Lesson 14: Transfer Functions Of Dc Motors

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10/28/2015lesson14et438a.pptxLESSON 14: TRANSFERFUNCTIONS OF DC MOTORS1ET 438a Automatic Control Systems TechnologyLEARNING OBJECTIVESAfter this presentation you will be able to: lesson14et438a.pptx Write the transfer function for an armaturecontrolled dc motor.Write a transfer function for a dc motor thatrelates input voltage to shaft position.Represent a mechanical load using a mathematicalmodel.Explain how negative feedback affects dc motorperformance.21

10/28/2015STEADY-STATE OPERATION OFSEPARATELY EXCITED DC MOTORSConsider steady-state modellesson14et438a.pptxia armature currenteb back emfea armature terminal voltagewm motor speed (rad/sec)T motor torqueTf static friction torqueRa armature resistanceLa armature inductanceJm rotational inertiaBm viscous frictionwmReview the steady-state relationshipsOf machine3STEADY-STATE OPERATION OFSEPARATELY EXCITED DC MOTORSRelationships of Separately Excited Dc MotorTorque-Current CurveKT DT/DiaebKT DT/DiaDTDia-TfSpeed-Torque Curvelesson14et438a.pptxTBack EMF CurveDTDiaiawmwmwnlwm wnl – (Dwm/DT)TDwmDTT42

10/28/2015STEADY-STATE MOTOR EQUATIONSDeveloped TorqueKVL in Armature Circuitea i a R a e b VT K T i a Tf N - mT motor torqueKT torque constantTf motor friction torqueia armature currentBack EMFDeveloped Powereb K e wm VP wm T Wwm shaft speed (rad/s)eb back emfKe back emf constantlesson14et438a.pptxea armature voltageeb back emfRa armature resistanceP shaft power5STEADY-STATE MOTOR EQUATIONSCombining the previous equations gives:wm (1)wm ea i a R aKe(2)lesson14et438a.pptxK T ea (T Tf ) R aKT KeIf the load torque is zero (T 0) then the above equation (1) gives theno-load speedwnl K T e a (Tf ) R aKT Ke63

10/28/2015STEADY-STATE MOTOR OPERATIONlesson14et438a.pptxExample 14-1: An armature-controlled dc motor has thefollowing ratings: Tf 0.012 N-m, Ra 1.2 ohms, KT 0.06 N-m/A,Ke 0.06 V-s/rad. It has a maximum speed of 500 rad/s with amaximum current of 2 A. Find: a) maximum output torque, b)maximum mechanical output power, c) maximum armaturevoltage, d) no-load speed at maximum armature voltage.7EXAMPLE 14-1 SOLUTION (1)Define given variableslesson14et438a.pptxa) Tmax occurs at Imax so .Answerb) Find Pmax8Answer4

10/28/2015EXAMPLE 14-1 SOLUTION (2)c) Find maximum back emflesson14et438a.pptxAnswerd) Find no-load motor speedAt no-load, T 0. Load torque is zero.T 09TRANSFER FUNCTION OF ARMATURECONTROLLED DC MOTORWrite all variables as time functionsRa La ia(t)ea(t)eb(t)T(t)JmBmlesson14et438a.pptxWrite electrical equationsand mechanical equations.Use the electromechanicalrelationships to couple thetwo equations.Consider ea(t) and eb(t) as inputs and ia(t) as output. Write KVLaround armatureea ( t ) R a i a ( t ) L Mechanical Dynamicsdi a ( t ) e b ( t )dtT( t ) J m dwm ( t ) Bm wm ( t )dt105

10/28/2015TRANSFER FUNCTION OF ARMATURECONTROLLED DC MOTORElectromechanical equationse b ( t ) K E wm ( t )Find the transfer function between armature voltage and motor speed m (s) ?E a (s)lesson14et438a.pptxT( t ) K T i a ( t )Take Laplace transform of equations and write in I/O formE a (s) L s I a (s) R a I a (s) E b (s)E a (s) (L s R a ) I a (s) E b (s)E a (s) E b (s) (L s R a ) I a (s) 1I a (s) E a (s) E b (s) L s Ra 11TRANSFER FUNCTION OF ARMATURECONTROLLED DC MOTORLaplace Transform of Electromechanical EquationsE b (s) K E m (s)Laplace Transform of Mechanical System DynamicsT( t ) J m dwm ( t ) Bm wm ( t )dtlesson14et438a.pptxT(s) K E I a (s)T(s) J m s m (s) Bm m (s)Rewrite mechanical equation as I/O equation 1T s J m s Bm m s m s T s J s Bm m126

10/28/2015lesson14et438a.pptxBLOCK DIAGRAM OF ARMATURECONTROLLED DC MOTORDraw block diagram from the following equations 1I a (s) E a (s) E b (s) L s Ra Ea(s)1/(Las Ra) - 1T(s) K T Ia (s) m s T s J s Bm mIa(s)T(s)KT1/(Jms Bm) m(s)Eb(s)KeE b (s) K E m (s)Note: The dc motor has an inherent feedbackfrom the CEMF. This can improve system stabilityby adding a electromechanical damping13TRANSFER FUNCTION OF ARMATURECONTROLLED DC MOTORUse the feedback formula to reduce the block diagramH(s) K EG(s) is the product of all the blocks in the forward path1/(Las Ra)KTlesson14et438a.pptx m s G s E a (s) 1 G s H s 1/(Jms Bm) 11KTG s K T La s R a J m s Bm La s R a J m s Bm 147

10/28/2015SIMPLIFICATION OF TRANSFER FUNCTIONSubstitute G(s) and H(s) into the feedback formulaG(s)H(s)G(s)Simplify by multiplying numerator anddenominator by factors (Las Ra)(Jms Bm)lesson14et438a.pptxKT L a s R a J m s Bm m s E a (s) KT1 KE L s R J s Bamm a m s KT E a (s) La s R a J m s Bm K T K EExpand factors and collect like terms of sFinal Formula m s KT E a (s) La J m s 2 (R a J m Bm La ) s (K T K E R a Bm )15Roots of denominator effected by values of parameters. Can be Imaginary.DC MOTOR POSITION TRANSFERFUNCTIONMotor shaft position is the integral of the motor velocity withrespect to time. To find shaft position, integrate velocityTo find the motor shaft position with respect to armature voltage,reduce the following block diagramT(s)Ia(s) m(s)Ea(s)1/(Las Ra)1/(Jms Bm)1/s KTlesson14et438a.pptxd ( t ) w( t )dtd ( t ) dt dt w(t ) dt (t )Qm(s)Eb(s)Ke168

10/28/2015DC MOTOR POSITION TRANSFERFUNCTIONPosition found by multiplying speed by 1/s (integration in time)lesson14et438a.pptx 1 Q m (s) m (s) s Q m (s) 1 KT E a (s) s L m J m s 2 L a Bm R a J m s K T K E R a Bm Q m (s)KT E a (s) s (L m J m s 2 L a Bm R a J m s K T K E R a Bm )Q m (s)KT E a (s) L m J m s 3 L a Bm R a J m s 2 K T K E R a Bm sT.F.17REDUCED ORDER MODELDefine motor time constantsJm m andBmLa eRaElectrical time constant is much smaller than mechanical timeconstant. Usually neglected. Reduced transfer function becomes lesson14et438a.pptxWhere: m mechanical time constant e electrical time constantKs m (s) E a (s) 1 s sWhere K s R a JmKTand s K T K E R a BmK T K E R a Bm189

10/28/2015MOTOR WITH LOADConsider a motor with load connected through a speed reducer.Load inertia JLLoad viscous friction BL N wL 1 wm rad/sec N1 N 2 N2 N TL 2 Tm N - m N1 lesson14et438a.pptxMotor coupled to speed reducer, motor shaft coupled to smallergear with N1 teeth. Load connected to larger gear with N2 teeth.N1 N 2Gear reduction decreases speed but increases torquePmech constant. Similar to transformer action19MOTOR WITH LOADSpeed changer affects on load friction and rotational inertiaWithout speed changer (direct coupling)N - m - s/radJT Jm JLN - m - s 2 / rad2 N BT B m 1 B L N2 N - m - s/radlesson14et438a.pptxWith speed changerBT B m B L2 N JT Jm 1 JL N2 N - m - s 2 / radWhere: BT total viscous frictionJT total rotational inertiaBL load viscous frictionBm motor viscous frictionJm motor rotational inertiaJL load rotational inertia2010

10/28/2015MOTOR WITH LOAD BLOCK DIAGRAMT(s)Ia(s)Ea(s) -1/(Las Ra)KT1/(JTs BT) m(s)N1/N2 L(s)lesson14et438a.pptxEb(s)KeInertia andfriction ofload includedTransfer function with speed changer N KT 1 m s N2 2E a (s) L a J m s (R a J m Bm L a ) s (K T K E R a Bm )21MOTOR POSITION WITH LOAD BLOCKDIAGRAMIa(s)Ea(s) -1/(Las Ra)KTQL(s)1/(JTs BT)N1/N21/s L(s)Kelesson14et438a.pptxEb(s) m(s)T(s)Motor position transfer function with speed changer. Note:multiplication by s N KT 1 Q L s N2 3E a (s) L a J m s (R a J m Bm L a ) s 2 (K T K E R a Bm ) s2211

10/28/2015DC MOTOR TRANSFER FUNCTIONEXAMPLEa) Determine the voltage/velocity and voltage/position transferfunctions for this motorb) Determine the voltage/velocity and voltage/position transferfunctions for the motor neglecting the electrical time constant.lesson14et438a.pptxExample 14-2: A permanent magnet dc motor has the followingspecifications.Maximum speed 500 rad/secMaximum armature current 2.0 AVoltage constant (Ke) 0.06 V-s/radTorque constant (KT) 0.06 N-m/AFriction torque 0.012 N-mArmature resistance 1.2 ohmsArmature inductance 0.020 HArmature inertia 6.2x10-4 N-m-s2/radArmature viscous friction 1x10-4 N-m-s/rad23EXAMPLE 14-2 SOLUTION (1)Define all motor parameterslesson14et438a.pptxa) Full transfer function model2412

10/28/2015EXAMPLE 14-2 SOLUTION (2)Compute denominator coefficients from parameter valueslesson14et438a.pptxCan normalize constant by dividing numerator and denominator by 0.0037225EXAMPLE 14-2 SOLUTION (3)To covert this to a position transfer function, multiple it by 1/slesson14et438a.pptxb) Compute the transfer functions ignoring the electrical timeconstant2613

10/28/2015EXAMPLE 14-2 SOLUTION (4)Compute parameter SSON 14: TRANSFERFUNCTIONS OF DC MOTORSET 438a Automatic Control Systems Technology14

J J J N- m-s / rad B B B N- m-s/rad 2 T m L T m L Where: B T total viscous friction J T total rotational inertia B L load viscous friction B m motor viscous friction J m motor rotational inertia JL load rotational inertia With speed changer J1 N- m- s / rad N N J J B N m- s/rad N N 2 L 2 T m L 2 2 1 T m » ¼ º « ª » ¼ º .

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