📄 rootlocusexamples .mht
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will be=20
complex instead of real. Just ignore the imaginary part. =
These poles=20
are highlighted on the diagram with dots, the value of "s" that was =
originally=20
specified is shown by an asterisk.<BR><BR>Note: it is often difficult to =
choose=20
a value of s that is precisely on the locus, but we can pick a point =
that is=20
close. If the value is not exactly on the locus, then the =
calculated=20
value of K will be complex instead of real. Just ignore the the =
imaginary part=20
of K (which will be small).<BR><BR>Note also that only one pole location =
was=20
chosen and this determines the value of K. If the system has more than =
one=20
closed loop pole, the location of the other poles are determine solely =
by K, and=20
may be in undesirable locations.<BR></P>
<HR>
<DIV style=3D"FONT-SIZE: smaller; TEXT-ALIGN: center">
<P>=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> =
</DIV><!--webbot bot=3D"Include" i-checksum=3D"56395" endspan -->
<HR class=3DfullWidth>
<A name=3Dex3></A><!--webbot bot=3D"Include" =
u-include=3D"Example3/Example3.html" tag=3D"BODY" startspan -->
<H1>Root Locus: Example 3</H1>
<H2>Transfer function</H2>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><IMG=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple3/imgB0.gif"></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DBckGrnd></A>Xfer Function Info</H2>
<P>For the open loop transfer function, G(s)H(s):<BR>We have n=3D2 poles =
at s =3D 2,=20
-1. We have m=3D1 finite zero at s =3D -3. So there exists =
q=3D1 zero as s=20
goes to infinity (q =3D n-m =3D 2-1 =3D 1).<BR><BR>We can rewrite the =
open loop=20
transfer function as G(s)H(s)=3DN(s)/D(s) where N(s) is the numerator =
polynomial,=20
and D(s) is the denominator polynomial. <BR>N(s)=3D s + 3, and D(s)=3D =
s<SUP>2</SUP>=20
- 1 s - 2.<BR><BR>Characteristic Equation is 1+KG(s)H(s)=3D0, or=20
1+KN(s)/D(s)=3D0,<BR>or D(s)+KN(s) =3D s<SUP>2</SUP> - 1 s - 2+ K( s + 3 =
) =3D=20
0<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2>Completed Root Locus</H2>
<P></P>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><IMG=20
alt=3DRLTot=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple3/RLTotal.png"></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleSym></A>Root Locus Symmetry</H2>
<P>As you can see, the locus is symmetric about the real axis<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleNum></A>Number of Branches</H2>
<P>The open loop transfer function, G(s)H(s), has 2 poles, therefore the =
locus=20
has 2 branches. Each branch is displayed in a different color.<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleStart></A>Start/End Points</H2>
<P>Root locus starts (K=3D0) at poles of open loop transfer function,=20
G(s)H(s). These are shown by an "x" on the diagram above<BR><BR>As =
K→∞ the=20
location of closed loop poles move to the zeros of the open loop =
transfer=20
function, G(s)H(s). Finite zeros are shown by a "o" on the diagram =
above. Don't=20
forget we have we also have q=3Dn-m=3D1 zero at infinity. (We have n=3D2 =
finite poles,=20
and m=3D1 finite zero).<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleReal></A>Locus on Real Axis</H2>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><IMG=20
alt=3DRLAx=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple3/RLRealAxis.png"></P>
<P>The root locus exists on real axis to left of an odd number of poles =
and=20
zeros of open loop transfer function, G(s)H(s), that are on the real=20
axis. These real pole and zero locations are highlighted on =
diagram,=20
along with the portion of the locus that exists on the real =
axis.<BR><BR>Root=20
locus exists on real axis between:<BR>2 and -1<BR>-3 and negative=20
infinity<BR><BR>... because on the real axis, we have 2 poles at s =3D =
-1, 2, and=20
we have 1 zero at s =3D -3.<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleInf></A>Asymptotes as |s| goes to infinity</H2>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><IMG=20
alt=3DRLAsym=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
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