File:Second order transfer function.svg
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 |
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 |
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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)
);
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