18	Torque-speed curve for Design Class B motor	`Chapman_Ch07_p200_torque_speed_plot.jpg`
`prob10_7.m`	Ch 10	10-7	Split-phase vs. capacitor-start torque-speed comparison	`Chapman_Ch10_p281_torque_speed_plot.jpg`

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Books/Chapman/Chapman_ECE2207_Solutions_Index.md

25.6 KB · commit d2f3b6e

title	Stephen J. Chapman: Electric Machinery Fundamentals - Master Solutions Index
book	Electric Machinery Fundamentals
edition	4th Edition
author	Stephen J. Chapman
course	ECE 2207 - Electrical Machines
total_chapters	5
total_problems	88
total_diagrams	83
total_pdf_pages_digitized	111
format	Obsidian-compatible Markdown
created_date	2026-09-30

Stephen J. Chapman: Electric Machinery Fundamentals (4th Edition)

Master Instructor's Solutions Manual — ECE 2207 Course Companion

Chapter	Title	PDF Pages	Book Pages	Problems	Diagrams	Markdown File
Ch 01	Introduction to Machinery Principles	7–28	1–22	1-1 to 1-22 (22)	21	`Chapman_Ch01_Introduction_to_Machinery_Principles.md`
Ch 02	Transformers	29–68	23–62	2-1 to 2-23 (23)	32	`Chapman_Ch02_Transformers.md`
Ch 04	AC Machinery Fundamentals	109–114	103–108	4-1 to 4-8 (8)	3	`Chapman_Ch04_AC_Machinery_Fundamentals.md`
Ch 07	Induction Motors	177–209	171–203	7-1 to 7-25 (25)	24	`Chapman_Ch07_Induction_Motors.md`
Ch 10	Single-Phase and Special-Purpose Motors	276–285	270–279	10-1 to 10-10 (10)	3	`Chapman_Ch10_Single_Phase_Motors.md`
TOTAL	5 Core Chapters	111 Pages	111 Pages	88 Problems	83 Diagrams	Complete Course Coverage

ECE 2207 Syllabus & Course Topic Mapping

graph TD
    ECE[ECE 2207: Electrical Machines] --> M1[Module 1: Magnetic Circuits & Principles]
    ECE --> M2[Module 2: Transformers]
    ECE --> M3[Module 3: AC Machine Fundamentals]
    ECE --> M4[Module 4: Polyphase Induction Motors]
    ECE --> M5[Module 5: Single-Phase & Special Motors]

    M1 --> Ch1[Chapter 1: Problems 1-1 to 1-22]
    M2 --> Ch2[Chapter 2: Problems 2-1 to 2-23]
    M3 --> Ch4[Chapter 4: Problems 4-1 to 4-8]
    M4 --> Ch7[Chapter 7: Problems 7-1 to 7-25]
    M5 --> Ch10[Chapter 10: Problems 10-1 to 10-10]

Module 1: Electromechanical Energy Conversion & Magnetic Circuits

Reference: Chapman_Ch01_Introduction_to_Machinery_Principles.md
Key Concepts Covered:
- Ampere's Law, magnetomotive force ( $F = N i$ ), magnetic field intensity ( $H$ ), flux density ( $B = μ H$ ), and total flux ( $ϕ$ ).
- Magnetic reluctance ( $R = l / μ A$ ) and permeance ( $P = 1/ R$ ) in series and parallel magnetic circuits (Problems 1-1, 1-2, 1-3, 1-4, 1-5).
- Magnetic core saturation, non-linear $B$ - $H$ curves, and magnetization characteristics (Problems 1-9, 1-10, 1-11, 1-12, 1-13, 1-14).
- Core losses: Hysteresis loss ( $P_{h} = k_{h} f B_{ma x}^{n}$ ) and eddy current loss ( $P_{e} = k_{e} f^{2} B_{ma x}^{2}$ ).
- Faraday's Law of electromagnetic induction ( $e_{in d} = - d λ / d t$ ) and Lenz's Law.
- Lorentz force ( $F = i (l \times B)$ ) and induced voltage in moving conductors ( $e_{in d} = (v \times B) \cdot l$ ).
- Linear DC machines: Starting conditions, acceleration, steady-state operation, motor vs. generator mode, dynamic braking (Problems 1-15 to 1-22).

Module 2: Transformers (Single-Phase, Three-Phase & Autotransformers)

Reference: Chapman_Ch02_Transformers.md
Key Concepts Covered:
- Real transformer per-phase equivalent circuit models: exact model vs. approximate model referred to primary or secondary (Problems 2-1, 2-2, 2-7, 2-8).
- Open-circuit (no-load) test and short-circuit test parameter extraction ( $R_{C}, X_{M}, R_{e q}, X_{e q}$ ) (Problems 2-3, 2-6, 2-7, 2-18, 2-19, 2-22).
- Voltage regulation ( $V R$ ) and transmission efficiency ( $η$ ) under lagging, unity, and leading power factor loads (Problems 2-1, 2-2, 2-3, 2-8, 2-10).
- Non-linear core magnetization current distortion and harmonics (Problem 2-5).
- Power systems and transmission lines: Loss reduction using step-up and step-down transformer pairs (Problems 2-4, 2-14, 2-23).
- Autotransformers: Construction, voltage/current relations, power advantage ( $S_{I O} / S_{W}$ ), impedance conversion ( $Z_{e q}^{'} = \frac{N _{S E}}{N _{S E} + N _{C}} Z_{e q}$ ) (Problems 2-12, 2-15, 2-16, 2-17, 2-22).
- Three-Phase Transformer Connections:
  - $Y$ - $Y$ , $Y$ - $Δ$ , $Δ$ - $Y$ , $Δ$ - $Δ$ voltage, current, and kVA ratings (Problem 2-9).
  - Open- $Δ$ ( $V$ - $V$ ) and Open- $Y$ —Open- $Δ$ connections for rural distribution (Problems 2-9, 2-13).
  - Rigorous phasor proofs of standard $3 0^{\circ}$ phase shifts: $Y$ - $Δ$ secondary lags by $3 0^{\circ}$ (Problem 2-20); $Δ$ - $Y$ secondary leads by $3 0^{\circ}$ (Problem 2-21).
- Frequency derating: Operating 60-Hz transformers on 50-Hz power grids (Problem 2-19).
- Power factor correction using capacitor banks in transformer distribution networks (Problem 2-23).

Module 3: AC Machine Fundamentals & Rotating Magnetic Fields

Reference: Chapman_Ch04_AC_Machinery_Fundamentals.md
Key Concepts Covered:
- The Rotating Magnetic Field (RMF): Derivation of constant magnitude ( $B_{n e t} = 1.5 B_{M}$ ) and synchronous angular velocity ( $ω_{sy n c} = 2 π f_{e}$ , $n_{sy n c} = 120 f_{e} / P$ ) (Problems 4-1, 4-2).
- Stator winding distribution, pole pitch, coil pitch, chording, and elimination of space harmonics (Problem 4-6).
- Pitch factor ( $k_{p}$ ), distribution factor ( $k_{d}$ ), and winding factor ( $k_{w} = k_{p} k_{d}$ ).
- RMS induced voltage in distributed AC stator windings ( $E_{A} = 2 π N_{c} f ϕ k_{w}$ ) (Problems 4-3, 4-4, 4-5).
- Machine speed regulation and rotor flux distribution (Problems 4-7, 4-8).

Module 4: Three-Phase Induction Motors

Reference: Chapman_Ch07_Induction_Motors.md
Key Concepts Covered:
- Slip ( $s = \frac{n _{sy n c} - n _{m}}{n _{sy n c}}$ ), slip speed, rotor frequency ( $f_{r} = s f_{e}$ ), and speed regulation (Problems 7-2, 7-3, 7-4, 7-6).
- Complete power flow stages: $P_{in} \to P_{S C L} \to P_{cor e} \to P_{A G} \to P_{R C L} \to P_{co n v} \to P_{F & W} + P_{mi sc} \to P_{o u t}$ (Problems 7-4, 7-5, 7-7, 7-15).
- Induction motor per-phase equivalent circuit analysis with Thévenin equivalent stator reduction ( $V_{T H}, R_{T H}, X_{T H}$ ) (Problems 7-7, 7-8, 7-12).
- Torque-speed characteristics: Induced torque equation, pullout torque ( $τ_{ma x}$ ), and pullout slip ( $s_{ma x}$ ) (Problems 7-8, 7-9, 7-16, 7-19).
- Wound-rotor induction motors: Insertion of external rotor resistance ( $R_{e x t}$ ) for starting torque optimization and speed control (Problems 7-10, 7-23).
- Frequency scaling: Operation of 60-Hz induction machines on 50-Hz power supplies (Problem 7-11).
- Non-linear and quadratic loads: Centrifugal pumps and fan characteristics ( $τ_{l o a d} \propto ω_{m}^{2}$ ) (Problem 7-13).
- Parameter extraction from laboratory tests: DC test ( $R_{1}$ ), No-Load test ( $X_{1} + X_{M}, P_{r o t}$ ), and Locked-Rotor test ( $R_{2}, X_{1}, X_{2}$ ) across NEMA Design Classes A, B, C, D (Problems 7-1, 7-14, 7-16, 7-18).
- Starting methods and controllers:
  - Across-the-line starting and terminal voltage dip (Problem 7-20).
  - Reduced-voltage autotransformer starters (Problems 7-20, 7-22).
  - Wye-Delta ( $Y$ - $Δ$ ) starters: Reduction of line starting current by factor of 3 (Problems 7-21, 7-24).
  - NEMA starting code letters and locked-rotor kVA/hp (Problems 7-19, 7-24).
- Rapid electric braking: The Plugging technique ( $s > 1$ , reverse torque) (Problem 7-25).

Module 5: Single-Phase and Special-Purpose Motors

Reference: Chapman_Ch10_Single_Phase_Motors.md
Key Concepts Covered:
- Theory of single-phase induction motors: Double Revolving Field Theory ( $B = B_{f} + B_{b}$ ) and Cross-Field Theory.
- Zero starting torque of pure single-phase stator winding and operational requirement for auxiliary starting circuits.
- Split-Phase Motors: Auxiliary winding with high resistance-to-reactance ratio ( $R_{A} / X_{A} > R_{M} / X_{M}$ ) creating phase angle shift $α \approx 3 0^{\circ}$ (Problem 10-1).
- Capacitor-Start Motors: Sizing of starting capacitor $C_{s t a r t}$ for optimal $9 0^{\circ}$ phase displacement to maximize starting torque (Problems 10-5, 10-7).
- Capacitor-Run and Permanent Split Capacitor (PSC) Motors: Design for balanced forward revolving field at rated running condition, minimizing backward field and acoustic noise (Problem 10-8).
- Capacitor-Start Capacitor-Run Motors: Dual capacitor configuration with centrifugal switch for optimal starting torque and smooth running efficiency.
- Shaded-Pole Induction Motors: Copper shading coil creating delayed flux and sweeping magnetic field; efficiency and application analysis (Problem 10-6).
- Stepper Motors: Variable-reluctance and permanent-magnet stepper motor step angles, teeth configurations, and pulse stepping rates (Problems 10-9, 10-10).

Master Problem Directory (All 88 Problems)

Chapter 1: Introduction to Machinery Principles (22 Problems)

Problem 1-1: Ferromagnetic core with air gap; flux, flux density, and reluctance calculation.
Problem 1-2: Two-legged magnetic core with varying cross-sectional area and air gap.
Problem 1-3: Three-legged symmetric magnetic core with center air gap.
Problem 1-4: Three-legged asymmetric magnetic core with air gaps in outer legs.
Problem 1-5: Three-legged core with air gap in center leg; total reluctance and flux distribution.
Problem 1-6: Magnetic circuit with multiple coils and opposing MMFs.
Problem 1-7: Magnetic core with fringing effects at the air gap.
Problem 1-8: Fringing calculation with effective air gap dimensions.
Problem 1-9: Non-linear core magnetization using given $B$ - $H$ magnetization curve.
Problem 1-10: Current required to establish specified flux in saturation region.
Problem 1-11: Relative permeability $μ_{r}$ variation across operating range.
Problem 1-12: Air gap effect on total magnetization characteristic of non-linear core.
Problem 1-13: Core loss calculation: Hysteresis loss and eddy current loss separation.
Problem 1-14: Transformer core loss variation with frequency and voltage scaling.
Problem 1-15: Moving conductor in uniform magnetic field: Induced voltage and polarity.
Problem 1-16: Current-carrying conductor in magnetic field: Force magnitude and direction.
Problem 1-17: Elementary linear DC machine: Starting current and initial acceleration.
Problem 1-18: Linear DC machine no-load steady-state velocity and induced voltage.
Problem 1-19: Linear DC motor under mechanical load: Velocity drop and current increase.
Problem 1-20: Linear DC generator mode: Applied mechanical force and power conversion.
Problem 1-21: Dynamic braking of linear DC machine by reversing terminal connections.
Problem 1-22: Linear DC machine efficiency and power flow analysis.

Chapter 2: Transformers (23 Problems)

Problem 2-1: Approximate equivalent circuit referred to primary: $I_{P}$ , voltage regulation, efficiency.
Problem 2-2: 20-kVA 8000/480-V transformer: Equivalent circuits referred to HV & LV, VR, efficiency.
Problem 2-3: Parameter extraction from OC and SC tests; VR at lagging, unity, leading PF.
Problem 2-4: Power system with real transformers and transmission line: VR & efficiency.
Problem 2-5: Non-linear magnetization current simulation with MATLAB at 120 V / 60 Hz and 240 V / 50 Hz.
Problem 2-6: 15-kVA distribution transformer: Performance with inductive vs. capacitive load.
Problem 2-7: 5000-kVA power transformer OC test on LV, per-unit series parameters, full-load VR.
Problem 2-8: 200-MVA transformer per-unit model; MATLAB voltage regulation profiles across loads.
Problem 2-9: 600-kVA three-phase transformer bank ratings for Y-Y, Y- $Δ$ , $Δ$ -Y, $Δ$ - $Δ$ , Open- $Δ$ .
Problem 2-10: 13,800/480-V three-phase Y- $Δ$ bank: Per-phase model, MATLAB $V_{S}$ and VR curves.
Problem 2-11: 100-MVA $Δ$ - $Δ$ transformer bank: Phasor diagram, VR, and LV equivalent circuit.
Problem 2-12: Autotransformer 13.2-kV to 13.8-kV step-up: Turns ratio and power advantage ( $S_{I O} / S_{W} = 23$ ).
Problem 2-13: Open-Y—Open- $Δ$ connection serving mixed 3-phase and 1-phase rural loads.
Problem 2-14: Transmission loss comparison: Direct connection vs. step-up/step-down system ( $80 \times$ loss reduction).
Problem 2-15: 480/120-V transformer reconnected as 600/120-V step-down autotransformer.
Problem 2-16: 480/120-V transformer reconnected as 600/480-V step-down autotransformer (25 kVA rating).
Problem 2-17: Mathematical proof of autotransformer series impedance reduction factor.
Problem 2-18: Three 25-kVA transformers in $Δ$ -Y: Parameter extraction, per-unit circuit, VR, efficiency.
Problem 2-19: 20-kVA distribution transformer at 60 Hz and derating for 50-Hz power grid.
Problem 2-20: Rigorous phasor proof: Secondary voltage lags primary by $3 0^{\circ}$ in Y- $Δ$ connection.
Problem 2-21: Rigorous phasor proof: Secondary voltage leads primary by $3 0^{\circ}$ in $Δ$ -Y connection.
Problem 2-22: 10-kVA transformer conventional per-unit model vs. 50-kVA autotransformer performance.
Problem 2-23: Three-phase power transmission system: Per-unit model, capacitor bank power factor correction.

Chapter 4: AC Machinery Fundamentals (8 Problems)

Problem 4-1: Rotating magnetic field from two-phase currents; clockwise vs. counter-clockwise rotation.
Problem 4-2: Three-phase stator winding rotating magnetic field verification and MATLAB script.
Problem 4-3: Induced voltage in a 4-pole distributed AC stator winding.
Problem 4-4: Induced voltage calculation for 2-pole Y-connected machine.
Problem 4-5: Line-to-line vs. phase voltages in Y-connected and $Δ$ -connected AC machines.
Problem 4-6: Fractional pitch (chorded) windings: Pitch factor $k_{p}$ and elimination of 5th/7th harmonics.
Problem 4-7: AC generator speed regulation from no-load to full-load.
Problem 4-8: Non-sinusoidal rotor flux distribution: Fundamental and harmonic components.

Chapter 7: Induction Motors (25 Problems)

Problem 7-1: DC test on $Δ$ -connected stator; calculation of per-phase resistance $R_{1}$ .
Problem 7-2: Synchronous speed, rotor speed, slip speed, and rotor frequency at 5% slip.
Problem 7-3: 4-pole, 208-V, 60-Hz induction motor speed and slip calculations.
Problem 7-4: Power flow balance: Input power, copper losses, core losses, converted power, output power.
Problem 7-5: 50-kW motor shaft speed, output power, load torque, induced torque, rotor frequency.
Problem 7-6: No-load and full-load slip, rotor frequency, and speed regulation.
Problem 7-7: Complete per-phase equivalent circuit analysis: $I_{L}, P_{S C L}, P_{A G}, P_{co n v}, τ_{in d}, τ_{l o a d}, η$ .
Problem 7-8: Pullout torque $τ_{ma x}$ and pullout slip $s_{ma x}$ using Thévenin equivalent.
Problem 7-9: MATLAB simulation of torque-speed and output power-speed curves.
Problem 7-10: External rotor resistance for maximum torque at starting condition ( $s_{ma x} = 1.0$ ).
Problem 7-11: 50-Hz derating from 60-Hz design: Voltage reduction, equivalent circuit, performance.
Problem 7-12: Circuit model derivation with core loss resistance $R_{C}$ parallel to $X_{M}$ .
Problem 7-13: Quadratic fan/pump load torque ( $τ_{l o a d} \propto ω_{m}^{2}$ ); operating point calculation.
Problem 7-14: Parameter extraction from DC, No-Load, and Locked-Rotor tests for 75-kW motor.
Problem 7-15: Efficiency calculation for motor of Problem 7-14 at rated slip.
Problem 7-16: Parameter extraction for 208-V Design Class B motor; pullout torque calculation.
Problem 7-17: Comprehensive MATLAB analysis: $τ_{in d}, P_{co n v}, P_{o u t}, η$ vs. speed curves.
Problem 7-18: Test parameter extraction & MATLAB torque-speed curve for Design Class B motor.
Problem 7-19: Determination of rotor resistance $R_{2}$ from full-load point; pullout torque; NEMA Code D.
Problem 7-20: Starting current comparison: Infinite bus vs. line impedance vs. autotransformer starter.
Problem 7-21: Wye-Delta ( $Y$ - $Δ$ ) reduced-voltage starter: Voltage and current reduction by factor of 3.
Problem 7-22: Autotransformer starter design for 100-hp motor to limit starting torque to rated value.
Problem 7-23: Wound-rotor motor at 25% load with doubled rotor resistance: Comprehensive parameter shifts.
Problem 7-24: Starting current for Code Letter E motor with across-the-line, Y- $Δ$ , and autotransformer.
Problem 7-25: Rapid stopping by plugging: Slip ( $s = 1.962$ ), rotor frequency ( $117.7 Hz$ ), braking torque.

Chapter 10: Single-Phase and Special-Purpose Motors (10 Problems)

Problem 10-1: Split-phase motor starting currents, phase angle $α$ , and starting torque.
Problem 10-2: Starting torque comparison between split-phase motor and capacitor-start motor.
Problem 10-3: Rotor speed and slip calculations under forward and backward revolving fields.
Problem 10-4: Rotor copper losses under forward and backward magnetic fields ( $P_{R C L, f}$ and $P_{R C L, b}$ ).
Problem 10-5: Capacitor sizing for maximum starting torque in capacitor-start induction motor ( $C_{s t a r t}$ ).
Problem 10-6: Shaded-pole induction motor: Operating principle, efficiency, and applications.
Problem 10-7: Complete MATLAB simulation of torque-speed curve for split-phase and capacitor-start motors.
Problem 10-8: Permanent split-capacitor (PSC) motor run capacitor optimization for quiet operation.
Problem 10-9: Variable-reluctance stepper motor: Teeth, poles, and step angle calculation.
Problem 10-10: Permanent-magnet stepper motor: Step angle and pulse frequency for target shaft velocity.

Embedded MATLAB Computational Scripts Directory

The following full, executable MATLAB scripts are embedded directly within the markdown files:

Script Name	Chapter	Problem	Description	Output Plot Embedded
`mag_field.m`	Ch 04	4-2	Visualizes rotating magnetic field in 3-phase stator	`Chapman_Ch04_p109_fig4-1.jpg`
`prob2_5a.m`	Ch 02	2-5(a)	Core magnetization current at 120 V / 60 Hz	`Chapman_Ch02_p38_vr_plot.jpg`
`prob2_5b.m`	Ch 02	2-5(b)	Core magnetization current at 240 V / 50 Hz	`Chapman_Ch02_p39_eff_plot.jpg`
`prob2_8.m`	Ch 02	2-8(c)	Secondary voltage vs. load for varying power factors	`Chapman_Ch02_p44_vr_plot.jpg`
`prob2_10c.m`	Ch 02	2-10(c)	Three-phase bank secondary voltage vs. load current	`Chapman_Ch02_p47_v_sec_plot.jpg`
`prob2_10d.m`	Ch 02	2-10(d)	Three-phase bank voltage regulation vs. load current	`Chapman_Ch02_p49_vr_plot.jpg`
`prob7_9a.m`	Ch 07	7-9(a)	Torque-speed characteristic of 25-hp induction motor	`Chapman_Ch07_p184_torque_speed_plot.jpg`
`prob7_9b.m`	Ch 07	7-9(b)	Output power vs. speed curve of 25-hp induction motor	`Chapman_Ch07_p185_power_speed_plot.jpg`
`prob7_17.m`	Ch 07	7-17	Multi-curve simulation: $τ_{in d}, P_{co n v}, P_{o u t}, η$ vs. speed	`Chapman_Ch07_p196_.jpg`, `p197_.jpg`
`prob7_18.m`	Ch 07	7-18	Torque-speed curve for Design Class B motor	`Chapman_Ch07_p200_torque_speed_plot.jpg`
`prob10_7.m`	Ch 10	10-7	Split-phase vs. capacitor-start torque-speed comparison	`Chapman_Ch10_p281_torque_speed_plot.jpg`

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azazarfin60/Electrical_Machine_I_Semester_Final

Books/Chapman/Chapman_ECE2207_Solutions_Index.md

Stephen J. Chapman: Electric Machinery Fundamentals (4th Edition)

Master Instructor's Solutions Manual — ECE 2207 Course Companion

Modular Chapter Navigation

ECE 2207 Syllabus & Course Topic Mapping

Module 1: Electromechanical Energy Conversion & Magnetic Circuits

Module 2: Transformers (Single-Phase, Three-Phase & Autotransformers)

Module 3: AC Machine Fundamentals & Rotating Magnetic Fields

Module 4: Three-Phase Induction Motors

Module 5: Single-Phase and Special-Purpose Motors

Master Problem Directory (All 88 Problems)

Chapter 1: Introduction to Machinery Principles (22 Problems)

Chapter 2: Transformers (23 Problems)

Chapter 4: AC Machinery Fundamentals (8 Problems)

Chapter 7: Induction Motors (25 Problems)

Chapter 10: Single-Phase and Special-Purpose Motors (10 Problems)

Embedded MATLAB Computational Scripts Directory

Instructions for Obsidian & Markdown Readers

Report a problem

azazarfin60/Electrical_Machine_I_Semester_Final

Books/Chapman/Chapman_ECE2207_Solutions_Index.md

Stephen J. Chapman: Electric Machinery Fundamentals (4th Edition)

Master Instructor's Solutions Manual — ECE 2207 Course Companion

Modular Chapter Navigation

ECE 2207 Syllabus & Course Topic Mapping

Module 1: Electromechanical Energy Conversion & Magnetic Circuits

Module 2: Transformers (Single-Phase, Three-Phase & Autotransformers)

Module 3: AC Machine Fundamentals & Rotating Magnetic Fields

Module 4: Three-Phase Induction Motors

Module 5: Single-Phase and Special-Purpose Motors

Master Problem Directory (All 88 Problems)

Chapter 1: Introduction to Machinery Principles (22 Problems)

Chapter 2: Transformers (23 Problems)

Chapter 4: AC Machinery Fundamentals (8 Problems)

Chapter 7: Induction Motors (25 Problems)

Chapter 10: Single-Phase and Special-Purpose Motors (10 Problems)

Embedded MATLAB Computational Scripts Directory

Instructions for Obsidian & Markdown Readers

Report a problem