AMPLITUDE AND PHASE: FIRST ORDER

At left is a representation of a first order system controlled by the equation $$\dot{x} + kx = k \cos(\omega t)$$. 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 $$\omega$$ is the angular frequency of the sinusoidal input signal.

To the right, the input signal $$\cos(\omega t)$$ is graphed in cyan and the system response $$x$$ is graphed in yellow. Diamonds indicate the current values of $$\cos(\omega t)$$ and of $$x$$, and a vertical white line between them indicates the difference in their values. A gray 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 system window produces the horizontal line and a readout of the value of $$x$$. The horizontal crosshair line is continued in the window displaying the system.

Use the slider handle below the graphing window to select a value of $$t$$. Animate the system using the [>>] key. During the animation the key changes to [||], and selecting it will stop the animation. At the end of the animation the key changes to [<<], which resets $$t$$ to zero.

Grab the $$[k]$$, or $$[\omega]$$ 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 $$\omega$$. The window below it displays the negative of the phase lag $$\phi$$ as a function of $$\omega$$.

The [Nyquist plot] key toggles display of a window at lower right, showing a portion of the complex plane. On it, a gray curve traces the path traversed by the complex gain $$\frac{k}{p(i\omega)}$$ (where $$p(s) = s + k$$ is the characteristic polynomial) as $$\omega$$ varies over positive values. A yellow diamond marks the value of this complex number for the chosen value of $$\omega$$. 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 $$-\phi$$.

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 $$\log(\omega)$$ or $$-\phi$$ vs $$\log(\omega)$$. A Nyquist plot displays $$\frac{k}{p(i \omega)}$$ as $$\omega$$ ranges from $$-\infty$$ to $$+\infty$$ it has a portion above the real axis which is symmetric with what is drawn.

© 2001 H. Hohn and H. Miller