MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Depart,rnent of Electrical Engineering ar~rlCornpllt,er Science
Spring Term 2007
Prol~lernSet 7
Problem 1: .1ft,er rlozir~goff in l e c t , ~ r e 011e
,
nf 7.0111 <:lass~r~at,es
asks you the follor3-ing (111estion: '.Lag
acids negative phase. tl111s lir'cr.f:lis%ngt l ~ ephase
corr~pensatio~~
rr~argin.I1-11y rvol~lclme ever r3-ant t,o rlo
that?'' Explain l~orryou rnol~l(iarlswer hirn. Problem 2: Consicler the following operational amplifier <:ir<:l~it,
where R = 1 kCl ar~rlC = 1000 IIF:
Asslnning an irleal o11 amp:
(a) IJ-llat is
LM?
l',
(.9) '
(b) Sketch the o l ~ t p l response
~t
for 1.; =
rarnp
( c ) m a t is t l ~ ephase rnargin, $,,, of L(s)'!
(d) Use klatlat~t,o plot t l ~ eo11tp11t response for 1.; =
(e) 11-hat is $,,, of L(s)Y
( f ) Use klatlat~t,o plot t,lx o ~ t , p lresponse
~t
for 1.; =
% ramp
5 ramp
Cite as: James Roberge and Kent Lundberg, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Problem 3: .i fee(il)a<:k-toI)ologJr
power amplifier is sllo~vr~
t)elom. Ass~nneillat t,lle 01) amp is ideal. Select
a v a l ~ ~for
e t,lle capacitor C s11c11t,llat the syst,errl phase rr~argir~
is 4:". Ass~nneillat R = 10 ldl arlrl
t l ~ etrarlsfer filrlct,ior~of t l ~ eporrer st,age is Problem 4: Tlle fixed elerr~er~t
of a. llnit,y-gain fee(l1)ack systerrl is
To improve the lorr f r e q ~ ~ e n ct)el~t~vior
y
of t,l~issystern, a lag net,mork is a(irle(i t,o G ( s ) r%-it,hru = 5, so
illat the lonp t,rarlsfer ~ ~ I I I ~ r10\%~ ~ O1)ec0111es
I I
(a) Fir111T and IC so t,llat t,lle systern has a t least 30" of phase margin.
(b) If rre ir~creaserl<u rvitllo~~t
<:hanging any ot,ller parameters of t,lle system. what effect ~volllrlthat
have? Sl~pportyollr argllrrlerlt r%-it,hthe appropriat,e root-locus plot,s.
Problem 5: Tlle r n e a s ~ ~ rfreqllency
e~l
response of a ~list,ril)l~t,e~l
systrrn exhit)its t l ~ l~eharior
e
of the follor%-ing
trarlsfer fim<:tior~:
Corr~y)ens;~te
tllis systern wit11 ;L g ~ i a.rlrl
n a. leiul rlet\%-nrk\%-it,l~
o 110
greater than 10. L1;rz;irnize crossover
a phase
fre(l~lerl<:y
and rr~air~t,air~
rnargin of 4:". (Do riot use T\t%tLt%l)).
Cite as: James Roberge and Kent Lundberg, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Problem 6: Consi(1er the following loop trarlsfer fi~nct,ion:
L(.) = Gr: ( s )
100
( 3 + 1)(0.13 + 1)(0.015
1)
+
IJ-hat is t l ~ crossover
e
freqi~ency,phase margin, and steady stat,e error to a. st,ep of t l ~ o
e ~ l e loop
r ~ systern:
Gr:(s) = l?
For each of t l ~ efollor3-ingc o ~ n ~ ~ e n s achoi<:es,
t i o ~ ~ <:ornpensat,ethe sj'sterrl for 4: rlegrees of phase rrlargir~
and rrlaxirrlize the <:rnssnverfre(l11e11~:y.111 rack1 case, rrhat, are t,he rlecessarji cornpensator pararnet,ers,
rrklat, is tlle rlew <:rossoverfreqilency, and rrklat is t,kle stearly state error to a. step'!
(a) Re(hl<:edDC gain cornpensation:
1
C:(: ( s ) = I(H
(b) Dorrlir~ar~t
pole cornpensation:
Ii
Gr: ( s ) = 2
3
(c) Lag corr~pen%~tiorl:
7s
+1
Gr:( s ) = ors + 1
(d) Lead cornpensation (IISPa11 o
110
larger t,han 10): Gr: ( 9 ) = ICL
ors + 1 7s+l
Cite as: James Roberge and Kent Lundberg, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].