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PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
C H A P T E R
5
Transient Analysis
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1
GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.1 Examples of transient response
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1.0
1.2
t (s)
(a) Transient DC voltage
1.4
1.6
1.8
2.0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
t (s)
(b) Transient sinusoidal voltage
1.6
1.8
2.0
1
0.5
0
–0.5
–1
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GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.2 Circuit with switched DC
excitation
t =0
R
Switch
12 V
C
L
Complex load
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GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.3 A general model of the transient
analysis problem
RS
Switch
t=0
Vs
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Circuit
containing
R L /RC
combinations
4
GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.5 Circuit containing energy-storage
element
A circuit containing energy-storage
elements is described by a
differential equation. The
differential equation describing the
series RC circuit shown is
diC
dt
vS ( t )
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+_
+
1
RC
+ vR
R
iR
dv S
dt
iC =
_
iC
C
5
GIORGIO RIZZONI
+
v C (t )
_
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.9 Differential equations of first-order
circuits
R
+
+
_
v S (t)
C
v C (t)
_
RC circuit:
dv C
dt
–
1
RC
vC –
1
RC
vS = 0
R
+
_
v S (t)
RL circuit:
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iL ( t )
di L
dt
–
L
R
1
iL –
vS = 0
L
L
6
GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.10 Decay through a resistor of
energy stored in a capacitor
t= 0
t= 0
Switch
vC
VB
Switch
C
i(t)
R
Exponential decay of capacitor current
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time, s
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GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.15
t=0
iL
IS
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L
8
GIORGIO RIZZONI
R
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.16
iL
(0 + )
L
iL(t)
–
v
+
R
( t)
iL( t)
10 mA
0
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t
9
GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.22 A more involved RC circuit
t= 0
R1
R2
+
V1
R3
vC
C
V2
_
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GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.23 The circuit of Figure 6.45 for t > 0
+
R
R2
1
vC
R3
V1
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V2
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GIORGIO RIZZONI
C
_
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.24 Reduction of the circuit of Figure
5.23 to Thevenin equivalent form
R1
V1
R1
V
R
1
1
R3
R2
V2 /R 2
+
V
R
2
R
R
T
= R
1
R
R
V
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T
2
T
2
R
3
T
+
_
12
GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.25 The circuit of Figure 5.22 in
equivalent form for t > 0
RT
+
VT
+
_
VC
C
_
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GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.39 Second-order circuits
R
_+
v
T
(t)
T
C
L
Parallel case
(a)
R
T
L
+
_
v
T
(t)
C
Series case
(b)
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GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.43 Response of overdamped
second-order circuit
2
1.5
xN ( t)
1
e–
0.5
e–
2t
1t
0
0
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0.1
0.2
0.3
0.4
0.5 0.6
t (s)
15
GIORGIO RIZZONI
0.7
0.8
0.9
1
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.44 Response of critically damped
second-order circuit
1
0.8
0.6
xN (t)
0.4
e–
0.2
te–
t
t
0
0
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0.1
0.2
0.3
0.4
0.5 0.6
t (s)
0.7
16
GIORGIO RIZZONI
0.8
0.9
1
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.45 Response of underdamped
second-order circuit
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GIORGIO RIZZONI
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.46
t =0
+ v C ( t) _
R
C
+ v R ( t) _
+
+
_
V
S
R = 5000
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vL ( t)
_
i(t)
L=1H C =1
V S = 25 V
18
GIORGIO RIZZONI
L
F
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.48
t =0
+
v( t)
C
R
iC ( t )
L
iR ( t )
i L( t)
IS
_
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L =2H
C =2
R = 500
IS = 5 A
19
GIORGIO RIZZONI
F
© The McGraw-Hill Companies, Inc. 2000
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING
THIRD EDITION
Figure 5.52
N2
= 100
N1
N1
N2
= 100
N1
N2
N1
LP,RP
+
VB
–
VB
spark
plug
C
–
switch
closed
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L P, R P
+
i
20
GIORGIO RIZZONI
N2
i
spark
plug
C
switch
closed
© The McGraw-Hill Companies, Inc. 2000