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 $$ I_s(t) = 0.5 sin(1000t + 45^{\circ})$$, find all other voltages, currents and power of the sources.



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