Plotting and charting: 2-D function plotting

This example shows a pair of associated function plots. These illlustrate the second-order response of a harmonic or control system. The response equation is

$$F_c(s) = \frac{\omega_n^2}{s^2 + 2c\omega_n s + \omega_n^2}$$ (1)

The plots are the magnitude and phase of this function against $\omega$ where $s=j\omega$ and the function is normalised with $\omega_n=1$.

The six curves are for damping factors, $c$, of $0.01$, $0.2$, $0.5$, $1/\sqrt{2}$, $1$, and $25/7$. The asymptotes and tangents are shown for $c=0.5$ and $c=25/7$. In the latter overdamped case the magnitude response is close to the asymptotes, the phase response is not close to the tangents. When overdamped, a second-order system is essentially a combination of two separate first-order systems.

Second-order response curves. The magnitude and phase response of a system for the damping factors $0.01$, $0.2$, $0.5$, $1/\sqrt{2}$, $1$, and $25/7$. Creation method: Octave, Gnuplot, XFig, ps output.

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