File:Second order transfer function.svg

From Wikipedia, the free encyclopedia

$Step responses for a second order system defined by the transfer function: H ( s ) = ω n 2 s 2 + 2 ζ ω n s + ω n 2 {\displaystyle H(s)={\frac {\omega _{n}^{2}}{s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2}}}} where ζ {\displaystyle \zeta } is the damping ratio and ω n {\displaystyle \omega _{n}} is the undamped natural frequency. The equations were obtained from here, plotted using maxima and edited in a text editor to insert the Greek alphabets in the plot. The equations are: h u n d e r ( t ) = 1 − 1 β ( ζ ) e − ζ ω n t sin ⁡ ( β ( ζ ) ω n t + θ ( ζ ) ) , β ( ζ ) = 1 − ζ 2 , θ ( ζ ) = tan − 1 ⁡ ( β ( ζ ) ζ ) {\displaystyle h_{under}(t)=1-{\frac {1}{\beta (\zeta )}}e^{-\zeta \omega _{n}t}\sin \left(\beta (\zeta )\omega _{n}t+\theta (\zeta )\right),\beta (\zeta )={\sqrt {1-\zeta ^{2}}},\theta (\zeta )=\tan ^{-1}\left({\frac {\beta (\zeta )}{\zeta }}\right)} h u n ( t ) = 1 − cos ⁡ ( ω n t ) {\displaystyle h_{un}(t)=1-\cos \left(\omega _{n}t\right)} h c r i t ( t ) = 1 − e − ω n t ( 1 + ω n t ) {\displaystyle h_{crit}(t)=1-e^{-\omega _{n}t}\left(1+\omega _{n}t\right)} h o v e r ( t ) = 1 + ω n 2 ζ 2 − 1 ( e − s 1 t s 1 − e − s 2 t s 2 ) , s 1 = ( ζ + ζ 2 − 1 ) ω n , s 2 = ( ζ − ζ 2 − 1 ) ω n {\displaystyle h_{over}(t)=1+{\frac {\omega _{n}}{2{\sqrt {\zeta ^{2}-1}}}}\left({\frac {e^{-s_{1}t}}{s_{1}}}-{\frac {e^{-s_{2}t}}{s_{2}}}\right),s_{1}=\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)\omega _{n},s_{2}=\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)\omega _{n}}$

Size of this PNG preview of this SVG file: 631 × 356 pixels. Other resolutions: 320 × 181 pixels | 640 × 361 pixels | 1,024 × 578 pixels | 1,280 × 722 pixels | 2,560 × 1,444 pixels.

Original file ‎(SVG file, nominally 631 × 356 pixels, file size: 80 KB)

This is a file from the Wikimedia Commons. Information from its description page there is shown below.
Commons is a freely licensed media file repository. You can help.

Summary

DescriptionSecond order transfer function.svg	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

Source code using w:python (programming language) with numpy and matplotlib toolboxes
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(-zwt)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(z2-1.)) s2 = lambda z: (z - sqrt(z2-1.)) 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 using w:python (programming language) with numpy and matplotlib toolboxes

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)

The maxima source code
beta(zeta) := sqrt(1-zeta^2); theta(zeta) := atan(beta(zeta)/zeta); h_under(wt) := 1 - beta(0.5)^-1exp(-0.5wt)sin(wtbeta(.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) );

The maxima source code

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

I, the copyright holder of this work, hereby publish it under the following license:

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

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	Dimensions	User	Comment
current	15:57, 7 June 2011	631 × 356 (80 KB)	Krishnavedala	figure correction after correcting the equations
	19:25, 26 May 2011	800 × 400 (36 KB)	Krishnavedala	erroneous under-damped curve corrected.
	18:28, 26 May 2011	800 × 400 (36 KB)	Krishnavedala

File usage

The following pages on the English Wikipedia use this file (pages on other projects are not listed):

Linear control

Global file usage

The following other wikis use this file:

Usage on am.wikipedia.org
- አር ኤል ሲ ዑደት
Usage on vi.wikipedia.org
- Hệ thống điều khiển
Usage on zh.wikipedia.org
- 控制系統

Retrieved from "https://en.wikipedia.org/wiki/File:Second_order_transfer_function.svg"

From Wikipedia, the free encyclopedia

Summary

Licensing

Captions

Items portrayed in this file

depicts

creator

some value

copyright status

copyrighted

copyright license

Creative Commons Attribution-ShareAlike 3.0 Unported

source of file

original creation by uploader

inception

26 May 2011

File history

File usage

Global file usage