MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Depart,rnent of Electrical Engineering arlrl Cornpllt,er Science
Spring Terrr~2007
Prol~lernSet 2
Problem 1: For tlle follor3-ingfor\%-arrl-pathtrarlsfer fim<:tionG ( s )arlrl feerll~ack-pat11trarlsfer fim<:t,ionH ( s )
find t,lle st,anclar(i trarlsfer filrlct,ior~sT ( s ) ,L ( s ) , Lo(s), ancl C / R ( s ) ,ancl simplify tllerr~ir~t,oratios of
polynomials. Also. find t l ~ ecllaracteristic eql~at,ionP ( J )and sirnplify it ir~toa. sirnple polynomial.
Definit,ions:
T(s)=-GH
Loop Trar~srr~issior~
L(s)=GH
Loop Trar~sferFim<:tion
Lo(s)=L(s)/K
C/R(s)=-
Norrrlalizerl Loop Trarlsfer Fim<:tion
Closecl Loop Transfer Fl~nction
Problem 2: For t,lle systerrl s110rnr11)elorr. rleterrr~ir~e
hor3- t l ~ efract,iorlal change in <:lose(i-loo11
gain (iepenrls
on fractiorlal changes in tlle pr%rr%rrleters
A, B. ancl C .
Problem 3: Sirr~plifjrt,lle follorrir~gt~lockrliagrarr~ir~t,oa single l~lock:
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 4: Derive t,lle l~lockrliagrarr~sfor t l ~ e01) arnp circuits l)elor%-..issllrr~ingillat t,lle ol)-arrlp gair~A(.?)
is cor~st,ar~t
and goes to infinity, (in t l ~ earlsrrers clleck r r i t l ~expectations'!
Problem 5: For t,lle follorrir~g01) arnp circuit, dram the l)lo<:k(iiagrarn arlrl rnanipillat,e it i r ~ t othe ilnity
feerll~ackforrr~..ilso, r r l ~ a tis t l ~ eideal trarlsfer filnction?
Problem 6: Sal Lektra wants to t)ililrl an electric car ilsirlg a DC rrlotor t,llat he foimcl on Jllnl<yar(i11-ars.
He rrlnilr~tst,lle rrlot,or in a. car, r r i t l ~the shaft, clirectly driving t l ~ ewheels. The wheels l ~ a v ea (liarr~eter
n tor%-st l ~ ec ar 1)ehincl his pickilp a t 50 rnph, he rrleasures 80 volts a1:ross the rrlotor
of 22 inches. W ~ e he
terminals. tV11e11 80 volts is connect,e<lt,o t,l~errlot,or terrninals, it t,akes 40 secon(is t,o reach 49 rnph
frorr~a standing start,. Ignore friction and in(ii~ctancr,
arlrl rerr~ernt)erillat there are S280 feet in a rnile.
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 7: .i sq~are-rooting<:ir<:~~it,
~ l s i r ~agtechniqlle sirrlilar to t,llat of t l ~ edivider sllo~vr~
in l e c t , ~ r eis,
for this circuit (ass~lrningan ideal 011 arr~p)?
shor3-n t)elom. 11-hat is t,lle ideal i n p l l t , - o ~ ~ t relationship
p~~t,
11-hat is t,lle i n p ~ ~ t - o ~ ~t,rar~sfer
t p l ~ t filrlct,ior~if t l ~ eo11 arr111 trarlsfer fim<:tion is
Det,errrlirle t,lle range of i n p ~ voltages
~t
for w l ~ i c lthe
~ sqllare rooter is st,at)le for this A(.?). How does
i n p ~ level
~ t affect t,lle speed of response?
R
VA
+
vo
R
vx -
Multiplier conn,ed?d
as squarer clrcult
r;y
= 'I:,?] /lO
-,
-
Problem 8: .i DC rrlot,or is t,o l ~ 11sec1
e
in a posit,ion servornr<:hanis~r~,
l111t illere is a choice as t,o llorv t l ~ e
rr~acllir~e
car1 1)e (:nrlrle<:ted. Tlle lnr%(iis r%r~i nertia conne<:tr(ito the rrlot,or throllgh a flexil~le(sprirlgy)
shaft. Tlle rrlot,or car1 t)e connect,e<las an arrnat~~re-controllprl
fielrl
rrlacl~irlesupplied with cor~star~t
c ~ l r r e r frorr~
~ t r%(:11rre11tsollr(:e. nr r%sa field <:ontroller1rrlacl~irles~lpplieclr%-it,h<:onstantarrrlat,llre <:~lrrent,
frorr~a <:~lrrent
source, as shor3-n in Figures 1 arlrl 2. These figures shnr3- t,lle arrr~at~~re-~:or~trolle~l
arlrl
fielcl-controllecl <:onnections,respect,ively.
11-hen operate11 in t,lle arrnat~~re-controllprl
mode (Figure l ) , a. rrrxrial~lrvolt,age is applierl to the arrrlatllre as an i n p ~ ~
t t, the
~ I I I O ~ Ot~~ cllar~get,lle shaft ~ ~ n s i t , i n r.ilternatively,
~.
the r~rxrial~le
<:ontrol
t,o r-ary the sllaft posit,ion rvller~t,lle rrlotor is operated in t l ~ e
voltage is applierl to t,lle fielrl trrrr~ir~als
fielcl-controllecl rrlorle (Figure 2).
Derive the trarlsfer filr~ctior~s
t,llat relate the arlg~llarrlisplacernent of the load inert,ia. t o the input,
voltage for t l ~ etrro (lifferent cor~fig~lratior~s
illl~st,raterlin Figures 1 and 2. Make 1111 variat)le rlarrles
(~~referat)ly
for q~lantitiesyou need in t,lle l)lo<:krliagrarns. Carefillly
corlsist,er~tr r i t l ~the la11 hanrlo~~ts)
specify ~ ~ 0 1 1<:hoi(:es.
1
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].
Motor Armature Motor Field
Winding Figure 1: Arr11at11recc111t,ro1
+-=-;I
Ra
if
La
;
~$2.
GmIfw = vm
v
Motor Field
Winding I
a
0
T
J
Jload
Motor Armature Figure 2: Fielrl cor~trol
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].