Circuit Theory/Phasors/Examples/example14

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

Complex 3R, 2L, 2C, voltage and current source circuit

Circuit marked up for analysis

Knowns, Unknowns and Equations

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

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

Equations:

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

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

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

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

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

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

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

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

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

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

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

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

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

Solutions and Analysis

numeric solution calculated in the phasor domain

simulation at circuitlab