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<H1>Example 4</H1>
<HR>
<P class=3Dequation><A href=3D"javascript:history.go(-1)"><IMG =
alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<P>Draw the Bode Diagram for the transfer function:</P>
<P class=3Dequation><IMG height=3D39 alt=3D"Transfer function"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example4/B=
odeEx4Eq.gif"=20
width=3D196></P>
<H2>Step 1: Rewrite the transfer function in proper =
form. </H2>
<BLOCKQUOTE>
<P>Make both the lowest order term in the numerator and denominator=20
unity. The numerator is an order 1 polynomial, the denominator =
is order=20
3.=20
<P class=3Dequation><IMG alt=3D"Rewritten Transfer Function"=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example4/i=
ndex.14.gif">=20
</P></BLOCKQUOTE>
<H2>Step 2: Separate the transfer function into its constituent =
parts.</H2>
<BLOCKQUOTE>
<P>The transfer function has 4 components:</P>
<UL>
<LI>
<P>A constant of -10 </P>
<LI>
<P>A pole at s=3D-10 </P>
<LI>
<P>A doubly repeated pole at s=3D-1 </P>
<LI>
<P>A zero at the origin </P></LI></UL></BLOCKQUOTE>
<H2>Step 3: Draw the Bode diagram for each part.</H2>
<BLOCKQUOTE>
<P>This is done in the diagram below.</P>
<UL>
<LI>
<P>The constant is the cyan line (A quantity of 10 is equal to 20 =
dB). =20
The phase is constant at -180 degrees (constant is negative). </P>
<LI>
<P>The pole at 10 rad/sec is the blue line. It is 0 dB up to =
the break=20
frequency, then drops off with a slope of -20 dB/dec. The =
phase is 0=20
degrees up to 1/10 the break frequency then drops linearly down to =
-90=20
degrees at 10 times the break frequency. </P>
<LI>
<P>The repeated pole at 1 rad/sec is the green line. It is 0 =
dB up to=20
the break frequency, then drops off with a slope of -40 dB/dec. The =
phase is=20
0 degrees up to 1/10 the break frequency then drops linearly down to =
-180=20
degrees at 10 times the break frequency. The magnitude and =
phase drop=20
twice as steeply as those for a single pole. </P>
<LI>
<P>The zero at the origin is the red line. It has a slope of =
+20=20
dB/dec and goes through 0 dB at 1 rad/sec. The phase is 90 =
degrees.=20
</P></LI></UL></BLOCKQUOTE>
<H2>Step 4: Draw the overall Bode diagram by adding up the results =
from=20
step 3.</H2>
<BLOCKQUOTE>
<P>The overall asymptotic plot is the translucent pink line, the exact =
response is the black line. </P></BLOCKQUOTE>
<P class=3Dfigure><IMG height=3D660 alt=3D"Bode Plot"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example4/i=
ndex.15.gif"=20
width=3D648></P>
<HR>
<P class=3Dequation><A href=3D"javascript:history.go(-1)"><IMG =
alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A> &n=
bsp; &nb=
sp; =20
<A=20
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example5/=
index.html"><IMG=20
alt=3DNext=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/RightArr=
ow.gif">Example=20
5<IMG alt=3DNext=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/RightArr=
ow.gif"></A></P>
<HR>
<P>Please <A href=3D"mailto:erik_cheever@swarthmore.edu">contact me</A> =
with any=20
comments, positive or negative. <A=20
href=3D"http://www.swarthmore.edu/NatSci/echeeve1">Erik Cheever's home=20
page.</A></P>
<P> </P><!-- #EndEditable -->
<HR class=3DfullWidth>
<DIV id=3DsiteInfo>
<P class=3Dequation>=A9 Copyright 2005-2007 <A=20
href=3D"mailto:erik_cheever@swarthmore.edu?subject=3DLinear Physical =
System Analysis (Copyright notice)">Erik=20
Cheever</A> This page may be freely used for =
educational=20
purposes.<BR></P>
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</P><A href=3D"http://www.swarthmore.edu/NatSci/echeeve1">Erik=20
Cheever</A> <A=20
href=3D"http://www.engin.swarthmore.edu/">Department of=20
Engineering</A> <A=20
href=3D"http://www.swarthmore.edu/">Swarthmore College</A> =
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<A name=3Dex5></A><!--webbot bot=3D"Include" =
U-Include=3D"Example5/index.html" TAG=3D"BODY" startspan -->
<DIV id=3DoverDiv=20
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absolute"></DIV><!-- #BeginEditable "MainText" -->
<H1>Example 5</H1>
<HR>
<P class=3Dequation><A href=3D"javascript:history.go(-1)"><IMG =
alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<P>Draw the Bode Diagram for the transfer function:</P>
<P class=3Dequation><IMG height=3D47 alt=3D"Transfer Function"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example5/i=
ndex.3.gif"=20
width=3D130></P>
<H2>Step 1: Rewrite the transfer function in proper form.</H2>
<BLOCKQUOTE>
<P>Make both the lowest order term in the numerator and denominator=20
unity. The numerator is an order 1 polynomial, the denominator =
is order=20
2.=20
<P class=3Dequation><IMG height=3D77 alt=3D"Rewritten Transfer =
Function"=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example5/i=
ndex.4.gif"=20
width=3D361> </P></BLOCKQUOTE>
<H2>Step 2: Separate the transfer function into its constituent =
parts.</H2>
<BLOCKQUOTE>
<P>The transfer function has 4 components:</P>
<UL>
<LI>
<P>A constant of 6 </P>
<LI>
<P>A zero at s=3D-10 </P>
<LI>
<P>Complex conjugate zeros at the roots of=20
s<SUP>2</SUP>+3s+50, <BR>with <IMG height=3D43 alt=3D"omega_0, =
zeta"=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/eqs/BodeRu=
37.gif"=20
width=3D221> </P></LI></UL></BLOCKQUOTE>
<H2>Step 3: Draw the Bode diagram for each part.</H2>
<BLOCKQUOTE>
<P>This is done in the diagram below.</P>
<UL>
<LI>
<P>The constant is the cyan line (A quantity of 6 is equal to 15.5=20
dB). The phase is constant at 0 degrees. </P>
<LI>
<P>The zero at 10 rad/sec is the green line. It is 0 dB up to =
the=20
break frequency, then rises with a slope of +20 dB/dec. The =
phase is 0=20
degrees up to 1/10 the break frequency then rises linearly to +90 =
degrees at=20
10 times the break frequency. </P>
<LI>
<P>The plots for the complex conjugate poles are shown in =
blue. They=20
cause a peak=20
=
of <BR> &=
nbsp; =20
<IMG height=3D58 alt=3D"Peak Height"=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example5/i=
ndex.5.gif"=20
width=3D331><BR>at a frequency of 7.07 rad/sec. This is shown =
by the=20
blue circly. The phase goes from the low frequency asymptote =
(0=20
degrees)=20
=
at<BR> &=
nbsp; =20
<IMG height=3D86 alt=3D"Phase Transition, low"=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example5/i=
ndex.6.gif"=20
width=3D226><BR>to the high frequency asymptote=20
=
at<BR> &=
nbsp; =20
<IMG height=3D89 alt=3D"Phase Transition, High"=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example5/i=
ndex.7.gif"=20
width=3D226> </P></LI></UL></BLOCKQUOTE>
<H2>Step 4: Draw the overall Bode diagram by adding up the results =
from=20
step 3.</H2>
<BLOCKQUOTE>
<P>The exact response is the black line. </P></BLOCKQUOTE>
<P class=3Dequation><IMG height=3D664 alt=3D"Bode Plot"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example5/i=
ndex.8.gif"=20
width=3D694></P>
<HR>
<P class=3Dequation><A href=3D"javascript:history.go(-1)"><IMG =
alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG alt=3DBack=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A> &n=
bsp; &nb=
sp; =20
<A=20
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/Example6/=
index.html"><IMG=20
alt=3DNext=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/RightArr=
ow.gif">Example=20
6<IMG alt=3DNext=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/RightArr=
ow.gif"></A></P>
<HR>
<P>Please <A href=3D"mailto:erik_cheever@swarthmore.edu">contact me</A> =
with any=20
comments, positive or negative. <A=20
href=3D"http://www.swarthmore.edu/NatSci/echeeve1">Erik Cheever's home=20
page.</A></P>
<P> </P><!-- #EndEditable -->
<HR class=3DfullWidth>
<DIV id=3DsiteInfo>
<P class=3Dequation>=A9 Copyright 2005-2007 <A=20
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