2-chapters chapter 6 electrical and electromechanical systems 2

8/18/2019 2-Chapters Chapter 6 Electrical and Electromechanical Systems 2

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ME 413 Systems Dynamics & Control Chapter 6: Electrical Systems and ElectromechanicalSystems

Chapter 6

Electrical Systems andElectromechanical Systems

A.Bazoune

6.1 INTRO!CTION

The majority of engineering systems now have at least one electricalsubsystem. This may be a power supply, sensor, motor, controller, or an acousticdevice such as a speaker. So an understanding of electrical systems is essential tounderstanding the behavior of many systems.

6." E#ECTRICA# E#E$ENTS

Current and %olta&e Current and 'olta&e are the primaryvariables used to describe a circuit’s behavior.

Current is the ow of electrons. It is the time rate of change of electronspassing through a dened area, such as the cross!section of a wire. "ecauseelectrons are negatively charged, the positive direction of current ow is oppositeto that of electron ow. The mathematical description of the relationship between

the number of electrons # called chargeq

$ and current i is

dqi

dt =

or ( )q t i t = ∫ d

The unit of charge is the coulomb #%$ #in recognition of %harles &ugustin

%oulomb, 'rench physicist and mathematician, ()*+!(-+$, which represents186.24 10× electrons.

The unit of current is the ampere, or simply, amp #in recognition of &ndre’ arie&mpere, 'rench physicist and mathematician, ())/!(*+$ which is dened as acoulomb per second0

&mpere 1 coulomb 2 second

Thus, ( amp is186.24 10× electrons moving from one body to another in (

second.

3nergy is re4uired to move a charge between two points in a circuit. Thework per unit charge re4uired to do this is called 'olta&e. The voltage di5erence

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between two points in a circuit is a measure of the energy re4uired to movecharge from one point to the other.

The unit of voltage is 'olt #6$ #in recognition of the Italian physicist &lessandro6olta, ()7/!(8)$, which is dened as a charge of ( joule of energy per coulombof charge.

& (oule #named in recognition of the 3nglish physicist 9ames 9oule, ((!(:$ isa unit of energy or work and has the units of ;ewton < meter. Thus,

volt 1 joule 2 coulomb

joule 1 ;ewton = meter

Acti'e and )assi'e Elements. %ircuit elements may beclassied as active or passive.

• Passive Element 0 an element that contains no energy sources #i.e.the element needs power from another source to operate$> theseinclude resistors, capacitors and inductors

• Active Element 0 an element that acts as an energy source> theseinclude batteries, generators, solar cells, and op!amps.

Current Source and %olta&e Source & voltage sourceis a device that causes a specied voltage to e=ist between two points in a circuit.

The voltage may be time varying or time invariant #for a su?ciently long time$.'igure +!(#a$ is a schematic diagram of a voltage source. 'igure +!(#b$ shows a

voltage source that has a constant value for an indenite time. @ften the voltage

is denoted by E or V . & battery is an e=ample of this type of voltage.

& current source causes a specied current to ow through a wirecontaining this source. 'igure +!(#c$ is a schematic diagram of a current source

'igure +.( #a$ 6oltage source> #b$ constant voltage source> #c$ current source

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Resistance elements. The resistance R of a linearresistor is given by

Re

Ri

=

where Re is the voltage across the resistor and i is the current through the

resistor. The unit of resistance is the ohm( )A

, where

voltohm1

ampere

Besistances do not store electric energy in any form, but instead dissipate it asheat. Beal resistors may not be linear and may also e=hibit some capacitance andinductance e5ects.

)RACTICA# E*A$)#ES0 Cictures of various types of real!world resistors are

found below.

+ire,ound Resistors+ire,ound Resistors in )arallel

+ire,ound Resistors in Series and in )arallel

Capacitance Elements. Two conductors separated by anonconducting medium form a capacitor, so two metallic plates separated by a

very thin dielectric material form a capacitor. The capacitance C is a measure of the 4uantity of charge that can be stored for a given voltage across the plates.

The capacitance C of a capacitor can thus be given by

c

qC

e=

where

q

is the 4uantity of charge stored andc

e

is the voltage across thecapacitor. The unit of capacitance is the farad

( )', where

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ampere!second coulombfarad

volt volt= =

;otice that, since d di q t = and ce q C = , we haved

dcei C

t =

or

1d dce i t

C =

Therefore,

( )0

10d

t

c ce i t e

C = +∫

&lthough a pure capacitor stores energy and can release all of it, real capacitorse=hibit various losses. These energy losses are indicated by a poer !actor ,which is the ratio of energy lost per cycle of ac voltage to the energy stored percycle. Thus, a small!valued power factor is desirable.

)RACTICA# E*A$)#ES0 Cictures of various types of real!world capacitors arefound below.

Inductance Elements. If a circuit lies in a time varying magnetic

eld, an electromotive force is induced in the circuit. The inductive e5ects can beclassied as sel! inductance and mutual inductance.

Sel! inductance" or simply inductance, L is the proportionality constant

between the induced voltage Le

volts and the rate of change of current #or

change in current per second$d di t

amperes per second> that is,

d d Le L

i t =

The unit of inductance is the henry #D$. &n electrical circuit has an inductance of (henry when a rate of change of ( ampere per second will induce an emf of ( volt0

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volt weberhenry

ampere second ampere= =

The voltage Le

across the inductor L is given byd

d L

L

ie L

t =

Ehere Li

is the current through the inductor. The current( ) Li t can thus be given

by

( ) ( )0

10d

t

L L Li t e t i

L= +∫

"ecause most inductors are coils of wire, they have considerable resistance. The

energy loss due to the presence of resistance is indicated by the #uality !actor Q ,

which denotes the ratio of stored dissipated energy. & high value of Q generallymeans the inductor contains small resistance.Mutual $nductance refers to the inuence between inductors that results frominteraction of their elds.

)RACTICA# E*A$)#ES0 Cictured below are several real!world e=amples ofinductors.

T&"F3 +!(. Summary of elements involved in linear electrical systems

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The following set of symbols and units are used0 v%t 1 6 #6olts$, i%t 1 & #&mps$, 4#t$ 1G #%oulombs$, % 1 ' #'arads$, B 1 A #@hms$, F 1 D #Denries$.

6.- !NA$ENTA#S O E#ECTRICA# CIRC!ITS

Ohm/s #a,. 'hm(s la states that the current in circuit isproportional to the total electromotive force #emf$ acting in the circuit and

inversely proportional to the total resistance of the circuit. That ise

i R

=

were i is the current #amperes$, e is the emf #volts$, and R is the resistance#ohms$.

Series Circuit. The combined resistance of series!connectedresistors is the sum of the separate resistances. 'igure +!8 shows a simple seriescircuit.

i&ure 60" Series %ircuit

The voltage between points & and " is

1 2 3e e e e= + +where

1 1 2 2 3 3, ,e i R e i R e i R= = =

Thus,

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1 2 3

e R R R

i= + +

The combined resistance is given by

1 2 3 R R R R= + +

In general,

1

n

ii

R R=

=∑

)arallel Circuit. 'or the parallel circuit shown in gure +!*,

i&ure 60- Carallel %ircuit

1 2 3

1 2 3

, ,e e e

i i i R R R

= = =

Since 1 2 3i i i i= + +

, it follows thatH

1 2 3

e e e ei

R R R R= + + =

where R is the combined resistance. Dence,

1 2 3

1 1 1 1

R R R R= + +

or1 2 3

1 2 2 3 3 1

1 2 3

1

1 1 1

R R R R

R R R R R R

R R R

= =+ +

+ +

In general

1

1 1n

i i R R== ∑

irchho2/s Current #a, 3C#4 3Node #a,4. & node inan electrical circuit is a point where three or more wires are joined together.irchho5’s %urrent Faw #%F$ states that

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The algebraic sum of all currents entering and leaving a node is Jero.

or

The algebraic sum of all currents entering a node is e4ual

to the sum of all currents leaving the same node .

i&ure 605 ;ode.

&s applied to 'igure +!7, kirchho5’s current law states that

1 2 3 4 5 0i i i i i+ + − − =

or

K1 3 4 523ntering currents Feaving currents

i i i i i+ +=+(7 8 7*

irchho2/s %olta&e #a, 3%#4 3#oop #a,4. irchho5’s6oltage Faw #6F$ states that at any given instant of time

The algebraic sum of the voltages around any loop in an electrical circuit is Jero.

or

The sum of the voltage drops is e4ual to the sum of the voltage rises around a loop.

'igure +!/ Liagrams showing voltage rises and voltage drops in circuits. #;ote0 3achcircular arrows shows the direction one follows in analyJing the respective circuit$

A rise in voltage Mwhich occurs in going through a source of electromotive forcefrom the negative terminal to the positive terminal, as shown in 'igure +!/ #a$, or

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in going through a resistance in opposition to the current ow, as shown in 'igure+!/ #b$H should be preceded by a plus sign.

A drop in voltage Mwhich occurs in going through a source of electromotive forcefrom the positive to the negative terminal, as shown in 'igure +!/ #c$, or in goingthrough a resistance in the direction of the current ow, as shown in 'igure +!/

#d$H should be preceded by a minus sign.

'igure +!+ shows a circuit that consists of abattery and an e=ternal resistance.

Dere E is the electromotive force, r is theinternal resistance of the battery, R is the

e=ternal resistance and i is the current.'ollowing the loop in the clockwise direction

( ) A B C A→ → →, we have

0 A B BC CAe e e+ + =or

0 E iR ir − − ='rom which it follows that

E i

R r =

+

'igure +!+ 3lectrical %ircuit.

6.5 $ATE$ATICA# $OE#IN7 O E#ECTRICA#

S8STE$S

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Trans9er unctions o9 Cascade Elements. %onsider

the system shown in 'igure +.(. &ssume ie

is the input and oe

is the output. The

capacitances 1C

and 2C

are not charged initially. Fet us nd transfer function

( ) ( )o i E s E s .

'igure +!( 3lectrical system

The e4uations of this system are0

Foop(

( )1 1 1 21

1i

R i i i dt eC

+ − =∫ #+!

()$

Foop8

( )1 2 2 2 21 2

1 10i i dt R i i dt

C C − + + =∫ ∫

#+!($

@uter Foop2

2

1oi dt e

C =∫

#+!(:$

Taking FT of the above e4uations, assuming Jero I. %’s, we obtain

( ) ( ) ( ) ( )1 1 1 21

1i

R I s I s I s E sC s

+ − = #+!8-$

( ) ( ) ( ) ( )1 2 2 2 21 2

1 10 I s I s R I s I s

C s C s− + + =

#+!8($

( ) ( )22

1o I s E sC s = #+!88$

'rom 34uation #+!8-$

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( ) ( ) ( ) ( )

( )( ) ( )

( ) ( )

1 1 1 2

1 1

2

1 211

1 1 1 1

1

1 1

1

1 1

i

i

i

R I s I s I s E sC s C s

E s I sC sE s I sC s

I s

R C s R C sC s

+ − =

++

= =+

+

Substitute( )1 I s into 34uation #+!8($

( )

( ) ( )

( )

1 1

2

1 1 2 2 1

1 1 2 2

1 2

2 2

1 1 2

1 2

1 1 2 2

1

2

2 2

2

1

1

1

1

o

i

R C R

E s

E s R C R C s R C

R C R C R C

C

R C R C R C R

R C R C s

C s s

=+ + + +

=

+ ++ +

which represents a transfer function of a second order system. The characteristicpolynomial #denominator$ of the above transfer function can be compared to that

of a second order system2 22 n n s sζω ω + + . Therefore, one can write

( )

( )

( )

( )

1 1 2 2 1 22

1 1 2 2 1 1 2 2

1 1 2 2 1 2 1 1 2 2 1 2

1 1 2 2 1 1 2 2

12

2 2

and

or

n n

n

R C R C R C

R C R C R C R C

R C R C R C R C R C R C

R C R C R C R C

ω ζω

ζ ω

+ += =

+ + + += =

Comple: Impedance. In deriving transfer functions forelectrical circuits, we fre4uently nd it convenient to write the Faplace!transformed e4uations directly, without writing the di5erential e4uations.

Table +!( gives the comple= impedance of the basics passive elements

such as resistance R , an inductance L " and a capacitance C . 'igure +!(: shows

the comple= impedances 1 Z and 2 Z

in a series circuit while 'igure +!(: shows thetransfer function between the output and input voltage. Bemember that theimpedance is valid only if the initial conditions involved are all Jeros.

The general relationship is

( )( ) ( ) E s Z s I s=

corresponds to @hm’s law for purely resistive circuits. #;otice that, likeresistances, impedances can be combined in series and in parallel$

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'igure +!(: 3lectrical circuit

eri'in& Trans9er unctions o9 Electrical Circuits!sin& Comple: Impedances. The T' of an electrical circuit can beobtained as a ratio of comple= impedances. 'or the circuit shown in 'igure +!8-,

assume that the voltages ie

and oe

are the input and output of the circuit,respectively. Then the T' of this circuit can be obtained as

( )

( ) ( )2 2

1 2 1 2

( )( ) ( )

( ) ( ) ( ) ( ) ( )i

Z s I s E s Z s

E s Z s I s Z s I s Z s Z s

ο = =+ +

'or the circuit shown in 'igure +!8(, Figure 6-20 !ectrica! circuit

1 2

1, Z Ls R Z

Cs= + =

Dence, the transfer function

( )

( )i

E s

E s

ο

, is

2

2

1 2

1( ) ( ) 1

1( ) ( ) ( ) 1i

E s Z s Cs

E s Z s Z s LCs RCs Ls R

Cs

ο = = =+ + +

+ +

Figure 6-21 !ectrica! circuit

6.; ANA#O7O!S S8STE$S

Systems that can be represented by the same mathematical model, but that arephysically di5erent, are called analogous systems. Thus analogous systems are

described by the same di5erential or integrodi5erential e4uations or transferfunctions. The concept of analogous is useful in practice, for the followingreasons0

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(. The solution of the e4uation describing one physical system can be directlyapplied to analogous systems in any other eld.

8. Since one type of system may be easier to handle e=perimentally thananother, instead of building and studying a mechanical system #or a

hydraulic system, pneumatic system, or the like$, we can build and study itselectrical analog, for electrical or electronic system, in general, much easierto deal with e=perimentally.

$echanical0Electrical Analo&ies echanical systemscan be studied through their electrical analogs, which may be more easilyconstructed than models of the corresponding mechanical systems. There are twoelectrical analogies for mechanical systems0 The 'orce!6oltage &nalogy and The'orce %urrent &nalogy.

orce %olta&e Analo&y %onsider the mechanical system of 'igure +!87#a$ and the electrical system of 'igure +!87#b$.

'igure +!87 &nalogous mechanical and electrical systems.

The e4uation for the mechanical system is

2

2

d x dxm b kx p

dt dt + + =

#+!87$

where x

is the displacement of massm

, measured from e4uilibrium position. Thee4uation for the electrical system is

1di L Ri idt e

dt C + + =∫

In terms of electrical charge q , this last e4uation becomes

2

2

1d q dq L R q e

dt dt C + + =

#+!8/$

%omparing e4uations #+!87$ and #+!8/$, we see that the di5erential e4uations forthe two systems are of identical form. Thus, these two systems are analogoussystems. The terms that occupy corresponding positions in the di5erentiale4uations are called analogous 4uantities, a list of which appear in Table +!8

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T&"F3 +!8 'orce 6oltage &nalogy

echanical Systems 3lectrical Systems

'orce p #Tor4ue T $ 6oltage e

ass m #oment of inertia J $ Inductance L6iscous!friction coe?cient b Besistance R

Spring constant k Beciprocal of capacitance, 1 C

Lisplacement x #angular displacement θ $ %hargeq

6elocity x #angular velocity θ $ %urrenti

orce Current Analo&y The student is advised to read thissection from the te=tbook Cage 8)8!8)*.

6.6 $ATE$ATICA# $OE#IN7 OE#CTRO$ECANICA# S8STE$S

To control the motion or speed of dc servomotors, we control the eldcurrent or armature current or we use a servo!driver as motor!driver combination.

There are many di5erent types of servo!drivers. ost are designed to control thespeed of dc servomotors, which improves the e?ciency of operating servomotors.Dere we shall discuss only armature control of a dc servomotor and obtain itsmathematical model in the form of a transfer function.

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2-chapters chapter 6 electrical and electromechanical systems 2

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