Observations: With transformers, we need to convert the voltage magnitudes, impedances, currents, etc. repeatedly.

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1 1 ntductin t Pe Unit Calculatins (ECE 459 nte) Obsevatins: With tansfmes, we need t cnvet the vltage magnitudes, impedances, cuents, etc. epeatedly. Questins: Can we bypass these cmplicatins? Answe: By using pe unit calculatins that diffeent vltage s (but the same pwe ) ae used in diffeent tansfme sectins. The cnvesins amng tansfme vltages ae absbed by the elatinships. Thus p.u. epesentatins f system elements (tansfmes, impedances, lads, geneatins, etc.) becme me unifm and the fixed- tap tansfmes disappea fm the p.u. epesentatin. The same situatins apply t wye-delta cicuits, and they will have the same p.u. epesentatin. Again, the cnvesin is absbed int the elatinship. The same f single-phase and theephase epesentatins. Cnside the simple cicuit f Figue 1 in which a lad impedance f L 60 + j70 Ω Ω is cnnected t a vltage suce. The n lad vltage f the suce is E The intenal esistance f the suce is j10 Ω. The ttal impedance f the cicuit is T 60 + j80 Ω. The lad cuent thugh the cicuit is E L 10A j80 T The appaent pwe pduced by the suce is E L ,000A 53 1 j10ω E 1000 L 60+ j70ω Figue 1 Basic Cicuit: Example f p.u. calculatins n this cicuit, we will designate the magnitude f each f these values as the efeence value f that quantity, i.e. the efeence value will be ne unit f that quantity. That is,

2 1000 is equal t ne unit f vltage ( 1000 ) 100 Ω is equal t ne unit f esistance ( 100 Ω) 10 A is equal t ne unit f cuent ( 10 A) 10,000 A is equal t ne unit f pwe ( 10,000 A) These basic units ae called the values. Theefe, the values f u cicuit ae 1000, 100 Ω, 10 A, and 10 ka. Nw we can efe any value f any quantity in the cicuit t these values. F example, the magnitude f the vltage acss the lad is L 9. L E T n tems f u chsen vltage, the lad vltage is the ati f the actual vltage and L 9 the chsen vltage, i.e f the chsen vltage. We say 1000 that the vltage is 0.9 pe unit f vlts. The designatin is 0.9 p.u. (). We can als say that the lad vltage is 9.% f the vltage. imilaly, in tems f u impedance, the lad impedance is L p.u.( Ω). 100 Just like the magnitude f the lad impedance, we can expess any hmic value in tems f the impedance. the pe unit esistance f the lad is R R L p.u.( Ω) X L 70 The pe unit eactance f the lad is X 0.7 p.u.( Ω) 100 Ntice hee that bth, the esistance and the eactance ae cmpaed t impedance. 1 L is the cnjugate f the lad cuent. n this case L 10A + 53

3 3 The pwe dissipated by the esistance is P R W. Expessed as pe unit value n u f 10,000 A, the dissipated pwe is P P ,000 p.u. 0.6p.u.(W) The eactive pwe t the lad is Q L X L A (inductance). Expessed as pe unit value n u f 10,000 A, the lad eactive pwe is Q Q L Lp.u 0.7 p.u.(a) (ind) imilaly as in handling the pe unit values f hmic quantities, bth eal pwe and eactive pwe ae cmpaed t the appaent pwe, usually called MA. Nte that the p.u. values ae much me unifm than the iginal values! All values ae nly magnitude. They ae nt assciated with any angle. The pe unit values, hweve, ae phass. The phase angles f the cuents and vltages and the pwe fact f the cicuit ae nt affected by the cnvesin t pe unit values. The pe unit values can be witten as phass, and all calculatins that can be dne with the actual quantities can be als dne with pe unit values. F example, in u cicuit, the pe unit lad impedance and the pe unit lad cuent ae: p.u.( Ω) and 1 53 p.u.(a) L Thus the vltage acss the lad is p.u.() n geneal, the pe unit value is the ati f the actual value and the value f the same quantity. actual value pe unit value value

4 4 Manufactues give impedance f equipment in pecent n wn. The pecent value is the pe unit value multiplied by 100: % p.u. 100%. The expessin wn means that the vltage is the ated vltage f the equipment, and the pwe is the ated appaent pwe (in A) f the equipment. The cuent and the impedance ae calculated fm the vltage and the A: and The pecent impedance is then in Ω in Ω actual % 100% Geneats and Pe Unit ystem A geneat ated 1000 A and 00 has intenal impedance f j10 Ω. Figue Basic Cicuit: Example f p.u. calculatins f geneats The atings f the geneat ae chsen as the values. n this example, 1000 A A Ω 1000 Nte that the magnitude de diffeences ae absbed in the!

5 5 The geneat impedance is stamped n the nameplate tgethe with the the atings. This geneat has impedance f j5%. This means that the pe unit impedance is % j5 p.u. j0.5 p.u. ( Ω) The actual impedance in Ω is actual p.u. j j10 Ω Example: The geneat abve is sht cicuited at its teminals. Find the sht cicuit cuent and the sht cicuit pwe deliveed by the geneat in p.u., in %, and in the actual units. lutin: Figue 3 Example: ht Cicuit n Geneat Teminals sc p.u. E G p.u. p.u. 1 j0.5 j4.0 p.u.(a) sc % sc p.u. 100% j400 % * sc p.u. E G p.u. sc p.u. 1 ( j4) j4.0 p.u. (A) % sc p.u. 100% sc j400 % The actual values calculated fm the pe unit values ae: sc sc sc p.u. sc p.u. j j0.0 A j j4000 A Tansfmes and Pe Unit ystem A tansfme is ated 000 A, 00/400, and has an intenal impedance f j4.0 Ω as seen fm the lw vltage side.

6 6 (a) (b) Figue 4 Basic Cicuit: Example f Pe Unit Calculatins f a Tansfme (a) Tansfme mpedance Refeed t the Lw ltage ide (b) Tansfme mpedance Refeed t the High ltage ide The intenal impedance f the tansfme as seen fm the high vltage side is H 400 L j4.0 j16. 0 Ω 1 00 The ated values f pwe and vltage ae used as the s f the calculatins. t means that the vltage is diffeent n each side f the tansfme. Cmpaisn f the s and the pe unit value n bth sides f the tansfme is in Table 1

7 7 lw vltage side 000 A 00 10A high vltage side 000 A 400 5A p.u. L 0Ω j4.0 0 j0.p.u( Ω) L 80Ω j16 80 j0.p.u( Ω) Table 1 Bases f tansfme lw side and high side Ntice in Table 1 that the tansfme pe unit impedance is the same, egadless f t which side f the tansfme it is efeed. Again, the cnvesin cmplicatins ae absbed int the elatinships. n the tansfme equivalent cicuit the diffeent vltage levels disappea and the tansfme equivalent cicuit is educed t a single impedance: p.u. Figue 5 Equivalent Cicuit f a Tansfme in Pe Unit Analysis Thee Phase ystem and Pe Unit Calculatins n thee phase systems the ttal appaent pwe is given by 3φ 3 LL * L whee LL is the line-t-line vltage and L is the line cuent.

8 8 imilaly, whee 3φ 3 L L L 3φ is the chsen thee phase pwe, usually the ated pwe f the equipment, the chsen pwe used f all system calculatin by the pwe utility LL is the chsen line-t-line vltage, usually the ated vltage f the equipment the nminal vltage level f a pat f the pwe system L is the line cuent, usually calculated fm the pwe and the Nte that vltage. and 3φ 3 1φ ; thus the p.u. epesentatin emains the same! 3φ 31φ and Y Cnnectin in Pe Unit Calculatins L Y Figue Y Cnvesin The values ae elated thugh the same elatinships as the actual quantities: 3 LL LN LN 3 3 L L 3 Y Y The pe unit values f the cnnected impedances and the Y cnnected impedances ae

9 9 p.u. Y p.u. Y Y Fm that it is easy t shw that the pe unit value f the cnnectin is the same as the pe unit value f the Y cnnectin. 3 Y Y p.u. 3Y Y Y p.u. Change f Base When pieces f equipment with vaius diffeent atings ae cnnected t a system, it is necessay t cnvet thei impedances t a pe unit value expessed n the same. The that we ae cnveting fm will be dented by subscipt M, the we ae cnveting t will be dented by subscipt N. The impedance f the s M and N ae, espectively, M M M N N N The pe unit impedances n the s M and N ae, espectively M p.u. M N p.u. N whee is the actual hmic value f the impedance f the equipment. t fllws that M p.u. M N p.u. N ubstituting f the impedances we get

10 10 M p.u. M M N p.u. N N N p.u. M p.u. N M M N Using the MA and k ntatin, N p.u. M p.u. MA MA N M (k (k M N ) ) Accdingly, we can cnclude that the pe unit system has the fllwing advantages: Nmalized values against a have me unifm values in the same system. Thus, it is easie t spt es. F example, 1000% p.u. will tigge a waning and thus is wth lking int f ptential es. Pwe is the same thugh the system; vltage s ae changing accding t the tansfme tun ati. Accdingly, ideal tansfmes tun atis disappea fm the pe unit system, the cnvesin cmplicatins ae absbed by the elatinships. and Y quantities have the same p.u. values in these tw diffeent s; again the cnvesin cmplicatins ae absbed by the elatinships. Thee-phase and single-phase quantities have the same p.u. values in these tw diffeent s; again the cnvesin cmplicatins ae absbed by the elatinships. Abnmal peating cnditins can be easily sptted fm the p.u. values.

11 11 The s f diffeent sectins in the system can be calculated, caefully veified and sted nce f all. When geneatin/lad changes, we nly need t change its pe unit values and calculated the pe unit values accdingly. The cnvesin will becme me eliable. The advantages ae me pnunced f lage system applicatins and systems that have many lad/geneatin changes!

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