Two coils, one mounted on a stator and the other on a rotor, have self- and mutual inductances of L11 = 3.5mH L22 = 1.8mH L12 = 2.1cos θ mH where θ is the angle between the axes of the coils. The coils are connected in series and carry a current i = √2I sin wt

a. Derive an expression for the instantaneous torque T on the rotor as a function of the angular position θ.

b. Find an expression for the time-averaged torque T avg as a function of θ.

c. Compute the numerical value of T avg for I = 10 A and θ = 90°.

d. Sketch curves of T avg versus θ for currents I = 5, 7.07, and 10 A.

e. A helical restraining spring which tends to hold the rotor at θ = 90° is now attached to the rotor. The restraining torque of the spring is proportional to the angular deflection from θ = 90° and is – 0.1 N ∙ m when the rotor is turned to θ = 0°. Show on the curves of part (d) how you could find the angular position of the rotor-plus-spring combination for coil currents I = 5, 7.07, and 10 A. From your curves, estimate the rotor angle for each of these currents.

f. Write a MATLAB script to plot the angular position of the rotor as a function of rms current for 0 < I < 10 A. (Note that this problem illustrates the principles of the dynamometer-type ac ammeter.)