capcalc

1 Created: 5/11/2003. Last update: 20/4/2005 Capacitance Calculations Antonio Carlos M. de Queiroz [email protected] Abstract —This document describes calculation methods for distributed capacitances of objects wit h several particular shapes, and methods for the evaluation of the electric field around them. It’s fundamentally a collection of formulas, some not very easy to find in the literature. The algorithms were implemented in the Inca program, available at http://www.coe.ufrj.br/~acmq/programs. I. I NTRODUCTION Most of the formulas below are known since long tim e, most dating from works in the XIX century. Some appear in Maxwell’s book [1], and some in other collections of explicit formulas for electromagneti c problems, as [2], or in other early works as [3]-[5 ]. In most cases I have just adapted the notation, but so me derivations not found in other works are presented too. In most of the early works, capacitance is expresse d in units of length. For example, the capacitance of a sphere of radius a in free space is listed in [1] and [2] as C=a. To convert this unit to Farads, it’s necessary to m ultiply the value by 4 πε0, where ε0 is the permissivity of vacuum, ε0 = 8.8541878 x 10 -12 . ε0 can be calculated from the speed of light c and from the magnetic permeability of vacuum, µ0 = 4 π x 10 -7 (a definition), from the relation: 001 εµ=c (1) The capacitance of a sphere of radius a meters is then: Csphere = 4 πε0a = 111.26501 a pF (2) Other figures that have simple expressions for the free- space capacitance are: A thin flat disk with radius a [2]: Cdisk = 8 ε0a = 70.833503 a pF (3) An open hemisphere with radius a [2]: Copen hemisphere = 4 πε0a(1/2+1/ π) = 91.049254 a pF (4) A closed (with a flat disk) hemisphere with radius a [2]: Cclosed hemisphere = ()3/ 1180−πε a = 94.052249 pF (5) Two spheres with radius a in contact [1]: Ctwo spheres = 8 πε0aLn(2) = 154.24505 a pF (6) An “oblate spheroid” is the figure generated by the rotation of an ellipse around its minor axis. A “pr olate spheroid” is generated by the rotation fo an ellips e around its major axis. The capacitances of these fi gures are, considering the major axis with length 2a and the minor axis with the length 2b [2]: ababaC 22 122 0 oblate sin 4 −−πε = − (7) bbaabaC 2222 0 prolate ln 4 −+−πε = (8) Note the limits when a = b reducing to (2), and the reduction to (3) when b = 0 in (7). For bodies embedded in materials with other permissivities, it’s just a question of multiplying ε0 by the relative permissivity ε of the material. The case when different dielectrics are present on the structure will be not discussed here. II. C APACITANCE OF A TOROID From [2] (the same formula appears in [3], that is probably the origin of this formula, but in a somew hat different notation) the capacitance of a toroid wit h major diameter D and minor diameter d , d<D/2, (fig. 1) is: aAxdadDAn nxPxQaA C nnnn n ==−=> = =σσ−ε=∑∞ =−− ;2 ;2; 0for 1 , 0 for 21;)()(16 02121 0 (9) where )(21xPn− and )(21xQn− are Legendre functions, or in this case, “toroidal functions”. 2 Created: 5/11/2003. Last update: 20/4/2005 D da A Fig. 1. Toroid with major diameter D, minor diameter d, center radius A, and tube radius a.. These functions can be evaluated in the following w ay: The first two terms can be obtained from their rela tions with the complete elliptic integrals of first and s econd kinds:   −π=π=−−== −− ''22)(; '2)(; 2)(;)( 21212121 kK kExPkK xPkK kEKxQkK xQ (10) The modulus for the elliptic integrals K and E is: aAak+=2 (11) And for the elliptic integrals K’ and E’ (evaluated in the same way, with modulus k’ ): 21'kk−= (12) This is enough for the evaluation of the two first terms of the series (enough for thin toroids). The other ter ms can be obtained using the recursion for Legendre functi ons, identical for both functions: () ( ) 21)(21)( 2)(;21)(21)( 2)( 121 21 21121 21 21 +−−=+−−= −− − +−− − + mxQmx mxQ xQmxPmxmxP xP m m mm m m (13) where m=n-1. All the terms can then be easily computed, starting with n=2 in the series (7), or m=1. The complete elliptic integrals are the irreducible functions: ( ) ( ) ϕϕ−==π=ϕ−ϕ==π= ∫∫ ππ dk k kEkdk kK sin 1 E) 2 /,E( sin 1F) 2 /,F( 2 0222 022 (14) They can be quickly and precisely evaluated using t he arithmetic-geometric mean method, below implemented in a Pascal routine: { Complete elliptic integrals of first and second classes - AGM method. Returns the global variables: Ek=E(c) and Fk=F(c) (E and K) Doesn’t require more than 7 iterations for c between 0 and 0.9999999999. Reference: Pi and the AGM, J. Borwein and P. Borwein, John Wiley & Sons. } procedure EF(c:real); var a,b,a1,b1,E,i:real; begin a:=1; b:=sqrt(1-sqr(c)); E:=1-sqr(c)/2; i:=1; repeat a1:=(a+b)/2; b1:=sqrt(a*b); E:=E-i*sqr((a-b)/2); i:=2*i; a:=a1; b:=b1; until abs(a-b)<1e-15; Fk:=pi/(2*a); Ek:=E*Fk end; III. A PPROXIMATE CALCULATIONS FOR PARTIAL TOROIDS A partial toroid can be described as a surface gene rated by the revolution of a partial circle of radius a centered at a distance A along the radial axis r from the revolution axis z. The circle limits are defined by two angles θ1 and θ2. See fig. 2. θ1θ2 Aa rz Fig. 2: A partial toroidal surface is generated by the rotation of a partial circle around the vertical ax is. 3 Created: 5/11/2003. Last update: 20/4/2005 With this formulation several figures can be genera ted, as a regular toroid when θ2-θ1 = 2 π and a<A, a sphere when A = 0, θ2 = −θ1 = π/2, an open hemisphere, etc. Even overlapping toroids, with a>A, can be generated. This surface can be decomposed in a set of n infinitely thin circles with axles at the z axis, positioned at heights zi, and with and radii ri , uniformly spaced at angles Δθ along the surface: i ii ii azaArni in θ=θ+==θΔ−+θΔ+θ=θθ−θ=θΔ sin cos ... 1 ,) 1(2112 (15) Each of these rings has an uniform charge distribut ion, with a total charge qi. The potential Ψ due to each ring i at any given position r0,z0 is given by: 2 02 0 110102 22 / 1 00200 )()(2212 4),( i iii ii i zzrrRRrr kKRq kQ rrqzr −++==επ=  − επ= Ψ− (16) The absolute values allow correct treatment of the cases when some radii are negative. Considering then the mutual influences among all th e rings, a matrix P can be computed, that allows the calculation of the potentials vi at each ring, once the charges qi are known [1]: i iii ii ijji ji ij qRzrPqzrPP / ),(/ ),( +Ψ=Ψ===Pq v (17) For the calculation of the “self-potentials” Pii , something must be assumed about the radius of the rings, R. The formulation calculates then the potentials at a dis tance R above the rings. The maximum physically possible va lue of R would be when adjacent rings touch: 2sin max θΔ=aR (18) Any reasonable fraction of this value can be used w ith similar results, but there is one that produces bet ter results in the next calculation, that was found (by trying!) to, curiously, be: 2sin θΔ π=aR (19) This radius makes the area of the surface of the ri ng to be identical to the flat area represented by it, at least in the cases when θ = nπ/2 and small Δθ (as in the equator and poles of a sphere split in many rings). The charge distribution for uniform potential V at all the rings can be calculated by inverting the matrix P. The total charge in each ring is then obtained from a s um of the corresponding lines of the inverse of P, C. The coefficients of C are the influence coefficients kij : 11 −= ==∑ PCn jij ikVq (20) And the capacitance of the whole assembly is simply the sum of all the elements of C: ∑∑ = ==n in jij total k C 1 1 (21) Surface electric field The electric field at any point of the surface is n ormal to it and can be calculated by Gauss’ law as proportio nal to the charge density at that point of the surface: 0ερ=i iE (22) where ρi is the surface charge density, uniform around the ring i. For a closed surface, the electric field is entirely at the outer surface. In this case, it can be calculated directly from the charge distribution al one. Assuming constant voltage at the surface of the obj ect, the charges at the rings can be calculated by (20). The ring i has a length 2 πri and a total charge qi. The ring represents a thin belt with width equal to aΔθ. The charge density and the electric field in a small le ngth l are then: 0221 2 θε Δπ=θΔπ=θΔπ=ρ arqEarq la lrq ii iii ii i (23) An important application of this calculation is the determination of the breakout voltage of the object , the voltage that causes ionization of the air around it when the electric field reaches about 3 MV/m: kV /Max 3000 max iE V= (24) For a toroid, this value occurs at the maximum diam eter. In the case of open objects, it’s not possible to c alculate the surface electric field in this way, because it is split in an unknown way between the two sides of the surface . The calculation is also meaningless if the object h as an edge, where the electric field is ideally infinite. A strange problem with (23) is that it fails when the rings are close to the center of a spherical surface. The last 4 Created: 5/11/2003. Last update: 20/4/2005 ring appears to have significantly less charge than it should have (around 92%). The calculations for capacitance, however, continue to result in good va lues. IV. G ENERAL TRUNCATED CONES Any other figure with circular symmetry can be anal yzed by the same method. A simple case is the revolution of a straight line around the central axle, that generat es figures ranging from a flat disk with a possible ce ntral hole to a cone or an open cylinder. r rz 1r2h Fig. 3. A line that rotates around the vertical axi s. The coordinates or the rings are then: nirnrrrnizizznrrrnhz ii ... 1 ,) 1(2... 1 ,) 1(2 112 =Δ−+Δ+==Δ−+Δ=−=Δ=Δ (25) The radius to use in the calculation of the self-po tentials would be, still using the maximum divided by π: ( ) π+−=nhrrR222 12 (26) With this radius, the surface charge density and th e surface electric field (for a closed object) can be calculated considering that the surface area of the ring is identical to the belt area represented by it, what is approximately valid also for the case or curves, us ing R given by (19): 022 4421 2 επ=π=ππ=ρ RrqERrq Rllrq ii iii ii i (27) V. E LECTRIC FIELD FROM A RING The electric field anywhere can be calculated by ad ding the electric fields due to the rings. From (16), th e radial and axial components of the electric field can be calculated by differentiation, resulting in:        −−−++−− επ−==Ψ−= )(2 '')( 2sign 00 220 3 1020 rrkkr kk KkEKrr Rrqdr dE ii i ii radial (28) 23 1020 0 ' 2)( kREzzq dz dEiii axial επ−=Ψ−= (29) 2 2 axial radial total EEE + = (30) where the derivative of the elliptic integral K in relation to the modulus k was used (the derivative of E is listed below too for reference, but was not necessary): 2 222 1' ;'' kkkKE dk dE kk KkE dk dK −=−=−= (31) VI. GENERAL CASE WITH AXIAL SYMMETRY The capacitance matrix and the potential and electr ic field around a series of objects with axial symmetr y decomposed in thin rings can then be easily calcula ted. The objects are decomposed in series of partial tor oids conical sheets, and other shapes (as ellipses) and these parts are decomposed in rings. To obtain the capaci tance matrix, it’s just a question of adding the terms of the total capacitance matrix that correspond to the rin gs that belong to the objects, instead of adding them all t o obtain the capacitance of the entire object. The ch arges in all the rings can be obtained from the complete equation q=CV , with the assigned voltages in the objects arranged in V in correspondence with the rings that belong to the objects. The potential anywhere aroun d the objects is obtained by adding (16) for all the ring s, and the electric field by adding (28) and (29) and usin g (30). The terms at the diagonal of the capacitance matrix correspond to the capacitances of the objects to gr ound when all the other objects are grounded too. The influence coefficients out of the diagonal measure the relation between the charge induced in one object a nd the voltage in another, when all the other objects are grounded. From the capacitance matrix, a model of t he circuit using lumped capacitors can be derived, by observing the equivalence: 5 Created: 5/11/2003. Last update: 20/4/2005     +++ − −− +++ −− − +++ ==     = − n nn n n nn nn nnn nnnn C CC C CC C CC CC C C CCk kkk kkk kk 1 1 2 12 2 12 2 12 1 12 1 12 1212 22 12 1 12 11 ... ... ... LM O M MLLLMOMMLL C (32) C1, C2, …, Cn are direct capacitances between the elements and the ground, and the other elements are the negative of the floating capacitances between the objects. The direct capacitance to ground for the o bject i is just the sum of the elements in the line, or col umn, i of C. VII. M AXIMUM ELECTRIC FIELD BETWEEN TWO SPHERES A good test for these field calculations is the kno wn formula for the maximum electric field between two different spheres [4]. The expression comes directl y from the method of images developed by Lord Kelvin. For two spheres of radii a and b, a<b, with distanc e between centers c, at potentials v1 and v2, the maximum electric field at the surface of the smaller sphere (assumed as being where the surface of the smaller sphere intercepts the line between the centers of t he spheres) is given by: ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )( ) α=+α+α<=αα+=ηα+=ξ      + ηα +ηα −ηα+ ηα +ηα −αη + ηα +ηα −η−  + ξα +ξα −α+ ξα +ξα −α+ ξ+ξ− ξ−ξ+== 2255 2 233 2 2244 2 222 2 12max of 1root ; ;; 11 11 1111 11 11 11 cabbacab cbavv aE KL (33) This formula converges slowly when the spheres are at small distance, but the speed is acceptable. [4] de velops a better expression for the case of spheres at smal l distance too. VIII. CAPACITANCES OF TWO SPHERES Similar formulas, due to Kirchhoff, lead to the capacitance matrix of two spheres [5]. For two sphe res with radii a and b and distance between centers c: ( )( )( )( ) cbacbacbacbacbac aakkk 2; ; ; 1;1118; 1 118;1 118 2263 42 2 0 12 242 22 2 0 22 242 22 2 0 11 +−−+−−++==−=−+= ++++++−= + ++ ++ += ++++++= λξαηξαλλξαα αα ααλπε ηαηα ηααη ηηλπε ξαξα ξααξ ξξλπε LLL (34) These formulas also converge slowly when the sphere s are at small distance. [5] shows a better formula f or small distances. The coefficients of the capacitance matrix represen t the ratio between the induced charges and the voltages. k11 and k12 represent capacitances to ground from a sphere with the other sphere grounded, and - k12 is the floating capacitance between the spheres. The differential capacitance between the spheres is obtained by assu ming opposite charges ±q on them: 12 22 11 2 12 22 11 21 2kkkkkk qvvCdiff ++−=−= (35) The capacitances to ground with the other sphere floating can also be calculated, by assuming zero c harge in the floating sphere: 11 2 12 22 11 20 22 2 12 22 11 10 ;kkkkCkkkkC−=−= (36) IX. P OTENTIAL AND ELECTRIC FIELD AROUND A TOROID The solution of this problem can be traced to [3]. The formula for the potential also appears in [2]. The potential around an isolated toroid in free space, with central radius A and tube radius a , at a radial distance r and axial distance z from the center, is found as: ( ) ( ) ( ) ;2tan cos sin tan sinh cosh cos ; sinh sin ;)()(ln 21; ;; 0for 1 , 0 for 21; cos cosh )()( cos cosh 22, 2221 12 22 2222 / 1 02121 czrcz rc rzcrzcrzaAcaAxn nn PxPxQ V nn nnn n −+=αα=αβ−β=αβ=α−+++=β−==> = =σαβ σα−βπ=βαΨ − −−∞ =−−∑ (37) The surface electric field can be found by the differentiation of (37). The maximum occurs when co sh 6 Created: 5/11/2003. Last update: 20/4/2005 β = x = A/a (toroid surface) and α=0 (major diameter). The result, hinted in [3] but not developed, is the series: ( ) ( )∑∞ =−ασ−πα−= 02122 / 3 )( cos 1 cos 24 nnnxPn xdxVE (38) The ideal exact breakdown voltage can then be obtai ned as in (24). This series converges somewhat more slo wly than (9) but still can achieve high precision. The series (37) may lose precision due to errors in the evalua tion of Qn+1/2 (x) by the recursion (13). X. E XAMPLES Some toroids analyzed by the methods above. Vmax was obtained from (38) and (24), except for the “holele ss” toroid, where (23) and (24) were used. All the capacitances (in this and the other examples) in pF : D x d Cexact 20 rings 200 rings Vmax (kV) 0.2x0.1 9.6877342 9.6862459 9.6877328 226.2 0.3x0.1 13.527991 13.526517 13.527990 282.9485 0.4x0.1 17.200315 17.198812 17.200313 328.9148 0.5x0.1 20.738038 20.736480 20.738037 367.4999 Open hemispheres ( D = diameter): D Cexact 20 rings 200 rings 0.2 9.1049254 9.0451871 9.0989244 0.3 13.657388 13.567781 13.648387 0.4 18.209851 18.090374 18.197849 0.5 22.762314 22.612968 22.747311 Flat disks ( D = diameter): D Cexact 20 rings 200 rings 0.2 7.0833502 7.0067052 7.0757027 0.3 10.625025 10.510058 10.613554 0.4 14.166701 14.013411 14.151405 0.5 17.708376 17.516763 17.689257 Hollow cylinders ( D = diameter, h = height): D h 20 rings 200 rings 0.2 1 27.2508153 27.5562772 0.3 1 32.7125753 33.0502066 0.4 1 37.6883716 38.0515957 0.5 1 42.3659124 42.7508210 Hollow cones ( D = diameter, h = height): D h 20 rings 200 rings 0.2 1 20.6332474 20.8219907 0.3 1 45.0755428 24.5554255 0.4 1 27.7305014 28.0027331 0.5 1 30.9951485 31.3004301 In the last two cases no explicit formulas were fou nd in the literature, although very probably they are kno wn. The general algorithm for objects with axial symmet ry was implemented in the Inca program and used to generate the next examples: A closed hemisphere can be generated by the combination of an open hemisphere and a flat disk ( half of the rings for each element, D = diameter): D Cexact 20 rings 200 rings 0.2 9.4052249 9.3751321 9.4038325 0.3 14.1078374 14.0626982 14.1057488 0.4 18.8104499 18.7502642 18.8076651 0.5 23.5130623 23.4378303 23.5095813 Two spheres in contact ( D = diameter, half of the rings for each sphere): D Cexact 20 rings 200 rings 0.1 7.7123025 7.7105894 7.7123007 0.2 15.4246050 15.4211788 15.4246014 0.3 23.1369075 23.1317682 23.1369021 0.4 30.8492100 30.8423576 30.8492028 0.5 38.5615125 38.5529470 38.5615035 A toroid with the central hole closed by a thin dis k. Note the small difference to a regular toroid. A toroid where the closure of the central hole doubles the capacit ance would have an aspect ratio of about 1x0.0004. Half of the rings for each element, D = major diameter, d = diameter of the tube: D x d 20 rings 200 rings 400 rings 0.3x0.1 13.5176679 13.5296046 13.5296149 0.4x0.1 17.2225678 17.2348074 17.2348180 0.5x0.1 20.8623320 20.8748605 20.8748714 Maximum electric field between spheres with opposit e voltages. Half of the rings to each sphere. Dimensi ons as in (33). Fields in V/m/V: a, b, c Exact 40 rings 400 rings 0.1, 0.1, 0.5 14.7654541 14.654655 14.762658 0.1, 0.2, 0.5 20.7165237 20.434842 20.711307 0.1, 0.3, 0.5 32.2318226 31.394734 32.219293 Capacitance matrix for two spheres. Half of the rin gs to each sphere. Dimensions as in (33): k11 (radius a) a, b, c Exact 40 rings 400 rings 0.1, 0.1, 0.5 11.6112177 11.6108704 11.6112174 0.1, 0.2, 0.5 12.3051750 12.3047650 12.3051745 0.1, 0.3, 0.5 13.7605384 13.7603742 13.7605373 k22 (radius b) a, b, c Exact (pF) 40 rings 400 rings 0.1, 0.1, 0.5 11.6112177 11.6108704 11.6112174 0.1, 0.2, 0.5 24.3154312 24.3146700 24.3154303 0.1, 0.3, 0.5 38.6334041 38.6326963 38.6334025 7 Created: 5/11/2003. Last update: 20/4/2005 k12 a, b, c Exact (pF) 40 rings 400 rings 0.1, 0.1, 0.5 -2.3264588 -2.3263316 -2.3264587 0.1, 0.2, 0.5 -4.9456676 -4.9454137 -4.9456673 0.1, 0.3, 0.5 -8.3626059 -8.3626805 -8.3626051 The problem with this approach is that as the numbe r of rings increases it becomes more and more difficult to invert the matrix P with precision and in reasonable time. In the next page is a table of exact toroid capacit ances calculated by (9). Note that it would be enough to have a single column with normalized aspect ratios, since for a fixed aspect ratio the capacitance is directly prop ortional to the major (or minor) diameter. Acknowledgments : Thanks to Paul Nicholson for discussions and verifications, and to Godfrey Loudn er for several papers and the derivation of (38). This document is not a published paper. REFERENCES [1] James Clerk Maxwell, “A Treatise on Electricity and Magnetism, Dover Publications Inc, New York, 1954 (reprint from the original from 1873). [2] Chester Snow, “Formulas for Computing Capacitance and Inductance,” National Bureau of Standards Circular #544. [3] W. M. Hicks, “On Toroidal Functions,” Proceedin gs of the Royal Society of London, 1, 31, March 1881. [4] Alexander Russel, “The Maximum Value of the Electrical Stress between Two Unequal Spherical Electrodes,” Proceedings of the Physical Society of London, November 1911, pp. 22-29. [5] Alexander Russel, “The Capacity Coefficients of Spherical Electrodes,” Proceedings of the Physical Society of London, June 1911, pp. 352-360. 8 Created: 5/11/2003. Last update: 20/4/2005 Exact toroid capacitances (diameters in meters, cap acitances in pF) Minor d. 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.1 20 0.130 0.140 0.150 0.160 0.170 0.180 Major d. 0.100 3.707 4.148 4.431 4.653 ------- -- ----- ------- ------- ------- ------- ------- ----- -- ------- ------- ------- ------- ------- ------- 0.125 4.468 5.001 5.340 5.598 5.816 6.010 ------- ------- ------- ------- ------- ----- -- ------- ------- ------- ------- ------- ------- 0.150 5.205 5.827 6.221 6.518 6.764 6.979 7.174 ------- ------- ------- ------- ----- -- ------- ------- ------- ------- ------- ------- 0.175 5.923 6.630 7.080 7.417 7.691 7.929 8.143 8.339 ------- ------- ------- ----- -- ------- ------- ------- ------- ------- ------- 0.200 6.625 7.414 7.919 8.295 8.600 8.861 9.094 9.306 9.503 ------- ------- ----- -- ------- ------- ------- ------- ------- ------- 0.225 7.313 8.182 8.740 9.156 9.492 9.778 10.029 10.258 10.469 10.667 10.854 ----- -- ------- ------- ------- ------- ------- ------- 0.250 7.990 8.937 9.547 10.002 10.369 1 0.679 10.951 11.196 11.422 11.632 11.831 12.0 19 ------- ------- ------- ------- ------- ------- 0.275 8.657 9.679 10.340 10.834 11.232 1 1.567 11.860 12.122 12.362 12.586 12.796 12.9 95 13.184 ------- ------- ------- ------- ------- 0.300 9.316 10.411 11.121 11.654 12.082 1 2.443 12.757 13.037 13.292 13.528 13.749 13.9 59 14.158 14.349 ------- ------- ------- ------- 0.325 9.966 11.133 11.892 12.462 12.921 1 3.307 13.642 13.940 14.211 14.460 14.693 14.9 13 15.122 15.322 15.513 15.698 ------- ------- 0.350 10.609 11.846 12.653 13.260 13.749 1 4.160 14.517 14.833 15.119 15.383 15.628 15.8 58 16.077 16.285 16.485 16.678 16.863 ------- 0.375 11.246 12.551 13.405 14.048 14.567 1 5.003 15.381 15.716 16.019 16.296 16.553 16.7 94 17.023 17.240 17.449 17.649 17.842 18.029 0.400 11.876 13.250 14.149 14.828 15.376 1 5.838 16.237 16.590 16.909 17.200 17.470 17.7 22 17.961 18.187 18.404 18.612 18.812 19.006 0.425 12.502 13.941 14.886 15.600 16.177 1 6.663 17.084 17.456 17.791 18.096 18.379 18.6 43 18.891 19.126 19.351 19.567 19.775 19.976 0.450 13.122 14.627 15.616 16.365 16.970 1 7.481 17.922 18.313 18.664 18.984 19.280 19.5 55 19.814 20.059 20.292 20.516 20.731 20.938 0.475 13.737 15.306 16.340 17.122 17.756 1 8.291 18.753 19.162 19.530 19.865 20.173 20.4 60 20.729 20.984 21.226 21.457 21.680 21.894 0.500 14.348 15.981 17.057 17.874 18.535 1 9.093 19.577 20.005 20.389 20.738 21.060 21.3 58 21.638 21.902 22.153 22.392 22.622 22.844 0.525 14.955 16.650 17.769 18.619 19.308 1 9.890 20.394 20.840 21.240 21.604 21.939 22.2 50 22.540 22.814 23.074 23.322 23.559 23.787 0.550 15.558 17.315 18.476 19.358 20.075 2 0.680 21.204 21.668 22.085 22.464 22.812 23.1 35 23.436 23.720 23.989 24.245 24.489 24.725 0.575 16.157 17.975 19.177 20.092 20.836 2 1.464 22.009 22.491 22.924 23.317 23.678 24.0 13 24.326 24.619 24.897 25.162 25.414 25.657 0.600 16.753 18.631 19.874 20.821 21.591 2 2.242 22.807 23.307 23.756 24.164 24.539 24.8 86 25.209 25.513 25.800 26.073 26.334 26.584 0.625 17.346 19.284 20.567 21.546 22.342 2 3.016 23.601 24.118 24.583 25.006 25.393 25.7 52 26.087 26.401 26.698 26.979 27.248 27.505 0.650 17.935 19.932 21.256 22.265 23.088 2 3.784 24.388 24.924 25.405 25.842 26.243 26.6 14 26.959 27.284 27.590 27.880 28.157 28.421 0.675 18.522 20.577 21.940 22.981 23.829 2 4.547 25.171 25.724 26.221 26.672 27.086 27.4 69 27.826 28.161 28.477 28.776 29.060 29.333 0.700 19.105 21.218 22.621 23.692 24.566 2 5.306 25.949 26.520 27.032 27.498 27.925 28.3 20 28.688 29.033 29.358 29.666 29.959 30.239 0.725 19.686 21.856 23.298 24.399 25.298 2 6.060 26.723 27.310 27.838 28.318 28.759 29.1 66 29.545 29.900 30.235 30.552 30.853 31.141 0.750 20.265 22.492 23.971 25.103 26.027 2 6.810 27.492 28.097 28.640 29.134 29.588 30.0 07 30.397 30.763 31.107 31.433 31.742 32.037 0.775 20.841 23.124 24.641 25.803 26.751 2 7.556 28.257 28.879 29.438 29.946 30.412 30.8 43 31.245 31.621 31.974 32.309 32.627 32.930 0.800 21.414 23.753 25.308 26.499 27.472 2 8.299 29.018 29.657 30.231 30.753 31.232 31.6 75 32.088 32.474 32.837 33.181 33.507 33.818 0.825 21.985 24.379 25.972 27.192 28.190 2 9.037 29.775 30.430 31.020 31.556 32.048 32.5 03 32.926 33.323 33.696 34.048 34.383 34.702 0.850 22.554 25.003 26.633 27.882 28.904 2 9.772 30.529 31.200 31.805 32.355 32.859 33.3 26 33.761 34.167 34.550 34.912 35.255 35.581 0.875 23.121 25.625 27.292 28.569 29.615 3 0.504 31.279 31.967 32.586 33.150 33.667 34.1 46 34.591 35.008 35.400 35.771 36.122 36.457 0.900 23.686 26.244 27.947 29.253 30.323 3 1.232 32.025 32.729 33.364 33.941 34.471 34.9 61 35.417 35.845 36.246 36.626 36.986 37.328 0.925 24.249 26.860 28.600 29.934 31.027 3 1.957 32.768 33.489 34.138 34.728 35.271 35.7 73 36.240 36.677 37.089 37.477 37.846 38.196 0.950 24.810 27.474 29.250 30.613 31.729 3 2.679 33.508 34.245 34.908 35.513 36.068 36.5 81 37.059 37.507 37.927 38.325 38.702 39.060 0.975 25.369 28.086 29.898 31.288 32.428 3 3.398 34.245 34.997 35.676 36.293 36.861 37.3 86 37.874 38.332 38.762 39.169 39.554 39.920 1.000 25.927 28.696 30.543 31.962 33.124 3 4.114 34.979 35.747 36.440 37.071 37.650 38.1 87 38.686 39.154 39.594 40.009 40.403 40.777 1.025 26.482 29.304 31.186 32.632 33.818 3 4.827 35.709 36.494 37.201 37.845 38.437 38.9 85 39.495 39.973 40.422 40.846 41.248 41.630 1.050 27.036 29.910 31.827 33.300 34.509 3 5.538 36.437 37.237 37.959 38.616 39.220 39.7 79 40.300 40.788 41.246 41.680 42.090 42.480 1.075 27.588 30.514 32.466 33.966 35.197 3 6.246 37.163 37.978 38.714 39.384 40.000 40.5 71 41.102 41.600 42.068 42.510 42.929 43.327 1.100 28.139 31.116 33.103 34.630 35.883 3 6.951 37.885 38.716 39.466 40.149 40.778 41.3 60 41.901 42.409 42.886 43.337 43.764 44.170 1.125 28.688 31.716 33.737 35.291 36.567 3 7.654 38.605 39.452 40.215 40.912 41.552 42.1 45 42.698 43.215 43.701 44.161 44.596 45.010 1.150 29.236 32.315 34.370 35.950 37.248 3 8.355 39.323 40.185 40.962 41.671 42.324 42.9 28 43.491 44.018 44.513 44.982 45.425 45.847 1.175 29.782 32.911 35.001 36.608 37.927 3 9.053 40.038 40.915 41.706 42.428 43.092 43.7 08 44.281 44.818 45.323 45.800 46.252 46.681 1.200 30.327 33.506 35.629 37.263 38.604 3 9.749 40.751 41.643 42.448 43.183 43.859 44.4 85 45.068 45.615 46.129 46.615 47.075 47.513 1.225 30.871 34.100 36.256 37.916 39.279 4 0.443 41.461 42.368 43.187 43.935 44.622 45.2 59 45.853 46.409 46.933 47.427 47.895 48.341 1.250 31.413 34.692 36.882 38.567 39.952 4 1.134 42.169 43.091 43.924 44.684 45.383 46.0 31 46.635 47.201 47.734 48.237 48.713 49.166 1.275 31.953 35.282 37.505 39.216 40.623 4 1.824 42.875 43.812 44.658 45.431 46.142 46.8 01 47.415 47.990 48.532 49.043 49.528 49.989 1.300 32.493 35.871 38.127 39.864 41.292 4 2.511 43.579 44.531 45.390 46.175 46.898 47.5 68 48.192 48.777 49.327 49.848 50.341 50.809 1.325 33.031 36.458 38.747 40.510 41.959 4 3.197 44.281 45.247 46.120 46.918 47.652 48.3 32 48.966 49.561 50.121 50.649 51.150 51.627 1.350 33.568 37.044 39.365 41.154 42.624 4 3.880 44.980 45.961 46.848 47.658 48.403 49.0 94 49.739 50.343 50.911 51.448 51.957 52.442 1.375 34.104 37.628 39.982 41.796 43.287 4 4.562 45.678 46.674 47.574 48.395 49.152 49.8 54 50.509 51.122 51.699 52.245 52.762 53.254 1.400 34.639 38.211 40.598 42.436 43.949 4 5.241 46.374 47.384 48.297 49.131 49.899 50.6 12 51.276 51.899 52.485 53.039 53.565 54.064 1.425 35.172 38.792 41.212 43.075 44.609 4 5.919 47.068 48.092 49.019 49.865 50.644 51.3 67 52.041 52.674 53.269 53.831 54.364 54.872 1.450 35.705 39.373 41.824 43.713 45.267 4 6.595 47.760 48.799 49.738 50.596 51.387 52.1 20 52.805 53.446 54.050 54.621 55.162 55.677 1.475 36.236 39.952 42.435 44.349 45.923 4 7.270 48.450 49.503 50.456 51.326 52.128 52.8 72 53.566 54.216 54.829 55.408 55.957 56.480 1.500 36.766 40.530 43.045 44.983 46.578 4 7.942 49.138 50.206 51.171 52.053 52.866 53.6 21 54.324 54.984 55.606 56.193 56.751 57.280