# Module 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur

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1 Module 5 Three-hse A iruits Version EE IIT, Khrgur

2 esson 8 Three-hse Blned Suly Version EE IIT, Khrgur

3 In the module, ontining six lessons (-7), the study of iruits, onsisting of the liner elements resistne, indutne nd itne, fed from single-hse suly, hs een resented. In this module, whih my lso e termed s n extension of the revious one, ontining three lessons (8-0), the solution of urrents in the lned iruits, fed from three-hse suly, long with the mesurement of ower, will e desried. In this (first) lesson of this module, the genertion of three-hse lned voltges is tken u first. Then, the two tyes of onnetions (str nd delt), normlly used for the ove suly, followed y line nd hse quntities (voltges nd urrents) for the onnetions, in oth suly nd lod sides (oth eing lned), re desried. Keywords: Three-hse lned voltge, str- nd delt-onnetions, lned lod. After going through this lesson, the students will e le to nswer the following questions:. How to generte three-hse lned voltges?. Wht re the two tyes of onnetions (str nd delt) normlly used for three-hse lned suly?. Wht re ment y the terms line nd hse quntities (voltges nd urrents), for the two tyes of onnetions in oth suly nd lod sides (oth eing lned)? Genertion of Three-hse Blned Voltges In the first lesson (No. ) of the revious module, the genertion of single-hse voltge, using multi-turn oil led inside mgnet, ws desried. It my e noted tht, the sheme shown ws shemti one, wheres in mhine, the windings re distriuted in numer of slots. Sme would e the se with norml three-hse genertor. Three windings, with equl no. of turns in eh one, re used, so s to otin equl voltge in mgnitude in ll three hses. Also to otin lned three-hse voltge, the windings re to e led t n eletril ngle of 0 with eh other, suh tht the voltges in eh hse re lso t n ngle of 0 with eh other, whih will e desried in the next setion. The shemti digrm with multi-turn oils, s ws shown erlier in Fig.. for single-hse one, led t ngle of 0 with eh other, in -ole onfigurtion, is shown in Fig. 8.. The wveforms in eh of the three windings (, Y & B), re lso shown in Fig. 8.. The windings re in the sttor, with the oles shown in the rotor, whih is rotting t synhronous seed of N s (r/min, or rm), to otin frequeny of f (( N s ) /0) (Hz), eing no. of oles [ ] (see lesson no. ). Version EE IIT, Khrgur

4 0º Y' N B' 0º B S ' 0º Y Fig. 8. () Shemti digrm of three windings of sttor for the genertion of three hsed lned voltge (-ole rotor). Three-hse Voltges for Str onnetion The onnetion digrm of str (Y)-onneted three-hse system is shown in Fig. 8., long with hsor reresenttion of the voltges (Fig. 8.). These re in ontinution of the figures 8.-. Three windings for three hses re () & ( ),Y () & Y ( ), nd B () & Y ( ). Tking the winding of one hse, sy hse s n exmle, then with sign () is tken s strt, nd with sign ( ) is tken s finish. Sme is the se with two other hses. For mking str (Y)-onnetion,, Y & B re onneted together, nd the oint is tken s neutrl, N. Three hse voltges re: e sin θ ; e sin ( θ 0 ) ; e N E m BN YN E m E sin ( θ 40 ) E sin ( θ 0 ) m It my e noted tht, if the voltge in hse ( m ) is tken s referene s stted erlier, then the voltge in hse Y( eyn ) lgs e N y 0, nd the voltge in hse B( ebn ) lgs e YN y 0, or leds e N y 0. The hsors re given s: e N Version EE IIT, Khrgur

5 E N 0 E (.0 j0.0) : 0 E ( 0.5 j0.866) ; E BN 0 E ( 0.5 j0.866). E YN E BN E YN B B - N B - Y E N - Y Y () -V N V BN V Y V B -V YN V N 0 V YN -V BN V YB () Fig. 8. () Three-hse lned voltges, with the soure str-onneted (the hse sequene, -Y-B) () Phsor digrm of the line nd hse voltges The hse voltges re ll equl in mgnitude, ut only differ in hse. This is lso shown in Fig. 8.. The reltionshi etween E nd Em is E E m /. The hse sequene is -Y-B. It n e oserved from Fig. 8. tht the voltge in hse Y ttins the mximum vlue, fter θ ω t 0 from the time or ngle, fter the voltge in Version EE IIT, Khrgur

6 hse ttins the mximum vlue, nd then the voltge in hse B ttins the mximum vlue. The ngle of lg or led from the referene hse, is stted erlier. eversl of hse sequene from -Y-B to -B-Y If the hse sequene is reversed from -Y-B to -B-Y, the wveforms nd the orresonding hsor digrm re shown in figures 8. (-) resetively. It n e oserved from Fig. 8. tht the voltge in hse B ttins the mximum vlue, fter θ 0 from the time (or ngle), fter the voltge in hse ttins the mximum vlue, nd then the voltge in hse Y ttins the mximum vlue. The ngle of lg or led from the referene hse, is stted erlier. The sme sequene is oserved in the hsor digrm (Fig. 8.), when the hse sequene is reversed to -B-Y. Version EE IIT, Khrgur

7 V YB V YN 0 0 V N V YN V B V BN VY () Fig. 8. () Three-hse lned voltge wveforms with the soure str-onneted (the hse sequene, -B-Y) () Phsor digrm of the line nd hse voltges eltion etween the Phse nd ine Voltges for Str onnetion Three line voltges (Fig. 8.4) re otined y the following roedure. The line voltge, E Y is E E E E 0 E 0 E [( j0) ( 0.5 j0.866)] Y N YN E (.5 j0.866) E 0 The mgnitude of the line voltge, E Y is times the mgnitude of the hse voltge EN, nd E Y leds E N y 0. Sme is the se with other two line voltges s shown in rief (the stes n esily e derived y the roedure given erlier). E E E E 0 E 0 E 90 YB YN BN EB EBN EN E 0 E 0 E 50 So, the three line voltges re lned, with their mgnitudes eing equl, nd the hse ngle eing disled from eh other in sequene y 0. Also, the line voltge, sy E, leds the orresonding hse voltge, E y 0 Y N Version EE IIT, Khrgur

8 eltion etween the Phse nd ine Voltges for Delt onnetion The onnetion digrm of delt ( Δ )-onneted three-hse system is shown in Fig. 8.4, long with hsor reresenttion of the voltges (Fig. 8.4). For mking delt ( Δ )-onnetion, the strt of one winding is onneted to the finish of the next one in sequene, for instne, strting from hse, is to onneted to Y, nd then Y to B, nd so on (Fig. 8.4). The line nd hse voltges re the sme in this se, nd re given s B ' - E B E Y B ' - B - Y ' Y E YB Y B () E B E Y E YB () Fig. 8.4 () Three-hse lned voltges, with the soure delt-onneted (the hse sequene, -Y-B) () Phsor digrm of the line nd hse voltges E Y E 0 ; E E 0 ; E E 0 YB B Version EE IIT, Khrgur

9 If the hsor sum of the ove three hse (or line) voltges re tken, the result is zero (0). The hse or line voltges form lned one, with their mgnitudes eing equl, nd the hse eing disled from eh other in sequene y 0. urrents for iruit with Blned od (Str-onneted) I N V Y N ' Y B I YN I BN () V BN Φ I BN 0 I YN 0 Φ 0 Φ I N V N Y YN () Fig. 8.5 () iruit digrm for three-hse lned str-onneted lod () Phsor digrm of the hse voltges, nd the line & hse urrents A three-hse str (Y)-onneted lned lod (Fig. 8.5) is fed from lned three-hse suly, whih my e three-wire one. A lned lod mens tht, the mgnitude of the imedne er hse, is sme, i.e.,, nd their ngle is lso sme, s φ φ N φyn φb N N YN BN. In other words, if the imedne er hse is given s, φ j X, then N YN B N, nd lso Version EE IIT, Khrgur

10 X X N X YN X BN re:. The mgnitude nd hse ngle of the imedne er hse X, nd φ tn ( X / ) hse voltges, V VN VYN VB N. For lned lod, the mgnitudes of the re sme, s those of the soure voltges er hse V V V, if it is onneted in str, s given erlier. So, this mens tht, N YN BN the oint N, str oint on the lod side is sme s the str oint, N of the lod side. The hse urrents (Fig. 8.5) re otined s, VN 0 VN I N φ φ φ I I YN BN N N VYN 0 VYN 0 φ ) (0 φ ) φ ( YN YN VBN 0 V 0 φ ) φ BN ( BN BN (0 φ ) In this se, the hse voltge, V N is tken s referene. This shows tht the hse urrents re equl in mgnitude, i.e., ( I I N I YN I BN ), s the mgnitudes of the voltge nd lod imedne, er hse, re sme, with their hse ngles disled from eh other in sequene y 0. The mgnitude of the hse urrents, is exressed s I ( V / ). These hse urrents re lso line urrents ( I I IY IB ), in this se. Totl Power onsumed in the iruit (Str-onneted) In the lesson No. 4 of the revious module, the ower onsumed in iruit fed from single-hse suly ws resented. Using the sme exression for the ove stronneted lned iruit, fed from three-hse suly (Fig. 8.4-), the ower onsumed er hse is given y W V I os φ V I os V, I ( ) It hs een shown erlier tht the mgnitude of the hse voltge is given y V V /, where the mgnitude of the line voltge is V. The mgnitudes of the hse nd line urrent re sme, i.e., ower onsumed is otined s W V / I os φ V I os φ ( l ) I I. Sustituting the two exressions, the totl Plese note tht the hse ngle, φ is the ngle etween the hse voltge hse urrent, I. V, nd the Before tking u n exmle, the formuls for onversion from delt-onneted iruit to its str equivlent nd vie vers (onversion from str to delt onnetion) using imednes, nd lso idel indutnes/itnes, re resented here, strting with iruits with resistnes, s derived in lesson #6 on d iruits. Version EE IIT, Khrgur

11 Delt( )-Str(Y) onversion nd Str-Delt onversion Before tking u the exmles, the formul for Delt( Δ )-Str(Y ) onversion nd lso Str-Delt onversion, using imednes s needed, insted of resistne s elements, whih is given in lesson #6 in the module on D iruit, re resented. The formuls for delt-str onversion, using resistnes (Fig. 8.6), re, The formuls for delt-str onversion, using resistne, re, The derivtion of these formuls is given in lesson #6. If three equl resistnes ( ) onneted in delt, re onverted into its equivlent str, the resistnes otined re equl, its vlue eing ) / (, whih is derived using formuls given erlier. Similrly, if three equl resistnes onneted in str, re onverted into its equivlent delt, the resultnt resistnes, using formuls, re equl ( ) / ( ). The formul for the ove onversions using imednes, insted of resistnes, re sme, reling resistnes y imednes, s the formul for series nd rllel omintion using imednes, insted of resistnes, remin sme s shown in the revious module on single hse iruits. Version EE IIT, Khrgur

12 The formuls for delt-str onversion, using imednes (Fig. 8.7), re, The formuls for delt-str onversion, using imedne, re, Plese note tht ll the imednes used in the formul given here re omlex quntities, like, φ, φ,, hving oth mgnitude nd ngle s given. The formuls n e derived y the sme roedure s given in lesson #6. An exmle is tken u, when three equl imednes onneted in delt re to e onverted into its equivlent str. The imednes re equl, oth in mgnitude nd ngle, suh tht, nd φ φ φ φ. The imednes onneted in delt re of the form φ ± j X. Using the formul given here, the imednes of the str equivlent re lso equl, hving the mgnitude s ( / ) nd ngle s φ φ φ φ. The ngles of the equivlent imedne onneted in str re equl to the ngles of the imednes onneted in delt. The imednes onneted in delt re lso equl, oth in mgnitude nd ngle, nd re of the form φ ( / ) φ ( / ) ± j ( X / ). Similrly, if three equl imednes onneted in str re onverted into its equivlent delt, the mgnitude nd ngle of the imednes using the formuls given here, re ( ) nd φ φ φ φ resetively. This shows tht three imednes re equl, oth in mgnitude nd ngle, with its vlue eing φ ( ) φ [ ( / )] ± j[ ( X / )] ± j X whih n lso e otined simly from the result given erlier. Version EE IIT, Khrgur

13 Now, let us use the ove formul for the iruits (Fig. 8.8), using indutnes only. The symols used for the indutnes re sme ( ). The imednes of the indutnes onneted in delt, re omuted s,,, X j ω φ, the ngles in three ses re. The mgnitudes of the imednes re roortionl to the resetive indutnes s 90 X. onverting the omintion into its equivlent str, the indutnes using the formuls given here, re These reltions n lso e derived. Further, these re of the sme form, s hs een erlier otined for resistnes. It my e oserved here tht the formuls for series nd rllel omintion using indutnes, insted of resistnes, remin sme, s shown in the revious module on single hse iruits, nd lso n e derived from first riniles, suh s reltionshi of indued emf in terms of indutne, s omred with Ohm s lw for resistne. The indutnes re ll idel, i.e. lossless, hving no resistive omonent. The formuls for str-delt onversion using indutnes (onversion of stronneted indutnes into its equivlent delt) re, These re of the sme form s derived for iruits with resistnes. If three equl indutnes ( ) onneted in delt, re onverted into its equivlent str, the indutnes otined re equl, its vlue eing, whih is derived using formuls given erlier. Similrly, if three equl indutnes onneted in str, re onverted into its equivlent delt, the resultnt indutnes, using formuls, re equl ( ) / ( ) / ( ). Version EE IIT, Khrgur

14 The formuls for the iruits (Fig. 8.9) using itnes re derived here. The symols used for the itnes re sme (,,, ). The imednes of the indutnes onneted in delt, re omuted s 0 X φ.0 jx 90, the ngles in three ses re ( 90 ). The mgnitudes of the imednes re inversely roortionl to the resetive itnes s, X X / ω ) (/ ). ( onverting the omintion into its equivlent str, the resultnt itnes using the formuls given here, re (/ ) (/ ) / / / / or Similrly, The itnes in this se re ll idel, without ny loss, seilly t ower frequeny, whih is true in nerly ll ses, exet otherwise stted. The formuls for str-delt onversion using itnes (onversion of str-onneted itnes into its equivlent delt) re, / or Similrly, (/ ) (/ ) (/ )(/ ) (/ )(/ ) / If three equl itnes ( ) onneted in delt, re onverted into its equivlent str, the itnes otined re equl, its vlue eing Version EE IIT, Khrgur

15 ( ), whih is derived using formuls given erlier. Similrly, if three equl itnes onneted in str, re onverted into its equivlent delt, the resultnt itnes, using formuls, re equl ( / ( ) / ). The formuls for onversion of three equl indutnes/itnes onneted in delt into its equivlent str nd vie vers (str-delt onversion) n lso e otined from the formuls using imednes s shown erlier, only y reling indutne with imedne, nd for itne y reling it reirol of imedne (in oth ses using mgnitude of imedne only, s the ngles re equl ( 90 for indutne nd 90 for itne). Another oint to note is left for oservtion y the reder. Plese hve lose look t the formuls needed for delt-str onversion nd vie vers (strdelt onversion) for itnes, inluding those with equl vlues of itnes, nd then omre them with the formuls needed for suh onversion using resistnes/indutnes (my e imednes lso). The rules for onversion of itnes in series/rllel into its equivlent one n e omred to the rules for onversion of resistnes/indutnes in series/rllel into its equivlent one. The reder is referred to the omments given fter the exmle 8.. Exmle 8. The str-onneted lod onsists of resistne of 5 Ώ, in series with oil hving resistne of 5 Ώ, nd indutne of 0. H, er hse. It is onneted in rllel with the delt-onneted lod hving itne of 90 μ F er hse (Fig. 8.0). Both the lods eing lned, nd fed from three-hse, 400 V, 50 Hz, lned suly, with the hse sequene s -Y-B. Find the line urrent, ower ftor, totl ower & retive VA, nd lso totl volt-meres (VA). I -jx Y I Y jx N B I B () Version EE IIT, Khrgur

16 Y I Y I / N Solution B I B () Fig. 8.0 () iruit digrm (Exmle 8.) () Equivlent lned str-onneted iruit f 50 Hz ω π f π rd / s For the lned str-onneted lod, 5 Ω For the indutne oil, r 5 Ω X ω Ω with the ove vlues tken er hse. The imedne er hse is, ( r j X ) 5 (5 j 6.8) (0 j 6.8) Ω For the lned delt-onneted lod, 90 μf onverting the ove lod into its equivlent str, / 90 / 0 μf 6 X / ω /( ) 06. Ω The imedne er hse is j In the equivlent iruit for the lod (Fig. 8.0), the two imednes, & re in rllel. So, the totl dmittne er hse is, Y Y Y [(4.6 j 4.46) j 9.45] 0 (4.6 j 5.0) Ω The totl imedne er hse is, / Y /( ) (99.0 j08.7) Ω The hsor digrm is shown in Fig Tking the hse voltge, s referene, V N VN V V / 400 /. 0 V Version EE IIT, Khrgur

17 I BN V BN V Y Φ I YN Φ Φ V N V YN I N Fig. 8.0 () Phsor digrm The hse voltges re, V N.0 0 ; VYN.0 0 ; VBN.0 0 So, the hse urrent, is, I N V N I N ( j.6) The two other hse urrents re, I YN ; I BN As the totl iruit (Fig. 8.5) is tken s str-onneted, the line nd hse urrents re sme, i.e., I I. 575 A Also, the hse ngle of the totl imedne is ositive. So, the ower ftor is os φ os lg The totl volt-meres is S V I kva The totl VA is lso otined s S V I kva The totl ower is P V I os φ W The totl retive volt-meres is, Q V I sin φ.575 sin VA This exmle n e solved y onverting the str-onneted rt into its equivlent delt, s shown in Exmle 9. (next lesson). A simle exmle (0.) of lned str-onneted lod is lso given in the lst lesson (#0) of this module. After strting with the genertion of three-hse lned voltge system, the hse nd line voltges, oth eing lned, first for str-onnetion, nd then for deltonnetion (oth on soure side), re disussed. The urrents (oth hse nd line) for A Version EE IIT, Khrgur

18 lned str-onneted lod, long with totl ower onsumed, re lso desried in this lesson. An exmle is given in detil. In the next lesson, the urrents (oth hse nd line) for lned delt-onneted lod will e resented. Version EE IIT, Khrgur

19 Prolems 8. A lne lod of (6j)Ω er hse, onneted in str, is fed from threehse, 0V suly. Find the line urrent, ower ftor, totl ower, retive VA nd totl VA. 8. Find the three voltges V n, V n, & V n, in the iruit shown in Fig. 8.. The iruit omonents re: 0 Ω, jx j7. Ω. Version EE IIT, Khrgur

20 ist of Figures Fig. 8. Fig. 8. Fig. 8. Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 8.8 Fig. 8.9 () Shemti digrm of three windings of sttor for the genertion of three hsed lned voltges (-ole rotor). () Three-hse lned voltge wveforms with the soure stronneted (the hse sequene, -Y-B) () Three-hse lned voltges, with the soure str-onneted (the hse sequene, -Y-B) () Phsor digrm of the line nd hse voltges () Three-hse lned voltge wveforms for the hse sequene, -B-Y () Phsor digrm of the line nd hse voltges () Three-hse lned voltges, with the soure delt-onneted (the hse sequene, -Y-B) () Phsor digrm of the line nd hse voltges () iruit digrm for three-hse lned str-onneted lod () Phsor digrm of the hse voltges, nd the line & hse urrents esistnes onneted in () Delt nd () Str onfigurtions Imednes onneted in () Delt nd () Str onfigurtions Indutnes onneted in () Delt nd () Str onfigurtions itnes onneted in () Delt nd () Str onfigurtions Fig. 8.0 () iruit digrm (Exmle 8.) () Equivlent lned str-onneted iruit () Phsor digrm Version EE IIT, Khrgur

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