Circuit Theory/Phasors/Examples/example14

Given that the voltage source is defined by ${\displaystyle V_{s}(t)=10cos(1000t+30^{\circ })}$ and the current source is defined by ${\displaystyle I_{s}(t)=0.5sin(1000t+45^{\circ })}$, find all other voltages, currents and power of the sources.

Knowns: ${\displaystyle v_{1},R_{1},R_{2},R_{3},L_{1},L_{2},C_{1},C_{2},i_{1}}$

Unknowns: ${\displaystyle v_{2},v_{3},v_{4},v_{5},v_{6},v_{7},v_{8},v_{9},i_{2},i_{3},i_{4},i_{5},i_{6}}$

Equations:

${\displaystyle v_{2}=r_{1}*i_{2}}$

${\displaystyle i_{3}=c_{1}{d \over dt}v_{3}}$

${\displaystyle v_{9}=r_{3}*i_{6}}$

${\displaystyle v_{6}=L_{1}*{d \over dt}i_{2}}$

${\displaystyle i_{4}=c_{2}{d \over dt}v_{4}}$

${\displaystyle v_{5}=L_{2}*{d \over dt}i_{5}}$

${\displaystyle v_{7}=r_{2}*i_{1}}$

${\displaystyle i_{2}+I_{1}-i_{3}=0}$

${\displaystyle i_{3}-I_{4}-i_{6}=0}$

${\displaystyle i_{4}-i_{1}-i_{5}=0}$

${\displaystyle v_{1}+v_{2}+v_{3}+v_{9}+v_{6}-V_{1}=0}$

${\displaystyle v_{4}+v_{5}-v_{9}=0}$

${\displaystyle v_{7}+v_{3}+v_{4}-v_{8}=0}$

numeric solution calculated in the phasor domain

simulation at circuitlab