Bài giảng Electromechanical energy conversion - Chapter III: Introduction to Rotating Machines - Nguyễn Công Phương

Nguyễn Công Phương ELECTROMECHANICAL ENERGY CONVERSION Introduction to Rotating Machines Contents I. Magnetic Circuits and Magnetic Materials II. Electromechanical Energy Conversion Principles III.Introduction to Rotating Machines IV. Synchronous Machines V. Polyphase Induction Machines VI. DC Machines VII.Variable – Reluctance Machines and Stepping Motors VIII.Single and Two – Phase Motors IX. Speed and Torque Control sites.google.com/site/ncpdhbkhn 2 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10.Leakage Fluxes sites.google.com/site/ncpdhbkhn 3 Elementary Concepts (1) • In rotating machines, voltage are generated in windings or groups of coils by rotating these windings mechanically through a magnetic field: – By mechanically rotating a magnetic field past the winding, or – By designing the magnetic circuit so that the reluctance varies with rotation of the rotor. • A set of such coils connected together is typically referred to as an armature winding. • In AC machines (e.g. synchronous or induction), the armature winding is typically on the stationary – portion of the motor (referred to as the stator ). • In DC machines, the armature winding is on the rotating portion of the motor (referred to as the rotor ). sites.google.com/site/ncpdhbkhn 4 Elementary Concepts (2) • Second winding(s) carrying DC currents and used to produce the main operating flux in the machine is called field winding – For a DC machine, it is on the stator. – For a synchonous machine, it is on the rotor. – Sometimes it is a permanent magnet. • The time – varying flux tends to induce currents, known as eddy currents , in the electrical steel. • There are no windings on the rotor of some machines, such as variable reluctance machines and stepper motors . sites.google.com/site/ncpdhbkhn 5 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines a) AC Machines i. Synchronous Machines ii. Induction Machines b) DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 6 Synchronous Machines (1) Armature – winding magnetic axis • A simple, two – pole, single – θ phase synchronous generator. a • The field – winding, Field producing a single pair of winding magnetic poles, is excited by −a direct current conducted to it by means of stationary carbon a brushes which contact rotating slip rings or collector rings . Rotor N – turn • The single, low – power field armature Stator winding on the rotor; the high winding Flux paths – power, typically multiple – phase, armature winding on the stator sites.google.com/site/ncpdhbkhn 7 Synchronous Machines (2) Armature – winding magnetic axis • The two coil sides (of the θ armature winding) a & –a a Field placed in diametrically winding opposite narrow slots on the −a inner periphery of the stator • The conductors forming the a coil sides are parallel to the Rotor shaft of the machine N – turn armature Stator • The rotor is turned at a winding Flux paths constant speed by a source of mechanical power connected to the shaft sites.google.com/site/ncpdhbkhn 8 Synchronous Machines (3) Armature – winding magnetic axis • The flux – linkages of the armature winding change with θ time. a • If the flux distribution is sinusoidal & the rotor speed is constant, then − the resulting coil voltage will be a sinusoidal in time. • The frequency (Hz, cycles per a second) of the coil voltage is the same as the speed of the rotor (revolutions per second). • The electric frequency of the Stator generated voltage is synchronized with the mechanical speed
the name “synchronous”. • 3000 rpm
50 Hz. sites.google.com/site/ncpdhbkhn 9 Synchronous Machines (4) − a1 θ= poles θ ae2 a =poles × n f e a a 2 60 1 2 B 1 0.8 0.6 − − a1 a1 a2 a2 0.4 − a2 0.2 θ π 2π a , mechanical radians 0 π 4π θ -0.2 2 ae , electrical radians -0.4 -0.6 -0.8 -1 0 2 4 6 8 10 12 sites.google.com/site/ncpdhbkhn 10 Synchronous Machines (5) Salient/projecting/concentrated Nonsalient/cylindrical/distributed windings windings poles n −a f = × 1 e 2 60 N a1 a2 S − a2 Hydroelectric generator Steam/gas turbin generator sites.google.com/site/ncpdhbkhn 11 Synchronous Machines (6) − c b − −c a a b S c N a N −b −b′ S N a′ − −a ′ S b c a −c′ −a′ c b′ −a a′ −c′ −a′ ′ −c −b′ b c′ −b c b sites.google.com/site/ncpdhbkhn 12 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines a) AC Machines i. Synchronous Machines ii. Induction Machines b) DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 13 Induction Machines • Synchronous machines: – Stator winding: AC current – Rotor winding: DC current • Induction machines: – Stator winding: AC current – Rotor winding: AC current – The rotor does not itself rotate synchronously sites.google.com/site/ncpdhbkhn 14 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines a) AC Machines i. Synchronous Machines ii. Induction Machines b) DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 15 DC Machines (1) Armature coil, • A very elementary two N turns – pole DC generator. Rotation N • The two coil sides a & –a are placed at −a diametrically opposite − points on the rotor with Carbon the conductors parallel brush to the shaft. • The rotor is normally Copper turned at a constant + commutator speed by a source of a segments mechanical power connected to the shaft. S sites.google.com/site/ncpdhbkhn 16 DC Machines (2) Armature coil, • The voltage induced in an N turns individual armature coil is AC
rectification is required. Rotation N • Commutator: a cylinder formed of copper segments insulated −a from each other by mica or some other highly insulating − material & mounted on (but Carbon insulated from) the rotor shaft. brush • Stationary carbon brushes held against the commutator surface connect the winding to the Copper external armature terminals. + commutator • Commutation is the reason why segments the armature windings of DC a machines are placed on the rotor. S sites.google.com/site/ncpdhbkhn 17 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10.Leakage Fluxes sites.google.com/site/ncpdhbkhn 18 MMF of Distributed Windings Flux lines N – turn coil carrying current i Magnetic axis of stator coil θ a Ni 2 Fundamental Fag 1 π θ 0 a − Ni 2 Rotor surface Stator surface sites.google.com/site/ncpdhbkhn 19 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings a) AC Machines b) DC Machines 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 20 AC Machines (1) =ωϕ ++ ωϕ ++ ωϕ ++ FtMag ( )1 cos( t 12 ) M cos(2 t 23 ) M cos(3 t 3 ) ... = + + + Ftag1() Ft ag 2 () Ft ag 3 ()... 4 Ni F( t )= cos θ ag1 π 2 a Ni 2 Fundamental Fag 1 π θ 0 a − Ni 2 Rotor surface Stator surface sites.google.com/site/ncpdhbkhn 21 AC Machines (2) a θ a Space – fundamental mmf wave Axis of phase a Axis of phase a 2N i c a − −π π a 0 θ a sites.google.com/site/ncpdhbkhn 22 AC Machines (3) a θ a Magnetic axis of stator coil Axis of phase a θ a −a 4 Ni 4 kw N ph poles  F( t )= cos θ F( t )= i cos θ  ag1 π 2 a ag1 π poles a2 a  sites.google.com/site/ncpdhbkhn 23 AC Machines (4) 4 2 Nc i a o ()F =cos(θ − 22.5 ) o ag1 22.5 o π 2 a 97.5 o 82.5 67.5 o 112.5 o a 4 2 Nc i a o θ ()F =cos(θ + 22.5 ) a ag1 −22.5 o π a 2 22.5 o 4 2 N i ()F =c a cos(θ − 7.5o ) ag1 7.5 o π 2 a Axis of phase a 4 2 N i ()F =c a cos(θ + 7.5o ) ag1 −7.5 o π 2 a (F) =( F) + ( F ) ag1total ag 122.5o ag 1 − 22.5 o − o −a −67.5 o +()() + 112.5 Fag1o F ag 1 o 7.5− 7.5 −97.5 o −82.5 o = θ 4.88Nc i a cos a sites.google.com/site/ncpdhbkhn 24 AC Machines (5) = θ (Fag1 ) 4.88 N ca i cos a o total 97.5 o 82.5 67.5 o 112.5 o a →(F) = 4.88 N i θ ag1 peak c a a 22.5 o 4 kw N ph poles  F( t )= i cos θ  ag1 π poles a2 a  Axis of phase a 4 k N i →()F = w ph a ag 1 peak π poles N=8 N ; poles = 2 − o ph c −112.5 o a −67.5 o − o → = −97.5 82.5 kw 0.96 sites.google.com/site/ncpdhbkhn 25 θ AC Machines (6) r 4 k N I poles  F = r r r cos θ ag1 π r  poles 2  9 10 8 mmf 7 1 2 6 3 N I N I 10 r 1 r 5 4 N9 I r N2 I r N8 I r N3 I r θ 0 r N7 I r Space N4 I r fundamental mmf wave N6 I r N6 I r sites.google.com/site/ncpdhbkhn 26 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings a) AC Machines b) DC Machines 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 27 DC Machines (1) Magnetic axis of armature winding a1 Magnetic axis of field winding − a1 sites.google.com/site/ncpdhbkhn 28 DC Machines (2) 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 6Nc i c 4Nc i c 2Nc i c π 0 − 2Nc i c − 4Nc i c − 6Nc i c Fundamental component Current mmf wave sites.google.com/site/ncpdhbkhn 29 DC Machines (3) S C i ()F = a a ag peak (2)(m poles ) N N N i C =a a, N = a polesa 2 m 8 N i S ()F = a a ag 1 peak π 2 poles sites.google.com/site/ncpdhbkhn 30 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery a) Machines with Uniforms Air Gaps b) Machines with Nonuniform Air Gaps 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 31 Machines with Uniform Air Gaps (1) N – turn coil θ F a = ag H ag g r g a µ → ∞ Fag 1 4 Ni Magnetic axis H = = cos θ rr ag1 gπ 2 g a of stator coil µ → ∞ 4 Ni ()H = Fag ag 1 peak π 2g Ni 2 Fundamental Fag 1 0 π θ − Ni a 2 sites.google.com/site/ncpdhbkhn 32 Machines with Uniform Air Gaps (2) a N – turn coil θ a θ a r g a µ → ∞ Axis of phase a r Magnetic axis r of stator coil µ → ∞ −a k N = 4 Ni θ = 4 w ph poles θ  Hag1( t ) cos a H( t ) i cos   π 2g ag1 π g( poles ) a 2 a  sites.google.com/site/ncpdhbkhn 33 Machines with Uniform Air Gaps Ex. (3) Given a four – pole synchronous AC generator with a smooth air gap has a distributed rotor winding with 263 series turns, a winding factor of 0.94, and an air gap of length 0.7mm. Find the rotor – winding current to produce a peak, space – fundamental magnetic flux density of 1.6T in the machine air gap? 4 k N 4 k N ()H= r r i →()B = µ r r i ag1 peak π g( poles ) r ag1peak 0 π g( poles ) r (π g ) poles →i = () B rµ ag 1 peak 4 0kr N r π ×0.7 × 10−3 × 4 = 1.6 4×× 4π 10−7 × 0.94 × 263 = 11.33A sites.google.com/site/ncpdhbkhn 34 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery a) Machines with Uniforms Air Gaps b) Machines with Nonuniform Air Gaps 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 35 Machines with Nonuniform Air Gaps • × • × • × Rotor Rotor • × • × • × Stator Stator sites.google.com/site/ncpdhbkhn 36 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines a) MMF Wave of a Single – Phase Winding b) MMF Wave of a Polyphase Winding c) Graphical Analysis of Polyphase MMF 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 37 MMF Wave of a Single – Phase Winding (1) 4 k N poles  F= w ph i cos θ  ag1 π poles a2 a  = ω ia I acos e t → = poles θ  ω Fag1 F max cos ae  cos t 2  4 k N =Fcosθ cos ω tF , = w ph I max ae e max π poles a sites.google.com/site/ncpdhbkhn 38 MMF Wave of a Single – Phase Winding (2) = θ ω Fag1 F max cos ae cos e t 1 cosαβ cos=[] cos( αβ −+ ) cos( αβ + ) 2 1 →=FF[]cos(θ −+ ω t ) cos( θ + ω t ) ag12 max aee aee + 1 F= Fcos(θ − ω t ) ag12 max aee − 1 F= Fcos(θ + ω t ) ag12 max ae e sites.google.com/site/ncpdhbkhn 39 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines a) MMF Wave of a Single – Phase Winding b) MMF Wave of a Polyphase Winding c) Graphical Analysis of Polyphase MMF 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 40 MMF Wave of a Polyphase Winding (1) Axis of phase b θ ia ib ic a a −c t −b Axis of phase a b c −a = ω Axis of phase c ia I mcos e t =ω − o ib I mcos( e t 120 ) =ω + o ic I mcos( e t 120 ) sites.google.com/site/ncpdhbkhn 41 MMF Wave of a Polyphase Winding (2) =+ + − Fa1 F a 1 F a 1 + 1 F= Fcos(θ − ω t ) a12 max ae e − 1 F= Fcos(θ + ω t ) a12 max ae e 4 k N F= w ph I max π poles m =+ + − =+ + − Fb1 F b 1 F b 1 Fc1 F c 1 F c 1 + 1 + 1 F= Fcos(θ − ω t ) F= Fcos(θ − ω t ) b12 max ae e c12 max ae e − 1 − 1 F= Fcos(θ + ω t + 120o ) F= Fcos(θ + ω t − 120o ) b12 max ae e c12 max aee sites.google.com/site/ncpdhbkhn 42 MMF Wave of a Polyphase Winding (3)  + F  + F  + F F=max cos(θ − ω t ) F=max cos(θ − ω t ) F=max cos(θ − ω t )  a1 2 aee  b1 2 ae e  c1 2 ae e    − F − F − F F=max cos(θ + ω t ) F=max cos(θ + ω t + 120o ) F=max cos(θ + ω t − 120o )  a1 2 aee  b1 2 ae e  c1 2 ae e θ = + + F(ae , tF ) a1 F b 1 F c 1 −θ = −−− + + F(ae , tFFF ) a1 b 1 c 1 − 1 FtF(θ , )= cos( θωθω ++ t ) cos( +−+ t 120o ) cos( θω ++ t 120 o )  ae2 max  ae e ae e ae e  = 0 +θ = +++ + + F(ae , tFFF ) a1 b 1 c 1 + 1 FtF(θ , )=[ cos( θω −+ t ) cos( θω −+ t ) cos( θω − t ) ] ae2 max ae e ae e ae e 3 =Fcos(θ − ω t ) 2 max ae e 3 3 poles  →=FtF(θ , ) cos( θω −= tF ) cos  θω − t  ae2max aee 2 max  2 ae  sites.google.com/site/ncpdhbkhn 43 MMF Wave of a Polyphase Winding (4) 3 3 poles  FtF(,)θ= cos( θω −= tF ) cos  θω − t  ae2max ae e 2 max  2 a e  2 synchronous angular velocity: ω= ω spoles e 120 synchronous speed: n= f spoles e sites.google.com/site/ncpdhbkhn 44 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines a) MMF Wave of a Single – Phase Winding b) MMF Wave of a Polyphase Winding c) Graphical Analysis of Polyphase MMF 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 45 Graphical Analysis of Polyphase MMF MMF of phase a MMF of phase b MMF of phase c The total MMF sites.google.com/site/ncpdhbkhn 46 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage a) AC Machines b) DC Machines 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 47 AC Machines (1) N – turn coil Phase b magnetic axis Rotor – wing 4 kf N f a B= µ i magnetic axis peak0 π g( poles ) f + b′ c′ θ ω t r m e Phase a magnetic = poles θ  B B peakcos  r  b c axis 2  − a′ Nf – turn field coil Phase c magnetic axis π Φ = / poles poles θ  θ = 2 pl∫ B peakcos  r  rd r 2Bpeak lr −π / poles 2  poles sites.google.com/site/ncpdhbkhn 48 AC Machines (2) N – turn coil Phase b magnetic axis Φ = 2 p2B peak lr a Rotor – wing poles + magnetic axis ′ λ = Φ ′ b peakk w N ph p c θ ω t r m   e λ= Φ poles  ω ak w N ph pcos   m t  Phase a 2   magnetic b c axis = Φ ω kw N ph pcos me t , − ′ Nf – turn ω= poles ω a me m field coil 2 Phase c magnetic axis dλ dΦ e==a kNp cosω tkN −Φ ω sin ω t adt wph dt me mewphp me Φ= =−ω Φ ω Ifp constthene a mewphp kN sin me t (electromotiveforce,emf) sites.google.com/site/ncpdhbkhn 49 AC Machines (3) N – turn coil Phase b magnetic axis a Rotor – wing magnetic axis = −ω Φ ω + emew kN ph psin me t b′ c′ θ ω t r m e Phase a E=ω kN Φ=2 π fkN Φ magnetic max mew ph p mew ph p b c axis − a′ Nf – turn field coil Phase c magnetic axis 2 E=π fkN Φ=2 π fkN Φ rms2 mewphp mewphp sites.google.com/site/ncpdhbkhn 50 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage a) AC Machines b) DC Machines 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 51 DC Machines e Armature coil, N turns Rotation N ωt − 0 π 2π a π − =1 ω Φ ω ω Ea∫ me N psin( me tdt )( me ) π 0 Carbon 2 brush =ω N Φ π me p poles ω= ω + me2 m a →=poles Φ=ω Φ n Ea N pm( poles ) N p π 30 Copper N= C/(2 m ) commutator a a S poles C poles C segments →=Ea Φ=ω a Φ n a2π m pm 60 m p sites.google.com/site/ncpdhbkhn 52 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines a) Coupled – Circuit Viewpoint b) Magnetic Field Viewpoint 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 53 Coupled – Circuit Viewpoint (1) s ω= poles ω me2 m r Magnetic θ= θ axis of s Lsr( me ) L sr cos( me ) −r θ λ=+LiL() θ iLiL =+ cos() θ i m  s sss sr mer sss sr mer λ=L( θ ) iLiL += cos( θ ) iLi + Magnetic  r sr mes rrr sr mes rrr −s axis of r λ  L L (θ )  i  →s = ss srme s λ θ    r  Lsr( me ) L rr  ir  + λ −  d s r + s v= R i + −  s s s dt θ  m dλ v= R i + r  r r r dt sites.google.com/site/ncpdhbkhn 54 Coupled – Circuit Viewpoint (2) λ=+ θ =+ θ  sLiL sss sr() mer iLiL sss sr cos() mer i λ= θ += θ +  rL sr() mes iLiL rrr sr cos() mes iLi rrr  dλ v= R i + s  s s s dt  dλ v= R i + r  r r r dt  di di d θ vRiL=++s Lcos(θ ) r − Li sin( θ ) me  s ss ssdt sr me dt srr me dt →  di di d θ vRiL=++r Lcos(θ ) s − Li sin( θ ) me  r rr rrdt sr me dt srs me dt sites.google.com/site/ncpdhbkhn 55 Coupled – Circuit Viewpoint (3) 1 1 W′ = Li()θ2 + L () θ iL 2 + () θ ii fld 211 1 2 22 2 12 12 1 1 =Li2 + Li 2 + Lii cos θ 2sss 2 rrr srsr me 1 1 poles  =Li2 + Li 2 + Lii cos θ  2sss 2 rrr srsr 2 m  ∂W′ (,, i i θ ) T = fld s r m ∂θ m is, i r are const poles poles  = − L i i sin θ  2sr s r 2 m  poles = − L i i sin θ 2 srsr me sites.google.com/site/ncpdhbkhn 56 Coupled – Circuit Viewpoint (4) poles s T= − Lii sin θ 2 srsr me = r Magnetic poles 2 axis of s →T = − Lii sin θ −r θ sr s r m m θ= ω + δ m m t Magnetic −s axis of r = ω is I scos e t →=−ω ω + δ T LIIsr s rcos e t sin( m t ) 1 sinαβ cos=[] sin( αβ ++ ) sin( αβ − ) 2 1 →=−TLII{}sin([][]ωωδ +++ ) t sin( ωωδ −+ ) t 2 srsr me me sites.google.com/site/ncpdhbkhn 57 Coupled – Circuit Viewpoint (5) 1 TLII=−{}sin([][]ωωδ +++ ) t sin( ωωδ −+ ) t 2 srsr me me ω= ω m e 1 →=−T LII[]sin(2ω t ++ δ ) sin δ 2 srsr e 1 T= − LII sin δ avg2 srsr 1 ωω=−→=−T LII[]sin δωδ − sin(2 t − ) me2 srsr e 1 T= − LII sin δ avg2 sr s r sites.google.com/site/ncpdhbkhn 58 Coupled – Circuit Viewpoint (6)  di di d θ vRiL=++s Lcos(θ ) r − Li sin( θ ) me  s ss ssdt sr me dt srr me dt  di di d θ vRiL=++r Lcos(θ ) s − Li sin( θ ) me  r rr rrdt sr me dt srs me dt = ω is I scos e t θ= θ = ω + δ e m m t =−ω ωω − ωδ + es esss LIsin e tLI esrr sin( e t )  →=−ω[ ω ωδ ++ ωωδ + ] eLIr esrssin( e t )cos( e t ) cos( e tt )sin( e )  = −ω ω + δ  esrsL Isin(2 e t ) sites.google.com/site/ncpdhbkhn 59 Coupled – Circuit Viewpoint (7) = θ = = Laf L afcos2 m Laa L bb L cc = = =θ − o Lab L bc L ca Lbf L bfcos(2 m 120 ) =θ + o I = const Lcf L cfcos(2 m 120 ) ω f s =ω + δ ia I acos( e t ) =ω −o + δ ib I bcos( e t 120 ) =ω +o ++ δ ic I ccos( e t 120 ) 1 1 Wii′ (,,)θ= L () θ i2 + L () θ iL 2 + () θ ii fld 1 22 11 1 2 22 2 12 1 2 ′ θ =1 2 +++++ 2 2  + Wiiiifldabcf(,,,,) m ( Li aaa Li bbb Li ccc ) Lii abab Lii bcbc Lii caca  2  θ independent of m +θ + θ + θ Laf() maf iiL bf () mbf iiL cf () mcf ii =1 2 +++++ 2 2  +  (Liaa a Li bb b Li cc c ) Lii ab a b Lii bc b c Lii ca c a  2  θ independent of m cos2θωδ cos(t++ ) cos(2 θ − 120o )cos( ω t −++ 120 o δ )  + L I I  me m e  af a f +θ +o ω ++ o δ  cos(2m 120 )cos( e t 120 )  sites.google.com/site/ncpdhbkhn 60 Coupled – Circuit Viewpoint (8) ′ θ =1 2 +++++ 2 2  + Wiiiifldabcf(,,,,) m ( Li aaa Li bbb Li ccc ) Lii abab Lii bcbc Lii caca  2  θ independent of m cos2θωδ cos(t++ ) cos(2 θ − 120)cos(o ω t −++ 120 o δ )  + L I I  me m e  af a f +θ +o ω ++ o δ  cos(2m 120)cos( e t 120 )  =1 2 +++++ 2 2  +  (Liaaa Li bbb Li ccc ) Lii abab Lii bcbc Lii caca  2  θ independent of m 3 +LIIcos(2θ − ω t − δ ) 2 af a f m e ∂W ′ T = fld ∂θ m ia, i bc , i , i f are const =−θ −− ω δ 3LIIaf a f sin(2 m e t ) → = δ ω T3 LIIaf a f sin If θ= ω t = e t m s 2 2 120 The synchronous speed:ω= ω (rad/s), orn = f (r/min) spoles e s poles e sites.google.com/site/ncpdhbkhn 61 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines a) Coupled – Circuit Viewpoint b) Magnetic Field Viewpoint 8. Linear Machines 9. Magnetic Saturation 10. Leakage Fluxes sites.google.com/site/ncpdhbkhn 62 Magnetic Field Viewpoint (1) Fr δ • Currents in the machine windings create sr magnetic flux in the air gap. F • Torque is produced by the tendency of s the two component magnetic fields to Stator line up their magnetic axes. • Mutual flux : produced by the stator & rotor winding, crosses the air gap & links both windings. • Rotor leakage flux & stator leakage flux : some of the flux that does not cross the air gap. • Only the mutual flux is of direct concern in torque production. sites.google.com/site/ncpdhbkhn 63 Magnetic Field Viewpoint (2) Fr = δ δ Hag g F sr( sr ) sr F =2 + 2 + δ s Fsr F s F r2 FF sr cos sr Stator F ()H = sr ag peak g Fr Fsr δ δ Frsin sr δ sr µ r 0 2 = δ Coenergy density of the air-gap : H Fsrsin s δ 2 ag s Fs F sin δ µ (H ) 2 s sr 0 ag peak = δ Average coenergy density : Fsrsin r 2 2 2 µ F  → Average coenergy density : 0 sr  4 g  sites.google.com/site/ncpdhbkhn 64 Magnetic Field Viewpoint (3) Fr µ   2 0 Fsr δ Average coenergy density :   sr 4 g  ′ = × Fs W fld (averagecoenergydensity) (volumeof air gap) 2 Stator µ F   µ π Dl =0 sr × ()π = 0 2    Dlg Fsr 4 g   4g =2 + 2 + δ Fsr F s F r2 FF sr cos sr µ π Dl →=W′ 0 ( FFFF2 ++ 2 2 cosδ ) fld4g s r sr sr ∂W′ µ π Dl T=fld = − 0 F F sin δ two− pole ∂δ s r sr sr 2g Fs, F r are const   µ π  = − poles0 Dl δ Tmultipole     Fs F rsin sr 2   2 g  sites.google.com/site/ncpdhbkhn 65 Magnetic Field Viewpoint (4) Fr δ poles  µ π Dl  sr T= − 0 F F sin δ multipole     sr sr F 2   2 g  s Stator δ • sr is the electrical space – phase angle between the rotor & stator mmf waves. • The torque T acts in the direction to accelerate the rotor. δ • When sr is positive, the torque is negative & the machine is operating as a generator. δ • When sr is negative, the torque is positive & the machine is operating as a motor. • The torque is proportional to the peak values of the stator– & rotor–mmf δ waves Fs & Fr, and to the sine of the electrical space – phase angle sr between them. • Minus sign: the fields tend to align themselves. • Equal & opposite torques are exerted on the stator & rotor. sites.google.com/site/ncpdhbkhn 66 Magnetic Field Viewpoint (5) Fr poles  µ π Dl  δ = − 0 δ sr Tmultipole     Fs F rsin sr 2   2 g  Fs poles T= − Lii sin θ Stator multipole2 sr s r me poles  µ π Dl  = − 0 δ Fr T    Fs F srsin s Fsr 2   2 g  δ δ Frsin sr δ sr r poles  µ π Dl  = F sin δ δ = − 0 δ sr s s     Frsr F sin r F 2   2 g  s F sin δ µ s sr Fsr0 F sr gB sr = F sin δ H=→= B →= F sr r sr sr sr µ g g 0 poles π Dl  →T = −    B F sin δ 2  2  srr r sites.google.com/site/ncpdhbkhn 67 Magnetic Field Viewpoint (6) Ex. A 2400 r/min, four – pole, 50 Hz synchronous motor has an air – gap length of 1mm. The average diameter of the air – gap is 27 cm, & its axial length is 32 cm. The rotor winding has 800 turns & a winding factor of 0.976. The maximum rotor current is 18A, the maximum Bsr = 2T, find the maximum torque & power output? 4 k N I poles  F = r r r cos θ  r π poles 2 r  4k N ( I ) 4 0.976× 800 × 18 →()F = r r r max = = 4474A r max π poles π 4 poles π Dl  T= −    B F sin δ 2  2  sr r r π π × × → = poles  Dl  =4 0.27 0.32 × = Tmax    Bsr( F r ) max 2 4474 2429Nm 2  2  2 2 π π P= ω T = n T = 2400 2429 = 610kW maxm max s 30 max 30 sites.google.com/site/ncpdhbkhn 68 Magnetic Field Viewpoint (7) Φ = × p (average value of B over a pole) (pole ar ea) π  =2  ×Dl = 2 Dl Bpeak    B peak π  poles  poles poles π Dl  T= −    B F sin δ 2  2  sr r r π poles  2 = −  Φ F sin δ 2 2  sr r r sites.google.com/site/ncpdhbkhn 69 Torque in Nonsalient – Pole Machines poles T= − Lii sin θ 2 srsr me   µ π  = − poles0 Dl δ T    Fs F rsin sr 2   2 g  poles π Dl  T= −    B F sin δ 2  2  sr r r π poles  2 T= −  Φ F sin δ 2 2  sr r r The torque is proportional to the product of the magnitudes of the interacting fields, and to the sine of the electrical space angle between their magnetic axes sites.google.com/site/ncpdhbkhn 70 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10.Leakage Fluxes sites.google.com/site/ncpdhbkhn 71 Linear Machines (1) • The most widely known use of linear motors is in the transportation field: – The moving vehicle: the AC “stator”, and – The rails: the conducting stationary “rotor”. • Also in the machine tool industry & in robotics. • The analysis of linear machines is quite similar to that of rotating mchines: – Angle
displacement, and – Torque
force. sites.google.com/site/ncpdhbkhn 72 Linear Machines (2) 4 Ni F 4 Ni F = cos θ →H =ag 1 = cos θ ag1 π 2 a ag1 gπ 2 g a π θ = 2 z a β 4Ni 2 π z →H = cos ag 1 π2g β Fag Ni 2 Fundamental Fag 1 0 β / 2 z − Ni 2 g sites.google.com/site/ncpdhbkhn 73 Linear Machines (3) 4Ni 2 π z H = cos (concentrated) ag 1 π2g β 4k N i 2 π z H = w ph cos (distributed) ag 1 π2 pg β = ω ia I mcos e t  =ω − o ib I mcos( e t 120 )  =ω + o ic I mcos( e t 120 ) π  →+ =3 2 z − ω FztF( , )max cos e t  2 β  4 k N ω β F=w ph Iv, =e = f β max π2p m 2 π e sites.google.com/site/ncpdhbkhn 74 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10.Leakage Fluxes sites.google.com/site/ncpdhbkhn 75 Magnetic Saturation • Magnetic materials are less than ideal. As their magnetic flux is increased, they begin to saturate. • Therefore saturation may influence the characteristics of the machines. • With saturation, it is more difficult to obtain analytical results. • Saturation characteristics of rotating machines are typically presented in the form of an “open – circuit characteristic” or ”magnetization curve” or ”saturation curve”. sites.google.com/site/ncpdhbkhn 76 Introduction to Rotating Machines 1. Elementary Concepts 2. Introduction to AC and DC Machines 3. MMF of Distributed Windings 4. Magnetic Fields in Rotating Machinery 5. Rotating MMF Waves in AC Machines 6. Generated Voltage 7. Torque in Nonsalient – Pole Machines 8. Linear Machines 9. Magnetic Saturation 10.Leakage Fluxes sites.google.com/site/ncpdhbkhn 77 Leakage Fluxes + − λ λ 3 − + 2 Coil 3 Coil 2 ϕ ϕ 12 13 ϕ 123 I Coil 1 1 ϕ 1l + λ − 1 sites.google.com/site/ncpdhbkhn 78