L6 Energy Conversion. EIEN20 Design of Electrical Machines, IEA, Previous lecture. L6: Energy conversion. Today s goal

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1 L: Enegy convesion a potentia fo causing a change R Ω u i L σ e R μ N i Pevious ectue R δ N i e L σ R Ω u i E x(y), i eectomagnetism eecto- and magnetomotive foce Induction action tansfomed o motiona votage Eectomagnetic enegy convesion z p y ec x B z t E x(y), i w d db dt z d Lund Univesity, Sweden Avo R Design of Eectica achines Today s goa. a potentia fo causing a change omuation of enegy convesion and fied couping eaning of shea stess in design of eectica machines Eectomechanica and eectomagnetic enegy convesion -. inductance enegy toque Avo R Design of Eectica achines Avo R Design of Eectica achines EIEN Design of Eectica achines, IEA,

2 Enegy convesion athematic fomuation Enegy is the capacity of a system to do wo Enegy cannot be ceated o destoyed, but ony conveted fom one fom into anothe Couping between the diffeent fieds obeys to the pincipe of enegy convesion B E t H J D B tm ds S W m aaday s aw Ampee s cicuita aw Gauss s Law, Eecticity Gauss s Law, Eecticity agnetic stess pe unit of aea Change of system enegy Eectomagnetism by axwe s equations Eecto-mechanism by eectomagnetic stess tenso o vitua wo Avo R Design of Eectica achines Avo R Design of Eectica achines Eectomechanica enegy convete Enegy convesion pincipe Eectomechanic enegy convete via intemediate magnetic fied Equivaent cicuit epesentation of the diffeent fieds The enegy consevation pincipe says that the sum of eectica and mechanica enegy input to a device at each time instant has to be equa to the sum of accumuated eectomagnetic and mechanica enegy and osses. dx N ph i u dt dw mag dw mech dw oss Avo R Design of Eectica achines 7 Avo R Design of Eectica achines 8 EIEN Design of Eectica achines, IEA,

3 echanica enegy Eectica enegy In a feey and ineay moving pat with a mechanic suppot we ae consideing ony inetic enegy and negecting potentia enegy d x dw mech dx dt In case the moving body is connected to the eastic sping the stiffness of the sping has to be taen into consideation Consideing Kichhoff s votage equations the diffeentia of the eectic enegy consists of esistive votage tem and eectomotive votage tem i u dt i i Rdt edt dwconducto id Accoding to aadays aw the opposing eectomotive votage can be eithe a tansfomed votage o/and a motiona votage d e dt d dt L i di L dt i dx dl dt dx Avo R Design of Eectica achines 9 Avo R Design of Eectica achines Losses Enegy baance Losses in an eectomechanica enegy convete can be sepaated accoding the oss oigin: eectica, magnetic and mechanica dw oss m i j j R dt dw j coe dw fiction dw conducto dw Eectomagnetic osses shea often the same oss mechanism. Common fo the osses is a phenomenon of fiction that opposes (cuent) fow, magnetisation, motion, etc and causes (ievesibe) heat enegy oss. coe dw fiction By sepaating esistive votage tem and induced votage tem, and dividing oss tem between the eectic, magnetic and mechanic oigin the enegy baance can be ewitten m oss dx id dx dwmag dw coe d x dt Avo R Design of Eectica achines Avo R Design of Eectica achines EIEN Design of Eectica achines, IEA,

4 agnetic enegy agnetic enegy and osses The magnet enegy is distibuted in a passive eements The diffeentia of fux is expessed though the pemeance and the mmf dop in the magnetic eement The pemeances ae: inea, paametic noninea and inheenty noninea ones These pemeances independent and dependent of motion and dispacement x Ne Ne dg dwmag d dg Gd dwstat dx paam The diffeentia of magnetic enegy in non-paametic eements ae expessed in tems of cuents and fux inages dw stat N N e ph d Coe osses ae dissipated in inheenty noninea pemeances ony, hence they ae independent on dispacement dwcoe Wcoe Wcoex dx i d Avo R Design of Eectica achines Avo R Design of Eectica achines oce equation fo tansation oce components Eectomechanica enegy convesion is possibe if the motion is pesent dx The appied foce is oaded with the eaction foce of the system. d x dg oss m dt dx Ony the paametic noninea pemeances gives the foce and the foce does not depend in the state of coe o eaage paths, but on conditions in the ai-gap ony. Inteaction between feous mateia and magnetic fieds (pemanent magnet(s) o/and coi(s)) esuting in vaiation of the magnetic enegy due to a position dependent euctance Reuctance foce/toque Detent o cogging foce/toque Inteaction between eectomagnet(s) o/and pemanent magnet(s) cause magnetic foce/toque Avo R Design of Eectica achines Avo R Design of Eectica achines EIEN Design of Eectica achines, IEA,

5 Eectomagnetic vs Reuctance Eectomechanica enegy convesion The enegy convesion between the eectic and mechanic enegy taes pace in pesence of the magnetic fied The basic goa in the anaysis of evey eectomechanica enegy convete is to compute the amount of enegy tansfomed fom one fom to anothe. The common methods to cacuate foce (o toque) Loentz foce Vitua wo axwe s stess tenso Avo R Design of Eectica achines 7 Avo R Design of Eectica achines 8 Loentz foce Vitua wo i q B oce that is appied to a cuent conducting wie in a magnetic fied ib A E v B J BdV JBdV Ad Bi Avo R Design of Eectica achines 9 WC x x The foce computation, accoding to the vitua wo method, consides the ate of the change of the tota co-enegy against the vitua dispacement. Wo aea eated to incementa foce distance Avo R Design of Eectica achines EIEN Design of Eectica achines, IEA,

6 - - - L Enegy Convesion x [N] fux x [m] x - 8 Shea stess oce cacuation I foce coenegy change of coenegy cuent.. E c, [J] E c, [J] x, [m] enegy E c /x, [J/m] E c /x, [J/m].... x, [m] x, [m] change of enegy x, [m] d/dx coenegy Depth aong z-axis:. m d/dx enegy oce cacuation II d G G G g G d/π d o G9 x s t p d d/π G G G g G G7 d/π d o x W G x G x W G T G Avo R Design of Eectica achines Avo R Design of Eectica achines oce cacuation III ux density, B oce integation ines axwe s Stess Tenso B agnetic foce on a suface accoding to magnetic pessue tn Bn Bt Bn Bt t t B n t n α α B t t t t foces fo m ong move stato move Avo R Design of Eectica achines Avo R Design of Eectica achines EIEN Design of Eectica achines, IEA,

7 L Enegy Convesion agnetic shea stess Toque capabiity I B σ shea σ s =/A, the aveage thust pe unit aea A of gap suface The vaue is geneay <7 N/m fo nomay cooed machines Gives the appoximate size of the machine adius, [cm] toque of R machine, T emr [Nm] h 9 8 toque of A machine, T ema [Nm] h Radia fux machine vs Axia fux machine T em E o T d i em A o i o.. Seected vaue of magnetic shea stess σ= N/m Coeess machines Avo R Design of Eectica achines Avo R Design of Eectica achines adius, [cm] adius, [cm] powe of R machine, P emr [W] 7 weight of R machine, emr [g] Low speed & high powe 7 powe of A machine, P ema [W] 7 weight of A machine, ema [g] Based on the pevious toque capabiity map the powe capabiity at pm is expessed By estimating the thicness fo the machine h= cm and aveage density of 7g/dm the weight is expessed Avo R Design of Eectica achines 7 7 T z w Design Taget B K Enegy convesion T η - efficiency Pe u t i t dt T P T Toque pe oto voume T VRT A gap Ai-gap shea stess BIz BKwz BK Agap Agap Agap Poduct of magnetic and eectic oadings σ=bk Avo R Design of Eectica achines 8 mech EIEN Design of Eectica achines, IEA, 7

8 Loentz foce Loenz foce on pemanent magnets Specific foce: f s =BK<N/m agnetic induction: B<.8T Sheet cuent density: K<A/m DC machine: f s <-N/m Synchonous machine: f s <-N/m Induction machine: f s <-N/m Specific foce: f s =BH C h m /τ p <N/m agnetic induction: B<.T Coecitive foce: H C <8A/m Poe ength to magnet thicness: τ p / h m = DC machine: f s <-N/m PS, BLDC: f s <-8N/m Avo R Design of Eectica achines 9 Avo R Design of Eectica achines Reuctance foce Summay Specific foce: f s =/i dl/dx B /μ δ/ τ p <N/m agnetic induction: B<.T Sheet cuent density: K<.A/m Poe ength to gap ength: τ p / δ = SR, stepping machines: f s <N/m axwe stess.. estimate B Inteaction Estimation of ining fux Detemination of cuent I Deiving toque fom I Attaction Estimation of gap pemeances G Estimation of ining fux Deiving toque fom enegy Enegy convesion, couped cicuits, wo and foce: eectomagnet exampe oce computation, axwe shea stess, types of machines Leaning though simuations the second home assignment Avo R Design of Eectica achines Avo R Design of Eectica achines EIEN Design of Eectica achines, IEA, 8