**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.**
**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** |