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Chapter 8Basic RL and RC Circuits

8.1 The Source-Free RL Circuit

8.2 Properties of the Exponential Response

8.3 The Source-Free RC Circuit

8.4 A More General Perspective

8.5 The Unit-Step Function

8.6 Driven RL Circuits

8.7 Natural and Forced Response

8.8 Driven RC Circuits

8.9 Predicting the Response of Sequentially Switched Circuits

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8.1 The Source-Free RL Circuit

• Natural response vs. Forced response

Natural Response : depends upon the general “nature” of circuit ( the type of elements, their sizes, the interconnection of the elements)Source-free response, transient response, complementary function

Forced Response : resemble the nature of the particular source ( or forcing function )Particular solution, steady-state response,

0 → 0

⇒ t⇒ ′

⇒ ln ′ ⇒ ln ln 0

⇒ ln ⇒

0

0

Energy is stored in the inductor at 0

Example 8.1

0 2 A20

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8.1 The Source-Free RL Circuit

• Alternative Approach

⇒ ln

0ln

→ ln ln

⇒ ln

⇒

0 → 0

→ 0 ⇒

Let

∴

⇒

0

• A More General Solution Approach

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8.1 The Source-Free RL Circuit

Consider 0

Assume

→ 1,

→ ,

→ ,⋮

→

0

→

⇒

• A Direct Route: The Characteristic Equation

Example 8.2 Find the voltage v at t=200ms

02410 2.4

→ 0 40 0 96

0t

10 5 0

10 40 5140 0

⇒ 10 0→ 10 0 ⇒ 10

40Li

Let

0 96⇒ 96

∴ 0.2 96 .

12.99

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8.1 The Source-Free RL Circuit

• Accounting for the Energy

212

Energy stored initially in the inductor

• There is no longer any energy stored in the inductor for at infinite time since its current eventually drops to zero

• All the initial energy is accounted for by dissipation in the resistor

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8.2 Properties of the Exponential Response

The initial rate of decay

Time constant

0.3679

⇒ 0.3679

→

1 →

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8.3 The Source-Free RC Circuit

0

→ 0

→1

0

⇒1

→ 0

Time constant

→

0

1

Example 8.3 Find the voltage v at t=200 μs

0t

0 90 9

⇒ 200 10 321.12 4 10 10 60

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8.4 A More General Perspective

• General RL Circuits

∥

,

Generally

Example 8.4 Determine both i1 and iL for t >0

0 ∶ 01850 360 ,

01890 200

1 2 ∥ 3 2.2

50 90 ∥ 120 60 110Ω

2.2 10110

20

0 0

0 0180

90 18023 360 240

0: 0 360

0 240

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8.4 A More General Perspective

• General RC Circuits

Example 8.5 Find v(0+) and i1(0+) if v(0-)=V0

0 0 0

0

⇒ 00

∥

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Example 8.6 Find vC(t) for t > 0 if vC(0-)=2 V

1.5 0.5

10 20

10

→ 0.5 10

→ 0.5 30 10

→1

60 10 0 → 1

60 10

→ 2 / 0 0 2

Another method for Example 8.6 Finding RTH on the left side of capacitor

1

1

1

301 1.5 30

30 45 30 1.560

x

x

x

x

Vi

V ii V

V V

601

xTH

VR

6 6

6

60 10 60 10

60 10

( ) 2 2 2t tt

C

RC s

t e e e V

8.4 A More General Perspective

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8.5 The Unit-Step Function

0

0

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8.6 Driven RL Circuits

0: 0

0:

0 0 0

→

→ ln

at 0 ∶ ln

ln ln → ⇒ , 0

∴ 1Example 8.7 Find i(t) for t =, 3-, 3+, and 3.0001 s.

For 12 ,

12 12

121000 12 ,

1 1050 10 20000

⇒ 12 12 33 3 0∞ 12 3.0001 10.38

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8.7 Natural and Forced Response

→

Consider general equation

forcing function

multiply : positiveconstant

integration

⇒

source-free case: 0

⇒ natural response

→ ∞, → 0

After the natural responsehas disappeared

: dc

complete response

forced response

,

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8.7 Natural and Forced Response

Example 8.8 Find i(t) for all values of time

2 ∥ 62 62 6 1.5

31.5 2

/ / , 0,

50 502 50

⇒ 50 / , 0

at 0 ∶ 502 50 ⇒ 25

Therefore,

50 25 , 025 , 0

25 25 1

2100 3 3 1 25 ( ) 50 0

2 6 2

tPt

di didi Qdt dt i i i t Ke Ke tdt P

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8.7 Natural and Forced Response

Example 8.9 Find i(t) for a rectangular voltage pulse forcing function.

1 , 0

1 ,

0, 0

1 , 0

1 1

1

0

2t

0t

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8.8 Driven RC Circuits

Example 8.10 Find vC(t) and i(t) for all time

0 050

10 50 120 100

60 ∥ 200 ∥ 50 24Ω → 1.2

20 / .

→ at 0: 0 20 100⇒ 80

/ / .

∞200 ∥ 50

60 200 ∥ 50 50 20

∴ 20 80 / . , 0

100, 0

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8.8 Driven RC Circuits

Example 8.10 Find vC(t) and i(t) for all time

0:

050

60 2000.1923

∞50

60 200 ∥ 50 50

200 50 0.1

0.1 / . 0

from 0 100

⇒ 0100200 0.5

0 0.1 0.5→ 0.4

0.1923 , 00.1 0.4 / . , 0

0.1923 0.1923 0.4 .

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8.8 Driven RC Circuits

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8.8 Driven RC Circuits

Example 8.11 Determine an expression for v(t) for t > 0

4.7 10 14.7Ω 14.7 22 10 323.4

→ 3.092 10 72.67 101

3.092 1072.67 10

→ 66.55

0

22 10 → 23.55 14.7 22 10

4.7200023.5 ( )te u t V

72.67 10

0 0 0 → 66.55

66.55

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8.9 Predicting the Response of Sequentially Switched Circuits

1

1k

1

s

• Case I : Time Enough to Fully Charge and Fully Discharge

/

∞ 9

9 1

0.01 9 18.9959 9

Capacitor has time to fully charge

8.9959 . 9 .

0 10 20 30

9

t

Vs

0

9 1 0 109 . 10 20

9 1 . 20 309 . 30 40

⋮

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8.9 Predicting the Response of Sequentially Switched Circuits

• Case II : Time Enough to Fully Charge but Not Fully Discharge

0 10 10.1 20.1

9

t

Vs

9 1

Capacitor has time to fully charge

.

9 .

9 0.856 .

10.1 9 8.144 → 0.856

→ 10.1 9 . . 8.144

. .

9 .

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8.9 Predicting the Response of Sequentially Switched Circuits

• Case III : No Time to Fully Charge but Time to Fully Discharge

0 10.10.1 10.2

9

t

Vs

. 9 1

→ 0.1 10 9 1 . 0.8565

. . .

0.1 10 0.8565 → 0.8565 .

• Case IV: No Time to Fully Charge or Even Fully Discharge

0 .2.1 .3

9

t

Vs. 9 1

. . 0.8565 .

. . 9 .

. . 1.558 .

8.225

0.2 10 0.7750

0.3 10 1.558

Homework : 8장 Exercises 7의 배수 문제

회로이론-І 2013년2학기 이문석

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