Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis and Dissertation Collection
1949
Gain margin, phase margin and transient response
of some simple servomechanisms.
Hunnicutt, William R.
Cornell University
http://hdl.handle.net/10945/6418
N PS ARCHIVE
1949
HUNNICUTT, W.
to
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93943-5101
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Naval Postgraduate School
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simple feedback system of
to eon* lit oC nn in^u% 6,,
tfea
error
*n
form tmowa
Rfta«Ul*iftg
funct .on KY(p) wUert* p ~ ju, an output 6
Let e
9-
/e.;
~
9
:
r
-
£
.
T*«n 9
ICY';j)
1
+
.
£ (KY(p))
-
a
A,
(r< f.
C
transfer
feedback: line.
)
or
fcfcat
form
tfee
A +A^p -fA^p
a
transfer
KHp)
-
A iaP m
+
transfer functions considered serein will be o~
(A)
t±.e
K
iTp^n
p(prl)(^ Ptl)(
KT(p) -
(C)
_
„K,.
.
fl
ptptl)(^- ptl)(
These transfer
()
h
KY(p)
function may be reduced to
Tfee
device
-^(p^-ej
been s;.ovm
"t
aai
by consiaeraal
aajr
Components? (a)
(»)
synchro transformer
motor vuta
6 o /X
f
'
_
p+1;
as follows:
r unctions ffllgkt
D,C.
J
-
*;;*unt
K(/V)/£
fjv)
field cori,r
\
iplifier
-
-
-
-
K^
form:
©'*—&
KYi
-.f/Rjff
',
Let
WT-
u
:rr:n u in, f's In orcur oX
7-being T; .
L «rg«st
K
n KY( ju)
,
si<i«
<ju4i7Tju^TI7
ere
c<
*
r
7
,
~'* e
»
w !•
**** °^
'i
nay be similarly obtained. Tt.us» in
treaeiisn (C)
giiavaaion o< and
will fc&va balnea
/3
Tie transient res
of these forma ma.
,:a
or
se
0— °^ -ij
h.
^ ~
l.
of eyateaa living t r ana fer function*
bf obtained by rhnuu of
Utt La
tr
I
*•
sfora.
far expression (A)l
-
for
follow,
ifce
ltKY(p)
a at;
itefp lunation end K -
Sm
JC
I,
5U
PW^rptl)
(for* 1.304), r ef ( d )
wfe en a e
e»C*>
.
;
g*'
tu
-f
^i"-it
r
u
ain(/T/2
t
-'/) vu.ere
trans lent reaponaea
of
ilent
r
so;
1
v,
:
'jo
pjTpti)' r f*J
ire
^
-
tan
re. /aired
-1
it
f
uej
p r
itor
"
*
-
n -,
" *o''£
fl©/« t.
Tor
*.;<->
tr
-
fun
r
•
pcpti;
,>+l)e
£
- 3i-6 c
=
Kt
For e i
(4
a
.
fe
8 r 4
/at) & =
constant (d^ /dt* r d /dt)d Q
-3-
-
-(A-jSiji
A ^
""
•
98
A
^^tt^Ti )£tt)~
*
Wi.enc©
£. (t)
m
»r expre
>
KLCt)
=
-,(t)
(8)
TUe solution to
1
-
(p)
-
©CO
K
*irrpti)T^rp?ii
solution
expression (C)
:r
•.
oi"
/r
»..
roa
£(t)
1-
-
t quail OB
,£7(1)
s
£{%), whence
tfcis
Y(p)
K
(a
p(pti)(Vp+i>(/3 p*i)
nee 9 (t) -
-
1
sient response to
tern
n:
-i
)
i
|
XY(p)
I
t._eae
will bo defined ae:
at u,e point w..ere
le of
.
las
(t.
#r*
,
P
©>
-i
-
step input Hill be discussed:,
a
ise
of f
.
£(t)
I
KYCp)
et*r\
i
.
angle of KY(p) diminished by 160° at the
|
- 1.
ismi lonsldered
r(p) -
£•»¥ funotl
#il]
k
pTpti;
KY(pJ -
K
P(prl)(c<ptl3
p(ptD(o<pti;(
-4-
/3
pti'3
Ir
sient responses to
r
*.
func'
largin
.'Ski
r
ins for v
nd
i
step Input will be discussed
«
t.„e
relations between gain
us Values of C<
r
,
keeti sen
iously stated,
p,
J
t
iier/is
e
error signal £cMll
lent response
To e&e*
ti.e
relations
r^uired for
eaeis
5f
.« =
i'(p),
a valu
n
-l.
m
input it is neeesa
U*e
\nu
to determine
r^
considered.
v»
is done by solving for
t^xce
.
tin and
stfcp
a
c .nnot
Die efieete of
era si.own in £urv«
transient *+ spoil sa to
K
signal
%
;ne.
•
K will be si*o*n.
will ue consicert
to be non-linear in tkat ti-e effective
li«d error signal £<
nu
ti
frequency u required to obtain
e
u*en substituting this
v
Ikis
t
to
tip)
K
r
expression (A) it Is
exist line
oint
.
a&ioJa
:t
does not
loeui &t KY(p) does not cross tae
i
KY(p)
[in
its
IS--
axis,
t
--.Qasttred,
K
-
K
i
FTF1T
K
-
inj)
6o+~
J>
-
A
-l
^ie KY(p) - -90 - tan *u
uju''ti
• u
-
/
KY(gQ -
It
-
-90 - tan^u
-
-c70 - t^n" 1 u or
JuJ +1
.
_
- It
*aa((f) t
c7G - -u
(
U*e
—
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/
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-
tefcsfc
v
^
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.
£
N
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6,
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lilii
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3
7
&££
/
Curve} a-1
/I
g
undgr sntfurpte d conditions*
S^tura>ed volute 4f pritor
(
mm* pdptDWpti )(3i>tD
-p^.i*. p
*~=-£e°
ffp^i <£-*
s
f
/
A
—£——
- > ^1
.;
r
.
I
I
1
Express
rgln ©
-
tan ($4 STO) S -
-90 -
•
I
fcaa
-
3?
cot
oCti
U (If °0
+
5)
u atitution of
Xu
- u(lf <*;
tan
-
(p
- i80
<x)
u(
14-
1-
©O?
fc
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z
1 ,£<
rious
tl
is
.iution for u ^.ives u
mrgin, Hien;
KYC ju
\
|
_^
a ~u(Uj*)
1-
cot
-
KY(p) * K
r.-in:
)|
=
for
=
1
and** a
ty
lire**
K
4
*
•
**0
K cos c^
or K -
u
mteere
©-,
v*
1
glv.
©o
<*
—
-
"**
tan
u_o
t n *<»©<
Ired
value of
t
»
(p)
</
:f'
1
~ -*Q
tsrT^u
u r
l
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rl u
""
-
^an'^u
•re
1- o<U
u df^)
"
i
»«
1
1
i
I-
mun
i
XU
i
™
*
refore
r
where
i-
P)
-9
taa
)
l-o<u^ -
*—
o<
u* - 1
:
il
-
,-i1 «uj
tan*
©, - tan
^ KY(p
)
- -870
4
%i.enc©
u,
3 6.
vrv^»t*T -i
- gain
<
OO
tan 90 or
In 9
-
tan u(l* op
- «70
g
u
_
- 1- KY(p)
go.;
1
u
©< u
;
coa
-
1
^r
v*
~
I
reaaion
((
Margins
..««•
^
(
"T'ju,< = 9 -
ju)
»
-90 - ta*~*u -tam c<ii
1
1
r |90 s -tan"* !! -tan"
u(^
u
V
-
3
4 <£)
«-
© - 180
1- \5U C
u(l~o<iSu c ) + W (O^f/^
oot $ - cot 8 -
u 5 - tt*(o0
- u(l ±*t +
to<:+/3 )cot$>
.
*
a
cofe8^
KY(ju)|?
c*',
and
-
tan
w^ere
0,
1
cos 0« sos ^3
w
revtously, cot
4)
- I\
.-1
cot
tf
- oa,
~
c<
xx
Q
fiu o
tar*
i
- u(l-<»</3 u*Q +
1.
^iv*s u Q
/3
1
cK,
^/
r
at u - u
1
Q
a;
uC^t/3
^(^Af^M
therefore
tt*
»
rkan, as for
I
J
<=*#
-f-
in)
<=<
-A/3
=
coa
jnr(p)
fc--,
u-
^r« 0* • 1
©J - tan
a*
}
,
u,
<=><
u,
/S
ui
1
cos
6-
)
)
-
^/3i
u,
to
3
cot
i
fc>)
olution for u for glvaa values of $,
3ince
ittfe
)
eo*
c;-
«<
fys
For expressions (B)
Urougll 1.0 nnd pn-ise Barging of 15°, 30°,
of 0.1
(3
values of c< of 0.1 ti*rougB l.J,
((J)
afti
75° ?rc consuiorsj. Sarvea of
of o<
and
K art given for (3)
v«j.
(Curvos
for (CO.
E
.
cns^nt
and of
60°,
-*5°,
I
.
-4;
-.l,.4 t
ex'
there-
,ed
upon -re turret of constant gain Margins. Fkoss curves snow
religion between gain and
consicered.
for tke trantf<
low ralnas of
T;.e
;C
and
ase margin as a function
vs. K for valuta of
through
C-l
T3-1,
|S
p.
sat*
aaaotJ
s
t*.e
notions
i
uneonpani
ay a ten ara evident.
\zm
prin
i
:«}t
RESPC
Lb
I
tr a
e-
;8ient response
toum of ov
input of noat aorvoi
and
time ^required for
t..e
or sufficiently
iioai
t,o
fcms
to
i
step
i
/rive
output of too syatea to
tae Talus of the input. Herein,
f
tirae
will Us the time reqnired for Ike output of tna systoa to arrive
one-tent
-
£
1, ~Y
Mia saturation value of
of
j
will
a In the sketoj
j<i
I
t..e
err«r signal.
3ince
.
ait)
£--r
t
Ourv
, --.
rei
the value o^
servo of axprosaion (B) as
lore
of
45
i
a
!,
function of<^.
ignt;
rgi
1
•"
is aaaontiallj
d"
i.
Ltfefl^*
t
d
rt
for
seen
t..e
l
ry
For paaoa Margins
iant of o< between oi- .1
and «^~ 1.0
i
error
*1
v i
-2
sko*s
a
'-,
tins
function of
to
c<
roast 1/10 aaturatioa vaiue of
for various
»
for
t..e
— —
—
r
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a
servo of expression (3).
when
re is evidently an optimum Value of
.;..
gives better reepoaee eaaraeterietles
servo is over*
appears tkat
30° - 50°,
for
la of 75°. From tuis curve It
p
tntially
i
From
'.
-
en,
tn
pkaae margin wklle fro
wit,
evereaeet
'Xinium
a
u -
jpproxi
shows aaxiaan overi
Curv
aes of Dttaae aar^in
(t>
z
id
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i
[3
expression (C). Here
met ion
but
f unction
*a
Lsftss
ovi
4
>t
wuert <x ~ C.I,
^C
d
ious v .lues
for
.8
la
sfei
eervo of
cf 0, Tkia \endency was alao s&own in Curve B-2 for
i.8
*Y
se aargia.
»t«
/"
.
funetloa
i
T~ is
oi
f^ior
•
$
fi
everaaeot
d is essentially independent of bo ta c< and/3
..-.->
^-0.1,;".
of X for
Lgkt line.
a]
(B).
•xpj
of
tee
As w$ultl b« expected
50 »45*i 60°.
data
to
60°.
turves versus ^< and K, tail curve is
of
o
wittiC< for
from the similarity of
rve C-5 shown
-eeociated
b;
4
time
Cwrve 8-3
assooi
aargia of
margins of
.
eat el
i.
settle fro
paaat
0° or 60°. The
fcfe
indent upon O^.out for
"f" i»
It
pkate aargia of 45°
considered. Here
/"is
i
.
ear fatal an.
-9-
for
-
:
a
se
for values
a* margin in a
b
—
-
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—
r—
e
i
p—t—
pra
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r——
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1-
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it
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l
.t
i
I
;
——
:i
J
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I"
;
:
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.-1
—-J
i
'tis plotted vs.
In Curve C-9
argln w^s
c* and
(3
taring
is
45
in all
sec
.
c:sto.
'.
,t
C
botta
Tkii depe^tt&nce was similarly noteci in t~e servo
(
le of
i
[
Pfcat*
.
dependence upon tkt valuta of
transfer fun* t ion of expression
t.ie
',j
fore*:- 0.1,0.4, 0.8
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r
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r
1
v
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fc-9
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0°
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of Run
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tramji
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7
!2i3Si
B44R
l^B^
-.
1~
T"
_
ksfe
in the servo of expression
t
(,",}
obtained in Run B-9 wi.ere
gives a response different from
a
pk
.
of 30° «i
.
so used.
his
mi
c*£
'md
it is
--.
git)
.
mar
action;
loa of
i
n«n
it ii
£
c-
reaped
wit,,
:c
la large (60°-75°)
-r
em
of pi
of o<
p
( :>r
of
s
)
atospe.
For aaall a^(orp)
in
vie laity
ttte
r
titis
;
•
I
i
curvts of g
Liar
eunres of
i
i
asnoci
is i eat
i»id
3b t tie
by <*
Q
^or^3
«*nd
iu« ^
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of o^or/3
reapos:
bals
I
tad
of
i
'
deternl
9
r,ex
fcala
t
or o<(or/3;
slailarltj eeaaes as
(&( or f3 \
For trinafer fuaetloa (G) $* o<
|5 }
.
jsti^r
....,-
-
o<ratidi
a
".
referenee
a y
-J
,
TT
R*H(j
Be
ww
»
ere
taay
..fi
o( j»v^
be obtained
<
MHi
[q^
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r
K
for a
^i
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tirae
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ill;
12627
Tnesii
wlliam R. Jr.
nase margin and
e of some simple
*
H93
Confidential
•**.
Catalog No. BGS-2507
Manufactured by
National FiberstoK Envelope Co.
Phila., Pa.
thesH93
Gam
w
margin, phase margin and transient
3 2768 001 03587 6
DUDLEY KNOX LIBRARY