me330 lecture7
TRANSCRIPT
ME 330 Control Systems
SP 2011
Lecture 7
ScanYZTest(dsc,bl=0.1,rr=1.0,yscan=10,zscan=0):ScanYZTest(dsc,bl=0.1,rr=1.0,yscan=10,zscan=0):
Electronic Systems Series Impedances
21
21
21
ZZZI
V
I
VI
VV
I
VZ
21
21
21
21
111
1111
ZZZ
ZZVI
VI
II
V
I
VZ
21
21
III
VVV
21
21
III
VVV
Parallel Impedances
Electronic Systems Operational Amplifiers
High input impedance, Zi = inf
Low output impedance, Zo = 0 High gain, A = inf
))()(()( tvtvAtv iio
A
vi+
vi-vo
)()( tAvtv io
If vi+ is connected to ground, vi+ = 0,
Electronic Example 1 Op-amp feedback of
the output voltage
0)()(1
)(
sVsVZ
sI ioampop
a
High impedance of op-amp
ampopZ
)()( 21 sIsI Thus, High gain of op-amp
)()( 1 sAVsVo 0)(1 sVA
,)(
)()(
11 sZ
sVsI i
)(
)()(
22 sZ
sVsI o
)(
)(
)(
)(
1
2
sZ
sZ
sV
sV
i
o
Electronic Example 2 Op-amp with different
impedance terms
C1
R1
R2 C2
21 ZZZ
21
111
ZZZ
Recall the rules about series and parallel impedances
)(
)(
)(
)(
1
2
sZ
sZ
sV
sV
i
o
11
22
11
1
)(
)(
RsC
sCR
sV
sV
i
o
211
2
2
112
1
)(
)(
CsRR
R
C
CCsR
sV
sV
i
o
Electrical Mechanical Systems
Combinations of electrical and mechanical systems
dc servomotor
Electrical Mechanical Systems Lorentz Force: magnetic field applies a
force proportional to current.
),()( tiKtT atm rBlNK t 2r = radius of motor armatureB = strength of magnetic fieldl = length of conductorN = number of coil windings
),()( tKtv mbb rBlNKb 2
Faraday’s Law: conductor moving in magnetic field creates potential voltage proportional to velocity.
“back EMF”
Va(t)
Transfer functionfor electrical system
)()()()( sVsIsLsIRsV baaaaa
Overall transfer function: Input = Va(s)Output = m(s)Internal = Ia(s), Vb(s), Tm(s)
Electrical Mechanical Systems
mmmm scsJsT 2)(
Transfer functionfor mechanical system
Electrical Mechanical Systems Lorentz Force and Faraday’s Law
t
ma K
sTsI
)()( )()( ssKsV mbb
),()( tiKtT atm ),()( tKtv mbb
Substitute into electrical and mechanical transfer functions
)()(
)()(
)(
2
ssKsK
scsJsLR
ssKK
sTsLRsV
mbmt
mmaa
mbt
maaa
Electrical Mechanical Systems Re-arranging the transfer function
a
btm
m
ma
t
a
m
RKK
cJ
ssJR
K
sV
s
1
1*
)(
)(
ssA
sV
s
a
m 1*
)(
)(
Intimidating coefficients, but simple general form
tm e
AAt
At
22)(
Suppose Va(t) = u(t)
Next Lectures
Direct transfer function manipulation.