Bài giảng Electric circuit theory - Chapter VII: First-Order Circuits - Nguyễn Công Phương
1. Write the general form
2. Find the initial condition
3. Find the forced response
4. Deactivate source(s), find the
natural response (with the
unknow integration
constant)
5. Find the integration constant
6. Write the complete response
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Electric Circuit Theory
First-Order Circuits
Nguyễn Công Phương
First-Order Circuits - sites.google.com/site/ncpdhbkhn 2
Contents
I. Basic Elements Of Electrical Circuits
II. Basic Laws
III. Electrical Circuit Analysis
IV. Circuit Theorems
V. Active Circuits
VI. Capacitor And Inductor
VII. First-Order Circuits
VIII.Second Order Circuits
IX. Sinusoidal Steady State Analysis
X. AC Power Analysis
XI. Three-phase Circuits
XII. Magnetically Coupled Circuits
XIII.Frequency Response
XIV.The Laplace Transform
XV. Two-port Networks
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 3
Introduction to Transient Analysis (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 4
+–
10 V 5 Ω
0.1 H
+–
t = 010 V 5 Ω
0.1 Hi
t
i
0
+–
10 V 5 Ω
0.1 mF
+–
t = 012 V 6 Ω v
0.1 mF
+
–
t
v
0
Steady-state Transient-state
Transient-stateSteady-state
Steady-state
Steady-state
Any change in an
electrical circuit,
which brings about
a change in energy
distribution,
will result in a
transient-state.
2 A
12 V
Introduction to Transient Analysis (2)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 5
Inductors in
DC circuits
Capacitors in
DC circuits
vL(t)
t0
Short-circuit Short-circuitNot short-circuit
iC(t)
t0
Open-circuit Open-circuitNot open-circuit
Old steady-state New steady-stateTransient
-state
Old steady-state New steady-stateTransient
-state
Introduction to Transient Analysis (3)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 6
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 7
Initial Conditions (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 8
10 V
5 Ω
0.1 Hi
t = 0
+–
t
i
0
Steady-state/Initial condition 1
2 A
10 V
5 Ω
0.1 Hi
t = 0
+–
+–
20 V
A
Steady-state/Initial condition 2 4
0–
Prior to switching
0+
After switching
Initial Conditions (2)
• 1st switching rule/law: the current (magnetic flux) in an
inductor just after switching is equal to the current (flux) in the
same inductor just prior to switching
iL(0+) = iL(0–)
λ(0+) = λ(0–)
• 2nd switching rule/law: the voltage (electric charge) in a
capacitor just after switching is equal to the voltage (electric
charge) in the same capacitor just prior to switching
vC(0+) = vC(0–)
q(0+) = q(0–)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 9
Initial Conditions (3)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 10
Ex. 1
The switch has been at A for a long time,
and it moves to B at t = 0; find I0?
(0 ) (0 )i i+ −=
(0 ) 0 Ai − = (0 ) 0Ai +→ = 0 0AI→ =
Ex. 2
The switch has been at A for a long time,
and it moves to B at t = 0; find I0?
(0 ) (0 )i i+ −=
20(0 ) 4 A
5
i − = =
(0 ) 4 Ai +→ = 0 4AI→ =
10 V
5 Ω
0.1 Hi
t = 0
+–
A
B
10 V
5 Ω
0.1 Hi
t = 0
+–
+–
20 V
A
B
Initial Conditions (4)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 11
Ex. 3
The switch has been at A for a long time,
and it moves to B at t = 0; find V0?
(0 ) (0 )v v+ −=
(0 ) 0 Vv − = (0 ) 0 Vv +→ = 0 0VV→ =
Ex. 4
The switch has been at A for a long time,
and it moves to B at t = 0; find V0?
(0 ) (0 )v v+ −=
(0 ) 20 Vv − =
(0 ) 20Vv +→ = 0 20VV→ =
10 V
5 Ω
0.1 mF
t = 0
+–
V0
+
–
A
B
10 V
5 Ω
t = 0
+–
+–
20 V
0.1 mF
V0
+
–
A
B
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 12
The Source-free RC Circuit (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 13
00(0) tv v V== =
0dv vC
dt R
→ + =
0R Ci i+ =
0dv v
dt RC
→ + =
1dv dt
v RC
→ = −
ln lntv A
RC
→ = − +
ln v t
A RC
→ = −
( )
t
RCv t Ae
−
→ =
00(0) tv v V== =
0
0
( )
t
RC
t
v t V e
V e τ
−
−
→ =
=
tτ
v
V0
0.368V0
0
R
t = 0
+–
E
C
R Cv
iR iC+
–
The Source-free RC Circuit (2)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 14
Ex. 1
R1 = 6 Ω; R2 = 12 Ω; vC(0) = 10 V;
C = 0.01F; find vC ? C
R1 R2vC
+
–
C
R12 vC
+
–
12
6 12 4
6 12
R ×= = Ω
+
12 4 0.01 0.04sR Cτ = = × =
250.04(0) 10 10 V
t t
t
C Cv v e e e
τ
− −
−
= = =
The Source-free RC Circuit (3)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 15
Ex. 2
E = 24 V; R1 = 8 Ω; R2 = 12 Ω; C = 0.01F;
the switch has been closed for a long time,
and it is opened at t = 0; find vC for t ≥ 0? C
R1
R2
vC
+
–
t = 0
+
–
E
R1
R2
V0
+
–
+
–
E
t < 0
R1
R2
V0 = 24 V
+
–
t > 0
0 24VV E= = 1 2( ) (8 12) 0.01 0.2sR R Cτ = + = + × =
50.2
0 24 24 V
t t
t
Cv V e e eτ
− −
−
= = =
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 16
The Source-free RL Circuit (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 17
00(0) ti i I== =
0R Lv v+ =
0diRi L
dt
→ + =
di R dt
i L
→ = −
ln lnRi t A
L
→ = − +
ln i R t
A L
→ = −
( )
R
t
Li t Ae
−
→ =
00(0) ti i I== = 0
0
( )
R
t
L
t
i t I e
I e τ
−
−
→ =
= tτ
i
I0
0.368I0
0
R
t = 0
+–
E
L R L
i
vR vL
+
– +
–
The Source-free RL Circuit (2)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 18
Ex.
E = 24 V; R1 = 5 Ω; R2 = 4 Ω; R3 = 12 Ω;
L = 0.01H; the switch has been closed for
a long time, and it is opened at t = 0;
find iL for t ≥ 0?
+
–
t = 0
E
R1
R2
R3
L
iL
+
–
E
R1
R2
R3
I0
t < 0
4 125 8
4 12eq
R ×= + = Ω
+ 1
24 3A
8
i→ = =
2
0 1
2 3
43 0.75A
4 12
RI i
R R
→ = = =
+ +
R2
R3
L
I0 = 0.75 A
t > 0
23
0.01 0.000625s
4 12
L
R
τ = = =
+
1600
0( ) 0.75 A
t
t
Li t I e eτ
−
−
= =
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 19
Step Response of an RC Circuit (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 20
E2
R
t = 0
+–
+–
E1
C
v
+
–
i
0 1(0 ) (0 )V v v E+ −= = =
2Ri v E+ =
2
dvRC v E
dt
→ + = 2
Edv v
dt RC RC
→ + =
2v Edv
dt RC
−
→ = −
2
dv dt
v E RC
→ = −
−
0
( )
2
0
ln( )
t
v t
V
t
v E
RC
→ − = − 2
0 2
ln v E t
V E RC
−
→ = −
−
2
0 2
t
RCv E e
V E
−
−
→ =
−
2 0 2 2 1 2( ) ( ) ( ) , 0
t t
RCv t E V E e E E E e tτ
− −
→ = + − = + − >
Step Response of an RC Circuit (2)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 21
E2
R
t = 0
+–
+–
E1
C
v
+
–
i
2 0 2
2 1 2
( ) ( )
( ) , 0
t
RC
t
v t E V E e
E E E e tτ
−
−
= + −
= + − >
t
v
E2
0
V0 = E1
t
v
E2
0 t
v
E2
0
V0 = E1
Forced response/steady-state response
Natural response/transient-state response
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 22
Step Response of an RL Circuit (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 23
1
0 (0 ) (0 )
EI i i
R
+ −
= = =
2
diRi L E
dt
+ = 2
E Ridi
dt L
−
→ =
2
di dt
E Ri L
→ =
− 2
di R dtE Li
R
→ =
−
0
2
0
ln
t t
I
E Ri t
R L
→ − = −
2
2
0
ln
E i RR tE LI
R
−
→ = −
−
2
2
0
R
t
L
E i
R eE I
R
−
−
→ =
−
2 2 2 1 2
0( ) , 0
tR
t
LE E E E Ei t I e e t
R R R R R
τ
−−
→ = + − = + − >
E2
R
L
i
t = 0
+–
+–
E1
Step Response of an RL Circuit (2)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 24
2 2
0
2 1 2
( )
, 0
R
t
L
t
E Ei t I e
R R
E E E
e t
R R R
τ
−
−
= + −
= + − >
Forced response/steady-state response
Natural response/transient-state response
E2
R
L
i
t = 0
+–
+–
E1
t
i
0
1
0
EI
R
=
2E
R
t
i
0
2E
R
t
v
0
1
0
EI
R
=
2E
R
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 25
The Classical Method (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 26
E2
R
L
i
t = 0
+–
+–
E12 1 2( )
R
t
LE E Ei t e
R R R
−
= + −
t
i
0
1
0
EI
R
=
2E
R
E2
R
L
i
t = 0
+–
+–
E1
E2
R
if
+–
R
L
in
2 1 2( )
R
t
LE E Ei t e
R R R
−
= + −
1 2
R
t
LE E e
R R
−
−
2E
R
The Classical Method (2)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 27
E2
R
L
i
t = 0
+–
+–
E1
E2
R
if
+– R L
in
2
f
Ei
R
→ =
2 0fRi E− =
' 0n nRi Li+ =
st
ni Ae=
0st stRAe LAse→ + =
( ) 0stR Ls Ae→ + =
0R Ls→ + =
R
s
L
→ = −
R
t
L
n
i Ae
−
→ =
2 1 2( )
R
t
LE E Ei t e
R R R
−
= + −
1
0 (0)
EI i
R
= =
2
R
t
LEi Ae
R
−
= +
02 2 1(0)
R
LE E Ei Ae A
R R R
−
= + = + =
1 2E EA
R R
= −
R
L
i(0)
+–
E1
The Classical Method (3)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 28
2fv E=
( )' 0n nv R Cv+ =
st
n
v Ae=
0st stAe RCAse→ + =
(1 ) 0stRCs Ae→ + =
1 0RCs→ + =
1
s
RC
→ = −
1
t
RC
n
v Ae
−
→ =
1
2 1 2( ) ( )
t
RCv t E E E e
−
= + −
0 1(0)V v E= =
1
2
t
RCv E Ae
−
= +
1 0
2 2 1(0) RCv E Ae E A E
−
= + = + =
1 2A E E= −
E2
R Ct = 0
+–
+–
E1
v
+
–
E2
R+–
vf
+
–
R
C vn
+
–
R C
+–
E1
v(0)
+
–
The Classical Method (4)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 29
E2
R
L
i
t = 0
+–
+–
E1
E2
R Ct = 0
+–
+–
E1
v
+
–
0 12. /I E R=
1. f nv v v= +
23. /fi E R=
4.
R
t
L
n
i Ae
−
=
1. f ni i i= +
2
05.
EI A A
R
= + →
2 1 26.
R
t
LE E Ei e
R R R
−
= + −
0 12. V E=
23. fv E=
1
4.
t
RC
n
v Ae
−
=
0 25. V E A A= + →
1
2 1 26. ( )
t
RCv E E E e
−
= + −
1. Write the general form
2. Find the initial
condition
3. Find the forced
response
4. Deactivate source(s),
find the natural
response (with the
unknow integration
constant)
5. Find the integration
constant
6. Write the complete
response
The Classical Method (5)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 30
Ex. 1
The switch has been at A for a long time,
and it moves to B at t = 0; find v for t ≥ 0?
f nv v v= +
0 12 VV =
36 Vfv =
3 200100 0.05 10
tt
tRC
nv Ae Ae Ae
−
−
−
−× ×
= = =
0
0 36 36 12 24V Ae A A= + = + = → = −
20036 24 Vtv e−→ = −
+
–
+
–
+
–12 V 36 V
50 Ω 100 Ω
0.05 mF
A B
v
t = 0
1. Write the general form
2. Find the initial condition
3. Find the forced response
4. Deactivate source(s), find the
natural response (with the
unknow integration
constant)
5. Find the integration constant
6. Write the complete response
The Classical Method (6)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 31
+
–
+
–
+
–12 V 36 V
50 Ω 100 Ω
0.05 mF
A B
v
t = 020036 24 Vtv e−= −
t
v
0
36V
12V
0 12 VV = 200tnv Ae
−
= 36 Vfv =
+
–
100 Ω
0.05 mF
B
vn
+
–
+
– 36 V
100 Ω
B
vf
+
–
+
–12 V
50 Ω
A
V0
The Classical Method (7)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 32
Ex. 2
The switch has been at A for a long
time, and it moves to B at t = 0;
find v for t ≥ 0?
f nv v v= +
0
12 10 2 V
50 10
V −= = −
+
80 36 16V
100 80f
v = =
+
380 100 0.05 10 45080 100
t
t
tRC
nv Ae Ae Ae
−
−
− ×
× ×
−+
= = =
0
0 16 16 2 18V Ae A A= + = + = − → = −
45016 18 Vtv e−→ = −
+
–
+
–
+
–12 V 36 V
50 Ω 100 Ω
0.05 mF
A B
v10 Ω 80 Ω
t = 0
1. Write the general form
2. Find the initial condition
3. Find the forced response
4. Deactivate source(s), find the
natural response (with the
unknow integration
constant)
5. Find the integration constant
6. Write the complete response
The Classical Method (8)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 33
Ex. 3
The switch has been at A for a long time,
and it moves to B at t = 0; find i for t ≥ 0?
1. Write the general form
2. Find the initial condition
3. Find the forced response
4. Deactivate source(s), find the
natural response (with the
unknow integration
constant)
5. Find the integration constant
6. Write the complete response
f ni i i= +
0 5AI =
120 6A
20f
i = =
20
400.5
R
tt tL
ni Ae Ae Ae
−−
−
= = =
0
0 6 6 5 1I Ae A A= + = + = → = −
406 Ati e−→ = −
+
–
5 A 120 V
50 Ω 20 Ω
0.5 H
A B
i
t = 0
The Classical Method (9)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 34
+
–
5 A 120 V
50 Ω 20 Ω
0.5 H
A B
i
t = 0406 Ati e−= −
t
i
0
6A
5A
0 5AI = 40tni Ae
−
= 6Afi =
5 A
50 Ω
A
I0
20 Ω
B
in0.5 H
+
–
120 V
20 Ω
B
if
The Classical Method (10)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 35
Ex. 4
The switch has been at A for a long time,
and it is opened at t = 0; find i for t ≥ 0? 5 A
50 Ω 100 Ω
0.25 H
40 Ω 30 Ω
t = 0i
f ni i i= +
0
1/ 50 5 1.28A
1/ 50 1/ 40 1/ 30
I = =
+ +
5 A
50 Ω 100 Ω
0.25 H
30 Ω
if
50 Ω 100 Ω
0.25 H
30 Ω
in
1/ 50 5 1.88A
1/ 50 1/ 30f
i = =
+
50 30
3200.25
R
tt tL
ni Ae Ae Ae
+
−−
−
= = =
0
0 1.88 1.28 0.60I Ae A= + = → = −
3201.88 0.6 Ati e−→ = −
First-Order Circuits
1. Introduction to Transient Analysis
2. Initial Conditions
3. The Source-free RC Circuit
4. The Source-free RL Circuit
5. Step Response of an RC Circuit
6. Step Response of an RL Circuit
7. The Classical Method
8. First-order Op Amp Circuits
First-Order Circuits - sites.google.com/site/ncpdhbkhn 36
First-Order Op Amp Circuits (1)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 37
Ex. 1
The switch has been at A for a long time,
and it moves to B at t = 0; find vo for t ≥ 0?
f nv v v= +
(0) 5Vv =
0fv =
6
3 3
01 10
50 10 50 10
n B C Bdv v v v
dt
−
− −
× = =
× ×
B nv v= −
0
0.05
n ndv v
dt
→ + = 20t
n
v Ae−→ =
0(0) 0 5 5v Ae A= + = → =
205 Vtv e−→ =
+–
+
–
5 V vo
50 kΩ
1 µF
A B
100 kΩ
+
–
C
v
–
+
t = 0
+–
von
50 kΩ
1 µF
100 kΩ
+
–
C
vn
–
+
B
First-Order Op Amp Circuits (2)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 38
Ex. 1
The switch has been at A for a long time,
and it moves to B at t = 0; find vo for t ≥ 0?
205 Vtv e−=
3
3
100 10 2
50 10o B B
v v v
×
= − = −
×
Bv v= −
2010 Vtov e
−→ =
+–
vi
vo
ii
io
R1 R2
2
1
o i
R
v v
R
= −
+–
+
–
5 V vo
50 kΩ
1 µF
A B
100 kΩ
+
–
C
v
–
+
t = 0
First-Order Op Amp Circuits (3)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 39
Ex. 2
Find vo(t)? f nv v v= +
(0) 0v =
+
–
vi
voii
io
R1 R2
2
1
1o i
R
v v
R
= +
20 6 4V
10 20Af
v = =
+
601 12V
30of Af
v v
= + =
4 12 8Vf Bf of Af ofv v v v v= − = − = − = −
360 10 0nv i+ × =
3 660 10 10 0n
n
dv
v
dt
−→ + × × =
16.67t
nv Ae
−→ =
+
–
+
–
t = 0
v
–+
30 kΩ
60 kΩ
1 µF
A
B
10 kΩ
20 kΩ
6 V
vo
+
–
16.67 16.674 ( 8 8 ) 12 8 Vt t
o Bv v v e e
− −
= − = − − + = −
First-Order Op Amp Circuits (4)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 40
Ex. 2
Find vo(t)? f nv v v= +
(0) 0v =
8Vfv = −
16.67t
n
v Ae−=
0(0) 8 0 8v Ae A= − + = → =
16.678 8 Vtv e−→ = − +
+
–
+
–
t = 0
v
–+
30 kΩ
60 kΩ
1 µF
A
B
10 kΩ
20 kΩ
6 V
vo
+
–
First-Order Op Amp Circuits (5)
First-Order Circuits - sites.google.com/site/ncpdhbkhn 41
Ex. 3
Find v(t)? f nv v v= +
0 0V =
20 20 5 10 V
10 10Bf Af
v v= − = − = −
+–
vi
vo
ii
io
R1 R2
2
1
o i
R
v v
R
= −
B
+–
10 kΩ
+
–
t = 0
20 kΩ
40 kΩ
60 kΩ
A
v
+
–
1 µF
5 V
60 60 ( 10) 6V
40 60 40 60f Bf
v v= = − = −
+ +
3 3
6
3 3
1
40 10 60 10 1 10
41.6740 10 60 10
t
t
n
v Ae Ae
−
−
× × ×
×
−× + ×
= =
0
0 (0) 6 0 6V v Ae A= = − + = → =
41.676 6 Vtv e−→ = − +
40 kΩ
60 kΩ vn
+
–
1 µF
Các file đính kèm theo tài liệu này:
- bai_giang_electric_circuit_theory_chapter_vii_first_order_ci.pdf