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(1) Show that the analysis of the electriccircuit shown leadstoamatrix eigenvalue
problem. (2) Determine the circular frequencies and the relative amplitudes of the
currents.
Solution of Part (1)
Kirchoff'sequationsfor the three loops are
L
di
1
q
1
−
q
2
dt
+
=
0
3
C
L
di
2
dt
q
2
−
q
1
q
2
−
q
3
+
+
=
0
3
C
C
2
L
di
3
dt
q
3
−
q
2
q
3
C
+
+
=
0
C
Differentiating and substituting
dq
k
/
dt
=
i
k
, we get
LC
d
2
i
1
dt
2
1
3
i
1
−
1
3
i
2
=−
LC
d
2
i
2
dt
2
1
3
i
1
+
4
3
i
2
−
−
i
3
=−
2
LC
d
2
i
3
dt
2
−
i
2
+
2
i
3
=−
These equations admit the solution
i
k
(
t
)
=
u
k
sin
ω
t
where
is the circular frequency of oscillation (measuredinrad/s) and
u
k
are the
relative amplitudes of the currents.Substitutioninto Kirchoff'sequations yields
Au
ω
=
λ
Bu
(sin
ω
t
cancels out), where
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1
/
3
−
1
/
3
0
1 00
0 1 0
002
2
A
=
−
1
/
3
4
/
3
−
1
B
=
λ
=
LC
ω
−
0
12
which represents an eigenvalue problem of the nonstandard form.
Solution of Part (2)
Since
B
is adiagonalmatrix,wecanreadily transformthe problem
into the standard form
Hz
=
λ
z
.FromEq. (9.26a) we get
⎡
⎣
⎤
⎦
1 00
0 1
L
−
1
=
0
/
√
2
001
and Eq. (9.26b) yields
⎡
⎣
⎤
⎦
1
/
3
−
1
/
3
0
/
√
2
H
=
−
1
/
3
4
/
3
−
1
√
2
0
−
1
/
1
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