Chapter 9
The RLC Circuits
9.1 The Source-Free Parallel Circuit
9.2 The Overdamped Parallel RLC Circuit
9.3 Critical Damping
9.4 The Underdamped Parallel RLC Circuit
9.5 The Source-Free Series RLC Circuit
9.6 The Complete Response of the RLC Circuit
9.7 The Lossless LC Circuit
회로이론-І 2013년2학기 이문석
1
9.1 The Source-Free Parallel Circuit
•
Natural response of parallel RLC circuit
1
1
1
1
2
1
1
0
1
→
1
⇒ 1
⇒ 1
0
1
1
1
1
Second-order
homogeneous
differential
equation
0
2
1
2
0
Assume solution :
0
1
→
→
′
1
0
2
Real or Complex numbers
⇒ 1
1
0
General form of the natural response
회로이론-І 2013년2학기 이문석
2
9.1 The Source-Free Parallel Circuit
•
Definition of Frequency Terms
1
2
s : real numbers of complex conjugates
1
1
unit:
2
1
1
→
→
2
exponential
damping
coefficient
resonant
frequency
complex frequency
overdamped response
→
0 ⇒ :
→
0⇒ :
→
0⇒ :
회로이론-І 2013년2학기 이문석
: frequency
critically damped response
underdamped response
3
9.1 The Source-Free Parallel Circuit
Example 9.1 Determine the resistor value for overdamped and underdamped responses.
1
1
2
10
100
1
10
overdamped case :
underdamped case :
회로이론-І 2013년2학기 이문석
→ 1
2
10 ⇒
10
1
100
2
10
10
1
100
10 10
/
5Ω
5Ω
4
9.2 Overdamped Parallel RLC Circuit
•
Overdamped parallel RLC circuit
⇒ 1
1
2
⇒ 4
: negative real number
,
0
⇒
→
→
→ 0
The constants A1 and A2 are determined by the initial conditions.
0
0 , 0
10
0
→
0
1
1
2
2
1
6
1
7
1,
6
1
42
1
42
회로이론-І 2013년2학기 이문석
3.5
6
0
,
6
,
0
6
420
10
1
42
⇒ 0
0
420 /
6
⇒
84,
∴
84
84
5
9.2 Overdamped Parallel RLC Circuit
•
Natural response of Overdamped parallel RLC circuit
회로이론-І 2013년2학기 이문석
6
9.2 Overdamped Parallel RLC Circuit
Example 9.2 Find an expression for vC(t) for t >0.
1
2
1
2
1
200 20 10
1
5
10
20
125,000
100,000
10
/
: overdampedresponse
50,000
200,000
,
,
0
⇒ 회로이론-І 2013년2학기 이문석
7
9.2 Overdamped Parallel RLC Circuit
Example 9.2 continued…
0
150
0.3
300 200
200
150 60
300 200
0
⇒
0
⇒
0
→
60
0
0
200
20 10
∴
⇒
∴
회로이론-І 2013년2학기 이문석
0
80,
80
0.3
0
60
0.3
200
20 10
200000
50000
0
0
20
20
,
0
8
9.2 Overdamped Parallel RLC Circuit
Example 9.3 Express iR for all time
1
2
1
2
1
30
10
1
2
10
12 10
2 10
: overdampedresponse
10 6.45
10 3.063
10 ,
13.60
10 ,
/
Let
0
0
8.33
4
0
32
10
⇒ 125
10 → 0
125
→
30
0
0
∴
3.063
10
10
30
10
∴
0
10
13.60
10
3.75
0
0
125 10 ,
.
161.3 10
.
36.34 10
회로이론-І 2013년2학기 이문석
125
0
0
0
0
0
,
0
9
9.2 Overdamped Parallel RLC Circuit
Example 9.4 For t>0, iC(t) =2e-2t-4e-t A. Sketch the current 0<t<5s and find settling time.
2
4
Maximumcurrent:
0
2
Settling time : 10% of maximum value.
2
4
0.02
5.296
회로이론-І 2013년2학기 이문석
10
9.3 Critical Damping
0
0 , 0
1
10
2
7 6 1
2 42
1
7
7 6
Ω
2
7
1
42
6
6
?
(8.57 Ω)
?
0
0
0→
0
0
0
6
→ ∴
1
42
6
1
0
10
1
42
420
420
회로이론-І 2013년2학기 이문석
11
9.3 Critical Damping
Example 9.5 Find R1 such that the circuit is critically damped for t>0 and R2 so that v(0)=2V
At
0
0
5
∥
2
5
2
→
∥
At
0: soureoff, 1
2
2
0.4Ω
isshorted
1
10
1
1
4
0
10
15,810
/
For critical damping case
→
회로이론-І 2013년2학기 이문석
1
2
10
15810
31.63 Ω
12
9.4 The Underdamped Parallel RLC Circuit
,
imaginary
1
+
+
cos
sin
cos
sin
cos
sin
cos
1
0
0 , 0
10
2
10.5
1
42
→ ∴
210 2
회로이론-І 2013년2학기 이문석
2
0
cos 2
→
1
2
7
2
0
sin
/
210 2
1
42
/
cos 2
→ 2
6
sin 2
sin 2
sin 2
2
0
10
1
42
420
sin 2
13
9.4 The Underdamped Parallel RLC Circuit
•
Graphical Representation of Critically Damped Response
210 2
sin 2
decreasing nature
2
oscillating nature
→ 2
→
0,
0
Damping
Oscillation!!
•
회로이론-І 2013년2학기 이문석
The Role of Finite Resistance
14
9.4 The Underdamped Parallel RLC Circuit
Example 9.6 Determine iL(t) and plot the waveform.
1
1
1.2
2
4.899
/
→ underdamped
4.750
.
100
3
48 100
0
0
48 ∥ 100
0
0
3
48
48
100
3
100
0
97.297
2.027
0
0
sin 4.75
2.027
0
0
→
cos 4.75
/
0
10
0
10
회로이론-І 2013년2학기 이문석
0
0
10
97.297
10
9.73
15
9.4 The Underdamped Parallel RLC Circuit
.
.
cos 4.75
4.75
4.75
⇒
9.73
.
sin 4.75
sin 4.75
1.2
4.75
1.2 2.027
4.75
2.027 cos 4.75
회로이론-І 2013년2학기 이문석
4.75
0
cos 4.75
1.2
1.2
2.027
.
cos 4.75
sin 4.75
9.73
2.5605
2.5605 sin 4.75
16
9.5 Source-Free Series RLC Circuit
1
′
1
→
0
0
Series RLC Circuit
1
′
0
1
→
1
0
Parallel RLC Circuit
1
Let
1
2
2
회로이론-І 2013년2학기 이문석
0
1
→
1
2
2
2
,
0
1
17
9.5 Source-Free Series RLC Circuit
회로이론-І 2013년2학기 이문석
18
9.5 Source-Free Series RLC Circuit
Example 9.7 Determine i(t)
0
,
1000
2
1
401
2 Ω
2
0
2
1
sin 20000
0
20000
2
→ 0
20025
/
10
/
2
10 2000
0 → 2
1
401
1
cos 20000
20000
1
20000
⇒
→ underdamped
2000
1
0
2
1000
2 ⇒ 10
회로이론-І 2013년2학기 이문석
0
20000
cos 20000 t
0
0
19
9.5 Source-Free Series RLC Circuit
Example 9.8 Determine vC(t) for t > 0.
At
0
0
0 ,
10
5 ,
2
3 0
0
0
10 3 5
0
10
→
11
1
1
8
2
ω
1
2
5
5
0.8s
1
2
3
10
/
5,
2 10 ,
8Ω
8
⇒ 5
0
.
,
10
1
1
0
1
10
2
cos 9.968
0
5
sin 9.968
5
→ underdamped
→
9.968
회로이론-І 2013년2학기 이문석
/
0
0
⇒ 0
2
0
10
0
20
9.5 Source-Free Series RLC Circuit
.
9.968
9.968
.
sin 9.968
0.8
5 cos 9.968
회로이론-І 2013년2학기 이문석
9.968
9.968
cos 9.968
4
0
0.8
⇒ .
4
9.968
cos 9.968
sin 9.968
0.4013
0.4013 sin 9.968
21
9.6 The Complete Response of the RLC Circuit
•
General Solution
1. Determine the initial conditions.
2. Obtain a numerical value for the forced response.
3. Write the appropriate form of the natural response with the necessary
number of arbitrary constants.
4. Add the forced response and natural response to form the complete
response.
5. Evaluate the response and its derivative at t = 0, and employ the initial
conditions to solve for the values of the unknown constants.
Forced response
Natural response
cos
sin
0
회로이론-І 2013년2학기 이문석
22
9.6 The Complete Response of the RLC Circuit
Example 9.9 Find the values of these six quantities at both t=0- and t=0+.
0
0,
0
0
5 ,
0
0
0
4
0
30
0
5
회로이론-І 2013년2학기 이문석
30
150 ,
5
5 ,
0
4
5
0
0
0
0
5
150 ,
0
0
0
0
0
0
0fordccurrentsoruce
30
5
0
1
0
150
1 ,
1
30 ,
4 ,
30
150
120
23
9.6 The Complete Response of the RLC Circuit
Example 9.10 Find the determination of the initial conditions
0
→ → 4
30
0
→ →
120
3
4
1/27
40,
108,
40,
30
1200,
→ 5
→ 회로이론-І 2013년2학기 이문석
1200
108
1092,
40
24
9.6 The Complete Response of the RLC Circuit
30
Find vC(t) for t > 0.
2
ω
1
2
3
3
5s
1
1
27
,
3
/
→ overdamped
,
Forced response
∞
1,
→
150
5
4
9
Complete response
150
0
→
9
150
0
13.5,
회로이론-І 2013년2학기 이문석
150
13.5
⇒
9
150
13.5
4
1
27
108
0
25
9.7 The Lossless LC Circuit
•
•
The resistor in the RLC circuit serves to dissipate initial stored energy.
When this resistor becomes 0 in the series RLC or infinite in the parallel RLC,
the circuit will oscillate.
2
0
0,
1
6
0
cos 3
0
1
4
1
, ω
4
1
36
3
/
→ underdamped
3
sin 3 ,
0
3
6
1
6
1
36
0
→
3 cos 3 →
⇒
0s
2 sin 3 0
6
Does not decay
2
0
1
36
0→
1
4
1
36
0 ⇒ 9
0 ⇒
sin 3
General solution
회로이론-І 2013년2학기 이문석
Homework : 9장 Exercises 6의 배수 문제
26
이 자료에 나오는 모든 그림은 McGraw·hill 출판사로부터 제공받은 그림을 사
용하였음.
All figures at this slide file are provided from The McGraw-hill company.
회로이론-І 2013년2학기 이문석