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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
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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
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(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
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/ /
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
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/
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
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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
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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
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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
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/
/
,
/
( )
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
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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
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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
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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
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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
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Example 7.15
Determine ሻ𝑣(𝑡 and ሻ𝑣0(𝑡 in the circuit
First Order Op-Amp Circuits
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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
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Applications – Photoflash
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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
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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
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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
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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
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Chapter 7
7.10, 7.23, 7.26, 7.43, 7.57, 7.71
Due Date :