Page - (000025) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

2.4 Frequency,Gain, andPhase 15 (a) (d) (b) (e) (c) (f) Fig.2.2 Dynamics,gain,andphaseofthelow-passfilterinEq.2.11inresponsetosinewaveinputs atvarying frequencies,ω.Detailsprovided in the text.a–cDynamicsgivenbyamultipliedby the transfer function on the right-hand side of Eq.2.10.dResponse of Eq.2.11 to unit step input. e The scalingof theBodegainplot is 20log10(gain). That scaling arises from the relationbetween themagnitude, M =|G(jω)|, and power, P =M2, of a signal at a particular frequency,ω, or equivalentlyM =√P. If we consider gain as themagnitude of the output signal, then the scale for the gain is given as 20log10( √ P)=10log10(P), the standarddecibel scaling for the relative powerofa signal. fBodephaseplot We can learn about a systemby studying how it responds to different kinds of fluctuatingenvironmentalinputs.Inparticular,howdoesasystemrespondtodifferent frequenciesof sinewave inputs? Figure2.2 shows the response of the transfer function in Eq.2.11 to sinewave inputs of frequency,ω. The left columnof panels illustrates the fluctuating output in response to thegreen sinewave input.Theblue (slow) andgold (fast) responses