電力系統(tǒng)分析_孫淑琴中文電子課件第3-4次課transer

1、Power Transformer Contents1. The ideal transformer2. Equivalent circuits for practical transformers3. Per unit system4. Three-phase transformer connections and phase shift5. Per-unit equivalent circuits of balanced Three-phase two-winding transformers6. Three-winding transformers7. Autotransformer8.

2、 Transformers with off-nominal turns ratios1.The ideal transformerFor an ideal transformer, the following are assumed:1）the windings have zero resistance r=0, therefore, the losses in the windings are zero. 2）there is no leakage flux, that is , the entire flux , is confined to the core and links bot

3、h windings. 1=2=3）the core permeability is infinite, which corresponds to zero core reluctance. 4） there are no core losses.tanenclosedHdlI The ideal transformerFrom Amperes and Faradays laws :Hc magnetic field intensity phasor1 12 2c cH lN IN IccccccBHB A 1 12 21 12 2/0ccccccccclN IN Il BRAN IN I T

4、he ideal transformer22222211222122/tttEZIa EENZa ZZIIaN*211 12222()()ttISE Ia EE ISa1122UNUN1221ININ1 12 20U IU I2.Equivalent circuits for practical transformersFor practical transformers:The windings have resistance.The core permeability is finite.The magnetic flux is not entirely confined to the c

5、ore.There are real and reactive power losses in the core.Equivalent circuits for practical transformers2112121111121211cccccmcmRRRNEIIjENNNj NNNIIIIGjBEN Ic: the core loss current,鐵芯損耗電流Im: the magnetizing current,磁化電流Ie=Ic+Im:the exciting current,勵磁電流， is less than 5% of rated currentTransformer pa

6、rametersFrom the short-circuit test, determine the equivalent series impedance Z=R+jX refered to winding 1. Neglect the shunt admittance.From the open-circuit test, determine the shunt admittance Y=G-jB refered to winding 1. Neglect the series impedance.Transformer parameters1. Two-winding transform

7、er:（1） resistance（電阻（電阻 R）（2） reactance(電抗電抗 X)（3）susceptance (電納電納 B)（4）conductance （電導(dǎo)（電導(dǎo) G） 2. Three-winding transformer3. AutotransformerEquivalent circuits for practical transformersSingle-phase two-winding transformer:為了簡化電力網(wǎng)的計算，雙繞組變壓器一般采用為了簡化電力網(wǎng)的計算，雙繞組變壓器一般采用型型等效電路。等效電路。RT jXTGT-jBTP0+jQ0RT j

8、XT變壓器空載損耗及勵磁功率變壓器空載損耗及勵磁功率對于地方電網(wǎng)及發(fā)展規(guī)劃中的電力系統(tǒng)通常不計對于地方電網(wǎng)及發(fā)展規(guī)劃中的電力系統(tǒng)通常不計變壓器的等值導(dǎo)納。變壓器的等值導(dǎo)納。Transformer parameters There are several transformer parameters ： 短路損耗短路損耗(short-circuit losses) ) Pk; ; 短 路 電 壓 百 分 數(shù)短 路 電 壓 百 分 數(shù) ( ( l o a d v o l t a g e percentage);); 空載損耗空載損耗( (open-circuit losses)P0 ; ; 空載電

9、流百分?jǐn)?shù)空載電流百分?jǐn)?shù)( (open-circuit current percentage) I0%。 %kUTransformer parameters 短路損耗短路損耗Pk：變壓器做短路試驗時，一次側(cè)繞組短接，變壓器做短路試驗時，一次側(cè)繞組短接，另一次側(cè)繞組利用調(diào)壓變壓器加電壓，電流達(dá)到該側(cè)另一次側(cè)繞組利用調(diào)壓變壓器加電壓，電流達(dá)到該側(cè)繞組的額定值后，所測得的有功功率即為短路損耗。繞組的額定值后，所測得的有功功率即為短路損耗。 空載損耗空載損耗P0 ：是變壓器一次側(cè)開路，在另一次側(cè)加：是變壓器一次側(cè)開路，在另一次側(cè)加額定電壓時測得的變壓器的有功損耗。額定電壓時測得的變壓器的有功損耗。 短路

10、電壓百分?jǐn)?shù)：短路電壓百分?jǐn)?shù)：短路電壓等于變壓器在額定電流下短路電壓等于變壓器在額定電流下產(chǎn)生的壓降，即產(chǎn)生的壓降，即 空載電流百分?jǐn)?shù)空載電流百分?jǐn)?shù)I0%：空載電流與額定電流之比?？蛰d電流與額定電流之比。33%*100NTNTkNNI ZI XUUU00%*100NIIITwo-winding transformerq 電阻（電阻（resistance）:在變壓器短路試驗中所測得的短路損耗在變壓器短路試驗中所測得的短路損耗PK近似等于近似等于額定電流流過變壓器時繞組中的總銅耗額定電流流過變壓器時繞組中的總銅耗，即，即cukPP 由銅耗與電阻之間的關(guān)系由銅耗與電阻之間的關(guān)系 得：得：222233(

11、 )3NNcuNTTTNNSSPI RRRwUU2222NNTcukNNUURPPSSTwo-winding transformer PK用用kW，SN用用MVA，UN用用kV，則上式可寫成：則上式可寫成： 221000kNTNPURSTwo-winding transformer電抗電抗(reactance) ： 在短路試驗中，短路電壓等于變壓器阻抗在短路試驗中，短路電壓等于變壓器阻抗（impedance）在額定電流下產(chǎn)生的壓降在額定電流下產(chǎn)生的壓降，即，即33%100NTNTkNNI ZI XUUU 2%1001003kNNKTNNUUUUXSI對于大中型變壓器，對于大中型變壓器，RT可可

12、忽略。忽略。SN用用MVA，UN用用kV。Two-winding transformer電導(dǎo)（電導(dǎo)（conductance）：變壓器空載試驗所得變壓器空載試驗所得變壓器空載損耗變壓器空載損耗 P0近似近似等于鐵耗等于鐵耗，因此，電導(dǎo)可由空載損耗求得：，因此，電導(dǎo)可由空載損耗求得：302210feTNNPPGUUPK用用kW，UN用用kV。Two-winding transformer電納（電納（ susceptance ）：）：變壓器的電納代表變壓器的勵磁功率。由于變壓器的變壓器的電納代表變壓器的勵磁功率。由于變壓器的空載電流包含有功分量和無功分量，與空載電流包含有功分量和無功分量，與勵磁功率

13、對應(yīng)勵磁功率對應(yīng)的是無功分量的是無功分量。由于有功分量很小，由于有功分量很小，無功分量無功分量（流經(jīng)（流經(jīng)導(dǎo)納支路）和導(dǎo)納支路）和空載電流空載電流在數(shù)值上幾乎相等在數(shù)值上幾乎相等。IBI0，則：，則：00003%*100*100*1003NbNNNNIU IQIIISU I所以：所以：0022%( )100NTNNQISBsUUSN用用MVA，UN用用kV。Two-winding transformer【例例】有一臺有一臺121/10.5kV、容量為、容量為31500kVA的三相雙的三相雙繞組變壓器，其短路損耗為繞組變壓器，其短路損耗為200kW，空載損耗為，空載損耗為47W，短路電壓百分?jǐn)?shù)為

14、短路電壓百分?jǐn)?shù)為10.5，空載電流百分?jǐn)?shù)為，空載電流百分?jǐn)?shù)為2.7，試計，試計算變壓器等值阻抗算變壓器等值阻抗(impedance)與導(dǎo)納與導(dǎo)納(shunt admittance)。解：計算變壓器阻抗解：計算變壓器阻抗, ,1 1）串聯(lián)電阻（歸算到）串聯(lián)電阻（歸算到121kV121kV高電壓側(cè)）高電壓側(cè)）22221212002.951000100031.5kNTNPURS222102222. 25 .315 .101000200低TR歸算到歸算到低壓側(cè)：低壓側(cè)：Two-winding transformer2 2）串聯(lián)電抗（）串聯(lián)電抗（ series reactance）2222121%0.1

15、0548.510031.510.5%0.1050.367510031.5kNTNkNTNUUXSUUXS高低Two-winding transformer3 3）勵磁回路（并聯(lián)）導(dǎo)納）勵磁回路（并聯(lián)）導(dǎo)納（shunt admittance ）電導(dǎo)（電導(dǎo)（conductance）：）：350223502247*100.32 1012147*100.426 1010.5TNTNPGSUPGSU高低Two-winding transformer 4022%31.50.0270.58 10100121NTNISBSU高高壓側(cè)電納高壓側(cè)電納( (susceptance) ： 低壓側(cè)的低壓側(cè)的電納電納30

16、2%.0.771 10100NTNISBSU低Two-winding transformer(2.9548.5)TTTZRjXj參數(shù)等效在高壓側(cè)參數(shù)等效在低壓側(cè)54(0.32 100.58 10 )TTTYGjBjS2(2.2222 100.3675)TTTZRjXj53(0.426 100.771 10 )TTTYGjBjSExample 3.23.Per unit systemPower-system quantities such as voltage, current, power, and impedance are often expressed in per-unit or pe

17、rcent of specified base values.For example, if a base voltage of 20kV is specified, then the voltage 18kV is (18/20)=0.9 per-unit or 90%.Per unit system One advantages of the per-unit system is that by properly specifying base quantities, the transformer equivalent circuit can be simplified. From on

18、e side of a transformer to the other, the per-unit quantities are the same. The per-unit system allows us to avoid the serious calculation errors.Per unit systemAnother advantage of the per-unit system is that the per-unit impedances of electrical equipment of similar type usually lie within a narro

19、w numerical range when the equipment ratings are used as base values. Because of this per-unit impedance data can be checked rapidly for gross errors by someone familiar with per-unit quantities.Per unit systemactual quantity（有名值、實際值有名值、實際值）：）：is the value of the quantity in the actual units.base va

20、lue of quantity ( (基準(zhǔn)值基準(zhǔn)值) )：has the same units as the actual quantity,the base value is always a real number.quantity in per unit(per-unit quantity) ( (標(biāo)么值標(biāo)么值) ) is calculated as follows: (dimensionless)基準(zhǔn)值實際值標(biāo)么值quantity in per unit = per-unit quantityactual quantitybase value of quantityPer unit s

21、ystemTwo independent base values can be arbitrarily selected as one point in a power system. Usually the base voltage UBLN and base complex power SB11 are selected for a single-phase or one phase of a three-phase circuit.The following relations must be used for other base values:IB= SB1 / UBLN, PB1

22、=Q BLN = SB1ZB = RB = XB = UBLN / IB = UBLN2 /S B1YB = GB = BB = 1 / ZB Per unit systemBy convention, we adopt the following two rules for base quantities:1.The values of SB1 is the same for the entire power system of concern. 2.The ratio of the voltage bases on either side of a transformer is selec

23、ted to be the same as the ratio of the transformer voltage ratings.Per unit system*UZ ISU IThe per-unit quantity have the same equations:In the single-phase circuit:1LNLNUZISUIBase values:1BLNB BBBLNBUZ ISUI*BIIIBZZZ*BUUU*BSSSper-unit quantity:S1 (per-phase) and ULN (line-to-neutral)Per unit system*

24、3*1 *L LL NL LL NUZ IUSUIUIS3133BBL LBL NBSSUU；Note：在標(biāo)幺制中三相功率與單相功率標(biāo)幺值相同，：在標(biāo)幺制中三相功率與單相功率標(biāo)幺值相同，線電壓與相電壓標(biāo)幺值相同。線電壓與相電壓標(biāo)幺值相同。3133L LL NSSUU；In the balanced three-phase circuit, usually S3(three-phase) and UL-L (line-to-line)are selected，there are relationship between S1(per-phase ) ,S3 ,UL-L ,UL-N (line-t

25、o-neutral ):Usually the base voltage UB and base complex power SB are selected for a circuit.Per unit systemIn the balanced three-phase circuit, the base current and impedance is:33BBL LBSIU23L LBBBUZSSB :(three-phase)UB :(line-to-line)23BBBBBBBBSIUUZRXS21BBBBBBSYGBZUPer unit system2BBUSZZ*3BBBU III

26、ISPer-unit quantity is calculated as follows:Per unit systemBase quantities for other parameters50BfBBf2BBBZLBBBZC1時間和衰減因數(shù)的基準(zhǔn)值時間和衰減因數(shù)的基準(zhǔn)值 3141211BBBftBBBtm1磁鏈的基準(zhǔn)值磁鏈的基準(zhǔn)值 BBBBBBBL IEEtPer unit system0*1000TNPGS0*%1 0 0TIB*%100kTUX*1000kTNPRS21121000kNTNPURS211%100kNTNUUXS02TNPGU02%*100NTNISBU2NTBNURS2

27、NTBNUXS2NTBNSGU2NTBNSBUper-unit quantitybase value of quantityactual quantityTransformerTransformerTransformerExample 3.3例題例題2-7：一個單相：一個單相雙繞組變壓器的額雙繞組變壓器的額定值為定值為20kVA，480/120V，50Hz，變壓器的等值阻抗變壓器的等值阻抗歸算到歸算到120繞組繞組為為 。采用變壓器額定值采用變壓器額定值為基準(zhǔn)值，確定歸為基準(zhǔn)值，確定歸算到二次繞組和一算到二次繞組和一次繞組的標(biāo)幺阻抗。次繞組的標(biāo)幺阻抗。 020.05 78.13eqZ（中文）P

28、er unit systemWhen only one component, such as a transformer, is considered, the nameplate ratings of that component are usually selected as base values. When several components are involved, however, the system base values may be different from the nameplate ratings of any particular device. It is

29、then necessary to convert the per-unit impedance of a device from its nameplate ratings to the system base values. To convert a per-unit impedance from “old” to “new” base values, use:()()*()()oldB oldactualnewB newB newZZZZZZPer unit system2BNBNBNUZS()()*()()2()()()()()oldB oldactualnewB newB newB

30、oldB newoldB newB oldZZZZZZUSZUS 2*NNNUSXX 22*2*BNNBNBBBUUSSXUSXXPer-unit reactance from “old”Per-unit reactance from “new”Per unit systemThe nameplate ratings of reactor(電抗器電抗器)are current and voltage: BNNBNBUUIIXX*NNNIXXUreactor Per-unit quantity (old)：reactor Per-unit quantity (new)：Example 3.4例2

31、-8，中文例例2-8：單相電路如圖所單相電路如圖所示，線路及變示，線路及變壓器參數(shù)圖中壓器參數(shù)圖中標(biāo)出，以容量標(biāo)出，以容量S=30kVA，第，第一段基準(zhǔn)值電一段基準(zhǔn)值電壓為壓為U1=240V。1）計算各元件標(biāo)）計算各元件標(biāo)幺值，并畫出幺值，并畫出標(biāo)幺值等值電標(biāo)幺值等值電路。路。2）計算網(wǎng)絡(luò)電流）計算網(wǎng)絡(luò)電流（標(biāo)么值和有（標(biāo)么值和有名值）。名值）。 例2-8：單相電路如圖所單相電路如圖所示，線路及變示，線路及變壓器參數(shù)圖中壓器參數(shù)圖中標(biāo)出，以容量標(biāo)出，以容量S=30kVA，第，第一段基準(zhǔn)值電一段基準(zhǔn)值電壓壓U1=240V。1）計算各元件標(biāo)）計算各元件標(biāo)幺值，并畫出幺值，并畫出標(biāo)幺值等值電標(biāo)幺值等

32、值電路。路。2）計算網(wǎng)絡(luò)電流）計算網(wǎng)絡(luò)電流（標(biāo)么值和有（標(biāo)么值和有名值）。名值）。 多電壓等級電力網(wǎng)絡(luò)的標(biāo)么值等值電路多電壓等級電力網(wǎng)絡(luò)的標(biāo)么值等值電路4.Three-phase transformer connections and phase shiftThree identical single-phase two-winding transformers may be connected to form a three-phase bank. Four ways to connect the windings are Y-Y , Y-, - Y, and -.The phases of

33、 a Y-Y or a - transformer can be labeled so there is no phase shift between correponding quantities on the low-and high-voltage winding. 4.Three-phase transformer connections and phase shiftHowever, for Y- and - Y transformers, there is always a phase shift between correponding quantities on the low

34、-and high-voltage winding. In either a Y- or - Y transformer, positive-sequence quantities on high-voltage side shall lead their corresponding quantities on the low-voltage side by 300.Setp1:draw the positive-sequence phasor diagram for these voltages.2.Move phasor A-N next to terminals A-N, similar

35、ly B-N and C-N.3. Draw a lne next to each low-voltage winding parallel to the corresponding line next to the high-voltage winding.4. Label the ends of the lines drawn in step 3 by inspecting the polarity marks.5.Bring the three lines labeled in step 4 together to complete the phasor diagram for the

36、low-voltage winding.5. Per-unit equivalent circuits of balanced three-phase two-winding transformersA common SB is selected for both the H and X terminals.The ratio of the voltage bases UbaseH/UbaseX is selected to be equal to the ratio of the rated line-to-line voltages UratedH/UratedX .Example 3.7

37、Example 3.83.6 Three-winding transformerIn actual units, for a basic single-phase three-winding transformer:Example 3.91UTGTjBRjX22RjX33RjX2U3U13:NN23:NNSingle-phase Three-winding transformer :三繞組變壓器短路試驗是在兩個繞組間進(jìn)行，三繞組變壓器短路試驗是在兩個繞組間進(jìn)行，另一繞組為開路另一繞組為開路，按雙繞組變壓器來做。另，按雙繞組變壓器來做。另外，額定容量不同，將受到較小容量繞組的外，額定容量不同，將

38、受到較小容量繞組的額定電流的限制。額定電流的限制。Three-winding transformerThree-winding transformer三繞組變壓器三個繞組容量比按國家標(biāo)準(zhǔn)有三三繞組變壓器三個繞組容量比按國家標(biāo)準(zhǔn)有三種類型：即種類型：即第第類類：100/100/100100/100/100，三繞組容量都等于變，三繞組容量都等于變壓器的額定容量；壓器的額定容量；第第II II 類類：100/100/50100/100/50，第三繞組容量僅為變，第三繞組容量僅為變壓器額定容量的壓器額定容量的50%50%；第第IIIIII類類：100/50/100100/50/100，第二繞組容量為變

39、壓，第二繞組容量為變壓器額定容量的器額定容量的50%50%。Three-winding transformer第第類類：各繞組額定容量相同時：各繞組額定容量相同時： 若假設(shè)三繞組間的短路損耗分別為：若假設(shè)三繞組間的短路損耗分別為：Pk1-2，Pk1-3，Pk2-3131332322121kkkkkkkkkPPPPPPPPP213231331322123213211212121kkkkkkkkkkkkPPPPPPPPPPPP則各繞組的短路損耗分別為：則各繞組的短路損耗分別為：Pk1，Pk2，Pk3。Three-winding transformer211222222332100010001000

40、kNTNkNTNkNTNPURSPURSPURS電阻（電阻（resistance）:Three-winding transformer第第II 類類：100/100/50100/100/50第第III類類100/50/100100/50/100 ： 各繞組額定容量不同時，短路實驗時，只能各繞組額定容量不同時，短路實驗時，只能是容量小的繞組達(dá)到它的額定電流值，所測是容量小的繞組達(dá)到它的額定電流值，所測得的短路損耗與按額定容量計算的額定電流得的短路損耗與按額定容量計算的額定電流流過時的短路損耗有差別因此需要進(jìn)行折算。流過時的短路損耗有差別因此需要進(jìn)行折算。Three-winding transfo

41、rmerkNNkkkNNkkPIIPPPIIPP42/42/2)32(322)21(21例如對于一臺例如對于一臺100/50/100100/50/100類型變壓器類型變壓器 進(jìn)行折算。短進(jìn)行折算。短路損耗路損耗 ， 都是在第二繞組中流過變壓器額定都是在第二繞組中流過變壓器額定電流的電流的1/21/2時測得的，而時測得的，而P Pk(1-3)k(1-3)仍是在流過變壓器額定仍是在流過變壓器額定電流時測得的。有：電流時測得的。有：為銘為銘牌數(shù)據(jù)。牌數(shù)據(jù)。(1 2)kP(2 3)kP(2 3)kP(1 2)kPThree-winding transformer則三繞組間的短路損耗分別為：則三繞組間的

42、短路損耗分別為：Pk（1-2），Pk（2-3） ， Pk (1-3) ，計算各繞組的短路損耗分別為：計算各繞組的短路損耗分別為：Pk1，Pk2，Pk3。131332322121kkkkkkkkkPPPPPPPPP213231331322123213211212121kkkkkkkkkkkkPPPPPPPPPPPPThree-winding transformer對于短路電壓百分?jǐn)?shù)，按國標(biāo)規(guī)定制造廠提對于短路電壓百分?jǐn)?shù)，按國標(biāo)規(guī)定制造廠提供的短路電壓是已經(jīng)歸算到變壓器額定電流供的短路電壓是已經(jīng)歸算到變壓器額定電流時的數(shù)值。因此，時的數(shù)值。因此，第第II、III類變壓器對于短類變壓器對于短路電壓不

43、需要再進(jìn)行歸算了。路電壓不需要再進(jìn)行歸算了。Three-winding transformer%313132322121kkkkkkkkkUUUUUUUUU%21%21%21%213231331322123231211kkkkkkkkkkkkUUUUUUUUUUUUThree-winding transformer211222233%100%100%100kNTNkNTNkNTNUUXSUUXSUUXS電抗（電抗（reactance）：）：3.7 AutotransformerFor the usual transformer the two windings are coupled magn

44、etically via the mutual core flux. For the autotransformer the windings are both electrically and magneticlly coupled. The autotransformer has smaller per-unit leakage impedances than the usual transformer, this results in both smaller series-voltage drops(an advantage) and higher short-circuit curr

45、ents (a disadvantage).3.7 Autotransformer自耦變壓器的等值電路及參數(shù)求取與普通自耦變壓器的等值電路及參數(shù)求取與普通變壓器相同。變壓器相同。計算等值電阻時要對短路試驗的數(shù)據(jù)進(jìn)行計算等值電阻時要對短路試驗的數(shù)據(jù)進(jìn)行折算。折算。1 31 332 32 33%NkkNNkkNSUUSSUUS短路損耗折算方法如：短路損耗折算方法如：短路電壓百分?jǐn)?shù)應(yīng)按下式短路電壓百分?jǐn)?shù)應(yīng)按下式折算：折算：kNNkkkNNkkPIIPPPIIPP42/42/2)32(322)21(21 Equivalent circuits of transformers前面計算中，將變壓器的阻抗折

46、算到高壓側(cè)，因此通前面計算中，將變壓器的阻抗折算到高壓側(cè)，因此通過計算得到的低壓側(cè)電壓、電流均為折算到高壓側(cè)的過計算得到的低壓側(cè)電壓、電流均為折算到高壓側(cè)的數(shù)值。若在變壓器電路中增加只反映變比的理想變壓數(shù)值。若在變壓器電路中增加只反映變比的理想變壓器，則可直接求出低壓側(cè)的實際電壓、電流數(shù)值。器，則可直接求出低壓側(cè)的實際電壓、電流數(shù)值。以雙繞組變壓器為例：阻抗歸算到一次側(cè)。以雙繞組變壓器為例：阻抗歸算到一次側(cè)。GT Equivalent circuits of transformers1I1122122/TUI ZUkUIIIk2I2I2UZT/kZT/(1-k)ZT/k(k-1)1U1I2U2I Equivalent circuits of transformers1U2U1I121112212211221()(1)()TTTTTTUUkkIUUUZZZkUk Ukk kIkIUUUZZZ2IZT/kZT/(1-k)ZT/k(k-1) Equivalent circuits of transformers11122122(1)()()(1)TTTTIk Y UkY UUIkY UUk kY UU1U2I1I2kYT(1-k) Y