Chapter 11

Chapter 11 - 11-1 Chapter 11 Induction Motors: Balanced,...

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Unformatted text preview: 11-1 Chapter 11 Induction Motors: Balanced, Sinusoidal Steady State Operation Exit 2001 by N. Mohan Print TOC " ! 11-2 Induction Motors ! Adjustable speed drives ! Servo drives • • • Exit 2001 by N. Mohan Induction motors under balanced sinusoidal steady state (Rated voltage at rated frequency) Speed control using V/f Field oriented control TOC " ! 11-3 Line-fed Induction Motors ! Structure ! Principle of operation ! Equivalent circuits ! Performance characteristics Exit 2001 by N. Mohan TOC " ! 11-4 Structure b − axis ib 2π / 3 a − axis 2π / 3 2π / 3 ia ic c − axis Simple representation of three phase stator windings Exit 2001 by N. Mohan Squirrel-cage rotor TOC " ! 11-5 Stator Representation Vc − − − va vb vc + + vb ib − + + ia − n − va + Va vc + ic Vb ! Assumptions : Rs, Ls,leakage = 0 va (t ) = 2 E cos(2π ft ) 2π ) 3 4π vc (t ) = 2 E cos(2π ft ) 3 (for a 2-pole machine) ω syn = ω = 2π f vb (t ) = ω syn = Exit 2001 by N. Mohan 2 E cos(2π ft - 2 2 ω = (2π f ) p p (for a p-pole machine) TOC " ! Electrically Open-circuited Rotor 11-6 Vc − va − vb − vc + + vb ib + − ia I mc I mb Va − n − va + vc + + ic Vb I ma ! Only magnetizing currents are present because rotor is inert ! Magnetizing currents set up rotating flux ˆ Im = ˆ V ω Lm ˆ ima (t ) = I m cos(ω t - π / 2), etc. Exit 3 ˆ ! vs (t ) = V ∠ω t 2 2001 by N. Mohan ! π 3ˆ ims (t ) = I m ∠(ω t - ) 2 2 3ˆ ˆ I ms = I m 2 3 ˆ ˆ Vs = V 2 TOC " ! 11-7 Electrically Open-circuited Rotor Fields at t = 0 I ma ω syn ! vs """" """ ! ! Bms , ims ! Exit ω syn """" ! Bms phase a − axis 90 o Va + Va I ma − is a constant magnitude, rotating flux ! 3ˆ ˆ ims (t ) = I m ∠(ω t - π / 2) = I ms ∠(ω t - π / 2) 2 ! ! N s ims H ms (t ) = 2# g ! ! ! µ0 N s ˆ ∴ I ms ∠(ω t - π / 2) Bms (t ) = µ0 H m (t ) Bms (t ) = 2# g 2001 by N. Mohan jω Lm TOC " ! Short-circuiting the Rotor (Rs, Ls,leakage = 0) " Transformer Analogy 11-8 φm ′ im + i2 i2 + v1 − N1 N2 Load ! Assuming no resistances or leakage inductance in the stator windings, the stator voltages completely determine the motor flux regardless of any rotor currents ! Flux Φm is unaffected by the load Exit 2001 by N. Mohan TOC " ! Induced Voltages on Rotor 11-9 at t = 0 − − − − ωm − + + + + + ω syn "" a − axis ! vs ω syn """" ! Bms θ Flux rotating at speed ωsyn Rotor rotating at speed ωm Rotor conductors cutting flux at speed: ωsyn – ωm = ωslip (slip speed) Cutting flux generates voltage across rotor conductors: ebar(θ) = Bms(θ) # r ωslip Exit 2001 by N. Mohan TOC " ! Induced Currents in Rotor 11-10 at t = 0 − − − ω m − − + + + + − − ω syn "" ! vs ω syn θ + + Rbar − − − a − axis θ """" ! Bms ebar front + + + + − + − ibar (θ ) + + + − − − back end-ring end-ring ! Rotor conductors (bars) shorted together by end rings ! Because of symmetry of induced bar voltages, end rings are at same potential, therefore bar voltage is dropped across bar resistance (assuming Lr,l = 0) generating currents by Ohms Law ibar Exit 2001 by N. Mohan Bm #rω slip ebar = = Rbar Rbar TOC " ! Rotor MMF – Reflected Rotor MMF – Reflected Rotor Current 11-11 φm,ir ' φm,ir at t = 0 at t = 0 net flux = 0 ωm "" ! vs ima + ira ' ω syn "" ! Fr "" ! Fr′ "" ! vs "" ! ir ' a − axis """ ! ims """" ! Bms "! is a − axis ! ! ! Fs (t ) = Fms (t ) + Fr′(t ) ! ! ! is (t ) = ims (t ) + ir′ (t ) ˆ ˆ I′ = k B ω r i ms slip """" ! Bms ! F ! ! r produced by rotor currents ! Fr′ produced by additional stator currents to keep total flux unchanged (transformer analogy) ! ! These currents are viewed as a current space vector ir′ ! Total stator current is magnetizing current plus this reflected rotor current Exit TOC " 2001 by N. Mohan ! 11-12 Slip frequency (fslip) in the rotor circuit ω slip ω syn - ω m = slip : s = ω syn ω syn f slip ω slip = f = sf ω syn ω slip + ω m = ω syn ! Slip is rotor speed normalized to synchronous speed ! Slip generally small (< 3%), therefore rotor current frequency is very low Exit 2001 by N. Mohan TOC " ! Electromagnetic Torque Production 11-13 # at t = 0 "" ! Fr """ ! Fr ' "" ! vs a − axis "" ! ir ' """" ! Bms ! ! ! Current ir′ , in field Bms , produces torque Tem Exit 2001 by N. Mohan N ˆ ˆ ˆ2 Tem = π s r # Bms I r′ = keω Bmsω slip $&' %2 % kt TOC " ! Torque – Speed Characteristics (slip small ; ωm near ωsyn) 11-14 ωm Tem,rated ω syn,rated ω slip, rated ω m,rated 0 Tem,rated Tem 0 ( ω ω syn,ratedm ω m,rated ω slip, rated ! Linear relationship ! These curves are valid up to rated torque Exit 2001 by N. Mohan TOC " ! Generator (Regenerative Braking) Mode 11-15 ωm """ ! Fr ' "" ! Fr a − axis ω syn """" ! Bms ! For generation or for braking – in either case rotor speed exceeds synchronous speed, ωm > ωsyn ! ωslip < 0 " Bar voltage polarities reversed ! " Rotor currents and mmf ( Fr ) reversed ! " Reflected rotor currents and mmf ( Fr′) reversed " Torque reversed Exit TOC " ! 2001 by N. Mohan Rotor Leakage Inductance φm,ir′ net=0 φm,ir at t = 0 "" ! Fr θr θr "" ! vs (t ) a − axis θr "" ! ir ' "" ! vs """ ! Fr ' a − axis """! ims """" ! Bms """" ! Bms "" ! Fr ω syn at t = 0 θr ""! Br 2001 by N. Mohan φ# r "" ! Fr θ r ω syn Exit at t = 0 θ ω syn """ ! F ' """!r " ims 11-16 θr 90 o "" ! Fr′ "" ! vs a − axis "! ′ ir """" ! 90 o Bms """ ! Blr TOC " ! Rotor Leakage Inductance (cont…) 11-17 ! Effect of rotor leakage inductance is to reduce Tem at high slip ! Rotor leakage inductance is often neglected when motor is operating near synchronous speed (below the rated torque) Exit 2001 by N. Mohan TOC " ! 11-18 Per-Phase Equivalent Circuit at t = 0 ′ I ra Ia θr "" ! Fr ωm """! " ims """ ! Fr ' "" ! ir ' "" ! vs a − axis + ω at f = V 2π a jω Lm I ma Req I ra ' I ma − ′ Req = Rr " ! is """" ! Bms Va jω Leq ω syn ω slip ′ Leq = Llr Ia ω syn Space Vectors Equivalent Circuit Phasor Diagram ! Includes rotor leakage inductance ! Does not include stator leakage inductance or resistance ! Req depends on slip Exit TOC " 2001 by N. Mohan ! Power Into Rotor – Power Lost In Rotor – Power Out Of Rotor 11-19 jω Llr ' + Va jω Llr ' ′ I ra jω Lm ′ I ra + Rr ' ω syn ω slip − Power in resistor is power into the rotor circuit Va − jω Lm ω syn Rr ' ω slip Rr ' Rr ' ⇒ Pr , loss ωm ⇒ Pem ω slip Resistor split to indicate rotor loss and mechanical power ! Power in equivalent resistance represents power entering rotor across air gap ! Depending on slip, some or all of this power becomes losses in the rotor Exit TOC " 2001 by N. Mohan ! Stator Winding Resistance and Leakage Inductance 11-20 Rs + Va (at ω ) I ra ' Ia jω Lls Va Ema − jω Llr + − I ma jω Lm Equivalent Circuit Exit 2001 by N. Mohan ' Rr ' ω syn ω slip I ma Ia ′ I ra Ema jω Lls I a Rs I a Phasor Diagram TOC " ! 11-21 Motor Tests ! DC – Resistance Test ( Rs ) ! No Load Test ( Lm ) ′ ! Blocked Rotor Test ( Rr′ , Lls , Llr ) Exit 2001 by N. Mohan TOC " ! No Load Test (Lm) Rs + Va (at ω ) I ra ' Ia jω Lls + Ema − 11-22 Rs jω Llr ' I ma jω Lls I ra ' ≈ 0 + Rr ' ω syn Va ω slip − Ia jω Lm − Equivalent Circuit under no load conditions (ω slip = 0 ∴ Req = ∞ ) Equivalent Circuit jω Lm Approximate Circuit ( Lm >> Ll ) (Rs negligible) ! Under no load conditions the equivalent circuit is dominated by the magnetizing inductance Exit 2001 by N. Mohan TOC " ! Blocked Rotor Test (Ll ) Rs + Va (at ω ) I ra ' Ia jω Lls + Ema − Rs jω Llr ' I ma − Equivalent Circuit + Rr ' ω syn ω slip Va − Ia jω Lls 11-23 I ra ' ′ jω Llr Rr ' ω syn I ma ≈ 0 ω slip =1 Approximate Circuit With Rotor Blocked (R eq << ωL m ) and (ωL#r << ωL m ) ! With the rotor blocked, the magnetizing inductance is nearly shorted out and can be neglected ! Measurements give real power (into Rs and Rr′ ) and reactive ′ power (into Lls and Llr ) ! Rr′ can be found since Rs was previously determined through the DC test 2 ′ ′ ! To find Lls and Llr we can often assume that Lls = Llr 3 TOC Exit " ! 2001 by N. Mohan Characteristics at Rated Voltage and Rated Frequency 11-24 ′ I ra Tem Tem,rated 2.0 ′ I ra ,rated pull-out torque ˆ Bms decreases 6.0 5.0 1.5 ˆ Bms,rated 1.0 3.0 (rated) I ma 2.0 0.5 0 4.0 0.2 0.4 0.6 ω slip 0.8 1.0 ωm ω syn ω slip ,rated 1.0 0 0.2 0.4 0.6 ω slip 0.8 ωm 1.0 ω syn ω slip,rated ! Nearly linear near ωsyn ! At higher slip (ωm smaller) leakage inductances and stator resistance reduce torque ! High currents at low speeds (start-up condition) Exit 2001 by N. Mohan TOC " ! Motor Currents, Efficiency, Power Factor As a Function of Load 11-25 100 90 Efficiency 80 70 Power Factor 60 Current (A) Efficiency (%) Power Factor (%) 50 40 Current 30 20 10 0 25 75 100 50 Load (%) 125 Typical for design B 10 kW, 4 pole, three-phase induction motor Exit 2001 by N. Mohan TOC " ! 11-26 Line Start ! When started directly off the line, induction motor draws a very large current (approx. 8 x rated) ! At the same time the torque available to accelerate the motor/load is limited ! Motor can quickly overheat – Solution: Reduced voltage soft start ′ I ra ′ I ra ,rated 6.0 Tem 5.0 Tacc = Tem − TL Tem 4.0 Tem 3.0 I ma 2.0 1.0 0 0.2 0.4 0.6 0.8 ω slip Current vs. Speed Exit 2001 by N. Mohan ωm 1.0 ω syn ω slip, rated 0 TL ωm steady state speed Accelerating Torque TOC " ! Reduced Voltage Starting (Soft Start) Energy Savings in Lightly – Loaded Machines 11-27 ia ia a 3-phase ac input 0 n t ea 0 t Tem Three-phase induction motor 0 t ! Circuit applies reduced voltage to motor during start-up to avoid large currents and over heating ! Circuit also used to reduce voltage to motor under light load steady state conditions. This improves efficiency Exit TOC " ! 2001 by N. Mohan 11-28 Summary/Review ! Describe the construction of squirrel-cage induction machines. ! With the rated voltages applied, what does the magnetizing current depend on? Does this current, to a significant extent, depend on the mechanical load on the motor? How large is it in relation to the rated motor current? ! Draw the space vector diagram at t = 0, and the corresponding phasor diagram, assuming the rotor to be open-circuited. ! Under a balanced, three-phase, sinusoidal steady state excitation, what is the speed of the rotating flux-density distribution called? How is this speed related to the angular frequency of the electrical excitation in a p-pole machine. Exit 2001 by N. Mohan TOC " ! 11-29 Summary/Review ! In our analysis, why did we initially assume the stator leakage impedance to be zero? How does the analogy to a transformer, with the primary winding leakage impedance assumed to be zero, help? Under the assumption that the stator leakage ! impedance is zero, is the flux-density space vector Bms (t) completely independent of the motor loading? ! What is the definition of the slip speed ωslip ? Does ωslip depend on the number of poles? How large is the rated slip speed, compared to the rated synchronous speed? ! Write the expressions for the voltage and the current (assuming the rotor leakage inductance to be zero) in a rotor bar located at an angle θ from ! peak of the flux density distribution the represented by Bms (t ) . Exit 2001 by N. Mohan TOC " ! 11-30 Summary/Review ! The rotor bars located around the periphery of the rotor are of uniform cross-section. In spite of this, what allows us to represent the mmf produced by the rotor bar currents by a space ! vector Fr (t ) at any time t? ! Assuming the stator leakage impedance and the rotor inductance to be zero, draw the space vector diagram, the phasor diagram, and the per-phase equivalent circuit of a loaded induction motor. ! In the equivalent circuit of Problem 9, what quantities does the ˆ′ rotor-bar current peak, represented by I ra , depend on? ! What is the frequency of voltages and currents in the rotor circuit called? How is it related to the slip speed? Does it depend on the number of poles? Exit 2001 by N. Mohan TOC " ! 11-31 Summary/Review ! What is the definition of slip s, and how does it relate the frequency of voltages and currents in the stator circuit to that in the rotor circuit? ! What is the speed of rotation of the mmf distribution produced by the rotor bar currents: (a) with respect to the rotor? (b) in the air gap with respect to a stationary observer? ! Assuming L′ to be zero, what is the expression for the torque lr ˆ Tem produced? How and why does it depend on ωslip and Bms ? Draw the torque-speed characteristic. ! Assuming L′r to be zero, explain how induction motors meet l load-torque demand. Exit 2001 by N. Mohan TOC " ! 11-32 Summary/Review ! What makes an induction machine go into the regenerativebraking mode? Draw the space vectors and the corresponding phasors under the regenerative-braking condition. ! Can an induction machine be operated as a generator that feeds into a passive load, for example a bank of three-phase resistors? ! How is it possible to reverse the direction of rotation of an induction machine? ! Explain the effect of including the rotor leakage flux by means of a space vector diagram. ! How do we derive the torque expression, including the effect ′ of Llr ? Exit 2001 by N. Mohan TOC " ! 11-33 Summary/Review ! ! ! ! ! Exit ! ! ! What is Br (t ) and how does it differ from Bms (t ) ? Is Br (t ) ! perpendicular to the Fr (t ) space vector? Including the rotor leakage flux, which rotor bars have the highest currents at any instant of time? What clue do we have for the vector control of induction machines, to emulate the performance of brush-type and brushless dc motors discussed in Chapters 7 and 10? Describe how to obtain the per-phase equivalent circuit, including the effect of the rotor leakage flux. ˆ′ What is the difference between I ra in Fig. 11-18c and in Fig. 1119c, in terms of its frequency, magnitude, and phase angle? 2001 by N. Mohan TOC " ! 11-34 Summary/Review ! Is the torque expression in Eq. 11-41 valid in the presence of the rotor leakage inductance and the stator leakage impedance? ! When producing a desired torque Tem, what is the power loss in the rotor circuit proportional to? ! Draw the per-phase equivalent circuit, including the stator leakage impedance. ! Describe the tests and the procedure to obtain the parameters of the per-phase equivalent circuit. ! In steady state, how is the mechanical torque at the shaft different than the electromechanical torque Tem developed by the machine? Exit 2001 by N. Mohan TOC " ! 11-35 Summary/Review ! Do induction machines have voltage and torque constants similar to other machines that we have studied so far? If so, write their expressions. ! Plot the torque-speed characteristic of an induction motor for applied rated voltages. Describe various portions of this characteristic. ! What are the various classes of induction machines? Briefly describe their differences. ! What are the problems associated with the line-starting of induction motors? Why is the starting currents so high? ! Why is reduced-voltage starting used? Show the circuit implementation and discuss the pros and cons of using it to save energy. Exit 2001 by N. Mohan TOC " ! ...
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This note was uploaded on 02/06/2012 for the course EE 4002 taught by Professor Scalzo during the Fall '06 term at LSU.

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