Circuit Theory/Complex Frequency Examples/example15

Find everything give that:

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

Terminal Relations:

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

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

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

${\displaystyle \mathbb {V} _{4}=R_{2}*\mathbb {I} _{1}}$

Junction Equation:

${\displaystyle \mathbb {I} _{1}-\mathbb {I} _{2}-\mathbb {I} _{3}=0}$

Loop Equations:

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

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

Complex Frequency:

${\displaystyle s=-100+6000j}$

Complex frequency source:

${\displaystyle \mathbb {V} _{s}=10}$

Solving in the complex frequency domain: