Communications and Power Systemocw.snu.ac.kr/sites/default/files/NOTE/812.pdf · RLC circuit...
Transcript of Communications and Power Systemocw.snu.ac.kr/sites/default/files/NOTE/812.pdf · RLC circuit...
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Lecture 9Lecture 9--11Circuit Theory ICircuit Theory I
- 통신 시스템 (예: Morse code, radio)에서는 전압 신호 등과 같은 입력 신호가 전원이 된다.
- 변환기는 전파 매체에 적당하도록 신호를 변환한다.
- 변환기의 출력은 수신기에 도착할 때까지 매체를 진행한다.
- 수신기는 사용자가 쓰기에 적절한 형태로 변환시킨다.
- 전력 시스템 에서는 발전기가 30 - 70 MW의 전력을 발생시킨다.
- 전선을 통해서 전력을 효율적으로 수용가에 수송한다.
- 통신시스템: undistorted transmission, 전력시스템: efficient power transmission.
- 전기 회로는 electrical signal 또는 electrical power를 전송하는 데 사용한다.
Communications and Power SystemCommunications and Power System
Electrical system
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Lecture 9Lecture 9--22Circuit Theory ICircuit Theory I
KVL KCL
LCv
LCv
dtdv
LR
dtvd
vvdt
dvRCdt
vdLC
dtdvCi
vvRidtdiL
sccc
sccc
c
sc
=++
=++
=
=−++
2
2
2
2
0
LCvf
LCLR s===
,
20
,
12
ωα
)(2 202
2
tfxdtdx
dtxd
=++ ωαLCi
LCi
dtdi
RCdtid
idt
dvCiRv
dtdiLvvv
dtdvCi
Rvi
nLLL
nL
LL
LLLc
cL
cn
=++
=++
==
=+++−
1
0)(
2
2
,
LCif
LCRCn===
,
20
,
12
1 ωα
R_vs
L
_ +
i+vc
+
_vcR L iL
+
_C
vL
in
Natural Response of SecondNatural Response of Second--Order CircuitsOrder Circuits
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Lecture 9Lecture 9--33Circuit Theory ICircuit Theory I
20
22
20
21
20
220
2
202
2
202
2
,020)2(
02
,2
ωααωααωαωα
ωα
ωα
−−−=−+−=
=++⇒=++
=⇒=++
+==++
ssssssKe
Kexxdt
dxdt
xd
xxxfxdtdx
dtxd
st
sthh
hh
ph
특성방정식
로 가정.
damping ratio
1. Over damped
2. Critically damped
3. Under damped
0/ωαζ =
),1( 0ωαζ >> s1, s2 : negative real
tstsh etKeKx 11
21 +=s1 : negative real),1( 0ωαζ ==
),1( 0ωαζ << s1, s2 : complex with negative real
teKteK
eKeKx
dt
dt
dtjtj
hdd
ωω
αωωαα
ωαωα
sincos
)(
43
220
)(2
)(1
−−
−−+−
+=
−=+= damped resonant frequency
SecondSecond--Order Differential EquationOrder Differential Equation
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Lecture 9Lecture 9--44Circuit Theory ICircuit Theory I
- RLC 회로에 step function의 전압을 가했다.
)(2 202
2
tAuxdtdx
dtxd
=++ ωα
− ζ=α/ω0의 값에 따라 출력 값에
변화가 있음. - 특히 ζ = 1 을 경계로
overshoot 보인다.- 응용의 특성에 따라 damping ratio을 변화시킨다.
t = 0
+
_v(t)+_A
RLC
circuit
v(t)=Au(t)
t_
Normalized Step Response of SecondNormalized Step Response of Second--Order Systems Order Systems
Underdamped, critically damped, and overdamped response curves for parallel RCL circuit of example 9.7.
DeCarlo 책 351쪽 Fig. 9.9
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Lecture 9Lecture 9--55Circuit Theory ICircuit Theory I
LCIi
LCdtdi
RCdtid
Idtdi
RLi
dtidLCv
dtdiLI
Rvi
dtdvCc
dtdivv
dtdib
iiiia
LLL
LL
Lc
LcL
c
t
L
t
L
LLLL
=++
=++==++
====
====
++ =
+−
=
−+−+
11
.,,)
.00)0(0)0(?)
.0)0()0(0)0(?)0()
2
2
2
200
∴이므로
이므로
Parallel RLC Circuit (I)Parallel RLC Circuit (I)
t = 0 인 순간 switch가 열리고 전류 24 mA가 회로에 가해진다.
저항 값은 400 Ω 이다.
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Lecture 9Lecture 9--66Circuit Theory ICircuit Theory I
특성방정식
80000,20000,103105
1041,1052
1
01016100000,011
2144
40
4
822
−=−=×±×−=
×==×==
=×++=++
sssLCRC
ssLC
sRC
s
ωα
32
31
800002
200001
3
1032,108
0)0(,0)0(
1024
−−
++
−−−
×−=×=⇒
==
++×=
AAdtdii
eAeAi
LL
ttL
04
04
104 , 500102.3
, 625
ωα
ωα
=×=
Ω=<×=
Ω=
RIf
RIf
Parallel RLC Circuit (II)Parallel RLC Circuit (II)
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Lecture 9Lecture 9--77Circuit Theory ICircuit Theory I
전압 v를 구하라.
Complete Response of RLC Circuits (I)Complete Response of RLC Circuits (I)
(a) 초기조건을 구하라.
- t = 0- 의 회로
A V, 1)0( 6)0( == −−LC iv
- t = 0+ 의 회로
6 V
1 A
0 0)0(00
=∴==++
+
dtdi
dtdiLv LL
L
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Lecture 9Lecture 9--88Circuit Theory ICircuit Theory I
Complete Response of RLC Circuits (II)Complete Response of RLC Circuits (II)
(b) KCL 로 식을 세워라.
LL
C
CL
sC
ii
i
dtdv
dtdvvv
61
025.04
+=
=+−+
)(6107
0)6(4)6(
32
2
tuevdtd
dtd
dtd
dtdv
dtd
tsL
LL
LL
LsLL
iii
ii
iii
−==++
=+++−+
3610)21(9
set 3
−==+−+
=
+=−
AAAA
Aei
iiit
Lp
LpLhL
ttLh
stLh
eKeKis
ss
Kei
52
21
2
5,20107
set
−− +=
−−==++
=
9520)0('
31)0(
3
21
21
352
21
+−−==
−+==
−+=
+
+
−−−
KKi
KKi
eeKeKi
L
L
tttL
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Lecture 9Lecture 9--99Circuit Theory ICircuit Theory I
Complete Response of RLC Circuits (III)Complete Response of RLC Circuits (III)
3/1 ,3/11952
4
21
21
21
===+
=+
KKKK
KK
tttL eeei 352 3
31
311 −−− −+=
ttt
tttttt
ttttttC
LL
C
eee
eeeeee
eeeeeedtdv
dtdv i
i
352
352352
352352
931
344
1836
3669
35
322
)331
311(6)3
31
311(
61
−−−
−−−−−−
−−−−−−
−+=
−+++−−=
−++−+=
+= 이므로
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Lecture 9Lecture 9--1010Circuit Theory ICircuit Theory I
- State variable method : 회로의 전체 응답을 구하기 위해서 state variable의
1계 미분방정식을 이용한다.- Inductor의 전류나 capacitor의 전압을 state variable로 이용한다.
dtd
Rvv
Rvv
dtdvCnode
Rvv
Rvv
dtdvCnode
KCL
b
a
0:2
0:1
3
2
2
1222
2
21
1
111
=−
+−
+
=−
+−
+
를 operator s로 써서 정리하면,
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡++−
−++
b
a
vv
RR
vv
RRsCRRRRsC
3
1
2
1
3222
2211
1,00,1
11,11,11
v1과 v2는 아래의 꼴로 쓰여진다.이것의 해를 구하는 방법은 14장 Laplacetransform에서 다루겠다.
( ) ( )BAss
vFEsvDCsvv ba
+++++
= 221,
State Variable Approach to Circuit AnalysisState Variable Approach to Circuit Analysis
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Lecture 9Lecture 9--1111Circuit Theory ICircuit Theory I
- 2계 미분 방정식 시스템
( parallel RLC 회로 )
djjsif
sLCRC
ss
ωααωααω
ωαα
±−=−±−=⟩
−±−=
=++
2200
20
2
2
,
01
Over damped
Critically damped
Under dampedUndamped
21, rrs −−=
11, rrs −−=
djs ωα ±−=djs ω±=
Roots in the Complex Plane (I)Roots in the Complex Plane (I)
- 해를 복소 평면에 그릴 수 있다.
- 실수 축과 허수 축, σ and jω
Natural response of a parallel RLC circuit
Form of response for v(0) = 1 V and i(0) = 0.
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Lecture 9Lecture 9--1212Circuit Theory ICircuit Theory I
i) Over damped
s = -r1 , -r2 (음의 실수 축 위의 두 점)
ii) Critically damped
s = -r1 , -r1 (음의 실수 축 위의 한 점)
iii) Under damped
s = -α ± j ωd (음의 실수 평면 위의 두 점)
iv) Undamped
s = ± j ωd (허수 축 위의 두 점)
The complete s-plate showing the location of the two roots, s1and s2,, of the characteristic equation in the left-hand portion
of the s-plane. The roots are designated by the × symbol.
Roots in the Complex Plane (II)Roots in the Complex Plane (II)
해의 각 경우를 복소 평면에 그려 보면
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Lecture 9Lecture 9--1313Circuit Theory ICircuit Theory I
- Airbag은 운전자의 안전을 위해서 이용된다.
- Pendulum이 capacitor energy를 igniter로 보내도록 스위치한다.
- 저항 R에서 받은 에너지로 화약 등을 폭발시켜 airbag을 팽창시킨다.
- 저항 R에서 1 J의 에너지를 소모해야 하고, 0.1 초 이내에 트리거해야 한다.
- L, C의 값을 정하라.
An automobile airbag ignition device
Auto Airbag Igniter (I)Auto Airbag Igniter (I)
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Lecture 9Lecture 9--1414Circuit Theory ICircuit Theory I
K
M
K
M
- Box 안에 pendulum 이 매달려 있다. - 가속도가 가해지면 관성력에 의하여
pendulum 과 box벽의 거리가 변화한다.
Auto Airbag Igniter (II)Auto Airbag Igniter (II)
Δxmkaxkma =⇒Δ=
[ ]
xdd
A
dxdxdA
dxdxdA
xdA
xdACCΔC
Δ=
Δ−−Δ+≈
Δ+−
Δ−=
Δ+−
Δ−=
−=
2
)1()1(
)1
11
1(
0
0
0
00
21
ε
ε
ε
εε
xdAC
xdAC
Δ+=
Δ−= 0
20
1 , εε
- Box의 위, 아래 벽에 전극을 설치하고, pendulum을 전극으로 이용한다.- Box의 위, 아래 벽 전극과 pendulum전극에 전압을 인가한다.- pendulum의 위치 변화에 따라 위 아래
전극과의 정전용량이 변화한다.- 이들 값의 차분을 구하면 위치변화분을
알 수 있다.
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Lecture 9Lecture 9--1515Circuit Theory ICircuit Theory I
Auto Airbag Igniter (III)Auto Airbag Igniter (III)
Electroplated gold cantilevers
Electroplated nickel and aluminum micromirrorAluminum micromirror array
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Lecture 9Lecture 9--1616Circuit Theory ICircuit Theory I
MEMS AccelerometerMEMS Accelerometer
ADXL 50 accelerometer From Analog Devices, Inc.
ADXL 50 accelerometer. The sensing element in the center is surrounded by active electronics. Chip size is 3 mm x 3 mm.
Top Electrode
Bottom ElectrodeRelative Mass
Displacement=x=y=z
General accelerometer structure and is mechanical lumped model.
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Lecture 9Lecture 9--1717Circuit Theory ICircuit Theory I
NonNon--Vacuum Sealed MEMS GyroscopeVacuum Sealed MEMS Gyroscope
Sensing springs
Stopper
Sensing electrodesTuning electrodes
Inertial mass
Gimbal
Driving springs
Driving electrodes
Comb electrodes after deep RIE(DRIE) From Lab. for MiSA, SNU
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Lecture 9Lecture 9--1818Circuit Theory ICircuit Theory I
Auto Airbag Igniter (IV)Auto Airbag Igniter (IV)
)0( 0)0(
)0( 12)0(
A
V+−
+−
=
=
=
=
LL
CC
ii
vv
- 0.1 초 이내에서 트리거되려면 under damped response를 보여야 한다.
- 0.1 초가 주기의 ¼ 정도가 되어야 한다.
0 1
0
,0
2
2
2
2
=++
=++∴
==++
LCi
dtdi
RCdtid
idtdi
RL
dtidLC
dtdiLv
Rvi
dtdvC
LLL
LLL
LL
0
0 /1,2/1ωα
ωα<
== :condition dunderdapme
LCRC
)(
sincos22
0
21
αωω
ωω αα
−=
+= −−
d
dt
dt
L teKteKi
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Lecture 9Lecture 9--1919Circuit Theory ICircuit Theory I
Auto Airbag Igniter (V)Auto Airbag Igniter (V)
teKteKi dt
dt
L ωω αα sincos 21−− +=
- 빠른 응답을 위하여 α = 2 로 선택한다.
mH
F
65)2/4.0(16/)2/4.0(
4.022116/12/1
/1,2/1
220
0
===
=≈==
====
ππ
ωπ
ωπ
αωα
CL
fT
RCLCRC
d
이므로
이고
이므로
)57.15(
57.15sin57.15cos22
0
22
21
=−=
+= −−
αωωd
ttL teKteKi
+
=
=
=
=
+
+−
0
12)0(
)0(0)0(
dtLdi
Lv
ii
L
LL
V
A
t = 0+
12 V
0 A
86.1157.15
12
57.15120
2
20
1
==
==
=
+
LK
KLdt
diK
L
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Lecture 9Lecture 9--2020Circuit Theory ICircuit Theory I
Auto Airbag Igniter (VI)Auto Airbag Igniter (VI)
teRvpte
tetedtdiLv
tei
t
t
tt
L
tL
57.15cos3657.15cos12
)57.15cos57.1557.15sin2(86.11065.0
57.15sin86.11
242
2
22
2
−
−
−−
−
==
≈
+−×=
=
=
J1 J >=×=
== ∫∫71.1095.036
21
1.0
0
1.0
0vidtpdtW
- 저항의 전압과 전류를 보여준다.- 0 초에서 0.1 초 까지 거의 직선적으로 변하기
때문에 전력을 삼각형으로 여기고 에너지를 구
한다.