An Introduction to Electrical Machines P. Di Barba, University of Pavia, Italy Academic year 0-0
Contents Transformer. An overview of the device. Principle of operation of a single-phase transformer 3. Power (three-phase) transformer, signal transformer, autotransformer Rotating electrical machines (REM). Ampère law, Faraday principle, Lorentz force. An overview of REM 3. Rotating magnetic field Asynchronous machine: induction motor Synchronous machine: alternator DC machine: commutator motor, dynamo Permanent magnet motor Stepping motor
Transformer: static electrical machine operating in AC regime. It converts electric power into electric power, by varying the power factors (V,I) and keeping the power (approximately) constant. : primary circuit (power in) : secondary circuit (power out) Single-phase Three-phase
Applications Power transformer Production systems operate at a reduced voltage in order to decrease dielectric material size Transmission systems operate at a low current in order to decrease conducting material size User systems operate at a low voltage in order to increase safety Signal (and power) transformer Adapts the voltage of two systems to be connected Adapts the impedance of two systems to be connected Couples two systems, keeping them electrically disconnected
Step up transformer: increases V and decreases I Step down transformer: decreases V and increases I Power range: from VA to 000 MVA Voltage range: from V to 400 kv Basically, a transformer is composed of: a magnetic circuit; (at least) two magnetically-coupled electric circuits.
limb Single-phase power transformer (5 Hz < f < 400 Hz) yoke windings high-voltage terminal lamination package (core) low-voltage terminal A<5 kva natural cooling A>5 kva forced cooling
Three-phase power transformer (f=50 Hz) : three pairs of windings a b c Three primary windings a,b,c and three secondary windings a,b,c Used connections: star-delta, delta-delta, delta-star. closed magnetic circuit Signal transformer single phase 0 Hz < f < 0 khz open magnetic circuit (f>0 khz, core made of Fe O 3 )
Operating principle (ideal case) Model assumptions: infinite core permeability, zero losses in the magnetic circuit and in the two electric circuits. : N turns : N turns I V V I A A I k I N N k, V k V Transformer equations I I Z F
Transformer laws (ideal model) I No-load operation (I =0) Let a sinusoidal voltage source v (t) supply winding Then, an infinitesimal magnetizing current I 0 flows in winding and gives rise to a magnetomotive force (mmf) Hopkinson law: magnetic flux N I 0 F infinitesimal mmf infinitesimal reluctance non-infinitesimal sinusoidal flux
Transformer laws (ideal model) II Flux F links both windings and Faraday-Lenz law: induced emf e N df dt e N df dt KVL v =-e, v =-e In phasor notation: V jnf V jn F In terms of RMS values, it turns out to be: V V N N k transformer voltage law
Primary circuit: supplied by V Secondary circuit: open circuited F I 0 Kirchhoff voltage law V E 0 V E 0 I 0 magnetizing current Terminal marking: currents entering the marked terminals originate like fluxes in the magnetic core.
Transformer laws (ideal model) III On-load operation Current I is delivered by winding to load Z mmf N I originates a secondary flux F opposing the primary flux F = F M, but F M V 4.44 N f cannot vary due to the applied primary voltage V
Transformer laws (ideal model) IV Consequently, voltage source V is forced to deliver an additional current I to winding in order to compensate the flux mismatch: N I -N I =N I 0 =0 It turns out to be: I N I N k transformer current law Finally, the complex power conservation holds: V I VI
Vector diagram (Kapp diagram) A resistive-inductive load is assumed
A more realistic model - In a real transformer, power losses are not zero: P 0 =GV active power in the magnetic core (eddy current and hysteresis) P c =RI active power in the electric circuits due to the Joule effect Moreover, the finite magnetic permeability of the core causes a non-zero magnetizing power: Q 0 =BV reactive power in the magnetic circuit of main flux (inside the core) Q C =XI reactive power in the magnetic circuit of stray flux (outside the core) P 0 and Q 0 dominate in the no-load operation (measured in the open-circuit test) P c and Q c dominate in the on-load operation (measured in the short-circuit test)
A more realistic model - As a consequence, in a real transformer: V is different from E due to voltage drops caused by R and X of the windings F is not constant with respect to E because E varies with I for a given V V is different from kv due to on-load voltage drops I is different from I /k due to no-load primary current A is different from A due to active and reactive power losses Transformer equations based on the ideal model are valid as a first approximation only
A more realistic model - 3 main flux stray flux power loss (eddy currents, hysteresis) power loss (Joule effect)
Z R jx Y G jb BV XI Q Q GV RI P P YV I k I ZI V k V A more realistic model - 4 Primary winding admittance Secondary winding impedance
A more realistic model - 5 Modified Kapp diagram
Impedance adaption - Problem: how to modify the value of an impedance Z to be connected at a terminal pair A-B? I V Zk
Impedance adaption - V V I I k Z k k Z k k V Z
Signal transformer: frequency response -
Signal transformer: frequency response - X m X c
Voltage drop - V V 0 V V V 0 V ( R jx ) I V cos k R jx I ji sen
Voltage drop - V V 0 V RI XI cos sin
Insulation transformer - Measurement
Insulation transformer - Safety
Insulation transformer - 3 Damping
k N N N V V k N N N I I A A V I V I A ) ( ) ( V I I I V V A d Auto-transformer A A d
Rotating Electrical Machine Basically, it consists of: an electric circuit (inductor) originating the magnetic field; a magnetic circuit concentrating the field lines; an electric circuit (armature) experiencing electrodynamical force or electromotive force.
Ampère law i b
e Faraday law transformer effect b(t) e(t), i(t)
Faraday law (motional effect) b v e
Lorentz force b i F
stator inductor circuit air-gap rotor armature circuit
Magnetic field of a circular coil dl r db n B i B 0 i r n
B B j axis r axis B Rotating magnetic field (two-phase) I t r I n t r I B cos cos 0 0 t sen r I j n t sen r I n t r I B cos 0 0 0
j axis Rotating magnetic field (two-phase) II r axis B B B B B B 0I r I r 0 j t cos t j sent e
r axis B j axis 3 t j e r I B 3 0 Rotating magnetic field (three-phase): Ferraris field 0 cos n t r I B 0 3 cos n t r I B 3 0 3 3 4 cos n t r I B
Three-phase asynchronous motor (induction motor) graphic symbol As a motor, it converts electric power into mechanical power
air gap shaft laminated stator and rotor ventilating fan three-phase inductor circuit
Wound rotor Squirrel-cage rotor W bar slip-ring commutator (single-phase) end ring
Induction motor: operating principle Inductor terminals are supplied by a three-phase symmetric voltage V A three-phase balanced current I is delivered to inductor windings. An induction field B rotating at speed N 0 = 0 f p - is originated (f current frequency, p number of poles, speed in rpm). In stator and rotor windings, subject to a sinusoidal magnetic flux F, electromotive forces E =. m f F and E =. m f F are induced, respectively. F max value of flux, E, rms value of electromotive force m, number of conductors in a winding
E three-phase balanced current I in the rotor windings Magnetic effect of I : a synchronous rotating field B (speed N 0 ) is originated, which tends to decrease the inductor rotating field B B depends on the applied voltage V and cannot vary voltage source V is forced to deliver an additional current I 3 to the inductor, such that rotating field B 3 due to I 3 compensates field B Mechanical effect of I : the axially-directed conductors placed in the radially-directed field B experience tangentially-directed forces F t The rotor experiences a torque C=RF t : if unconstrained, it rotates.
Having defined the slip factor s=(n 0 -N)/N 0 as the speed of the rotating field with respect to the rotor, the frequency of E and I is f =sf If the opposing torque is equal to zero then, the running torque C M =0 and f 0 I =0 E =0 f =0 s=0 N=N 0 the rotor is synchronous wrt the rotating field If the opposing torque is different from zero then, the running torque C M 0 and f 0 I 0 E 0 f 0 0<s< N<N 0 the rotor is asynchronous wrt the rotating field (the rotor follows the field)
Torque-speed curve Torque C is proportional to both flux F of the rotating field and real part of the armature current I cos f (active component): Current I and power factor cos f depend on emf E and armature circuit impedance: On the other hand, electromotive force E depends on both relative rotor speed N 0 -N and flux F It turns out to be: cos I k C F L s R Z resistive-inductive impedance R +jsl 0 ' F N N k E 0 0 0 0, V C N C C L N N R N R N N N k Z R Z E k C F F
Torque-speed curve C M self-starting C s = k F N 0 R /(R + L ) C Max torque C M = k F N 0 /(L ) when N = N 0 (-R /L ) C S operating point C R 0 N 0 N mechanical power P m = CN electric power P e = 3 VI cosf synchronous speed (rpm) 0f N 0 p
Speed control decreasing V decreasing f increasing R (wound rotor only) C C C N N N
Synchronous generator turbine alternator Three-phase output
Alternator structure Three-phase armature circuit DC inductor circuit (two-pole) In low-power alternators it can be replaced by a permanent magnet. As a generator, it converts mechanical power into electric power.
Alternator operating principle The inductor circuit is supplied by a DC current I E A magnetic flux F R sinusoidally distributed in the air-gap is originated A speed W is applied to the rotor shaft In the armature circuit a three-phase symmetric electromotive force E = kwf R at an angular frequency = Wp/ is induced If the armature circuit is on load, a three-phase (balanced) current I is delivered, so providing an active power P = 3EI cos f An equal amount of mechanical power must be supplied to the shaft Current I generates a synchronous rotating field (speed ), which originates a flux F S opposing flux F R In order to keep E constant, excitation current I E must be increased
electric frequency W p number of poles rotor speed I I Two-pole DC winding Four-pole DC winding
DC machine graphic symbol From electric to mechanical power: motor From mechanical to electric power: dynamo
air gap salient pole shaft laminated stator and rotor ventilating fan sliding brushes with commutator inductor armature
DC machine - Principle of operation I MOTOR Inductor circuit is supplied by DC voltage V E Current I E is originated Magnetic field B, and so flux F, take place Armature circuit is supplied by DC current I A Armature conductors, carrying axial current I A and placed in radial field B, are subject to tangential force F Main effect F C N P m = CN Secondary effect Due to the rotor motion, in the armature conductors an AC emf is induced, which is subsequently rectified by the multi-sector commutator. Therefore, a back emf E = knf appears at the commutator terminals. It turns out to be: E = V (V voltage across armature terminals) P e = EI A = CN = P m (power balance)
DC machine - Principle of operation II DYNAMO Inductor circuit is supplied by DC voltage V E Current I E is originated Magnetic field B, and so flux F, take place Rotor is given speed N, externally applied to the shaft In the armature conductors, an AC emf is induced At the commutator terminals a rectified emf E = knf is available Main effect If armature terminals are connected to a resistive load E I A P e = EI A Secondary effect Armature conductors, carrying axial current I A and placed in radial field B, are subject to tangential force F opposing the motion F C N P m = CN To maintain the motion, power P m must be supplied to the shaft (P m = P e ).
Multi-sector commutator: principle of operation A A a b b + b t t t B a a Each armature conductor is electrically connected to a sector. The polarity of each sector is reversed when crossing the A-B line joining the brushes (orthogonal to the pole axis). B with two conductors/sectors V AB t V AB The voltage V AB between brushes exhibits always the same polarity. with several conductors/sectors As a result, the original AC signal is mechanically transformed into a DC signal, and viceversa. t
I DC motor excitation schemes V E M Series excitation (motors for traction) When C res varies, both N and C vary in such a way that CN const (constant power motor) I E I A I E I A V E M V A V M Independent excitation Derived excitation When C res varies, excitation current I E varies N const (constant speed motor)
Equivalent circuit of the DC motor (with independent excitation) inductor circuit Excitation flux Induced emf (rectified) F F I E E knf armature circuit (R a resistance of armature conductor) k voltage constant Torque C k' FI A k' F V E R A k torque constant
Torque-speed curve (independent-excitation motor) self-starting F F(I E ) N 0 C S V k F k' F V R A C k 'F R A V knf CN
Speed control
AC to DC conversion - Ideal diode: current-voltage curve
AC to DC conversion - i R i i R Ri R Transformer + single phase rectifier
AC to DC conversion - 3 Single-phase rectifier (full-wave): principle of operation A rectified voltage v R is originated: non-zero average value
Wound synchronous motor Permanent magnet synchronous motor Magnets Excitation windings SPM Magnets IPM
Synchronous motor: torque-speed curve
Three-phase inverter Control system feedback Reference signal Rotor position sensor (Hall probe) Brushless DC motor: principle of operation
Stepping motor: principle of operation
Permanent-magnet stepping motor Variable-reluctance stepping motor