$(2)(6$ points$)$

Consider the circuit below. You may assume it has reached steady state. The phasor diagram shows the phasors of $i_1$ and $i_2($the angles are as shown; the magnitudes are not drawn to scale$).$

Each rectangular box represents a connection of mystery elements in series$/$parallel$.$ The impedances of the two boxes are $Z_a=aj$ and $Z_b=b(1+j)$ but you are not told which one is which.

Note: you are given the values for $a,b$ and $|$V$|.$

$($a$)$ We want to build the impedance $Z_b$ using two elements in series. What elements would you use and what are their values?

$($b$)$ Consider the source $v_S(t).$ Find the value of $x.$

$($c$)$ Between A and B $,$ we add a third impedance $Z_3,$ such that the current $i_s(t)$ through the voltage source becomes equal to $i_1(t).$ What is the value of this impedance $Z_3?$