Electrical Machines

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This section covers the fundamentals of magnetic circuits — flux linkage, inductance and stored energy — the properties of magnetic materials, and how EMF and mechanical force are induced by magnetic action.

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Flux Linkage, Inductance & Energy Quantity Description Magnetic flux (Φ) Total magnetic field passing through a surface, measured in webers (Wb) Flux linkage (ψ) Total flux linking all turns of a coil, ψ=NΦ for N turns Inductance (L) Property relating flux linkage to current, L=NΦ/I=ψ/I, measured in henries (H) Energy stored in a magnetic field W = ½LI², the energy stored in the magnetic field of an inductor carrying current I Magnetic Materials & Their Properties Material Class Key Property Ferromagnetic (e.g. iron, steel, nickel) Very high permeability, strongly attracted to magnetic fields, retains magnetization (used in cores of machines/transformers) Paramagnetic (e.g. aluminium, platinum) Weakly attracted to magnetic fields, permeability slightly greater than that of free space Diamagnetic (e.g. copper, bismuth) Weakly repelled by magnetic fields, permeability slightly less than that of free space Hysteresis: the lagging of magnetization (B) behind the applied magnetizing force (H) in a ferromagnetic material, traced as a B-H loop; the area of the loop represents energy lost per cycle as hysteresis loss.

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Retentivity is the flux density remaining after the magnetizing force is removed; coercivity is the reverse magnetizing force needed to reduce residual flux density to zero.

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Induced EMF & Force Faraday’s law:

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EMF is induced in a circuit whenever the flux linkage through it changes; e = −N(dΦ/dt) (Lenz’s law gives the negative sign, indicating the induced EMF opposes the change producing it).

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Statically induced EMF: induced due to changing current/flux with the coil stationary (e.g. transformer action).

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Dynamically induced EMF: induced due to relative motion between a conductor and a magnetic field (e.g. generator action, e=Blv).

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Force on a current-carrying conductor in a magnetic field:

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F=BIl (motor action), the basis of torque production in electric motors.

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This section covers transformer construction and components, no-load/on-load operation, equivalent circuits, testing, losses and efficiency, three-phase connections, parallel operation, capacity, and temperature rise.

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Construction & Components Type/Component Description Core-type transformer Windings surround (encircle) the core limbs; core forms a rectangular frame Shell-type transformer Core surrounds (encloses) the windings, providing better mechanical protection and lower leakage flux Conservator An expansion tank mounted above the main tank, accommodating oil expansion/contraction with temperature and keeping the main tank full Breather Contains a desiccant (e.g. silica gel); dries air drawn into the conservator as oil level changes, preventing moisture ingress Explosion vent A safety device that relieves excess internal pressure (from a fault) by venting oil/gas, preventing tank rupture Bushings Insulated terminals allowing conductors to pass safely through the earthed tank wall Transformer oil Provides both electrical insulation and cooling (heat dissipation) for the windings and core No-Load & On-Load Operation No-load operation: with the secondary open, the primary draws only a small no-load (exciting) current, supplying core (iron) losses and the small magnetizing component to set up the mutual flux.

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On-load operation: as load current flows in the secondary, the primary current increases to balance the secondary MMF (approximately maintaining constant mutual flux), and both windings carry current related by the turns ratio.

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Equivalent Circuit & Tests Test Purpose Open-circuit (no-load) test Performed with secondary open, rated voltage on primary (usually LV side); determines core (iron) losses and no-load/magnetizing branch parameters Short-circuit test Performed with secondary shorted, reduced voltage applied to primary (usually HV side) to circulate rated current; determines copper (I²R) losses and equivalent series resistance/reactance Losses, Efficiency & Maximum Efficiency Hysteresis loss and eddy current loss together form the core (iron) loss — essentially constant regardless of load, since it depends on flux (approximately constant with constant applied voltage) and frequency; eddy current loss is minimized using laminated cores.

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Copper (I²R) loss in the windings varies with the square of the load current, so it changes with loading.

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Transformer efficiency = output power / input power = output/(output+losses); maximum efficiency occurs at the load where copper loss equals iron (core) loss.

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Three-Phase Connections, Parallel Operation, Capacity & Temperature Rise Topic Description Three-phase transformer connections Common configurations: star-star, star-delta, delta-star, delta-delta — chosen based on required voltage ratio, neutral availability, and harmonic/unbalance handling Parallel operation Requires matching voltage ratio (turns ratio), same phase sequence/polarity, and (ideally) equal per-unit impedance for proper load sharing between transformers Transformer capacity (rating) Expressed in kVA (not kW), since the rated capacity is limited by heating from both copper and core losses, independent of load power factor Temperature rise Determined by total losses (copper+core) and the cooling method used (e.g.

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ONAN, ONAF, OFAF); excessive temperature rise accelerates insulation aging and reduces transformer life

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This section covers the construction, working principle, and operating characteristics of DC generators and motors, armature reaction, losses and efficiency, applications, and starting/speed control of DC motors.

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Construction & Working Principle A DC machine consists of a stationary field (stator) producing the main magnetic field, a rotating armature (rotor) with windings, and a commutator with brushes that mechanically rectifies the AC generated in the armature into DC at the terminals (generator) or supplies DC to the armature coils in the correct sequence to sustain rotation (motor).

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As a generator, a DC machine converts mechanical energy to electrical energy via Faraday's law (rotating conductors in a magnetic field).

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As a motor, it converts electrical energy to mechanical energy via the force on current-carrying conductors (F=BIl).

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Types & Operating Characteristics Type (by Field Connection) Key Characteristic Shunt Field winding connected in parallel with armature; approximately constant field current/flux — gives fairly constant speed (motor) or moderately drooping voltage with load (generator) Series Field winding connected in series with armature, carrying full armature current; high starting torque (motor) but speed varies widely with load — must never be started/run unloaded Compound Combines both shunt and series field windings; cumulative compounding adds series and shunt MMF for a compromise characteristic between shunt and series machines Armature Reaction, Losses & Efficiency Armature reaction is the distorting/weakening effect of the armature's own magnetic field (produced by armature current) on the main field flux, which shifts the magnetic neutral axis and can cause sparking at the brushes if not compensated (e.g. via interpoles/compensating windings).

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Losses in a DC machine include copper losses (armature, field), core (iron) losses (hysteresis, eddy current), and mechanical losses (friction, windage); efficiency is maximum when variable (armature copper) losses equal constant losses.

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Starting & Speed Control of DC Motors A DC motor requires a starter (e.g. a face-plate/rheostatic starter) to limit the high inrush armature current at starting, since the back-EMF (which normally limits current) is zero at standstill.

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Speed control methods include: armature (rheostatic) control (varying armature circuit resistance, speed below normal), field control (varying field current/flux, speed above normal, inversely with flux), and armature voltage control (varying applied voltage, wide range, used with variable-voltage supplies/drives).

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This section covers the construction and working principle of synchronous machines (alternators and synchronous motors), excitation systems, the induced EMF equation, speed-frequency correlation, voltage/frequency regulation, losses and efficiency, and parallel operation/synchronizing.

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Construction & Working Principle A synchronous machine has a field winding (usually on the rotor, excited by DC to produce the main field) and an armature winding (usually on the stator, where AC is induced/supplied), with the rotor driven at synchronous speed by a prime mover (as an alternator) or synchronized to the supply frequency (as a motor).

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Rotor types: salient-pole rotors (protruding poles, used for low-speed machines like hydro alternators) and cylindrical (non-salient/round) rotors (uniform air gap, used for high-speed machines like turbo-alternators).

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Excitation Systems Excitation Type Description DC excitation system A separate DC generator (exciter) supplies field current via slip rings AC excitation system An AC exciter with rotating rectifiers (brushless-style AC source) supplies field current Static excitation system Field current is obtained directly from the machine's own AC terminals via static (thyristor- based) rectifiers and slip rings Brushless excitation system A rotating AC exciter armature and rotating rectifier supply the field winding directly, eliminating slip rings and brushes entirely An Automatic Voltage Regulator (AVR) automatically adjusts the exciter's output (hence field current) to maintain the alternator's terminal voltage at the desired setpoint despite load changes.

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Induced EMF, Speed-Frequency Correlation & Regulation Relation Formula/Meaning Speed- frequency relation f = PNₛ/120, where P is the number of poles and Nₛ is synchronous speed in rpm Induced EMF equation E = 4.44 fΦNKₖ (per phase), where f is frequency, Φ is flux per pole, N is turns per phase, and Kₖ is the winding factor Voltage regulation The percentage rise in terminal voltage when full load (at a given power factor) is removed, keeping speed and field excitation constant; indicates how much terminal voltage varies with load Losses, Efficiency, Parallel Operation & Synchronizing Losses comprise copper losses (armature, field), core losses, and mechanical (friction/windage) losses; efficiency is output/(output+losses), maximized under a similar variable-vs-constant loss balance as other rotating machines.

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Synchronizing: connecting an alternator to the grid (or to another alternator/infinite bus) requires matching voltage magnitude, frequency, phase sequence, and phase angle before closing the connecting switch, to avoid damaging inrush currents.

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Once connected to an infinite bus (a bus whose voltage and frequency are fixed, unaffected by the machine), an alternator's speed/frequency is fixed by the bus; adjusting its prime-mover input changes its power output (not speed), and adjusting its field excitation changes its reactive power output (not terminal voltage).

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Synchronous Motors Working principle: a synchronous motor runs at exactly synchronous speed, locked in step with the rotating stator field, by the magnetic attraction between the rotor's DC-excited poles and the rotating stator field — it develops no net starting torque from a standstill and needs an external means to reach near- synchronous speed.

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Starting methods: using an auxiliary/pony motor to bring it up to near-synchronous speed, or (most common) an induction-motor-type starting via a damper (amortisseur) winding, after which DC excitation is applied and the rotor pulls into synchronism.

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Applications: constant-speed industrial drives, and specifically as a synchronous condenser (over- excited, running unloaded) to supply leading reactive power for power-factor correction/voltage support on a system.

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This section covers the construction, working principle, equivalent circuit, torque-slip characteristics, starting methods, testing, and speed control of three-phase induction motors.

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Construction & Working Principle Rotor Type Construction Squirrel cage rotor Rotor conductors (bars) are permanently short-circuited by end rings; simple, rugged, low cost, no external rotor connection possible Phase wound (slip- ring) rotor Rotor winding is brought out to slip rings, allowing external resistance to be connected for starting/speed control Working principle: the stator's three-phase winding produces a rotating magnetic field at synchronous speed; this field induces currents in the rotor conductors (transformer action), and the interaction between rotor current and the rotating field produces torque that drags the rotor to follow the field (always at a speed slightly below synchronous speed).

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Slip (s): s = (Nₛ−Nr)/Nₛ, where Nₛ is synchronous speed and Nr is actual rotor speed; the rotor never reaches synchronous speed under load, since zero slip would mean zero relative motion, zero induced EMF, and hence zero torque.

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Equivalent Circuit & Torque-Slip Characteristics The equivalent circuit resembles that of a transformer, with the rotor circuit referred to the stator side, and the rotor resistance term modified by slip (R₂/s) to represent both real rotor resistance and the mechanical power developed.

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The torque-slip curve starts at zero torque at s=0 (synchronous speed), rises to a maximum (pull- out/breakdown) torque at a critical slip, then falls toward zero as slip approaches 1 (standstill can still develop a finite starting torque, depending on rotor resistance).

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Effect of increasing rotor resistance: increases the slip at which maximum torque occurs (shifting it toward standstill, increasing starting torque) without changing the value of maximum torque itself — the basis of rotor-resistance starting/speed control in slip-ring motors.

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Starting Methods & Testing Starting Method Application Direct-on-line (DOL) starting Simple, but draws high starting current (5-7× full load); used for small motors Star-delta starting Starts the motor in star (reduced voltage/current), switches to delta for normal running; reduces starting current to about 1/3 Auto-transformer starting Applies reduced voltage via a tapped auto-transformer during starting, then switches to full voltage Rotor resistance starting For slip-ring motors only: external resistance in the rotor circuit reduces starting current while increasing starting torque; resistance is cut out as the motor accelerates No-load test: determines core loss and magnetizing branch parameters (motor runs at very low slip).

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Blocked-rotor (locked-rotor) test: rotor held stationary, reduced voltage applied to circulate rated current; determines copper losses and equivalent series impedance — analogous to a transformer's open-circuit and short-circuit tests respectively.

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Speed Control Methods Methods include: changing the number of stator poles (pole-changing, discrete speed steps), varying supply frequency (variable frequency drive, V/f control — the most widely used modern method), varying supply voltage (limited range), and rotor resistance control (slip-ring motors only, reduces efficiency).

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This final section of the chapter covers the construction, operation and characteristics of single-phase induction motors, including split-phase and shaded-pole types, their applications, and speed control.

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Construction & Operation A single-phase stator winding alone produces a pulsating (not rotating) magnetic field, which can be resolved into two equal counter-rotating fields; these produce equal and opposite torques at standstill, so a single-phase induction motor has no starting torque and cannot self-start — it requires an auxiliary means to create an initial rotating field.

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Once running (given an initial push in either direction), the forward-rotating field's torque dominates over the backward field's torque, and the motor continues to run and accelerate in that direction (this is why, once started, it can run on the single main winding alone using the double revolving field theory).

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Split-Phase & Shaded-Pole Motors Type Starting Mechanism Split-phase (resistance-start) motor Uses a main (running) winding and an auxiliary (starting) winding with higher resistance/lower reactance, creating a phase difference between the two winding currents that produces a rotating field for starting; the auxiliary winding is disconnected (via a centrifugal switch) once the motor reaches sufficient speed Capacitor-start motor Similar to split-phase, but a capacitor is added in series with the auxiliary winding, giving a larger phase angle difference and much higher starting torque Shaded-pole motor A small portion of each pole is 'shaded' by a short-circuited copper ring, which delays flux in that part of the pole, producing a weak rotating field sufficient to self-start (low starting torque, simple and cheap construction, no centrifugal switch needed) Applications & Speed Control Split-phase motors are used for moderate starting torque loads (e.g. fans, blowers, small pumps, washing machines); capacitor-start motors for higher starting torque loads (e.g. compressors, small conveyors); shaded-pole motors for very low-torque, low-cost applications (e.g. small fans, hair dryers, record players).

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Speed control of single-phase induction motors is typically achieved by varying the supply voltage (e.g. via a tapped autotransformer or triac-based control, giving a limited range, commonly used for fans) or by pole- changing (discrete speed steps, less common for these small motors).