ECE 1311
First Order Circuits
Chapter 7
Dr. AHM Zahirul Alam
ECE Dept., IIUM
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The Step
Step-Response
Response of a RC Circuit (2)
„ Integrating
g
g both sides and considering
g the initial
conditions, the solution of the equation is:
⎧V0
v(t ) = ⎨
−t / τ
V
(
V
V
)
e
+
−
0
s
⎩ s
Final value at
t -> ∞
Complete Response =
=
Natural response
(stored energy)
V0e–t/τ
+
Initial value at
t=0
+
t<0
t >0
Source free
Source-free
Response
Forced Response
(independent source)
Vs(1–e–t/τ)
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The Step-Response
p
p
of a RC Circuit ((3))
Three steps to find out the step response of an
RC circuit:
1. The initial capacitor voltage v(0).
2. The final capacitor voltage v(∞) — DC voltage
across C.
3. The time constant τ.
−t /τ
v (t) = v (∞) + [v (0+) − v (∞)]e
Note: The above method is a short-cut
short cut method
method. You may also determine the
solution by setting up the circuit formula directly using KCL, KVL , ohms law,
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capacitor and inductor VI laws.
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The Step-Response
p
p
of a RC Circuit ((4))
Example 5
Find v(t) for t > 0 in the circuit in below. Assume the
switch has been open for a long time and is closed at t
= 0.
0
Calculate v(t) at t = 0.5.
Answer:
v (t ) = 60e −22tt − 50
and v(0.5) = V
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The Step-response
p
p
of a RL Circuit ((1))
„ The step response of a circuit is its behavior when the excitation
is the step function, which may be a voltage or a current source.
•
Initial current
i(0-) = i(0+) = Io
•
Final inductor current
i(∞) = Vs/R
•
Time constant τ = L/R
−
t
Vs
Vs τ
i (t ) = + ( I o − )e u (t )
R
R
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The Step-Response
p
p
of a RL Circuit ((2))
Three steps to find out the step response of an
RL circuit:
1. Th
1
The initial
i iti l iinductor
d t currentt i(0) att t = 0+
0+.
2. The final inductor current i(∞).
3 The time constant τ.
3.
i (t ) = i (∞ ) + [i (0+ ) − i (∞ )] e
− t /τ
Note: The above method is a short-cut
short cut method
method. You may also determine
the solution by setting up the circuit formula directly using KCL, KVL ,
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ohms law, capacitor and inductor VI laws.
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The Step-Response
p
p
of a RL Circuit ((4))
Example 6
The switch in the circuit shown below has been closed
for a long time. It opens at t = 0.
Find i(t) for t > 0.
Answer:
i (t ) = 2 + e −10 t
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3
•Read Chapters 6
•Required Problems
-Ch. 6: 17, 46, 48, 54, 64
•Recommended Problems
-Ch.
Ch 6: 13
13, 19
19, 53,
53 65,
65 73
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Read Chapters 9 & 10
Appendix B
Required Problems
Ch. 7: 47, 64, 69
Ch. 9: 8, 11, 14, 16,
17 18
17,
Read Chapters 10 & 11
Required Problems
Ch. 10: 19, 36, 42, 56, 80
Recommended
R
d dP
Problems
bl
Ch. 10: 16, 39, 66, 77, 76
Recommended Problems
Ch. 7: 50,, 57,, 71
Ch. 9: 9, 10, 19
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