.3 1|H|N||W||H|| J « MIUHII'IIHWIHU STABJLITY (if TRAJ‘QSMJSSION ClRCUITS Thesis for the Degree of M. S. H. J. Kurtz. 192.8 l J 3 .‘ 3 3 I ¥ 2 I 1 ? Bryon” If‘fiw ‘3‘ -:%:.' ,5" 2‘“ : ‘- . I I... 1 I .'. _, I .M h. " 3 I.‘ ‘4 Ii, 5] &§ ‘_ um“ " v. ’ . i" - , 3‘. ~‘ .; . ”4‘4 .6, I I . r. ‘ . . m I . '.’ _‘ .. 2 ’.. V lo ‘ l‘.’ 4.— '4. V ‘~_..k~ « =1'”'"r1'3 " ' ~, .. f‘ v ' ‘ . ~ '. 33-17'fi.‘ «.5431er '3 ‘ “i. I,‘ * l ' .1. I ' .l -‘i ' b. "a .'»‘ v ‘ . . -— . . {1‘4 "J“jAI' .‘ ‘1 ‘ i; '5'.) ,.' ' ' if" ‘- 7 I‘ .'--'; -; .—' t- a" . y. is. 3"", "', .»_’-. 1' ‘A |_ l . ' ‘ . ‘1' Q. -' ‘ ‘ _ _ .> . o 5 '4 I ’ ' P If .‘ k ' . . , .. ‘9‘: _ y L f.‘;g _ ‘ ;:\.' 1, - ‘ f.) . . ' 2.. ‘f‘ -.:'. ,5': --|;\ . I. v u’ b ' ‘3 c' “ _ _ ‘. .. - ‘ ‘ A. Y -'.7 3 _ ' ‘. ' ' | . ‘—1 . - . ‘t-; 5 "'1‘, w) 4 _ , I . ‘ , l r \.- “‘ I ‘r . ' ‘ .' . y ‘ L c 1 v -“7- ‘ f... i . '. -:‘i , {gm-5 4;}: 1.: ‘15: . .f ;- ”7:54: I: . .' 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[in-‘AW STABILITY OF TRANSMISSION CIRCUITS A Thesis Submitted to The Faculty Of The HICPHGAN SEATS GLLEGB 551’ (w , _ ‘t‘ H; J? Kurtz Candidate for the Degree Of Master of Science. June 1928. TH E88 CONTENT Page Introduction -------------------------- 1 General Considerations --------------------- 2 Theoretical Considerations ------------------- 5 Tests and Oscillogrems ..................... 20 102115 INTRCDUCTION The purpose of this thesis is to develop the theory for the condi- tion of maximum stability in a transmission circuit under varying con- ditions of’load, power factor, line constants, etc. and to represent by means of charts the synchronizing power correSponding to any rational combination of generator terminal voltage, generator resistance and reactance, transmission circuit resistance and reactance, load power factor, and effective load resistance and reactance, The sc09e of this work is necessarily limited, hence the stability of circuits during transient periods is not discussed and steady-state conditions are assumed in the following discussion. Several oscillograms were taken of the voltages and currents of two generators Operating in parallel, under varying conditions of excitation, phase displacement and line constants, and Operating just within the stability limit. mm; L consrmmr IONS Until a very few years ago the loads on power systems had a comparatively low power factor causing poor generator and transmission line regulations. However, the size of the loads and the length of the lines were small, so that the generator field rhcostats were entirely adeguate to control the voltage without loss of synchronism. Recent demand for customer paver factor improvement and increase in length and voltage of transmission lines have combined to create a general tendency toward better power factor. Hence during the light load period on a long transmission line the generator load becomes decidedly leading. As a result of this, unstable voltages are produced and field rheostat control alone is no longer sufficient. If the transmission line is of considerable length the generators may even become self-exciting and reverse field excitation necessary to control the voltage. This is, in effect, a resonant circuit between the capac- ity of the transmission line and the inductance of the generator. When the field is reversed the generator is liable to slip a pole WhiCM will cause abnormal voltages. From the above considerations it is evident that it is desirable to make a study of transmission stability sapecially when systems of high voltages are interconnected. In general, there are three main types of instability in electric systems: (1) The transients of readjustment to changed circuit conditions. (2) Unstable electrical equilibrium, the condition in which the effect of a cause increases the cause. (3) Permanent instability resulting from a combination of circuit constants which cannot co-exist. Transients are the phenomena by which, at the change of circuit conditions, current, voltage, load, etc. readjust themselves from the values corresponding to the initial conditions to the values correspond- ing to the new conditions of the circuit. For example, when a switch is closed and a load put on a circuit, the current cannot increase instantly due to inductance of the circuit andssome time must elapse so that electro~magnetic energy can be stored. Also when a switch is closed in a motor circuit the transient period corr65ponds to the period of acceleration of the motor. The characteristic of transients is that they are of very short duration and exist between two periods of steady state conditions. It should be kept in mind when dealing with transients that resistance, inductance, capacitance, and leakance, the ordinary con- stants of a circuit, are no longer constants, and that the results obtained when they are used as constants are only approximations. If the effect brought abOut by a cause is such as to reduce the cause, the effect limits itself and stability results. If, however, the reverse is true, the effect continues with increasing intensity and instability results. This applied equally well to all phenomena. Instability may manifest itself in three/[differentj’formsz (1) Instability leading up to stable conditions, (2) Instability causing permanent interruption to service, (3) Instability leading again to stability, and thus periodically repeating itself. Whether instability results, and what form it assumes depends, not only on the cause, but on the circuit taken as a whole. In connection with transmission line stability, the effect of the armature of the generator on the field excitation must be considered. For a unity power factor load the field set up by the ampere-turns of the armature is at right angles to the field set up by the ampere-turns of the field and causes cross-magnetization. This results in a shifting of the field flux and under heavy load conditions actually demagnetizes the field which, in turn, reduces the induced voltage of the generator. If the load is zero power factor lagging, the field caused by the ampere-turns of the armature is directly Opposed to the main field of the generator, and hence, causes a considerable reduction in the effec- tive field and induced voltage of the generator. If the load is zero power factor leading, the field set up by the unpere—turns of the armature is in phase with.the main field and results in a field and induced voltage much larger than normal. From these considerations it is evident that, for all possible load conditions on a generator, it is necessary to have a wide range of field excitation if constant terminal voltage is to be maintained. THEORETICAL CONSIDERATIONS When two or more alternators are Operating in parallel, the natural reactions which result from a departure from synchronism are such as to reestablish it. This is the principle which makes possible the parallel Operation of generators. If two equal generators are considered Operating in parallel and their induced voltages not quite equal, there will be a circulating current between the machines caused by the difference between the two voltages and flowing thru the series circuit consisting of the synchro- nous impedances of the two machines and the impedance of the circuit between the machines. This circulating current may be caused by the two voltages being either out of phase or differing in magnitude. When it is produced by a difference in phase it prodices synchronizing acting. When resulting from an inequality of voltages, it equalizes the terminal voltages, mainly by the effect of armature ampere-turns on the main field flux. For two equal generators the circulating current can be expressed by the formula Ic = E] -32 (1) Z where Z represents the synchronous impedances of the two machines and the line between them in series. I,x. I"; V1 K 1 M #1 Fig. 1 Fig. 1 represents two equal generators with their armatures taken as a series circuit and with equal excitations and equal loads. V represents the terminal voltage and B represents the induced voltage. The terminal voltages are equal and in phase when considered with ' respect to the parallel circuit but are opposite in phase when con— sidered with respect to the series circuit. EI+EZ 1"}1‘ E: I. . . ‘1 c “J E; ....,“..I.K *A“ L; I V: ‘4 1% IhfIc‘ I. Fig. 2 Fig. 2 represents two equal generators with equal terminal volt- ages but with the induced voltages out of phase. Since 31 and E2 are not in Opposition it follows that their sum is not zero and hence there must be a circulating current flowing thru the two armatures represented by the formula Ic = E] + E? where Z (Z = r +hix) represents the 22 synchronous impedance of one machine. This circulating current lags behind the voltage (El + 32) by the angle tan'l.;§_, where X is the r synchronous reactance and r the effective resistance. Since the ratio of the reactance to resistance is usually large, this angle is nearly 90°. The circulating current has a component in phase with E1 and a component in Opposition to E2. Hence it produces generator action with reSpect to machine No. l and motor action with respect to machine No. 2, so that the result is to slow down machine No. l and to accelerate machine Ho. 2. thereby tending to bring the voltages El and E2 in phase. The current carried by the armatures is the vector sum of the circulating current and the load current. The current delivered by the two generators is the vector difference between the two armature currents. The change in the output of the generators when the voltages are out of phase is partly due to the pewer develOped by the circulating current considered with respect to the induced voltages in the gener- ators. and partly due to the change in phase and magnitude of the generated voltages with respect to the load current supplied by each generator. Referring to Figs. 1 and 2. the change in the power develOped by generator No. 1 due to a phase diaplacement is E1(Il + Io) cos a} - Ell1 cos a1. Assuming constant tenninal voltage, 11 will not change and l _ E1(Il+Ic) cos aléfill cos al’E1Il cos pleElIl cos al+E11c cos p1 (2) I0 is sometimes called synchronizing current since a large part of the synchronizing power is caused by this current directly. I is the only current which tends to restore synchronism when c there is no load on the system. and this occurs only when Ic lags behind (El + E2). If E1 and.Ez are equal andlc is in phase with their sum. there would be equal positive projections on E1 and BB and an equal generator action would be produced on each machine, and conse- quently no synchronizing power develOped. The synchronizing power develOped by Ic is dependent upon its lag behind (El + E2), hence in- ductance is necessary for the parallel operation of generators. By inserting capacity between the generators in parallel, the circulating current can be made to lead (31 + E2). The action of Ic under this condition is to change the 12 drop thru the machines to an 12 rise, thereby reducing the induced voltages. Generators inherently have inductance in their armatures. so that the natural tendency is for prOper parallel Operation and their stability depends upon the amount of resistarce and inductance in their armatures and to some extent upon the constants of the circuit and load. If the constants of two generators are not in the inverse ratio of their ratings they may be Iaralleled by properly adjusting the in- puts and field excitations so they will assume any desired load. There will be a circulating current between the arwmturefi however. which is desirable.and necessary since it is in this way that the terminal voltages are equalized. Changing the input to a generator operating in parallel with others changes its load and phase position, but does not change the value of its generated voltage appreciably. Changing the excitation changes the induced voltage and its phase positi on. but does not change the load appreciably. [1‘51 cx E: L.- -n‘ r' L (9 =< \ 4)- I Fig. 3 Fig. 3 shows the vector diagram for two equal generators drawn with respect to their parallel circuit. I represents the value of current in each gmerator which is in phase with the load current 10' lo the circulating current equals E12; $2 . If P1 represents the power develOped by generator No. 1 when the induced voltages are displaced by an angle a, and 11 the armature current , then P1 - 11E1 cos B s (lo + E] -E2;E1 cos B - (I + Io) E1 cos B (2 22 10 P1 s IlEl cos 9 cos §_- 131 sin 6 sin‘g + Igr + sz sin.g cos-g (3) 2 2 I 2 2 Similarly 2 9 P s IE cos 9 cos a + IE sin s sin a + I r ~ZE“x sin a cos a (4) 2 2 E 2 2 The change in power developed by each generator due to phase displace- ment is found by subtracting Ell cos 9 andIBZI cos 8 from equations (3) and (4) respectively. Making this subtraction gives the change in power developed by generator’No. 1 equal to RI cos 8 (cos 3 - 1) - IE sin s sin; + Igr + 32:: sing cos 5 (5) 2 2 Z2 2 2 Similarly the change in power develOped by generator No. 2 is equal to El cos 0 (cos 3 - l) + El sin e sing + Iir -sz sing cos 2 (6) 2 2 Z2 2 2 The first and third terms of equations (5) and (6) are equal and of the same sign so that they must represent generating action in each machine and hence do not cause synchronizing action but tend to slow down the system frequency. The second and fourth tenns are equal but of Opposite signs and hence must represent the cynchronizing power acting between the two machines and is equal to P8 = E) E5 cos.§ - I sin 9} sin-g . (7) L Z 2 2 This value of power is one half the difference between the powers develOped by the two generators. From equation (7) it can readily be seen that for everything constant except the generator power factor. the synchronizing power is a maximum when 9 is equal to 90° and nega- tive. This means that maximum stability is obtained for a zero power factor leading load. ll Equating cos.§ sin.§ to sin a and rewriting (7) PS becomes 2 2 2 P8 3 El Ex sin a - I sin 9 Sin.§] (8) 22 2 By differentiating (8) with reSpect to a, equating to zero and solving for a, the value of a for maximum synchronizing poser is obtained d? . Er Ex cos a - I sin 9 cos a} a 0 ___JL. L"_TT' .________. _. da L 22 2 2 ‘1 2 2 + . —— ._fi ‘2?2 a = cos [fl sinfi Q, - I sinfie VIZ 31n49 + SE x (9) 434x“ 43~x4 Similarly, dP - Bx sin a - I sin 9 sin a - 0 __.JL_ ._ dB 22 2 _ 2 E — IZ sin g (10) x 003.; 2 This gives the value of E for maximum synchronizing power. The above statements regarding maximum synchronizing power and hence maximum stability, were made assuming the induced voltage con- stant. However, in practice the excitation is usually varied to main- tain constant terminal voltage and under these conditions the synchron- izing power becomes greatest for inductive loads for a given phase displacement between the induced voltages, If RC and X0 are the resistance and reactance reSpectively of the load, than .£ .2 I = E cos 2 a E cos 2 . I ——-—--—-—-- —- ,i , __a_ total Z VIZRo+r)‘+(2X°+X)d 2 and sin 6 = total X a ZXn,+ X total 2 VIZRo+r)3+(zxo+x)3 Substituting these values in equation (7) gives for the value of 12 synchronizing power PS: :22 x - raga+x Wisina (11) 2 8‘ (230+r)4+(2K0+X)“ If a short circuit should occur with no impedance between it and the generators the equation for synchronizing power becomes equal to zero. This means that instability would result and the generators would fall out of synchronism. By expressing the induced voltage in terms of the tenninal voltage, an equation for synchronizing power can be written in terms of the phase displacement, the various resistances and reactances, and the terminal voltage, E = V + I(r + 3x),1 V=ZIZO=EIW since both generators are supplying equal loads. E 3- I(total Z) 3 I yrmo+r)z+(mo+xvfz E a V (283:3)2+(2XA+X)2 2 R§7+ Kg Substituting this value for E in equation (11) and simplifying gives 2 2 2 . P t VEL +(X -r)( XQ )+ :3ng Sins (12) s 2 z ( 223 (so + x0 ) 2 (so + 110)) When the load is zero the equation reduces fa P8 = VZX sin a = ng sin a 222 2(r3 + XE) The maximum synchronizing power that can be obtained for the zero load condition by separately varying the resistance and reactance is shown by the following, by setting the first derivative equal to zero and solving for the variable quantity. 9 O ‘) O I hr 0 dPs = 2(rH + K“) V“ Sln a - QV‘RZ Sln a = o 'J ‘T 'T dK 4(r~ + X‘)“ Solving for X gives X . r (14) The other values for maximum synchronizing can be obtained by inspection of equation (13) as well as by differentiation. The synchronizing power varies directly as the square of the tenninal voltage; it is maximum when a = 90°; and it is maximum when r = o. The maximum synchronizing power when the system is carrying a load is shown by the following. Rewriting equation (12) in four terms gives P8 .. v21: sin a + vzxz sin ax - vars sin ax + Ver sin a (15) W) W; Wis Was )(R°+xo) Differentiating equation (15) with reapect to r, equating to zero gives 2 dPa = VgX sin a (- 2r ) +Izr' zgisin a (- 2r ) dr 2 ( (r2+x2)e”) 4mg + x3) ( (r4+x2)2) - Yaxn_sinAa (2r(r§:§g)- rzLZr),+ VZX sin a (r2+ z-r 2r )=o 4(s§+x§) ( (r2+x2) J magma ( (r +x~H Solving for r gives rs a§+x§+x§aivlag+x§+xxaz+siix7 (15) R 0 This represents not only the effective resistance of the generator armature, but also the resistance of the circuit up to the point of synchronizing. Differentiating equation (15) with respect to X gives 2 2 2 2 2 2 gas - V sin a (Ir +x -x(2x;I- X. 2x r +x —x 2x dx 2 ( (r + x“ 2(a%+x§) (r + 12) - rZXQ - 2x + rs r2+x2-x 2x ) e o 2(B°+Xo) E (r2+Xfi)1 n§+x§ E (r2 + x2 B) l4 Solving for x gives x - r (17) This value of x also includes the reactance of the circuit up to the point of synchronizing, as well as the synchronous reactance of the generator. Differentiating equation (15) with respect to R0 gives 2 2 dP a V sin a (Xox [- flu -l -X°r 23a —L . r7 ' dno 2(r2+xz) ( 2 [ (a + x3)~ J 2 (as + KHZ) O O + rx‘ 2.20 + 320 - #9330 i) a o (a + x3)“ ) L o Solving for R0 gives R0 -= Xor or - Xgr (18) x r This gives the value of the effective resistance of the load for the maximum synchronizing power. Differentiating equation (15) with respect to X0 gives dP - V 2 2 2 2 2. 2 2 m a sina(gEgo+xo-zxg —;[§E+xa-%§ dxo 2(r“+x~) (2 (R3 + X3)“ 1 2 (R8 + 10) i +E- 2r ngg 1) = 0 (R8 + x3) J) Solving for X0 gives X0 = -Bo r - x or BO 1 - r (19) 1‘ +1 1+2 Differentiating (15) with respect to a gives 2 d? s V ([x cos a]+1 it cos ai-llrzgq cos gi+ —_..a— _ .. da 2(r2+x~5) ( 2E R§+xé 2 B$+X3 [r132 cos a])=o no+xo ) Solving for 8 gives a = 90° for maximum synchronizing power, (20) Substituting the values R0 for cos 6 and X for sin 6 in 20 o 1 equati on (15), the value of synchronizing power becomes P3 . V2 sin a (x + x2 sin e cos 9 - r2 sin 9 cos 8 + rx cosze!(21) 2(r2+xz) ( 23° 2'80 Bo Assuming the effective resistance of the load constant, different- iating with respect to 6 gives 15 2 En. ' LUOSZG - 811126] - 1'2 [00826 - sin'ae] + rx de 28° 23° Bo [-2 sin 9 cos 6] s 0 Solving for 6 gives 9 ‘ 005-1.;L;LJ£_ or cos"1 r - X (23) r +1 +x Rewriting equation (15) again, keeping in mind that the power factor is to vary and that the effective reactance of the load is to remain constant, gives PS - V2 sin a [x + (32 - rzlsinza + r x cos Qgsin_Q] (23) 2(r +x ) 2X0 X0 Differentiating with respect to 9 gives Q33 -sLx2 - r2) 2_cos e sin e, + r x c 32 - n"B - o X. d6 2X0 0 Solving for 6 gives (24) 9 n cos"1 1 r ___£;__,o * x W Far—£7“ The above discussion has been devoted to the consideration of two equal generators operating in parallel. In the follo ing discussion there will be considered the general case, that of a generator Operat- ing in parallel with.a large inter-connected system, or, what amounts to the same thing, two generators of unlike characteristics Operating in parallel. Fig. 4 shows the relations existing between the various voltages and currents for this condition, drawn with reSpect to the parallel circuit. V is the voltage at the point where the effective load is taken off and is used as the reference vector. E1 and E2 are the induced voltages reapectively in the two machines, displaced‘by the angle a, and unequal in magnitude. I1 and I2 are the armature 16 E I I 5‘6; ‘ 1.x. at. E, A V Ir- T ’ I a; " lhx: 02 49. -Ic I“ II=IOI+I€ 1., 'I‘=131“1E Fig. 4 currents respectively of the two machines. 101 and 102 are the components of the two armature currents which are in phase with the load current Io. 1c is the circulating current caused by the differ- ence between the induced voltages of the two uaduims (Bl - 33), the armature currents being the vector sun of the circulating current and the components in phase with the load current. The difference between V and the induced voltage of the mchines is the impedance drop in the circuits from the point of paralleling back to the machines plus the impedmce drew in the machines themselves. The values for the induced voltages can be eXpressed as follows: E1 = V + 1121 s V + 111'1 cos 0‘1 + I1X1 sin 0'1 + 3(1111 cos 0’1 - Ilrl sin 0—1) = el + jei‘ (25) 17 32 s V + I222 a V + Igrz cos 0'2 + IZXZ sin 0—2 + 3(1212 cos 0'2 - Igr2 sin 0‘2) = e2 + 3e; (26) The voltage which causes the circulating current can be expressed similarly, E1 - 32 a Ilrl cos 0'1 + lel sin 0'1 - Igrg cos 0-2 - 121(2 sin 0’2 + 1(11K1 cos 6'1 - Ilrl sin 0‘1 - ISXZ cos 0‘2 + Igrz sin 0-2) (27) The angle/Y is equal to tan"1 x + x and , in most cases will r1 * 1‘2 be nearly ninety degrees. When the circuit between the generators is of comparatively high resistance, the angled’ will be somewhat less than ninety degrees. By representing (31 -232) as ec + Jeé, the value of the circulating current can be expressed as Ic - r + r - x + x e + e1 (28) (r1 + r2) + (x1 + xz) The total power develOped'by generator No. 1 is P1 = 3111 cos B1. The power develOped'by generator‘No. l which supplies the load, circuit losses and machine losses, is V101 cos 6 + Ifirl. Hence the synchroniz- ing power developed by generator No. 1 must be Pa s Elll cos 51" (VI01 cos 6 + lirl) (29) EXpressing‘E1 in terms of V, equation (29) becomes P8 - ‘(V+I1r1 COS 0—1*‘IIXI 8111 0-1 )z+(11X1 COS 0—1—111'1 sin 0—1)3 11 cos Bl - (V101 cos 9 + lirl) (30) 81 = tan-1.g% + tan-1.1} & 0'1 = tan"1 ii, where B1 = 91 + 3e} °1 i1 11 1 18 If the values of Bl and 0'1 are substituted in equation (30) the eXpression becomes quite cumbersome and, unless the two generators have constants which deviate considerably from the inverse ratio of their ratings, it would be advisable to use equation (21) instead of equation (30) for an approximate result. Rearranging equation (21) in the form Ps 8 P1 + P2 - P8 + P4 where = vgxz sinAQ cos 8 sin a, Plavzx sin a, P 480 (r2 + :2) 2(r +1 ) 2 -P3 - rzvz sin 6 cos 9 singg, P4 . V“r x cosge sin a, 48° (r2 + 15) BRO (r'5 + 12-3- and choosing proper values for the variables, the value of synchroniz- ing power can be put in chart form. From Chart I the value of P1 can be obtained by placing a straight edge from the value of a to the value of x, rotate the straight edge about the intersection of pivot line (8) to the value of Z, rotate the straight edge about the intersection of pivot line (5) to the value of V and read the value of kilowatts on the P1 scale. In a similar manner the values of P2, P3 and P4 can be put in chart form. However, P1 represents the synchronizing power due to the circudating current and.is most important. If the load is zero P1 represents the total synchronizing power. If the load is not zero, the values of P2. P3 and P4 depend upon the effective values of the resistance and reactance of the load. If the reactance of the load is zero, P2 and P3 become zero. CHART] (6 j 5 2 VW/s . ' X/Uth) mooo rfifl - 80 5000 , 3 C60 3000 _ 50 5355 . -40 /000 7 _ 30 4 500 ~20 300 MM _ ,5 z/Ohms} 200 [50 \ Maw / _ _ l5 , \ MM ° / P 6 . \ Mo K / “ 5 * '2 \ / ~ 4 \/0 / b 3 -.3 / /.0 / v.4 a/ ' /\ — 2 - ,5 M/ >‘ \ \ -/5 I" ‘0‘ . ' / \ \ / E '30 70 / \ \ :— -— . 50:3 / \ \ _-_ ,g b If 40'.‘ / \ \ .- ‘6 o °" \ - .5 — z 25'.“ \h P .4 ' 20.4 \ — 3 _ 3 /5‘- \ ' _ 4 /0'-J . y-\Z\ — 5 -,/5 \ C i __,/ \ -'-/a '— \ J \ U T —- 20 ~30 - 40 ~50 : 60 80 Ema 20 TESTS AND OSCILLOQ’aEE In connection with the study of stability of transmission circuits several oscillograms were taken of the voltages and currents of the two laboratory generators GB and GS operating in parallel near the point of instability. Fig. 5 is a connection diagram showing a typical set of conditions under which the oscillognams were taken. Fig. 5 The following two pages show the short circuit and saturation curves of G2 and G3 respectively. The synchronous impedance to neutral as calculated from these curves is 0.238 ohm for 62 and 1.045 ohms for G3. The average D.C. resistance to neutral of G2 is 0.10 ohm. The average D.C. resistance to neutral of G3 is 0.113 ohm. As can be seen from Fig. 5, all the readings and oscillograms were taken of values of current and voltage in the X and z phases. ‘ fl . i t . ; )O Myst-.1. ‘_ f . p u 5"- c976 1.; \ I. fil.‘ O t 21- Shartf (fray/f d: Safurd/I'M Carl/€5- fbr flr/ Wayne Genera/fir A/a. /052 5 Kw. 25' Amps/20 l/o/f-J FL /2& Speed A300 P/z use 3 (ye/es 60 ’ £5591; 9.1m #3; 1.. gig-2'.- L Q l t .2!» =3: z-. - .1 '1’...“ . I. o _‘:. S J u: If} ' .995“ F'- . I .A" ‘ 3* O ‘ . 0' 4’ - . .- ‘ I _‘ O a I, I” '- “L; .‘ab. ‘. - -‘ ' 9‘... r Erika» ENC fitflwsesmnowwnmiq S a. 3 me am «a .I .,r . . .. ,. . u . -, - . .-. i . V$‘\ ‘83 \mwkm; WNQV “MEG 3 km A w S mm Sewimw gxsxsxmsmm Nw «Q . . N. . $38 as castle 3.: _ , u _. l . . W» 339 as 3%sz uos xSuxC kmfi. . . . . 8 - L —c . 6‘ O ..‘ I a. 3 .f v}. .5 .. . _. 3 o .H I . _ . .:... f... .1“: ...~ .5 . .n . .sb- . . . .... 0L”...Zu &..»o_,~_rr—.....E.uwk~, . . ‘v I I | 1 . .. _ . 1.. - .. ...... #5....” ‘L. c. _. .. a h b 23 OsciIIOgrams 1A. 1B. 10, 2A, 23, and 20 were taken with constant generator excitations on both machines, the driving D.C. motor of GS overexcited and 2.63 ohms resistance in each of the lines between the generators. For these conditions the hunting was very bad and with a prolonged period so that it was necessary to take a slow oscillogram to include the part showing the maximum circulating current. This is shown by 1A. The central portion of 13 shows a reduced circulating current corresponding to the period when G3 was supplying synchronizing power to 02. The approximate meter readings for la, 18 and 1c are as follows: V2 V3 I z W, 1 75-110 65-120 0-40 0-2000 57.2-58.2 Oscillograms 3A and SB represent the same conditions that oscillo- grams 1A, 13 and 10 do except that there is 4.15 ohms resistance in the lines. The approximate meter readings are: E2 E3 Iz W2 f 55-115 82-113 0-25 0-1370 57.9-58.9 Oscillogram.30 represents the same conditions as the previous ones except that the resistances in the lines were 5.84 ohms. The approximate meter readings are: E2 E3 Iz W2 f 106-108 99 10-11 200-240 58.5 This condition has considerably less hunting than the previous cases, however. the resistance in the circuit is too large to be prac- tical. Oscillogram 4A represents the condition of constant driving speeds 24 for both machines, G3 underexcited and 2.63 ohms resistance in each line. The meter readings are: E2 E3 12 W2 f 92.5 56.5 20.5-21.5 90- 20 59.1 Oscillogram 43 represents the same conditions as 4A except G3 is overexcited. The meter readings are: Ea E3 Iz W2 1 9O 95 7-7.5 50-100 58.5 Oscillogram 40 represents the same conditions as 4A except that the line resistance is 5.84 ohms. The meter readings are: E2 E3 Iz W2 f 104-105 20-70 8-12 200-1500 59 Oscillogram 5A represents the condition of constant driving Speeds, GS under excited and reactance coils in the lines. The value of re- actance and resistance of the coils is approximately 14.25 ohms and 0.61 ohm respectively. There was no hunting and the meter readings are: E2 E3 Iz ‘ wz f 122 80 4.1 30 59.9 OscilIOgram 53 represents the same condition as 5A except that G3 is overexcited. Tne meter readings are: E2 E3 Iz W3 f 82 120 5.4 250 61.5 Oscillogram 50 represents the same condition as 5A except that the driving motor of G3 was accelerated. The meter readings are: 25 E2 E3 Iz W2 f 105.5 60 4.3 210 62.0 Oscillogram 6A represents the sane conditions as 50 except that the driving'motor of G3 was retarded. The meter readings are: E2 E3 Iz wz r 107 123 4.7 275 60.9 Oscillogram GB represents the same conditions as 5A except that 63 is over excited and 15.1 ohms resistance was connected in parallel with the reactance coils. The meter readings are: E2 E3 Iz W2 f 73 113 4.0 150 63.5 Oscillogram 60 represents the same conditions as 63 except that G3 was underexcited. The meter readings are: E2 E3 Iz W2 f 100.5 (50-55) 2.7 20 62.4 Oscillogram 7A represents GZ and G3 paralleled as single phase generators, their Y-phases being open. A resistance of 2.8 ohms was connected in the line, a load of 7.7 ohms resistance and 14.2 ohms reactance was connected across the line and the driving motor of G3 accelerated. The meter readings are: Ea as 1. ‘W(11ne) ' W(10ad) f 86 ? 1s 80 640 60.2 Oscillogram 73 represents the same conditions as 75 except that G3 was underexcited. The meter readings are: I W(line) W(load) f 1.“ 1.4 B2 3 106 83 15 20 600 60.2 Oscillogram 70 represents the same conditions as 7; except that the reactance was used alone as the load and the driving motor of G3 was retarded. The meter readings are: E2 E3 I W(11ne) 17(10541) 103 103 15 800 40 In order to check the experimental results with the value of synchronizing power which can be obtained from Chart I, suppose the conditions represented by oscillogram 5A are tested. The angle a or the displacement between the induced voltages of the two gauerators can be assumed to be approximately the same as the phase difference between the voltages shown on the oscillogram since the current is comparatively small and the reactance is practically all in the line. This angle by measurement is approximately 120°. The total reactance in series to neutral is the sum of the two synchronous impedances (14.25 + 1.04 + .24 = 15.53) plus the reactance in the line. The total resistance (.10 + .113 + .61 s .823) is negligible in comparison with the reactance. Using these values together with the voltage to neutral for this case, the value of synchronizing power is 40 watts as compared with 30 watts read on the wattmeter. ....._.;::__ 1‘ :.. : : f : .. i i_ .. _. :::....: Z... 31 3 37 39 44 O . 4 .. . [fill-lullrablwwka . BIBLIOGRAPHY Studies of Transmission Stability. Journal of A.I.E.B., April, 1926, by R. D. Evans and C. F. Wagner. Practical ASpects of Svstem Stability, Journal of A.I.E.E., February, 1926, By Roy Wilkins. Stored Mechanical Energy in Transmission Systems,-Journal of A.I.E.E., September, 1925. Steady-State Stability in Transmission Systems. Calculations by Means of Equivalent Circuits or Circle Diagrams, Journal of A.I.E.E., April, 1926, by Edith Clark. Theory and Calculation of Electric Circuits, by C. P. Steinmetz. Alternator Characteristics Under Conditions Approaching Instability, Journal of.A.I.E.B., January, 1928, by J. 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