Inductance Calculation Techniques --- Part II: Approximations and Handbook Methods

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1 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Marc T. Thmpsn, Ph.D. 1 INDEX OF SYMBOLS A Area enclse by cil a Cil mean raius b Cil axial thickness c Cil raial thickness D Sie length f square cil Wire-t-wire spacing K Nagaka cil cnstant l Cil length p Perimeter enclse by cil P, F Grver isk cil cnstants R Circular wire raius s Sie length f generic plygn w Trace with x, y Mean sie lengths f rectangular cil µ Magnetic permeability f free space 4π 10-7 H/m I. INTRODUCTION In the secn part f this tw part series n inuctance calculatin techniques, apprximatin techniques an hanbk meths are shwn fr air-cre structures that nt easily len themselves t clse-frm slutins. A set f references is als given which is useful fr fining the inuctance f many ifferent lp shapes. Inclue are inuctance calculatins fr plygns, isk cils, finite-length slenis an flat planar spirals. In sme cases results are given withut significant explanatin, as the calculatins are very cmplicate an the full calculatins may be fun in the references given. Many f the ler references wrk ut inuctance prblems in English units; t ease esign these results have been cnverte t MKS units with inuctance in Henries an length scales in meters. II. REFERENCE REVIEW Inuctance calculatin references necessarily start with Maxwell's seminal wrk [1], first publishe in Maxwell wrke ut sme interesting inuctance prblems, incluing fining the mutual inuctance between circular caxial filaments [1, pp. 339], an fining the size an shape f a cil which maximizes inuctance fr a given length f wire [1, pp. 345]. 1 The authr is an inepenent cnsultant at 5 Cmmnwealth Ra, Watertwn Massachusetts, USA 047. Business phne: (617) Fax: (617) Website: [email protected] an is Ajunct Assciate Prfessr f Electrical Engineering, Wrcester Plytechnic Institute, Wrcester, MA Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 1 f 11

2 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website A seminal reference fr inuctance calculatin (which, unfrtunately is nw ut f print) is the Freerick Grver bk []. Prfessr Grver spent mst f his prfessinal life calculating inuctance f ifferent kins f wire lps, an many useful examples an tables are given fr lps f interesting shapes an sizes. The reference is especially useful as it gives crrectin factrs t accunt fr high frequency peratin an ther effects such as the insulatin thickness f wires. He was als a cnsultant fr the U.S. Bureau f Stanars, an the results f sme f his wrk are fun in [3]. The Wheeler references [4,5] give simple results fr circular cils, an are particularly useful where analytic expressins are neee. The empirically-erive analytic expressins in [5] are remarkably accurate fr multiple-turn, circular cils. Anther g reference is the Rai Engineers' Hanbk, eite by Terman [6]. This reference shws the inuctance f many ifferent cil shapes such as single-turn lps, rectangles, multiple-layer cils, an slenis f varius lengths. The Dwight reference [7] is very useful fr calculating high-frequency effects in inuctrs, an cvers skin an prximity effects. The Rters reference [8] shws a useful meth by which leakage inuctance in transfrmers can be estimate. The Smythe reference [9] is a great wrk cvering classical fiel an inuctance calculatins. Sleni Magnet Design [10] by D. Bruce Mntgmery, cvers esign aspects f sleni-shape cils frm a supercnucting magnet pint f view. Thermal aspects f inuctr esign are well cvere here. Other references, as neee, are given in the text. A mre cmprehensive bibligraphy f inuctance calculatin references is given n the authr's business website, given in the REFERENCES sectin. III. APPROXIMATE AND CLOSED-FORM RESULTS FOR COMMON STRUCTURES 1. Circular Wire Lp The self-inuctance f a straight cnuctr f length l an raius R, neglecting the effects f nearby cnuctrs (i.e. assuming that the return current is far away) is given by [, pp. 35]: µ l l 3 [1] L ln π R 4 Many ther magnetic structures can be mele as simpler structure, since inuctance is in general a weak functin f lp shape fr lps with lp raius much larger than wire crss sectin. Surprisingly, there is n clse-frm slutin fr the inuctance f a filamentary lp (since the expressin fr inuctance blws up if the wire raius ges t zer). A circular lp f run wire (Figure 1) with lp raius a an wire raius R has the apprximate lw frequency inuctance [, pp. 143], [4]: 8a [] L = µ aln 175. R Using this frmula, the inuctance f a 1 meter circumference lp f 14 gauge wire is 1.1 µh; fr 16 gauge wire it's 1.17 µh; an fr 18 gauge wire it's 1.1 µh. Nte the weak epenence f inuctance n wire iameter, ue t the natural lg in the expressin. Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page f 11

3 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website A crue apprximatin fr circular lps is L µ πa [11, pp. 56] which preicts an inuctance f 0.63 µh fr a 1 meter circumference wire lp. a R Figure 1. Circular wire lp. Parallel-wire line Fr tw parallel wires whse length l is great cmpare t their istance apart, (Figure ) the inuctance f the lp is [, pp. 39]: l L = + µπ ln 1 R 4 l [3] Fr l = 0.5 meter an a wire-wire spacing = 1 cm, results are: L = µh fr 14 gauge; L = µh fr 16 gauge an L = µh fr 18 gauge. Therefre, fr the parallel-wire line with clsely-space cnuctrs, the inuctance is apprximately 0.5 micrhenries per meter f ttal wire length. R Figure. Parallel-wire line 3. Square lp The self inuctance f a square cil mae f rectangular wire (Figure 3), with epth b (int the paper) small cmpare t sie length D an trace with w is a cmplicate expressin fun in the Zahn reference [1, pp. 343]. Hwever, fr w << D a relatively nasty expressin can be apprximate by: L µ D 1 D sinh ( ) 1 π w [4] Fr instance, a square PC bar trace 1cm 1cm with trace with 1 millimeter has an inuctance f apprximately 16 nanhenries (assuming that the grun plane is far away, f curse). Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 3 f 11

4 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website w D D Figure 3. Square cil with rectangular crss sectin 4. Rectangle f run wire The inuctance f a rectangle f run wire with rectangle sie lengths x an y is [, pp. 60]: µ x y x y [5] L = xln + y ln + x + y xsinh 1 y sinh ( x+ y) R R y x π 5. Plygn f run wire These results suggest an interesting result fr a plygn f wire. The inuctance f a generic plygn f wire with perimeter p an area A may be apprximate by: p p p L + µ R [6] ln 05. ln π A Nte that this functin is strngly epenent n the perimeter an weakly epenent n lp area an wire raius. Fr this reasn, the inuctance f cmplicate shapes can ften be well apprximate by a simpler shape with the same perimeter an/r area. N clse-frm exists t calculate the inuctance f a generic plygn f wire. Grver [, pp. 60] has wrke ut a variety f cases fr plygns with sie length s an wire raius R: Equilateral Triangle Square Pentagn Hexagn Octagn 3µ s s L = ln π R µ s s L = ln π R 5µ s s L = ln π R 3µ s s L = ln π R 4µ s s L = ln π R [7] [8] [9] [10] [11] Hwever, it is fun in practice [, pp. 61], [15] that the inuctance f a multi-face plygn may be apprximate by replacing a plygn by a simpler plane figure f either Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 4 f 11

5 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website equal area f equal perimeter. If the perimeter f the cil is p an the area enclse is A then the inuctance takes the general frm: p p p L + µ R ln 05. ln π A [1] This result is apprximate, an an empirical fit t the clse-frm calculatins fr plygns abve. In general, cils enclsing the same perimeter with similar shapes will have apprximately the same inuctance. This is a very useful result fr strange-shape cils. 6. Use f filaments Cnuctrs can be replace by filaments in rer t calculate inuctances, ften with very accurate results. Fr straight filaments mae f parallel cnuctrs, with length l an filament-filament spacing (Figure 4a), the mutual inuctance is [, pp. 31]: µ l M = l ln + π l l + l [13] z a 1 a h l r (a) (b) Figure 4. Mutual inuctance between filaments. (a) Straight cnuctrs (b) Lps Fr the caxial lps, the mutual inuctance between lps as fun by Maxwell is: M k 4a = ( ) ( ) 1a µ a1a 1 K k E, k = k ( a1 + a) + h 1 k where K(k) an E(k) are elliptic integrals. Mutual inuctance is a very imprtant parameter t calculate, as if the mutual inuctance M 1 is fun the frce between lps can be fun as f = I I [15] 1 1 M 1 IV. HANDBOOK METHODS 1. Disk cil A useful gemetry fr which tabulate results exist is the run lp with rectangular crss sectin, with mean raius a, axial thickness b, an trace with c (Figure 5). The self-inuctance f this single lp is calculate using techniques utline in Grver [, pp. 94], where the inuctance is shwn t be: [14] Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 5 f 11

6 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website L = µ [16] apf 4π This result is in MKS units, with a in meters an L in Henries. P an F are unitless cnstants; P is a functin f the cil nrmalize raial thickness c/a (Figure 5c) an applies t a cil f zer axial thickness (b = 0), an F accunts fr the finite axial length f the cil. Fr b << c an c <<a (cils resembling thin isks) the factr F 1, an imprtant limiting case. Therefre, fr a thin isk cil with uble the mean raius, there is a crrespning ubling f the inuctance. If the cil is mae f multiple turns f wire, an if c << a the inuctance can be apprximate by multiplying the abve expressin by N. a a a 1 a c c b (a) Crss sectinal view (b) Tp view (c) Functin P, fr isk cils Figure 5. Circular cil with rectangular crss sectin Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 6 f 11

7 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website An interesting result pps ut here... magnetic scaling laws [13] shw that large magnetic elements are mre efficient in energy cnversin than smaller nes. Fr this isk cil gemetry, this effect can be quantifie by cnsiering the rati f inuctance t resistance. The inuctance L is apprximately prprtinal t a as shwn abve. The resistance f the cil is prprtinal t a/bc, the rati f current path length t cil crsssectinal area. Therefre, the rati f inuctance t resistance is prprtinal t bc, r the crss-sectinal area f the cil. If all cil lengths are scale up by the same factr l, this rati increases by the factr l, r the length square.. "Brks" Cil An interesting prblem is t maximize the inuctance with a given length f wire. Maxwell [1, pp. 345] fun that the ptimal cil has a square crss sectin with mean iameter 3.7 times the imensin f the square crss sectin, r a = 3.7c. Brks an thers [14] later refine this estimate an recmmen a/c = 3 as the ptimum shape, with b = c. The result fr the Brks cil is: L = µ an [17] The inuctance is a rather weak functin f a/c s the exact gemetry isn't s imprtant. 3. Finite-length sleni The inuctance f a thin-wall finite-length sleni f raius a an length l mae f run wire can be calculate with g accuracy by using ne f the Wheeler frmulas [4, 11]. If the length l f the sleni is larger 0.8a, the accuracy is better than 1%: 10πµ N a [18] L 9a + 10l Fr the shrt sleni with l < a, a hanbk meth may be applie by using the Nagaka frmula [, pp. 143], [6, pp. 53] where (in MKS units): L = πµ a ln K [19] K is a cnstant (Figure 6) epening n the rati f the iameter t the length f the cil. This calculatin shws that a 1 meter lng cil, with raius 1 meter an a single turn has an inuctance f.075 micrhenries, which is in g agreement with the Wheeler frmula abve. Fr very shrt cils interplatin f the factr K frm the graph becmes ifficult an a series frmula is available, in Grver [, pp. 143]. Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 7 f 11

8 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website Figure 6. Crrectin factr fr finite-length sleni 4. Run Planar Spirals Planar spiral cils have increasing applicatin in miniature pwer electrnics an in PC-bar RF inuctrs. A number f meths are available fr the calculatin f the inuctance f a run spiral cil. Using the Grver meth [, pp. 110] we fin L = 5 µ apn π [0] where a is the mean cil iameter in meters an P is the factr epening n c/a, as state befre. This equatin is applicable if the inner an uter raii f the cil are nt t ifferent. Fr a circular cil with uter raius R an number f turns N, Schieber [16] calculates: L = µπ R N [1] where R is in meters an L is in Henries. This equatin is suitable if winings are use ver the entire area. Wheeler gives [5]: L = µ N a 8a+ 11c [] where a is the cil mean raius, an c is the thickness f the wining. Wheeler states that the frmula is crrect t within 5% fr cils with c > a. Other wrkers have reprte errrs using Wheeler s equatin f < 0% [17]. Errrs ccur when there are few turns, r if the spacing between the turns is t great. Fr a spiral cil with uter raius R = 0.15 an inner raius R i = 0, with N=5 calculatin frm the Grver meth gives L Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 8 f 11

9 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website 55 nh, the Schieber meth gives L 55 nh an the Wheeler frmula gives L 67 nh. It appears that the Wheeler frmula is mre accurate. 5. Planar Square Cil Fr the square cil, the effects f mutual cupling are nt as simple t calculate as fr the spiral case an the inuctance is mre ifficult t calculate analytically. An empirical apprximatin fr an N-turn square spiral (Figure 7) is given in [18]. It is reprte in the same paper that a rati f D/D i f 5 ptimizes the Q f the cil. D D i Figure 7. Planar Square Cil In the case when the wining area is cmpletely fille, r D i = 0, the inuctance is: L DN 3 [3] The expnent f 5/3 is thught t be ue t en effects in the square cils. Nte that this simplifie expressin esn't take int accunt the trace with. Mre etaile inuctance calculatins fr square planar spirals are given in references by Bryan [19], Greenhuse [0], Crkhill [1] an Saleh []. The Greenhuse reference is f special use as it cmpares varius meths f calculatin f planar spiral cils (bth run an square) with experimental results. V. CONCLUSIONS Part II f this wrk has shwn a variety f hanbk an apprximate inuctance calculatin results fr ifferent shape cnuctrs. A cmprehensive set f references is liste if the reaer wishes t elve int the tpic in mre etail. A bibligraphy cvering inuctance calculatin techniques may be fun at the authr's business website [3] at: The authr welcmes cmments n this article an aitins t the references at [email protected]. Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 9 f 11

10 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website VI. REFERENCES [1] James C. Maxwell, Electricity an Magnetism, vls. 1 an, reprinte in Dver Publicatins, 1954 [] Freerick W. Grver, Inuctance Calculatins: Wrking Frmulas an Tables, Dver Publicatins, Inc., New Yrk, 1946 [3] U. S. Bureau f Stanars, Rai Instruments an Measurements, Gvernment Printing Office, 194 [4] Harl A. Wheeler, "Frmulas fr the Skin Effect," Prceeings f the I.R.E., September 194, pp [5] Harl A. Wheeler, "Simple Inuctance Frmulas fr Rai Cils," Prceeings f the I.R.E., vl. 16, n. 10, Octber 198 [6] Freerick E. Terman, Rai Engineers Hanbk, McGraw-Hill, New Yrk, 1943 [7] Herbert B. Dwight, Electrical Cils an Cnuctrs, McGraw-Hill, 1945 [8] Herbert C. Rters, Electrmagnetic Devices, Jhn Wiley, 1961 [9] William R. Smythe, Static an Dynamic Electricity, e., McGraw-Hill, New Yrk, 1950 [10] D. Bruce Mntgmery, Sleni Magnet Design, Wiley-Interscience, 1969 [11] Thmas H. Lee, The Design f CMOS Rai Frequency Integrate Circuits, Cambrige University Press, 1998 [1] Markus Zahn, Electrmagnetic Fiel Thery: A Prblem Slving Apprach, Jhn Wiley, 1979 [13] Marc T. Thmpsn, High Temperature Supercnucting Magnetic Suspensin fr Maglev, Ph.D. Thesis, Department f Electrical Engineering an Cmputer Science, Massachusetts Institute f Technlgy, May 1997 [14] P. Murgatry, "The Brks Inuctr: A Stuy f Optimal Sleni Crss- Sectins," IEE Prceeings, vl. 133, part B, n. 5, September 1986, pp [15] V. J. Bashenff, "Abbreviate Meth fr Calculating the Inuctance f Irregular Plane Plygns f Run Wire," Prceeings f the I.R.E., vl. 15, 197, pp [16] D. Schieber, On the inuctance f printe spiral cils, Archiv fur Elektrtechnik, 68 (1985) pp [17] D. Daly, S. Knight, M. Caultn, an R. Ekhlt, Lumpe Elements in Micrwave Integrate Circuits, IEEE Transactins n Micrwave Thery an Techniques, vl. MTT-15, n. 1, December 1967, pp [18] H. Dill, Designing Inuctrs fr Thin-Film Applicatins? Electrnic Design, Feb. 17, 1964, pp [19] H. E. Bryan, "Printe Inuctrs an Capacitrs," Tele-Tech an Electrnic Inustries, vl. 14, n. 1, December 1955, pp. 68 [0] H. Greenhuse, Design f Planar Rectangular Micrelectrnic Inuctrs, IEEE Transactins n Parts, Hybris, an Packaging, vl. PHP-10, n., June 1974, pp [1] J. Crkhill an D. Mullins, The prperties f thick film inuctrs, Electrnic Cmpnents, vl. 10, n. 5, 1969, pp. 593 Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 10 f 11

11 THOMPSON: Inuctance Calculatin Techniques --- Part II: Apprximatins an Hanbk Meths Pwer Cntrl an Intelligent Mtin, December 1999, website [] N. Saleh, "Variable Micrelectrnic Inuctrs," IEEE Transactins n Cmpnents, Hybris, an Manufacturing Technlgy, vl. CHMT-1, n. 1, March 1978, pp [3] Marc T. Thmpsn, business website AUTHOR'S BIOGRAPHY Marc T. Thmpsn was brn in Vinalhaven Islan, Maine. He receive the B.S.E.E. egree frm the Massachusetts Institute f Technlgy (MIT) in 1985, the M.S.E.E. in 199, the Electrical Engineer s egree in 1994, an the Ph.D. in Presently he is an engineering cnsultant an Ajunct Assciate Prfessr f Electrical Engineering at Wrcester Plytechnic Institute, Wrcester MA. At Wrcester Plytechnic Institute, he teaches intuitive meths fr analg circuit, magnetic, thermal an pwer electrnics esign. His main research at MIT cncerne the esign an test f high-temperature supercnucting suspensins fr MAGLEV an the implementatin f magnetically-base rie quality cntrl. Other areas f his research an cnsulting interest inclue planar magnetics, pwer electrnics, high spee analg esign, inuctin heating, IC packaging fr imprve thermal an electrical perfrmance, use f scaling laws fr electrical an magnetic esign, an high spee laser ie mulatin techniques. He has wrke as a cnsultant in analg, electrmechanics, mechanical an magnetics esign, an hls patents. Currently he wrks n a variety f cnsulting prjects incluing high pwer an high spee laser ie mulatin, ey-current brake esign fr amusement applicatins, flywheel energy strage fr satellites, an magnetic tracking fr inter-by catheter psitining an is a cnsultant fr Magnemtin, Inc., Magnetar, Inc., Unite States Department f Transprtatin an Plari Crpratin. Marc T. Thmpsn, 1999 file: Inuct.c 4/6/01 1:16 PM Page 11 f 11