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DescriptionSecond order transfer function.svg	English: Step responses for a second order system defined by the transfer function: $H(s)={\frac {\omega _{n}^{2}}{s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2}}}$ where $\zeta$ is the damping ratio and $\omega _{n}$ is the undamped natural frequency. The equations were obtained from here, plotted using maxima and edited in a text editor to insert the Greek alphabets in the plot. The equations are: $h_{under}(t)=1-{\frac {1}{\beta (\zeta )}}e^{-\zeta \omega _{n}t}\sin \left(\beta (\zeta )\omega _{n}t+\theta (\zeta )\right),\beta (\zeta )={\sqrt {1-\zeta ^{2}}},\theta (\zeta )=\tan ^{-1}\left({\frac {\beta (\zeta )}{\zeta }}\right)$ $h_{un}(t)=1-\cos \left(\omega _{n}t\right)$ $h_{crit}(t)=1-e^{-\omega _{n}t}\left(1+\omega _{n}t\right)$ $h_{over}(t)=1+{\frac {\omega _{n}}{2{\sqrt {\zeta ^{2}-1}}}}\left({\frac {e^{-s_{1}t}}{s_{1}}}-{\frac {e^{-s_{2}t}}{s_{2}}}\right),s_{1}=\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)\omega _{n},s_{2}=\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)\omega _{n}$
Date	26 May 2011
Source	Own work
Author	Krishnavedala
SVG development InfoField	The SVG code is valid. This Unicode character was created with an unknown SVG tool. Category:Valid SVG created with Other tools:Unicode#Second%20order%20transfer%20function.svg

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from matplotlib.pyplot import *
from numpy import *

wt = linspace(0,15,100)
b = lambda z: sqrt(1. - z**2)
t = lambda z: arctan(b(z)/z)
h1 = lambda wt,z: 1. - exp(-z*wt)*sin(b(z)*wt+t(z))/b(z)
h2 = lambda wt: 1. - cos(wt)
h3 = lambda wt: 1. - exp(-wt)*(1.+wt)
s1 = lambda z: (z + sqrt(z**2-1.))
s2 = lambda z: (z - sqrt(z**2-1.))
h4 = lambda wt,z: 1. + ( (exp(-s1(z)*wt)/s1(z)) - \
	(exp(-s2(z)*wt)/s2(z)) ) / (2.*sqrt(z**2-1.))

fig = figure(figsize=(8,4))
ax = fig.add_subplot(111)
ax.grid(True)
ax.plot(wt,h2(wt),'g',label=r"undamped $(\zeta=0)$")
ax.plot(wt,h1(wt,.5),'b',label=r"under $(\zeta=0.5)$")
ax.plot(wt,h3(wt),'r',label=r"critical $(\zeta=1.0)$")
ax.plot(wt,h4(wt,1.5),'m',label=r"over $(\zeta=1.5)$")
ax.set_ylim(0,2)
ax.minorticks_on()

leg = ax.legend(frameon=False,handletextpad=.05)
setp(leg.get_texts(),fontsize=10)
ax.set_xlim(0,15)
ax.set_xlabel(r"$\omega t$",fontsize=15)
ax.set_ylabel("Step response",fontsize=12)
fig.savefig("Second_order_transfer_function.svg",bbox_inches="tight",\
	pad_inches=.15)

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beta(zeta) := sqrt(1-zeta^2);
theta(zeta) := atan(beta(zeta)/zeta);
h_under(wt) := 1 - beta(0.5)^-1*exp(-0.5*wt)*sin(wt*beta(.5)+theta(0.5));
h_un(wt) := 1 - cos(wt);
h_crit(wt) := 1 - exp(-wt) * (1+wt);
s1(zeta) := zeta+sqrt(zeta^2-1);
s2(zeta) := zeta-sqrt(zeta^2-1);
h_over(wt) := 1 + ((exp(-s1(1.5)*wt)/s1(1.5))-(exp(-s2(1.5)*wt)/s2(1.5)))/(2*sqrt(1.5^2-1));
load(draw);
draw2d(dimensions=[800,400],terminal=svg,
  user_preamble="set mxtics; set mytics;",
  grid=true, yrange=[0,2], xlabel="omega t",
  line_width=1.5, ylabel="Step response",
  key="under (zeta=0.5)",color=blue,explicit(h_under(wt),wt,0,15),
  key="critical (zeta=1)",color=red,explicit(h_crit(wt),wt,0,15),
  key="over (zeta=1.5)",color=magenta,explicit(h_over(wt),wt,0,15),
  key="undamped (zeta=0)",color=green,explicit(h_un(wt),wt,0,15)
 );

Licensing

Krishnavedala, the copyright holder of this work, hereby publishes it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Attribution:

Krishnavedala

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attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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