He steady-state response of a stable system to a sinusoidal input function will, itself, bea sinusoid of the same frequency as the input function, with the only difference being the ratio of input to output amplitude, and a phase shift between the input and output sinusoids. Using the transfer function for the system (T(s) relating the input, u, to the output, y), how can you mathematically determine the amplitude ratio and the phase shift for a given input sinusoidal frequency, ω?

An external consultant recommends that a plant installs a bank of capacitors for power factor correction. this will reduce the peak electrical demand charges by an average of 93 kw every month. the plant current pays $13 per kw in peak demand charges. the capacitor bank will include 223 kw of fixed capacitors, and 183 of variable capacitors. the fixed capacitors cost $59 per kw, and the variable capacitors will cost $65 per kw. the consultant charges 21% of the equipment costs to install the capacitors. because this project will reduce the demand for the electric utility, they are prepared to provide a one-time rebate of $42 per kw of reduced demand. what is the simple payback period for this project (in years)?