[<< wikibooks] Circuit Theory/Complex Frequency Examples/example15
Find everything give that:

V

s

=
10
∗

e

−
100
t

cos
⁡
(
6000
t
)

{\displaystyle V_{s}=10*e^{-100t}\cos(6000t)}
Terminal Relations:

V

2

=

R

1

∗

I

2

{\displaystyle \mathbb {V} _{2}=R_{1}*\mathbb {I} _{2}}

I

3

=

C

1

∗
s
∗

V

3

{\displaystyle \mathbb {I} _{3}=C_{1}*s*\mathbb {V} _{3}}

V

1

=

L

1

∗
s
∗

I

1

{\displaystyle \mathbb {V} _{1}=L_{1}*s*\mathbb {I} _{1}}

V

4

=

R

2

∗

I

1

{\displaystyle \mathbb {V} _{4}=R_{2}*\mathbb {I} _{1}}
Junction Equation:

I

1

−

I

2

−

I

3

=
0

{\displaystyle \mathbb {I} _{1}-\mathbb {I} _{2}-\mathbb {I} _{3}=0}
Loop Equations:

V

3

+

V

1

+

V

4

−

V

s

=
0

{\displaystyle \mathbb {V} _{3}+\mathbb {V} _{1}+\mathbb {V} _{4}-\mathbb {V} _{s}=0}

V

3

−

V

2

=
0

{\displaystyle \mathbb {V} _{3}-\mathbb {V} _{2}=0}
(trivial loop)Complex Frequency:

s
=
−
100
+
6000
j

{\displaystyle s=-100+6000j}
Complex frequency source:

V

s

=
10

{\displaystyle \mathbb {V} _{s}=10}
Solving in the complex frequency domain: