damped second-order systemsocw.mit.edu/.../video-lectures/6002_l15b.pdfloop. 2kΩ. 50Ω. 2kΩ....
TRANSCRIPT
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6.002 Fall 03 1
6.002 CIRCUITS ANDELECTRONICS
Damped Second-Order Systems
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6.002 Fall 03 2
Damped Second-Order Systems
CA B
5V
+–
5V
CGS
largeloop
2KΩ50Ω
2KΩ
Remember this Demo
Our old friend, the inverter, driving another.The parasitic inductance of the wire and the gate-to-source capacitance of the MOSFET are shown
[Review complex algebra appendix in Agarwal & Lang for next class]
S
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6.002 Fall 03 3
Damped Second-Order Systems
CA B
5V
+–
5V
CGS
largeloop
2KΩ50Ω
2KΩ
+–5V CGS
2KΩB
LRelevant circuit:
S
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6.002 Fall 03 4
Now, let’s try to speed up our inverter by closing the switch S to lower the effective resistance
tvA
5
0
vB
0 t
vC
0 t
Observed Output
2kΩ
2kΩ
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6.002 Fall 03 5
tvA
5
0
vB
0 t
vC
0 t
Observed Output ~50Ω
50Ω
Huh!
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6.002 Fall 03 6
+– CL +
–)(tv
)(tvI
)(ti
In the last lecture, we started by analyzing the simpler LC circuit to build intuition
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6.002 Fall 03 7
And for initial conditionsv(0) = 0 i(0) = 0 [ZSR]
IvIV
0 t
We solved
For input
IvLC
vLCdt
vd 112
2
=+
In the last lecture…
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6.002 Fall 03 8
toVVtv II ωcos)( −= where
LC1
o =ω
Total solution
In the last lecture…
LC
)(tv
t
IV
IV2
0Iv
+– CL +
–)(tv
)(tvI
)(ti
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6.002 Fall 03 9
See A&L Section 13.6
+– C
R L
+
–)(tv)(tvI
)(ti
Damped sinusoids with R – remember demo!
Today, we will close the loop on our observations in the demo by analyzing the RLC circuit
LC
)(tv
t
IV
IV2
0Iv add R
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6.002 Fall 03 10
v, i state variables
+– CL +
–)(tv
)(tvI
)(ti
Node method:Recall
element rules
Let’s analyze the RLC network
R
dtdiLvL =
idtvL
t
L =∫∞−
1
L:
dtdvCi C
C =C:
Av
IvLC
vLCdt
dvLR
dtvd 112
2
=++
Rvvdt)v(v
L1 A
t
AI−
=∫ −∞−
dtdvC
RvvA =
−
:Av
:v
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6.002 Fall 03 11
+– CL +
–)(tv)(tvI
)(ti
Node method:
Let’s analyze the RLC network
R
Av
2
2
)(1dt
vdvdtdvRCv
LC I =−−
IvLC
vLCdt
dvLR
dtvd 112
2
=++
Rvvdt)v(v
L1 A
t
AI−
=∫ −∞−
dtdvC
RvvA =
−
:Av
:v
vdtdvRCvA +=
2
2
)(1dt
vdCvvL AI =−
2
2
)(1dt
vdvvLC AI =−
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6.002 Fall 03 12
Recall, the method of homogeneous andparticular solutions:
( ) ( )tvtvv HP +=
1
2
3
Find the particular solution.
Find the homogeneous solution.
4 steps
The total solution is the sum of theparticular and homogeneous.Use initial conditions to solve for theremaining constants.
Solving
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6.002 Fall 03 13
And for initial conditionsv(0) = 0 i(0) = 0 [ZSR]
Iv
0 t
IV
Let’s solve
For input
IvLC
vLCdt
dvLR
dtvd 112
2
=++
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6.002 Fall 03 14
1 Particular solution
IP Vv = is a solution.
IPPP V
LCv
LCdtdv
LR
dtvd 112
2
=++
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6.002 Fall 03 15
0vLC1
dtdv
LR
dtvd
HHH =++2
2
Solution to
Homogeneous solution2
Recall, vH : solution to homogeneousequation (drive set to zero)
Four-step method:
Dtsts
H eAeAv 2121 +=
General solution
Assume solution of the formA?s,A,Aev st
H ==
B Form the characteristic equationf(s)
Find the roots of the characteristic equationC
21 ss ,
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6.002 Fall 03 16
LR
2=α
0vLC1
dtdv
LR
dtvd
HHH =++2
2
Solution to
Homogeneous solution2
Dtt
Hoo
eAeAv⎟⎠⎞⎜
⎝⎛ −−−⎟
⎠⎞⎜
⎝⎛ −+−
+=2222
21
ωααωαα
General solution
LC1
o =ω
Assume solution of the formA?s,A,Aev st
H ==
characteristicequationB 02 =++
LC1s
LRs
0=++ o22 s2s ωα
RootsC o1s22 ωαα −+−=
os 222 ωαα −−−=
so, 012 =++ ststst AeLC
AseLReAs
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6.002 Fall 03 17
Total solution3
Find unknowns from initial conditions.
)()()( tvtvtv HP +=
:0)0( =v 210 AAVI ++=
:0)0( =i
dtdvCti =)(
tt
Ioo
eAeAVtv⎟⎠⎞⎜
⎝⎛ −−−⎟
⎠⎞⎜
⎝⎛ −+−
++=2222
21)(ωααωαα
( ) +−+−=⎟⎠⎞⎜
⎝⎛ −+− t
oo
eCA22
221
ωααωαα
( ) to
o
eCA⎟⎠⎞⎜
⎝⎛ −−−
−−−22
222
ωααωαα
so, ( ) ( )oo AA 222
2210 ωααωαα −−−+−+−=
Mathematically: solve for unknowns, done.
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6.002 Fall 03 18
Let’s stare at this a while longer…
ttttI
oo
eeAeeAVtv⎟⎠⎞⎜
⎝⎛ −−−⎟
⎠⎞⎜
⎝⎛ −− ++=
2222
21)(ωααωαα
3 cases:
ttI eAeAVtv 21
21)( αα −− ++=
tjttjtI
oo
eeAeeAVtv⎟⎠⎞⎜
⎝⎛ −−−⎟
⎠⎞⎜
⎝⎛ −− ++=
2222
21)(αωααωα
tjttjtI
dd eeAeeAV ωαωα −−− ++= 2122 αωω −= od
t
IvIV
vOverdampedoωα >
Underdampedoωα <
teKteKV dt
dt
I ωω αα sincos 21−− ++= tjte dd
tj d ωωω sincos +=
Critically dampedoωα =Later…
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6.002 Fall 03 19
Let’s stare at underdamped a while longer…
Underdamped contd…oωα <
teKteKVtv dt
dt
I ωω αα sincos)( 21−− ++=
:0)0( =v
:0)0( =i dtdvCti =)(
IVK −=1
dKK ωα 210 +−=
d
VKωα1
2 −=
teVteVVtv dt
dId
tII ω
ωαω αα sincos)( −− −−=
tVVtv oII ωcos)( −=
Same as LC as expected
Note: For 00 == ⇒αR
teCKteCK
teCKteCK
dt
ddt
dt
ddt
ωωωα
ωωωααα
αα
cossin
sincos
21
21−−
−−
+−
−−=
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6.002 Fall 03 20
Underdamped contd…oωα <
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= −−
dd
t
d
oII teVVtv
ωαω
ωω α 1tancos)(
Remember, scaled sum of sines (of the same frequency) are also sines! -- Appendix B.7
teVteVVtv dt
dId
tII ω
ωαω αα sincos)( −− −−=
add R
Let’s stare at underdamped a while longer…
LC
)(tv
t
IV
IV2
0Iv
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6.002 Fall 03 21
Underdamped contd…oωα <
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= −−
dd
t
d
oII teVVtv
ωαω
ωω α 1tancos)(
Remember, scaled sum of sines (of the same frequency) are also sines! -- Appendix B.7
teVteVVtv dt
dId
tII ω
ωαω αα sincos)( −− −−=
LC
)(tv
t
IV
IV2
0Iv add R
tCritically dampedoωα =
Section 13.2.3
underdamped
overdamped
criticallydampedv
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6.002 Fall 03 22
tvA
5
0
vB
0 t
vC
0 t
Remember this? Closed the loop…
50Ω
See example 12.9 on page 664 of the A&L textbookfor inverter-pair analysis
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6.002 Fall 03 23
Underdamped
:2αωoQ = Quality factor (approximately
the number of cycles of ringing)
:IV Final value :)0(v Initial value
)(tv
t
IV
0
:dω Oscillation frequency
dωπ2
te α−
:α Governs rate of decay
“ringing”
02 =++LC1s
LRs
0=++ o22 s2s ωα
Characteristicequation
22 αωω −= od
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= −−
dd
t
d
oII teVVtv
ωαω
ωω α 1tancos)(
Intuitive AnalysisSee Sec. 12.7 of A&L textbook
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6.002 Fall 03 24
Intuitive Analysis
)(tv
t
IV
0
IV
dωπ2
period
QRinging stopsafter cycles
02 =++LC1s
LRs
0=++ o22 s2s ωα
Characteristicequation
αω2
oQ =
22 αωω −= od
)0(v ?
+– CL +
–)(tvIv
)(ti
R
)0(iis –veso v(t) must drop
)0(i)0(v
given-ve+ve
See Sec. 12.7of A&L textbook