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Chapter 9

# Chapter 9 - 9-1 Chapter 9 Introduction to AC Machines Exit...

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Unformatted text preview: 9-1 Chapter 9 Introduction to AC Machines Exit 2001 by N. Mohan Print TOC " ! 9-2 Introduction ❏ Primary AC motor drives x Induction motors x Permanent Magnet Brushless (Synchronous Motors) Both have similar stators but different rotor construction b − axis ❏ Stator windings produce sinusoidal field distribution ib 2π / 3 2π / 3 2π / 3 ia a − axis ic c − axis Exit 2001 by N. Mohan TOC " ! 9-3 Sinusoidally-distributed Stator Windings dθ θ 3' 4' 5' ia 6' 2' 1' 7' 1 7 6 θ =0 magnetic axis of phase a θ ia 2 5 4 3 [no. of Conductor density ˆ ns (θ ) = ns sin(θ ) Total N s = ∫0 ns (θ ) dθ = π ⇒ ns (θ ) = Exit 2001 by N. Mohan π ∫0 Ns sin(θ ) 2 conductors / rad ] 0 < θ < π ˆ ˆ ns sin(θ ) dθ = 2ns 0 <θ <π TOC " ! 9-4 Air-gap Field Distribution ξ H a (θ ) = − H a (θ + π ) H a (θ ) l g − H a (θ + π ) l g = 2H a (θ )l g (negative sign because line of integration points inwards at θ + π ) 2H a (θ ) l g = dξ θ θ =π π ∫0 θ =0 µ =∞ Ha ns (θ + ξ ) ia dξ N π 2 H a (θ ) ! g = s ia ∫0 sin(θ + ξ ) dξ = N s ia cos (θ ) 2 N Fa , H a , Ba (θ ) ⇒ H a (θ ) = s ia cos (θ ) 2! g θ +π at time 't' stator surface Ns ia cos (θ ) 2! g µo N s Ba (θ ) = µo H a (θ ) = i cos (θ ) 2! g a N Fa (θ ) = ! g H a (θ ) = s ia cos (θ ) 2 H a (θ ) = − π 2 rotor surface Fa , H a , Ba (θ ) θ 0 π 2 t1 3π 2 π t2 t3 2π t4 θ Field quantities have different magnitudes and units but same shape Exit 2001 by N. Mohan TOC " ! 9-5 b − axis ∠120 o Three-phase sinusoidally-distributed stator windings b ib b ib b' Ba (θ ) 0 a − axis o a ' i a ∠0 a ic 1 ia 1 a Bb (θ ) 0 c' c θ -1 0 60 120 180 240 300 360 420 0 60 120 180 240 300 360 420 0 60 120 180 240 300 360 420 0 60 120 180 240 300 360 420 -1 ic c 1 Bc (θ ) c − axis ∠240 o 0 -1 1 ❏ Example Bs (θ ) µ Ni Ba (θ ) = o s a cosθ = 0.628 cosθ Wb / m 2 2! g Bb (θ ) = −0.440 × cos θ − 1200 Wb / m 2 θ θ 0 -1 θ ( ) Bc (θ ) = −0.188 × cos (θ − 2400 ) Wb / m 2 Bs (θ ) = Ba( θ ) + Bb( θ) + Bc θ = 0.967 cos θ − 13.03° ( ) ( ) Exit 2001 by N. Mohan TOC " ! 9-6 Space Vector to Represent Sinusoidal Distributions b − axis b − axis ib = − ""# Fc (t ) ""# Fa (t ) ic = − ""# Fb (t ) At time ‘t’ θ Fs (t ) ""# Fs (t ) ""# Fc (t ) ""# a − axis Fa (t ) ""# Fb (t ) a − axis ia = + c − axis c − axis ❏ Properties of sine(cosine) Fs (θ ) x sum of two sine = sine x integral/derivative of sine = sine 0 ❏ Complex number representation π 2 π 3π 2 stator surface θ 2π 5π rotor surface 2 ""# Ns N i a (t ) cos( θ ) ⇔ Fa ) = s i a (t ) ∠0 o Fa (θ , t ) = (t 2 2 "" # "" # N Ns o ; Similarly, Fb ( t ) = ib (t ) ∠120 Fc ( t ) = s ic (t ) ∠240 o 2 "" ""# 2 "" "" # # # ˆs ∠θ F And Fs = Fa + Fb + Fc = F ❏ Similar expressions for B and H Exit 2001 by N. Mohan s TOC " ! 9-7 Example N Three-phase, sinusoidally-distributed stator with s = 50 turns 2 At time t , ia = 10 A, ib = −10 A and ic = 0 A "# Find F "# N F (t ) = s ia ∠00 + ib∠1200 + ic ∠2400 2 ( ) = 50 {10 + ( −10 ) [cos 1200 + j sin 1200 ](+ )0 "# F (t ) = 50 × 17.32∠ − 30 o = 866 ∠ − 30 o A ⋅ turns Fs (θ ) } [cos 2400 + j sin 2400 ] at t phase 'a' magnetic axis −30 o 30 o θ θ =0 ""# Fs (t ) " # If l g = 1.5 mm and µ m =∞, find B (t ). " # B(t ) = 0.73∠ − 30 o T Exit 2001 by N. Mohan TOC " ! 9-8 Space Vectors Representation of Combined Phase Currents and Voltages ❏ Mathematical concept b − axis ∠120 o b At time t " # is (t ) = ia (t )∠00 + ib (t )∠1200 + ic (t )∠2400 ˆ = I s (t )∠θ i (t ) b' a − axis o a ' i a ∠0 a s # v s (t ) = va (t )∠00 + vb (t )∠1200 + vc (t )∠2400 ˆ = Vs (t )∠θ vs (t ) Exit 2001 by N. Mohan ib ic c' c c − axis ∠240 o TOC " ! " # Physical interpretation of is (t ) # "" # Ns " Ns 0 Ns 0 Ns 0 is (t ) = ia (t )∠0 + ib (t )∠120 + ic (t )∠240 = Fs (t ) 2 2 2 % \$%&%' \$% ""# &%% \$% ""# ' 2 % &%% ' ""# Fa (t ) F (t ) Fc (t ) at time t # ˆs (t ) b # F (t ) F ˆ ⇒ I s (t ) = is (t ) = s Ns 2 Ns 2 magnetic axis of hypothetical winding ˆ with current I s and θ is (t ) = θ Fs (t ) "" # " # Fs (t ) and is (t ) are collinear "" # # N s µo " Bs (t ) = is (t ) 2l g ❏ Magnetic filed is produced by combined effect of ia , ib and ic but could equivalently be produced by hypothetical " # winding current is (t ) at θ is ❏ helps in obtaining expression for torque Exit 2001 by N. Mohan 9-9 a − axis ˆ Is phase b magnetic axis o phase a magnetic axis 30 "# is ""# Bs phase c magnetic axis TOC " ! 9-10 Space Vector Components: Finding Phase Currents from Current Space Vector b − axis ∠120 o # 3 Re is ∠00 = ia + Re ib ∠1200 + Re ib ∠2400 = ia % % \$% &%% \$% &%% 2 ' ' 1 − ib 2 1 − ic 2 1 − ic 2 c − axis ∠240 o b − axis " # o 2 2ˆ θ − 120 o ⇒ ib (t ) = Re(is ∠ − 120 ) = I s cos is 3 3 # is ∠ − 2400 = Re ia ∠ − 2400 + Re ib∠ − 2400 + ic = 3 ic Re \$%% &%% \$%% &%% 2 ' ' 1 − ia 2 1 − ib 2 " # o 2 2ˆ θ − 240 o ⇒ ic (t ) = Re(is ∠ − 240 ) = I s cos is 3 3 Exit 2001 by N. Mohan at time t ib b' " o # 2 2ˆ ⇒ ia (t ) = Re(is ∠0 ) = I s cosθ is 3 3 # 3 Re is ∠ − 1200 = Re ia ∠ − 1200 + ib + Re ic ∠1200 = ib \$%% &%% % ' \$% &%% 2 ' 1 − ia 2 b a − axis o a ' i a ∠0 a ic c' c ia + ib + ic = 0 projection × 2 = ic (t ) 3 θ "# is (t ) "# " is \$%&%' a − axis 2 projection × = ia (t ) 3 projection × c − axis 2 = ib (t ) 3 TOC " ! 9-11 Balanced Sinusoidal Steady-State Excitation (Rotor Open-Circuited) b imb ima (t ) ωt 0 a @ ( ) @ ωt = ωt = π ωt = 0 c ❏ imc (t ) ˆ Im ima imc imb (t ) 2π @ωt = 3 ( @ ωt = π 3 ωt = 0 4π 3 @ ) ωt = 5π 3 ˆ ˆ ˆ ima = I m cos ωt; imb = I m cos ωt − 1200 ; imc = I m cos ωt − 2400 ""# ˆ ims (t ) = I m cos ω t∠00 + cos(ω t − 1200 )∠1200 + cos(ω t − 2400 )∠2400 ""# 3ˆ ˆ ˆ ⇒ ims (t ) = I ms ∠ω t where I ms = I m """# # 2 Ns " ˆ ˆ Rotating MMF Fms (t ) = is (t ) = Fms ∠ω t where Fms = 3 N s Iˆm = N s Iˆms 2 2 2 2 """" # µo N s ""# & Flux density Bms (t ) = ims (t ) !g 2 ❏ Constant amplitude Exit 2001 by N. Mohan TOC " ! 9-12 Relation Between Space Vectors and Phasors ima (t ) ❒ Time domain 0 imb (t ) ˆ Im imc (t ) ωt ωt ˆ ia (t ) = I m cos (ω t − α ) α ωt = 0 ❒ Phasor ˆ I a = I m∠ − α ˆ I ma = I m ∠ − α ❒ Space Vector ""# ims t =0 ""# ims Exit 2001 by N. Mohan @ t =0 3ˆ ˆ ˆ = I ms ∠ − α ; I ms = I m 2 ❒ Space Vector t =0 ref α α a − axis """ # ˆ ims = I ms ∠ − α ⇔ phasor ⇔ 3 I ma 2 TOC " ! 9-13 Voltages in the stator windings b Emc imb + eb − − − e a ec + + Ema a ema (t ) = Lm """ # ims I ma + Ema jω Lm − imc c I mc I mb ima @ t =0 """# ems I ma Emb d ima (t ) etc. dt Where the three phase magnetizing inductance (2 pole), Lm = 3 πµo rl N s 2 lg 2 """# ""# # N s """" 3 ⇒ ems (t ) = jω Lm ims (t ) = jω π rl Bms (t ) 2 2 Exit 2001 by N. Mohan TOC 2 " ! 9-14 Example va (t ) = 120 2 cos ω t v phase b magnetic axis "" # vs (t ) vb (t ) = 120 2 cos (ω t − 120° ) 30 o vc (t ) = 120 2 cos (ω t − 240° ) # 3 vs = × 120 2∠300 = 254.56 ∠300 V 2 ""# ims = "" # im (t ) phase c magnetic axis phase a magnetic axis 60 o """ # Bm (t ) # vs 254.56 ∠(300 − 900 ) = = 0.869∠ − 600 A jω Lm 2π × 60 × 0.777 # 0 −7 """" µ N i # o s ms = 4π × 10 × 50 × 0.869 ∠ − 60 = 0.055∠ − 600 Wb / m 2 Bms = 2! g 10 −3 Exit 2001 by N. Mohan TOC " ! 9-15 Summary ❏ Draw the three-phase axis in the motor cross-section. Also, draw the three phasors Va , Vb and Va in a balanced sinusoidal steady state. Why is the phase-b axis ahead of the phase-a axis by 120 degrees, but Vb lags Va by 120 degrees? ❏ Ideally, what should be the field (F, H, and B) distributions produced by each of the three stator windings? What is the direction of this field in the air gap? What direction is considered positive and what is considered negative? ❏What should the conductor-density distribution in a winding be in order to achieve the desired field distribution in the air gap? Express the conductor-density distribution ns (θ ) for phase-a. Exit 2001 by N. Mohan TOC " ! 9-16 Summary ❏ How is sinusoidal distribution of conductor density in a phase winding approximated in practical machines with only a few slots available to each phase? ❏ How are the three field distributions (F, H, and B) related to each other, assuming that there is no magnetic saturation in the stator and the rotor iron? ❏ What is the significance of the magnetic axis of any phase winding? ❏ Mathematically express the field distributions in the air gap due to ia as a function of θ . Repeat this for ib and ic . ❏ What do the phasors V and I ""# denote? What are the ""# meanings of the space vectors Ba (t ) and Bs (t ) at time t, assuming that the rotor circuit is electrically open-circuited? ❏ What is the constraint on the sum of the stator currents? Exit 2001 by N. Mohan TOC " ! 9-17 Summary ❏ What are physical interpretations of various stator winding inductances? ❏ Why is the per-phase inductance Lm greater than the single -phase inductance Lm,1− phase by a factor of 3/2? ❏ What are the characteristics of space vectors which represent the field distributions Fs (θ ), H s (θ ) and Bs (θ ) at a given time? What notations are used for these space vectors? Which axis is used as a reference to express them mathematically in this chapter? ❏ Why does a dc current through a phase winding produce a sinusoidal flux-density distribution in the air gap? ❏ How are the terminal phase voltages and currents combined for representation by space vectors? ❏ What is the physical interpretation of the stator current "# space vector is (t ) ? Exit 2001 by N. Mohan TOC " ! 9-18 Summary ❏ With no excitation or currents in the rotor,#are """" the # all # """" of """ space vectors associated with the stator ims (t ), Fms (t ), Bms (t ) collinear (oriented in the same direction)? "" # "# ❏ In ac machines, a stator space vector vs (t ) or is (t ) consists of a unique set of phase components. What is the condition on which these components are based? ❏ Express the phase voltage components in terms of the stator voltage space vector. ❏ Under three-phase balanced sinusoidal condition with no rotor currents, and neglecting the stator winding resistances Rs and the leakage inductance Lls for simplification, answer the following questions: (a) What is the speed at which all of the space vectors rotate? Exit 2001 by N. Mohan TOC " ! 9-19 Summary (b) How is the peak flux density related to the magnetizing currents? Does this relationship depend on the frequency f of the excitation? If the peak flux density is at its rated value, then what about the peak value of the magnetizing currents? (c) How do the magnitudes of the applied voltages depend on the frequency of excitation, in order to keep the flux density constant (at its rated value for example)? ❏ What is the relationship between space vectors and phasors under balanced sinusoidal operating conditions? Exit 2001 by N. Mohan TOC " ! ...
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