1
Summary of Lecture #7
9/08/2010
•
L and C with zero initial conditions in
frequency domain
L
→
Ls,
C
→
1/(Cs)
•
L and C with nonzero initial conditions
–
Eq. voltage source in series
–
Eq. current source in parallel
•
Transfer
functions
•
Cramer’s rule and matrix inversion
•
Nodal
analysis
•
Loop or mesh analysis
•
Thevenin and Norton equivalents
•
Superposition
•
Examples
2
•
Frequency shift
•
L
[e

at
f (t)] = F (s+a)
•
1st and 2nd order time differentiation
•
Nthorder time differentiation
See Eq. 13.45

=




=

=
0
t
2
2
2
dt
)
t
(
df
)
0
(
sf
)
s
(
F
s
]
dt
f
d
[
)
0
(
f
)
s
(
sF
]
dt
df
[
L
L
initial value
initial derivative
3
Cap. with nonzero initial voltage
v
C
(0

)
Eq. voltage source in series
Eq. current source in parallel
s
)
0
(
v
)
s
(
I
Cs
1
)
s
(
V
c
C
C

+
=
)
s
(
CsV
)
0
(
Cv
)
s
(
I
C
c
C
=
+

)]
0
(
v
)
s
(
sV
[
C
)
s
(
I
,
dt
)
t
(
dv
C
)
t
(
i
c
C
C
C
C


=
=
4
Ind. with nonzero initial current
i
L
(0

)
Eq. current source in parallel
Eq. voltage source in series
s
)
0
(
i
)
s
(
V
Ls
1
)
s
(
I
L
L
L

+
=
)
0
(
Li
)
s
(
LsI
)
s
(
V
L
L
L


=
)]
0
(
i
)
s
(
sI
[
L
)
s
(
V
,
dt
)
t
(
di
L
)
t
(
v
L
L
L
L
L


=
=
5
C
v
c
(t)
i
c
(t)
+

v
c
(0

)
Cv
c
(0

)
I
c
(s)
V
c
(s)
+

I
c
(s)
V
c
(s)
+

+

Cs
1
v
c
(0

)
s
i
L
(t)
v
L
(t)
+

L
i
L
(0

)
I
L
(s)
V
L
(s)
+

Ls
i
L
(0

)
s
+

Ls
I
L
(s)
V
L
(s)
+

Li
L
(0

)
Cs
1
Circuit elements having
nonzero initial
conditions
in time and frequency domains
Current source
in parallel
Voltage source
in series
6
Transfer function of twoport circuits
Assumptions: No independent source
Zero initial conditions
where
F
j
(s)
and
G
i
(s)
stand for
V
1
(s), I
1
(s)
V
2
(s)
or
I
2
(s)
. Subscripts
i
and
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 Spring '06
 Mesh Analysis, Invertible matrix

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