DAVID H. SHOEMAKER - LIGO - Caltech

Th6e eqlationt 3rc necdcd ro dascrih€ !h. lransf$ function
of th! fccdback ioop. They shou thai a! high frequ.nci6
(lrl > q) lb. difrcrcntial motion b.tw.cn !h.
ma3r add thc pla!. followr thc gfound morio$. whcrcsl
$c inccial motion of rhe rDass gocs !o zclo ir thi, liDir.
E. Sarvo aqutUona
Having dc.ivcd tha difrcrcn(id equetions which rclalc
pcndulum modon to thc ground noi$ rrd rha damping
c.apacitoa potcnlial ia i! irrsightiorward to dmw rh€
signal flow graph for thc cnrirc sarvo syslcrn (Fig. l0).
Th€ BFplt cln rhen bc u!.d ro dlrivc rh. chataclcristic
cqualion of th. scn,o md any transfd fuocrioni of in.
rar6t. llrc charact.risric .quatron of thc scr.vo i5
Article: Lou-noise rf capa,citance btidge transducet
Ttis cqu.lion givcs !h. r.lation bclwccn thc o$i-pLatc
scparatioo. the siBnal voltig?, rnd thc lnotion of thc aupport
poinr which is idcntical wirh the grourd modon. Thc
incrtial positio[ of lhc mast i3 rlia|.d ro rh6c vaaiabl.s
br/
XJr)'
T T ,,, \
{ do,,:d(r) + ld!. t '^-'
o(r)
.((s)
(rJ
.v(r,
Co i(r + J,)
,/ (t - o.)(r - -,)(r - "J.,'
(r_ _ lcolelr _ dh:Xr + dt,
--
(r - ,r,){r - ,,Jr(r - ,,i.)
*, f.r,(rt
u t
F{ r) \/
+ :-!:.1 ll r: -
"o .I-'
and
-,-,- o(r)
=-dY|
s-"
'^-'
--dxM
F,(r) (t - d.)ls - ur)lt - d-)
,o.
- uo -:--__t-=:'--""..i = 6t
J - uJr r'+ tu!/?lr
- i.ro'
Gt'ab
^ d r
rr, = -_ aE'
ax
rrc thc dc loop gain! in tha sysGrn. Ttc bchavior lnd
slibili(y of thc svltcm a! a fundion of C? and 6o crn
b€ srudiad by lpplying roo!.locus tccbniquca !o this cquation.
Tha nosr itnpollaat tmnsfcr functioni aclatc lhc pendultrm
rnotrcn io thc grcurd noisc [.Y,(j)|.
-,. XJr)
r'tJr'-
- q,o{ I + co) r :+
'.r'tl .t (cr1!1 + cq)l
(t-d.J(J-'r,rxr-.r.,,
to thc si.crrooic !oi!. [/V(r)],
-,. X