2008 Fall 1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringUniversity of British ColumbiaUniversity of British ColumbiaLecture 8Lecture 8Modeling of DC motorsModeling of DC motors2008 Fall 2Course roadmapCourse roadmapLaplace transformLaplace transformTransfer functionTransfer functionModels for systemsModels for systems••electricalelectrical••mechanicalmechanical••electromechanicalelectromechanicalBlock diagramsBlock diagramsLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime responseTime response••TransientTransient••Steady stateSteady stateFrequency responseFrequency response••Bode plotBode plotStabilityStability••RouthRouth--HurwitzHurwitz••NyquistNyquistDesign specsDesign specsRoot locusRoot locusFrequency domainFrequency domainPID & LeadPID & Lead--laglagDesign examplesDesign examples((MatlabMatlabsimulations &) laboratoriessimulations &) laboratories2008 Fall 3What is DC motor?What is DC motor?An actuator, converting electrical energy into An actuator, converting electrical energy into rotational mechanical energyrotational mechanical energy(You will see DC motor during Lab 1 and 4.)(You will see DC motor during Lab 1 and 4.)2008 Fall 4Why DC motor?Why DC motor?Advantages: Advantages: high torquehigh torquespeed controllabilityspeed controllabilityportability, etc.portability, etc.Widely used in control applications: robot, tape Widely used in control applications: robot, tape drives, printers, machine tool industries, radar drives, printers, machine tool industries, radar tracking system, etc.tracking system, etc.Used for moving loads whenUsed for moving loads whenRapid (microseconds) response is not requiredRapid (microseconds) response is not requiredRelatively low power is requiredRelatively low power is required2008 Fall 5How does DC motor work?How does DC motor work?ArmatureArmature(from Dorf and Bishop book)2008 Fall 6Model of DC motorModel of DC motoreeaa(t(t))iiaa(t(t))RRaaLLaaeebb(t(t)) JBBmmArmature circuitArmature circuitMechanical loadMechanical loadInputInputOutputOutput2008 Fall 7Armature circuitArmature circuitConnection between mechanical/electrical partsConnection between mechanical/electrical partsMotor torqueMotor torqueBack EMFBack EMFMechanical loadMechanical loadAngular positionAngular positionModeling of DC motor: time domainModeling of DC motor: time domainLoad torqueLoad torque2008 Fall 8Armature circuitArmature circuitConnection between mechanical/electrical partsConnection between mechanical/electrical partsMotor torqueMotor torqueBack EMFBack EMFMechanical loadMechanical loadAngular positionAngular positionModeling of DC motor: sModeling of DC motor: s--domaindomain2008 Fall 9DC motor: Block diagramDC motor: Block diagramFeedback systemFeedback systemTachometer Tachometer (Page 46)(Page 46)Encoder Encoder (Page 44)(Page 44)2008 Fall 10Useful formula for feedbackUseful formula for feedbackNegative feedback systemNegative feedback systemEEaaMemorize this!Memorize this!2008 Fall 11Ex: Derivation of transfer functionsEx: Derivation of transfer functionsEEaaEEaaCompute transfer functions from Compute transfer functions from R(sR(s) to ) to Y(sY(s).).EEaaEEaa2008 Fall 12DC motor: Transfer functions (TF)DC motor: Transfer functions (TF)22ndndorder systemorder system2008 Fall 13DC motor: Transfer functions (contDC motor: Transfer functions (cont’’d)d)Note:Note:In many cases La<<Ra. Then, an In many cases La<<Ra. Then, an approximated TF is obtained by setting La=0.approximated TF is obtained by setting La=0.11ststorder systemorder system22ndndorder systemorder system2008 Fall 14Summary and ExercisesSummary and ExercisesModeling of DC motorModeling of DC motorWhat is DC motor and how does it work?What is DC motor and how does it work?Derivation of a transfer functionDerivation of a transfer functionBlock diagram with feedbackBlock diagram with feedbackNextNextStabilityStabilityof linear control systems, one of the most of linear control systems, one of the most important topics in feedback controlimportant topics in feedback controlExercisesExercisesRead Section 2.7, 2.8.Read Section 2.7, 2.8.Go over the derivation for DC motor transfer functions Go over the derivation for DC motor transfer functions by yourself. Obtain by yourself. Obtain T(s)/ET(s)/Eaa(s(s).).2008 Fall 15Main message until this pointMain message until this pointMany systems Many systems can be represented can be represented as as transfer functionstransfer functions!!Using the transfer functions, Using the transfer functions, ……. . (to be continued)(to be
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