chapter 9 the rlc circuits

회로이론-І 2013년2학기 이문석 1

Chapter 9The 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



1′ 0

→1 1

0

→ 1 1

0

Assume solution :

→ 1 1

0

12

12

1

12

12

1

Second-order homogeneous differential equation

⇒ 1 1

0

⇒ 1 1

0

⇒ 1 1

0

General form of the natural response

• Natural response of parallel RLC circuit

Real or Complex numbers



• Definition of Frequency Terms

12

12

1

12

1→

→

s : real numbers of complex conjugates

unit: : frequency

exponential dampingcoefficient

resonant frequency

complex frequency

→ 0 ⇒ : overdamped response

→ 0 ⇒ : critically damped response

→ 0 ⇒ : underdamped response



Example 9.1 Determine the resistor value for overdamped and underdamped responses.

10 100

12

1

1 110 10 100 10 10 /

overdamped case : → 1

2 10 ⇒1

2 10 100 10 5Ω

underdamped case : 5Ω


9.2 Overdamped Parallel RLC Circuit

⇒ 1

21

⇒ 4

⇒ 0→ →

→ 0

• Overdamped parallel RLC circuit

, : negative real number

12

1

2 6 142

3.5 ,

1 1

7 142

6 ,

1, 6

0 0

6 ⇒ 6

0 0 , 0 10

→0 0 0

10142

420 /

06 420

⇒ 84, 84

∴ 84

The constants A1 and A2 are determined by the initial conditions.



• Natural response of Overdamped parallel RLC circuit



Example 9.2 Find an expression for vC(t) for t >0.

0

⇒

12

12 200 20 10 125,000

1 15 10 20 10

100,000 /

: overdampedresponse

50,000 ,

200,000 ,



Example 9.2 continued…

0150

300 200 0.3

⇒ 0200

300 200 150 60

⇒ 0 60

→ 0 0 0

0

200 0.320 10

60200 0.320 10 0

0

∴ 50000 200000 0

⇒ 80, 20∴ 80 20 , 0



Example 9.3 Express iR for all time

12

12 30 10 2 10 8.33 10

1 112 10 2 10

6.45 10 /

: overdampedresponse

3.063 10 ,

13.60 10 ,

Let 0

0

0 04

32 10 125 10 → 0 30 10 125 10 3.75

⇒ 0 125 10 ∴ 0 0

→ 0 0 0

0∴ 3.063 10 13.60 10 0

30

125 10 , 0161.3 10 .

36.34 10 . ,0



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

2 4 0.02

5.296

Settling time : 10% of maximum value.



0 0 , 0 10

7 62 Ω

1

27 62

142

1

7 142

61

7 142

6

6

??

0 0 0 → 0 6

→

0 0 0 10142

∴ 420

420

(8.57 Ω)



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

→ ∥25 0.4Ω

At 0: soureoff, isshorted1

21

2 10

1 14 10

15,810 /

For critical damping case

→1

2 10 15810 31.63 Ω

0



,

imaginary1

+

+ cos sin cos sin

cos sin cos sin

0 0 , 0 101

2 10.5 142

21

7 142

6 /

2 / cos 2 sin 2

0 0

→

2 cos 2 2 sin 2 20 10

142

420

∴ 210 2 → 210 2 sin 2

→ sin 2



• Graphical Representation of Critically Damped Response

210 2 sin 2

decreasing nature oscillating nature

2 → 0,

2→ 0 Damping

Oscillation!!

• The Role of Finite Resistance



Example 9.6 Determine iL(t) and plot the waveform.

12 1.2

14.899 /

→ underdamped

4.750 /

. cos 4.75 sin 4.75 0

0

0

0100

48 1003 2.027

0 48 ∥ 100 348 10048 1003 97.297

0 0 2.027

00

100

→0 0

10010

97.29710 9.73



. 4.75 sin 4.75 4.75 cos 4.75 1.2 . cos 4.75 sin 4.75

4.75 1.2 4.75 1.2 2.027 9.73

⇒9.73 1.2 2.027

4.75 2.5605

. 2.027 cos 4.75 2.5605 sin 4.75

. cos 4.75 sin 4.75 0


9.5 Source-Free Series RLC Circuit

1′ 0

→1

0

1′ 0

→1 1

0

Series RLC Circuit

Parallel RLC Circuit

Let 1

0 → 1

0

2 21 2 2

12 ,

1



Example 9.7 Determine i(t)

1

1401

2 Ω

0 2 , 0 2

2 1000 1

1 1401 10

20025 /

→ underdamped

20000 /

cos 20000 sin 20000 → 0 2 10

20000 1000

20000 2 2 ⇒ 0

20000 0

0 → 2000

⇒0 2000 0

1 2

2 10 cos 20000 t 0



Example 9.8 Determine vC(t) for t > 0. At 0

0 0 ,

0102 5 ,

10 3 0 0→ 0 10 3 5 5

5,2 10 ,1

11 1 31 8Ω

10 ⇒ 5 8

12 10 0

28

2 5 0.8s ,

ω1 1

5 2 1010 /

→ underdamped

9.968 /

. cos 9.968 sin 9.968

0 0 5

→ ⇒ 0

0 0

2 10 0



. 9.968 sin 9.968 9.968 cos 9.968 0.8 . cos 9.968 sin 9.968

9.968 0.8 9.968 4 0 ⇒4

9.968 0.4013

. 5 cos 9.968 0.4013 sin 9.968



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.

• General Solution

Forced response

Natural response

0

cos sin



Example 9.9 Find the values of these six quantities at both t=0- and t=0+.

0 0, 0 0fordccurrentsoruce

0 5 , 0 30 5 150 ,

0 0 150 ,

0 0 5

0 0 5 , 0 0 150

0 4 0 4 5 1 ,

0 30 0 30 1 30 ,

0 5 0 5 1 4 ,

0 0 0 30 150 120



Example 9.10 Find the determination of the initial conditions

→ 0 120

3 40,

→ 0 4

1/27 108,

4 → 40,

30 → 30 1200,

→ 1200 108 1092,

5 → 40



2302 3 5s ,

ω1 1

3 127

3 /

→ overdamped

, 5 4

→ 1, 9Forced response

∞ 150

Complete response

150

0 150 150→ 0

9

⇒ 94127

108

13.5, 13.5

Find vC(t) for t > 0.

150 13.5 0


9.7 The Lossless LC Circuit

Homework : 9장 Exercises 6의 배수 문제

• 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.

→ underdamped

3

cos 3 sin 3 ,

0 0

0 0,

016

→0

16136

6

3 cos 3 → 3 6⇒ 2

2 sin 3 0 Does not decay

2 0s , ω1

4 136

3 /

14 0

136 0 →

14

136 0 ⇒ 9 0 ⇒ sin 3

General solution

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chapter 9 the rlc circuits

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