File:Second order transfer function.svg
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Summary
| Description |
English: Step responses for a second order system defined by the transfer function:
where is the damping ratio and 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: |
| Date | |
| Source | Own work |
| Author | Krishnavedala |
| SVG development |
Source code
This media was created with Python (general-purpose programming language) and NumPy (numerical programming package for the Python programming language) and Matplotlib (comprehensive library for creating static, animated, and interactive visualizations in Python)
Here is a listing of the source used to create this file.
Here is a listing of the source used to create this file.
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)
Source code
This media was created with Maxima (computer algebra system)
Here is a listing of the source used to create this file.
Here is a listing of the source used to create this file.
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:
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- 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.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 15:57, 7 June 2011 | | 631 × 356 (80 KB) | wikimediacommons>Krishnavedala | figure correction after correcting the equations |
File usage
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