Electro-Mechanical SystemsMechanical Systems Notes...Electro-Mechanical SystemsMechanical Systems...

ElectroElectro--Mechanical SystemsMechanical SystemsElectroElectro Mechanical SystemsMechanical Systems

•• DC MotorsDC Motors–– Principles of OperationPrinciples of Operation

M d li (D i ti f G i E ti (EOM))M d li (D i ti f G i E ti (EOM))–– Modeling (Derivation of Governing Equations (EOM))Modeling (Derivation of Governing Equations (EOM))

•• Block Diagram RepresentationsBlock Diagram RepresentationsUsing Block Diagrams to Represent Eq ations inUsing Block Diagrams to Represent Eq ations in DomainDomain–– Using Block Diagrams to Represent Equations in Using Block Diagrams to Represent Equations in ss -- DomainDomain

–– Block Diagram Representation of DC MotorsBlock Diagram Representation of DC Motors

•• ExampleExample•• ExampleExample

School of Mechanical EngineeringPurdue University

ME375 ElectroMechanical - 1

DC MotorsDC MotorsDC MotorsDC Motors•• TerminologyTerminology

–– RotorRotor : the rotating part of the motor: the rotating part of the motorRotorRotor : the rotating part of the motor.: the rotating part of the motor.–– StatorStator : the stationary part of the : the stationary part of the

motor.motor.–– Field SystemField System : the part of the motor: the part of the motorField SystemField System : the part of the motor : the part of the motor

that provides the magnetic flux.that provides the magnetic flux.–– ArmatureArmature : the part of the motor : the part of the motor

which carries current that interacts which carries current that interacts with the magnetic flux to produce with the magnetic flux to produce torque.torque.

–– BrushesBrushes : the part of the electrical : the part of the electrical circuit through which the current iscircuit through which the current iscircuit through which the current is circuit through which the current is supplied to the armature.supplied to the armature.

–– CommutatorCommutator : the part of the rotor : the part of the rotor that is in contact with the brushes.that is in contact with the brushes.

Motors are actuation devices (actuators) Motors are actuation devices (actuators) that generate that generate torquetorque as actuation.as actuation.



DC MotorsDC Motors -- Principles of OperationPrinciples of OperationDC Motors DC Motors Principles of OperationPrinciples of Operation•• Torque GenerationTorque Generation

B B

dLd f

i

Force will act on a conductor in a Force will act on a conductor in a magnetic field with current flowing magnetic field with current flowing through the conductor.through the conductor.

iIntegrate over the entire length:Integrate over the entire length:d f i dLa

B

f Total torque generated:Total torque generated:

Coil B



DC MotorsDC Motors -- Principles of OperationPrinciples of OperationDC Motors DC Motors Principles of OperationPrinciples of OperationLet Let NN be the number of coils in the motor. The total torque be the number of coils in the motor. The total torque generated from the generated from the NN coils is:coils is:

m aN i L R

( )2 B

For a given motor, (For a given motor, (N, N, BB, L, R, L, R) are fixed. We can define) are fixed. We can defineK N L RT 2 B Nm / A

as the as the Torque ConstantTorque Constant of the motor.of the motor.The torque generated by a DC motor is proportional to the The torque generated by a DC motor is proportional to the armature current armature current iiaa ::

T

Large KLarge KTT ::–– Large (N, L, R).Large (N, L, R).

aa

For a DC motor, it is desirable to have a large KFor a DC motor, it is desirable to have a large KTT . However, size . However, size and other physical limitations often limits the achievable Kand other physical limitations often limits the achievable K

m T aK i (N, L, R) is limited by the size (N, L, R) is limited by the size and weight of the motor.and weight of the motor.

–– Large Large BB::



and other physical limitations often limits the achievable Kand other physical limitations often limits the achievable KTT .. Need to understand the Need to understand the methods of generating flux ...methods of generating flux ...

DC MotorsDC Motors -- Principles of OperationPrinciples of Operation•• BackBack--EMF GenerationEMF Generation

Electromotive force (EMF) is generated in a conductorElectromotive force (EMF) is generated in a conductor

DC Motors DC Motors Principles of OperationPrinciples of Operation

Electromotive force (EMF) is generated in a conductor Electromotive force (EMF) is generated in a conductor moving in a magnetic field:moving in a magnetic field:

Integrate over the entire length Integrate over the entire length LL::de v dLemf ( )

B v B

B

vg gg g

Since the Since the NN armature coils are in series, the total EMF is:armature coils are in series, the total EMF is:

eemf

E N R Lf 2 ( ) B

v

Define the Define the BackBack--EMF ConstantEMF Constant KKbb ::

Th B kTh B k EMF d d h i f h i i h li dEMF d d h i f h i i h li d

E N R Lemfv

2 ( )B

K N R Lb 2 B V / (rad / sec)

The BackThe Back--EMF generated due to the rotation of the motor armature is opposing the applied EMF generated due to the rotation of the motor armature is opposing the applied voltage and is proportional to the angular speed voltage and is proportional to the angular speed of the motor:of the motor:

E Kemf b



Note:Note: KKTT = K= Kbb is true only if consistent SI units are used is true only if consistent SI units are used !!

DC MotorsDC Motors -- Principles of OperationPrinciples of OperationDC Motors DC Motors Principles of OperationPrinciples of Operation•• Generating Magnetic FluxGenerating Magnetic Flux

–– Permanent MagnetPermanent MagnetPermanent MagnetPermanent MagnetPermanentPermanent--Magnet DC Motors (PMDC)Magnet DC Motors (PMDC)

–– Field Coil Induced Magnetic FieldField Coil Induced Magnetic Field(a)(a) Series Wound DC MotorSeries Wound DC Motor(a)(a) Series Wound DC MotorSeries Wound DC Motor

–– High starting torque and noHigh starting torque and no--load speedload speed–– UnidirectionalUnidirectional

(b)(b) Shunt Wound DC MotorShunt Wound DC Motor–– Low starting torque and noLow starting torque and no--load speedload speed–– Good speed regulationGood speed regulation–– UnidirectionalUnidirectional

(c)(c) Compound DC MotorCompound DC Motor–– High starting torque & good speed High starting torque & good speed

regulationregulation(d)(d) Separately Excited DC MotorSeparately Excited DC Motor



(d)(d) Separately Excited DC MotorSeparately Excited DC Motor

DC MotorsDC Motors -- ModelingModelingDC Motors DC Motors ModelingModelingSchematicSchematic

Element LawsElement Laws::+ e+ e Element LawsElement Laws::Electrical SubsystemElectrical Subsystem

iA

+ eLa

LA

+ eRa

RA+ei(t)

+Eemf

JA

mL

Mechanical SubsystemMechanical Subsystem

_ _

B

FBDFBD::yy

Interconnection LawsInterconnection Laws::



DC MotorsDC Motors -- ModelingModelingDerive I/O ModelDerive I/O Model::

DC Motors DC Motors ModelingModeling

I/O Model from eI/O Model from eii((tt) ) andand LL to angular speed to angular speed ::

I/O Model from eI/O Model from eii((tt) ) andand LL to angular position to angular position ::

L JK

L BK

R JK

R BK

K e tL R

KA A

T

A

T

A A

T

A

Tb i

A L A L

T

( )

FHG

IKJ FHG

IKJ

b g

F I F I b gSchool of Mechanical Engineering

Purdue UniversityME375 ElectroMechanical - 8

L JK

L BK

R JK

R BK

K e tL R

KA A

T

A

T

A A

T

A

Tb i

A L A L

T

( )

FHG

IKJ FHG

IKJ

b g

DC MotorsDC Motors -- ModelingModelingDC Motors DC Motors ModelingModelingTransfer FunctionTransfer Function::

( ) ( ) ( )s E s si L

QQ L t th l d t b (N L d) h t i th t d t t d (NL t th l d t b (N L d) h t i th t d t t d (N L d S d) f thL d S d) f th

( ) ( ) ( )i Ls E s s

Q:Q: Let the load torque be zero (No Load), what is the steady state speed (NoLet the load torque be zero (No Load), what is the steady state speed (No--Load Speed) of the Load Speed) of the motor for a constant input voltage Vmotor for a constant input voltage V ??

Q:Q: Let the load torque Let the load torque LL = T, what is the steady state speed of the motor for a constant input = T, what is the steady state speed of the motor for a constant input voltage V voltage V ??



Block Diagram RepresentationBlock Diagram Representation•• Signal Addition/SubtractionSignal Addition/Subtraction

Block Diagram RepresentationBlock Diagram Representation•• Differential Equation Differential Equation Transfer Transfer

Function (System & Signals)Function (System & Signals) Y s U s U s( ) ( ) ( ) 1 2( y g )( y g )

Y s G s U s( ) ( ) ( )

G(s)U(s) Y(s)

I O

Y s U s U s( ) ( ) ( )1 2

U1(s) Y(s)

+

ExEx:: Draw the block diagram for the following Draw the block diagram for the following DEDEExEx:: Draw the block diagram for the followingDraw the block diagram for the following

( )InputSignal

OutputSignal U2(s)

DEDE::ExEx:: Draw the block diagram for the following Draw the block diagram for the following DEDE::

J J B



Block Diagram RepresentationBlock Diagram Representation•• Multiple InputsMultiple Inputs

Block Diagram RepresentationBlock Diagram Representation•• Transfer Function in SeriesTransfer Function in Series

Y s G s Y s Y s G s U s( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 1 , Y s G s U s Y s G s U s1 1 1 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ,

Y s G s G s U s

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) 2 1 1 1

2 1

,

b gU(s) Y(s)

G2 (s)G1 (s)Y1(s)

Y s Y s Y sG s U s G s U s

1 1 1 2 2 2

1 2

1 1 2 2

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )

,

U (s) Y (s)

•• Transfer Function in ParallelTransfer Function in Parallel

InputSignal

OutputSignal

2 ( )1 ( ) U1(s)

U2(s)

InputG2 (s)

G1 (s)

Y(s)

OutputSignal

Y1(s)

Y2(s)X s G s U s X s G s U s

Y s X s X s

G s G s U s

1 1 2 2

1 2

1 2

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

,

b g

InputSignals

Y2(s)

U(s) Y(s)

Inputi l

OutputSi lG (s)

G1 (s) X1(s)



Signal SignalG2 (s)X2(s)

Block Diagram RepresentationBlock Diagram RepresentationBlock Diagram RepresentationBlock Diagram Representation

•• Feedback LoopFeedback Loop

U(s) Y(s)

InputSignal

OutputSignal

G (s) X (s)

H(s)



Block Diagram Representation of DC MotorsBlock Diagram Representation of DC MotorsBlock Diagram Representation of DC MotorsBlock Diagram Representation of DC MotorsSchematicSchematic ElectroElectro--Mechanical CouplingMechanical Coupling::

Take Laplace Transform of the Eqs.Take Laplace Transform of the Eqs.+ eLa + eRa

iA LARA+ei(t)

_

+Eemf

_JA

mL

Governing Equations:Governing Equations:B

d ( ) ( )

( )

A A A A emf idL i R i E e tdt

J B

( )

A m L

m T A

J B

K iE K



emf bE K

Block Diagram Representation of DC MotorsBlock Diagram Representation of DC MotorsBlock Diagram Representation of DC MotorsBlock Diagram Representation of DC Motors

Mechanical SubsystemMechanical Subsystem::Take Laplace Transform of the Eqs.Take Laplace Transform of the Eqs.

Electrical SubsystemElectrical Subsystem::Take Laplace Transform of the Eqs.Take Laplace Transform of the Eqs.

Q: Now that we generated a block diagram of a voltage driven DC Motor, can we derive the transferfunction of this motor from its block diagram ? ( This is the same as asking you to reduce the multi-



block diagram to a simpler form just relating inputs ei(t) and L to the output, either or .)

Block Diagram ReductionBlock Diagram ReductionFrom Block Diagram to Transfer FunctionFrom Block Diagram to Transfer Function•• Label each signal and blockLabel each signal and block

Block Diagram ReductionBlock Diagram Reduction

gg

1J + BKT

1L + R

JA s + BTLA s + RA

•• Write down the relationships between signalsWrite down the relationships between signals

Kb



Block Diagram ReductionBlock Diagram ReductionBlock Diagram ReductionBlock Diagram Reduction•• Solve for the output signal in terms of the input signalsSolve for the output signal in terms of the input signals

•• Substitute the transfer functions’ label with the actual formula and simplifySubstitute the transfer functions’ label with the actual formula and simplifySubst tute t e t a s e u ct o s abe w t t e actua o u a a d s p ySubst tute t e t a s e u ct o s abe w t t e actua o u a a d s p y

( )( ) ( )

( )( ) ( )

( )s KL J s BL R J s R B K K

E s L s RL J s BL R J s R B K K

sT

A A A A A A b Ti

A A

A A A A A A b TL

2 2

K L R



( )( ( ) ( ))

( )( ( ) ( ))

( )s Ks L J s BL R J s R B K K

E s L s Rs L J s BL R J s R B K K

sT

A A A A A A b Ti

A A

A A A A A A b TL

2 2

ExampleExampleExampleExample(A)(A) Given the following specification of a DC Given the following specification of a DC

motor and assume there is no load, find its motor and assume there is no load, find its (B)(B) Find the poles of the transfer function.Find the poles of the transfer function.

transfer function from input voltage to motor transfer function from input voltage to motor angular speed angular speed LLAA = = 10 mH10 mHRRAA == 1010 RRAA 10 10 KKTT == 0.06 Nm/A0.06 Nm/AJJAA = = 4.7 4.7 1010--66 Kg mKg m22

B = B = 3 3 1010--66 Nm/(rad/sec)Nm/(rad/sec)(C)(C) Plot the Bode diagram of the transfer Plot the Bode diagram of the transfer

functionfunctionfunctionfunction



ExampleExampleExampleExample

0

20

Mag

nitu

de (d

B)

-60

-40

-20

0

)

M

-80

60

0

Pha

se (d

eg)

-135

-90

-45

100

101 102 103 104 105

Frequency (rad/sec)

-180

Q If l i t t d i th t t 1000 d/ i lif d l ?



Q: If we are only interested in the system response up to 1000 rad/sec, can we simplify our model ?How would you simplify the model ?

Model ReductionModel ReductionModel ReductionModel Reduction•• Neglect Electrical DynamicsNeglect Electrical Dynamics

Derive the model for the DC motor if the armature inductanceDerive the model for the DC motor if the armature inductance LLAA is neglected:is neglected:Derive the model for the DC motor, if the armature inductance Derive the model for the DC motor, if the armature inductance LLAA is neglected:is neglected:

1J s + BKT

1L s + R

JA s + BTLA s + RA

Kb

( )( ) ( )

( )( ) ( )

( )s KL J s BL R J s R B K K

E s L s RL J s BL R J s R B K K

sT

A A A A A A b Ti

A A

A A A A A A b TL

2 2

( ) ( ) ( )s K E s sTi L

By neglecting the effect of armature inductance we reduced the order of our modelBy neglecting the effect of armature inductance we reduced the order of our model



By neglecting the effect of armature inductance, we reduced the order of our model By neglecting the effect of armature inductance, we reduced the order of our model from two to one.from two to one.

Model ReductionModel ReductionModel ReductionModel ReductionQ:Q: Physically, what do we mean by neglecting armature inductancePhysically, what do we mean by neglecting armature inductance ??

ByBy neglectingneglecting thethe armaturearmature inductanceinductance wewe areare assumingassuming thatthat itit takestakes nono timetime forforByBy neglectingneglecting thethe armaturearmature inductance,inductance, wewe areare assumingassuming thatthat itit takestakes nono timetime forforthethe currentcurrent toto reachreach itsits steadysteady statestate valuevalue whenwhen therethere isis aa stepstep changechange inin inputinputvoltage,voltage, ii..ee..,, aa suddensudden changechange inin inputinput voltagevoltage willwill resultresult inin aa suddensudden changechange ininthethe armaturearmature current,current, whichwhich inin turnturn willwill resultresult inin aa suddensudden changechange inin thethe motormotorthethe armaturearmature current,current, whichwhich inin turnturn willwill resultresult inin aa suddensudden changechange inin thethe motormotortorquetorque outputoutput.. ThisThis isis equivalentequivalent toto havinghaving directdirect controlcontrol overover thethe motormotor currentcurrent..

Mathematically:Mathematically: From Block Diagram:From Block Diagram:

dL ddt

i R i E e t

J BK i

A A A A emf i

A m L

U

( )

KT1

LA s + RA

K iE K

m T A

emf b

UVW



ExampleExampleExampleExampleMedia Advance System in InkJet PrintersMedia Advance System in InkJet PrintersThe figure on the right shows the mediaThe figure on the right shows the mediaThe figure on the right shows the media The figure on the right shows the media advance system of a typical inkjet printer. advance system of a typical inkjet printer. The objective of the system is to precisely The objective of the system is to precisely and quickly position the media such that ink and quickly position the media such that ink droplets can be precisely “dropped” on to the droplets can be precisely “dropped” on to the media to form “nice looking” images. The media to form “nice looking” images. The system is driven by a DC motor through two system is driven by a DC motor through two sets of gear trains. You, as the “new kid” onsets of gear trains. You, as the “new kid” onsets of gear trains. You, as the new kid on sets of gear trains. You, as the new kid on the development team, are given the task of the development team, are given the task of specifying a motor and designing the control specifying a motor and designing the control system that will achieve the desirable system that will achieve the desirable

f S ti illf S ti illperformance. Some time your manager will performance. Some time your manager will also walk by your desk and ask you if a also walk by your desk and ask you if a certain level of performance is achievable. certain level of performance is achievable. How would you start your first engineering How would you start your first engineering



y y g gy y g gproject?project?

ExampleExampleExampleExampleSchematicSchematic

NLLDC Motor Assumptions:

iA

+ eLa

LA

+ eRa

RA+ei(t)

+Eemf

JA

m

JL

N2• Gears and shafts are

rigid and massless.

Bl k di f th l d i tiBl k di f th l d i ti

_emf_

B

N1BL

Bl k di f th t iBl k di f th t iBlock diagram of the load inertiaBlock diagram of the load inertia:: Block diagram of the gear trainBlock diagram of the gear train::



ExampleExampleExampleExampleBlock diagram of the DC motor subsystem:

1JA s + BKT

1LA s + RA

Reduce the mechanical portion of the block diagram:

Kb



ExampleExampleExampleExampleSimplified block diagramSimplified block diagram::

KT1

LA s + RA

Transfer Function from input voltage ETransfer Function from input voltage Eii((ss)) to the angular position of the loadto the angular position of the load ((ss):):

Kb

Transfer Function from input voltage ETransfer Function from input voltage Eii((ss)) to the angular position of the loadto the angular position of the load ((ss):):

G s sE sE

ii

( ) ( )( )

QQ:: Is this system stableIs this system stable ??

QQ:: What command What command ((voltagevoltage) ) would you use to move the roller’s angular position by, say 60would you use to move the roller’s angular position by, say 60 oo ??

i



QQ (( gg )) y g p y, yy g p y, y

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