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A vecti fier Ka device Which Conaverin ae
pulsating dec
cu atodes:
§ .
Zaie Re sto, Voltage, Using ont OY"
p-n guneti gue pon sunctionw
directiou.. J+ Couducl
pract? cally it does
brased - mus i ue
p-“ spunctou
eAPit be
VAOTE
diode couduch euly ta
for ward brased tinile
when
not ¢ouduch bihen TeVewaR
2 epplieanoarees a
at ternative voltage
h aetPue palf cycl the ddcte
oun
atods , Aum
forward bioasd and will ceuduct successfully» einile
suring tht negative malt cych. FET wit) be revenge
prassd! aud with vot cde chp mies yes tata
Bo counecti 1 Suries tatth the cede 51h coutpul
yottage. across sue veststance witt be uniditrect oual.
| ned Cees TWhus =P sunctionw Ajocte Subsected
eo vars voltage acts jas @ wecti-fier couverti
+0 @ pulsatin eco val
alternating voltagt
J Ve sues
Neecit fiers (CAVimiee Classified ©
prodes
_ a Half wove vectifer
| { b) Ful! wave vectier == 1» Centre tap Fu wave
a ti fey
us origdae Reatiier
Half wave
>
SIN II
————e
—
v
Ss
z
-|
é
ans
Las
jie
\
°
\A
|
(exh of HoH, wou veetifer(Bleck aight of gue powor Pepply)oo
” ws
Inpu wage r~
wput ? ; \
= Vu Siu wt |
\ "I se
\ >, Ar
x Ax “i
© e
° 1 ‘ f;
dung Positive half Cycle ef Ving A Is posits
with yeapecl— +0 By. Making cod. For Ward! b taste!
The iecle forward biased and tty Curent-
Hows th the ar¥cuit- ta He Cleckwixe hrecteu
ne vegative hall Cvel then terminal A &
cu iM tNidh Veapect +o Germinal Byte chocte
br toiuex Yeverer biased anc no Cuwentl tenn
Ih othr Civeuil-, Shus the circuit curvenl— cinich
& Abo the load cuwenl— ( hy thn gia theme
of Sihusedal Pulses » Therefors
7 ru
outpul> volt oe = ae ee aed ae
ue “t Cee)
r oe Gees in lee
5 Of re <2epee!) ammmmmtian
Averagt D.C. Load Cuvyeunl— —
( ss ss
Mure ee ac valut of, eae cuneul
'
i, eH rey Netegreckions + Por fuidhng culm
iar QveVage =Valut of, aM alt+ernatin diallen
inn have Y+o determine +e area Yundey
Har Cuvvue Ovey OMe Comiplels Cycle that
th fvowr 0 to tr Gnch then chivids jl by
the bake QI «
| Math wi aticenty
| br Ceacribed aa ite 2 <
| Hee elie goal. ee
}
a soy 1 vt Ger
tJ hey a =
R Tuc = Peak valu of Curent! —
an
| rae ae IL A@E,
oo
|
|
» Cuwenl- waoaveforiun cau
Sthce theve
Bs
= Lf TnSin wt dob) pve cum
es cher Oy, half
cnc
eT al
ME [F conect | ft)
| 7 4
| es -3 \Contt = cox-o] = ee
| = Puan Vodur
Vas
Ra
rr
WW
$Vpc Lye
a“ Lia
*
= Via
Say 4 ea
TOR
¢ A+ uo load (se
/ M
ioa rhea
RMS valut ef Load cuwenl> —_ ams)
RMS. weauh ue ae Figg wmreau and
; Squove vool-. He
i a m og
vy ‘ij a J Qu Sinwt) ACwt)
- [at eeme en,
: Tie | at, Ue ne wt) Hest)
a | ui a ey
; J Iw :
2~ be Powey output -
ie oa
= ee Ne (Gey, Ri
ws Ao Ge Potter Wnweud-_ Caza)
Rc = Bag (ees) ee
> i oY. 6:
“GZ eas
» Rectifiey Efficiency .— AP
Dic: PoweY cufpul —
Teer Pour apel °
=
= () Re “4oG t
=. Se \
ae (+ “e)
iS pire)
4 Remmi ete chit 'b Jey Adie /ecoee 1p SEO RE
Re
men = O- 406 = — 406 +/
> Maxinauwme FHiciency hap W aur rect;
wo Peak IMvevss Voltags (iy) whew aiods ite
byeascl, vali ot vorihar ACVORA Aloe fs Vin thie peck
of Min
ius Piv raté of a tock. uxtal in o halt
wave ctifier ct Oy br higher than Vaa-
ste mateo an te z(Hews Rectifier ) S|
St ts Seew thal i
Putput— of a hatf wave vectitrer ts vok pure |
Gc but a Puleating ac. Yar oretpul~ coutatur —
Redecd rig ee ged Cattid wipples - Jae weasurs
Sy sue mpples Pyesert= ta tar —Gutpul— hk ith the
hetp ef @ -factoy catiach “pple factor: denolzd by,
Majnematicatly tipple foctor & dethed ast |
ratio Sh RMS Valut Of Hho a+. Cows povent—
yh He = Suttpul— to the ae oF A+ Ce Cotupovient
Present ty the cCutpals
c Y- Rms vali ef ac. Com porierl of outpt
Comipovel= of cutpul—
‘Teme = RMS value ef total culsul- cummed —
{ a) aa
Auer +*Tpoe
a ee
Y- Al Teme Tae
Tee (pees S
RMS ¢ Lae
Gy Y= EEE ie
® BGs
Whisk Qeneyak ex x t
. Pression for zp. ctor, Gt
wid for any rectifier, Cae aad
a Half wave Rectifier
1+ (ee 2
(eS)
Y= bait tian sur dice cong
True
JAG lndicates halt 4h vipple Contewlim (4 4
ypeeioo ll tates Sot lence Com pores I— shale
ac- Component Wrieh very mon. wa
>
pele Pl) wayeckE & 4 poor” coyee ee Vw
Tram-formey Utilisation factor — (r.u.F.)
howl wiuch “4
tie factor wihich tmolrcal on
al cK
ulBigahiouee® Gaye treus tartans
Catled Tue fe i by
wee TUR 6. Sei) ae gee cule cb alae
“ ee Poway radiin
@elivoved . tngithna tema tee ie P 3
oO, qn “Hrous-forasevi
Ac: Powey rating of traus-foruay
= Vame ® Tenge = Mu. Dy - Vuln
yas ase
Rewenmauber +thal-— Secondary Vottagt & pure
y purely
Suiusmdal iW RMS vale & tfygl tie, wae 1
whilk ane Cuvveut- & woh Suticerdal here ti =
vis Valur & EE tices ef ne warrramc, as ceri
Lares ic
Dec: Power clelivernl to +e load = Ip,* Ru
aR
“eve
e x
De ce Power delivercc) +otiu teach =
Ae Power Be Op trautforuay
es
(=) ae Negiectin 4h. arap =
Via Tua & “ ale Re we cant!
ee
ti
ath
= Nieee Tos RL)
a se . 2 fa lene
Tne Ss as fe ee ae
a Tue RL
Tr valu oY Tuk Bw Lou which show, t
a hort wove civeutt trausferuer %& nol:
~fully utilized a(4),
Ac: (a)
Maing
Sh two diodes ars
$0 Comnectsd that Canduetion
occurs thyreugh our lode cunng one har, ofthe inpab
Sea exch Yaud through su ees
nh
or Alode, Guring tne
alk oh Ate evela 5 Unidirectional teas 1 eae
How snrough ue toad vesistance Sune iy pu cycle
op ht im wl Veltage. Guch a Ckt Caklad ——
| <
Fuu wave rectifiér. This i, au improvement eovera
rt
next
rath wave rectifier wheve He cutpul- cuwent—
Howls cy for the positive half cycles:
Ckh diagrant of Pau waure rectifiey i gheun
above. oT d centre -tapped trausfoy wer, aud dard
D2 aye tio identical afedes - As cach hath of the
travaforwey secoudary has equal nuviber Of hurt, the
Vottagt induced Na each hath ef she Secoudary i equal
ta Waguituds ulm opposite tia phase. E
Let at auy fastaul— ef sh tupul— halt cycle,
dnt petai- A have a positive voltage wii vespect-
to ste centre pein O- Te pom “B nas au eyal
emai tute ef vegative voltags with vespect +0 0.
Se tt ciede By Comducts A D2 does net aust
5 RR haf “eyela Su the fone hart o
Couducts and Br yewatin ogee
4, aud £, carviecl by 1,
are Shown th gare -
cyeh, BD.
aud Or respec
'QUuyt+ She Custavou gh tu loach veaistance RL & the Suu
aud iy. She toad Currenl— f. ik .chaved alter
by tne +utlo diodes. Tw cutpul— voltage Vi(= Re) 2
hax ae Gaur wave shape a8 + lod Current-z)
Simce 42 Nodes aye fdeutical ,the averces valu
of Av eof @ full wavs vectifiey th deuble tral=
a hatf wavs rectifier Bag oO filuilar diodes
29
7
he cutpul- LoadBeumendt— i sll puts ahi
asc. and uot pure ec.
ay anBic Twverss VelAgt — 4g ore intevested *
F..: I i verpe Voltage
q fd Se Valet Of LOR yeveve2 VO 4
i diode When it 4
f (pv) Eppeani ag acvobs @& sod.
vou conducting » a
bet we © usicdey positive hart cyech of rapul
aud 22
fovwavol © laesd
Bea 4 or :
Br: Brosed! worriniu ul” Vortage acYokh Dr
ean bn Cal cuuiaiec Ag as follows :
= —_}}__—
a a top©
f col: Soe
‘ = Loop®
E Vow oop!
= Ly * ——
. Vp
writin kvi equatiou tn tewev tosp ©
Ua t Vn — Vp, = 2
y Vp, = 2Vine
: snus D2 wust be abl. to toterali ot least aun
between ita termina
{) vevover voltae 5 !
[Pom soak |
leads to a Conclusion that each ciecl
tap fu wave yectifier ghould
shan 2Um-
st
In
| navi
a centve-
PIVE2Vm Or were
<< weiss ob eta ume)a ow" , an
Dc. Load Cunen|— (ibe)
Cousidey our Cycle Of tnt load curenl—m
fyour oto w +0 obtain Hu eects Vou
tihich & ac. Vabup of load Curent
orth a
fle Tu Stilt odet Gr
ae iol NAN
\ +3)
oe Ri ol Te
Tav.= Tpe = Lf A ded
Or
5 Bo Duy Sth wt oF tot) = ees ie comet)”
w
= Dat (Con™ ~ Com oJ =2Q €
Fe / : i
Kay
for haf wave tt fa ps And Fut wave recta |
i, the Cewbination Se ing had, wave cireutts
acting atteynat th RAO Wot, Cycles ot inputs
Hence Sbvicutt, 4hi Ac: volur fox tut ware
cet iA fiw.
=
¥ pve D-C. Loach oes (pe) ——
Yr dec. Load Voltag, ts
us
Ty,
New lp Rus Sipe
eT Toon Biohe Ty. 2
3 = AVS eee Vw Ru
_(¥ssR)@),
S$ Load Cunenl CTene)
of curyenl— Cau bs obtarieal
“Me RMS vaolut
- arr ee
Leus = Al dz [tae
wectitiey are Similar
© °
Siwwce two hal wave
nee
opevation- we can
Shenae a(ata — sinwky erCeot )
on I ¢
Ma
lems
In J Emcee aot)
a Be
° eae
ES 78 foe awk Me
iB ee)
z %& /o
C di
lems = T™ Fae(ee
2
Spee
; pul oseye ea - = - -_
(B-Cs Pour outpul—— (foc)
Dc Pow ey cutpul = = Ve * Tee
4 x Re
(ee That Cee
A.C. Powter Inpuls— (Pac) —
fe 5. Eee) t
Fae Zz Teams
2 tae CrptRe)
os
Pane ste volut of Tu, We 33g.
nectifier
y=
Sega ed
nava
4 =
Efficievrey (|) —
Ye Vmax =
- (3)
Foe sutpul — |
Pace tupul—
_
a ene
7
“Tad (44 +Re)
fi
&RL
oS
KE a RL)
a
Seopa d Bees)
ee frour aaehetesc:
RU
8
xe
18 NOG. =e Oy 2 fl
ee
‘etc ee
QC) th % waximum efficiency ay \recti fier .
Ripple
by
Poy Full Wave
|
ce)
emer express(ou '
Ripple factor i ques
(eye
Recti fies Tey e~ Ee and Ip. = 2a
So 8 Tul ae
Y Jey eles [ea = 643
Dale ter
Dhia iwreli cates hal~ 4he ripple Contenle ia utp
Axl 48% Of ths AC. Com pouen|—uinich & J
nan drat for dine i halt. Lidve ous‘es THU
Pactor ce a
Utiligation
geeaudary cum
lravsyorn ey
Wave oreeti fern the
du Fw
from each halt geperate
ianile primory ob +trrauafor wey canes curve
comb murousl . Hence TOR w calculated for’
primar an Secondary windin sepevat
y y
i deterwiivred +
and shew ths soe TOR.
seconclary T.ULP. = Tbcbhpotcieyaste test lea
Ac. Powley rating Of Secovolay
2
a Se: Re ss (ee) Re
€
pee Suc
Vast lems 7 Var. Lun)
FN TE ey
& Tw Ru ENepestiin forward ae
ot co
a ae Z
Tut Re Vue Lue Re
eee
a@
ro —8& = -ot2
aes
Th. primary aye trausfoswer & feeding +0
maf wave yectifiers seperately. Thest tuo hall wave
vectifiers tole, Vndependently ef each othur buk dq
doad- tHe have already sevivach :
q common
cut. +o be equal to °286- Hence!
tuP. for yallhe ways
Tu.P for primary oe ieee
ee 28% = O° 51+ 4|
{ue ges So fee wave ert, WNL Be
oar suc = UF of primary +T.U-F. of Se
Sars
; = 0-644 4-812
T.Us = = e-age.
hus ty Fal wave ct a
ravstos ey Be
wort chau. hal} wavs Be ae ase utt lized
it dab Sie Bia oi ida De aimee- Ae. as - ' ee =—
(el)
dvant ag
ss OF FPuulwaus Rectifier
4. ate Load = vottage and Current arg wort
thay ath wave ect fier.
ib
; Pl ho icc Curment— ae er Gena
hence no peasibiltly ef sttturation +
3, TUP t& better as trausforwier losses are lesb +
4. The etficiency te higher.
5. The Vipple -foctor i leas.
Disadvantags
4. Une Ply rahng ef diode & higher .
2 Hither Piv effocles are larger yu sige avcl costtier.
3. Tht cost of Cenba-tap -trousforutey piapare
owe ae
: ae eae
:
'
Si bridge vectitier cet & esnertbiatly att
wave vettifier circuit, Using your atode:,
ormin she four amu of electical
t bridges To ou iagoual ef -Hu bridge. ti,
a veltagre & plied Tthrough a “trays—
Porney it veceasary , Qud rectified aec.
outpul= voliage Is Ppalkermst rows ima otter
aiagoual ee maa boric ES The main Savage
of Sin crcuit & +ha jt does vet require
{ a centve stap cu ht Secondary ulndin
- of she trave-former. Hence hey e@cuey
possible , G-c. Veltage Can we Afrectty ap so i
+o he Pas q atyn —— = 7 ‘o Ct Se
Ac. mpauls”
pa
(Cet, gaara se)
2 oe a tht positive has cycle
ap arc. opal Votta Win) terminal 4 %&
positive avd & th Svregative + Tate waales erg
B, aud D2 formard faasa aad Seucueting,
current: flowing ak this period, . HF
dy Shot obuagestness Howls ee
shroun dby,Re Qnd Pz andl we
aes Back +o
Ckt. opevation —
Dunn ative halk Ach of ane vel tH ags(ve)
a w& vegative and B&B & positives his wales aicddl
Bo aud : Ba Coud Be eo: whit a aud o2 how
coydtee y + Cunvenl= 12 2 aaa a
from au snk period ae &
trou “+o Bo -+heu
Ghycugk Rt aud BBancinc
Back to) fae t a
+olal toad Current = 42c= Mi t4r
— Ve,as
PIN =
Gp
i
A fr se
— toy
ee
‘i =
Wher fapul- voltage Mn)
attains if postttVe want
valur Vu, Bb; aud p2 are
Basing: Max eu Qcvore reverted biared
olrode (> Vor oD) qk AlyA
4.
Av iia of Brad
—<— re — _ w a
qi Rectifier crl—_—_
V
“The Curvvenl- ih both the Primary -ang ¥
Aecoudcry f “Ihe power t+ramforurer Hous,
for the entive Cyele and hence for o
Power orrtpul=, Power ausforuuy oh
See aud tess Cost way be Used,
(ven
ofall
The Currenl> tn the fecoudary Oh the
trausforuey & th opposite aitection in
Helo malf Cycles + Hence vet dec. coupouent
flown Ik Zero » tNhich Yecluces Correa ang, y
sear oR saturyation +
Bus to pum alternating cures tn Secouclasy
ef trausforuer, +e tral former gets utTEaed |
eftectively and hence tne circuit“ fs guitable
for applications Where Mote Oe apa.
required +
An -tulo cllodes Conduct th werica th each ©
halt Cycle» ivverse one Bee oe
Arodes gel flared: Henck the eu can be
usd for high veltaps applicobSiy. Such a»
Deak Yevevse me Appearing acrons chivcle |
& cated peak In
ef} trode
Over elaine roe (Pwcenteheretor node
each havin
tntinite buevss
loA +o @
Nuvuimeyicat
4 Used
sew conductor
a
RA a. bridge ecti ney etre
-fornsayd Uta cre She aiene anch
Restctovcens Tee oS dec cuvenl of
a Simesoidally ag.
The four
vestohve toad frout
sug supply of Bov crus) © Deter
i ne vest as wee Of tte loach aud efficiency af
se crvenu ls
Sol vps ott pee 104 Vinee 2 OY
HC Vice Mirco eels 30% Sie 42-4264
Te I 2a :
oe at Tae ptoc =) EX
as 2 aa
Lge = Bra 1570019 4
{ VY 4264
| Ht = ee aa ets eee
aygeRe 2xo-dt RU
} OT RL OLS rk or RL= 25
2 Pe
(eat Wayne = (oe ee ee
“2 0 ee ee
He) Wee = eer (2% +R Tighe ae at
| 2 Je Ie
| = (ie tent) [exert +25) en ste one
| u
a iy ge BS
| ie ee
|
| o eeR Fee ep.
{, EH (ciency = #S xloo = 250 * 160
Pace Basen
= 1505 ef,
|
|
I"ak ee
2 A Sn. wad % fed from 4 rida veut
—— Counected G@CVOsK Q TOUS er wiey aecoudiun|
Whose pyiwiary th Counected +o 460V, So Ha. 9
supply eth Yate of vtimibey of primary tung
+o &ecoudary spurs cs 2%)-
dead Curent”,
Cal cuikte =A a
d-c. toad ve ta ae » Tipple Volt
aud piv vating of aAtode
Selu: > Ru = Sk = SHIA 4 Nyt Ny 2
eae ie Eps 460V @ns vole)
ESM eine Es =p 2 Oescy
ies Ny 2 Ti
Vena MUA hence, Win =) IU c 4
Va or Vu= S2K 230V
a Wii Sesstcas) Vi
| eae oe ye
| Gamer eae Vines Vu heplect
i Bi Veas C# Re) RE ee
RLS ;
a) = : A
Tye 8 ae Cae Se ae om
7K 5% lo?
Veet | Ty ogni = 41.41 nie oS
| Ve 20 eee,
Ripph factor = Ripple vetta
e Vp e
ov Ripple voltae = Ripph factor *Voe
+ 4.82 x 204-042 = 99-8V
2
Prev. vay of each tiods = Vue
;
= B2 OF 27 A’
|ae Hi fe sea ce its tap
vertio +° half Secovdar
tard 2a0V ct Selly
ilo 3 |
2 eo Calcstate the
df vew
Pt
|
ig A fur wavs
[
ke toad yeath fay ce
| ie and —the ripple vole
Frausforuer Nose ums
eyo suppl zd
load voltoa* F
a capacitor of 470 DE use AL OF Cie
wecalculaly +e toad voltage, auc) she ripple
load currents
osauusls q pawn
moa tag? ’
Vp crus) = 230v, t= Gols NiziN, loz!
i Ru=se
© Vpcrmsd _~ Ni = Jo Get ier Up = 230.23
Megane a ie
Vs Crus) Ne oo 2 Oe
Moa = lid eV stern) ee 32-+5269V
Tike Meenas oe ae o: 6505 A
Ru So
| ue
Asie = fain 2 2% 6505 = 0 41414
| ae ae
aR 7
Cc Vince =e och es GOR 56 See es
4
0: 40 for Fein: uitthoul filter
Rippla factor =
vippl. factor *Vbe
Ripple Voltage Gus) =
0 48% noe yoy? = 993%
* Now Capacitor filter fh Ysac)
Vc Mia ee S202 691 oe
afc AXSok 1
Vpe = 20-1215 V 3
Ripple vattags Y= ! = 3-25 6sume
435 $e Re 4
” 4 ae
y (Yus) = Y*Mae = 3.256% 10 re
Vy = + 09156operated) frou Baie y
A full wavs yectifier :
iaovcrun): St & Connecite Forays
toad ayauing 60 WA and ug Joo 41 ¢ filtey Capacin;
CA eRe eto shicUm cep bg ote and ytu-s valiee
ob ai pph eine we. abu) Cluuloubeer ae vi ppl fachors
Supply with
5o0wA , Gale,
Vaca sce eae ta He, dae =9
Di PTE ats oes ee cre a vi
i ; ; Tac st
or Puu wove Rect frer Gk)
Yes fegnese = S0Ne— _¢ —1ey 2enee
4x 50x Joo *so
Vy CVU) = idea. Sie xia?
ala fe 43 % So Xlooxld
She vipple factor given by y Vr cys)
4. 44232 alae
167-2056
Determine the peak to Peak ripple vet
——— mppte factor for a@ bride yech firey uatns pacitor
fhput- fiter. dae toad veltatance t ak Uutnila the
cece voltage acyobs the load G& \2V + ASZUMt Supply
frequency to br Sot, ancl joleal roctex “Tht Capacitor
&- 63x%163
sade aud hag
Mi 4
ef joo He use) W tne cut:
Ris (et Bevee = (21 = Soles ctooare &
Benes ts. UA ay iene ae ov
4 ote Re B50 100 ¥10 8x 2%10 >
she Ripple factor = ee a4 \
4 Safe Re Le ne
= ©0144 anal— & 144%,“ Nes
+ lhe Cryouth, —
oo
J+ t% observed + hak eutpukE
of 9 vectifier (half wave oY Full wave) ; “ot pure
dec. SF coutatus fluctuatious or ripples winich
are undesived: Te winimiga tae ripple couterts
HW the cutpul—, Filter cireutts, are used Ofer”
city are Counectec) between tht rectifier aud
lead.
ae eal ta filtered
Rectifier ap Prlter ie ie
ee we i ti tL
: Le
jane
An Ac: twpul— ts applied to the vectifier. at thu
output — eh she yectifier, there Wt be aece and
vipple voltage present tihich & tpul— to thy filters
Pyacticaty Y4n filter cvcuil= Wilby to wining
str vipple at the cutpul— as fav as possible
Basically, +r ripple & Arc. thak ik varying Wit
time, whi alec. th Cousth tov. to tia: cine
(oan +o Aepavold tec: frou ripple; tne Filter
use, coucpouient CTuductors and Capacitors)
nave widely ctippevernt— tpaedaucs -for a-c. aud
Sdsatty ane inductaver acts as a ghort circ
for dec, bab ib hay a favgr Nw pedance
Similarly the Capacitor acts ¢. open for abe
aud aluost ghort fey a-c. dy the vatus of
hh Bubbi cieutly laaGapoctiy inp lee ae sitter )
prarpisaves Paul wove oy briclys
: 7
a
Arc cireadr
oc aml UN
Coe Cray )
a Block eragyaus oO
al} wavs Rectifier —
twhdh
epevation ocr lter
eee é
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at Vax ‘
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tie Coustonl unich ts vevy dae aud
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cus Cae Se very UHL frotu Vt
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eile
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ow
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euch io nae 2 hav forum i
pars ee
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punu tine = Hane «
: penoq T25+h a
i joo. cee ar
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de a fiat
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chy
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or Vy Loe Te
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ie 1 ae ‘
| [ ree fox fru wav
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i ee een i
a aa .
s vyectifiey With Capacitor IM pu
for agp wave a
flex eae oe
fi 2 factor = gate Re
€ i ae
Ne Pvoducl CRE '4 din tiv Ceovath st
Sh etter CIGe 1
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ae Ih Mow Couductiing Huu in lize:
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Ate Wee 2h cee B4e
She ce
Percentage Repulation a
Ro, Tae
ne a fumes ;
TM SF Re Re => fntentrestatance
joo ‘
ie 4 @& TacrRe a| iH (xTRévenn’®
ca ee de ; =n
s coe equivaterst cack.
ofa vecti fier >
\
Slopt =°
stops = RE
ie
(xeputotion ehavyeacten aie p)