Amplitude and Phase: First Order -- Help At left is a representation of a first order system controlled by the equation x' + kx = k cos(ωt). The input signal is represented by the cyan level, the output by the yellow level, and the coupling between them by a white diagonal. The equation governing this system is displayed in yellow at the top. k is the coupling constant and ω is the circular frequency of the sinusoidal input signal. To the right, the input signal cos(ωt) is graphed in cyan and the system response x is graphed in yellow. Diamonds indicate the current values of cos(ωt) and of x , and a vertical white line between them indicates the difference in their values. A grey vertical line measured by a red segment indicates the time lag t0 (which is also read out in red at the bottom of the screen, below a readout of the period P in cyan). Rolling the cursor over the graphing window produces crosshairs and a readout of the values of t and x. The horizontal crosshair line is continued in the window displaying the system. The time value is set using a slider under the window. The [>>] key starts an animation. The [<<] key resets t to t = 0. Grab the [k] , or [ω] slider to vary those parameters. The [Bode plots] key toggles display of two windows on the right side of the screen. The top window displays the amplitude A of the sinusodial response as a function of ω. The window below it displays the negative of the phase lag φ as a function of ω. The [Nyquist plot] key toggles display of a window at lower right, showing a portion of the complex plane. On it, a grey curve traces the path traversed by the complex gain k / p(iω) (where p(s) = s + k is the characteristic polynomial) as ω varies over positive values. A yellow diamond marks the value of this complex number for the chosen value of ω. A yellow line segment connects it to the origin. The length of this segment is the amplitude A, and the angle up from the positive real axis, marked by a a green arc, is -φ. Roll the cursor over the amplitude window to cause a horizontal yellow line to appear in that window and in the graphing window, marking the maximal displacement, and a readout of that maximal value. Roll the cursor over the phase shift window to cause a readout of the phase shift. Note: These are not quite truly Bode or Nyquist plots. A Bode plot graphs log(A) vs log(ω) or -φ vs log(ω). A Nyquist plot displays k/p(i ω) as omega ranges from -∞ to +∞ it has a portion above the real axis which is symmetric with what is drawn.   © 2001 H. Hohn