Magnetically Coupled Coil

Transcription

1 Mgnetilly Coupled Ciruits Overview Mutul Indutne Energy in Coupled Coils Liner Trnsformers Idel Trnsformers Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 Mgnetilly Coupled Coil i v L φ Frdy s Lw: where v = N dφ φ = NPi v = N d(npi) = N 2 P di = L di v = voltge in volts (V) N = number of turns φ = mgneti flux in webers (Wb) t = time in seonds (s) P = permene of the flux spe i = urrent in mperes (A) L = indutne in henrys (H) Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 3 Introdution to Mgnetilly Coupled Ciruits φ2 φ2 v φ φ22 Mgnetilly oupled oils re oneptully similr to two indutors tht hve shred (oupled) mgneti field Not ll of the mgneti field is shred Mgneti oupling is widely used in power systems Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 2 Mgnetilly Coupled Coil i v L φ v = N dφ =(N 2 P) di = L di The flux (& urrent) hve to hnge to indue voltge The reltionship between the flux nd the urrent is onstnt Consistent with wht we lredy know bout indutors L is proportionl to N 2 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 4

2 Mutul Indutne φ2 φ φ = φ φ2 We n deompose the mgneti flux indued in one oil into two omponents φ is the totl flux produed in oil φ is the portion of this flux tht links only oil φ2 links both oil nd oil 2 The oils re not onneted eletrilly Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 5 Mutul Indutne Continued 2 φ2 φ v =L v2 = ± M2 M2 =N2NP2 M2: the mutul indutne of oil 2 with respet to oil Note tht v2 is the open iruit voltge Wht if urrent ws pplied to oil 2 s well? Superposition pplies Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 7 Mutul Indutne Continued φ2 φ dφ v =N dφ2 v2 = ± N2 = ± M2 d(npi) =N d(np2i) = ± N2 =L = ± (N2NP2) d Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 6 Mutul Indutne: Two Soures M v v = L ± M di2 2 v2 = ±M2 We will ssume M2 = M2 = M If ssumption holds, the oils re lled liner trnsformer M is lled the mutul indutne di2 Like indutors, is mesured in units of henrys (H) Polrity of oupling term depends on how the oils re wound Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 8

3 Liner Trnsformer: The Dot Convention φ2 φ2 v φ φ22 The dot onvention determines the polrity of the oupling Dot Convention: If urrent enters dotted terminl, it indues positive voltge t the dotted terminl of the seond oil If urrent leves dotted terminl, it indues negtive voltge t the dotted terminl of the seond oil Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 9 Exmple 2: The Dot Convention M M M M Write the defining equtions for eh of the iruits shown bove. Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 Exmple : The Dot Convention M M M M Write the defining equtions for eh of the iruits shown bove. Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 0 Mutul Indutne & Self Indutne L = N 2 P P = P P2 L2 = N 2 2 P2 P2 = P22 P2 LL2 = N 2 N 2 2 PP2 LL2 = N 2 N 2 2 (P P2)(P22 P2) Sine M2 = M2 forlinersystem,p2 = P2 nd ( LL2 = N 2 N 2 2 P 2 2 P )( P ) 22 P2 P2 = M 2 k 2 M = k LL2 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 2

4 Coeffiient of Coupling (k) k 2 = M = k LL2 ( P )( P ) 22 P2 P2 k is lled the oeffiient of oupling Sine k, k 2 k is nonnegtive sine P > 0 If two oils hve no ommon flux, k =0 If both oils shre ll flux, k = It is physilly impossible for k =, but some mgneti ores hve k very lose to Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 3 Liner Trnsformers: Energy Continued The energy stored in the oils during the seond period is given by w2 = = t2 t t2 t = ±MI p2 dτ ( ±M di ) 2 dτ I di2 L2i2 dτ dτ I2 0 di2 L2 = ±MII2 2 I 2 2 I2 0 i2 di2 Then the totl energy stored in mgnetilly oupled oils fter the urrents hve been pplied is given by w = w w2 = 2 I 2 2 I 2 2 ± MII2 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 5 Liner Trnsformers: Energy Suppose no energy is stored in the oils t t =0nd over some period of time t the urrent in oil inreses from 0 to I while the urrent in oil 2 is zero, i2 =0. The energy stored in the oils over this period is given by w = t 0 vi dτ = t 0 ( L di ) I i dτ = L i = dτ 2 I 2 0 Now suppose the urrent in oil is held onstnt, i = I, while the urrent in oil 2 inreses from 0 to I2. v = L ± M d v2 = ±M d L di2 2 = ±M d di2 = L2 p2 = vi v2i2 = ±M d I L2 di2 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 4 Liner Trnsformers: Energy Comments M v w = ± Mii2 The polrity of the shred term depends on how the oils re wound Cn the energy stored ever be negtive? Rell tht M = k LL2 This limits the expression bove to nonnegtive vlues only Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 6

5 TimeDomin Anlysis M Time Domin v = L ± M d v2 = ±M d L di2 2 Wht is v if i = A os(ωt) nd i2 =0? Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 7 Sinusoidl StedyStte Anlysis Continued I jωm I 2 V jωl jωl2 V 2 Frequeny Domin (Phsors) V = jωli jωmi2 V2 = jωmi jωl2i2 The dot onvention still pplies (not shown) Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 9 Sinusoidl StedyStte Anlysis Wht is v if i = A os(ωt) nd i2 =0? v =L ± M d v =L v2 = ± M d L di2 2 v2 = ± M d v =ωla ( sin(ωt)) v2 = ± ωma ( sin(ωt)) v =ωla os(ωt 90 ) v2 = ± ωma os(ωt 90 ) Wht is the reltionship in the phsor domin? V =jωli V2 = ± jωmi Superposition pplies so if i2 = A2 os(ωt), Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 8 Exmple 3: Liner Trnsformers & Phsor Anlysis 0 Ω 4 mh i s i L v s 2 mh 8 mh 30 Ω Find the stedystte expressions for the urrents is nd il when vs = 70 os(5000t) V. Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 20

6 Exmple 3: Workspe Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 2 Liner Trnsformer Anlysis: Typil Configurtion Z s R R 2 jωm I I 2 V s jωl jωl2 b Liner Trnsformer d Trnsformers re typilly used in only few types of iruits The most ommon onfigurtion onnets soure to lod Useful to derese (or inrese) the voltge ross the lod Why not just use voltge divider? Should know how to nlyze this type of iruit thoroughly Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Phsor Anlysis: TEquivlent I jωm I 2 I I 2 jω(lm) jω(l2m) V V 2 jωl jωl2 jωm V V 2 Frequeny Domin (Phsors) V = jωli jωmi2 V2 = jωmi2 jωl2i2 The Tequivlent is only vlid if bottom terminls re onneted There is lso equivlent (see text) Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Liner Trnsformer: Soure Lod Anlysis Z s R R 2 jωm I I 2 V s jωl jωl2 b Liner Trnsformer d R = Resistne of primry winding R2 = Resistne of seondry winding L = Selfindutne of primry L2 = Selfindutne of seondry M = Mutul indutne Zs = Soure impedne ZL = Lod impedne Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 24

7 Liner Trnsformer: Soure Lod Anlysis Continued Z s R R 2 jωm I I 2 V s jωl jωl2 b Liner Trnsformer d Vs = (Zs R jωl)i jωmi2 0 = jωmi (R2 jωl2 ZL)I2 Z Zs R jωl R2 jωl2 ZL Vs = ZI jωmi2 0 = jωmi I2 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Liner Trnsformer: Soure Lod Internl Impedne Z s R R 2 jωm I I 2 V s jωl jωl2 b Liner Trnsformer d Zi V s I = Z ω 2 M 2 = Z ω2 M 2 Zb = Zi Zs = R jωl ω2 M 2 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Liner Trnsformer: Soure Lod Anlysis Continued 2 Vs = ZI jωmi2 0 = jωmi I2 I2 = jωm I (Z ω2 M 2 ) Vs = I = I Z ω 2 M 2 V s I2 = jωm Z ω 2 M V 2 s Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Liner Trnsformer: Soure Lod Refleted Impedne Z s R R 2 jωm I I 2 V s jωl jωl2 b Liner Trnsformer d Zb = R jωl ω2 M 2 ZR = Zb (R jωl) = ω2 M 2 = ω2 M 2 2 Z 22 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 28

8 Exmple 4: Liner Trnsformers 500 Ω 200 Ω 00 Ω j200 Ω 800 Ω I I V j3600 Ω j600 Ω j2500 Ω b d Find the following:. Selfimpedne of primry & seondry iruits 2. Impedne refleted into the primry winding 3. Impedne seen looking into the primry terminls of the trnsformer 4. Thévenin equivlent with respet to the terminls,d Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Exmple 5: Liner Trnsformers 8 Ω j56 Ω 20 Ω j50 Ω 3 Ω I I V j40 Ω j00 Ω (rms) b d Find the following:. Thévenin equivlent with respet to the terminls,d 2. If ZL is set equl to Z eq, whtisi? 3. Wht is I2? Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 3 Exmple 4: Workspe Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Exmple 5: Workspe () Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 32

9 Exmple 5: Workspe (2) Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Idel Trnsformer Anlysis N:N2 dφ v = N v2 = N2 dφ v2 v = N 2 = n N p = p2 vi = v2i2 i i2 = = n v Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Introdution to Idel Trnsformers N:N2 Idel/Liner trnsformers re similr to idel/rel models of opertionl mplifiers Both idel models mke ssumptions tht simplify nlysis Idel pproximtion: ll of the flux links both oils Idel Assumptions Lrge retne: L,L2,M Perfet oupling: k Primry nd seondry re lossless: R = R2 =0 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Idel Trnsformers: Comments N:N2 v2 v = N 2 = n N i2 i = N = N2 n Defining equtions for idel trnsformers do not inlude time The phsor domin equtions re identil to the time domin The idel trnsformer nnot store energy Note diretion of seondry urrent Sometimes only the turns rtio is given: N2/N = n Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 36

10 Idel Trnsformers: The Dot Convention N:N2 The dot onvention determines the polrity of the defining equtions Dot Convention: If v nd v2 re both positive or both negtive t the dotted terminls, use n. Otherwise, use n. If i nd i2 both enter or both leve the dotted terminls, use n. Otherwise, use n. Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Exmple 7: The Dot Convention for Idel Trnsformers :n :n :n :n Write the defining equtions for eh of the iruits shown bove. Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Exmple 6: The Dot Convention for Idel Trnsformers :n :n :n :n Write the defining equtions for eh of the iruits shown bove. Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Idel Trnsformer: Refleted Impedne Z s I I 2 :n V s V V 2 b d ZR = V I = V 2 n I2n = V 2 I2 n 2 = n 2 Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 40

11 Exmple 8: Idel Trnsformers 0.25 Ω 5 mh mω 0: v g 25 µh If vg = 2500 os(400t) V, find i, v, i2, ndv2. Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 4 Exmple 9: Idel Trnsformers 25 Ω 20 Ω 60 Ω : mv j50 Ω C (rms) Find the vlue of C tht mximizes the power bsorbed by the 60 Ω resistor. Wht is the verge power delivered for this vlue of C? Reple the resistor with vrible resistor nd find the vlue tht mximizes the power delivered? Wht is the mximum verge power tht n be delivered? Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Exmple 8: Workspe Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver Exmple 9: Workspe () Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 44

12 Exmple 9: Workspe (2) Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 45