Circuit Theory/2Source Excitement/Node and Mesh

The first step is to convert everything to phasors and impedances in symbolic form if possible:

$\mathbb {V} _{1}=5*({\sqrt {3}}+j)$

$\mathbb {I} _{1}={\frac {1-j}{2*{\sqrt {2}}}}$

$Z_{C1}={\frac {1}{j\omega C_{1}}}=-10j$

$Z_{C2}={\frac {1}{j\omega C_{2}}}=-5j$

$Z_{L}=j\omega L_{1}=L_{2}=1j$

Node Analysis

marked up for node analysis

${\frac {\mathbb {V} _{1}-\mathbb {V} _{a}}{R_{1}+j\omega L_{1}}}+\mathbb {I} _{1}-{\frac {\mathbb {V} _{a}-\mathbb {V} _{b}}{\frac {1}{j\omega C_{1}}}}=0$

${\frac {\mathbb {V} _{a}-\mathbb {V} _{b}}{\frac {1}{j\omega C_{1}}}}-{\frac {\mathbb {V} _{b}}{R_{3}}}-{\frac {\mathbb {V} _{b}-\mathbb {V} _{c}}{\frac {1}{j\omega C_{2}}}}=0$

${\frac {\mathbb {V} _{b}-\mathbb {V} _{c}}{\frac {1}{j\omega C_{2}}}}-\mathbb {I} _{1}-{\frac {\mathbb {V} _{c}}{j\omega L_{2}}}=0$

Results using matlab:

$\mathbb {V} _{a}=-6.5-7i\Rightarrow V_{a}(t)=9.55cos(1000t-2.32)$

$\mathbb {V} _{b}=-3.08-3.31i\Rightarrow V_{b}(t)=4.52cos(1000t-2.32)$

$\mathbb {V} _{c}=0.327+0.385i\Rightarrow V_{c}(t)=0.506cos(1000t+0.866)$

Mesh Analysis

marked up for mesh analysis

$\mathbb {I} _{1}R_{2}+{\frac {\mathbb {I} _{1}+\mathbb {I} _{2}}{j\omega C_{1}}}+{\frac {\mathbb {I} _{1}+\mathbb {I} _{3}}{j\omega C_{2}}}-\mathbb {V} _{Is}=0$

$\mathbb {V} _{1}+\mathbb {I} _{2}R_{1}+{\frac {\mathbb {I} _{2}+\mathbb {I} _{1}}{j\omega C_{1}}}+(\mathbb {I} _{2}-\mathbb {I} _{3})R_{3}+\mathbb {I} _{2}j\omega L_{1}=0$

${\frac {\mathbb {I} _{3}+\mathbb {I} _{1}}{j\omega C_{2}}}+\mathbb {I} _{3}j\omega L_{2}+(\mathbb {I} _{3}-\mathbb {I} _{2})R_{3}=0$

Results using Matlab:

$\mathbb {I} _{2}=-0.239+0.234i\Rightarrow I_{2}(t)=0.334cos(1000t+2.37)$

$\mathbb {I} _{3}=-0.238+0.235i\Rightarrow I_{3}(t)=0.334cos(1000t+2.36)$

$\mathbb {V} _{Is}=175-179i\Rightarrow V_{Is}(t)=250cos(1000t-0.795)$