Chapter 4 Transients
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Transcript of Chapter 4 Transients
![Page 1: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/1.jpg)
Chapter 4Transients
2008.9
Electrical Engineering and Electronics II Electrical Engineering and Electronics II
Scott
![Page 2: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/2.jpg)
•Main Contents
1. Solve first-order RC or RL circuits.
2. Understand the concepts of transient response and steady-state response.
3. Relate the transient response of first-order
circuits to the time constant.
4. Solve RLC circuits in dc steady-state
conditions.
![Page 3: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/3.jpg)
Introduction
Initial state and DC Steady State
First-order RC Circuits
First-order RL Circuits
Summary
•Main Contents
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t
E
Cu New steady statetransient
C
Old steady statesteady state
K R
E+
_ CuSwitch K is closed
4.1 Introduction
New steady steady statestate
R
Us+
_ Cu
Conception of steady state and transient state
When t=0 , uc(0)=0
When t=∞, uc(∞)=Us
Old steady state
![Page 5: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/5.jpg)
Why the transient response happens?
No transient
I
Resistance circuit
t = 0
E R
+
_
I
K
•Resistor is a energy-consumption element, current is
proportional to voltage, no transient response will happen
even if changing source
![Page 6: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/6.jpg)
Energy can not change instantly because of accumulating or decaying period.
CW Charging or discharging Cu Change
gradually
Electric field energy )( 2
2
1CCuWc
E
K R
+
_ CuC
E
t
Cu
![Page 7: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/7.jpg)
Magnetic field energy )( 2
2
1LL LiW
LWLi Change
gradually
K R
E+
_
t=0iL
t
LiE/R
Energy can not change instantly because of accumulating or decaying period.
![Page 8: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/8.jpg)
Transients
•The time-varying currents and voltages resulting from the sudden application of sources, usually due to switching.
•By writing circuit equations, we obtain integrodifferential equations.
![Page 9: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/9.jpg)
The causes of transients:
1. Energy storage elements -inductors and capacitors
change gradually;
2.Changing circuit, such as switching source.
LC iu ,
![Page 10: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/10.jpg)
4.2 Initial state and steady state
Assume changing circuit when t=0, then t=0– is end point of old steady state; t=0+ is the start point of transient state.
)0()0(
)0()0(
CC
LL
WW
WW
)0()0(
)0()0(
CC
LL
uu
iiFrom t=0–to t=0+,iL 、 uC
change continuously.
t=0tt=0
-
t=0+
The law of changing circuit
![Page 11: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/11.jpg)
DC Steady State Response
•The steps in determining the forced response or steady state response for RLC circuits with dc sources are:
1. Replace capacitances with open circuits.
2. Replace inductances with short circuits.
3. Solve the remaining circuit.
![Page 12: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/12.jpg)
Example 4.1 Find steady-state values of vx and ix in this circuit for t>>0.
Answer: vx =5V, ix = 1A t>>0
![Page 13: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/13.jpg)
Exercise 4.3 Find steady-state values of labeled currents and voltages for t>>0.
Answer: va =50V, ia = 2A
i1 = 2A, i2=1A, i3=1A
![Page 14: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/14.jpg)
How to get initial valueExercise 1: Assuming old circuit is in DC steady state
before switch K is closed. how to get uC(0+),iR(0+)?
iR
R14k
12V
K
t=08kR2
2FuC
Solution:
When t=0-, capacitor is considered as open circuit, we get equivalent circuit. R1
4k
12V uC(0–)8k
t=0-
8(0 ) 12 8V
4 8Cu
![Page 15: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/15.jpg)
R14k
12V uC(0–)8k
iR
R14k
12V
K
t=08kR2 2F
uC
8(0 ) 12 8V
4 8Cu
Vuu CC 8)0()0(
substituting voltage source for uC(0+)
iR(0+)
8kR2
+
– u
C(0+)
t=0 +2
(0 ) 8(0 ) 1m A
8C
R
ui
R
How to get initial value
![Page 16: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/16.jpg)
•Exercise 2: Given by Exercise 2: Given by RR11=4Ω, =4Ω, RR22=6Ω, =6Ω, RR33=3Ω, =3Ω, CC=0.1µF, =0.1µF,
LL=1mH, =1mH, UUSS=36V, switch S is closed for a long time. =36V, switch S is closed for a long time.
Open the switch S wOpen the switch S whenhen t=0, how to get the initial values t=0, how to get the initial values
of all elements?of all elements?
How to get initial value
![Page 17: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/17.jpg)
Equivalent circuit of First-order circuit
Two parts: one (equivalent) capacitor or inductor; a two terminal network with resistance and sources.
N L N Cor
First-order circuit
Only one (equivalent) capacitor or inductor is included in a linear circuit.
4.3 First-order RC Circuits
![Page 18: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/18.jpg)
According to Thevenin Law
N L N Cor
RU LuL
iL
+
-
RU CuC
iC
+
-
4.3 First-order RC Circuits
![Page 19: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/19.jpg)
Differential equation of first-order RC circuit
R
U LuL
iL
+
-
R
U CuC
iC
+
-
Uuu CR
Uudt
duRC C
C
Uuu LR
Udt
tdiLtRi L
L )(
)(
R
Uti
dt
tdi
R
LL
L )()(
![Page 20: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/20.jpg)
)0( tS
R
C
Ci
Cu
SU
2
1Ru
0)0( Cu
Solution:
0
Cu Ru i
0
t)(tf
0 0 0
0 SU
SU
R
U S
0 0
First-order RC Circuits•Example: to find the transient response after changing circuit when t=0.
![Page 21: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/21.jpg)
)0( tS
R
C
Ci
Cu
SU
2
1Ru
SCR Uuu
dt
duCiRiu C
R
SCC Uudt
duRC
0)0( Cu 0)0( Cu 0)0()0( CC uu
First-order RC Circuits
![Page 22: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/22.jpg)
)0( tS
R
C
Ci
Cu
SU
2
1Ru
"'CCC uuu
stC Aeu '
"Cu
——homogeneous solution
——particular solution
SCC Uudt
duRC
First-order RC Circuits
![Page 23: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/23.jpg)
homogeneous solution
)0( tS
R
C
Ci
Cu
SU
2
1Ru
SCC Uudt
duRC
01 RCs
RCs
1
tRC
C Aeu1
First-order RC Circuits
![Page 24: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/24.jpg)
)0( tS
R
C
Ci
Cu
SU
2
1Ru
SCC Uudt
duRC
Therefore
SCC Uuu )("
Then, the final solution is
Sst
CCC UAeuuu "'
Particular solution
First-order RC Circuits
![Page 25: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/25.jpg)
The solution of differential equation
Sst
CCC UAeuuu "'
Substituting the initial condition:
0)0( 0"' Ss
CCC UAeuuu
SCC UuuA )()0(
tRC
SS
tRC
CCCC
eUU
euuutu1
1
)]()0([)()(
First-order RC Circuits
![Page 26: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/26.jpg)
The solution of differential equation
RC ——Time constant
t
CCCC euuutu
)]()0([)()(
)(Cu
)0( Cu
——Steady state value
——Initial value
First-order RC Circuits
![Page 27: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/27.jpg)
Solution of other parameters
( ) ( ) (0 )
( ) [ (0 ) ( )]
t t
R S C S R
t
R R R
u t U u t U e u e
u u u e
( )
( ) (0 )
( ) [ (0 ) ( )]
t tSR
t
Uu ti t e i e
R R
i i i e
Three elements method
Three elements: 1.steady state value f(∞); 2.time constant τ; 3. initial value f(0+).
![Page 28: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/28.jpg)
Formula of Three element method :
t
effftf
)]()0([)()(
f(∞)——steady state value
τ——time constant
f(0+)——initial value
τ=RC ——time constant of RC circuitτ= ?? —— time constant of RL circuit
4.3 First-order RL Circuits
![Page 29: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/29.jpg)
4.3 First-order RL Circuits
![Page 30: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/30.jpg)
Time constant
τ=RC
τ=L/R
Uudt
duRC C
C
R
Uti
dt
tdi
R
LL
L )()(
RU LuL
iL
+
-
RU CuC
iC
+
-
4.3 First-order RL Circuits
![Page 31: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/31.jpg)
• Time constant reflects the length of transient period.
t 2 3 4 5 6 7
e-t/ 36.8% 13.5% 5% 1.8% 0.3% 0.25% 0.09%
•After about five time constants, the transient response is over.
•After one time constants, the transient response is equal to 36.8 percent of its initial value.
![Page 32: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/32.jpg)
The curves versus time
SU
)(tuC
)(tuR
R
U S
0t
)()( titu
)(ti
SU632.0
)0(368.0 RU
2
The initial slop intersects the final value at one time constant.
Mounting curve
Decaying curve
• Time constant reflects the length of transient period.
![Page 33: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/33.jpg)
•Three element method
Initial value: t=0-→t=0+ f(0+) Steady state value: t =∞ f(∞) Time constant : τ=RC τ=L/R Substituting three elements
Draw the curve versus time
t
effftf
)]()0([)()(
Steps
Limited Condition:
1) first-order circuit2) DC source
![Page 34: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/34.jpg)
•Example 4.2 Find voltage of v(t) and current i(t) in this circuit for t>0.
Answer:( ) 2 2 (A), ( ) 100 (V)
0.12(ms)
50
t t
i t e v t e
L
R
![Page 35: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/35.jpg)
( ) 2 2 (A), ( ) 100 (V)
0.12(ms)
50
t t
i t e v t e
L
R
![Page 36: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/36.jpg)
•Example 4.3 Find voltage of v(t) and current i(t) in this circuit for t>0.
Answer: 1 1
2
( ) , ( )t t
s SV LVi t e v t e
R R
L
R
![Page 37: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/37.jpg)
1 1
2
( ) , ( )t t
s SV LVi t e v t e
R R
L
R
![Page 38: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/38.jpg)
•Exercise 4.5 Find voltage of v(t) and current iR(t) , iL(t) in this circuit for t>0, assume that iL(0)=0.
Answer:
)(2.0
)(20)(),(22)(),(2)(
s
VetvAetiAetitt
L
t
R
![Page 39: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/39.jpg)
•Exercise 4.5 Find voltage of v(t) and current i(t), v(t) in this circuit for t>0, assume that the switch has been closed for a very long time prior to t=0.
Answer:
1, 0 1, 0( ) ( ) 5( )
0.5 0.5 , 0 100 , 0t t
t ti t v t ms
e t e t
![Page 40: Chapter 4 Transients](https://reader033.fdocuments.in/reader033/viewer/2022061500/568138a4550346895da05fc5/html5/thumbnails/40.jpg)
P4.8 P4.18 P4.26 P4.30
•Homework 4