Electrical Machines

From Fundamentals to GATE-Level Mastery — A Comprehensive Study Note

Author: Dr. Mithun Mondal | Associate Professor, Department of Electrical and Electronics Engineering, BITS Pilani, Hyderabad Campus | Category: EEE · Revision Notes

Learning Objectives On completing this study note, the reader will be able to:

Explain the principles of electromagnetic induction and magnetic circuits, and apply them to electrical machines.
Analyse the steady-state performance of DC machines, transformers, induction motors, and synchronous machines.
Derive the EMF, torque, power-flow, and equivalent-circuit equations for each machine class.
Compute efficiency, voltage regulation, slip, power factor, and starting performance using standard tests.
Select appropriate starting, speed-control, and braking techniques for a given application.
Solve GATE/ESE/PSU-level numerical problems with clear physical interpretation.
Appreciate modern developments — BLDC, PMSM, SRM, stepper, linear motors — and their role in EVs and automation.

Pre-requisites: Basic Circuit Theory · Electromagnetic Field Theory · Fundamentals of Electrical Engineering.

1. Fundamentals of Electromagnetism

Four Pillars of Electrical Machines Every electrical machine, however complex, rests on four foundations:

Magnetic-field production by current.
Force on current-carrying conductors in a field.
EMF induction by changing flux.
Energy conservation between electrical, magnetic, and mechanical domains.

Faraday's and Lenz's Laws

The induced EMF in a coil of \(N\) turns is given by Faraday's law: \[ e \;=\; -N\,\frac{d\Phi}{dt}. \] Lenz's law states that the induced EMF always opposes the change that caused it, ensuring energy conservation.

Ampere's Law and MMF

For a magnetic circuit with a mean path length \(\ell\), cross-sectional area \(A\), and permeability \(\mu\): \[ \mathcal{F} \;=\; NI \;=\; H\ell \;=\; \Phi\,\mathcal{R}, \qquad \mathcal{R} = \frac{\ell}{\mu A}. \] Reluctances in series add; reluctances in parallel combine as conductances (the dual of Kirchhoff's laws for electric circuits).

Hysteresis and Eddy-Current Losses

Ferromagnetic cores suffer two types of core loss. The B–H hysteresis loop shows residual flux density \(B_r\) retained when \(H \to 0\) and coercive force \(H_c\) required to reduce \(B\) back to zero. The area enclosed by the loop represents energy dissipated per cycle per unit volume.

Core Loss Formulas (Steinmetz) \[ P_h \;=\; K_h\,B_m^{1.6}\,f\,V \qquad \text{(hysteresis loss)} \] \[ P_e \;=\; K_e\,B_m^{2}\,f^{2}\,t^{2}\,V \qquad \text{(eddy-current loss)} \]

To reduce eddy-current loss, cores are made of thin silicon-steel laminations of thickness \(t\). To reduce hysteresis loss, materials with a narrow loop (CRGO, amorphous alloys) are used.

2. Electromechanical Energy Conversion

Energy Balance (Motoring Convention) \[ \underbrace{W_{\text{elec}}}_{\text{input}} \;=\; \underbrace{W_{\text{mech}}}_{\text{output}} \;+\; \underbrace{\Delta W_{\text{fld}}}_{\text{stored}} \;+\; \underbrace{W_{\text{loss}}}_{\text{dissipated}} \]

Field Energy and Co-energy (Linear, Singly-Excited)

For a linear magnetic system: \[ W_{\text{fld}} \;=\; \int_0^{\lambda} i\,d\lambda \;=\; \frac{\lambda^2}{2L} \;=\; \frac{1}{2}LI^2, \qquad W'_{\text{fld}} \;=\; \int_0^{i}\lambda\,di \;=\; \frac{1}{2}LI^2. \] The force at constant current (mechanical variable \(x\)) is: \[ F \;=\; \left.\frac{\partial W'_{\text{fld}}}{\partial x}\right|_i. \]

Torque Expressions (Rotational) Singly-excited: \[ T \;=\; \frac{1}{2}\,I^2\,\frac{dL(\theta)}{d\theta} \] Doubly-excited (general): \[ T \;=\; \frac{1}{2}I_1^2\,L_{11}' + \frac{1}{2}I_2^2\,L_{22}' + I_1 I_2\,M', \quad (\cdot)' \equiv d(\cdot)/d\theta \]

Singly-Excited vs Doubly-Excited Systems

In a singly-excited system (e.g., variable-reluctance stepper, switched-reluctance motor), only one winding carries current and torque arises purely from reluctance: \(T = \tfrac{1}{2}I^2\,dL/d\theta\). Non-zero torque requires rotor saliency (\(dL/d\theta \neq 0\)).

In a doubly-excited system (e.g., synchronous machine, DC machine, wound-rotor induction motor), both windings carry current. For a cylindrical rotor the self-inductance derivatives vanish, leaving only the mutual term; for a salient-pole rotor both reluctance and mutual torques contribute.

3. DC Machines

Construction

A DC machine consists of a yoke (outer flux path and mechanical support), pole cores and shoes carrying the field winding, a laminated armature core, an armature winding (lap or wave), a commutator (mechanical AC-to-DC rectifier), brushes, and optionally interpoles and a compensating winding for commutation improvement.

Lap vs Wave Winding
Property	Lap	Wave
Parallel paths \(A\)	\(P\)	\(2\)
Best for	High current, low voltage	Low current, high voltage
Equalizers	Yes	No
Number of brushes	\(P\)	\(2\)
Coil span	≈ pole pitch	≈ 2 × pole pitch

Memory Aid \(\text{Lap} \to A = P\) (high current); \(\text{Wave} \to A = 2\) (high voltage).

Cross-section of a 4-pole DC machine showing yoke, pole shoes, field winding, armature slots with conductors (dots out / crosses in), commutator brushes, and magnetic flux paths from north to south pole. — Cross-section of a 4-pole DC machine illustrating the yoke, field winding, pole shoe, air-gap, armature conductors (current directions shown by dot and cross symbols), brush contact in the magnetic neutral plane, and the air-gap flux path from north to south pole.

EMF and Torque Equations

The EMF in one conductor is \(e = B L v\). Summing over all series conductors per parallel path:

DC Machine EMF and Torque \[ E_a \;=\; \frac{\Phi\,Z\,N\,P}{60\,A} \;=\; \frac{\Phi\,Z\,\omega\,P}{2\pi A} \] \[ T \;=\; \frac{\Phi\,Z\,I_a\,P}{2\pi A} \;=\; \frac{60}{2\pi N}\,E_a I_a \quad [\text{N·m}] \]

Motor and Generator Conventions

Motor (back-EMF \(E_b\)): \(V = E_b + I_a R_a\), so \(E_b = V - I_a R_a\).
Generator (generated EMF \(E_g\)): \(E_g = V + I_a R_a\).

Mechanical power developed: \(P_m = E_b I_a = T\omega\). Shaft output: \(P_{\text{out}} = P_m - P_{\text{rot}}\).

Speed Equation \[ N \;=\; \frac{V - I_a R_a}{K\,\Phi} \;=\; K_1\,\frac{E_b}{\Phi} \] Speed varies directly with armature voltage and inversely with flux. Below base speed: keep \(\Phi\) fixed, vary \(V\) (constant-torque region). Above base speed: keep \(V\) fixed, weaken \(\Phi\) (constant-power / field-weakening region).

Armature Reaction and Commutation

Armature reaction is the effect of the armature MMF on the main field. It has two components: a cross-magnetising component that distorts the field along the q-axis, and a demagnetising component when brushes are shifted forward.

Demagnetising and Cross-Magnetising AT/Pole \[ AT_d \;=\; \frac{Z\,I_a}{2AP}\cdot\frac{2\beta}{180°}, \qquad AT_c \;=\; \frac{Z\,I_a}{2AP}\!\left(1 - \frac{2\beta}{180°}\right) \] (\(\beta\) = brush shift in mechanical degrees)

Commutation involves reversing the current in a coil passing under the brush. The reactance voltage \(e_L = L(2I_c/T_c)\) opposes reversal, causing delayed commutation and sparking. Remedies include interpoles (commutating poles), a compensating winding in the pole face for large machines, and high-resistance carbon brushes.

DC Generator: Types and Self-Excitation

DC generators are classified as separately excited (field from external source) or self-excited. Self-excited types include shunt (\(R_{sh}\) in parallel), series (\(R_{se}\) in series), and compound (both, cumulative or differential).

Shunt and Series Generator Equations Shunt: \(\;I_a = I_L + I_{sh},\;\; I_{sh} = V/R_{sh},\;\; E_g = V + I_a R_a + V_{br}\)
Series: \(\;I_a = I_{se} = I_L,\;\; E_g = V + I_a(R_a + R_{se})\)

Conditions for Self-Excitation (Shunt Generator) (i) Residual magnetism must be present; (ii) field connection must aid residual flux; (iii) \(R_{sh} < R_c\) (critical resistance); (iv) \(N > N_c\) (critical speed).

DC Motor: Types and Speed–Torque Characteristics

Property	Shunt	Series	Cum. Compound	Diff. Compound
Speed regulation	Good	Poor (highly variable)	Moderate	Very flat
Starting torque	Moderate	Very high	High	Low
No-load behaviour	Safe	Runaway	Safe	Safe
\(T\)–\(I_a\) shape	Linear	Parabolic (\(\propto I_a^2\))	Mixed	Mixed
Typical applications	Lathes, fans	Traction, cranes	Rolling mills, lifts	Rare

For a shunt motor, flux is constant, so \(T \propto I_a\) and speed is approximately constant. For a series motor, pre-saturation gives \(\Phi \propto I_a\), hence \(T \propto I_a^2\) and \(N \propto 1/\sqrt{T}\).

Speed Control and Braking of DC Motors

Shunt motor methods: (a) Flux control via variable field resistance — gives speeds above base, constant power; (b) Armature-resistance control — below base speed, constant torque, lossy; (c) Voltage control (Ward–Leonard) — smooth, four-quadrant operation.

Braking Methods

Regenerative: \(E_b > V\) — power returned to mains.
Dynamic (rheostatic): disconnect supply, dissipate \(E_b\) in a resistor.
Plugging: reverse supply polarity — fastest stop, very lossy.

Modern alternatives include chopper (DC–DC) drives for buck/boost/four-quadrant control, and fully-controlled rectifier drives where \(V_a = V_{dm}\cos\alpha\) for a full converter under continuous conduction.

DC Machine Losses, Efficiency, and Tests

Losses are classified as: copper losses (\(I_a^2 R_a\), \(I_f^2 R_f\), brush contact) — variable with load; iron losses (hysteresis and eddy current) — approximately constant; mechanical losses (friction, windage) — approximately constant; and stray losses (~1% of output).

Condition for Maximum Efficiency \[ P_{\text{variable}} \;=\; P_{\text{constant}} \;\Rightarrow\; I_a^2 R_a = P_{\text{const}} \]

Swinburne's test is a no-load test on a shunt motor that determines constant losses, enabling efficiency prediction at any load (not applicable to series motors). Hopkinson's (back-to-back) test uses two identical machines to find full-load losses without supplying full-load power.

4. Transformers

Definition A transformer is a static device that transfers electrical energy between two or more circuits via electromagnetic induction without change of frequency.

EMF Equation \[ E_1 \;=\; 4.44\,f\,N_1\,\Phi_m \;=\; 4.44\,f\,N_1\,B_m\,A \] \[ E_2 \;=\; 4.44\,f\,N_2\,\Phi_m \] Transformation ratio (ideal): \(\;K = N_2/N_1 = E_2/E_1 = I_1/I_2\)

The factor 4.44 arises from \(\sqrt{2}\pi \approx 4.44\), which appears when integrating \(V_m\sin\omega t\) over a half-cycle and converting to RMS.

Core-Type vs Shell-Type Construction

Side-by-side cross-sections of core-type transformer (two-limb rectangular core with LV and HV windings on each limb) and shell-type transformer (three-limb core with both windings concentrated on the central limb). — Comparative cross-section of core-type and shell-type transformer constructions, showing the laminated core arrangement, LV (inner) and HV (outer) winding placement, and the flux return path through the yoke.

In the core-type, each limb carries half the LV and half the HV winding, providing good leakage reactance control. In the shell-type, both windings sit concentrically on the central limb with the core enclosing them, providing better mechanical protection and lower leakage flux.

Equivalent Circuit and Phasor Diagram

Per-phase equivalent circuit of a single-phase transformer showing primary resistance R1 and leakage reactance X1 in series, shunt branch with core-loss resistance Rc and magnetising reactance Xm in parallel, followed by referred secondary resistance R2-prime and leakage reactance X2-prime. — Per-phase equivalent circuit of a single-phase transformer referred to the primary side, illustrating the series primary impedance, shunt magnetising branch, and referred secondary impedance.

Voltage Regulation

Voltage Regulation \[ \text{Reg\%} \;=\; \frac{V_{NL} - V_{FL}}{V_{FL}} \times 100 \;\approx\; \frac{I_2(R_{02}\cos\phi \pm X_{02}\sin\phi)}{V_2} \times 100 \] \[ \text{Reg (pu)} \;=\; v_r\cos\phi \pm v_x\sin\phi \;+\; \frac{(v_x\cos\phi \mp v_r\sin\phi)^2}{2} \] Use \(+\) for lagging pf, \(-\) for leading pf.

Zero regulation occurs at \(\tan\phi = R_{02}/X_{02}\) (leading load). Maximum regulation occurs at \(\tan\phi = X_{02}/R_{02}\) (lagging load): \(\text{Reg}_{\max} = \sqrt{v_r^2 + v_x^2}\).

Efficiency and All-Day Efficiency

\[ \eta \;=\; \frac{V_2 I_2\cos\phi}{V_2 I_2\cos\phi + P_i + P_{cu}} \]

Condition for Maximum Efficiency \[ P_{cu} \;=\; P_i \quad \text{(copper loss = iron loss)} \] \[ \text{kVA at }\eta_{\max} \;=\; \text{kVA}_{FL}\,\sqrt{P_i / P_{cu,FL}} \]

All-Day (Energy) Efficiency \[ \eta_{\text{all-day}} \;=\; \frac{\text{kWh output per day}}{\text{kWh input per day}} \] Important for distribution transformers energised 24 hours a day under partial load. Design rule: keep \(P_i < P_{cu,FL}\) to maximise daily efficiency.

Open-Circuit and Short-Circuit Tests

Open-Circuit (OC) Test Performed on the LV side with the HV side open at rated voltage. Measures iron loss \(W_0\) and yields the shunt-branch parameters: \[ R_0 = \frac{V_1^2}{W_0}, \quad X_0 = \frac{V_1}{I_\mu}, \quad I_w = \frac{W_0}{V_1}, \quad I_\mu = \sqrt{I_0^2 - I_w^2} \]

Short-Circuit (SC) Test Performed on the HV side with the LV side shorted at reduced voltage (to circulate rated current). Measures copper loss \(W_{sc}\) and yields the series-branch parameters: \[ R_{eq} = \frac{W_{sc}}{I_{sc}^2}, \quad Z_{eq} = \frac{V_{sc}}{I_{sc}}, \quad X_{eq} = \sqrt{Z_{eq}^2 - R_{eq}^2} \]

Sumpner's (Back-to-Back) Test Two identical transformers are connected back-to-back, allowing simultaneous determination of both iron and copper losses and full-load thermal testing without supplying full-load power from the mains.

Three-Phase Transformer Connections

Connection	Line Voltage Ratio	Phase Shift	Application
Y–Y	\(N_1/N_2\)	\(0°\)	Rarely used (3rd-harmonic issues)
\(\Delta\)–\(\Delta\)	\(N_1/N_2\)	\(0°\)	Large LV; fault-tolerant (open-\(\Delta\) possible)
Y–\(\Delta\)	\(\sqrt{3}\,N_1/N_2\)	\(30°\) lag	Generator step-down
\(\Delta\)–Y	\(N_1/(\sqrt{3}\,N_2)\)	\(30°\) lead	Distribution step-up; LV neutral available

Conditions for Parallel Operation (1) Same voltage ratio; (2) same per-unit impedance; (3) same polarity; (4) same phase sequence; (5) same vector group / phase displacement. Load sharing: \(S_A/S_B = Z_{B,\text{pu}}/Z_{A,\text{pu}}\) (inversely proportional to pu impedance).

Special Transformers

Auto-transformer A single winding is partly common — both conductively and magnetically coupled. Copper saving \(= (1-K) \times\) two-winding copper weight, where \(K = V_2/V_1\). Advantages: smaller, cheaper, better regulation and efficiency. Disadvantages: no isolation, higher fault current, full LV stress appears on a fault.

Scott Connection (3ϕ → 2ϕ) Uses two single-phase transformers — the main at 100% tap and the teaser at \(86.6\% = \sqrt{3}/2\) tap. Used for arc furnaces and electric traction.

Instrument Transformers CT: steps down current to 5 A or 1 A — never open-circuit the secondary! (dangerously high induced voltage).
PT: steps down voltage to 110 V — never short-circuit the secondary.

Magnetising Inrush Current and Harmonics

Magnetising Inrush Current Switching at certain instants (especially at voltage zero with adverse residual flux) drives the core into deep saturation. Inrush can reach 8–10× rated current, dominated by the 2nd harmonic (exploited in differential-relay restraint logic). Worst-case flux peak: \(\Phi_{\text{peak}} \approx 2\Phi_m + \Phi_r\).

Excitation-current harmonics arise from B–H non-linearity, predominantly the 3rd, plus 5th, 7th, etc. In a Y–Y connection without neutral, 3rd-harmonic flux has no path, causing peaked phase EMF and insulation stress. A tertiary delta winding provides a circulating path for 3rd-harmonic currents.

Tap Changers and Three-Winding Transformers

Taps on the HV side adjust the turns ratio to control voltage. An off-load tap changer (DETC) requires de-energisation; an on-load tap changer (OLTC) switches seamlessly using a diverter switch with resistor or reactor transition. Typical range: ±10% in steps of 1.25% or 2.5%.

A three-winding transformer adds a tertiary (usually delta) winding that suppresses 3rd-harmonic distortion, supplies auxiliary loads, or connects reactive compensators. An open-delta (V–V) connection delivers \(\sqrt{3}\) times the single-unit rating, i.e., 57.7% of the closed-delta capacity — used for emergency or small loads.

5. Three-Phase Induction Motors

Working Principle The 3-phase stator winding produces a rotating magnetic field; this field induces EMF and current in the rotor; the interaction produces torque. Because the rotor must lag the field to experience a changing flux, the machine is also called an asynchronous motor.

Synchronous Speed and Slip \[ N_s = \frac{120f}{P} \;[\text{rpm}], \qquad \omega_s = \frac{4\pi f}{P}\;[\text{rad/s}] \] \[ s \;=\; \frac{N_s - N_r}{N_s}, \qquad f_r = sf, \qquad E_{2r} = sE_2, \qquad X_{2r} = sX_2 \]

The squirrel-cage rotor uses short-circuited aluminium or copper bars — rugged and simple but with relatively low starting torque. The slip-ring (wound-rotor) type brings out the 3-phase rotor winding via slip rings, allowing external resistance for high starting torque and speed control.

Rotating Magnetic Field (Ferraris Principle)

Stator cross-section showing three sets of windings (Phase A in red, Phase B in green, Phase C in gold) placed 120 degrees apart in space, with the resultant rotating flux vector shown at a given instant. — Cross-section of a three-phase stator illustrating Phase A, B, and C windings displaced 120° in space. The three-phase supply currents (120° apart in time) produce a constant-amplitude resultant flux \(\vec{B}_{\text{net}} = \tfrac{3}{2}B_m\) that rotates at synchronous speed \(\omega_s\).

Three windings displaced 120° in space, fed by three-phase currents displaced 120° in time, produce: \[ \vec{B}_{\text{net}}(t) \;=\; B_A + B_B + B_C \;=\; \tfrac{3}{2}\,B_m \angle\omega t. \] The amplitude is constant at \(\tfrac{3}{2}B_m\) and the direction rotates at \(\omega_s = 2\pi f/(P/2)\). Swapping any two supply phases reverses the direction of rotation.

Torque Equation

Torque Developed (per phase) \[ T \;=\; \frac{3}{\omega_s}\cdot\frac{s\,E_2^2\,R_2}{R_2^2 + (sX_2)^2} \] Starting torque (\(s=1\)): \[ T_{st} = \frac{3}{\omega_s}\cdot\frac{E_2^2\,R_2}{R_2^2 + X_2^2} \]

Maximum Torque: Condition \(R_2 = s_m X_2\) \[ s_m \;=\; \frac{R_2}{X_2}, \qquad T_{\max} \;=\; \frac{3}{\omega_s}\cdot\frac{E_2^2}{2X_2} \] \(T_{\max}\) is independent of \(R_2\). Adding rotor resistance shifts \(s_m\) to a higher value without altering \(T_{\max}\). Maximum starting torque is achieved when \(R_2 = X_2\), giving \(s_m = 1\). \[ \frac{T_{st}}{T_{\max}} \;=\; \frac{2s_m}{1 + s_m^2} \]

Torque–Slip Characteristic

Graph of normalised torque T/Tmax versus slip from 0 to 1, showing three curves for small, moderate, and maximum rotor resistance. All curves share the same peak torque value but the peak shifts to higher slip as rotor resistance increases. — Torque–slip characteristics of a three-phase induction motor for three values of rotor resistance. The maximum torque \(T_{\max}\) is identical for all curves; increasing \(R_2\) shifts the peak to a higher slip value \(s_m\), enabling higher starting torque at \(s = 1\) without affecting \(T_{\max}\).

Three operating regions exist: the stable motoring region (\(0 < s < s_m\), where \(T \propto s\)); the unstable region (\(s_m < s < 1\), transited only during starting); and the plugging region (\(s > 1\), used for braking).

Kloss's Approximation \[ \frac{T}{T_{\max}} \;\approx\; \frac{2}{s/s_m + s_m/s} \] At low slip: \(T \propto s\). At high slip: \(T \propto 1/s\).

Power Flow

Key Power Relations — Very Important for GATE \[ P_{ag} : P_{cu2} : P_m \;=\; 1 : s : (1-s) \] \[ P_{cu2} \;=\; s\,P_{ag}, \qquad P_m \;=\; (1-s)\,P_{ag} \] \[ T \;=\; \frac{P_{ag}}{\omega_s} \;=\; \frac{P_m}{\omega_r} \;=\; \frac{P_{cu2}}{s\,\omega_s} \] Rotor efficiency \(\eta_r \leq (1-s)\); hence IMs run at low slip (typically 2–5%).

Equivalent Circuit and Tests

Per-phase equivalent circuit of a three-phase induction motor showing stator resistance R1 and leakage reactance X1 in series, shunt magnetising branch with Rc and Xm in parallel, followed by referred rotor resistance R2-prime over slip (decomposed into R2-prime plus the mechanical-power resistance R2-prime times (1-s)/s) and rotor leakage reactance X2-prime. — Per-phase equivalent circuit of a three-phase induction motor referred to the stator. The term \(R_2'(1-s)/s\) represents the mechanical power developed by the rotor.

The no-load test (rated voltage, no mechanical load) separates core loss and friction/windage loss and yields the shunt branch parameters \(R_c\) and \(X_m\). The blocked-rotor test (reduced voltage, rotor locked) gives the series branch parameters \(R_{eq}\) and \(X_{eq}\).

Starting Methods

Starter	Line Starting Current	\(T_{st}/T_{FL}\)
Direct-on-line (DOL)	\(I_{sc}\)	\((I_{sc}/I_{FL})^2\,s_{FL}\)
Stator resistor / reactor	\(x\,I_{sc}\)	\(x^2(I_{sc}/I_{FL})^2\,s_{FL}\)
Auto-transformer (\(K\) tap)	\(K^2 I_{sc}\)	\(K^2(I_{sc}/I_{FL})^2\,s_{FL}\)
Star–Delta	\(I_{sc}/3\)	\(\tfrac{1}{3}(I_{sc}/I_{FL})^2\,s_{FL}\)
Rotor-resistance (slip-ring)	As designed	Up to \(T_{\max}\)
Soft starter (TRIAC/IGBT)	Smoothly ramped	Adjustable
VFD	\(\leq I_{FL}\)	\(T_{\max}\) at start

Star–Delta starting requires the motor to be rated for delta-running; both line current and starting torque reduce to one-third of DOL values.

Speed Control

Stator-side methods: V/f control (most common — keeps \(\Phi\) constant, \(V/f = \text{const}\)); voltage control (lossy, \(T \propto V^2\)); pole-changing for discrete speed steps; variable-frequency drive (VFD).

Rotor-side methods (slip-ring only): external rotor resistance (simple but lossy); cascade connection; slip-power recovery via Kramer or Scherbius drives (efficient for large machines).

Cogging and Crawling Cogging: magnetic locking at standstill — prevented by skewing rotor slots.
Crawling: stable running at approximately \(N_s/7\) — caused by 7th-harmonic torque from the slotted stator.

Braking and Double-Cage Rotor

Braking methods include plugging (reverse two phases, slip becomes \(2-s\), high rotor heating), dynamic braking (disconnect AC, apply DC to stator), and regenerative braking (drive \(N_r > N_s\), slip becomes negative, power returned to mains).

Double-Cage Rotor The outer cage has high resistance and low reactance (brass/bronze); the inner cage has low resistance and high reactance (copper). At starting, high rotor frequency forces current into the outer cage, giving high starting torque. At rated speed, low frequency allows the inner cage to dominate, giving good efficiency. Deep-bar rotors achieve the same effect through skin effect.

Induction Generator

Induction Generator — Operating Principle Drive an induction machine above synchronous speed (\(N_r > N_s\), \(s < 0\)) — power flow reverses and the machine generates. In grid-connected mode the grid supplies magnetising VARs; in self-excited isolated mode a capacitor bank provides VARs, using residual magnetism to initiate build-up. Used in wind turbines (DFIG, SCIG) and mini-hydro.

6. Single-Phase Induction Motors

Double-Revolving Field Theory A pulsating single-phase field decomposes into two equal counter-rotating fields. At standstill the two torques cancel — a single-phase IM is not self-starting. Once running in either direction, the forward component dominates and sustains positive net torque.

Type	Starting Torque	Typical Application
Split-phase	\(1.5–2\,T_{FL}\)	Fans, small pumps
Capacitor-start	\(3–4.5\,T_{FL}\)	Compressors, refrigerators
Capacitor-start & -run	\(3–4.5\,T_{FL}\)	Better pf, quieter operation
Permanent capacitor (PSC)	\(\sim 0.5\,T_{FL}\)	Fans, blowers, indoor AC
Shaded-pole	Very low	Small fans, toys

A centrifugal switch disconnects the auxiliary winding once the rotor reaches approximately 75% of synchronous speed, after which the motor runs on the main winding alone. A universal motor (series DC type) operates on both AC and DC, achieving high speed; used in drills, mixers, and vacuum cleaners.

Side-by-side schematics of a split-phase induction motor (main and auxiliary windings on stator, centrifugal switch in series with auxiliary winding) and a capacitor-start induction motor (capacitor in series with auxiliary winding before the centrifugal switch). — Wiring schematics of (left) a split-phase induction motor with a centrifugal switch in the auxiliary winding circuit and (right) a capacitor-start induction motor with a starting capacitor in series with the auxiliary winding and centrifugal switch, achieving approximately 90° phase displacement for high starting torque.

7. Synchronous Machines

Construction

The stator (armature) carries the 3-phase AC winding. The rotor (field) carries DC excitation fed via slip rings or a brushless exciter.

Salient-Pole vs Cylindrical Rotor Salient-pole: low speed (<1500 rpm), multi-pole, large diameter, used in hydro-generators. Non-uniform air gap gives \(X_d \neq X_q\). Equipped with damper windings for stability.

Cylindrical (non-salient): high speed (1500–3000 rpm), 2 or 4 poles, small diameter and long axial length, used in turbo-generators (steam, gas). Uniform air gap gives \(X_d = X_q\).

EMF Equation

EMF per Phase \[ E_{ph} \;=\; 4.44\,K_p\,K_d\,f\,N_{ph}\,\Phi_m \] Distribution factor: \(\;K_d = \dfrac{\sin(m\beta/2)}{m\,\sin(\beta/2)}\) (\(m\) = slots/pole/phase, \(\beta\) = slot angle)
Pitch factor: \(\;K_p = \cos(\alpha/2)\) (\(\alpha\) = short-pitch angle)

Short-pitching reduces harmonic content, saves end-turn copper, and lowers slot-leakage reactance. The \(n\)th harmonic is eliminated when \(K_{p,n} = \cos(n\alpha/2) = 0\), i.e., \(\alpha = 180°/n\) (e.g., 5th harmonic at \(\alpha = 36°\)).

Armature Reaction in Synchronous Generators

Load	Armature Reaction	Effect on Terminal EMF
Unity pf (resistive)	Cross-magnetising	Distorts field
Zero pf lagging (inductive)	Demagnetising	Reduces \(E\)
Zero pf leading (capacitive)	Magnetising	Strengthens \(E\)

Synchronous Reactance and EMF Phasor \[ X_s \;=\; X_l + X_a, \qquad \vec{E} \;=\; \vec{V} + \vec{I}_a(R_a + jX_s) \] Magnitude (lagging pf, \(R_a\) neglected): \[ E \;=\; \sqrt{(V\cos\phi + I_a R_a)^2 + (V\sin\phi + I_a X_s)^2} \]

Voltage Regulation and SCR

\[ \text{Reg\%} \;=\; \frac{E_0 - V}{V} \times 100 \]

Methods of determining regulation: (a) synchronous-impedance (EMF) method — pessimistic; (b) MMF (ampere-turn) method — optimistic; (c) Potier (ZPF) method — most accurate, separates \(X_l\) and \(X_a\); (d) ASA method — accounts for saturation.

Short-Circuit Ratio (SCR) \[ \text{SCR} \;=\; \frac{I_{f,\,V_{\text{rated}}\,(\text{OC})}}{I_{f,\,I_{\text{rated}}\,(\text{SC})}} \;=\; \frac{1}{X_{s,\text{pu}}} \]

Typical SCR values: 0.5–0.7 for turbo-generators; 1.0–1.5 for hydro-generators. A higher SCR means better regulation and stability but a bulkier, costlier machine.

Power Equations

Cylindrical Rotor — Power and Reactive Power (per phase) \[ P \;=\; \frac{EV}{X_s}\sin\delta, \qquad Q \;=\; \frac{EV\cos\delta - V^2}{X_s} \] Steady-state stability limit: \(P_{\max} = EV/X_s\) at \(\delta = 90°\).

Salient-Pole Machine — Two-Reaction (Blondel) Power \[ P \;=\; \frac{EV}{X_d}\sin\delta \;+\; \frac{V^2}{2}\!\left(\frac{1}{X_q} - \frac{1}{X_d}\right)\!\sin 2\delta \] First term = excitation power; second term = reluctance power (exists even when \(E = 0\)). Maximum-power angle \(\delta < 90°\).

Parallel Operation and Synchronising

Synchronising Conditions (1) Same voltage magnitude; (2) same frequency; (3) same phase sequence; (4) same instantaneous phase. Synchronising power: \(P_{sy} = (EV/X_s)\cos\delta\).

Load Sharing on Infinite Bus Active power (kW) is controlled by the prime-mover governor (steam/water valve position). Reactive power (kVAR) is controlled by field excitation current.

Synchronous Motor: V-Curves and Power Angle

Two graphs: (left) V-curves showing armature current Ia versus field current If at full load, half load, and no load — each U-shaped curve has a minimum at unity power factor; (right) power-angle characteristics for cylindrical and salient-pole rotors — cylindrical peaks at 90 degrees, salient-pole peaks earlier due to reluctance power term. — (Left) V-curves of a synchronous motor: armature current plotted against field current at constant load; the minimum of each curve corresponds to unity power factor. (Right) Power-angle characteristics comparing cylindrical-rotor (sine curve, peak at \(\delta = 90°\)) and salient-pole (shifted peak at \(\delta < 90°\) due to reluctance power term) synchronous generators.

A synchronous motor runs at exactly \(N_s = 120f/P\) regardless of load. It is not self-starting (needs an auxiliary method). Power factor is adjustable via field excitation: under-excited draws lagging current; over-excited draws leading current (acts as a capacitor). Phasor equation: \(\vec{E}_b = \vec{V} - \vec{I}_a(R_a + jX_s)\).

Synchronous Condenser, Hunting, and Pull-Out Torque

Synchronous Condenser A synchronous motor running at no-load and over-excited draws leading current, acting as a variable capacitor for the system. Used for power-factor correction and voltage regulation on long transmission lines.

Hunting Oscillation of the rotor about its synchronous position after a sudden load disturbance. Cure: damper (amortisseur) windings induce braking currents opposing the oscillations. \[ f_n = \frac{1}{2\pi}\sqrt{\frac{P_{sy}\,P_{\text{poles}}}{2J\,\omega_s}} \quad \text{(natural frequency of oscillation)} \]

Pull-Out Torque \[ T_{po} = \frac{EV}{\omega_s X_s} \quad (\text{at } \delta = 90°) \] Beyond \(T_{po}\) the rotor slips out of synchronism and stalls. Higher excitation raises \(T_{po}\), improving the stability margin.

Two-Reaction Theory for Salient-Pole Machines

Blondel's Two-Reaction Theory In a salient-pole machine the non-uniform air gap means the d-axis (along pole axis) and q-axis (between poles) present different reluctances, giving \(X_d > X_q\). The armature current is decomposed as \(\vec{I}_a = \vec{I}_d + \vec{I}_q\). \[ E_0 = V\cos\delta + I_d X_d, \qquad I_q X_q = V\sin\delta \]

Slip Test (to Find \(X_d\) and \(X_q\)) Field open-circuited; apply reduced 3-phase voltage to armature; drive rotor at slightly below \(N_s\): \[ X_d \approx \frac{V_{\min}}{I_{\max}}, \qquad X_q \approx \frac{V_{\max}}{I_{\min}} \]

Capability Curve

P-Q capability chart of a synchronous generator showing three boundary curves: a large circle (stator current limit), an arc (field current limit at over-excited side), and a horizontal line (prime-mover real power limit). The permissible operating region is the area inside all three boundaries. — Capability (P–Q) chart of a synchronous generator defining the safe operating region bounded by the stator current limit (circle centred at origin), field current limit (arc on the overexcited side), and prime-mover real-power limit (horizontal line). The operating point must lie within all three boundaries simultaneously.

Park's (dq0) Transformation

Why dq0? The inductances of a salient-pole synchronous machine's stator windings vary with rotor position, making the equations time-varying. Park's transformation (1929) maps stator quantities \((a, b, c)\) onto a rotor-fixed reference frame \((d, q, 0)\), replacing time-varying coefficients with constants. It is the analytical foundation of vector control, FOC, and modern grid simulation.

Transformation matrix: \[ \begin{bmatrix} f_d \\ f_q \\ f_0 \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \cos\theta & \cos(\theta-120°) & \cos(\theta+120°) \\ -\sin\theta & -\sin(\theta-120°) & -\sin(\theta+120°) \\ \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \end{bmatrix} \begin{bmatrix} f_a \\ f_b \\ f_c \end{bmatrix} \] where \(\theta = \omega_e t\) and \(f\) represents any stator variable (\(v\), \(i\), or \(\lambda\)).

Voltage Equations in dq Frame \[ v_d = R_s i_d + \frac{d\lambda_d}{dt} - \omega_e\lambda_q, \qquad v_q = R_s i_q + \frac{d\lambda_q}{dt} + \omega_e\lambda_d \] \[ T_e = \frac{3P}{4}\,(\lambda_d i_q - \lambda_q i_d) \]

The \(\omega_e \lambda\) speed-voltage terms cross-couple the d and q axes; modern vector-control schemes decouple them. Applications include FOC of PMSM and IM, transient stability studies, and HVDC/wind-turbine controllers.

8. Special Machines

Stepper Motors

Stepper Motor A digital electromechanical device that rotates in discrete angular steps per input pulse. No position feedback needed — open-loop control is sufficient.

Step Angle \[ \beta \;=\; \frac{360°}{m\,N_r} \;=\; \frac{N_s - N_r}{N_s\,N_r}\times 360° \] (\(m\) = phases, \(N_s\) = stator teeth, \(N_r\) = rotor teeth)

Common step angles: 1.8°, 3.75°, 7.5°, 15°. Types: Variable Reluctance (VR) — no permanent magnet, high step rates; Permanent Magnet (PM) — higher torque; Hybrid — most popular, combines VR and PM. Applications include CNC machines, robotic axes, printers, and medical pumps.

Brushless DC Motors (BLDC)

BLDC Motor Permanent-magnet rotor with electronic commutation via Hall sensors or sensorless methods. High efficiency, long service life, and compact size. Used in EVs, drones, HVAC fans, and servo applications.

Switched Reluctance Motor (SRM)

SRM Doubly-salient stator and rotor — no rotor windings or permanent magnets. Extremely rugged and fault-tolerant. Requires dedicated power electronics for commutation. Well-suited to harsh environments and high-temperature applications.

Permanent Magnet Synchronous Motor (PMSM)

PMSM Produces a sinusoidal back-EMF and is fed by a variable-frequency drive. High power density and efficiency. Widely used in servo drives, machine tools, and EV traction (Tesla, BYD). Controlled using vector/FOC techniques (dq transformation).

Linear Induction Motor (LIM)

LIM An induction motor "unrolled" into a flat form — the rotating field becomes a linearly travelling field that exerts a direct thrust force on the secondary. Applications include Maglev trains (JR Maglev, Hyperloop concepts), conveyors, and airport baggage handling.

Servo Motors

High-performance motors (AC or DC type) with closed-loop position and speed control via encoders or resolvers. Used in CNC machining centres, robotic joints, and precision automation.

9. Machine Ratings, Cooling and Insulation

Name-Plate Rating and Duty Cycles (IEC 60034-1)

The name plate specifies rated voltage, current, power (kW/kVA/HP), frequency, speed, number of phases, power factor (AC machines), insulation class, IP enclosure rating, and duty class.

Class	Description
S1	Continuous duty (pumps, fans)
S2	Short-time duty (valves, gates)
S3	Intermittent periodic, no significant starting heat
S4	Intermittent with significant starting losses
S5	Intermittent with electric braking
S6	Continuous with periodic loading
S7	Continuous with starting and braking
S8	Continuous with periodic load/speed change

Insulation Classes

Class	Max Temperature (°C)
Y	90
A	105
E	120
B	130
F	155
H	180
C	> 180

The industry standard is Class F insulation with Class B temperature rise.

Cooling Methods and IP Enclosure

Common cooling codes: IC 0 (natural convection), IC 01 (shaft-mounted self-cooled fan), IC 06 (external forced-ventilation blower), IC 81W (water-jacket cooling). Hydrogen-cooled turbo-generators exploit H₂'s thermal conductivity (~7× that of air). Enclosures include TEFC (totally-enclosed fan-cooled), ODP (open drip-proof), and IP55 (dust-tight, jet-proof).

10. Worked Numerical Examples

Example 1: DC Generator — EMF and Load Current

Problem A 4-pole, lap-wound DC shunt generator has 840 conductors. Flux per pole = 25 mWb, speed = 1000 rpm, \(R_a = 0.25\,\Omega\), \(R_{sh} = 125\,\Omega\), terminal voltage = 220 V. Find the generated EMF and load current.

Generated EMF: \[ E_g = \frac{\Phi Z N P}{60 A} = \frac{0.025 \times 840 \times 1000 \times 4}{60 \times 4} = 350\,\text{V} \]

Field current: \(I_{sh} = 220/125 = 1.76\,\text{A}\)

Armature current: \(I_a = (E_g - V)/R_a = (350 - 220)/0.25 = 520\,\text{A}\)

Load current: \(I_L = I_a - I_{sh} = 520 - 1.76 = \mathbf{518.24\,\text{A}}\)

Example 2: Transformer — Efficiency and Load at Maximum Efficiency

Problem A 40 kVA transformer has iron loss 450 W and full-load copper loss 850 W. Find (a) efficiency at full load, 0.8 pf lag; (b) kVA at maximum efficiency; (c) maximum efficiency.

(a) \[ \eta_{FL} = \frac{40\,000 \times 0.8}{40\,000 \times 0.8 + 450 + 850} = \frac{32\,000}{33\,300} = \mathbf{96.10\%} \]

(b) \[ \text{kVA}_{\eta_{\max}} = 40\sqrt{450/850} = 40 \times 0.728 = \mathbf{29.11\,\text{kVA}} \]

(c) At \(\eta_{\max}\), \(P_{cu} = P_i = 450\,\text{W}\): \[ \eta_{\max} = \frac{29\,110 \times 0.8}{29\,110 \times 0.8 + 450 + 450} = \frac{23\,288}{24\,188} = \mathbf{96.28\%} \]

Example 3: Induction Motor — Slip, Speed, and Torque Ratios

Problem A 3-phase, 6-pole, 50 Hz IM has full-load slip = 4%, \(R_2 = 0.01\,\Omega\)/phase, \(X_2 = 0.1\,\Omega\)/phase. Find (a) synchronous speed, (b) rotor speed, (c) slip at maximum torque, (d) ratio \(T_{\max}/T_{FL}\).

(a) \(N_s = 120 \times 50 / 6 = \mathbf{1000\,\text{rpm}}\)

(b) \(N_r = (1-0.04)\times 1000 = \mathbf{960\,\text{rpm}}\)

(c) \(s_m = R_2/X_2 = 0.01/0.1 = \mathbf{0.1}\)

(d) \[ \frac{T_{\max}}{T_{FL}} = \frac{s_m^2 + s_{FL}^2}{2\,s_{FL}\,s_m} = \frac{0.01 + 0.0016}{2 \times 0.04 \times 0.1} = \frac{0.0116}{0.008} = \mathbf{1.45} \]

Example 4: Synchronous Generator — Voltage Regulation

Problem A 3-phase, star-connected, 10 MVA, 11 kV, 50 Hz cylindrical-rotor alternator has \(X_s = 5\,\Omega\)/phase, negligible resistance. Find the voltage regulation at full load, 0.8 pf lagging.

\(I_a = 10 \times 10^6 / (\sqrt{3} \times 11\,000) = 524.86\,\text{A}\), \(\quad V_{ph} = 11\,000/\sqrt{3} = 6350.85\,\text{V}\)

\[ E_0 = \sqrt{(V\cos\phi)^2 + (V\sin\phi + I_a X_s)^2} = \sqrt{(5080.68)^2 + (3810.51 + 2624.30)^2} = 8198.6\,\text{V} \] \[ \text{Reg\%} = \frac{8198.6 - 6350.85}{6350.85} \times 100 = \mathbf{29.10\%} \]

Example 5: Star–Delta Starter — Starting Torque Ratio

Problem A 3-phase IM has full-load slip = 5%. With DOL starting it draws 6× full-load current. Find the ratio of starting to full-load torque with a star–delta starter.

DOL: \[ \left.\frac{T_{st}}{T_{FL}}\right|_{\text{DOL}} = \left(\frac{I_{sc}}{I_{FL}}\right)^2\!s_{FL} = 36 \times 0.05 = 1.8 \]

Star–Delta: \[ \left.\frac{T_{st}}{T_{FL}}\right|_{Y\text{-}\Delta} = \frac{1}{3} \times 1.8 = \mathbf{0.6} \] Line starting current: \(I_{st,line}(Y\text{-}\Delta) = I_{sc}/3 = 2\,I_{FL}\).

Star–Delta reduces both starting current and starting torque to exactly one-third of DOL values.

11. GATE-Level Quick Revision Formulas

Topic	Formula / Key Relation
DC Machine EMF	\(E_a = \Phi Z N P / (60 A)\)
DC Machine Torque	\(T = \Phi Z I_a P / (2\pi A)\)
DC max efficiency	Variable loss = Constant loss
Transformer EMF	\(E = 4.44\,f N \Phi_m = 4.44\,f N B_m A\)
Transformer max \(\eta\)	\(P_{cu} = P_i\); \(\text{kVA} = \text{kVA}_{FL}\sqrt{P_i/P_{cu,FL}}\)
Voltage regulation (approx)	\(\varepsilon \approx v_r\cos\phi \pm v_x\sin\phi\)
Synchronous speed	\(N_s = 120f/P\)
IM slip	\(s = (N_s - N_r)/N_s\); \(f_r = sf\)
IM power split	\(P_{ag} : P_{cu2} : P_m = 1 : s : (1-s)\)
IM max-torque slip	\(s_m = R_2/X_2\); \(T_{\max}\) independent of \(R_2\)
IM torque ratio (Kloss)	\(T/T_{\max} = 2ss_m/(s^2 + s_m^2)\)
Star–Delta starting	Line current and \(T_{st}\) both reduce to \(1/3\) of DOL
Sync gen EMF	\(E = 4.44\,K_p K_d\,f N_{ph} \Phi_m\)
Distribution factor	\(K_d = \sin(m\beta/2)/[m\sin(\beta/2)]\)
Pitch factor	\(K_p = \cos(\alpha/2)\)
Power (cylindrical rotor)	\(P = (EV/X_s)\sin\delta\)
Power (salient pole)	\(P = \tfrac{EV}{X_d}\sin\delta + \tfrac{V^2}{2}\!\left(\tfrac{1}{X_q}-\tfrac{1}{X_d}\right)\!\sin 2\delta\)
Reluctance torque	Exists even at \(E = 0\) (salient pole only)
Stability limit	\(\delta = 90°\) (cylindrical); \(\delta < 90°\) (salient)
SCR	\(\text{SCR} = 1/X_{s,\text{pu}}\)
Auto-transformer saving	\((1-K)\times\) two-winding copper weight
Scott teaser tap	86.6% of main winding
Stepper step angle	\(\beta = 360°/(mN_r)\)
Hysteresis loss	\(P_h = K_h B_m^{1.6} f V\)
Eddy-current loss	\(P_e = K_e B_m^2 f^2 t^2 V\)

Common GATE Tricks and Pitfalls

DC motor runaway: series motor on no-load; shunt motor with an open field circuit.
Transformer polarity: additive vs subtractive affects whether parallel operation is possible.
IM maximum starting torque: occurs when \(R_2 = X_2\) (i.e., \(s_m = 1\)), not simply when \(R_2 = s_m X_2\).
Salient-pole machine: maximum power at \(\delta < 90°\) because of the reluctance-power term.
V-curve minimum: occurs at unity pf, not at maximum excitation.
CT secondary: must never be opened — huge induced voltage causes insulation failure.
Armature reaction direction: lagging pf → demagnetising in a generator but magnetising in a motor (and vice versa for leading pf).
Induction generator: always consumes reactive power; slip is negative (\(N_r > N_s\)).
Per-unit calculations: always keep base values consistent across all elements.

Pro Tip Draw phasor diagrams for every synchronous machine problem. Memorise the torque–slip and torque–speed curves. Practice numericals using both the EMF and MMF methods so you can cross-check answers.

12. Previous-Year GATE Questions

GATE PYQ 1: DC Machine (GATE EE 2016)

Question A 220 V, 1500 rpm, 11.2 kW separately-excited DC motor has armature resistance \(0.1\,\Omega\). It is driven by a single-phase fully-controlled rectifier with a 230 V, 50 Hz AC source. Assuming continuous conduction, find the firing angle for rated-torque operation at 1000 rpm.

Solution:

Rated \(I_a = 11200/220 = 50.9\,\text{A}\).

Back EMF at 1500 rpm: \(E_{b1} = 220 - 50.9 \times 0.1 = 214.91\,\text{V}\).

Back EMF at 1000 rpm (flux fixed, rated torque ⇒ rated \(I_a\)): \[ E_{b2} = 214.91 \times \frac{1000}{1500} = 143.27\,\text{V} \]

Required armature voltage: \(V_a = 143.27 + 50.9 \times 0.1 = 148.36\,\text{V}\).

For a single-phase full converter: \(V_a = (2\sqrt{2}/\pi)V_s\cos\alpha = 207.07\cos\alpha\). \[ \cos\alpha = 148.36/207.07 = 0.7164 \;\Rightarrow\; \boldsymbol{\alpha \approx 44.3°} \]

GATE PYQ 2: Transformer Voltage Regulation (GATE EE 2019)

Question A single-phase transformer has per-unit resistance \(v_r = 0.02\) pu and reactance \(v_x = 0.06\) pu on its own base. Calculate voltage regulation at full load, 0.8 pf lagging.

Given: \(\cos\phi = 0.8\), \(\sin\phi = 0.6\).

Approximate: \[ \varepsilon \approx v_r\cos\phi + v_x\sin\phi = 0.016 + 0.036 = \mathbf{0.052\,\text{pu} = 5.20\%} \]

Exact (with second-order correction): \[ \varepsilon = 0.052 + \frac{(v_x\cos\phi - v_r\sin\phi)^2}{2} = 0.052 + 0.000648 = \mathbf{5.26\%} \]

GATE Insight For lagging pf the correction term is additive; for leading pf it subtracts.

GATE PYQ 3: Induction Motor Speed at Maximum Torque (GATE EE 2018)

Question A 3-phase, 4-pole, 50 Hz induction motor has rotor resistance \(0.2\,\Omega\)/phase and standstill rotor reactance \(1.0\,\Omega\)/phase. The rotor is short-circuited. Find the speed at which the motor develops maximum torque.

Synchronous speed: \(N_s = 120 \times 50/4 = 1500\,\text{rpm}\).

Slip at maximum torque: \(s_m = R_2/X_2 = 0.2/1.0 = 0.2\).

Rotor speed: \(N_{r,T_{\max}} = (1 - 0.2) \times 1500 = \mathbf{1200\,\text{rpm}}\).

Insight: \(T_{\max}\) is independent of \(R_2\), but \(s_m\) scales linearly with \(R_2\). Inserting external rotor resistance can shift \(s_m\) all the way to 1 (maximum starting torque) without changing \(T_{\max}\).

GATE PYQ 4: Synchronous Generator Power (GATE EE 2020)

Question A cylindrical-rotor synchronous generator is connected to an infinite bus at 1.0 pu voltage. Excitation \(E_f = 1.5\) pu, \(X_s = 0.5\) pu, resistance neglected. Find the maximum real power (in pu) the generator can deliver.

Maximum power at \(\delta = 90°\): \[ P_{\max} = \frac{E_f V}{X_s} = \frac{1.5 \times 1.0}{0.5} = \mathbf{3.0\,\text{pu}} \]

GATE Insight For a salient-pole machine, \(P_{\max}\) occurs at \(\delta < 90°\) due to the reluctance-power term, and the total \(P_{\max}\) is typically higher.

GATE PYQ 5: Transformer Testing and Efficiency (GATE EE 2021)

Question A 10 kVA, 2000/200 V transformer gives: OC test (LV): 200 V, 1.2 A, 90 W; SC test (HV): 100 V, 5 A, 110 W. Find the efficiency at 75% full load, 0.8 pf lagging.

Iron loss (OC, constant): \(P_i = 90\,\text{W}\).

Full-load HV current: \(I_{FL,HV} = 10\,000/2000 = 5\,\text{A}\) ⇒ SC test at rated current ⇒ \(P_{cu,FL} = 110\,\text{W}\).

At 75% load: \(P_{cu,0.75} = 0.75^2 \times 110 = 61.875\,\text{W}\).

Output: \(P_{\text{out}} = 0.75 \times 10\,000 \times 0.8 = 6000\,\text{W}\).

\[ \eta = \frac{6000}{6000 + 90 + 61.875} = \frac{6000}{6151.875} = \mathbf{97.53\%} \]

13. References and Further Reading

Standard Textbooks

P. S. Bimbhra, Electrical Machinery, 7th ed., Khanna Publishers, 2011.
A. E. Fitzgerald, C. Kingsley, S. D. Umans, Electric Machinery, 7th ed., McGraw-Hill, 2014.
S. J. Chapman, Electric Machinery Fundamentals, 5th ed., McGraw-Hill, 2012.
I. J. Nagrath, D. P. Kothari, Electric Machines, 5th ed., Tata McGraw-Hill, 2017.
P. C. Sen, Principles of Electric Machines and Power Electronics, 3rd ed., Wiley, 2013.
B. L. Theraja, A. K. Theraja, A Textbook of Electrical Technology, Vol. II, S. Chand, 2014.
P. C. Krause, O. Wasynczuk, S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, 3rd ed., IEEE Press/Wiley, 2013.
R. Krishnan, Electric Motor Drives: Modelling, Analysis and Control, Prentice-Hall, 2001.

Standards and Codes

IEC 60034-1: Rotating electrical machines — Rating and performance.
IS 325: Three-phase induction motors — Specification.
IEEE Std 112: Standard test procedure for polyphase induction motors.
IEC 60076: Power transformers.

Online Resources

NPTEL: Electrical Machines (IIT Kharagpur, IIT Delhi).
MIT OCW 6.685: Electric Machines.
IEEE Xplore: Transactions on Energy Conversion; Industry Applications.

GATE-Specific Resources

GATE previous-year papers — EE stream, 2000–2025.
Made Easy / ACE Engineering Academy handbooks for Electrical Machines.