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.

Grab the [t] slider to set the time t, press the [>>] key to animate the system, or press the [>] or [<] key to increase or decrease t by 0.1.

Grab the [k] , or [ω] slider to vary those parameters.

The [Bode and Nyquist Plots] key toggles display of three windows on the right side of the screen. The top window displays the amplitude A as a function of ω. The middle window displays the negative of the phase lag φ as a function of ω. The bottom window displays the complex number k/p(iω) (where p(s) = s + k is the characteristic polynomial of the operator). The magnitude of this complex number, indicated by a yellow radial segment, is the amplitude A , and its angle, indicated by a green arc, is -φ.

Roll the cursor over the amplitude window to cause a horiziontal yellow line to appear relating the amplitudes in the three top windows, along with a readout of the amplitude. 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.


c 2001 H. Hohn