Calculate the following for a 2 hp and a 20 hp dc machine, each rated for 500 rpm. Use data from the Study Plan 1 data sheet, including "hot" armature resistance value for all calculations. Note that the value of K is proportional to the field flux, and the printed value is for rated (100%) flux. Both the load moment of inertia JL and the viscous friction coefficient B are zero unless stated otherwise.

a) Calculate the eigenvalues (real or complex) for operation at rated flux and at 50% of rated flux:

b) Calculate the dominant time constant τ of the 2 hp machine and the natural frequency ωN and damping factor ζ of the 20 hp machine (assume rated flux for both machines). Use them to determine the approximate percentage overshoot and settling time (within 2%) for the rotor speed's natural response for each machine following a step change in the armature voltage. Assume zero load inertia. Plot the transient response of the rotor speed ω (in rpm) for both machines for a step in the armature voltage from 50% to 100% rated voltage, assuming no steady-state load torque (i.e., TL=0) and an initial rotor speed corresponding to the no-load speed at 50% rated voltage. Calculate the initial and final speed values for both machines.

c) Find the value of an external series resistance for both machines that will limit the steady-state stall current (i.e., speed = 0) with rated voltage to 125% of rated current. With this resistor in the circuit, repeat the eigenvalue calculation of part a) for both machines. Assume rated field flux. Plot the migration of 20 hp machine's eigenvalues (i.e., root locus) as the additional series resistance Radd is increased from 0 to its final value.