Problem $3$: Synchronous machine

Consider a salient$-$pole synchronous machine having two identical stator windings a and $b,$ and a field winding $f.$ Neglecting the space harmonics,

The self$-$inductances are: Laa$=0\mathrm{.}5+0\mathrm{.}25cos(2r),Lbb=0\mathrm{.}5-0\mathrm{.}25cos(2r),$ and $Lff=0\mathrm{.}4,$

The mutual$-$inductances are: Lab$=0\mathrm{.}125sin(2r),$Laf$=0\mathrm{.}625cos(r),$ and $Lbf=0\mathrm{.}625sin(r),$ where, $r$ is defined as the angular position of the rotor with respect to the magnetic axis of Phase$-$A$.$ Also, consider the operating condition in which the field winding is excited by direct current $If=8\mathrm{.}35$A and the stator windings are connected to a balanced two$-$phase voltage source of frequency $.$ Under this condition, the stator currents are $ia=\sqrt[\phantom{2}]{2}(8)cos(t+\frac{}{6}),$ and $ib=\sqrt[\phantom{2}]{2}(8)sin(t+\frac{}{6}).$ Find the value of the average torque if the rotor revolves at synchronous speed, i$.$e$.r=120t-\frac{}{9}$