R a. R b. Figure 4B.1

Transcription

1 4B.1 Lecture 4B Manetc Crcuts The anetc crcut. Manetc and eectrc equvaent crcuts. DC exctaton. AC exctaton. Characterstcs. Deternn F ven φ. Deternn φ ven F (oad ne). Equvaent crcut of a peranent anet. The Manetc Crcut N turns csa A φ v suppy R a R R b ar ap Fure 4B.1 Consder a torod wth a core of ferroanetc atera. A sa ap s ade n the core. We know fro prevous anayss and by deonstraton, that the anetc fed nsde a torod s fary unfor. No anetc fed es outsde the torod. For ths case, the fux path s defned aost exacty. The ferroanetc atera can be consdered a ood "conductor" of fux, just ke a eta wre s a ood conductor of chare. The surroundn ar, because of ts ow pereabty, acts ke an nsuator to the fux, just ke ordnary nsuaton around a eta wre. Snce the fux path s we defned, and snce the anetc fed s assued to be unfor, Apère's Law w reduce to a spe suaton. The anetc fed n a thn torod s unfor An anaoy between anetc and eectrc crcuts

2 4B.2 Inore frnn of the fux n sa ar aps If the ar ap s sa, there w not be uch frnn of the anetc fed, and the cross secton of the ar that the fux passes throuh w be approxatey equa to the core cross secton. Let the cross sectona area of the torod's core be A. The ean enth of the core w be defned as the crcuference of a crce wth a radus the averae of the nner and outer rad of the core: R 2 2 a + R R π b 2 π (4B.1) Apère's Law around the anetc crcut ves: Apère s Law for a thn torod H d Hd N H U N (4B.2) The ntera can be spfed to a suaton, snce the fed H s n the sae drecton as the path. Ths s a drect resut of havn a ferroanetc atera to drect the fux throuh a we defned path. In a cases we w et the subscrpt ean ron (or any ferroanetc atera) and subscrpt ean ap (n ar). Apère's Law wrtten expcty then ves: turns nto a spe suaton N H + H B B + µ µ φ + µ A µ A φ (4B.3) that s anaoous to KVL n eectrc crcuts Ths ooks ke the anetc anao of KVL, taken around a crcut consstn of a DC source and two resstors. We w therefore expot ths anaoy and deveop the concept of reuctance and f.

3 4B.3 Defne reuctance as: R (4B.4) µ A Reuctance defned and anetootve force (f) as: F N (4B.5) Manetootve force (f) defned then Apère's Law ves: F ( R + R ) Rφ φ (4B.6) Apère s Law ooks ke a anetc Oh s Law for ths spe case Ths s anaoous to Oh's aw. It shoud be ephassed that ths s ony true where µ s a constant. That s, t ony appes when the atera s near or assued to be near over a partcuar reon. The nductance of the toroda co s ven by the defnton of nductance: L λ Nφ H N 2 N φ Rφ N R 2 (4B.7) The nductance of a torod usn physca characterstcs Eectroechanca devces have one anetc crcut and at east one eectrc crcut. The anetc atera serves as a coupn devce for power. Such devces ncude the transforer, enerator, otor and eter. Because anetc crcuts contann ferroanetc ateras are nonnear, the reatonshp where µ s a constant. F Rφ s not vad, snce ths was derved for the case Apère's Law, on the other hand, s aways vad, and the concept of anetc potenta w be used where µ s nonnear.

4 4B.4 Gauss Law s the anetc anao of KCL The anetc anao to KVL s Apère's Law. What s the anetc anao to KCL? In spe systes where the fux path s known, the fux entern a pont ust aso eave t. The anao to KCL for anetc crcuts s therefore Gauss' Law. The two aws we w use for anetc crcuts are: Apère s Law and Gauss Law are spe suatons for anetc crcuts F U, φ, around a oop at a node (4B.8a) (4B.8b) Manetc and Eectrc Equvaent Crcuts A eectroanetc systes shoud be reduced to spe anetc and eectrc equvaent crcuts To forase our probe sovn capabtes, we w convert every concevabe eectroanetc devce nto an equvaent anetc crcut and an equvaent eectrc crcut. We can anayse such crcuts usn technques wth whch we are faar. The anetc crcut for the toroda co s: R F φ U U R Fure 4B.2 There are varous ways to anayse the crcut, dependn on whether we know the current or fux, but a ethods nvove Apère's Law around the oop: F U H + U + H (4B.9)

5 4B.5 The eectrc crcut for the toroda co s: R v e L Fure 4B.3 KVL around the oop ves: v R+ e (4B.1) It s noray a dffcut crcut to anayse because of the nonnear nductance (whch ust be taken fro a λ - characterstc). The votae source apped to the co s sad to excte the co, and s known as votae exctaton. Two speca cases of exctaton are of partcuar nterest and practca snfcance DC exctaton and AC exctaton. DC Exctaton DC exctaton refers to the case where a source s apped to the co whch s constant wth respect to te. DC exctaton defned For DC exctaton, n the steady-state, the eectrc crcut s easy to anayse. Faraday s Law for the nductor s: ( L) d d e λ dt dt d L dt (4B.11) where L ay be nonnear. Intay, the crcut w undero a perod of transent behavour, where the current w bud up and raduay convere to a steady-state vaue. The crcut w be n the steady-state when there s no

6 4B.6 ore chane n the current,.e. d dt. Then Faraday s Law ves the votae across the nductor as vots (reardess of whether the nductance s near or not). KVL around the equvaent crcut then ves: V RI (4B.12) where we use a capta etter for I to ndcate a constant, or DC, current. Therefore, there s a drect reatonshp, n the for of Oh s Law, between the apped votae and the resutant steady-state current,.e. the votae source sets the current, so we need to ook up the resutant fux on the nductor s λ ~ characterstc. Ths fux s obvousy constant wth respect to te, snce the current s constant wth respect to te. AC Exctaton AC exctaton defned AC exctaton anaysed wthout consdern the effect of the resstance AC exctaton refers to the case where a source s apped to the co whch s contnuousy chann wth respect to te n ost cases the exctaton s snusoda. For AC exctaton, we spfy the anayss by assun the resstance s nebe. KVL then ves: ( t) v V ˆ cos ω e dλ dt (4B.13) and: t λ edτ Vˆ cos t Vˆ ω ( ωτ ) dτ sn( ωτ ) (4B.14) Therefore, there s a drect reatonshp between the apped votae and the resutant snusoda fux,.e. the votae source sets the fux, so we need to ook up the resutant current on the nductor s λ ~ characterstc. Even thouh the

7 4B.7 fux s snusoda, the resutn current s not snusoda, due to the hysteress characterstc of the ferroanetc atera used to ake the nductor. However, the current s perodc, and t does possess haf-wave syetry. Characterstcs The B-H characterstc can be converted to a φ -U characterstc for a ven atera: φ BA U H (4B.15) For ths partcuar case, there s ony one path that the fux takes, so the fux s the sae throuh each atera (ron and ar). We shoud, for a ven fux, be abe to ook up on each atera's characterstc how uch U there s because of ths fux. The tota U for the anetc crcut for a ven fux s just the addton of the two Us. For each vaue of fux, we can draw the correspondn tota vaue of U. The resut s a coposte characterstc. It s usefu f there s ore than one ferroanetc atera n the crcut. A B-H characterstc can be rescaed to ve a φ -U characterstc How a coposte B-H characterstc takes nto account the propertes of a the ateras n a anetc crcut Fux (Wb) Coposte Characterstc Manetc Potenta (A) Iron Gap Coposte Fure 4B.4

8 4B.8 Deternn F ven φ If we are ven the fux, then the potentas can be obtaned fro a φ -U characterstc. If ven φ, then ook up U Iron Characterstc Fux (Wb) Manetc Potenta (A) Fure 4B.5 For ar aps, we don't need a characterstc, snce t s near. We use: For ar aps, the characterstc s a straht ne, so use the equaton U R φ µ A φ (4B.16) Apère's Law s apped, and we et: F U + U N (4B.17) Gven F, the choce of N and s dctated by other consderatons The nuber of turns and current can be chosen to sut the physca condtons, e.. sa wre (ow current ratn) wth ots of turns or are wre (hh current ratn) wth a few turns.

9 4B.9 Deternn φ ven F (Load Lne) An teratve procedure ay be carred out n ths case. A better way s to use a concept caed the oad ne. (A ap s sad to oad a anetc crcut, snce t Gven φ, the oad ne deternes F has a hh U). Ths concept s used n raphca anayss of nonnear systes. The oad ne, ben near, ust be derved fro a near part of the syste. In a anetc crcut, the ar ap has a near reatonshp between φ and U. Usn Apère's Law, we et: F U 1 φ R + U U ( U F ) + R φ (4B.18) The oad ne equaton for a anetc crcut wth one ap Ths s the equaton of the oad ne. The unknown quanttes are φ and U. Ths equaton ust be satsfed at a tes (Apère's Law s aways obeyed). There are two unknowns and one equaton. How do we sove t? We need another equaton. The other equaton that ust be obeyed at a tes s one whch s ven n the for of a raph the atera s characterstc. It s nonnear.

10 4B.1 The oad ne and the ferroanetc atera s characterstc are both satsfed at the pont of ntersecton To sove the syste of two equatons n two unknowns, we pot the oad ne on the characterstc. Both raphs are satsfed at the pont of ntersecton. We can read off the fux and potenta. Fux (Wb) Iron B -H Characterstc souton oad ne Manetc Potenta (A) Fure 4B.6 If the atera s specfed n ters of a B-H characterstc, then the equaton of the ne becoes: F B H + µ B µ H F (4B.19)

11 4B.11 Equvaent Crcut of a Peranent Manet PM soft ron ar ap Fure 4B.7 In ths case, we know the f t s zero snce there s no apped current. The ethod of fndn the fux n a anetc crcut contann a peranent anet (PM) therefore foows the sae procedure as above. We nore the soft ron A peranent anet (PM) produces fux wthout f (t has nfnte pereabty copared to the ar ap): U + U U + R φ 1 φ R U (4B.2) Apère s Law (oad ne equaton) for a PM To sove for the fux, we need the PM's characterstc. We can see that for a postve fux, the anet exhbts a neatve potenta. Ths akes sense because we have aways assued that the anetc potenta s a drop. A neatve drop s equvaent to a rse a PM s a source of potenta and therefore fux. The oad ne ntersects the B-H hysteress oop (not the nora For a PM, ony one part of the hysteress oop s vad anetzaton characterstc) to ve the operatn pont (or quescent pont, or Q-pont for short).

12 4B.12 φ oad ne Q P φ Q reco pereabty ne -F U Q U anet characterstc Fure 4B.8 The operatn pont of a PM oves aon the reco ne for chanes n oad A PM exhbts hysteress, so when the ap s repaced wth a soft ron keeper, the characterstc s not traced back. The operatn pont oves aon another ne caed the reco pereabty ne (PQ n Fure 4B.8) to P. Subsequent openn and cosn of the ap w cause the operatn pont to ove aon PQ. A ood peranent anet w operate aon PQ aost contnuousy. If the operatn pont aways es between P and Q, then we can use the equaton for ths straht ne n the anayss. Ths s aso equvaent to oden the PM wth near eeents. A near ode of a PM hysteress oop φ sope 1 R Q P - F U Fure 4B.9

13 4B.13 The PM near ode s therefore: R φ φ A near crcut ode of a PM F U or φ R U Fure 4B.1 The near crcut ode for the PM s ony vad for oad nes that cross the reco ne between P and Q.

14 4B.14 Exape Deterne F ven φ Consder the foown eectroanetc syste: 5 Ia a 4 b c Ic Nc Na 2 1 Fure 4B.11 Gven: The core s anated sheet stee wth a stackn factor.9. φ a 18. Wb, φ b 8. Wb, φ c 1 Wb. I a s n the drecton shown.

15 4B.15 Sheet Stee B -H Characterstc 1.2 Manetc Fux Densty B, T Manetc Fed Intensty H, A/ Fure 4B.12 Draw the anetc equvaent crcut. Show the drectons of φ a, φ b and φ c. Deterne the antude of I a and the antude and drecton of I c.

16 4B.16 Souton: The anetc equvaent crcut s: R a R c F a φ a R b φ b φ c F c Fure 4B.13 As the cross sectona area s unfor, branches a and c are taken rht up to the dde of the centre b. Therefore: a 36., b 1. and.36. c Fro the B-H characterstc, snce B φ A and φ s ven, we just ook up: H a 2 A -1, H b 5 A -1 and H c 75 A -1. Appyn Apère s Law around the eft hand sde (LHS) oop ves: F a U a + Ub H aa + Hbb A Appyn Apère s Law around the rht hand sde (RHS) oop ves: F c Uc Ub A Therefore: I F N.385 A and I F N.22 A ( ). a a a c c c

17 4B.17 Exape Peranent Manet Operatn Pont Consder the foown peranent anet (PM) arraneent: PM soft ron ar ap or soft ron keeper Fure 4B.14 The PM has the foown B-H characterstc: PMs operate wth a neatve H Peranent Manet B -H Characterstc Manetc Fux Densty B, T Q ar ap ne reco ne Manetc Fed Intensty H, ka/ Fure 4B.15

18 4B.18 a) Reove keeper. Deterne ap fux densty B. b) Insert keeper. Deterne resdua fux densty. Inore eakae and frnn fux. Assue µ reco 2µ and µ soft ron. anet, soft ron, ap. Souton: a) As there s no externay apped current, Apère s Law ves: (snce µ ) U + U + U Therefore: U R φ or U The ar ap ne for a PM φ 1 R U (4B.21).e. the equaton of a straht ne (the ar ap ne or oad ne). If the B-H characterstc of the PM s re-scaed to ve a φ -U characterstc, the oad ne (sope 1 R ) ntersects t at the operatn pont Q. Otherwse, as φ BA and U H: rewrtten n a for sutabe for pottn on a B-H characterstc B µ A A H (4B.22) Fro the densons ven (and µ 4π 1 B 7 H -1 ) we et 72π 1 7 H. Therefore, at H A -1, B T. We draw the oad ne throuh ths pont and the orn. At the operatn pont Q, B 112. T. Therefore B A A B 2 B T.

19 4B.19 b) When the keeper s renserted: We draw a ne fro the Q-pont, wth sope 2µ to the B axs. The ntersecton ves the resdua fux densty B. or: The chane n anetc fed ntensty s δh The fux densty chane s δ B 2µ δh 13. T. Therefore resdua B T. Exape Peranent Manet Mnu Voue Consder the PM and characterstc of the prevous exape. The fux n the PM s: B A B A and the potenta across the PM s: H H The voue of the PM can therefore be expressed as: V A A H B HB (4B.23) The voue of a PM can be nsed by carefu choce of the Q-pont whch s a nu f H B s a axu.

20 4B.2 If we pot B- HB for the anet we et: 1.6 Peranent Manet B -H, B -BH Characterstc Manetc Fux Densty B, T Q 1.2 ar ap ne H (eft), ka/ and -HB (rht), kat/ HB ax Fure 4B.16 Fure 4B.16 shows how the pot s constructed fro the PMs B-H characterstc. The pot shows that f we choose a PM wth nu voue (sa cost PMs are expensve) then we shoud choose an operatn pont ven by Q. The prevous exape was therefore a ood desn.

21 4B.21 Suary We defne the reuctance of a anetc atera as: R. µ A Wed defne the anetootve force (f) of a co as: F N. For unfor anetc feds and near anetc atera, Apère's Law s the anetc anao of Oh s Law: F Rφ. The nductance of a toroda co s ven by: 2 N L. R Apère's Law and Gauss Law are the anetc anaos of Krchhoff s Votae Law and Krchhoff s Current Law, respectvey: U F and φ. We can convert every eectroanetc devce nto an equvaent anetc crcut and an equvaent eectrc crcut. For ferroanetc ateras, we use the nonnear B-H characterstc anayse the anetc crcut. A oad ne represents a near reatonshp between crcut quanttes and s usuay raphed on a crcut eeent s characterstc to deterne the operatn pont, or Q-pont. A peranent anet exhbts an nterna neatve anetc potenta and can be used to create fux n a anetc crcut wthout the need for an externa f. The operatn pont of a anetc crcut that uses a peranent anet can be optsed to nse the voue of peranent anetc atera. References Ponus, Martn A.: Apped Eectroanetcs, McGraw H Koakusha, Ltd., Snapore, 1978.

22 4B.22 Probes 1. Consder the foown anetc structure: The nora anetsaton characterstc of the core atera s, H ( A 1 ) ( ) B T Deterne the fux densty n the centre b and the necessary f for a wndn on the centre b: (a) (b) For a fux densty of 1.2 T n each ar ap, For a fux densty of 1.2 T n one ar ap when the other s cosed wth a anetc atera of the sae pereabty as the core atera.

23 4B Consder the anetc structure of Q1. The centre b has a wndn of 5 turns, carryn 1 A. Draw the equvaent anetc crcut and deterne the tota reuctance of the crcut and the fux densty n the RHS ar ap for: (a) (b) Both ar aps open, LHS ar ap cosed. Assue a constant pereabty µ H -1.

24 4B Consder the anetc core and anetzaton characterstc shown. a b c I turns 2 I 2 2 turns The core has a unfor cross sectona area (csa) A , a 88. and b 16.. Co 1 has I 1 5. A (DC). c (a) Deterne the antude and drecton of I 2 needed to ve φ b. (b) Deveop expressons for L 21 and L 12, and cacuate ther vaue for the currents and fuxes deterned n (a).

25 4B Refer to the exape Peranent Manet Operatn Pont. The ar ap fux was assued to be confned wthn the ar ap. Consder now the eakae fux between the upper and ower horzonta sectons of the soft anetc atera, and assue that the eakae fux densty s unfor. (a) (b) Deterne the fux densty n the PM when the keeper s reoved. Copare the ar ap fux densty wth that cacuated n the exape. The eakae fux ay be reduced by pacn the PM coser to the ar ap. Sketch an proved arraneent of the syste.

26 4B The PM asseby shown s to be used as a door hoder (keeper attached to the door, reander attached to the frae). 2 PM 2 soft ron 2 x 5 soft ron keeper 1 2 deased B-H ch H(A ) B (T).4.15 Assue that for soft ron µ. (a) (b) Derve a nearsed anetc equvaent crcut (neect eakae and frnn), and deterne the axu ar ap enth x ax for whch t s vad. The near ode used n (a) assues that the anet w be deanetsed when x > x ax. Show that the eakae reuctance between the upper and ower soft ron peces s ow enouh to prevent deanetsaton fro occurrn.