Resonance Is a Bargain with Terms: Enhancement, Bandwidth and Ring-Up in Phase Driving

Driving a phase coordinate resonantly multiplies a weak drive by the quality factor Q, at the price of bandwidth. This paper computes the trade for a damped-oscillator model of the phase response. At Q = 100 the on-resonance enhancement is 100, but a detuning of 1 percent cuts the response to 44.72; at Q = 1000 a detuning of only 0.1 percent already cuts the response to 447.21, and the fractional bandwidth is 0.001. Ring-up at a 1 MHz natural frequency takes 3.183e-06 s at Q = 10 but 3.183e-04 s at Q = 1000. Resonance is therefore a bargain with strict terms: enhancement is bought with frequency discipline and with time, and the optimum Q for a transit drive is set by the stability of the available frequency reference, not by the desire for gain. Computed; not measured. SIMULATED, NOT MEASURED.