RL and RC circuits first- order response Electric circuits ENT 161/4.
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Transcript of RL and RC circuits first- order response Electric circuits ENT 161/4.
![Page 1: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/1.jpg)
RL and RC circuits first-order response
Electric circuitsENT 161/4
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RL and RC circuit original response
A first-order circuit is characterized by a first-order differential equation. This circuit contain resistor and capacitor or inductor in one close circuit.
![Page 3: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/3.jpg)
The natural response of a circuit refers to the behaviour ( in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
RL circuit: circuit that have resistor and inductor.
RC circuit: circuit that have resistor and capacitor.
![Page 4: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/4.jpg)
Natural response RC circuit
![Page 5: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/5.jpg)
Consider these three condition :
1. At initially, t=0 -, switch doesn’t change for some time
2. At initial, t=0 +, switch doesn’t change for some time
3. At final condition, t→∞, switch doesn’t change for some time
![Page 6: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/6.jpg)
Known t ≤ 0, v(t) = V0.
dtRCtv
tdvRC
tv
dt
tdvRC
tv
dt
tdvR
tv
dt
tdvC
ii Rc
1
)(
)(
)()(
0)()(
0)()(
0
RCt
eVtv
RC
t
V
tv
tRC
Vtv
dvRC
duu
dvRCu
du
tv
V
t
0
0
0
)(
0
)(
)(ln
)0(1
ln)(ln
11
1
0
voltage
Therefore t ≥ 0:
![Page 7: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/7.jpg)
For t > 0,
RCt
eVtv 0)(RCt
eR
V
R
tvtiR
0)()(
RCt
eVCtvCtW2
20
2
2
1)(
2
1)(
![Page 8: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/8.jpg)
Natural response RC circuit graph
0
0)(
0
0
teV
tVtv
RCt
![Page 9: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/9.jpg)
This show that the voltage response of the RC circuit is an exponential decay of the initial voltage. constant, τ = RC
t
eVtv 0)(
![Page 10: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/10.jpg)
Constant τ define how fast voltage reach stable condition :
![Page 11: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/11.jpg)
Natural response RL circuit
![Page 12: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/12.jpg)
Consider these three condition :
1. At initially, t=0 -, switch doesn’t change for some time
2. At initial, t=0 +, switch doesn’t change for some time
3. At final condition, t→∞, switch doesn’t change for some time
![Page 13: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/13.jpg)
Known at t ≤ 0, i(t) = I0
Therefore t > 0,
dvL
R
u
du
dtL
R
ti
tdi
tiRdt
tdiL
tiRdt
tdiL
tiRtv
)(
)(
)()(
0)()(
0)()(
LRt
tti
i
eiti
L
Rt
i
ti
tL
Riti
dvL
Rduu
)0()(
)0(
)(ln
)0()0(ln)(ln
10
)(
)0(
Current
![Page 14: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/14.jpg)
For t > 0,
LRteIti 0)(
LRteRI
Rtitv
0
)()(
LRteLI
tiLtw
220
2
2
1
)(2
1)(
![Page 15: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/15.jpg)
EXAMPLESwitch in circuit for some time before
open at t=0. Calculate
a) IL (t) at t ≥ 0
b) I0 (t) at t ≥ 0+
c) V0 (t) at t ≥ 0+
d) Total energy percentage that stored in inductor 2H that absorb by 10Ω resistor.
![Page 16: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/16.jpg)
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Answer
a) Switch close for some time until t=0, known voltage at inductor should be zero at t = 0-. Therefore, initial current at inductor was 20A at t = 0-. Thus iL (0+) also become 20A, because immidiate changes for current didn’t exist in inductor.
![Page 18: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/18.jpg)
Equivalent resistance from inductor and constant time
1010402eqR
saatR
L
eq
2.010
2
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Therefore, current iL (t)
020
)0()(5
tAe
eitit
L
t
![Page 20: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/20.jpg)
b) Current at resistor 40Ω could be calculate by using current divider law,
4010
100 Lii
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This current was at t ≥ 0+ because i0 = 0 at t = 0-. Inductor will
become close circuit when switch open immediately and produce changes immediately at current i0.
Therefore, 04)( 5
0 tAeti t
![Page 22: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/22.jpg)
c) V0 could be calculate by using Ohm’s Law,
0160
40)(5
00
tVe
itVt
![Page 23: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/23.jpg)
d) Total power absorb by 10Ω resistor
02560
10)(
10
20
10
tWe
Vtp
t
![Page 24: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/24.jpg)
Total energy absorb by 10Ω resistor
J
dtetW t
256
2560)(0
1010
![Page 25: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/25.jpg)
Initial energy stored at 2H inductor
J
iLW
40040022
1
)0(2
1)0( 2
![Page 26: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/26.jpg)
Therefore, energy percentage that absorb by 10Ω resistor
%64100400
256
![Page 27: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/27.jpg)
Step response RC circuit The step response of a circuit is its
behaviour when the excitation is the step function, which may be a voltage or a current source.
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Consider these three condition :
1. At initially, t=0 -, switch doesn’t change for some time
2. At initial, t=0 +, switch doesn’t change for some time
3. At final condition, t→∞, switch doesn’t change for some time
![Page 29: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/29.jpg)
Known at t ≤ 0, v(t)=V0
For t > 0,
s
s
s
s
Vtv
tdvdt
RC
tvV
tdvdt
RC
dt
tdvRCtvV
tRitvV
)(
)(1
)(
)(1
)()(
)()(
s
s
ss
s
VV
Vtv
RC
t
VVVtvRC
t
Vu
dudv
RC
0
0
)(ln
ln)(ln
1
t
RCt
eVVV
eVVVtv
ss
ss
0
0)(
voltan
![Page 30: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/30.jpg)
Current for step response RC circuit
t
t
t
eR
V
R
V
eVVR
eVVC
dt
dvCti
s
s
s
0
0
0
1
)(1
)(
t
eiti )0()(
![Page 31: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/31.jpg)
Then, for t >0
t
t
eVVV
VV
VV
eVVVV
sn
sf
nf
ss
0
0
Where
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Vf = Force voltage or known as steady-state response
Vn = known as transient response is the circuit’s temporary response that will die out with time.
![Page 33: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/33.jpg)
Step response RC circuit graph
force
Natural
total
![Page 34: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/34.jpg)
Step Response RL circuitStep Response RL circuit
![Page 35: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/35.jpg)
Consider these three condition :
1. At initially, t=0 -, switch doesn’t change for some time
2. At initial, t=0 +, switch doesn’t change for some time
3. At final condition, t→∞, switch doesn’t change for some time
![Page 36: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/36.jpg)
known i(t)=I0 at t ≤ 0. For t > 0,
RV
RV
s
s
s
s
s
i
didt
L
R
ti
tdidt
L
Rdt
tdi
R
Lti
R
Vdt
tdiLtiRV
tvtiRV
)(
)(
)()(
)()(
)()(
RV
RV
RV
RV
ti
IR
V
t
RV
s
s
ss
s
s
I
ti
L
Rt
ItiL
Rt
u
dudv
L
R
u
dudv
L
R
0
0
)(
0
)(ln
ln)(ln
0
LR
ss tR
VR
V eIti 0)(
Current
![Page 37: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/37.jpg)
Finally,
0
0)(
0
0
teI
tIti
LR
ss tR
VR
V
0
0)(
)(
0
teIRV
tdt
tdiLtv
LRt
s
![Page 38: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/38.jpg)
Question Switch in those circuit was at x
position for some time. At t=0, switch move to position y immediately. Calculate,
(a) Vc(t) at t ≥ 0(b) V0 (t) at t ≥ 0+(c) i0 (t) at t ≥ 0+(d) Total energy absorb by 60kΩ resistor.
![Page 39: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/39.jpg)
![Page 40: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/40.jpg)
Answer (a)
Constant for circuit
ms40)1080)(105.0( 36
VC (0)=100V
equivalent resistor = 80kΩ.
![Page 41: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/41.jpg)
Then, VC(t) for t ≥ 0:
0100)( 25 tVetV tC
![Page 42: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/42.jpg)
Answer (b)V0 (t) could be calculate by using
voltage divider law.
060
)(80
48)(
25
0
tVe
tVtV
t
C
![Page 43: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/43.jpg)
Answer (c)
current i0 (t) can be calculated by using ohm’s law
01060
)()( 25
30
0 tmAetV
ti t
![Page 44: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/44.jpg)
Answer (d)
Power absorb by 60kΩ resistor
060
1060)()(50
32060
tmWe
titpt
k
![Page 45: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/45.jpg)
Total energy
mJ
dttiW k
2.1
1060)( 3
0
2060
![Page 46: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/46.jpg)
Second-order RLC circuit
RLC circuit : circuit that contain resistor, inductor and capacitor
Second-order response : response from RLC circuit
Type of RLC circuit:1. RLC series circuit2. RLC parallel circuit
![Page 47: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/47.jpg)
Original response for parallel RLC circuit
![Page 48: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/48.jpg)
Take total current flows out from node
01
0 0 t
dt
dvCIvd
LR
V
![Page 49: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/49.jpg)
differential of t,
01
2
2
dt
vdC
L
v
dt
dv
R
01
2
2
LC
v
dt
dv
RCdt
vd
![Page 50: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/50.jpg)
Take steAv
02 ststst eLC
Ae
RC
AseAs
012
equationsticcharacteri
st
LCRC
sseA
![Page 51: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/51.jpg)
Characteristic equation known as zero :
012
LCRC
ss
![Page 52: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/52.jpg)
The root of the characteristic equation are
LCRCRCs
1
2
1
2
12
1
LCRCRCs
1
2
1
2
12
2
![Page 53: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/53.jpg)
Response for RLC parallel circuit
tsts eAeAv 2121
![Page 54: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/54.jpg)
The root of the characteristic equation are
RC2
1
20
21 s
20
22 s
where:
LC
10
![Page 55: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/55.jpg)
summarize
20
22
20
21
s
s
RC2
1
0 LC
10
Parameter Terminology Value in natural response
s1, s2characteristic
equation
α frequency Neper
resonant radian frequency
![Page 56: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/56.jpg)
Roots solution s1 and s2 depend on α and
Consider these cases saperately: 1. If < α , voltage response was
overdamped 2. If > α , voltage response was
underdamped3. If = α , voltage response was
critically damped
0
0
0
0
![Page 57: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/57.jpg)
Overdamped voltage responseOverdamped voltage response
Solution for overdamped voltage
tsts eAeAv 2121
![Page 58: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/58.jpg)
constant A1 and A2 can be determined from the initial conditions v(0+) and
Known, dt
dv )0(
21)0( AAv
2211
)0(AsAs
dt
dv
![Page 59: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/59.jpg)
Here v(0+) = V0 and initial value for
dv/dt was
C
i
dt
dv C )0()0(
![Page 60: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/60.jpg)
Solution for overdamped natural response, v(t) :
1. Calculate characteristic equation, s1 and s2, using R, L and C value.
2. Calculate v(0+) and
using circuit analysis. dt
dv )0(
![Page 61: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/61.jpg)
3. Calulate A1 and A2 by solve those equation
4. Insert s1, s2, A1 and A2 value to calculate overdamped natural response for t ≥ 0.
21)0( AAv
2211
)0()0(AsAs
C
i
dt
dv C
![Page 62: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/62.jpg)
Example for overdamped natural response for v(0) = 1V and
i(0) = 0
![Page 63: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/63.jpg)
Underdamped voltage response
At > α2, root of the characteristic equation was complex number and those response called underdamped.
0
![Page 64: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/64.jpg)
Therefore
ωd : damped radian frequency
dj
j
s
220
2201 )(
djs 2
![Page 65: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/65.jpg)
underdamped voltage response for RLC parallel circuit was
teB
teBtv
dt
dt
sin
cos)(
2
1
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constant B1 and B2 was real number.
10)0( BVv
211
)0()0(BB
C
i
dt
dvd
C
Solve those two linear equation
to calculate B1 and B2,
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Example for underdamped voltage response for v(0) = 1V and i(0) = 0
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Critically Damped Voltage Response
Second-order circuit was critically damped when = α . When circuit was critically damped, two characterictic root equation was real and same,
RCss
2
121
0
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Solution for voltage tt eDetDtv 21)(
21
20
)0()0(
)0(
DDC
i
dt
dv
DVv
C
•Linear equation to calculate D1 and D2 value
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Example for critically damped voltage response at v(0) = 1V and i(0) = 0
![Page 71: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/71.jpg)
Step response RLC parallel Step response RLC parallel circuitscircuits
Step response RLC parallel Step response RLC parallel circuitscircuits
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From Kirchhoff current law
Idt
dvC
R
vi
Iiii
L
CRL
![Page 73: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/73.jpg)
Known
Therefore
dt
diLv
2
2
dt
idL
dt
dv L
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Have,
Idt
idLC
dt
di
R
Li LLL
2
2
LC
I
LC
i
dt
di
RCdt
id LLL 1
2
2
![Page 75: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/75.jpg)
There are two solution to solve the equation, direct approach and indirect approach.
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Indirect approachIndirect approach
From Kirchhoff’s current law:
Idt
dvC
R
vvd
L
t0
1
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Differential
01
2
2
dt
vdC
dt
dv
RL
v
01
2
2
LC
v
dt
dv
RCdt
vd
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Depend on characteristic equation root :
tsts eAeAv 2121
teB
teBv
dt
dt
sin
cos
2
1
tt eDetDv 21
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Insert in Kirchhoff’s current law eq:
tstsL eAeAIi 21
21
teB
teBIi
dt
dt
L
sin
cos
2
1
ttL eDetDIi 21
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Direct approach
It’s simple to calculate constant for the equation
directly by using initial value response function.
212121 ,,,B,, DDBAA
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Constant of the equation could be calculate from
and dt
diL )0()0(Li
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The solution for a second-order differential equation with a constant forcing function equals the forced response plus a response funtion identical in form to the natural response.
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If and Vf represent the final value of the response function. The final value may be zero,
responsenaturaltheas
formsametheoffunctionIi f
responsenaturaltheas
formsametheoffunctionVv f
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Natural response for RLC Series circuit
The procedures for finding the natural or step responses of a series RLC circuit are the same as those used to find the natural or step responses of a parallel RLS circuit, because both circuits are described by differential equations that have the same form.
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RLC series circuit
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Summing the voltages around the closed path in the circuit,
01
00 Vdi
Cdt
diLRi
t
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differential
02
2
C
i
dt
idL
dt
diR
02
2
LC
i
dt
di
L
R
dt
id
![Page 88: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/88.jpg)
Characteristic equation for RLC series circuit
012 LC
sL
Rs
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Characteristic equation root
LCL
R
L
Rs
1
22
2
2,1
@
20
22,1 s
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Neper frequency (α) for RLC series circuit
sradL
R/
2
and resonant radian frequency was,
sradLC
/1
0
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Current response
Overdamped
Underdamped
critically damped
220
220
220
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Three kind of solution
tsts eAeAti 2121)(
teB
teBti
dt
dt
sin
cos)(
2
1
tt eDetDti 21)(
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Step response for RLC series circuit
The procedures for finding the step responses of series RLC circuit are the same as those used to find the step response of a parallel RLC circuit.
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RLC series circuit
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Cvdt
diLiRv
Using Kirchhoff’s voltage law,
![Page 96: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/96.jpg)
Current known as,
dt
dvCi C
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Differential for current
2
2
dt
vdC
dt
di C
![Page 98: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/98.jpg)
Insert in Voltage current law equation
LC
V
LC
v
dt
dv
L
R
dt
vd CCC 2
2
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Three solution that possibly for vC
tstsfC eAeAVv 21
21
teB
teBVv
dt
dt
fC
sin
cos
2
1
ttfC eDetDVv 21
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Contoh 1 Tenaga awal yang disimpan oleh litar
berikut adalah sifar. Pada t = 0, satu punca arus DC 24mA diberikan kepada litar. Nilai untuk perintang adalah 400Ω.
1. Apakah nilai awal untuk iL?2. Apakah nilai awal untuk ?
3. Apakah punca-punca persamaan ciri?4. Apakah ungkapan numerik untuk iL(t)
pada t ≥ 0?
dt
diL
![Page 101: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/101.jpg)
![Page 102: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/102.jpg)
Jawapan1. Tiada tenaga yang disimpan
dalam litar sebaik sahaja punca arus digunakan, maka arus awal bagi induktor adalah sifar. Induktor mencegah perubahan yang serta-merta pada arus induktor, oleh itu iL (0)=0 sebaik sahaja suis dibuka.
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2. Nilai awal voltan kapasitor adalah sifar sebelum suis dibuka, oleh itu ia akan sifar sebaik sahaja suis dibuka. Didapati:
dt
diLv L maka 0
)0(
dt
diL
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3. Dari elemen-elemen dalam litar, diperolehi
812
20 1016
)25)(25(
101
LC
srad
RC
/105
)25)(400)(2(
10
2
1
4
9
82 1025
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Oleh kerana , maka punca-punca persamaan ciri adalah nyata
srad
s
srad
s
/00080
103105
/00020
103105
442
441
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4. sambutan arus induktor adalah overdamped dan persamaan penyelesaian adalah
tstsfL eAeAIi 21
21
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Dua persamaan serentak:
0)0(
0)0(
2211
21
AsAsdt
di
AAIi
L
fL
mAAmAA 832 21
![Page 108: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/108.jpg)
Penyelesaian numerik:
0
8
3224)(
80000
20000
tuntuk
mAe
eti
t
t
L
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Contoh 2
Tiada tenaga disimpan dalam inductor 100mH atau kapasitor 0.4µF apabila suis di dalam litar berikut ditutup. Dapatkan vC (t) untuk t ≥ 0.
![Page 110: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/110.jpg)
![Page 111: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/111.jpg)
JawapanPunca-punca persamaan ciri:
sradjs
sradj
s
/48001400
/48001400
4.01.0
10
2.0
280
2.0
280
2
62
1
![Page 112: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/112.jpg)
Punca-punca adalah kompleks, maka sambutan voltan adalah underdamped. Oleh itu, diperolehi voltan vC :
04800sin
4800cos48
14002
14001
tteB
teBv
t
tC
![Page 113: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/113.jpg)
Pada awalnya, tiada tenaga tersimpan dalam litar, maka:
12
1
140048000)0(
480)0(
BBdt
dv
Bv
C
C
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Selesaikan untuk dan 1B
2B
VB
VB
14
48
2
1
![Page 115: RL and RC circuits first- order response Electric circuits ENT 161/4.](https://reader038.fdocuments.net/reader038/viewer/2022102719/5697c0301a28abf838cda8d3/html5/thumbnails/115.jpg)
penyelesaian untuk vC (t)
0
4800sin14
4800cos4848)(
1400
1400
tuntuk
Vte
tetv
t
t
C