Single-Phase Transformers

Transcription

1 Trnsforers MT 30 Spring 006.F. L-Sdny Single-Phse Trnsforers t should be entioned here tht the eletri power is generted t voltge in the rnge of the -30K. Trnsitting given ount of power ruires fixed produt of voltge nd urrent. Thus the higher the voltge, the lower the urrent n be. Lower line urrents re ssoited with lower losses ( Z). - Trnsitting the power in low voltge rnge for long distne is not fesible sine ll the power will be lost s voltge drop on the trnsission line. An obvious solution to this proble is to rise the voltge level to soe hundred K (rnging fro 0 to 700 K). Here oes the funtion of the trnsforer. This devie is one of the ost iportnt inventions of ll ties. Without doubt, they re the ost iportnt piee of uipent long the power trnsission nd distribution systes. Trnsforers ke it possible to onvert the energy tken fro genertors into usble, trnsittble for. Without the, it would be next to ipossible to use the energy produed by the utility opnies. Trnsforers re used indifferent pplitions, suh s:. Step-up trnsforers for trnsission. Step-down trnsforers for distribution 3. High voltge esureents (potentil trnsforers) 4. High urrent esureent (urrent trnsforers) 5. nsulting one iruit fro the other 6. nsulting DC iruits fro AC iruits. Trnsforer Constrution A single-phse trnsforer onsists bsilly fro two or ore windings oupled by gneti ore s shown in Fig.. When one of the windings (priry) is onneted to n AC soure, tie vrying flux is produed in the ore. This flux is onfined within the gneti ore nd it links the seond

2 Trnsforers MT 30 Spring 006.F. L-Sdny winding (seondry). When ny eletri lod is onneted to the seondry winding, urrent will flow. Fig. Single-phse trnsforer onstrution φ Fig. Single-phse trnsforer iruit

3 Trnsforers MT 30 Spring 006.F. L-Sdny. del Trnsforer For idel trnsforer, the following ssuptions re vlid:. o lekge flux (ll the flux produed by the priry winding links the seondry). o winding resistne ( nd ) 3. The ore relutne is zero ( 0, nd μ ) 4. o ore losses (eddy + hysteresis) Let the utul flux linking both windings φ be sinusoidl, suh tht: φ φ sinω t. Aording to p Frdy's lw, the indued ef n be expressed s: dλ dφ e ωφ p osω t dλ dφ e ωφ p osω t The MS vlues of the indued ef re: ωφ p ωφ p fφ p fφ p The polrities of the indued ef re given by Lenz's lw, tht is, the efs produes urrents tht tend to oppose the flux hnge. The rtion between the priry nd seondry indued ef is:, whih is known s the trnsforer Turns tio. Sine the trnsforer is idel nd there is no losses, then the input power uls the output power,, then: 3

4 Trnsforers MT 30 Spring 006.F. L-Sdny Fro the bove ution, we n show tht: nd Dividing the bove two utions we obtin: where Z Z Z is known s the seondry winding ipedne referred to the priry winding. n ddition, both the urrent nd the voltge in the seondry iruit n be referred to the priry iruit s follows: Z, then, then The uivlent iruit of the trnsforer eferred to the priry is given s shown in Fig. 3. Z z Fig. 3 Trnsforer uivlent iruit referred to the priry side The uivlent iruit of the trnsforer eferred to the seondry is given s shown in Fig. 4. 4

5 Trnsforers MT 30 Spring 006.F. L-Sdny Fig. 4 Trnsforer uivlent iruit referred to the seondry side xple A 40/0 volt, 60Hz, idel trnsforer is rted t 5 KA. ) Clulte the turns rtio b) Clulte the rted priry nd seondry urrents ) Clulte the priry nd seondry urrents when the trnsforer delivers 3.KW t rted seondry voltge nd 0.8 power ftor lgging. Solution: 40 ) b) rted 0. 83A rted 4. 67A 0 or 300 ) For the given lod, A nd 6. 67A 5

6 Trnsforers MT 30 Spring 006.F. L-Sdny 3. on-del (Atul) Trnsforer The ssuptions de in the previous setion for idel trnsforer re no longer pplible when nlyzing the perforne of n tul trnsforer. Chrteristis of non-idel trnsforers. The priry nd seondry windings hve resistnes. ot ll the flux produed by one winding will link the other winding beuse of flux lekge 3. The ore of n tul trnsforer hs finite perebility. 4. There re ore losses (Hysteresis nd ddy urrent) due to the presene of lternting flux in the ore; (iron losses). Consider the tul trnsforer iruit shown in Fig Lod Where: φ Fig. 5 Atul trnsforer iruit The utul flux (linking flux) φ Priry lekge flux φ Seondry lekge flux Priry winding nuber of turns Seondry winding nuber of turns Priry winding resistne Seondry winding resistne 6

7 Trnsforers MT 30 Spring 006.F. L-Sdny For the priry winding, the flux linking the winding is given s: φ + φ φ The voltge ution for the priry loop n be written s: dλ i + i + dφ Thus i + dφ + dφ Sine dφ di φ il, L di dφ di i + L + i + L + e (A) n the seondry iruit, the voltge ution y be written s follows: dλ dφ i + i + Fro the flux diretion, φ φ φ, Thus: i + dφ dφ di dφ di i L + i L + e where e nd e re the indued ef in the priry nd the seondry windings, respetively. t n be shown tht: e e (B) qutions A & B n be written in frueny doin s follows: ω + i + j Li 7

8 Trnsforers MT 30 Spring 006.F. L-Sdny ω + i j Li To odel the ore losses of the trnsforer prllel iruit onsists of n indutor is dded usully to the priry side of the trnsforer uivlent iruit, L nd resistor where: L epresents the ore gnetiztion epresents the ore losses (Hysteresis & ddy urrent losses) The ore relted iruit eleents L & re usully deterined t the rted voltge nd referred to the priry. They re ssued onstnt when the trnsforer is operting t or ner the rted onditions. 4. quivlent Ciruits The uivlent iruit of the trnsforer is shown in Fig. 6. L L L Fig. 6 quivlent iruit The uivlent iruit of the trnsforer in the frueny doin is given s shown in Fig. 7. 8

9 Trnsforers MT 30 Spring 006.F. L-Sdny j j j Where: e Fig. 7 quivlent iruit in frueny doin Priry indued voltge Seondry indued voltge Priry terinl voltge Seondry terinl voltge Priry urrent Seondry urrent xittion urrent Mgnetizing urrent Core loss urrent Mgnetizing retne Priry lekge retne Seondry lekge retne Core loss resistne Priry winding resistne Seondry winding resistne 9

10 Trnsforers MT 30 Spring 006.F. L-Sdny 4. Trnsforer uivlent iruit phsor digr: The lod is, nd θ i + j i + e i // e + + j i + j e θ j Fig. 8 Trnsforer uivlent iruit phsor digr 4. eferred Trnsforer quivlent Ciruit 4.. eferred to priry side 0

11 Trnsforers MT 30 Spring 006.F. L-Sdny j j e j j del Trnsforer Fig. 9 Trnsforer uivlent iruit referred to the priry side j e θ j Fig. 0 Trnsforer uivlent iruit, referred to the priry side, phsor digr 4.. eferred to seondry side

12 Trnsforers MT 30 Spring 006.F. L-Sdny j j e j del Trnsforer Fig. Trnsforer uivlent iruit referred to the priry side xple A 5 KA, 440/0, 60 Hz trnsforer hs the following preters: 0. 6Ω, 0. 04Ω, 70Ω, 0. 3Ω, 0. 08Ω, 00Ω. The trnsforer delivers 0kW t 0.8 power ftor lgging to lod on the low voltge side with 0 ross the lod. Find the priry terinl voltge. Solution Step # Deterine the lod voltge nd urrent: The voltge ross the lod is tken s referene in this se nd is ul to 0 0. For lod of 0 kw t power ftor of 0.8 lg, the lod urrent is ul to: P 0,000 os osθ Step # efer the iruit to the priry side: A

13 Trnsforers MT 30 Spring 006.F. L-Sdny Ω A 0. 3 Ω Step #3 Solve the uivlent iruit j j e j + ( ) + j Fig. quivlent iruit ( j0.3) j The shunt brnh urrent is: j j4.58 j j00 e +.79 j4.55 Thus the priry urrent is given s: e A 3

14 Trnsforers MT 30 Spring 006.F. L-Sdny The priry voltge is: ( + j ) ( j9.07) + ( )( j0.3) Approxite quivlent Ciruits The pproxition is bsed on the ft tht the gnetiztion (no lod) urrent e is sll opred to the full lod priry input urrent. n prtie e 3 5%. Moreover, sine the priry nd the seondry winding resistnes nd lekge retnes re very sll, then the internl voltge drop is ( ) ( ) very sll suh tht Δ + j 3 5. % + + j Sine + e, then: ( ) ( ) e j + j 4.3. First pproxition f e is very sll opred to nd & re lso very sll, then e ( + j) is very sll nd negleting it will hve negligible effet on both nd. The pproxite uivlent iruit n thus be given in the for: j j j Fig. 3 Approxite uivlent iruit 4

15 Trnsforers MT 30 Spring 006.F. L-Sdny j j Fig. 4 Approxite uivlent iruit Where: The phsor digr of the pproxited uivlent iruit is given s: e θ j Fig. 5 Approxite uivlent iruit phsor digr 4.3. Seond Approxition f the urrent e n be negleted, then the iruit is redued to the seond pproxition s shown in the figure below: 5

16 Trnsforers MT 30 Spring 006.F. L-Sdny j Fig. 6 Seond pproxition uivlent iruit nd re the uivlent resistne nd retne referred to the priry side. The phsor digr of this iruit will be s shown below: θ j Fig. 7 Seond pproxition uivlent iruit phsor digr 5. oltge egultion Distribution nd power trnsforers re often used to supply lods tht re designed to operte t essentilly onstnt voltge. The ount of the seondry urrent drwn by the lod depends on the lod gnitude. As this urrent hnge, the lod voltge will hnge onsuently. This hnge is due to the voltge drop on the trnsforer internl ipedne. A esure of how uh the voltge will hnge s the lod is vried is lled voltge regultion Definition The voltge regultion is defined s the hnge in the gnitude of the seondry voltge s the urrent hnges fro full lod to no lod with the priry voltge held onstnt. 6

17 Trnsforers MT 30 Spring 006.F. L-Sdny oltge regultion ε nd no lod nd full lod nd full lod ε 6. ffiieny The perentge effiieny of the trnsforer is defined s the rtio of the power output to the power input. P η P output input 00 η P output Poutput 00 + losses where the losses re the ore nd opper losses. losses P losses P ore ore + P + losses P L + P s opper fl 6. ffiieny t ny lod x f the trnsforer is loded with x% of its full lod, then the opper losses t this loding level will be x P s. For ny loding perentge x, x x fl. The output power t ny perentge x of the full lod is: P output xp x output fl x fl osθ The effiieny for ny loding ondition is: η Poutput Pinput x Poutput + losses Pinput losses xp PL + x Ps + x P + P output s L 7

18 Trnsforers MT 30 Spring 006.F. L-Sdny 6. Mxiu ffiieny η For xiu effiieny, 0. This will led to PL x P s. n other words, for the xiu dx effiieny will our t the loding level where the no-lod losses is ul to the opper losses. Consuently, the loding level t whih the xiu effiieny ours is given by: x P P L s 7. Trnsforer Preters Deterintion The trnsforer preters re,,,, nd,. These preters n be deterined experientlly by two tests; the open iruit nd the short iruit tests. 7. Open iruit test This test gives infortion regrding the losses in the trnsforer ore. t n be used to deterine nd. f we onnet the iruit s shown in Fig. 8. φ Fig. 8 Open iruit test onnetion The uivlent iruit of the trnsforer will be s shown in Figs. 9 nd 0. Mesuring the voltge, urrent nd power, then: 8

19 Trnsforers MT 30 Spring 006.F. L-Sdny 0 j j j Fig. 9 Open iruit test onnetion uivlent iruit 0 j Fig. 0 Open iruit test onnetion uivlent iruit The esured vlues re: P L, L nd. Fro the bove iruit: C nd P L Y o L Y o j Y o 7. Short iruit test This test provides the vlues of the totl lekge ipednes nd the vlue of the losses in the winding t full lod. 9

20 Trnsforers MT 30 Spring 006.F. L-Sdny n short iruit test, we short-iruit the low voltge winding nd we put the input voltge on the high voltge winding. nrese the input voltge in steps until we reh the full lod urrent (bout 0-30% of input voltge) nd esure the voltge, urrent nd power. The iruit onnetion nd the uivlent iruits will be s shown in Figs. nd, respetively. H L φ A W Fig. Short iruit test onnetion j j Fig. Short iruit test onnetion uivlent iruit P s Z Z f the trnsforer is designed to hve ul losses on the priry nd seondry iruits, then: 0

21 Trnsforers MT 30 Spring 006.F. L-Sdny xple A 50 KA, 400/40, 60 Hz, single-phse trnsforer hs short iruit nd open iruit tests perfored on the high-voltge nd low-voltge sides respetively, nd the following results were obtined: oltge () Current (A) Power (W) Open iruit test Short iruit test ) Deterine the pproxited uivlent iruit referred to the priry side. b) Deterine the voltge regultion nd the effiieny t rted lod, 0.8 power ftor lgging nd rted voltge t the seondry terinls. Solution ) The preters of the trnsforer referred to the priry side re:,,,, nd. Fro the short iruit test Z s 48. 3Ω 0.8 s Ps Ω s ( 0.8) (.3) (.43). Ω Z 8 Fro the open iruit test Sine the open iruit test ws perfored on the low-voltge side, then the deterined preters re going to referred to seondry side. L 5.4 Yo S 40 θ o P L os 8. o L o 8 Y o S Lgging Y o G jb ( 3.3 j.3) 0 S

22 Trnsforers MT 30 Spring 006.F. L-Sdny Ω G 44. 8Ω B Ω G, ( 0) 30. kω ( 0) 4. kω b) At the rted seondry onditions nd 0.8 power ftor lgging, ,000 os A 400 ( + j ) ( )(.43+ j.8) The perentge voltge regultion is: ε % 400 The trnsforer effiieny is: Output η nput Output power nput power Losses η 00 50,000 * ,000 Wtt Output power + Losses Core losses + Copper losses Wtt 40, % 40,