Inductance Calculation for Conductors of Arbitrary Shape


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1 CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors of arbtrary shape. The conductor s dscretzed usng soparametrc brcks wth unform current densty. The calculaton s numercally stable and automatcally adaptve up to a predefned accuracy. A convergence acceleraton based on the study of the numercal error propertes s used to acheve fast and accurate results.. Introducton In the electromagnetc analyss of magnetc confguratons, n crcutal analyss and n general n electrcal engneerng t s often necessary to have a precse value of the self and mutual ndutance of conductors. In the case of smple confguratons the nductance calculaton s often performed usng analytcal approxmatons. Once the confguraton becomes complex, or tends to lmts of small dstances or small conductor sze, analytcal approxmaton can become very naccurate, and n the lmt fal to provde physcally consstent values. In ths note we descrbe a numercal method sutable for artrary, threedmensonal conductor shape that gves accurate results n all lmtng cases. For the calculaton of the nductance we start from the followng expresson of the magnetc energy E stored n the nteracton of the current flowng n two conductors and of arbtrary shape and volume V and V : E µ J J 0 = dv dv (.) 2 4 r V V PQ where J s the current densty n the volume V, J s the current densty n the volume V and r PQ s the vector dstance between a pont Q centered n the
2 volume element dv, and a pont P centered n the volume element dv. We show ths stuaton n Fg. for the case of two conductors of arbtrary shape and poston n space. J Q dv r PQ dv P J Fgure. Defnton for the calculaton of nductance coeffcents among conductors of arbtrary shape. ote that E s only the porton of magnetc energy assocated wth the nteracton of conductors and. The total magnetc energy E stored n the system can be obtaned by the sum: E = E + E + E + E (.2) where all possble nteractons must be consdered. In the absence of current sources or snks, each conductor carres a total current I and I constant along ts length, and the correspondng nteracton energy can be also wrtten usng the nductance L as follows: E = L I I (.3) 2 We defne now two versors and orented as the current densty n the conductors, but wth module normalsed to provde unt current n the conductor. The current densty n the conductor can then be wrtten as: J = I (.4) 2
3 and smlarly for the conductor. Wth the above defnton we can dentfy terms n Eqs. (.) and (.3) to obtan: L µ 0 = dv dv (.5). 4 r V V PQ The order of the scalar product and ntegraton n Eq. (.5) can be changed as follows: L = µ 0 dv (.6) 4 dv V V rpq where we recognze that the product of the multplcatve constant and of the term n parantheses corresponds by defnton to the vector potental A generated by a current densty of strength n the conductor, or: A = µ 0 dv (.7). 4 r V PQ Accordngly, the nductance coeffcent can also be wrtten as follows: L = dv (.8). V In the above form the nductance calculaton can therefore be reduced to the volume ntegral n conductor of the scalar product of the current densty versor and the vector potental A generated by a current densty of strength n the conductor. Ths form s the most convenent for numercal ntegraton. 2. umercal calculaton 2. Dscretzaton For the numercal ntegraton of Eq. (.8) we start dscretsng the system of conductors usng 8nodes brck elements wth plane faces. The ntegral of the scalar quantty, the product, over the volume V of conductor s then broken nto the sum of the ntegrals over ts elements of volume V : L = V dv = dv (2.). V 3
4 To perform the volume ntegraton n each element of conductor we need the value of the vector potental generated at an arbtrary pont by the elements belongng to conductor. For ths calculaton we use the analytc expressons establshed n [2]. As explaned there, the calculaton s performed transformng the volume ntegral n a sum of lne ntegrals over the edges of the soparametrc brck, and s exact n the case that the faces of the brck are plane. The ntegral over the volume V s performed numercally usng Gauss quadrature [3]. To ths am we use the coordnate transformaton propertes of an soparametrc element to transform the volume ntegral n Eq. (2.) as follows: V ( ) dv = Dddd (2.2) where the ntegraton s transformed from the volume V of the element n physcal space to the rght prsm wth vertces placed at coordnates + and  n the parent space (,,) as shown n Fg. 2. The transformaton s defned as conventonally done n fnte element theory usng the determnant D of the Jacoban matrx[3]. Fgure 2. Transformaton of an soparametrc brck n physcal space to the rght prsm n the parent space. The numercal ntegraton of Eq. (2.2) s smplfed by the fact that t takes place n the parent plane, where the ntegraton nterval s [ ]. For the ntegraton n 3D space we use the same number of Gauss ponts wth coordnates k = l = m and assocated weght w n the three drectons and we can wrte the ntegral as follows: ( ) Dd dd wkwlwm ( ( k, l, m ) D( k, l, m) k= l= m= (2.3) 4
5 where we have evdenced the fact that the current densty s constant whle the vector potental and the Jacoban must be evaluated at each Gauss pont. 2.2 Convergence acceleraton and adaptvty The sum on the r.h.s. of Eq. (2.3) provdes an approxmaton u to the exact ntegral u exact over each element. The qualty of the approxmaton mproves, and the assocated error = u  u exact decreases, ncreasng the number of Gauss ponts used for the evaluaton,.e. the order of the ntegraton. In general we can wrte that the result of the ntegral s affected by an error that depends asymptotcally on the ntegraton order as follows: R = C (2.4) where C s a constant and R s the convergence rate of the numercal approxmaton. Provded that the relaton Eq. (2.4) holds, t s possble to use the result of the numercal ntegral obtaned usng dfferent ntegraton orders to estmate the ntegraton error and thus mprove the result. We refer to ths process as convergence acceleraton. Assumng n partcualr that the numercal ntegral has been performed for two dfferent orders and , leadng to the two results u and u  respectvely, we can calculate the multplcatve constant C: C u u (2.5). R = R The asymptotc value of the ntegral for an nfnte ntegraton order s then estmated as follows from the result obtaned usng an ntegraton order : u u C R (2.6). The estmate thus obtaned for the ntegraton order can be compared to a prevous estmate, obtaned for an order M wth M < to verfy the convergence by computng the varaton among the two estmates: u um 2 (2.7). u + u M 5
6 The parameter provdes a smple mean to adapt the ntegraton order. The ntegraton s started wth a sngle Gauss pont,.e. =. The order s augmented, recomputng at each successve ntegraton the estmate u and montorng the varaton between successve ntegratons (.e. M = ). Convergence s acheved when the varaton becomes smaller than a preset value. The procedure outlne here requres the knowledge of the convergence rate R, dscussed n the next secton. 2.3 Convergence rate In order to determne the conmvergence rate R to be used n the adaptve procedure descrbed, we have performed tests of nductance calculatons on brck elements placed dfferently n space. The frst test was performed usng as conductors rght prsms of sze 2 m placed as shown n Fgure 3(a). The current was assumed to flow along the z axs of the prsms. The second convergence test was performed on the brcks used to dscretzed two coaxal solenods of nner radus 0.5 m, outer radus.5 m and heght m and placed as shown n Fg. 3(b). The solenods were dscretzed usng 50 soparametrc brcks each. 0 Fgure 3. 2 (a) Geometry used for the convergence tests. (b) Calculatons of the self and mutual nductance of the brcks were performed usng to 0 Gauss ponts n each space drecton. The error was computed assumng that the exact value of the ntegral u exact s the asymptotc value u estmated for the hghest ntegraton order (.e. = 0). In addton we have computed the error on the asymptotc estmate gven by the dfference between the asymptotc value u at a gven order and the exact value u exact,.e. = u  u exact. In Fgs. 4 and 5 we report the results of ths analyss. The errors 6
7 have been normalsed to the exact soluton, and are plotted n absolute value. We see that for both calculatons that the relatve error on the computed nductances of closely placed brcks s at the ppm level for an ntegraton of order 6 and hgher. For brcks wth large spacng compared to the dmensons the requred ntegraton order to reach ppm accuracy s much smaller, and typcally 2 to 3 Gauss ponts per drecton are enough. The typcal convergence rate R s of the order of L(,) = C (/) 7.5 L(,2) L(,3) L(,4) L(,) = C (/) 8.5 L(,2) L(,3) L(,4) relatve error () relatve error () / () / () Fgure 4. Convergence rate of the computed self and mutual nductance coeffcents for the frst test case (rght prsms). On the left the relatve error on the volume gven by Eq. (2.3), on the rght the relatve error on the asymptotc estmate provded by Eq. (2.6). In both cases the power law s plotted for C=. 0. self mutual = C (/) self mutual = C (/) 8.5 relatve error () relatve error () E / () E / () Fgure 5. Convergence rate of the computed self and mutual nductance coeffcents for the second test case (coaxal solenods). On the left the relatve error on the volume gven by Eq. (2.3), on the rght the relatve error on the asymptotc estmate provded by Eq. (2.6). In both cases the power law s plotted for C=. 7
8 The asymptotc estmate of the ntegrals converges faster, typcally wth a rate (R + ) and hgher. The gan n the convergence rate allows to reach hgher accuracy for a gven ntegraton order or the same accuracy wth a reduced order. Ths prooves that the acceleraton procedure devsed works as expected. 3. Example As a fnal test of the procedure descrbed here we have tested the convergence of the calculaton of the nductance of the solenod confguraton reported n Fg. 3(b) and we have compared t to known analytcal solutons. The self nductance of each solenod computed wth the analytcal approxmatons gven n [4] s.2666 µh, whle the mutual nductance s µh. The numercal calculaton was performed for a dscretzaton of the frst solenod usng 00 brcks, whle for the second a varable number of brcks n the range of 5 to 80 has been used n order to establsh the convergence rate. The result for the largest number of brcks s a self nductance of.2655 µh and a mutual nductance of µh. Both values agree well wth the analytcal approxmatons quoted above. 0. = C (/ brcks ) 2 relatve error () self mutual / brcks () Fgure 5. Convergence of the calculaton of self and mutual nductance for a set of two coaxal solenods. The power law s plotted for C=0.5. The convergence of self and mutual nductance calculaton s shown n Fg. 6. In ths case the convergence appears to be quadratc. The reason for the apparent loss of convergence speed s that n ths test an addtonal error s generated by the approxmaton of the solenod geometry through brcks. The curved geometry s approxmated better at ncreasng number of brcks, and t can be shown that the geometrcal error of ths approxmaton process has a quadratc convergence. Ths error domnates the overall performance of the calculaton, thus hdng the hgher convergence rate that s acheved on a brckbybrck 8
9 bass. It s therefore clear that care should be exerted when approxmatng curved geometres wth brcks and n estmatng the assocated error. In spte of ths the calculaton acheves remarkable accuracy wth moderate number of brcks. References [] F. Barozz, F. Gasparn, Fondament d Elettrotecnca, UTET, 989. [2] L. Bottura, Analytcal Calculaton of Vector Potental n an Isoparametrc Brck, CryoSoft Internal ote CRYO/97/003, 997. [3] O.C. Zenkewcz, The Fnte Element Method, 4 th Edton, McGraw Hll, 99 [4] F.W. Grover, Inductance Calculatons, Dover Publcatons,
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