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**Unformatted text preview: **SMALL for Big Things
University at Buffalo
SMALL for Big Things
University at Buffalo
nanobioSensors & MicroActuators Learning Lab
The State University of New York
nanobioSensors & MicroActuators Learning Lab
The State University of New York
EE 203 Circuit Analysis 2
Lecture 26
Chapter 13.8
Impulse Function
in Circuit Analysis
Kwang W. Oh, Ph.D., Assistant Professor
SMALL (nanobioSensors and MicroActuators Learning Lab)
Department of Electrical Engineering
University at Buffalo, The State University of New York
215E Bonner Hall, SUNY-Buffalo, Buffalo, NY 14260-1920
Tel: (716) 645-3115 Ext. 1149, Fax: (716) 645-3656
E-mail: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 1/12
Impulse Function by a Switching Operation
Capacitor Circuit (1)
C1 is charged to an initial voltage of V0 at the time
the switch is closed. The initial charge on C2 is zero.
Q: find the expression for i(t) as R 0.
Vo
1
1
= I(
+R+
)
s
sC1
sC2
As R gets smaller, the current starts from a larger
initial value and then drops off more rapidly
idl
Vo
Q R(↓) → Initial Current (↑)
R
R (↓) → Time Constant RCe (↓)
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 2/12
SMALL for Big Things
University at Buffalo
SMALL for Big Things
University at Buffalo
nanobioSensors & MicroActuators Learning Lab
The State University of New York
nanobioSensors & MicroActuators Learning Lab
The State University of New York
Impulse Function by a Switching Operation
Capacitor Circuit (2)
As
As R gets smaller, the current starts from a larger
initial value and then drops off more rapidly
as R zero, i is approaching an impulse function
R 0: i VOCeδ(t)
Impulse Function by a Switching Operation
Series Inductor Circuit (1)
Q:
Q: find the time-domain expression for
vo after the switch has been opened.
Opening the switch forces an
instantaneous change in the current of
L2, which causes vo to contain an
impulsive component.
Initial condition at t = 0-
When R = 0, a finite amount of charge is
transferred
transferred to C2 instantaneously.
Set R = 0
The current in the 3 H inductor = 10 A
The current in the 2 H inductor = 0 A
100
+ 30
+ 30
=s
10 + 3s + 15 + 2 s 5s + 25
20
+ 6 6 s + 20 4
2
=s
=
=+
s + 5 s( s + 5) s s + 5
I=
100
s
i = ( 4 + 2 e −5 t )u ( t )
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 3/12
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 4/12
SMALL for Big Things
University at Buffalo
SMALL for Big Things
University at Buffalo
nanobioSensors & MicroActuators Learning Lab
The State University of New York
nanobioSensors & MicroActuators Learning Lab
The State University of New York
Impulse Function by a Switching Operation
Series Inductor Circuit (2)
Vo = (15 + 2 s ) I
= (15 + 2 s )
6s + 20
s( s + 5)
12( s 2 + 5s ) − 60s + 90s + 40s + 300
s 2 + 5s
70s + 300
= 12 +
s( s + 5)
K1
K
60 10
= 12 +
+ 2 = 12 +
+
s s+5
s s+5
300
70 s + 300
= 60
K1 = [12 s +=
s+5
5
s =0
=
K1 = [12( s + 5) +
100
+ 30
+ 30
=s
10 + 3s + 15 + 2 s 5s + 25
20
+ 6 6 s + 20 4
2
=s
=
=+
s + 5 s( s + 5) s s + 5
I=
70s + 300
− 350 + 300
= 10
=
s
−5
s = −5
vo = 12δ (t ) + (60 + 10e −5t )u (t )
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
100
s
i = ( 4 + 2 e −5 t )u ( t )
Lecture 27 | Chapter 13 | 8/8 | 5/12
Impulsive Sources
Impulsive
Impulsive functions can occur in sources as well as
responses
A mechanical analogy is striking a bell with an impulsive
clapper blow. After the energy has been transferred to the
After
bell, the natural responses of the bell determines the tone
emitted (that is, the frequency of the resulting sound waves)
and
and the tone’s duration.
Initial condition
When the impulse voltage source is applied, the initial
energy
energy in the inductor is zero
Therefore the initial current is zero.
There is no voltage drop across R, so the impulsive
voltage source appears directly across L.
An impulsive voltage at the terminals of an inductor
establishes an instantaneous current.
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 6/12
SMALL for Big Things
University at Buffalo
SMALL for Big Things
University at Buffalo
nanobioSensors & MicroActuators Learning Lab
The State University of New York
nanobioSensors & MicroActuators Learning Lab
The State University of New York
Internally Generated and Externally Applied Impulses Simultaneously (1)
100
+ 50 + 30
+ 80 20 + 16
=s
=s
10 + 3s + 15 + 2 s 5s + 25
s+5
16s + 20 K1
K
4 12
=
=
+ 2= +
s( s + 5)
s s+5 s s+5
Initial
Initial Condition at t = 0-
I=
i1(0-) = 10 A
i2(0-) = 0 A
SMALL for Big Things
University at Buffalo
nanobioSensors & MicroActuators Learning Lab
The State University of New York
Impulse Function by a Switching Operation
100
s
i = ( 4 + 12e −5t )u(t )
Series Inductor Circuit (1)
Vo = (15 + 2 s ) I = (15 + 2 s )
Q: find the time-domain expression for
vo after the switch has been opened.
Opening the switch forces an
instantaneous change in the current of
L2, which causes vo to contain an
impulsive component.
Initial condition at t = 0-
16s + 20
s( s + 5)
32( s 2 + 5s ) − 160s + 240s + 40s + 300
s 2 + 5s
120s + 300
K
K
= 32 + 1 + 2
= 32 +
s( s + 5)
s s+5
60 60
= 32 +
+
s s+5
vo = 32δ (t ) + (60 + 60e −5t )u(t )
=
The current in the 3 H inductor = 10 A
The current in the 2 H inductor = 0 A
100
100
+ 30
s + 30
=s
10 + 3s + 15 + 2 s 5s + 25
20
+ 6 6 s + 20 4
2
=s
=
=+
s + 5 s ( s + 5) s s + 5
I=
i = ( 4 + 2 e − 5 t )u ( t )
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Kwang Oh
Internally Generated and Externally Applied Impulses Simultaneously (2)
Lecture 27 | Chapter 13 | 8/8 | 4/12
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 7/12
K1 = [32 s +
120s + 300
300=
= 60
s+5
5
s =0
K1 = [32( s + 5) +
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
120s + 300
− 600 + 300
=
= 60
s
−5
s = −5
Lecture 27 | Chapter 13 | 8/8 | 8/12
SMALL for Big Things
University at Buffalo
SMALL for Big Things
University at Buffalo
nanobioSensors & MicroActuators Learning Lab
The State University of New York
nanobioSensors & MicroActuators Learning Lab
The State University of New York
Internally Generated and Externally Applied Impulses Simultaneously (3)
Summary (1)
Initial
Initial Condition at t = 0i1(0-) = 10 A
i2(0-) = 0 A
Switching Off (t ≥ 0+)
i = ( 4 + 12e −5t )u (t )
it =0+ = 16 A
i1 = 10 A → 16 A
Δi1 = 6A
i2 = 0 A → 16 A
Δi2 = 16A
it =∞ = 4 A
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 9/12
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 10/12
SMALL for Big Things
University at Buffalo
SMALL for Big Things
University at Buffalo
nanobioSensors & MicroActuators Learning Lab
The State University of New York
nanobioSensors & MicroActuators Learning Lab
The State University of New York
Summary (3)
Summary (2)
input
output, response
h(t)
δ(t)
impulse
source
system,
circuit,
black box…
impulse
response
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 11/12
EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo
Lecture 27 | Chapter 13 | 8/8 | 12/12
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