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.