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EEM 209
CIRCUIT ANALYSIS
• Transient Response
• The Simple RL Circuit
• The Simple RC Circuit
• Unit‐Step Function
• Response of an RL Circuit to a Step Function
• RLC Circuits : Passive Series RLC Circuit
• Complex Frequency
• Frequency Response
• Poles and the Natural Response
• Complete Response
• Resonance:The Simple Series RLC Circuit
• Scaling
• Bode Diagrams
• The Operational Amplifier
• Two Port Networks
• The Fourier Transform
• The Laplace Transform
Transient Response
A change in the source output, or a change in the circuit or
element values will cause the currents and voltages in the circuit to
change.The question is how these changes will take place.
Ex:
1
I 
2
3
2 42
1 21
4
1
I 
:
0
t
:
0
t

 A
 A
Ex:
12
 2A
24
t  0 : ? (The currents do not show sudden jumps)
t  0 : I1 
t   : I1 
12
 3A
4
The Simple RL Circuit
o
S
.
y
l
n
e
d
d
u
s
e
g
n
a
h
c
t
o
n
n
a
c
iL
t
n
e
iL r
r
t u
n
e c
r
e
r
h
u
c t
t
e u
h b
t
, 0
f y
o
l
e n t
s e r
u d o
a d f
c u i
t
e s u
b s c
r
L n i
n e c
p
i
g
d o n
e h i
r
w
o c o
t
t
i
s
l
w l
s s o
i
f
y e e
g h h
r t t
,
e
0
e
n
v
E
: t a
0 t h
A e
w
t


I0
)
0
(
iL
t
n
0
e
r
r
0
u
c
e R
h V
t
Lt
i
t Ld
d
e
L V L



iL
d
t
iL d
d
)
(
 RiL 
)
(
n
l
iL I

t
)
0
R t
 t
L
(
R L
0
)
I0
(
n
l
)
t
0
(
iL
n
l
iL t
)
(
)
t
(
I0
iL
n
l
0
R
 t
R
  t  iL t  I e L
L
IL t

I
0
R
R
  iL 
  dt 
iL
L
L
 R
  dt
 L
Note that, the power delivered to the resistor is
s
i
r
o
t
s
i
s
e
r
e
h
t
n
i
t
a
e
h
o
t
d
e
t
r
2
e
v
e n
R o
2
0c
y
g
) r
( e
2 n
e
e
h
t
) d
( n
a
PR t  RiL t  I
r
o
t
c
u
d
n
i
e
h
t
n
i
d
0
e
r
o
2
t
s
y
g
2
r
e
s
n
i
e
t
l
a
2
2
a
0
e
i
t
h
i
n
o
i
1 t
e
d 20 h
e
t
2
2
t
0 r
o
2
1
e
t
0 1 2 v
l
a
n
2
u
o
0
q
1 c
e
y
y
g
l
r 1 t
)
e
c
2
(
n 2 a
0
x
e
e
l
2
1
0
a
s
2
i
0 t
o
h
)
t
c
1 2 e
(R h
i
h
T W w
t   PR t dt  I R  e
)
(R
W
I L
I L  e   
t 
R
 t

L

I L  e



  LR t

  I L e
  


t
R
 L  Lt
dt  I R 
e
R


R
t
L

t
t
R
t
L

Ex:
•
•
•
•
•
The switch has been connected
to the terminal a for very long
time and then switched to b at t=0.
a) Find i(t)
b) Plot i(t)
c) In how many seconds after t=0 does the current i drop down to 50% of its initial
value?
d) In how many seconds does i drop down to 10% of its inital value
e)In how many seconds does the power in become 75%of its initial value
f) In how many seconds does the energy in the L become 60% of its initial value?
0
0 5 1
4
5
)
a t
i
i
3
)
0
(

)
0
(

)
0
(
 Ai

 A
0

  ta 
s
1
4
.
0
t
2


6
.
0
 R
td 

 I L 
I L  e L   




 e  td 
 td 
s
0
1

 tc 
1
2
0
0
4
.
1 2 0
)
0
1
(
)
1
td
s
9
2
0
.
0
1
2
0
5
5
0
.
7 0 1 2 0
.
1
0
0
n
4
l
.
0


0
1
0
9
tc

e
)
0
9
(
5
7
.
0
0
1
e
f WR td 
 e

W
0
9
PR tc  
 t
2

e 
5
PR
 tb 
3
0
1
2
0
9
)
)
( 0
(
)
e PR t  Ri 
)
3
(
0
1
.
0
5
3
)
(
)
d i tb  e  tb 
0
1 5
.
0
n
l
ta 

3
9
6
.
0
ta

5
.
0
n
l
e
ta
5
A
c i ta  e 
)
3
(
0
5
.
0
 t
5
t
0 9
3 5 3
.
0
1
)
.
0
(
) 5
it  e

5
3 3
) )
( (
it  e
0
12
t
3
i
s
Properties of the Exponential Response
)
(
Normalized current expression :
0
R
 t
it
e L
I
When would the current become zero if it had continued to decrease at its initial rate? The line tangent to the curve at (0,1) point will give as the answer.
The slope of the line : )
(
0
0
0

1
τ
)
t
n
a
t
s
n
o
C
e
m
i
T
:
(
f
o
s
e
l
p
i
t
l
u
m
t
a
t
n
e
r
r
u
c
d
e
z
i
l
a
m
r
o
n
e
h
t
f
o
s
e
u
l
a
V
L
τ
R


1
)
t
(
y
:
e
n
i
l
e
h
t
f
o
n
o
i
t
a
u
q
e
e
h
T
t
τ  τ
R L
0
:
o
r
e
z
e
b
o
t
)
t
(
y
r
o
f
e
m
i
T
t
R L
R
L

d i t 
R  RL t
 e


dt  I  t 
L
=
2τ
=0.1353
<5%
3τ
=0.0498
<5%
=0.3679
=
<1%
=0.0067
5τ
~
τ
<1%
0.0183
4τ
Appoximate Value
İ(t)/I0
t
e
b
y
a
m
t
i
s
n
o
i
t
a
l
u
c
l
a
c
t
s
u
m
r
o
f
t
u
b
,
t
t
a
0
o
t
s
e
o
g
e
s
n
o
p
s
e
r
e
h
T
:
e
t
o
N
τ
)
.
d
e
r
i
s
e
d
s
i
τ
n
o
i
s
i
c
e
r
p
e
r
o
m
n
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h
w
5
r
o
(
3
r
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t
f
a
e
u
l
a
v
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t
a
t
s
y
d
a
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s
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h
c
a
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r
t
i
t
a
h
t
d
e
m
u
s
s
a

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