Ghosh, Chandra Sekhar

Rao, P Venkata

English

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SINGLE-PHASE OPERATION OF THREE-PHASE MOTORS*69B y  P .  V E N K A T A  R A O  and C H A N D R A  SK K H A R  ('.HOSII{Received for ptiblicalion, N o v e m b e r  to, ig^i)ABSTRACT. Single-phase operation of tliree-phase inot<jrs has been a subject of in te res t to  both design and operating engineers, A large amount of lite ia tu re  exists, in most o f which the m ethod of sym m etrical components analysis has been used, lo r  the analysis o f performance o f three-phrse motors w ith  unbalanced voltages. In  th is  paper the method o f Dyadic c ircu it analysis developed by Sah is applied to the problem of operation o f three-phase motors from  single-phase supply and the necessary relationships have been developed of the complete performiince characteristic.s from standstill to syn­chronous speed w ith  a u x ilia ry  impedances in c irc u it.A  considerable amount of work has been done on the su!)jecl of single- phase induction motors which use split-phase starting. The existing litera­ture on these machines deals with the auxiliary impedances required and the torque developed. The use of auxiliary impedances in the single-phase operation of polyphase motors is also well known.When a polyphase motor is operated from a single-phase supply, and auxiliary impedances are used to shift the phase currents and the motor terminal voltages so that the currents and voltages of the motor are properly balanced both as to phase and maguitude, the motor will operate as though it were supplied from its normal polyphase line. The two common methods in use are the series impedance method and the shunt impedance method. By a judicious choice of the resistance and reactance values of the auxiliary impedances it is possible 1o have leasouably good balancing, both as regards phase and magnitude of the currents and voltages. It happens quite often that the external impedances are used only for starting purposes, the motor being kept running with two of its terminals connected direct to the single-phase Hue, the third terminal being left open.Due to the fact that the motor current and power factor both change as the speed changes, the auxiliary impedances required also must change with the motor speed. Of interest, therefore, is the determination of the perfor­mance characteristics of a three-phase motor on single-phase supply over the complete speed rauge from standstill to synclironous speed with the auxiliary impedance in circuit.Lumm ^1936) ill his paper, outlines a method that is quite general but the utility of his method is limited by the tedious algebra involved. Reed and Koopman (1936) have presented an analysis which is neither so tedious nor highly involved. They use the exact equivalent circuit and the analytical * Com m unicated by Prof. P . C. M ahantl.566 P. V. Rao and C. S. Ghoshdetcnniuation of tlie characteristics which are familiar to all designers of induction motors. In their paper they have also applied the analysis to the performance of some motors under various conditions of starting and operating with auxiliary capacitance.Tracy and W yss (1935) have analysed the problem of split-phase starting of three-phase motors and have arrived at some very useful conclusions. They determined the best values of resistance and reactance for series and shunt methods of split-phase starting. In the first method, the starting current is not restricted to any particuiar value as in the second to obtain the maximum starting torque. In the analysis, a direct solution is visualised whereby the optimum values of resistance, reactance or capacitance can be arrived at for maximum starting torque with an arbitrarily fixed starting current in the line. Thacker and Gcpalakrishna (194S), in their paper have extended the work of Tracy and Wyss using the same method of symmetrical components. In their paper, methods of calculations are presented for the determination of the complete performance characteristics from standstill to synchronous speed of a three-phase motor on a single-phase supply with the auxiliary impedances in circuit. They have set up equations taking into account generalized impedances so that the equations may be applicable to any combination of resistances, reactances and capacitances. However, the final relations for the line currents, voltages and torque are expressed in terms of the sequence impedances. The vector diagrams drawn for the equivalent circuit of the motor with the auxiliary impedances are also incomplete. Further, the steps required in arriving at the final relation­ships are also long and inv'olved.The Dyadic circuit analysis method developed by Sah (1939) offers an extremely simple and short method of calculating the performance of three phase motors under unbalanced voltages. In this paper, simple calculations are given by the method of Dyadic algebra for the exact detemination of the complete performance characteristics from standstill to synchronous speed of a three-phase motor run from single-phase supply with auxiliary impedances in circuit. The correct vector diagrams for the citrrents and voltages for conditions with series impedance and shunt impedance in circuit arc given. Expressions for the starting torque and starting current have been obtained in terms of phase impedances. This is an advantage compared to the other formula developed previously in which the relationships have been obtained in terms of the sequence impedances. By simple substitution of the sequence impedance values in the final formula for the current, the relationships developed here are shown to agree with those developed by other authors.Finally, as a matter of interest, the method has been applied to the most commonly employed system in which a capacitance is placed in shunt across two of the phases, while no other impedance is used in the other branch. Expressions for the line current, sequence currents and torque are developed for this arrangement.-S E R I E S  I INI P E D A N C E :\1 E T II O D O V o  P E R \ T I O NFigure i{a) gives the equivalent circuit with series impedances and the vectoi diagiaiii for the currents and voltages with this anangcineut is given in figure ivb).Singlc-phosc OpefQtion of Thtce-^phase Motors 567F i g . i ( b )V e c to r  d ia g r a m  o f  s e iie s  im p e d a n c e  c i r c u i tI ... (i)The fiindaincntal volta|>e equations areV„==AIa + CTb + B[<+ A J t + C l ,W = C l a + B I i + A l e*where r „ , F t, and ! ’ « are the voltages to the neutral in the respective phases.Referring to figure i (a) and by use of Kirchoff’s laws,i  o — I a +  1 1t b\ ... (a)• •• (3 'Since power input is VA,570 P. V. Rao and C. S. GhoshEfficiency = r ( i - s ) - ( F +  iV)VAS H U N T  I M P E D A N C E  M E T H O D  O F  ( O P E R A T I O NThe schematic circuit arrangement is shown in figure z a^). Figure 2<fc) gives the vector diagram for the currents and voltages in the circuit of figure 2(a).F k ; 2(6)Vector diagram of shunt impedance circuitI c = I a  + - - I .  \X3(17)(iSJand V=^VcaHence, by using the fundamental voltage equations (i)V a ^ I j A - C ) + I A B - C )  ^V i ^ l A B - A )  + I A C - A )  j- . V c = I a { C - B }  + l A A - B )  JUsing the same notations K,  M and N  as before, from equations (18)(19)and (19), it followsI' = F ,„  =  / „ ( - N ) + 7 ,cMalsol^ba’^lafC + l A - N)Fc* =  7«M + / J - ■ K)From equation (17) thereforeZi I „ =Z^I a+ V b a = Z J a + I a K  + I e i - N ) = I „ i K + Z ^ ) - I e N  Similarly, Z A o = ^ I a M - I A K  + ZA(i8a)(20)(21)Single-phase Operation of Three-phase Motors Solving for I*  and 7,57/(K + Z,)' K-i-Z: ,^-MN>j ,  =  I qS M Z , - ^ ( K  _J {K + “z7) (K  - rZ- ^' -MN)} , Substituting these values of /„ and /, in (j8a)(22)V = l oHence- N \ Z ^ i K  -i-ZJ M {M Z ,  -  (K + Z J }\''K  +  z , ) { i < - ^ z ^ ) - m n ]  {(ycVZi)(A'  + Z a ) - m MF|(Ar + Z j ) (IC +  Z2 ) - M N i[ {N=Za -  N Z J K  t- -  M z J k  + Zi)}]I t  is th u s possib le  to  pred eterm in e the valu e  of the lin e cu rren t from  th e kn o w n  v a lu e s  of the a u x ilia ry  im pedances and the ijhase im pedance. S in c e  A ', M . N  can be easily  d eteim in ed  for any va lu e  of slip , it is possible to  d e term in e  the cu rren t 7c at an y slip.T h e  co n d itio n s  at stan d still are,Z ,  + Z -  =  Z ; B =  r  =  A  =  - Z  K  =  2ZH e n c e , b y  su b stitu tio n  in (23)^ ___  l ' { ( K ^ - Z , y K - i - Z , ) - M ’‘ }‘ ‘ {A'^Za -  N Z V k  -f Z A ^  M ^ Z / -  M Z ^ ( K  + Z,)}In  term s of sym m etrica l com ponentsM N = { o ^ Z ^  + a Z . ) ( a Z ^  +  a ^ Z -) =  Z  +  Z .   ^-  Z , Z . M^  =  CLZ^  ^ +  a^Z'^- +  2Z , Z  7V" =  a®Z^ * + aZ-* + 2Z+Z A=* =  Z *“ + Z_" + 2Z+Z_F ro m  (23)r _  T f Z ^ +  (Z, -f ZJ(Z^ +  Z -)  + 3^ ^ ^ -](24)and from (24)alsoT(z+ + zlyz.z^  + 3-^+z-^z, z,) ]r ^  f^rZ,Z,4- 2Z f2Z, + Z.) + 3 ^ ’’ 3(23a)(24a;7„ =*7. i:,(z*+ 2 Z ) ^-zz.(Z, + 2Z)(2Z + Z a ) - Z *  - 2 Z ,-Z a f Z g  + 2Z)(Z| +  2Z / { Z s +  2Z) — z*and the positive and negative sequence currents are7  =  J -  a) +  J — o)  I7 + 1c^a — a^)572 P. V. Rao and C. S. Ghosh• •• (25)v^ 3as before, torque A particular caseT ^ 3-/ 2 //rurf 2 — 5 )jPI'-Ij (26)The sp>ecial case in which a capacitance is connected across two of the phases \A'hile there is no impedance connected across the other side is shown diagraminatically in figure 3la) and the corresponding vector diagram for currents and voltages is given in figure 3(b).VcAVector diagram of capacitor in shunt circuit As shown in figure 3(a), the conditions in this case areZ, =  oo and Za =  —j XHence from equation (24a), after dividing the numerator and denominatorby Zir _ y f  Zj-i- 2Z+ (ZIZi'j{;iZ +  2Za) "  [  2ZZ, + 3^“ +32='Zs/Z;Now, substituting the values of Zj and Za in this casej"  Z (3 Z -2 ,X ) ••• '27)In the same way, substituting the values for Zj and Za in eqn. (23a) after proper transformation, the current during running condition isSingle-phase Operation of Three-phase Motors 5732^8)The values of /-, and /_ are the same as given in equation '25,'. The currents in the other blanches at standstill condition are also determined as below.Since 7n = 7„, and — jX,  in this case, and reniembeiing that A/= - /  and J\ =  2/. at standstill, from equation (21 !7 = 7  ^ 1  i - ' •y ±^ -2 '  ! - ( / - j X )  !"(7C + Z2' "i "! izZ— iX)After substituting for 7„, the exjuessioa olitained in 27'r.' (S u b s titu tin g  for /„ and I   ^ in equations (25'iSimilarlytorque,— ) { 2 / . j X ) { \  — a)  ,— ]/J (u 1 ) I) / (3 Z -2 /X ) '/ 'r  _ +y + jX)\a^ — a) } /{Jy~2j x^ \r = 3rsI . (2 — s)- i 7 .(29 >(30b)Symbols used in the papeiVa, Vb, I’ c — Voltages of the three respective jihases to neutral ^1, y  ^ — Kxternal auxiliary impedances7" —Torque developed in synchronous watts P  —Power converted to mechanical form F  + W — Friction and windage loss in watts Z —Machine phase impedance Z^, Z_ — Positive and negative sequence impedances /J, B, C — Respective jdiase impedance parameters of the machine — Single-jthase line current I t , 7« —Sequence components of current P’ — Applied single-phase voltage.The steady state values for the machine constants are :574 P. V, Mao and C. S, Ghosh^iz=r + i X  +  3-^w  ^ j  __ __ 4 ,________ 2 5_____4 \ r^j; + j s X t )  \rg + j {2 - s ) X x \f - 3A V ]4 ( \ r x + ] s X ia(2 — s'‘ ^ —  “  jA  „ f, +V   ^ ({r.f+i^2- a)X^} a®(2-s)4 ((r^ + j s X x )  {rx +  i (2 - a-)A'x>Dfel'ARTWKNT OK POWK.K KNGINKERINO I ndian Institutk ok Sciknir, B a m .alori',.R K V I<: R E  N  C E  SEunini, li O , J936, Train. A. 1. E. F.,  85, 387.Reed, H. R . and Kyopnian, R J W., 1936 Trans. A, 1. E. T\, 86, 1206.Tracy, G. V., and Wyss, W. E., 1935, Trans. A. 1. E  F. 54. 1068.Thacker, M. vS., and Gopalakrishna, H V., 1948, Elcctrotechnics , 20, 82. vSah, A. Pen-Tung, 1939 , ‘ ‘Dyadic Circuit Analysis” International Text Book Co., Scranton, Pa.

Single-Phase Operation of Three-Phase Motors

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/1648/1/Single-Phase%20Operation%20of%20Three-Phase_By%20P.%20Venkata%20Rao.pdf

Single-Phase Operation of Three-Phase Motors

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