Estimation of Inductance for Sine and Cosine Shape Spiral Coils

Transcription

1 Estiation of Inductance for Sine and Cosine Sape Spiral Coils A. Darwin Jose Raju Assistant Professor, St. Xavier s Catolic College of Engineering, Nagercoil, Tail Nadu, India S. Solai anoar Professor, Anand Institute of Higer Tecnology, Kalasalinga Nagar, Kazipattur, Cennai, Tail Nadu, India.. Azagar Raj Researc Scolar, College of Engineering, Guindy, Tail Nadu, India. Abstract In tis paper, we focus on te developent of Grapical User Interface (GUI) tool to deterine te accurate inductance value for sine and cosine sape of spiral coils. Te inductance calculations of spiral inductors used in non-contact type position sensors are very critical for te accuracy of te sensor. Te existing etod of inductance calculations are based on anual coputation and are derived fro te basic epirical forulas of self and utual inductance. In tis tool, te inductance of sine and cosine coils are estiated by approxiating sape of coil troug line segents and te ateatical odel for te sae is also given to validate te results. Tis etod of coputation gives results wit iniu 95% accuracy tereby eliinate te trial and error based selection of inductance value for te sensor and easureent applications. In tis paper, te coputed results are verified wit actual easured value. Keywords: Coputer Siulation, Electroagnetic Induction, Inductance easureent, Sensor, Sine and Cosine sape coils, Spirals. Introduction Te inductance calculation of spiral inductors is a researc topic of very great interest in wide range of applications. any researc activities ave focused on te calculation, design and odelling of spiral inductors of various regular sapes like square, exagonal, octagonal and circular. Spiral inductors play a very iportant role in integrated circuits (ICs). Spiral type planar inductors are widely used in te design of power aplifiers, oscillators, icrowave switces, cobiners and splitters, etc. In te recent years tey ave evolved in various diensions and sapes. Te evolution of tese spiral inductors as led to treendous canges tat affect teir perforance. In order to iprove teir perforance it is absolutely necessary to analyze using appropriate etods. In tis paper we focus on te calculation of inductance of spiral inductors. Harold as discussed soe general inductance siple teoretical forulas for circular coil and square coil. For any sape, te relative error is very sall. [1] Papananos is used a generic luped eleent odel for analysing te perforance of any arbitrary sape inductors. An independent algorit for te evaluation of any inductor circuit is presented in detail. Octagonal, square and two-layer spiral inductors are designed and te easured results are copared wit siulation results[]. Hurley as been establised a new set of forulas for te self and utual ipedance calculation of planar coils on ferroagnetic substrates. Te forulas are derived fro axwell s equation by considering te current distribution and eddy current losses. [3] Zierofer considerably enanced te coupling coefficient between two agnetically coupled coils, if te turns of te coils are not concentrated at te circuferences, but distributed across te diaeters. Eac of te two coils is assued to be coposed of concentric circular loops. [4] Greenouse as been introduced and discussed in detail about te concept of negative utual inductance. It is usually saller in agnitude tan overall inductance in electronic circuits and it can be neglected. However, in icro circuits, its negligence can result in too ig values of inductance. Inductance equations are derived for planar tin-or tick-fil coils and include negative utual inductance. A coputer progra is developed for square and rectangular geoetry inductors and provides grapical output. [5] Robertson ave been analyzed wide range of inductor sapes in te literature. Spiral inductors are used to realize ig value inductance (>1nH). [6] ukerjee as been discussed about te design of spiral inductors using layout syntesis and optiization tecnique on a crystalline polyer substrate. [7] Lu stated tat te perforance of a circular spiral inductor is superior to tat of a square or rectangular spiral. [8] After tis introductory part te following sections describe te basic principles of self and utual inductances, te estiation of inductance for spiral coil using basic epirical forulas of self and utual inductances for sine and cosine sape coils and te coputation of inductance troug GUI using ATLAB. Te coparison between te analytical or 3663

2 coputed values and te easured values also discussed in te subsequent section. Finally, conclusions are drawn. Basic Principles of Self and utual Inductances Self Inductance In general, te self inductance of te circuit is a reaction to cange in current wic flow troug it. di e L (1) dt Wen a current i flows in circuit, i produces a agnetic field B; te agnetic flux troug circuit due to current i is Φ. Since B is proportional to i, so is Φ. Te constant of proportionality is te self-inductance L of te circuit. It is defined by te equation Li () Figure 1: Self Inductance If te coil is aving N turns, te flux linkage is N Li (3) So, te self inductance of te circuit is L (4) i Te self inductance can be calculated fro agnetic field energy stored also. utual Inductance Te utual inductance is ainly due to te cange in current in a nearby circuit. It is a easure of coupling between te two circuits. Let us consider a pair of two coils kept adjacently, were tere is soe flux linkage wit second coil wen tere is a current flow i1 troug te first coil due to agnetic coupling. Figure : Flux linkage wit te second coil Let te flux linkage wit te second coil be (5) 1 1i1 Were 1 & 1 are constant, wic can be written as 1 = 1 =, were is called as utual inductance of two coils. So 1 (7) i i 1 Estiation of Inductance for Spiral Coil Te analytical calculations of te inductance of Spiral coils ave several advantages copared to te conventional etods like lab experients or using siulation packages. Te analytical calculation of inductance will reduce te design cycle tie and cost. In tis paper te analytical etod for finding te inductance of sine and cosine coil sapes are discussed. Te inductance of any sape of Spiral inductor can be calculated according to te Greenouse teory. Basically, te Spiral inductor is split up into sections consisting of straigt conductors, and te self-inductance of te sections is calculated and sued up. utual-inductance Besides te consideration of te self-inductance of eac straigt conductor, te utual inductance (positive or negative) between parallel and unparallel segents depends on te sape of spiral needs to be included for te calculation to predict te overall inductance. Te general equation for a coil or a part of a coil of any sape is L 0 (8) L T Were L T, is te total inductance, L 0, is te su of te selfinductances of all te straigt segents, and is te su of all te utual inductances, bot positive and negative. Since utual inductance is positive wen current flow in two parallel conductors is in te sae direction and negative wen current flow is in opposite directions, te above equation can be rewritten as L L T 0 (9) Were is te su of te positive utual inductances and is te su of te negative utual inductances. Te utual inductance between two parallel conductors is a function of te lengt of te conductors and of te geoetric ean distance between te. Self-Inductance Consider a single segent of te conductor of lengt l, widt w and tickness t as sown in te fig. 4. [9] Siilarly, te second coil carries a current i wic creates flux linkage wit first coil wic can be given as (6) 1 1i Figure 3: Flux linkage wit te first coil Figure 4: Single conductor 3664

3 l w t L lln (10) w t 3l Were L is te segent inductance in nh. l is te segent lengt in c w is te segent widt in c t is te segent tickness in c Tus in total te total positive utual inductance is pos 8 nh (13) Negative utual inductance exists between segents 1-8, -7, 3-6, 4-5 Estiation of Inductance for Sine sape Spiral Coil In tis capter te analytical etod for finding te inductance of sine and cosine coil sapes are discussed. Figure 7: Segent 1 and 8 Figure 5: Approxiation of Sine Coil Self inductance Sine coil structure is approxiated as sown in te fig. 5. It is divided into 8 equal segents 1 to 8. Te current direction in eac segent is sown wit arrow direction. Te self inductance of te coil is found by adding te self inductance of eac of te segents using te forula. (10) utual inductance Positive utual inductance exists between segents 1-, 3-4, 5-6, 7-8. Consider segent 1 and, wic are equal in lengt It can be found by te forula(11), l cos yl tan tan nh (14) l Here l tus l 4l cos y tan nh (15) l Were y sin l Tus te total negative utual inductance is neg 8 nh (16) Te segents 1-76, 8-3, 5-3, 4-67 are parallel and ave negative utual inductance wic is found by te forula. Consider any two parallel conductors as sown in fig. 8. [9] Figure 6: Segent 1 and Figure 8: Parallel conductors It can be found by te forula, [9] l cos xl tan tan nh (11) l R R Here l tus l 4 l cos x tan nh (1) l R Were x cos l R l l Te forula for coputing te inductance of te above parallel conductors is given by sin sin sin d d d sin d d (17) d d d H Were, l ; l ; Here, l l; l ; d l and l 3665

4 Hence, l; 0; l Te above equation becoes, 0.001l[sin 5 ] H (18) Tus par 8000 nh (19) So te total inductance of Sine coil is calculated by adding Self Inductance, Positive utual Inductance, Negative utual Inductance and inductance between parallel conductors. and x cos l Positive utual Inductance pos 8 nh (1) Total Positive utual Inductance () pos pos1 pos Negative utual inductance also exists between segents 1-8, -7, 3-6, 4-5 Estiation of Inductance for Cosine Sape Spiral Coil Siilar logic is used to obtain te inductance of a cosine coil. Approxiation of te cosine coil is sown in te fig. 9. Figure 9: Approxiation of Cosine coil Te coil is divided into 10 equal segents 1 to 10. Te current direction in eac segent is sown. Te self-inductance of te coil is found by adding te self-inductance of eac of te segents found by te forula. (10) Positive utual inductance exists between segents 1-34, and It can be found by te forula, (11) l cos xl tan tan nh l R R Here, l l ; R Tus Positive utual Inductance pos 1 4 nh (0) Positive utual inductance also exists between segents 9-34, and 56-9, 78-10, 10-1 Figure 10: Segents 9-34 It can be found by te forula, (11) l cos xl tan tan nh l R R Here, l l ; Figure 11: Segent 1 and 8 It can be found by te forula, (14) l cos yl tan tan nh l Here l, tus l 4l cos y tan nh (3) l Were y sin l Tus in total te total negative utual inductance is neg 8 nh (4) Te segents 1-56, are parallel and ave negative utual inductance wic is found by te forula. Consider any two parallel conductors as sown in fig. 8. Te forula for coputing te inductance of te above parallel conductors is given by(17) sin sin sin d d d sin d d d d d H Were, l ; l ; Here, d l ; Hence l l; l; l; 3 l; l; l 3666

5 l [3sin 3 sin 1 10 ] H Segents 9 and 10 are parallel, tus forula (17) sin sin sin d d d sin d d d d d H Were, l ; l ; Here, d R; l ; ; 0; ; ; Hence sin sin sin R 4R 4R 4 par (5) (6) 6R 6R H 4000 nh (7) Tus 1 So te total inductance of Cosine coil is calculated by adding Self Inductance, Positive utual Inductance, Negative utual Inductance and inductance between parallel conductors. wic as ig flexibility to design any coplicated design wit its inbuilt functions. It allows one to perfor coplicated calculations wit visual effect of results. It allows te users to accurately solve probles and to write te code efficiently. Te front end can be designed using te GUI and te tool works in less execution tie tan any oter prograing languages(c or C++). Eac section of input is designed in suc a way tat te erroneous inputs, if entered, gives error essages. Te required inputs are te coil geoetry, its diensions. Tis is used to find te inductance. A snap sot of te tool is given in fig. 13 for sine and cosine coils. Depending upon te sape of te coil te inputs are given and inductance value is displayed on te screen. Coputation of Inductance Troug GUI Te flow cart for te coputation of inductance is given in fig. 1. OPEN AN APPLICATION Figure 13: GUI created in ATLAB SINE COIL SELECT THE COIL 1. SINE. COSINE CHECK THE SELECTION COSINE COIL Results & Validation Te inductances of sine and cosine coils are calculated using te proposed GUI etod and are analysed. Te analysis is done by calculating inductance of various saples wit varying widt, tickness and lengt of segent ten coparing tese values wit te easured ones. Te error between te calculated and easured values are obtained to deterine te accuracy of te proposed etod. ENTER THE INPUT DIENSIONS CALCULATE INDUCTANCE ENTER THE INPUT DIENSIONS CALCULATE INDUCTANCE Sine coil: Total lengt of te sine coil is dividing into 8 segents. So te lengt of one segent can be calculated. Te Inductance is calculated in nanoenries. DISPLAY THE RESULT EXIT THE APPLICATION Figure 1: Flow Cart for Coputation of Inductance Te calculation of inductance of te Spiral coils as been ipleented using ATLAB tool along wit Grapic User Interface (GUI) progra. atlab is an Engineering package Figure 14: Sine coil 3667

6 Table 1: Coparison of analytical and easured values of sine coil Specifications Saple1 Saple Saple 3 Widt in c Tickness in c Lengt of one segent in c L Analytical in nh L easured in nh %error Cosine coil: Total lengt of te cosine coil is dividing into 10 segents. So te lengt of one segent can be calculated. Te inductance is calculated in nanoenries. Conclusion Tis paper as presented a etod to calculate te inductance for sine and cosine sape coils troug analytical etod using Grapical User Interface (GUI). It as a very friendly interface and easy to use. We ave presented a siple, approxiate expression for spiral inductors of sine and cosine sapes. We consider tis progra is very useful and accurate for te faster design of spiral inductors and its inductance coputation. It can also be used for optiization of spiral inductors for various diensions and wide range of applications. Tis etod can be extended for dynaic inductance calculation also. Te error can be furter reduced by going for ore nuber of segents to represent te actual sape of te coil. easureents confired te feasibility of te analytical forula. References Figure 15: Cosine coil Table : Coparison of analytical and easured values of cosine coil Specifications Saple1 Saple Saple3 Widt in c Tickness in c Lengt of one segent in c L Analytical in nh L easured in nh %error Te sine and cosine coil spiral tracks are developed in PCB and te easureents are carried out to easure te values of inductances using Agilent 484A Precision LCR eter. Figure 16: Snapsot of spiral tracks in PCB Te inductance for sine and cosine sape spiral coil is calculated using above etod and copared wit easured value. Te axiu of 5% error is calculated and found tat errors is acceptable. Tis etod is very siple and te obtained results are iniu 95% accurate. [1] Harold A. Weeler Inductance forulas for circular and square coils proceedings of te IEEE, vol. 70, no. 1, pp , Deceber 198. [] Y. Papananos, Y. Koutsoyannopoulos, C. Aleanni, S. Bantas A Generic CAD odel for Arbitrarily Saped and ultilayer Integrated Inductors on Silicon Substrates. National Tecnical University of Atens, icroelectronic Circuit Design Group, 9 IroonPolitecniou, GR-15773, Atens, Greece Sgs- Toson icroelectronics, RF Division, DSG, Catania, Italy. [3] W. G. Hurley and. C. Duffy Calculation of Self and utual Ipedances in Planar agnetic structures leee transactions on agnetics, vol 3 1, no 4, pp July [4]. Zierofer and E. S. Hocair Geoetric Approac for Coupling Enanceent of agnetically Coupled Coils IEEE transactions on bioedical engineering, vol. 43, no. 7, pp , July [5] H.. Greenouse, Design of Planar Rectangular icroelectronic Inductors IEEE Transactions On parts, Hybrids, and Packaging, Vol. Pp-10, No., pp June1974. [6] I. D. Robertson and S. Lucyszyn, RFIC and IC design and tecnology, IEEE publising, London, UK, 001, ISBN [7] ukerjee, S., B. utnury, S. Dalia, and. Swainatan, Layout-level syntesis of RF inductors and filters in LCP substrate for Wi-Fi applications, IEEE Trans. on icrowave Teory and Tec., Vol. 53, , 005. [8] H-C. Lu, T. B. Can, C. C-P. Cen, C- Liu et al, LTCC spiral inductor syntesis and optiization wit easureent verification, IEEE Trans. Adv. Packag., vol. 33, no. 1, pp Feb 010. [9] Frederick W. Grover, Inductance Calculations Working Forulas and Tables Dover Publications, INC., ineola, New York, 013. [10] Heng-ing Hsu, Analytical Forula for Inductance of etal of Various Widts in Spiral Inductors, IEEE 3668

7 Transactions on Electron Devices, vol. 51, No. 8, pp , August 004. [11] arie Y vanoff, Generalised Analytical odel for Planar Inductors Using a Segentation Tecnique Tesis, Departent of Electrical Engineering, College of Engineering, Rocester Institute of Tecnology, Rocester, NY, ay 004. [1] Katrin Steinberg, arc Sceffler, and artin Dressel, icrowave inductance of tin etal strips Journal of Applied Pysics 108, _010. Bibliograpy A. Darwin Jose Raju received is B.E degree in Electrical and Electronics Engineering fro Hindustan College of Engineering, Cennai and.e in Applied Electronics fro Hindustan College of Engineering, Cennai. He is currently working as an Assistant Professor in St. Xavier s Catolic College of Engineering, Nagercoil and also doing P.D degree in Electrical Engineering. S. Solai anoar received is B.E degree in Electrical and Electronics Engineering fro Kuaraguru College of Tecnology, Coibatore and.e degree in Applied Electronics fro Hindustan College of Engineering, Cennai and Copleted P.D degree in Electrical Engineering. Currently e is working as a Professor in Anand Institute of Higer Tecnology, Cennai.. Azagar Raj received is.e. degree in Electrical Engineering wit few patents. Worked for GE, Honeywell and suzlon. He is currently a Researc Scolar in College of Engineering Guindy, Cennai. 3669