**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 t**_{0} (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** |