Problem $1$: AC Max Power Transfer UMF $8\mathrm{.}31($modified$)$

a$)$ First do the text problem. Find $S_{\text{L}\text{o}\text{a}\text{d}}$ the load power factor.

Now we are going to replace the load $Z_{\text{L}\text{o}\text{a}\text{d}}$

b$)$ Find the phasor domain Thevenin equivalent that the load sees a terminals a and $b.(Z_{\text{L}\text{o}\text{a}\text{d}}$ is removed$).$

c$)$ What load impedance $Z_{\text{L}\text{o}\text{a}\text{d}}$ will result in maximum power transfer to the load and what is that power?

$8\mathrm{.}31$ In the phasor$-$domain circuit shown in Fig. P$8\mathrm{.}31,V=\frac{15}{{?}_{45}}V,R_1=5,R_2=2,Z_C=-j1,Z_L=j2,$ and $Z_{\text{l}\text{o}\text{a}\text{d}}=6+j4.$ Determine the complex power of the load.

Figure P$8\mathrm{.}31$ Circuit for Problem $8\mathrm{.}31.$