MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Depart,rnent of Electrical Engineering arlrl Cornpllt,er Scien1:e
Spring Terrr~2007
Prol~lernSet 1
Tlle prereq~~isit,es
for 6.302 are lirlear systerns, circuits, physics. and rnath. Officially, illis req~~irerr~ent
is
r%-it,hcircl~itsis rxss~lrrled,s11c11as i r ~6.002 nr 6.071
6.003 or 2.003 or 16.040. P r e v i o ~ ~lat~orat,(>r~.
s
. experiet~c:e
nr 16.0.40. lye p.;~lw)
tlssnrne t,ll;~tyou IGLV~
not f<,rg<>tt,rn
8.01 p.;Lrlrl8.02, p.;Lrlrlttfi~tyon 1~rvelmrl sorr~eexposure
t,o cornl~lexrxrial~les(18.01 is belpfi~l,l111t overkill). This prol~lernset sho111clremind yo^^ of what yo11 already
knorr. You sl~ol~lrl
feel <:ornforfal)lerritl~t l ~ efollowir~g~ ~ r o t ~ l eifr nyo^^
s intend t,o take 6.302.
Problem 1: Sketcl~(1)y hand) t l ~ eBncle plot (rnagnit~ldeand phase) for each of t,lle following trarlsfer
fimctio~~s.
1
(a) H I (s) = -s +1
3
(1)) Hz (s) = -,3 + 1
(1:) H:3 (s) =
(,?
+ I)(,?+ 10)
,s(O.O1s
+ 1)(0.001s+ 1)
Problem 2: For a single-pole syst,ern of t l ~ eforrr~
express t l ~ efollor3-ing qllantit,ies in t,errrls of the t,irr~rcor~st,ar~t
(7): f h , t l ~ e3dB i)an~lr3-i~lth
in llertz (the
freqllencjr at r3-hich t,lle rrlagr~it~lrle
of t l ~ efreq~~ency
response is 0.707 of the DC gain). arlrl t,., t,lle 10%-90%
rise tir~leof t,kle step resporlse.
1;I
Problem 3: Find the trarlsfer fimction v(,s) of the follorrir~go11 arnp circuit,. dss~lrnet l ~ e011 arnp is ideal.
Use tllr forrr~(7,s + 1) for t l ~ epoles arlrl zeros.
cF
vI
R
R
Cl
.Vo 1OR
Problem 4: Use tile Firm1 Value Tlleorerrl ti) find t,l~e
s t ~ i ~ (stat,e
I y I P S ~ O I I S Pd the follimir~gt,rar~sferfilr~l:tior~
t,o an inpllt urlit rarnp.
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 5: hIatc11 e a c l ~step response to t,lle appropriate pole-zero plot in t,lle figure 1)elo~v
I
I
Problem 6: D r a v a pole-zero plot for F ( J )
Using geornetry alone, fir111the arlgles i F ( - 3 ) , i F ( - 1 ) . L F ( l ) , and i F ( 3 )
Problem 7: l i 1 1 1 are giver1 t,lle following pole-zero plot for G ( J )and tolcl t,llat IG(0) = 2. Usir~ggeornet,ry
alone. find G ( j 2 )1.
I
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 8: Find a sirr~r~le
rx111efnr
Problem 9: '4 ssolirl disc: of radius r. ar~rlrrlass rrr, ir~itiallyat rest, is acted on I1y a. tar~ger~tial
force F. IJ-hat
is t,lx ang~llarvelocity d(t)'!
Problem 10: Drarr~t l ~ efolio\%-ing plot,s I1y l~anclfor the fim<:tionL(.s). Do not use \Iat,lal)
(a) The pole-zero plot,.
(I)) The Bncle plot (rnagnit~~rle
versus freqllency and ar~gleversus freq~~ency).
I3e sure to rlote ~ r h e r et l ~ e
real plot deviates frorr~t l ~ estraight-line asyrr~ptot,es. (c) The polar plot (\%-l~ere
t,lx radius r. = L ( j d ) is t l ~ erlistar~cefrorr~the origin, t,lx ar~gleH = L L ( j d ) is t l ~ ea r~gle\%-it,hrespect to t,lx positive real =is. ar~rld is the parameter). ((i) The gain/phase plot (a. plot of loglO L ( j d ) versus LL(jw): t,l~isplot car1 t)e cor~str~~cterl
I1y exarr~ir~ir~g
t l ~ eal)ove Bncle plot and r ~ o t i r ~l g
~ o ~t,lx
v rnagnit,l~(leand angle <:hangeas freqllency increases).
(e) Finally, or11:e you have cor~str~~cterl
t,lx atlove plots I1y hancl, gerlerate t l ~ e r r\%-it,h
~ hlatlal~.You rnay
find t,lx follorrir~ghlatlal~cornrnan(1s helpfi~l:h e l p , p l o t , p o l a r , abs , a n g l e , and bode. Corr~rr~er~t
on t l ~ e~1ifferen~:es
1)etrveen your plots and the hlatlal~plot,s ((lifferences are okay. in fact, if
illere aren't (iifferences, rre'll assurrle that you used klatlat) for parts (a)-((i) and rlr(iu<:tpoints...).
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].