Magnetostatcs grd G E dual grd G J Ch = j SM h = T wth h= h + S Φ curl = J Ch = j T dv grad ϕ = dv = q SM S Φ = SM h = q m m The basc equatons n magnetostatcs are smlar to the equatons n electrostatcs. The dfference s that now our prmary varable s the magnetc feld strength nstead of the electrc feld strength E. In electrostatcs, we allocated the electrc feld strength between two adjacent grd ponts. In order to reach an essental analogy (and thus exchangeablty of program parts) we can allocate the magnetc feld strength, too, n these locatons,.e. on the prmary grd. In consequence, we can use the formulas derved for electrostatcs just exchangng electrc by magnetc grd voltages and fluxes, respectvely. Ths yelds a dfferent set dscrete Ampère s law as dsplayed above. Pluggng n our approach for the soluton fnally yelds the two equatons shown on the bottom. Now, t only remans to fnd an algorthm to determne the arbtrary, non-physcal auxlary feld.
Magnetostatcs ds = J da Ch = j A A 1. Buld the sum I of all currents passng through area A and dvde t by the crcumference of A. Regard 2D example wth Neumann b.c. on one sde: a I b t = a+ a+ b= I I = 2 a + b 1. Ampère s law demands that the contour ntegral over the magnetc feld strength equals the current passng through ths area. Ths demand can be easly fulflled: The sum of all currents passng through the area s dvded by the crcumference of the area. Ths value s placed n the correspondng entry of the auxlary feld. Let s regard a 2D example for the rght hand sde where the Neumann boundary condton has been ncorporated. In ths case, the effectve crcumference s obtaned from the other three sdes as shown n the sketch above.
Magnetostatcs (1) Possble cut for step 2 (1) b a I t = 2. Dvde A n A1 and A2. s known at all but one edge. (1) Repeat ths step untl all edges have been assgned some value of. Easy generalzaton to 3D (volume wth I sub-volumes) 2. Next, the area s dvded nto two halves and each of the partal areas s treated as n step 1. For both sub-areas three edges are known already, each, such that only the remanng one has to be assgned some value. Ths procedure s successvely repeated untl the nhomogeneous feld component has been assgned to all edges n the grd. 3. Ths method can easly be generalzed to 3D volumes. To do so, we search for a sub-volume as small as possble, frst, whch contans all currents. Then, feld values are assgned approprately to the edges of ths sub-volume. By each step through the volume or the arsng sub-volumes four new edges appear whch have to be assgned some nhomogeneous feld value by choosng approprate ntegratons paths. Ths successve subdvson s contnued untl all edges n the grd have been assgned an nhomogeneous feld value. All edges are acqured by cyclc change of the cuttng drecton n x, y and z to obtan the sub-volumes. Ths cyclc exchange of cuttng drecton also decreases the resultng roundng error.
Magnetostatcs Profle of a possble feld Profle of the B and feld Source of pctures: T. Weland, CAD-Skrpt, Darmstadt 22 ere, a smple 2D example s presented.
Magnetostatcs h 1. -algorthm yelds: I = h + L L+ h B B ds = I L+ h I 2. = grad ϕ yelds for the magnet: I L I Bar = B B= L/ + h/ ar = = h I I h + L + I I grad ϕ = grad ϕ h+ L h+ L 1 L Wth the -algorthm descrbed before we can compute the nhomogeneous feld values very fast n all grd cells, Yet, there s the problem of numercal cancellaton n the feld value computaton n flux carryng metal. Usng the algorthm gven before, the computed -contrbuton s nearly dentcal wth the contrbuton by grad ϕ. Ths example shows a smple horseshoe magnet wth the mean feld lne length L and a gap of wdth h. Accordng to the sketch we receve the exact values of the magnetc feld strength n the and n the ar gap as gven above. Usng the -algorthm as above and computng accordng to = grad ϕ We can estmate grad ϕ as gven above. Snce the magnet s made of ferromagnetc materal t s permeablty s large compared to leadng to close values of grad ϕ and. Computng now the magnetc feld n the by = grad ϕ may lead to large errors due to numercal cancellaton.
Numercal example: Magnetostatcs = L= h= m I = A 4 1, 1, 2 I A = = 1. h+ L m I I A grad ϕ.9998 h+ L h L m + 4 1 A A ( 1.9998) =.2 m m For a computer wth 5 dgts accuracy the result s flawed by an error of up to ±5%! As the example demonstrates the -approach n the present form leads to numercal problems by cancellaton for >>.
ar Magnetostatcs mproved algorthm : ( ) ( ) ( ) α L+ h + 1 α L+ h = I wth arbtrary α We choose α = I A =.2 / h+ L m I A = 1.9998 h+ / L m 1/ L / + h / I = grad ϕ = grad ϕ / h+ L I / h+ L grad ϕ. The weghted -algorthm gven above presents an mproved alternatve. α s arbtrary but wth the above choce gets materal-dependent: small n materals wth large permeablty and vce versa. Applyng ths -algorthm the -components n and ar read as dsplayed above and the cancellaton n the computaton of the magnetc feld n the s avoded. The -algorthm presented here s a smple, non-teratve method. The -feld dstrbuton results from purely mathematcal consderatons wth the secondary condton to need computatonal effort as low as possble. In general, ths -feld s non-physcal. A computaton of the current-descrbng feld wth help of Bot-Savart s law would be possble alternatve to ths strategy. In absence of ths feld would be physcal. Yet, t s dsadvantage s that the computed feld has barely any smlarty wth the real soluton n presence of whch agan would lead to numercal cancellaton, the computatonal effort s ncomparably hgher compared wth the prevous - algorthm: For the feld value n each grd pont a summaton over the coeffcents n all current-carryng grd areas s necessary. These coeffcents have to computed agan and agan. Ths means that the computaton of the -feld dstrbuton accordng to Bot-Savart ncreases quadratcal (αn p2 ) whle the one of the prevous -algorthm ncreases only lnearly ( βn p ). Emprcally, effort Bot-Savart >> effort -approach holds for N p 1 5.
Magnetostatcs Iteratve -algorthm - the -update method: 1. Compute wth the -weghted -algorthm. 2. Coarse soluton of Maxwell's equatons accordng to the -scheme. 3. Compute = grad ϕ. 4. Set =. 5. If max grad ϕ > δ (accuracy lmt): Go to step 2. 6. Precse soluton of Maxwell's equatons accordng to the -scheme. An even better approxmaton to the feld dstrbuton may be obtaned usng the socalled -update-method. It s supposed that a startng dstrbuton of the nhomogeneous feld s obtaned by the -weghted -algorthm. Next a coarse soluton of Maxwell s equatons accordng to the -scheme descrbed on page 2 of lecture CEM-5 s computed yeldng a somewhat better feld dstrbuton already. Ths feld dstrbuton s taken as new -feld mprovng step-by-step the feld dstrbuton. Ths algorthm has no problems wth numercal cancellaton, of course the computaton tme s somewhat ncreased.
Lamnated Work Peces Transformer sheet d h d h Bar, tan + d B, tan < Btan > = ; Bn s contnuous! h+ d h ar, n + d, n < n > = ; tan s contnuous! h+ d Dealng wth lamnated transformers dfferent permeablty values occur wth regard to the tangental and normal drecton wth respect to the lamnaton. Between the plates of thckness d there s always a slce of ar and/or varnsh coatng. These dfferent permeablty values have no negatve nfluence of the formulas derved above snce the dagonal matrx of the permeablty tensor can easly be adjusted.
Lamnated Work Peces ( + ) B Bn h d n < n > : = = h + d n ar, n, n h+ d h / + d / B h + d tan < tan >: = = h d tan = = ar, tan, tan tan h + d h+ d ( + ) d h d < > effectve permeablty < > < n = tan tan The mean permeablty s defned as gven above. The effectve permeablty s collected n a dagonal tensor. Such drectonal permeablty values do not only occur wth lamnatons but also for mechancal deformatons such as rollng or punchng.
Non-lnear Ferromagnetc Materals ( ) B = f ( ) For B = we may plot a hysteress curve: flux densty B feld ntensty In the computaton of electromagnetc felds s gets necessary to consder non-lnear dependences of materal parameters f materals such as, nckel, cobalt, steel and ther alloys are used. Ths materal class s of great mportance n engneerng and appled physcs. They are employed as ferromagnetc materals n the manufacturng of electrcal machnes and n the electrcal power engneerng or as ferroelectrc delectrcs n non-lnear optcs. These materals nteract non-lnearly wth strong electromagnetc felds. In contrast to da- and paramagnetc materals, the characterstc curve B = f() of the ferromagnetc materals do not show a lnear shape but partly they steeply ncrease n the front curve. In case that the permeablty depends on the magnetc feld strength the functonal relaton s gven by B = ( ). The computaton of ( ) affords to know the relaton between B and whch, n general, s not sngle-valued but depends not only of the magnetc feld but also on the magnetzaton hstory of the materal. All possble relatons B() for some materal are plotted n the so-called hysteress curve. One example s shown above. In the followng we wll assume sotropc materal and restrct ourselves to the treatment of non-lnear hysteress-free permeablty snce a unt relaton between B and s needed for a unque soluton of the magneto-statc problem. For ths purpose we choose a curve nsde the hysteress (the dashed lne) whch corresponds to a mean permeablty or the ntal magnetzaton curve, respectvely.
Non-lnear Ferromagnetc Materals B Relatve permeablty : B = = r r Dfferental permeablty : d d = 1 db d In materal-flled space the two magnetc feld quanttes and B are proportonal to each other and lnked by the (total) permeablty : B = = r. The permeablty splts nto the product of the vacuum permeablty = 4π 1-7 /m and the permeablty constant r whch s often denoted as relatve permeablty, too. The latter one ndcates by whch factor the permeablty of some materal ncreases compared to the vacuum permeablty. Thus, the relatve permeablty s dmenson-less. The hgher the saturaton of the materal, the lower the relatve permeablty. Fnally, the so-called dfferental permeablty d descrbes the permeablty n some arbtrary pont of the hysteress loop for small values of B and for db and d, respectvely. Thus the dfferental permeablty d ndcates the local slope of the hysteress curve.
Praxs-relevant Magnetsaton Curves 2 db d B Monotone-convex magnetzaton curve > and < : 2 d d B contnuously decreasng permeablty = B 2 2 S-lke magnetzaton curve, concave d B / d > for small B ( ) Magnetzaton curves practcally used may be characterzed by two fundamental classes: monotone-convex magnetzaton curves wth contnuously decreasng permeablty and S-lke magnetzaton curves where low feld values show a concave curve such that the permeablty starts wth relatvely low values, rses to a maxmal value and falls down agan. Both typcal curve shapes are dsplayed above.
Interpolaton of Measured Values Tabulated values need to be nterpolated Oscllatons may occur n flat parts of the hysteress dm whch could lead to volaton of d where B = = + M wth the magnetzaton M ( ) Interpolaton methods for tabulated values: Lnear nterpolaton of ( B, )-values Lnear nterpolaton of (, )-values Cubc splnes Frolch-Kennely nterpolaton Interpolaton wth ratonal functons For praxs-relevant problems ths non-lnear relaton between materal values and local feld strength values s only avalable n form of dscrete characterstc curves. Generally, these curves are obtaned emprcally. Often they are dffcult to reproduce. The curves are used n form of dscrete tables wth B()-values. The typcally relatve small number of only 2-5 tabulated values needs to nterpolate the B- and -values at hand. Especally n flat parts of the hysteress curve the nterpolaton can cause oscllatons leadng to a volaton of the physcal condton that the dervatve of the magnetzaton M s always postve,.e. the magnetzaton always rses wth ncreasng feld strength, t never descends for ncreasng feld strength. Ths holds for all materals and all curves nsde the hysteress loop. Thus, the choce of the nterpolaton method s crucal and strongly nfluences the accuracy of the soluton. Ths s a possble reason for dscrepances between measured data and numercal results. A varety of nterpolaton methods whch may be used to approxmate the materal curve can be found n lterature: Lnear nterpolaton of (B, )-values Lnear nterpolaton of (, )-values Cubc splnes Frolch-Kennely nterpolaton Interpolaton wth ratonal functons.
Non-lnear Magneto-statc Algorthm algorthm wth -update: 1. = start 2. Determne. 3. Determne q. m new ( ) 4. Compute ϕ only a few teratons suffce 5. Compute = grad ϕ. ( ) 6. Compute from the characterstc curve. Usually an nhomogeneous materal dstrbuton follows. new old δ ( ) ( ) ϕ ( ) 7. If > accuracy lmt : startng wth step 3. 8. Now r s known. Compute precse teraton 9. = grad ϕ. Snce the non-lnear relaton between materal values and feld values s usually avalable as dscrete characterstc curve t s not possble n general to represent ths relaton by some smple operator whch then could be ntegrated nto the dfferent soluton methods for the lnear problems. Thus the non-lnear problem can only be solved by soluton of a seres of lnear problems wth an ntermedate update of the materal propertes. In order to mplement the non-lnear permeablty nto the numercal relatons we choose the prevously ntroduced -algorthm and add some -update. The sngle steps of ths new algorthm for non-lnear magneto-statc problems are shown above. The convergence propertes of ths algorthm can be mproved by the choce of optmal nterpolaton parameters whch take nto account the shape of the magnetzaton curve. Fnally we would lke to note that the convergence of the soluton of such non-lnear problems as sequence of lnear problems can only be guaranteed for strongly monotonous functonal dependences between and B.
Non-lnear Magneto-statc Algorthm ( ) Cycle 1 Cycle 2 { () 1 { ( 2) lnear problem magnetzaton curve lnear problem magnetzaton curve () 1 ( 2) B Consderaton of non-lnear permeablty () Cycle M { ( M) lnear problem ( M +1) Ths sketch schematcally llustrates the procedure of the non-lnear algorthm.
Convergence of cg- and SOR-solver Relatve resdual norm Precondtoned cg-algorthm Relatve resdual norm SOR-algorthm Number of teratons Number of teratons As example a non-lnear C-magnet s studed. ere two dfferent soluton methods are used to solve the lnear problems. Obvously the precondtoned needs much less teratons and converges to a more accurate result. In the convergence curve of the cg-algorthm one can recognze the sequence of lnear problems solved: At the begnnng of each cycle the relatve error somewhat ncreases frst.
Results for Non-lnear C-Magnet Vector potental Permeablty ere the fnal soluton of the non-lnear C-magnet problem s shown Left the vector potental s plotted and on the rght hand sde we see the permeablty dstrbuton.
C-Magnet (lnear) 2 symmetry planes Computaton for 1/4 of the structure 98,6 grd ponts strongly varyng step sze nhomogeneous materal dstrbuton Ths shows a smple lnear model of a C-magnet.
3.5 Tme-varyng feld d Ce = b dt d Ch = d+ j dt full MGE set Sb = Sd = q Representaton of harmonc feld va complex ampltude: () = Re{ jωt } = cos( ω + ϕ) mt ϕ = ( ) f t f e f t f Tme-dervatve of harmonc feld: d f t j f e j t dt ω () = Re{ ω } For the statc felds treated before the partal dfferental equatons for electrc and magnetc feld were decoupled. For tme-varyng electromagnetc felds the complete set of Maxwell s equatons s needed snce electrc and magnetc feld are coupled whch s reflected n ths system of coupled dfferental equatons. If such felds shall be computed wth the Fnte Integraton Technque (FIT) the complete set of the Maxwell-grd-Equatons (MGE) has to be taken nto account. The two curl-equatons buld a system of ordnary dfferental equatons wth regard to the tme varable t. For a smulaton of felds wth arbtrary tme-dependence (socalled transent felds) they have to be ntegrated by an approprate tme-ntegraton scheme. Ths s treated n detal n the next chapter. Often one s nterested n felds whch have a sne-lke behavour for a fxed frequency so-called harmonc felds. For such felds n steady-state all tme-dependent quanttes may be represented by ther complex ampltudes as gven above. Then the tme-dervatve reduces to an algebrac multplcaton of the complex ampltude wth the factor jω. Usually, computatons are done wth the complex quantty nstead of t s real part snce ths eases the computaton. Only, one has to keep n mnd that the complex quantty and t s magnary part have no physcal meanng and that the physcal felds are only obtaned after buldng the real part!
Tme-varyng feld d Ce = b dt = d Ch d dt + j Sb = Sd = q d f t j f e j t dt ω () = Re{ ω } Ce = jω b Ch = jω d + j For the Maxwell-Grd-Equatons ths means that nstead of dfferental equatons n tme doman now complex algebrac equatons n frequency doman result. The unknown dscrete varables whch now hold the complex ampltudes of all sngle voltage and flux components, respectvely, are called dscrete phasors. In the followng we wll denote these (and all other complex quanttes) by an underscore. Above we show the frst two equatons, the curl equatons, as resultng usng the complex notaton and after shortenng both sdes by the complex phase factor e jωt. All other equatons, the dvergence equatons and the materal equatons of the lnear case, reman formally the same we just need to exchange the grd voltages and fluxes, respectvely, by the phasors snce they do not contan explct tme dervatves. Yet, non-lnear materal relatons cannot so easly be treated n the frequency doman snce the non-lnearty usually causes hgher harmonc contrbutons to the wave.
3.5.1 Dscrete Wave Equaton of FIT 1. Regard the r.h.s. of Ch = jω d + j : j = M e+ j σ S 1 Ch = jω d + j = jω Mε + Mσ e + j = jωm S ε e + j jω 2. Insert h= M 1 b nto the l.h.s. 3. Use Ce = jω b to replace b 4. Re-orderng yelds the dscrete wave equaton of FIT: ( 2 1 ω ε ) CM C M e = jωj S S Regardng the two dscrete curl equatons and resolvng them for the electrc and magnetc grd voltage, respectvely, then nsertng one nto the other yelds a sngle equaton holdng only one feld quantty (ether the electrc or the magnetc grd voltage). We regard the case where the magnetc quanttes are elmnated. In that case, usng the materal relatons we frst derve an equvalent for the rght hand sde (r.h.s.) of the dscrete Ampère law wth the complex permttvty matrx M ε whch comprses the permttvty and the conductvty as t s usual n frequency doman. Usng also the materal relaton for magnetc grd voltage and flux we can replace the left hand sde (l.h.s.) of the dscrete Ampère law. The dscrete Faraday law s used to replace the magnetc grd flux. Then, we get the dscrete wave equaton of FIT.
Dscrete Wave Equaton of FIT Dscrete wave equaton or dscrete Curl-Curl-Equaton 2 ( C 1 C ω ε ) M M e= jωj S ω ε = ω -1 2 curl curl E E j JS 1. Buld C h d j by dfferentaton. 2 d d d = + 2 dt dt dt d d 2. Insert Ce = bafter resolvng for h dt dt + 1 dt + dt = CM Ce M e M e j dt 2 d d d ε 2 σ S Because of the double applcaton of the curl operator ths equaton s also denoted as dscrete Curl-Curl-Equaton. Of course, ths equaton corresponds to the analytcal wave equaton. A smlar dervaton could also be carred out n tme doman. For ths purpose we frst dfferentate the dscrete Ampère law wth respect to (w.r.t.) the tme. Next, we resolves the dscrete nducton law for the tme dervatve and nsert t. Ths yelds the equaton as shown above. In the followng we wll frst study n more detal the wave equaton n frequency doman. We wll see that analysng ths equaton we can deduce mportant results for the tme doman method.
3.5.2 Propertes of the Wave Equaton Choose x= y = z = 1 and homogeneous materal fllng. σ 1 smpler materal matrces: Mε ε I and M I j ω = + 1 = system matrx 1 σ CC ω ε I jω 2 + ( I = unt matrx) wth PP+ PP PP PP CC = PP PP+ PP PP T T T T w w v v v u w u T T T T u v w w u u w v T T T T PP u w PP v w PP v v PP u u In order to study the structure of the system matrx we choose the most smple case wth equdstant step sze and homogeneous materal fllng. Then, the materal matrces get a very smple form (only near the boundares there may be slght devatons from ths form). In consequence, also the system matrx gets a somewhat smpler form as shown above. It has a symmetrc block structure.
Propertes of the Wave Equaton of FIT M w M v M u M w M v -1-1 4-1 -1 M v M u -1 1 1-1 ( ) T T ( ) T CC = P P + P P uu w w v v CC uv = P P v u structure of CC : Snce the multplcaton of P-matrces has been treated before we can drectly wrte down the result for the uu- and the uv-block of the system matrx. Thus the system matrx has a man dagonal wth the value 4 and 12 sde dagonals wth the values ±1. The complete structure s dsplayed n the sketch above. In the general case (varyng step sze and materals) only the matrx entres but not the band structure of the system matrx s changed. The same also holds for the permttvty matrx of the general dscrete wave equaton as gven prevously. Some small modfcatons (entres whch are set to zero) only result for components on the boundary of the computatonal doman.