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1 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.451 Dynamic Systems – Electrical

Electrical Systems

Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell


Electrical Systems

Passive Electrical ElementsResistor, Inductor, Capacitor

Elemental Relations for Voltages

ResistorR

RVRi+ −

RR RiV =Voltage Resistance Current

)ampere/voltohm( =

Resistors dissipate heat – no energy storage


Elemental Relations for Voltages cont.

Inductor

Capacitor

Inductors generate considerable resistance

dtdiLV L

L =

Voltage Inductance current of change of Rate

sec)/( amponeofchangeofrateHenry =)( voltoneofemfaninducewill

L

LV+ −

dtic1V cC ∫=

VoltageeCapacitanc

Current

)sec(volt

ampFarad =Note C=q/vc q-charge

Note also i=dq/dt and vc=q/c

cV+ −

ci

Li


Elemental Relations for Currents

ResistorR

Vi RR =

Inductor

Capacitor

dtVL1i LL ∫=

dtdVci c

C =


Electrical Systems

Rei =

321 eeee ++=

Ohm’s Law states that the current in a circuit is proportional to the total electromotive (emf) force acting in the circuit and inversely proportional to the total resistance in the circuit.

Series

i – current (amps); e – emf (volt); R – resistanace (Ohm)1R 2R 3R

B

1e 2e 3eAi

effective321 RRRRie

=++=

Ve ≈


Electrical Resistance

Parallel

33

22

11 R

ei;Rei;

Rei ===

Since

Then

1R2R 3R

1i 2i 3i

e

321 iiii ++=

effective321 Re

Re

Re

Rei =++=

321eff R1

R1

R1

R1

++=133221

321eff RRRRRR

RRRR++

=∴


Kirchoff’s Law

Two Law’s exist: Current Law (node law)Voltage Law (loop law)

Current Law (node law)The algebraic sum of all currents entering and leaving a node is zero.

0iii 321 =++ 0)iii( 321 =++− 0iii 321 =−+

3i

2i

1i 3i

2i

1i 3i

2i

1i


Kirchoff’s Law

Consider the circuit

The instant the switch is closed, the current i(o)=zero because the inductor cannot change from zero to a finite value instantaneously

RL VVV +=

VRidtdiL =+

i LR

+−RV

LV+

−V+

−


Kirchoff’s Law

Take the Laplace Transform

[ ]SE)s(RI)0(i)s(sIL =+−

+−=

+=

)LR(S

1S1

RE

)RLS(SE)s(I

SE)s(I)RLS( =+

Since i(0)=0

OR

Note that E and V are often used for voltage


Kirchoff’s Law

The inverse laplace gives

−=

− t)LR(

e1RE)t(i

RLat

RE

=τ

Prove that the slope of the function intersects the final value

RE

RE63.0

RL


Voltage Law (Loop Law)

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

0VVVV CRL =−++

L

LV+ −

V

+

−

C

R+− CV RV

+

−


Voltage Law (Loop Law) cont.

Continuing with this loop, the diff. eq. isdti

C1V;RiV;

dtdiLV CCRR

LL ∫===

Then

Taking a derivative to eliminate the integral∫ =++ aVdti

ot

c1Ri

dtdiL

dtdVi

C1

dtdiR

dtidL a2

2=++

Thus

ORdt

dVL1i

LC1

dtdi

LR

dtid a2

2=++

LCR

21

2LR;

LC1 2

nn =

ω=ς=ω



Taking the Laplace Transform (with IC=0)

)S(VLS)S(I)

LC1S

LRS( a

2 =++

The Transfer Function is

Rearranging

++

=)

LC1(S)

LR(SL

S)S(V)S(I

2

Recalling the relation

)S(VCS)S(I)1RCSLCS( a2 =++

∫= dtic1V cc



The Transfer Function relating the input voltage to the voltage drop across the capacitor is

Recalling the relation ∫= dtic1V cc

++

==

)LC(1S)

LR(SLC

1)S(V

)S(ICS1

)S(V)S(V

2aa

c


Operational Amplifier

An OP-AMP is an electronic amplifier with a single output and very high voltage gain (105 to 106) –low current!

The OP-AMP amplifies the difference in voltages and and is often referred to as a differential amplifier

In this configuration, an OP-AMP is inherently unstable –to stabilize it a negative feedback is often used.

+e( )

GeneralIn eeKeo −+ −=

−

+

−e

+e −e


Operational AmplifierAn OP-AMP is the most widely used analog electronic device

VoltagesInput e,e BA ⇒

Bi+

Ground Signal

oioe

oz

osV

AiBe+

−+

Dz

+

Ae

GroundChassisDefinitions:

Currents Bias i,i BA ⇒ImpedanceInput alDifferenti ZD ⇒

Gain LoopOpen A ⇒

ImpedanceOutput Zo ⇒VoltageOffset Vos ⇒

Supply VoltageVs ⇒±

oe

osV

( )mVee BA −

( )AVeee osBAo −−=



Voltage Follower( )AVeee osBAo −−=

∞→A 0Vos →Assume and

oio eee

−=∞ io ee =∴

Output follows input

ieSource supplying works into an “infinite”impedance thus no current is drawn

oe does not draw current or consume power

Sometimes called a buffer amplifier



Consider an inverting amplifier shown

Since only negligible current flows into the amplifier, i1 must equal i2

1

2RR is Gain ⇒i

1

2o e

RRe −=

( )1

i1 R

'eei −=

'e

+−

ie +−

+

−

1R

1i

2R

2i

oe

( )2

o2 R

e'ei −=

Since then . Thus the approximate model is:1K >> 0'e ≈

21 Re

Re oi −=



Consider a non-inverting amplifier shown'e

+−

ie+

−

+

−

1R 2R

oe

Since

i1

2o e1

RRe

+=

1

2RR is Gain ⇒

1K >>

And ifK1

RRR

21

1 >>+

Then an approximate model is

021

1i

021

1i0

e)K1

RRR(e

or

)eRR

Re(Ke

++

=

+−=

For this circuit


Low-Pass Filter

Transfer Function

( ) ( )sXXs

ee

i

o

i

o =

ω= js

oe

RC≈τ

ie

Electrical

Frequency response function is evaluated atMechanical

iX

K B

oX

KB

≈τ

1j1+τω

=

1s1+τ

=

R C


Low-Pass Filter

Passive styled filters have various advantages:

Very low noise output

No power required

Versatile

Wide dynamic range


High-Pass Filter

Transfer Function

( ) ( )sXXs

ee

i

o

i

o =

1ss+ττ

=

oe

RC≈τ

ie

Electrical

RC

ω= jsFrequency response function is evaluated at

1sj+ττω

=Mechanical

iX

KB

oX

KB

≈τ


Cascade a low pass and high pass filter together

Band-Pass Filter

1s1

1 +τ 1ss

2

2+τ

τ

12 τ>τ


Integration (Approximation Using Low Pass)

+−

ie +−

+

−

R

C

oe

( )1j

1jee

i

o+ωτ

=ω

Now if then1>>ωτ

( )ωτ

=ωj1j

ee

i

o

Or( )

s1s

ee

i

oτ

= dte1e io ∫τ≈∴


Differentiation (Approximation Using High Pass)

+−

ie +−

+

−

RC

oe

( )1j

jjee

i

o+ωτ

ωτ=ω

Now if then1<<ωτ

( ) ωτ=ω jjee

i

o

Or

dtdee i

o τ=∴( ) ssee

i

o τ≈

These circuits are sensitive to noise upon differentiating

BEWARE!!


Differentiation (w/Low Pass Filter)

( )1CsR

CsRsee

1

2

i

o+

−=

Accurate for 1CR1 <<ω

1

2RR

But amplifies high frequency noise by

+−

ie +−

+

−

2RC

oe

1R


Differentiation (w/Low Pass Filter & Noise Atten.)

( )( )( )1sCR1sCR

sCRsee

1122

12

i

o++

−=

+−

ie +−

+

−

2R

1C

oe

1R2C


ElectroMechanical SystemsArmature-controlled DC motors are popular. Field Controlled DC motors are not as popular.

Elemental RelationsElectrical and mechanical parts are coupled. The mechanical motion of the rotor relative to the stator creates an electromotive force voltage (EMF effect). The back EMF voltage across the DC motor is proportional to angular speed of the rotor.

while the torque developed by the motor is proportional to the current

θ=ω= &eeb KKe

iKT T=


ElectroMechanical Systems

where

iKT T=

oltageback emf veb −ntdisplacemeangular−θ

velocityangular−ω=θ&motor)by (exertedrotor toapplied torqueT −

)(motor ofconstant emf KRPMKV

e −

)(motor ofconstant torque K Ampoz

tf−

Conversion:

RPM

fK

V741.0Aoz-in1 =


Armature-Controlled DC Motors

shaftof dampingc −disk of dampingcd −

current armatureia − (constant)current field if −

disk of inertia mass Jd −

rotor of inertia mass JR −

dcbe

RJ

aR aL

+

−−

+ai

dJ

θ=ωθ &,

c,k )t(TL

shaft stifftorsionalk −inductance armature La −resistance armature R a −

voltagearmature a −ν

aν


Armature-Controlled DC MotorsRigid Shaft – Lump the rotor, shaft and disk together so that

dr JJJ +=

aν

0e0V abLRdrop aa=ν−+ν+ν=∑

One input,

Two outputs, θ and ia

For the electrical circuit, the loop method gives

Then, ( )θ=ω=ν=++ &eebab

aaaa KKee

dtdiLiR

So thataeaa

aa KiR

dtdiL ν=θ++ &


Armature-Controlled DC MotorsFor the mechanical part

θ=θ−+ &&& JBTT dL

Then

Latd TiKBJ =−θ+θ &&&

J

θ

θ&dBLTT +

(Note: Voltage is input to system but current is input to mechanical part)

System Equations

Latd TiKBJ =−θ+θ &&&

aeaaa

a KiRdt

diL ν=θ++ &

OR

ν

=

θ

−+

θ

+

θ

a

L

aa

ta

ae

d

2a

2 TiR0

K0

dtdi

LK0B

dtid

000J &&&


Armature-Controlled DC MotorsExpressing the equations using (since )θ=ωω &

Latd TiKBJ =−ω+ω&

aeaaa

a KiRdtdiL ν=ω++

ν

=

ω

−+

ω

a

L

aae

tda

a

TiRK

KB

dtdi

L00J &


Armature-Controlled DC Motors – Laplace TransformThe Laplace Transform gives

( ) 0KKRBsJRLBsJL teadaad2

a =++++

=

Ω

+

−+)s(V)s(T

)s(I)s(

RsLKKBJs

a

L

aaae

td

After some manipulation, the characteristic equation is given by

( )a

tead2n JL

KKRB +=ω

After putting this in standard form,

( )a

aadn JL

JRLB2 +=ζω


Armature-Controlled DC Motors – Laplace TransformThe transfer functions of interest are

( ) teadaad2

a

t

a KKRBsJRLBsJLK

)s(V)s(

++++=

Ω

++

++

=θ

a

tead

a

aad2

a

t

a

JLKKRBs

JLJRLBss

JLK

)s(V)s(

Note: A more complex model which includes a flexible shaft can also be developed.


Example - Circuit AttenuationA low frequency signal (at 3 Hz) is to be measured but is contaminated by a higher frequency component such as 60 Hz noise for instance.

Filter

R C

in sig

outV

scope

outVinV

The filter must be selected to attenuate the 60 Hz noise without significantly affecting the actual signal attenuation by no more than a factor of 0.9.

A first order low pass filter (RC circuit)


Example – Circuit AttenuationFilter Transfer Function

1RCs1

)s(V)s(V)s(H

in

out+

==

( )9.0

1

1VV

)j(H2in

out =+ωτ

==ω

Since

The magnitude of the frequency response function is

s 0.026 constant timeRCτ ==

( )( ) 13Hz2

10.92 +π⋅τ

=

At 60 Hz, the amplitude is ( )[ ] 093.016020257.0 21

2 =+π⋅⋅⋅−

with RC - possible values s026.0= F2.0C µ=Ω= k130R


Force Voltage Analogy

CurrentCurrentiVelocityVelocity

ChargeChargeqxDisplacementDisplacement

Reciprocal of CapacitanceReciprocal of Capacitance1/CkSpring ConstantSpring Constant

ResistanceResistanceRcViscous FrictionViscous Friction

InductanceInductanceLmMassMass

VoltageVoltagevfForceForce

ElectricalElectricalMechanicalMechanical

x&

)t(fkxxcxm =++ &&& ν=++ qC1qRqL &&&


Force Current Analogy

VoltageVoltageVelocityVelocity

Magnetic FluxMagnetic FluxxDisplacementDisplacement

Reciprocal of InductanceReciprocal of Inductance1/LkSpring ConstantSpring Constant

Reciprocal of ResistanceReciprocal of Resistance1/RcViscous FrictionViscous Friction

CapacitanceCapacitanceCmMassMass

CurrentCurrentifForceForce

ElectricalElectricalMechanicalMechanical

)t(fkxxcxm =++ &&& iL1

R1C =ψ+ψ+ψ &&&

x& ν

ψ

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