ME451 L5 CJR ModelElectrical

8/12/2019 ME451 L5 CJR ModelElectrical

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ME451: Control SystemsME451: Control Systems

Prof. Clark Radcliffe and Prof.Prof. Clark Radcliffe and Prof. Jongeun ChoiJongeun Choi

Department of Mechanical EngineeringDepartment of Mechanical Engineering

Michigan State UniversityMichigan State University

Lecture 4Lecture 4

Modeling of electrical systemsModeling of electrical systems

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Course roadmapCourse roadmap

Laplace transformLaplace transform

Transfer functionTransfer function

Models for systemsModels for systemselectricalelectrical

mechanicalmechanical electromechanicalelectromechanical

Block diagramsBlock diagrams

LinearizationLinearization

ModelingModeling AnalysisAnalysis DesignDesign

Time responseTime responseTransientTransient

Steady stateSteady state

Frequency responseFrequency responseBode plotBode plot

StabilityStabilityRouthRouth-Hurwitz-Hurwitz

NyquistNyquist

Design specsDesign specs

Root locusRoot locus

Frequency domainFrequency domain

PID & Lead-lagPID & Lead-lag

Design examplesDesign examples

((MatlabMatlabsimulations &) laboratoriessimulations &) laboratories

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Controller design procedure (review)Controller design procedure (review)

plantplantInputInput OutputOutputRef.Ref.

SensorSensor

ActuatorActuatorControllerController

DisturbanceDisturbance

1. Modeling

Mathematical modelMathematical model

2. Analysis

ControllerController

3. Design

4. Implemenation

!! What is theWhat is the mathematical modelmathematical model??

!! Transfer functionTransfer function

!! Modeling of electrical circuitsModeling of electrical circuits

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!! Representation of the input-output (signal)Representation of the input-output (signal)relation of a physical systemrelation of a physical system

!! A model is used for theA model is used for the analysisanalysisandand designdesignofofcontrol systems.control systems.

Mathematical modelMathematical model

PhysicalPhysical

systemsystem

ModelModel

ModelingModeling

InputInput OutputOutput

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!! Modeling isModeling is the most important and difficult taskthe most important and difficult task

in control system design.in control system design.

!! No mathematical model exactly represents aNo mathematical model exactly represents a

physical system.physical system.

!! Do not confuseDo not confuse modelsmodelswithwith physical systemsphysical systems!!

!! In this course, we may use the termIn this course, we may use the term systemsystemtoto

mean a mathematical model.mean a mathematical model.

Important remarks on modelsImportant remarks on models

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Transfer functionTransfer function

!! Defined byDefined by

!! Zero initial condition are assumed becauseZero initial condition are assumed because

!! We are interested in the response we can designWe are interested in t he response we can design

!! IC response disappears with timeIC response disappears with time

!! FFinal system response depends only on inputsinal system response depends only on inputs

Laplace transform of system outputLaplace transform of system output

Laplace transform of system inputLaplace transform of system input


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Impulse responseImpulse response

!!

Suppose that u(t) is the unit impulse functionSuppose that u(t) is the unit impulse functionand system is at rest.and system is at rest.

!! The output g(t) for the unit impulse input is calledThe output g(t) for the unit impulse input is calledimpulse responseimpulse response..

!! When U(s)=1, the transfer function can also beWhen U(s)=1, the transfer function can also bedefined as thedefined as the Laplace transform of impulseLaplace transform of impulseresponseresponse::

SystemSystem

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Models of electrical elements:Models of electrical elements:

(constitutive equations)(constitutive equations)

v(t)v(t)

i(t)i(t)

RR

ResistanceResistance CapacitanceCapacitanceInductanceInductance

i(t)i(t)

LLv(t)v(t) v(t)v(t)

LaplaceLaplacetransformtransform

i(t)i(t)

CC

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ImpedanceImpedance

!! Generalized resistance to a sinusoidalGeneralized resistance to a sinusoidalalternating current (AC) current I(s)alternating current (AC) current I(s)

!! Z(s): V(s)=Z(s)I(s)Z(s): V(s)=Z(s)I(s) V(s)V(s)I(s)I(s)Z(s)Z(s)

Memorize!Memorize!

Time domain Impedance Z(s)

ResistanceResistance

CapacitanceCapacitance

InductanceInductance

Element

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KirchhoffKirchhoffs Voltage Law (KVL)s Voltage Law (KVL)

!! The algebraic sum of voltage drops around anyThe algebraic sum of voltage drops around any

loop is =0.loop is =0.

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KirchhoffKirchhoffs Current Law (KCL)s Current Law (KCL)

!! The algebraic sum of currents into any junctionThe algebraic sum of currents into any junction

is zero.is zero.

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Modeling exampleModeling example

!! Kirchhoff voltage lawKirchhoff voltage law(with zero initial conditions)(with zero initial conditions)

!! ByBy Laplace transformLaplace transform, (easiest, (easiestto do directly)to do directly)

vv11(t)(t)

i(t)i(t)

RR22InputInput

RR11

vv22(t)(t) OutputOutput

CC


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Modeling example (contModeling example (contd)d)

!! ByBy Laplace transformLaplace transform, (easiest, (easiestto do directly)to do directly)


(first-order system)(first-order system)

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Modeling example (contModeling example (contd)d)


vv11(t)(t)

i(t)i(t)

RR22InputInput

RR11

vv22(t)(t) OutputOutput

CC


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Example: Modeling of op ampExample: Modeling of op amp

!! Op Amps: The Amplifier you want to know aboutOp Amps: The Amplifier you want to know about

!! Op Amp Behavior is simpleOp Amp Behavior is simple

!! Voltage:Voltage:

!! Current:Current:

Kv+

v!

vout

+

-

vout

= K v+! v

!( ) where K! 106 " #

means for finite vout

, v+= v

!

i+= i

!=

v!

R where R" 10

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means i+= i

!= 0

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vvdd


!! ImpedanceImpedanceZ(s): V(s)=Z(s)I(s)Z(s): V(s)=Z(s)I(s)

VVii(s)(s)

I(s)I(s)

InputInput

ZZii(s)(s)

VVoo(s(s)) OutputOutput

--

++

ii--

Rule 1 =>Rule 1 => vvdd=0=0

Rule 2 =>Rule 2 => ii--=0=0

ZZff(s)(s)IIff(s(s))

VO(s) !0( ) = Zf(s)If(s)

Vi (s)!0( ) = Zi (s)I(s)

0 = I(s)+If(s)

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!! ImpedanceImpedanceZ(s): V(s)=Z(s)I(s)Z(s): V(s)=Z(s)I(s)

!! Transfer functionTransfer functionof the above op amp:of the above op amp:

VO(s)!0( ) =Zf(s)If(s)

Vi (s)!0( ) =Zi (s)I(s)

0 = I(s)+If(s)

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Modeling example: op ampModeling example: op amp

!! By the formula in previous two pages,By the formula in previous two pages,

vvii(t(t))

i(ti(t))

RR22

InputInput

RR11

vvoo(t(t)) OutputOutput

CC

--

++

ii--

vvdd

VVdd=0=0

ii--=0=0



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Modeling exercise: op ampModeling exercise: op amp

Find the transfer functionFind the transfer function G (s) =vo(t)

vi(t)

i!

(t) =0

vd(t) =0

InputInputvi(t)

RR22

RR11

CC22

--

++

CC11

vd

i!

OutputOutputvo(t)

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!! WeWeneedneedthe impedancethe impedanceof the parallel RCof the parallel RCcircuitcircuit

!! Find the impedance transfer functionFind the impedance transfer function

v(t)

i(t)

R C

iR C

V(s) = R( )IR(s)

V(s) =1

Cs

!"#

$%&I

C(s)

I(s) =IR(s)+I

C(s)

I(s) = IR(s) +I

C(s) =

V(s)

R

!"#

$%&+ CsV(s)( ) =

1

R

+Cs!"#

$%&V(s)

=1+ RCs

R

!"#

$%&V(s)

Z(s) =V(s)

I(s)=

R

1+RCs

!"#

$%&

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!!With theWith the impedenceimpedence, we can now model the op amp circuit, we can now model the op amp circuit

!! The Transfer FunctionThe Transfer Function

InputInput vi(t)

RR11--++

RR22

CC22

CC11

vd

i!

OutputOutputvo(t)

Z1(s)

Z2 (s)

Z1(s) =R1

1+ R1C1s

!

"#$

%&

Z2(s) =

R2

1+R2C2s

!"#

$%&

G(s) =Vo(s)

Vo(s)

=!Z2(s)

Z1(s)=!

R2 1 + R1C1s( )R1 1 + R2C2s( )

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Summary & ExercisesSummary & Exercises

!! ModelingModeling

!! Modeling is an important task!Modeling is an important task!

!! Mathematical modelMathematical model


!! Modeling of electrical systemsModeling of electrical systems

!! Next, modeling of mechanical systemsNext, modeling of mechanical systems

!! ExercisesExercises

!! Read Sections 2.2, 2.3Read Sections 2.2, 2.3

!! Solve problems 2.1, 2.2, and 2.3 in page 62Solve problems 2.1, 2.2, and 2.3 in page 62

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ME451 L5 CJR ModelElectrical