$6\mathrm{.}18$ A three$-$phase, $6-$pole, $120kW,460-V$ aluminum$-$squirrel$-$cage induction motor has the following single$-$phase equivalent$-$circuit parameters in ohms$-$per$-$phase

$R_1=15\mathrm{.}3{10}^{-3},R_2=34\mathrm{.}5{10}^{-3}$

$x_1=0\mathrm{.}183,x_2=0\mathrm{.}219,x_m=13\mathrm{.}4$

You may assume that the motor friction and windage loss of $1370$ W remains constant over the normal range of operation and the core loss at $460$ V is $1100$ W $.$

a$.$ Make a table including the motor slip, speed, terminal current, power factor, and efficiency when operating at $460$ V and supplying rated power. Hint: It may be easiest to search for the desired operating points using MATLAB.

b$.$ The manufacturer proposes replacing the rotor of this motor with an otherwise identical rotor except that the squirrel cage is cast from copper instead of aluminum. Assuming the electrical conductivity copper to be $1\mathrm{.}5$ times that of aluminum, repeat the calculation of part $($a$)$ for the motor operating with this new rotor. Expand your table of part $($a$)$ to include the performance of the copper$-$rotor motor and compare the results.

c$.$ Compare the performance of this motor with the cast$-$aluminum and cast$-$copper rotors operating at rated voltage and $75$ percent, $50$ percent and $25$ percent of rated load.