Circuit Theory/Phasors/Examples/example11

Given that the voltage source is defined by $V_{s}(t)=cos(t)$ and the current source is defined by $I_{s}(t)=2sin(t-{\frac {\pi }{3}})$, find all other voltages, currents and check power.

Complex 2R, L, C, voltage circuit

Circuit marked up for analysis

Knowns, Unknowns and Equations

Knowns: $V_{s},R_{1},R_{2},L,C,I_{s}$

Unknowns: $v_{1},v_{2},v_{C},v_{L},i_{1},i_{2},i_{C},i_{L}$

Equations:

$v_{1}=R_{1}*i_{1}$

$v_{2}=R_{2}*i_{2}$

$i_{C}=C*{d \over dt}v_{C}$

$v_{L}=L*{d \over dt}i_{L}$

$i_{1}+I_{s}-i_{2}-i_{C}=0$

$i_{c}-I_{s}-i_{L}=0$

$v_{1}+v_{2}-V_{s}=0$

$v_{C}+v_{L}-v_{2}=0$

Solutions and Analysis

numeric solution calculated in the phasor domain

attempt at a Laplace solution

simulation at circuitlab