Ch.7 First Order CircuitsPart 2 – 1차회로 Step Response

Step Response of an RC Circuit

3

• Step response : the behavior of a circuit when the

voltage or current excitation is the step function

1) Initially charged to 𝑉0

Pusan National University

0(0 ) (0 )v v V

0

0

0( )

0 ( )

( ) 0

st

RCs s

V tV u t vdv

C v tdt R V V V e t

Step Response of an RC Circuit

4

2) Initially charged to 0

Pusan National University

(0 ) (0 ) 0v v

0 0( )

0 ( )

(1 ) 0

( ) (1 ) ( )

( ) ( ) ( )

st

RCs

t

RCs

t t

sRC RCs

tV u t vdv

C v tdt R V e t

v t V e u t

Vdv Ci t C V e u t e u t

dt RC R

Step Response of an RC Circuit

5

• Step response의 성분

Pusan National University

/ /

0 where , (1 )t t

n f n f sv v v v V e v V e

Stored energy External force

0

0

0

( )

( ) 0t

RCs s

V t

v t

V V V e t

Step Response of an RC Circuit

6

• Step response의 성분

Pusan National University

/

0 where ( ) ,t

t ss t s ss sv v v v V V e v V

Temporary part(Transient)

Permanent part(Steady-state)

0

0

0

( )

( ) 0t

RCs s

V t

v t

V V V e t

Step Response of an RC Circuit

7

• 성분별 분석을 통한 Complete Step response의 계산

Pusan National University

0

0

/

( )/

0

( ) ( )

( ) ( ) (0) ( )

or ( ) ( ) ( ) ( )

t

RCs s

t

t t

v t V V V e

v t v v v e

v t v v t v e

Step Response of an RC Circuit

8Pusan National University

Example 7.10

The switch in the following figure has been in position A for a long time. At t=0, the switch moves to B. Determine ሻ𝑣(𝑡 for 𝑡 > 0 and calculate its value at 𝑡 = 1sec and 4sec.

Step Response of an RC Circuit

9Pusan National University

Example 7.11

In the following figure, the switch has been closed for a long time and is opened at t=0. Find 𝑖 and 𝑣 for all time.

Step Response of an RL Circuit

10

• RL회로의 Step response의 성분

Pusan National University

/

/

,

/

( )

t

t

ss s

t s

Li Ae

R

i V R

Vi t Ae

R

( ) t ssi t i i

0

0

/

0

(0)

( )

s

s

ts s

Vi I A

R

VA I

R

V Vi t I e

R R

Step Response of an RL Circuit

11

• 성분별 분석을 통한 Complete Step response의 계산

Pusan National University

0

/

0

/

( )/

0

( )

( ) ( ) (0) ( )

or ( ) ( ) ( ) ( )

ts s

t

t t

V Vi t I e

R R

i t i i i e

i t i i t i e

Step Response of an RL Circuit

12Pusan National University

Example 7.12

Find ሻ𝑖(𝑡 in the following circuit for 𝑡 > 0. Assume that the switch has been closed for a long time.

Step Response of an RL Circuit

13Pusan National University

Example 7.13

At 𝑡 = 0, switch 1 in the figure is closed, and switch 2 is closed 4 sec later. Find ሻ𝑖(𝑡 for 𝑡 > 0. Calculate 𝑖 for 𝑡 = 2s and 𝑡 = 5s.

Step Response of an RL Circuit

14Pusan National University

Example 7.14

For the following op amp circuit, find 𝑣0 for 𝑡 > 0, given that 𝑣(0ሻ =3V. Let 𝑅𝑓 = 80𝑘𝛺, 𝑅1 = 20𝑘𝛺 and 𝐶 = 5𝜇𝐹.

First Order Op-Amp Circuits

15Pusan National University

Example 7.15

Determine ሻ𝑣(𝑡 and ሻ𝑣0(𝑡 in the circuit

First Order Op-Amp Circuits

16Pusan National University

Example 7.16

Find the step response ሻ𝑣0(𝑡 for 𝑡 > 0 in the following op amp circuit. Let ሻ𝑣𝑖 = 2𝑢(𝑡 V, 𝑅1 = 20𝑘𝛺, 𝑅𝑓 = 50𝑘𝛺, 𝑅2 = 𝑅3 = 10𝑘𝛺 and

𝐶 = 2𝜇𝐹.

Example 7.19 (RC 시정수를 이용한 Delay Circuit)

In the following circuit, 𝑅1 = 1.5𝑀𝛺, 0 < 𝑅2 < 2.5𝑀𝛺. (a) Calculate the extreme limits of the time constant of the circuit. (b) How long does it take for the lamp to glow for the first time after the switch is closed when 𝑅2 is set at the largest value.

Applications – Delay Circuits

17Pusan National University

Applications – Photoflash

18Pusan National University

Example 7.20

An electronic flashgun has a current-limiting 6𝑘𝛺 resistor and 2000𝜇𝐹 electrolytic capacitor charged to 240V. If the lamp resistance is 12𝛺, find:

(a) the peak charging current,

(b) the time required for the capacitor to fully charge,

(c) the peak discharging current,

(d) the total energy stored in the capacitor

(e) the average power dissipated by the lamp

Applications – Relay

19Pusan National University

Example 7.21 (Relay : magnetically controlled switch)

The coil of a certain relay is operated by a 12V battery. If the coil has a resistance of 150𝛺 and an inductance of 30mH and the current needed to pull in is 50mA, Calculate the relay delay time.

Applications – Ignition Circuit

20Pusan National University

chargeSW: On 5 ,and

SS: 0

SW: Off large current high voltage induced

L dit v L

R dt

div L

dt

Applications – Ignition Circuit

21Pusan National University

Example 7.22

A solenoid with resistance 4𝛺 and inductance 6mH is used in an automobile ignition circuit. If the battery supplies 12V, determine: the final current through the solenoid when the switch is closed, the energy stored in the coil, and the voltage across the air gap, assuming that the switch takes 1𝜇𝑠 to open.

Assignment

22Pusan National University

Chapter 7

7.10, 7.23, 7.26, 7.43, 7.57, 7.71

Due Date :