Symmetrical Faults Part I.pdf

NIRMA UNIVERSITY INSTITUTE OF TECHNOLOGY B. Tech. SEM. - V (ELECTRICAL), JULY-2017 EE501: ANALYSIS OF ELECTRICAL POWER SYSTEM Unit 4 SYMMETRICAL FAULTS

Page 1 of 8

Page 2 of 8

Page 3 of 8

ALTERNATOR TYPES There are two types of alternator (i) revolving armature type and (ii) revolving field type Revolving Armature Type: It has stationary field poles and revolving armature. They have small capacity and low voltage rating. It resembles a DC generator on appearance accept that it has slip rings instead of commutator. The field excitation is supplied from an external dc source. Revolving Field Type: It has a stationary armature and field poles rotate. Most of the power system alternators are revolving field type. The excitation is given through slip rings and brushes from external source. The field voltage of around 100 to 220 volts is required. When the rotor rotates, the stator conductors are cut by the magnetic flux and hence emf is induced in it. Since the magnetic poles are alternatively north & south, the induced emf depends upon the number of north and south poles moving past a conductor in 1 second.

Revolving Armature Type Alternator

1.6

Revolving Field Type Alternator

CONSTRUCTION Page 4 of 8

Main parts of the alternator, obviously, consist of stator and rotor. But, the unlike other machines, in most of the alternators, field exciters are rotating and the armature coil is stationary. 1.6.1 Stator

The stator of a synchronous machine carries three phase armature winding. It consists of a laminated steel core to minimize the loss due to eddy currents. The laminations are stamped out in complete rings (for smaller machines) or in segments (for larger machines). The laminations are insulated from each other and have spaces between them for allowing the cooling air to pass through. The slots for housing the armature conductors lie along the inner periphery of the core and are stamped out at the same time when laminations are formed. The above figure shows a stator lamination of a synchronous machine that has a number of uniformly distributed slots. Coils are in these slots and connected in such a way that the current in each phase winding would produce a magnetic field in the air gap around the stator periphery. Each phase is able to produce a specified number of magnetic poles. The windings of the three phases are arranged uniformly around the stator periphery and are labeled in the sequence that phase ‘R’ is 120o (electrical) ahead of phase ‘Y’ and 240o (electrical) ahead of phase B. Different shapes of the armature slots are shown in figure.

Wide- open

Semi-closed

Closed

The wide open slot permits easy installations and removal of windings. The induced emf in the windings produce ripples because of distribution of air gap flux into bunches. This is somewhat overcome in semi closed type slots. The totally closed slots do not disturb the air gap flux but they tend to increase the inductance of the coil. The armature conductors have to be threaded through thereby increasing the initial labour and cost of winding. 1.6.2 Rotor, Salient Pole, Cylindrical Pole Page 5 of 8

There are two types of rotor used in an AC generator / alternator: (i) Salient and (ii) Cylindrical type 1. Salient pole type: Salient pole type rotor is used in low and medium speed alternators. Construction of AC generator of salient pole type rotor is shown in the figure above. This type of rotor consists of large number of projected poles (called salient poles), bolted on a magnetic wheel. These poles are also laminated to minimize the eddy current losses. Alternators featuring this type of rotor are large in diameters and short in axial length. 2. Cylindrical type: Cylindrical type rotors are used in high speed alternators, especially in turbo alternators. This type of rotor consists of a smooth and solid steel cylinder havingg slots along its outer periphery. Field windings are placed in these slots. The DC suppy is given to the rotor winding through the slip rings and and brushes arrangement.

1.6.3 Difference Between Salient Pole and Cylindrical Rotor Alternator Based on the rotor construction, synchronous machines are of two types: 1) Salient Pole or projecting pole type and 2) Smooth Cylindrical type

1 2

3 4

5

6

Smooth Cylindrical Rotor It is used for steam turbine driven alternators i.e. turbo alternators. The rotor is cylindrical in shape with DC field winding embedded in slots at the outer periphery. It does not uses damper winding.

Salient Pole Rotor It is used for low and medium speed (engine driven) alternators or hydro-generators. The rotor has projecting pole shape with field winding supplied through slip rings and damper winding to prevent hunting (momentary speed fluctuations). It is wound for 2 or 4 number of poles. It is wound for large number of poles (more than 100). These rotors are relatively long but have small They have small axial length but has large diameter to limit the centrifugal forces diameter. developed. It permits accurate dynamic balance and provides greater mechanical strength. Cylindrical rotor has uniform air gap and thus The salient pole rotor has non uniform air gap permeance offered to MMF acting on the and therefore the MMF varies considerably with magnetic circuit is independent of the angle the angle between the MMF axis and that of the between the MMF and that of the rotor poles. rotor pole. It has quieter operation and has less windage Its operation is noisy and had greater windage Page 6 of 8

losses. The rotor is simple and easy for analysis. Armature reaction is simple 8 Cylindrical rotor alternators are usually long, and the problem of air flow requires very special attention. 9 Hydrogen is normally used as cooling medium in all the turbine-driven alternators because hydrogen provides better cooling than air and increases the efficiency and decreases the windage losses. 10 Voltage regulation is good 7

losses. The rotor analysis is complex due to armature reaction. The slow speed salient pole alternators are ventilated by fan action of the salient poles which provides circulating air. Liquid cooling is used for the stators of cylindrical rotor generators.

Voltage regulation is poor.

1.7 (i)

ADVANTAGE OF STATIONARY ARMATURE Output of current can be drawn directly from fixed terminals of stator and fed to load without passing through gears and brushes. (ii) Stationary armature winding can be easily placed in a rigid frame. (iii) Armature winding can be cooled more easily because air space and cooling ducks can be large. (iv) If stator winding is stationary, it is easy to insulate for high AC voltages of 33 kVA or more. (v) Only two slip rings are required to supply DC to the field circuit. (vi) The brush and gears are on low power side i.e field current and therefore easy to insulate due to light construction. (vii) Due to light weight of field poles, it can be rotated easily. (viii) The maintenance of stationary armature winding is easy. 1.8 COMPARISON BETWEEN ROTATING AND STATIONARY FIELDS SYSTEM Sr. Rotating Pole Stationary Pole 1 Field system is rotating Field system is stationary 2 Armature is stationary Armature is rotating 3 Two slip rings are required Four slip rings are required. 4 Sparking is weak Sparking is very strong 5 Brush friction is low Brush friction is high 6 It is used for above 25 kVA It is used for below 25 kVA 1.9

DAMPER WINDINGS

Most of the alternators have their pole shoes slotted for receiving copper bars of a grid or damper winding (also known as squirrel cage winding). The copper bars are short circuited at both ends by heavy copper rings. These dampers are useful in preventing the hunting (momentary speed fluctuations) in generators and are needed in synchronous motors to provide the starting torque. Turbo generators do not have these damper windings (except in special case to assist in synchronizing) because the solid field poles themselves act as efficient dampers. It should be clearly understood that Page 7 of 8

under normal running conditions, damper winding does not carry any current because rotor runs at synchronous speed. Damper windings also tend to maintain balanced three phase voltage under unbalanced load conditions. Since a synchronous motor is not self starting, damper winding helps it acting as an induction motor at during the starting period. When the rotor attains a speed closed to synchronous speed, the DC field is excited.

Page 8 of 8

-Sh-0 C i.cJif, "f q ^*d ;

*

,f,*

.sh# c'rrdiP odn

Sd oc,Ayooo,; tvl" shl'q

coob, ?io")s.

-flre.

Co"r, ryo-

c".y

}Y'roh">< lr'.a*@1

**^:o 0?,r?* Jr^]*e' q )'"*6'*''d qs q fra

=i; , nW* is

^oDettJ

rocfo'2tn

i, \serd,r J,O -i\0- io),rD *rl -bis ltqclo-rt rOu cp-r bioD N,-t, -fu Y*\% )t o cloa ce A t + -fk g o clriv i s cpil D s;1,t'r,h*2 o,: 14 o el.ota Aa ( );x.r ott )nchrorrror ,*Y.lol* irr cAsa !,

g4_

S"lie,rl- ?dL aauhinc) . Js ot ry*v,e ?ts',slo-rtris 4vo ..*u*rl & hdn is ngnl'+$ q"b" FV.L.

+

Xa

9rooht*,o", L/ (rqcFoo )

fyt

cr-

s.c. Model + " S1,rchrvttow . rqlc oF No- {oo?-

-Stt.qEd \Shre

,As*to. tr* -lx.a - o grca

tu)otb'a!-

r ,4*L ntovhi"z "l'', :li 1-'* ".,*.4oll fi *t:,!\le ylrote" 'ln'+si l'

a o'r W)Wnr< d f, "h (rrl},"rg w i,t .s,t"L -sl'ok eg,),|'irar Qs )irrt,uQ n o t ...t 4- a hn,oVc',rr'rt, l;",-o -t1.F -fi\r c,nc,)t)- bx.,4a^ ryurl. ;obfL ,fv is "/;rD r')'tLb h4"* t'M o*,b 'shk v'#"ro, .

t #5.

finnd'"% "l+- s' c' l*" i9 olt Y1rl.e- nbne

Dc ,f,,"*

hoscr J

cvwoqt- u? ( eed

J

r

3

orrgou(, ( &? fr* corqrt chog ins1cqf e*x^f rffiryy' "i"H1':&5ffi11;

,4s

'ri#'I"

""nr"

-T.r,,t*t)

3-i^p 'q\liol go* 4 -t.r- .s.c. 7c^iD, -f\a ).yr^ ,,,D'f I '**f \lly* ,," $tn i\. 8r, .-f'a

)"r"

-\0-

^nrf

Jonuel

J,5o tuwl,qtr qu (in, .h, ),a. o{t

xt: T&d;t,:,,ffi)t.'4r 4 i'ry;,, Al- q l"t+, -l* . iy be to-cr ole4 c.rcJit

#flfu

.

iq)ntn"+

Dixcl- At is S:t&h.ctf si e-rk 8eo4*cr-

)If*''ryak c,rct/i}'t"l d"f,*

+

0

rSr/b

ju',O "f Xt+ is

rr|,o,t

C$l

y aio)

c:,tutiF .

sho+"

ct.

a -fhl

dnil

+

trq;,Lr* Vbhr+ sr,*fik

iriliol

ety'rtr.

X ol"

t. h)

rsi,,b l.rry+s'ta,r.

,%

-tlr- $y,

C'vrctt,P

bl -ft-r 9oc.hi^e b

pr"*D

-r\o- \*vh"*

ywnto."W

t*{ode,l

yuir)

h,.qsiaol

-rllnf,

,A f

l{gcJ+y$w -f -tL4 t)ac,l,it4 clLa>a*s

d;e ovr.

Xn+ /,a' G". *r) U D <1{ t>yqset,l- ;1o c-h.u + +Lr yot*irr.,L Xe"< xa'< x.l q tir* D"c4i^-e- -th.r "4,fr* l-€st rro c)\r76 Ln' ii,, + .\ot' 4 -f, 4"t I

IJi

-n^..1

-t\rl-

rf

t-P

I

cvwt,* ]t, e * .shoO- cir-ciir cvt*tl. -Thr -*\n

I

? g^iool

)

pblD

Nox

J

di'fJo)

c9n

befur. ivtiD

nblee

s-

) )ctrirl r.sr/rh"r.si-:t' j*:D ,t;o>u <w cvtu/'rv is l*rf4o s ,-tt{ Oq vh'r * "llu, s"A },, ysle,, } tt o c*r,r sa n" i,.Nala |voi.aol- y,iD u,hey 4+a- ,D ae-Lt'trt#*t lp2itutl uq+;r!. * "*b 'r.sl'ok luirl 'n\e't +t'rt,t? !,r",b;^ )ac)ir.t T*, c

/

t,

Od, s)lvoovr

"offii" *"j';i5

.

.l

"hrluihll"

o

'.s;+

U

orqeMc"l

-sFeq

/rqol">< Cvtltol

Et -

e S7nc4.

Hlc.

t.., -Slo[,q cwr,e,rl, 0

= 93 = 168lr* $N

o'lufkoNoloDl€

Eo

Ad ar.^',1;3:H"u.^q -fprr'ut rl- c-vr-,r,i} rn4r ercLoty Dc cg,.tr rc4L .

=- 9b =

Jt-

f" -

rtochnrc^e

[r!-.lrr,

^-,rr'..^).dcl.oart

xd'

=SvbIre'lsir-l- cur"tn')'

= $} : JU

xJ. J,"orr"

lE6

T,

1,,,

Trt\t

J ,.

Q,actafo,

A4,5! .oxis

ca*f{e^L

*UAt

5

Tbe {w,oy;e, <^, rn}s .sbr+ ctrrui} stm-,y ,[,frl { ti'qe. ce4 be es bdsa -ne 1*JD oUun^r' fittt,;,U f\,,a W.*.

.l*-t.d.

, P.T* ,'fif t,^r crafianl)

7o 'J stf"ft

J,"n" ,'Dff

Circtli

't,,*

v wJnn loodq?

ca,,U'l-i-,

ci.rcrjiI rDo.td W

*!*U

c.rcJ|

qeJo.l ca*l"Y'rtr l:.,.lr'*"Y

-s.Atr.rlu'r-

Cvn4-lP

6't7n

be-hi4)

-lhr- Stblr.r t|eaY

6ehiD

e{trYa!+ r< qc}ol

sSvb

C.Q-

= f" +ir{', *.

fi4

lvnt:tt-v "i.r.l-,tl-

{o, Ulc -le.,r9;ool tlorfolL {" - /D oC,bir.A, C-f,.ra,-rl!n* -fr\0_

t*

Crm;t'i'

a1n

6n"f

cp,ttknt-

lprl stoail- curc-,I-

.f " =

uf

?o.ff

tD

d
"f l""t-

'

.s.C.

"

fr,,

-T\o- ,fp4rrw,l ct,,'<,.y \*o o s.c. -,.'= X:t T\z gfq_s3,-st"k ct^r.t^rLG9 = T= /

jIl

Ef.

Z4

e

6{3

,

DPY)

A los M4, 2zW, 50 He )1nc&no,a o* $+t>tr*la< is o??7gfiry "l-10 1+-',10* vl?A c'rcuiO & is ea"i# b dfiyt*,{*" .Dc-

-UlqTglr

4 F;a

csp,a ft + \u+*ry t

cstvwi

u

.?ur,r. & , Fio) +{ ctlvtnV aF -{\0- e; + 4/'n ?)'W el,0 + ,o *sec.

)

-h..0- ,r"ldi{

g.

"r +74

G*4,

Bo,se' lgrr) MVA

.X,t=

l.oy.u. Ai

Tol*. o.o3

Srl+t

-sec

o .3

=

,J ,

i

1:.u.

X

i' = o .2 f.o.

/ sqc.

.

9 5nlktrl cr,ryal-

I"-

El Ad "

Bqst ctrnt''F

zs

0.2 =

lcroaro6 :r.fs 22

y'u.

; -?6 zg.3 , A1

^,o,

I

-,b

f

.{,"

ft' =

.S ,

T'=

A4

'+\1

\ -L/^,/ Taa

-T )e T,= Ep ='_ = (-"

AJ,

Af

x 26 zE.g Z

ea

+

G'- r) 3.s

I

O.3

g

1'

*

r-t

,

'

3.33;r ze2q,32- ? 8;Sg A,nt. rrTeriq,Y .,r7^,

= +- = =

"l

: l312l,6' a^y -!",

L ?.v. "ord olf,, e6 z(t.3 z_ A*f

/

+^,o cdtW

r gcU. "= h -s€c = Jo x,'ot-- o.o esec. Zgrbr a o.oz^z= o,vq -see.

7

,Tsc (zc7.lu

)o.oio ' ,) -,/ _

= (s- s.3, A

' $fu

, O.o1

-O.ot,

a-# + G.o, - rJ * L

2- +

"?.zga6 + ,

3.67S J.o. 9 es e.(s

)

-, \ - O.2o'o3

=[s

'33) e

-

.3 4.t zs3 lo+

Ay. _ o.L

+

(r.rt-)r'

1.9o76

+

n

I

t

4,30 97

) Tege Ary. -{". f r,r .'" .) -lo -lo =[s-t.s9"d r G.r.-,) e

+t

O + o.oooloqq+t

=

l.OoorotrV J.u.

=

t

l_

=

p.u.

o?ezr.se_ Arrn s*t!

J

er s}sk is <)

!eq.,rrC

-

t

Faults on Power System A fault occurs when two or more conductors that normally operate with a potential difference come in contact with each other. These faults may be caused by sudden failure of a piece of equipment, accidental damage or short-circuit to overhead lines or by insulation failure resulting from lightning surges. Most of the faults on the power system lead to a short-circuit condition. When such a condition occurs, a heavy current (called short circuit current) flows through the equipment, causing considerable damage to the equipment and interruption of service to the consumers. Short circuit studies provide the information about currents and voltages at different points in the system during various fault conditions. These are necessary in order to design the power system equipments, insulation level and conductor diameter. It is also essential to design effective and adequate relaying as well as switching system. Under fault conditions, the relay should trip and isolate the faulty part from the healthy system as quickly as possible without damaging itself. These studies also provide the information about short circuit MVA capacity required for circuit breakers. The circuit breaker installed at different locations must be capable of carrying short circuit MVA momentarily and open its contacts. Thus short circuit studies are required during planning, designing and at various operating stages. The faults in power system are categorized as (i) symmetrical faults, (ii) unsymmetrical faults. During symmetrical fault, system remains in balanced condition. The power system can be analyzed on single-phase basis. However, during unsymmetrical faults, system remains in unbalanced condition. Therefore such a power system cannot be analyzed on single-phase basis. It can be solved on three-phase basis or by using the symmetrical component transformation 0,1,2. The following table summarizes fault statistics. Causes of Faults Percentage Types of Fault Probability of failure Lightning 6% Single phase to ground 10% Sleet, wind, Mech* 10% Phase to phase 15% Apparatus Failure 10% Two phase to ground 10% Switching a fault 10% Three phase faults 05% Misc, Tree falling etc 10% *mechanical damage to the equipment, bridging of conductors by birds. For the effective design of protective scheme, it is very useful to have an idea of frequency of occurrence of these faults. Usually, generation plants are located far away from load centers. The distance between the two is few hundred kilometers that exposes large number of equipments to the varying atmospheric conditions. The chances of faults occurring due to storms, falling of external objects on the lines, flashover resulting from dirt deposits on insulators, etc, are greater for overhead lines than other parts of the power system. The occurrence of fault drive a very large current called short circuit currents to flow in the circuit. The determination of fault current is the basis for selection of appropriate protective devices such as relays and circuit breakers. It also helps in estimating the withstanding forces that arises out of fault currents during the period prior to the interrupting device clearing the fault. The short circuit currents may cause heavy damage to equipment and would also cause intolerable interruption of service to consumer. The modern power system is designed on two aspects: one is to design the system so that faults cannot occur and the other is to accept the possibility of faults and take necessary safety measures to safeguard the costly equipments from getting damaged temporarily or permanently. In an electric power system, a fault or fault current is any abnormal electric current. For example, a short circuit is a fault in which current bypasses the normal load. I Load Current

Generator

Load

G

Fig.1 Normal Operation of Power System Page 1 of 5

Generator

G

If >> I

Load

For Short circuit analysis, in most of the cases, the load currents are ignored

Fig.2 Power System during fault Symmetrical Faults A fault on the power system which gives rise to symmetrical fault currents (i.e. equal fault currents in the lines with 120o displacement) is called a symmetrical fault. The symmetrical fault occurs when all the three conductors of a 3-phase line are brought together simultaneously into a short- circuit condition as shown in Fig.3. This type of fault gives rise to symmetrical currents i.e. equal fault currents with 120o displacement. Thus referring to Fig. 3, fault currents IR, IY and IB will be equal in magnitude with 120o displacement among them. Because of balanced nature of fault, only one phase need be considered in calculations since condition in the other two phases will also be similar. The following points may be particularly noted: (i) The symmetrical fault rarely occurs in practice as majority of the faults are of unsymmetrical nature. However, symmetrical fault calculations are being discussed in this chapter to enable the reader to understand the problems that short circuit conditions present to the power system. (ii) The symmetrical fault is the most severe and imposes heavy duty on the circuit breaker.

Fig. 3

Current Limiting Reactors

Fig.4 When a short circuit occurs at any point in a system, the short-circuit current is limited by the impedance of the system upto the point of fault. Thus referring to Fig. 4, if a fault occurs on the feeder at point F, then the short circuit current from the generating station will have a value limited by the impedance of generator and transformer and the impedance of the line between the generator and the point of fault. This shows that the knowledge of the impedances of various equipment and circuits in the line of the system is very important for the determination of short-circuit currents.

Page 2 of 5

The fault MVA is mathematically expressed as:

𝐵𝑎𝑠𝑒 𝑀𝑉𝐴 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑐𝑒 The power demand is growing at the rate of 7 to 8% per annum. This needs addition of generators into the system. The fault current and hence fault level also increases on account of increased voltamperes and decreased impedance. It becomes necessary to replace the old switchgear with a new one with higher rating which is not economically viable. The alternative way is to cope up with this situation is to use the current limiting reactors. In order to limit the short-circuit currents to a value which the circuit breakers can handle, additional reactances known as reactors are connected in series with the system at suitable points. A reactor is a coil of number of turns designed to have a large inductance as compared to its ohmic resistance. The forces on the turns of these reactors under short-circuit conditions are considerable and, therefore, the windings must be solidly braced. It may be added that due to very small resistance of reactors, there is very little change in the efficiency of the system. Advantages 1) Reactors limit the flow of short-circuit current and thus protect the equipment from overheating as well as from failure due to destructive mechanical forces. 2) Troubles are localised or isolated at the point where they originate without communicating their disturbing effects to other parts of the power system. This increases the chances of continuity of supply. 3) They permit the installation of circuit breakers of lower rating. 𝑓𝑎𝑢𝑙𝑡 𝑀𝑉𝐴 =

Location of Reactors

Generator Reactor When the reactors are connected in series with each generator, they are known as generator reactors. In this case, the reactor may be considered as a part of leakage reactance of the generator; hence its effect is to protect the generator in the case of any short-circuit beyond the reactors. But these generator reactors causes a constant voltage drop and power loss in the reactors even during normal operation. If a bus-bar or feeder fault occurs close to the bus-bar, the voltage at the bus-bar will be reduced to a low value, thereby causing the generators to fall out of step. If a fault occurs on any feeder, the continuity of supply to other is likely to be affected. Due to these disadvantages and also since modern power station generators have sufficiently large leakage reactance to protect them against short-circuit, it is not a common practice to use separate reactors for the generators.

Page 3 of 5

Feeder Reactor

When the reactors are connected in series with each feeder, they are known as feeder reactors. Since most of short-circuits occur on feeder, a large number of reactors are used for such circuits. Two principal advantages are claimed for feeder reactors. Firstly, if a fault occurs on any feeder, the voltage drop in its reactor will not affect the bus-bars voltage so that there is a little tendency for the generator to lose synchronism. Secondly, the fault on a feeder will not affect other feeders and consequently the effects of fault are localised. Disadvantages There is a constant power loss and voltage drop in the reactors even during normal operation.

Ring system bus bar reactor

The above two methods of locating reactors suffer from the disadvantage that there is considerable voltage drop and power loss in the reactors even during normal operation. This disadvantage can be overcome by locating the reactors in the bus-bars. Under normal operating conditions, each generator will supply its own section of the load and very little power will be fed by other generators. This results in low power loss and voltage drop in the reactors. However, the principal advantage of the system is that if a fault occurs on any feeder, only one generator (to which the particular feeder is connected) mainly feeds the fault current while the current fed from other generators is small due to the presence of reactors. Therefore, only that section of bus-bar is affected to which the feeder is connected, the other sections being able to continue in normal operation.

Page 4 of 5

Tie bar system

Comparing the ring system with tie-bar system, it is clear that in the tie-bar system, there are effectively two reactors in series between sections so that reactors must have approximately half the reactance of those used in a comparable ring system. Another advantage of tiebar system is that additional generators may be connected to the system without requiring changes in the existing reactors. However, this system has the disadvantage that it requires an additional bus-bar i.e. the tiebar.

STEPS FOR SYMMETRICAL FAULT CALCULATION It has already been discussed that 3-phase short-circuit faults result in symmetrical fault currents i.e. fault currents in the three phases are equal in magnitude and have an inter-phase difference of 120o from each other. Therefore, problems involving such faults can be solved by considering one phase only as the same conditions prevail in the other two phases. The procedure for the solution of such faults involves the following steps: i) Draw a single line diagram of the complete network indicating the rating, voltage and percentage reactance (per unit) of each element of the network. ii) Choose a numerically convenient value of base kVA and convert all percentage reactances to this base value. iii) Corresponding to the single line diagram of the network, draw the reactance diagram showing one phase of the system and the neutral. iv) Indicate the % reactances on the base kVA in the reactance diagram. v) The transformer in the system should be represented by a reactance in series. vi) Find the total % reactance of the network upto the point of fault. For example, let the reactance upto fault point be X% or X p.u. Find the full-load current corresponding to the selected base kVA and the normal system voltage at the fault point. Let it be I. Then various short-circuit calculations are: Short-circuit current, 𝑃𝑟𝑒𝑓𝑎𝑢𝑙𝑡 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑉𝑓𝑜 𝐼𝑆𝐶 = 𝑅𝑒𝑎𝑐𝑡𝑎𝑛𝑐𝑒 𝑢𝑝𝑡𝑜 𝑡𝑕𝑒 𝑓𝑎𝑢𝑙𝑡 Percentage Reactance The machine reactance is expressed in percentage. All transformer, generator reactance have reactive voltage drop equal to IX. Hence percentage reactance = Or

𝑋=

(%𝑋)𝑉 100 𝐼

=

%𝑋

𝑉2

𝐼𝑋 𝑉

𝑥 100

100 𝑉𝐼

Page 5 of 5

(

Et.

For -tt* p.r,tf

F.

*,V

.rrdio) \eta*n5 , 6 39

OC:C\/t6' OJ-

$u,r 'tz\r- )i,v. Oet (htvt't'tQ. -th, N*OV c&ttY 04)

ol-

,tiu

,l KV lcw

-&r,

k

c ooJi )-;o",.

St,,:Tl ssroo

I

o MUA t3qse

f*

Our,W'fors

: llKv

basQ

F*< o)A linc = 33 xy Fq qoble __ 6.6xy

6lU line

laos@,

leqse

ab,r ,X5,

i<,

33hv

,

Sot
2? +Jo.e

e)tl4^

Tz 5 Mvn, 37. =.1

o. 15 ?.r)

X1.".jo. lzs

iT. + p.o . ;^7. "( SroitrilT

eqc to t

SKrnl q

l!

col:f

=

j

olH

a)nq

o.,G p.u.

: it = lo o't a /rh. to 2

lirro

liqt . (o,zT+jo.ze)zo

bosB ;*.y.

p.u.t^1. +

z = o.r3s+j o.ag

RV

?.1).

,Xt,-- Jo, I F.u. Xrz= jo.ogx r3osP

c.6

.

to8'9

"l- cs.lrlA -__ a.6'= (.sse .rt_

o.o74sgi o.oeet P.v.

lo

cabt2

: Ql,r:lo 6.gre "a)r-

o-o9e9tlo.osso9

cO

jo.o

99

I

Jo.o55og

o.0713 C

oblO

F-,r,

f

F^ H.1,n?-* y,r. $urr =

foitr

,'s /ljo.rz<J + j o.r t o.o7g3r j o.ogrr +jo-16 + o.ogz9 + jo.ossog

Qio

o,t6?2-

+jo.0gz:

P.u.

o.sroq6 Lro.88'(,v.

Novo $r't

-r1 = o

.rv ;y'I' ihs l3qx

t-ole c

d

in @bV =

qvvla'ok

6rh9pl

=

l.

l-+o.&S'

-t1ol +^r $uff

-rh %J,*

'lrlsrJ

.slo96

crn+ml

is

f

.9

sA4 +;qL

( Uoo) c we"u)

(Ar.nV

-S",t

W -sho# ei reuil- c tznsel- (O' "') l /-o' : t'tsW7-?o.Bg" p...

etryr,rml'

loxto6 .J3 x

ry -sb"'n- g\rtJ' I

=

Bro.77

=

lT ls.

x

:

6.6 rro3

8?9.7? A*p

(l

cvn'a'u

A*!)

t.qsg g /-?o.Bg'

rss /?o,ta" 2ny.

ol- ll xY bur 9s

*1 t7o.Bd'n Lj o 'r + o 'o?( s +j o.o9er +i o .09 z9

o.re

+jo.ossoi

,

il-*

nfa ?'v' &w l'.i F +D,l tv ,t k v bvr flo'.rL l'I " t.qs&? LTa'&8 x Io.,ut" +jo.4s4t r

.'' Aclr./ wt$a

ol- rrKy

bu

:

J $,,rD

l

t.ss8 q t?o.Bg

, [o.,rr 6e /,6g.o 0..8766 t*.8c72 ?.v.

+-l

I

,: lc o.B?( e /2.&c?? 1,e z /-a,gc7 z t
L

Ea.L f,6. 13

q

*Shotr,,r

2 KV SVsten. .D

fione.nl;ty ertur

6'folio")

n)n Q -jru- lofq

I

Fovlr lr/ul ^4 frurF cunt-l- .svppli^eJ q 3p foulr- at- -t^l. $ezraohx (; -rA, liru is 4oo Lsry lU.

&,"y

q

fotrrl .*?, l".y egc.l4 xca

\vif e od

.

Tr

'T'ii^H

X = o.

loo mv't

nf tlzlr-v

2nlf^

Lr

t

lQ

ed ezr

to.l.

Le-

Tz

X=o.2/?.fu-,lfh

fout)-

€)0 Mvn

ll kv I

o'/.

SO MYA tr

/tzz4v,

$onn ,- ,4 s",-r, I oo

ll KY

r.,t

g./.

vA

lcoso P"t

l3z t
W

bo

re

SAa Orohor:.

7 .L.

lb, = jo'ts p.v. ) lB. = J o.f x lg = \Jo.L p.rt. -St) Ar,=jo.rp.r. ) Xr. = -j o'oBnry --jo'16pu. S(>

t32

L

teo

,Xu, p.u.

= o.zx zoo t7 9.?g

Be

=

17g,zq JL

io. 229s

p.u.

qcro",ca ?fi?f "o-l J o'22q5

Jo.

z

3o.16

-j0.229S

Fqoll-

3

l /-o"

FouU- C'yrreol (s.c. clytry+^,y)

| /o"

j(

o' 2s // o'z c) r ro'ttq?s

=

I

/-o"

-J3.&l 2S;ee

Jo,zGzz Bosr cuw€zn l- 04 I"$n^

=w= loox

{sr

€.c.

Foull-

le

vol :

= 4zt.3Ax s.&lzs = l66?.ss A"y {*l

zs

--Z_

loet a. 2 623 Fqulr M!/A

= 3e1."q MyA

sc

l(b

Bq

'rbis

lor'

x e1.

=

4s2.33 ,4Y. Nole

Bqs? MyR

=

=

l3zxto3

Foutf' cwl,aqY i.l A"y

rqve -Fc,ollMv,

ro 6

.I.rq

MYA

= s.Bta9

b/ Oqk*q L io puwit: -js.&resx jG.z+o.te) j [o.e+ o.t0 *: (o.rs *o.D = _j:.gtas A jo.s6 = t x'2s P'u' 6r io,s io.es R.o ft curtf-'ll' og

6ulF cunarrl- co,rFl'b4€)

d€., Qasl*t

=

looxta e

ffi

= -5218'63 A^'f

Rulr etrwt,rl- co.^fyi b,h) lr/ ge rtuslor iq A.,yt ! = -j2' zs i szr'.6s I I &o9 .r,s

=

/-

oto,

Ay

@

l=aull-

b"fD lT do/). Z io f, = j(s.&r zs - 2-zs) = -3 l,sezs ' '. P'v'

cLry1+^F Co"rM

foulF cwre-rl-

co",hri

b,rtO ,y dt%. Z iq l'S6zs x

=

().

fl^1,

*SzqB.e e

Bzao.qS

lS"'A^7

Er.s

FD. sbotur o foucw Atlt", , Eoe;l6 4 +'* sttotlolcu, q -svi'h"1s1ons' C q O. is ^d.qlr) ol- loo r.r,rn , ^kv 6os *te'rr 4'rt"t 'fo teocl-o,o cl "l- 4o/. €oc'h 4' +1'L ^D q 2pqql-oa Q "( 55'/. ol- loo MvR , tt /lsz Rv & hq ' -fho llru L, hqs o, i'r),,cfoocl "l t-2-9U/fU/kq..,,) o lefh "{ loo h"r . Lin Q Lz !- L3 lr.on o4

^oJ.jooue 4eec.v\.

F ro

d J*

jorrr ol-

lqn/pnfu*t

4 q lryn "l s

$utr M /4 4 &r,i

ctrntol-

lrur 5 '

k't

fu

q

L3 lnxlltbly,, o Fl/ph/r"r

t.

loor.lvA ,l t4y

4

o'/.

$ato'-

z ,.wlph

trlwzkv l aol ua

rc

loo N4 ya

5/.

,Z1o"^lo lao qvn bo s4 lt Rv bo5p W EoA erol-a6. l3e Kv bqse W Tro4sgiSsio.,r

z/nkv

3./.

ttnC_

,oo MvA I

ll(y

4o'/.

7

/At-- b"= io'z p.u. Xt, = Ar. = jo'oS t3

P'o.

o5a iny

Xr,

=

.z loc>

l.2xr63n loo x 2xr x5o t+q.zg

= JO.Zt63 prl.

lx lO-3 X 5-g x

2t ,rx so t74.zg

Jo.ogors

P.u.

P.u.

(e ecl",

u

.D iuD'o,o.

jo.oiors

jo.o9ots

jo.z

j o.os

-jo.os

J o.z

jo.e16t

lo'o gszg Jo.ogqzg

(eochncL upts Favtl- pct,ol= iJo'esto'og9zB * o .o no=J = u o't?o p.u.

rh

I

FoulF

ld

= - Js.8?&j p.o.

o-t Zat

I

.s .87 &9 p.u.

Fqutl-

MvR = hulr,MvNJ

6jol7or

Foulf

i-t a^y.

Q:.c

Curyr-,rF

MI/A

= :jS.S7 &g

=

A

loo,rr,o6

-/3 a lZ Zoso

-r .asr, '3 s Ar"tl

/-1r'

Ex.Q A 3p

-slro#

oo

33 kv

-s\t oaro -lrsline

C iy

S7sle,rrl

ciycJil

o +

ctiiF

A"-y

cc\rs o l- -il.A 6ili f N"foP .shodJ lie il\o;t) -tt'!Re)q. O

.

bmoI5e-

o^r,

I"ttr,

cpo MvA

E./.

lo tttlR 8./.

I5 MYA

LrL Lr

tz.l.

L

lo Mve

E

/.

o .lo

N{ya C,1,

san,

-A

xb, A8r,

--

=

-?o

{!.

=

plvn q, bore & ss ku

io.oE x

j o.rz x*

# a g0.16,,.u. = jo.rdlr.r.;ftp{r=

lro

rg.

ld, -- jo.o6 F. 0. x6. - jo'o8 P.u. t3q s+

Xt,

;nx' +

l;q-t-

=Jry s9.9s

20

Z

=

= j0.os$E p.o.

\,

ALz -_ J t.6 3

5i.9s

3s

=

=

s4'9s

tt

) Xt"= j..z,e

.os98 u. P. s(.4s =Jo

jo.oe p.u. ) Xt4 = j43s

S9.9s

-_

jo.r8p.o.

.Fcotl-eD B,.r

j o.or

jo'osi8

x8r

.to'o8 o.o

o.o

t?ig

9lg

jo.

10.o398

'[9.

jo.ossg O.O rAz

io.r

x6r -aio.16

Foull

Fqutl-

io.ot??E

O

io.oo81 J

o'otfil

jo.olszg

Jo.os98

O

i o'08

o ro'or?7

o

jo.oZeT

jo.ot3e8

jo .ot 7s8

j o .ro6'6

Xq

r

I

=

Jo.oo7

o.o2 =)('o??s8s++ o.o

.:,,

l8 lo. o.o G.orJe8l

+ o.oo? .o o?

a (o .,osrr$E ll llco. - J 0. o, 6'o29

:

+ o .oo] U.

o

&oll(1,

MvR

c. M/A)

.llrs

4o

xg.

Cw

-rhL

Bqsc Mv4

=

cr,A

l-

FD..hrqrr

br+

672.2s

ac',+/ -slr ovlJ

"\e"

.tS,q6lo

t

o. o zq?s

liac- J'Y. 4 a Sp

lza_

MvA

6?

z .?

s tlvA

syste-,-t.-Ttl-

fb"+rryissior l\,ue e1?ao?s ol- loKv lrn5l q t€sist-o4 q! 4 lJr 4 reo.lo4cl "4 4t is ig-a e+D f" M ta sR{ -firg. r\er exofi>P *.5;l-oh'o'; hw'frort -firrog"l'" s \gritr o 23ocl-oqcr- "1 3'/' %.,0q, Xrtf*; b+ l>rn bor-r o)-( .stFplLa) by a. ,o MvA ollevqofar [oV,,.-A lo /. TtoctooQa- . Coleurlolq i\4 -Sll'{l ;,-.9 v Xv a ft a +D t/ ^o ett, cal $u* bt't.'oec-4

4 iF occvrr' ;) ol' 'frt loo) enJ 4 frolsn0issioo lirta 49 o* -l-\, hU\ votYgx 'lQ.ve\nols + 4t*"ttf^'orap'hose"

'

Srr,-

lsou',o-

lO M/A bqse

loRtt 1", T'L. lot BqsQ ;y. P* T-L. =

&

X rrnu =

(=+i9 - lo - = o.r+1o.g

4)--io'' ?u

l*

&utr

F

)

ro-rr

=

lo

p.u.

X-i 3 JO.oS x ,o - Jo.f 3

P.v.

I

Ag, -9 c. KuA,

--

6.rQ l(vA xa(,

A?. =i(o'lro.)+

lo ooo

o.r + ro,q

o.z

=

o

5o Ooo K yA

.t +j o. G

a.6o82

P'u c

#2 (/U.s; ?.v-

EtG A -3rD 4o navn, lo kv olhpwrola',' ho' irtl4-l-qo I aeoeloaqt g, n grlig' bre t<sist oo c'- ' Fi'rd J1.'0- Qxft 'qo I a't 4 q.e^.)4e.F) /, -sQ/lQ-s ,J,-rb ?tecfo4q.o ir" phose 'lo be ou> .sl,l n* +u o l Fe rllJlu -:ro -th ol- =k" D/ Curyf-,}--}tu $Af cttw+-1- ' c '''rru'.)- boet qdT exc e e) B |i'nls 5an,- Ld- 4omvR / lo Kv Qs bqs{s. L

at

+,\0..e&*

4o,

O,0S

S. C. ctrwarl- =

I

alV>toclq

a l-

ort\,r{

{c"r'n'ta

of

"

Mv A.

/c;

o -o5

s. e.

Ctrnrc^r\-

5. c.

Cynzarf 1

^t I p.u.

aw

t\d1- +o

exc

oD 8 fi,rtps U,

| /o, 6 .os + Xexy.

8 o

Xer| = j

'os

f

o'o zs

xe*y p.v

.

lo"

4o

" trV .

l&qe-Far

+D lrt c9.,4n;) -fu s. C. cct'..-rl-

-tho- $* +D liryith"kJ" /o.) Cvtl+-,tl-

fU

=

5 11'0,o75 :

w)

-saalsr ^[fr\

1D

. _10 3?5: Jl_

B l,^<,

,;

€r.6.

-rh^e. ploo)- Co{ac'ly

-l' q 39 da/'^"V^ r'sf-ol-i oa

a€oclonco-l- {rai lo,ooo l
cwtsrsl-s

rQqcto4ci- 4 5/..be*O"d)i"1'!-*1-7au -f.,.L c'vrc,' l- lc *alt e<s Lb /A M \ox.r)wl J-s;ot-o $r,, "'l\'' o{ e ol A'"rh' o{^) (,-) f- r/ a, H v *si ol-e' rttT hq x t'

eec.l4 [sviryo

looaol(uA

looooKr/A

Sooo KvA t 8./.

1.,

sooo KVA s'/.

Sar:-

Sooo

X:r,

s,l.

-- X1, = )ts

--

5,opo

l*,

5'/.

Jrrrnr

AUA

/ sst"4a lo oo, X6, = jo.tz p.o. Xt.-- jo't z P'o'

ulhlu

RVA

t Ky

A

X,ys

I

X4,t Avrr o cc,",rr ol-

bqse =

o.o5

jo'tB x

x

loooo

_s oao

loooo v soao = -io.36p.u = .-j o'l

P'o

Fl

X.1, = o.ostg p.o. Foutl'MvA

Af

fou

=

ffi-_nq+1g

tl-

Mv

jo.se Xve

rr o eckn q l- Fe

Ay" -- jo'rsrq p.u. pou

A=

ooo o'rs l9 to

.

jo.rr

-- 66oso xvn

ex.7 Fr6. slloorrr q -see*io^4 br,c- t-aorrs A 4 B Qx l;nkD lT o brrr-bovr ctar rr.Fe)oPSoool
-fru

1iho"n,

u,ln ,z'/ -

louva

8

to'/. A

SOOO K uA

l,ssvaz

Sotr,-

,r\6, -- j o.l

/At

=

lo MvA

1,.o

jo.tzx

)

.

XB. =

=j

ro

B

J0,'

,'!1

2r

lo 5

o''s =

-1

o.

P'o ;

Jo'

I P.tt Xgc=jo'rsp.ur.

z P'o

.

{orrr = ^Y = IJ[o't

lz

*o,]il("+)

=G ,25 // 6,07s) :Q

.o5 ?69

P.t)

-9.c. M,/A or Noulr Mva o .o

s76g

173.s3 MyA

{€

l7g.s r lo

:

17.3s p. u .

E,r.

8

q

Fil. Shot rr

I

4 olF€rnolc.'< e.a le o ch T"kD ol- ll tcY z 50 ,YYA r,i rfrl .Sdcfoonsi .ueqclqcl. ls 7. "lFinol ("-.r pourt- lQvr) f* a &''- o,) o"n-'I- "4- i7"4-Sys{e.^.1

v/

oVrBp

&n Cn e*" +119 lrn) l.rrrD .€qcf ryc x. Sl -ru- ttqcJ-o.,, c4 . + -tTt- cllrtizrl- lia;+;y )€o c+a^,/ le wl +o &oo rqv| lry o (.u x +D lieir -t1a Syir or) oq,o, 4 +..0 ' f+ule-rr Croor -tt\L b9 &u

q

6=€'==o',,

nl*rro I I &s,/

I I

.3S'n''-

Assuq'e

4,=

t.

=

ll

I

kv, So s r4 Ule-

4J.= 4lq = jo.rs f.o.

I=o

*

Xq Fqu tl-

= Jo.o3Zs p.o. Le ne-.|

=

Lo'

6u* j Gtaf MvA =

Qtl nr},y x Faurl

-hw

MurA qdr

8so

o.os7s =

Qe 'G? ?'o

o.o32r

-J ze.e7 P.o-

5o o.o37s

fu

=

41. -

exce

.

lass'?3 nva.

D

&e-o

so

Wd o'o 6es p .u.

lqv

A

lVltrl.l

i

\

F.r+

Mv/tr

- BoseHyn)

x?/

,I\l.gO

o.06zs--fo,t

(+ +x)/t (,+)

J 0.15

j o.ts

=G.o?s+x)xo.o?s o'o?s r X +-o.ois o'o6es

u

=

-1r (o.o?s + x) 0.rS

+X

o'oo93?s + o.o6zsX = o.00562S + o.o?5x o.o tzs x = o'oo3?S x = O. 3 p.o. l3osa ib[r.

+ fl.fu. : -+ = 2'4za-5o r.1

Xorn"l = 4"qzxo.3: o.T..6JL Ex.1

A

I A%Axl;yp Sl-oh"-r hcs +^n o ller,l ol-ars 45oo" RvA snd Pq'?W x'{acl-o4ca 'f 4 3ooo r-tA "4eeliw\ ' --Ar- ci"'c'ri l- bteo \ut ho ve q i t & I '/. *'l \ttvt,t6t c7,-oc',17 + t$ M vg . .Tf is o{e-siyep lD ex@td +ho q *p1tl fl"", -f\a d"/E viq )ste--"1 b) o +ry "fur-- ,l' Z-s* n.v; e, T,s./.>eocforcc, . fi"r) a" ?€s cfoaqL "l +1a ]esotry ca.necj+] n Jr-a b^ _ b or se c,fio- ,. v>rl -ft.^o- ch.c,li l- b *o \ u, f< beiy Sgot

ot?A|r.?q!

4 a ? r,

e)rtu) --sh.r} cioc,,tF

-ffi*?"Y#"#":';;Yh SOoo

KvA

7./.

33ao

rrcll

KuA *l* ,rffrnF-soo T.s'/.

it

Sr)nr- ,A ssu^or-

Tsoo

Ku

4

A

4,io'o? ^ Z-5"". S@o X5, = jo.o{,

4 soo P.o

Kv

qs

bose

.jo't?sP-0.

x TSoo

Xr = jo.o?s

ss ao

=

j

0. ,333

P.0.

.

jo.r+s

j o'o7s Jo.r333

,X1.= (o.ns// o.raa) tr (x + 0.079)

to or, 8'o7.eeiG + O,lSoe et x

-o

.q -11c $r* M trn is ool' tD €xr e.) l50 6".,r tdi l- bato lSe-r ayacilT 4 'q+n'U

ltr

vA

i4,

.

Fo,rtf M vA

Bo se MyA

xej. l-So

=

7's

xg

X
r"og o'o5

Ay

"q

efo,

h/r

lrore

o'o?sefx+ o.oos67sr X + o.tro@ O-o5 + ?.SSsz\163= O.o?SGCX + 0.oosCTqS ^ O.oesf6A = o.oorgsl? t3n

P

=

.,, x

i'^1

)c+"1 x

7.s

=

jo.olzg

p.u-

J

?x. to

*f,f'4 Te r\e-rtfs o

lQ

loid ovA' qs

*sholio-r

O^*^r'U

qcenYodl

f

r.?oc f oocQs

qs ilolicole). f+

1.Q

.Shoar4 'fi

"(- di ((sr"-y

a sP s.c.

o ecws ol- on/ qt f, eJe^ DQor 'lao n sfarocr -saeo,rday Cq. poir t F) , F *eol .ll..r .sh",+ circt/rl-

Mue

fta

-fi\L par.

A

lo Mvl ao

loMvA

/.

loF4 vA 3o./.

3o 7.

lotvrv6

to'1.

S MUN 5./.

s

rquR

3./.

6outf

Sanr-

,A sa

Xd,

v{,1a lo M v A

=

18" =

I!,

=

bqse-

J0.3 P0

jo.os x lo 5

=

J 0. t P.u.

(-

XE l[.10.s+rc,.r *

l,,i'.1

'

[---

j o.z s ?.v.

J 0'3

RutF rquR : lO o.zs

=

(o

r\auR

.

J 0'l

'tJ o't

-jo.g

J 0'3

J 0.r

-.t.o.r

*Sho^l *tot q OeiernoH5l_ plo"rt- f "rX l) 5ec tivt bw- [ovs -eoc,h *"1.)' ol- d K;A rJ,frr

Ex tl

_

"fff "'n:

;to tol

.sh

off

;

'*."ffi 7'f

"f ""-";'"*,,: l- I'
C'yrcdr

-s e c{-io''l

ry,

"(

+ lru+"J" rg_ l^,^ x L oo *+y is SoooolflvA )

T0 +u_

-seefi

x"=

4 t = lo I ,

I

4u

/

KVA r.5rJb (, 3 .u

Frrd

(;;-l q -sec'tio,,ts

;na 'slr oa- c,'u1nl-

& Q'p'sboA tn-or h'rf')

q uu) "t,o-,, l"EPa nr,rqbQa "f -secti4s,--il"!'-sh'+c*diP K vA r^0ou lE lvL e.xc e IJ ? ,Sorooo wv A ' dar,Q tcve r
LoF d be

Bosc RVA

--B x'/ +

x'/,

h

e( d

L./

8", ,5

Gr-

I

Do, +

x%+

b%

Ly,

r , -ruX+b .

-s@cf,'6.2',

(t-t- ,)

^/

/o

e3t,r-\n-,V % a+sofo^ W

,tfidtt il- is n

*t4+bl 6 Ir"

-rt1w

aLo* Xt

-f\L

=s0xUs

c't+tr

$h

x e -/ tq.y+u(tt CN-') )

-sec+i ,.'\ L

-J ,l

w is ^4

-T\,^ qviwU-,| ><s c.n,')w

L/, 4

y otdl

4 J,+n

>c/,

"

uplv

y*t?' is

I",*

"/

t

l'C +) +,, " LT

+b + x )

z

x + bttx

-) .' .

?

x(x

+ rrrs)

t\ (x+ b) 0A

wl,Y

r.toc,[o9

s1

eto !^t p,* -x

N [* + t") tSb

off

c.,rql!- Kva

w forrr tqvA qJ x[x+ rNT-

ft5<xj-

% is (

foo

l'' tt'

Q,

lao

=[E-* [_x

(>t+ t )

d(r.r-,)

l^

bN+ x )

tn

TD rU,1n N = 3

s,c

P

*

&r't

KvA

= Goow f :

SoaooC.)").

('s: .;;*="/xto' r ar*') too Gt* \ 4so,ooo

Rv A

(,D w,'tn N = 9

: fzs^ rs636.g0,* \ =

f''l t i,; r.I,rr LAS. C. or

P,y lov( l\ = 4w

h"#*+= (ry+

r^ a

p

)

KvA

v

= H

6136s6 .t

ro r-eJ,

ffixtw

7,425,79'z

kvA