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MCEN 467Control Systems

Chapter 2:

Dynamic Models

Part D: Electromechanical Systems

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Material covered in the PRESENT LECTURE

is shown in yellow

I. DYNAMIC MODELING

Deriving a dynamic model formechanical, electrical,

electromechanical, fluid- & heat-flow systems Linearization the dynamic model if necessary

II. DESIGN OF A CONTROLLER: Several design methods exist

1. Classical control or Root Locus Design:

Define the transfer function; Apply root locus, loop shaping,

2. Modern control or State-Space Design:

Convert ODE to state equation; Apply Pole Placement, Robust control,

3. Nonlinear control: Apply Lyapunovs stability criterion

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Dynamic ModelingElectro - mechanical

system

xmFext IMlga

ext

0

jlg,a

je0jlg,a

ji

KCL (or KVL) Newtons law

Translation Rotation

Here voltages are denoted by e to avoid confusion with velocities

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Dynamic Model

)ior(e

)IMor(xmF

jj

extext

00

The dynamic model is derived from:

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A Electrical Energy can be converted

in Mechanical Work

and vice versa !

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Converting Electrical Energy into

Mechanical Work

Law of Motors

Learning Examples:Loudspeaker & DC Motor Operation

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Loudspeaker to provide

translational motion

Overall configuration Permanent magnet

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DC Motor to providerotary motion

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Electric and Magnetic Fields Interactin Two Ways

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Effect of Mechanical motion & Magnetic

Field interaction on Magnetic Field

A current-carrying wire of

length lplaced at right angle in

a magnetic field B experiencesa magnetic force F,

perpendicular to the wire.

Law of motors:

BL iF

B)t(iLF

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Loudspeaker

Operation

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Laws of MotorsRotational Motion

Law of motors:

)t(iK)t(Tt

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Example 1: DC Motor OperationDC Motor

Operation

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DC Motor

Operation

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Converting Mechanical Work into

Electrical Energy

Mechanical Motion & Magnetic Field Interaction:

Law of Generators

Learning Examples: Loudspeaker & DC Motor Operation

&Mechanical Motion & Electric Field Interaction.

Learning example: Capacitor Microphone Operation

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Effect of Motion & Magnetic Field Interaction

on Electrical Field

A current-carrying wireof

lengthL moving in a magnetic

field B at a velocity generatesa voltage eacross the wire.

Law of generators:

xBLe

B)t(xL)t(e

x

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Loudspeaker

Operation

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Laws of GeneratorsRotational Motion

Law of generators:

)t(K)t(ee

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Magnetic & Electric Field Interaction

- Summary -Rotational motion

Torque developed in rotor:

Back emf (= voltagegenerated as a result of

rotation of a rotor):

Example: DC motor

Translational motion

Law of motors:

Law of generators:

Example: Loudspeaker

Bl iF

xBle

iKT t

e

Ke

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Dynamic Modeling Examples

DC Motor

&

Loudspeaker

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Example 1: Loudspeaker with circuit

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Example 1: Loudspeaker (contd)

Assumptions:

The effects of the air can be modeled as if the cone hadequivalent massMand viscous friction constant b.

The magnet establishes a uniform fieldB of 0.5 Tesla andthe bobbin has 20 turns at a 2 cm-diameter.

Required: Write the dynamic model

Note: the bobbin length is such as:

m..

L 2612

020220

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Dynamic model of the loudspeaker:

Newtons law states:

i..xbxM

BLixbxM

xMxbBLi

26150

i.xbxM 6280

FBD:

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Dynamic model of the electric circuit:

KVL states:

x..vRidt

diL

evRidt

diL

Rivedt

di

L

a

coila

acoil

26150

0

x.vRidt

diL a 6280

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x.vRidt

diL

i.xbxM

a

6280

6280

Dynamic Model for the loudspeaker

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Example 2: DC Motor

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Example 2: DC Motor (contd)

Dynamic model of the loudspeaker:

Newtons law states:

at

mm

iKTwhere

JbT

FBD:

atmm iKbJ

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Dynamic model of the electric

circuit:

KVL states:

0 aaamea

a iRvKdt

diL

meaaaa

a KviRdtdiL

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meaaaa

a

tmm

KviRdt

diL

iKbJ

Dynamic Model for the DC Motor

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Converting Mechanical Work into

Electrical Energy

Mechanical Motion & Magnetic Field Interaction:

Law of Generators

Learning Examples: Loudspeaker & DC Motor Operation

&Mechanical Motion & Electric Field Interaction.

Learning example: Capacitor Microphone Operation

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Effect of Mechanical Motion of One Part of a

Capacitor on Electrical FieldAssume that:

the electric circuit includes a capacitor.

x(t)is the system variable describing the displacement of

one part of the capacitor, that has a capacitance C(x) An mechanical external force is applied to the plant.

Reminder:

A capacitoris a circuit element that consists of

two conducting surfaces separated by

a dielectricmaterial.

)x(C

)t(q)t,x(e;

x

A)x(C 0

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Capacitor Microphone Operation

)ior(e

)IMor(xmF

jj

extext

00

x)t,x(e)x(Cf

)x(C/)t(q)t,x(e

dt

)t(dq)t,x(e)x(Cdt

d)t(i

electric2

2

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Mechanical Work & Electric field

Translational motion

Mechanical external force: Force due to the electric field:

Voltage generated as a result of displacement of one plate:

Example: Capacitor microphone

)t,x(ex)x(C

A)t(q)t(fe 2

0

2

22 )t(fm

)x(C/)t(q)t,x(e

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Example 3: Capacitor Microphone to

Provide Electric Field

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Modeling Capacitor Microphone

See homework!

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