Compact Form of Expressions for Inductance Calculation of Meander Inductors

Transcription

1 SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 1, No. 3, November 2004, Compact Form of Expressions for Inuctance Calculation of Meaner Inuctors Goran Stojanovic 1, Ljiljana Živanov 2, Mirjana Damjanovic 3 Abstract: The quality of moelling an analysing of RF IC with planar inuctors extremely epens on accuracy of expression for inuctance. In this paper efficient methos for total inuctance calculation of meaner inuctor, are given. By using propose algorithm, we are able to preict correctly all inuctance variations introuce by varying geometry parameters. Our evelope program gives possibility for fast an accurate inuctance calculation for meaner inuctor, while new expression in monomial form is useful for optimisation of this inuctor. The results valiations, given by propose software tool are conforme by comparison with measure ata from literature. Keywors: Equivalent electroes metho, Electric file, Cub electroe. 1 Introuction Rapi evelopment of raio-frequency integrate circuits (RF IC) an increible evelopment an sprea of market wireless-cell, i.e. mobile communications, have also mae a hasty interest for monolithic inuctors. The inuctors of spiral shape (square, octagonal, circular) are most frequently applie, although for their fabrication more metal levels are neee. In earlier publishe papers [1], [2] are presente the simulators for analytical calculation of inuctance of spiral inuctors, or inuctance calculation in interconnect structures [3], [4], but there is no a software tool for quick an accurate calculation of meaner's inuctance. Layout of meaner inuctor is given in Figure 1. Although in comparison to inuctors of other shapes it gives weaker performances, i.e. it gives smaller inuctance per equal chip surface an has smaller quality factor (Q-factor) for the equal inuctance ue to longer conuctor (greater DC resistance), meaner inuctors are use because of simple process of prouction. Hence, in aition to a simple layout, it is not neee to make metal contacts of inuctors in two levels, so proviing way to avoi two levels of photolithography what simplifies technological process [5]. 1 Faculty of Technical Sciences, Trg Dositeja Obraovi}a, Novi Sa, Serbia an Montenegro, Phone: , Fax: , sgoran@uns.ns.ac.yu 2 Faculty of Technical Sciences, Trg Dositeja Obraovi}a, Novi Sa, Serbia an Montenegro, lilaziv@uns.ns.ac.yu 3 Faculty of Technical Sciences, Trg Dositeja Obraovi}a, Novi Sa, Serbia an Montenegro, mira@uns.ns.ac.yu 57

2 G. Stojanovi}, M. Damjanovi} Commercial electromagnetic simulators can be use to calculate inuctance of meaner inuctors. However, they are computationally intensive an time consuming. To quickly calculate inuctance for esign guiance, simple analytical formulae are esire. In orer to be implemente in circuits, meaner inuctors have to be supporte by moels. The major problem for implementation of a compact lumpe inuctor moel was the lack of an accurate an simple expression for meaner inuctance. In this work, we have explaine how we have generate such an expression, which can be irectly implemente in the RF circuit simulator. In this paper are presente the results of our software tool INDCAL (INDuctance CALculation) that serves for fast an accurate inuctance calculation of the meaner inuctors. In the section 2 are given analytical expressions for the self-inuctance an mutual inuctance of meaner an in clear (tabular) way are summarize expressions suitable for implementation into a computer program. The simple monomial expression for inuctance that is suitable for application in the optimisation proceures is given in the Section 3. In the Section 4 are presente the most important results of propose program while in the Section 5 are given conclusions of this work. 2 Inuctance Calculation The starting point for the erivation of propose formula is Greenhouse theory [6]. Greenhouse ecompose inuctor into its constituent segments. Basically, meaner inuctor is ivie into straight conuctive segments. Then the total inuctance of the meaner inuctor is a sum of self-inuctances of all segments an the negative an positive mutual inuctances between all combinations of straight segments. In aition to Greenhouse's, the great contribution to the inuctance calculation is given by Grover [7]. He has consiere the concept of partial inuctance, i.e. contribution of iniviual segments to the overall inuctance, an he also has introuce the iea of the geometric mean istance, calle GMD. In this paper are stuie inuctors manufacture by means of MEMS technology, having a final thickness (8µm) an with of several tenths of µm, so that in calculation of self-inuctance as one of the factors also appears GMD. When stuying of mutual inuctance as filament we take the mile lines of the observe meaner as in Fig. 1. The filaments are the straight segments of the meaner inuctor having neglectable cross section in relation to the length. Fig. 1 - Meaner inuctor with characteristic imensions. 58

3 Compact Form of Expressions for Inuctance Calculation of Meaner Inuctors 2.1 Self-inuctance Expression for self-inuctance of the conuctive segment, as in Fig. 2, is given by the following equation L = l [ln(2l /GMD) AMD/ l + ( µ / 4) T], (1) with the following: L inuctance in micro henries, l length of conuctive segment in centimetres, G M D an AMD represent geometrical an arithmetical mean istance of the conuctor's cross-section, µ conuctor's permeability, an T is frequency epenant correction factor. Fig. 2 Cross-section of the conuctive segment of the meaner inuctor. If the expressions given in literature [5] for GMD an AMD for the conuctor of rectangular cross-section are replace, then the expression for self-inuctance is L = 0.002l { ln[2l / ( w + t)] [( w + t) / 3l] + ( µ / 4) T}, (2) i.e. L = l { ln[2l / ( w + t)] [( w + t) / 3l] }, (3) where w an t are imensions of the cross-section, as it is shown in Fig. 2. The last equation is vali in case when the magnetic permeability is 1 an when the frequency epenant factor T is not taken into account, i.e. T = 1. Fig. 3 Presentation of characteristic geometrical imensions of the meaner inuctor. 59

4 G. Stojanovi}, M. Damjanovi} The expression (3) will be use as an analytical expression for calculation of selfinuctance of iniviual straight segments (shown in Fig. 3), an by that also for calculation of total inuctance of the meaner inuctor. The total self-inuctance L selftot is calculate as a sum of self-inuctances of all iniviual line segments, which form meaner inuctor, ( N + ) L Lselftot = 2 La + 2 Lb + N Lh + 1. (4) This is a generalize equation where L a,b,h, are self-inuctances of segments where the length is l = a, b, h, (respectively) an are calculate by means of the expression for self-inuctance (3). In the equation (4), N represents a number of segments of the greatest length h (for the example given in Fig. 3 it is evient that N = 6 ). 2.2 Mutual Inuctance In orer to get the final expression for the total mutual inuctance, we are observing two cases that appear in the process of manufacture of inuctors, with even number an o number of the longest segments (length h ). In Fig. 4 there is presente characteristic situations of the position of two segments that can appear in meaner inuctors, i.e. two parallel conuctors of the equal length at the corresponing mutual istance. The equation (5), taken from the book of Grover [5] calculates mutual inuctance (let sign it with M c ) of the segments with equal length an l, at istance r an are place opposite one to another as in Fig µ l l r r M + + c ( l,r) = ± 0 l ln + 1+ π r r 1. (5) 2 l l Fig. 4 Two equal parallel straight filaments. Mutual inuctance of another segment positions can be expresse by linear combination of expression for calculation of mutual inuctance of segments with equal length, at istance r an which are positione one opposite the other (expression (5)). Equation for the segment combinations as given in Fig. 5 (a), for unequal parallel filaments, is M a 1( l1, l2, r, s) = 0.5 [ M c ( l1 + l2 + s,r) + M c ( s,r) M c ( l1 + s,r) M c ( l2 + s, r). (6) 60

5 Compact Form of Expressions for Inuctance Calculation of Meaner Inuctors For the situation in the Fig. 5(b), where filaments with their ens in the common perpenicular, vali equation is as the following Ma2( l1, l2, r) = 0.5 [ Mc( l1, r) + Mc( l2, r) Mc( l1 l2, r) ]. (7) (a) (b) () (c) Fig. 5 Presentation of four typical segment combinations in meaner inuctors. For the situation in Fig. 5 (c), appearing only in case when N -o for the couple of filaments with length b, the following equation is use, M a3 ( l1, l2, r) = 0.5[ M c ( l1 + l2, r) M c ( l1, r) M c ( l2, r)]. (8) For the position as in Fig. 5 () when the two segments are in the same axis (segments pairs a a an ) at the corresponing istance between each other the expression given in the equation (9) esigne with M b is given, µ 0 M b ( l1, l2, s) = [( l1 + l2 + s) ln( l1 + l2 + s) ( l1 + s) ln( l1 + s) ( l2 + s) ln( l2 + s) + s ln( s)]. (9) 4π Thus, now we can present the final equations for self-inuctance an mutual inuctance by which we calculate accurate values for inuctances of meaner inuctors, an which are suitable for programming. In orer to make it clearer, in Table I are given the characteristic situations of all obtaine expressions for calculation of the total inuctance. It is necessary to emphasize that formulae given in Table I make a close form of the expression for inuctance of the meaner. Their avantage is also in that to the esigner of circuit they give information on how the geometrical parameters of the inuctor influence upon the self-inuctance, positive or negative mutual inuctance. 61

6 G. Stojanovi}, M. Damjanovi} 3 Fitting Technique In reference [8] there alreay exist expressions in a simple monomial form for the square an octagonal planar inuctors. This form of the expression (10) for calculation of inuctance is very suitable an necessary for application in the optimisation proceure of the inuctor by means of geometrical programming metho. While in reference [8] this expression has been obtaine by fitting accoring to a great number of fabricate inuctors, in this paper fitting techniques has been mae accoring a great number of numerical values of inuctance using the expressions escribe in section 2. We have taken five variables because there are five characteristic geometrical imensions that in fact etermine inuctance of the inuctor itself. Let's take the following a x1, h x2, N x3, x4, ω x5. In Fig. 1 all these values can be seen, α1 α2 α3 α4 α5 1 x2 x3 x4 x5 L = β x. (10) However, this simple form of expression for inuctance oes not exist in literature for the meaner inuctor. The expressions for calculation of the total inuctance, presente in the previous section we have implemente into the computer program which serve for generating of a great number of results, base on the given input geometrical an technological parameters. The fitting was one by means of the Lsm2000 program [9]. The fitting using the metho of the least squares fins the parameters of the equation by which are minimize the sum of squares of the error between the accurate ata an fitte equation. This program serves for fitting (ajusting) parameters, analytically given equations (curves, surfaces), so to suit the given points. Here, will be presente only final expression in the monomial form, which is suitable for optimisation via geometric programming. Table I Presentation of all expressions for calculation of total inuctance for N -even an N - o. L = 0.002l { ln[2l / ( ω+ t)] [( ω+ t) / 3l] } L selftot = 2 La + 2 Lb + N Lh + ( N + 1) L N / 2 M 1 = (2N + 4 4i) M ul (,,h, ( 2 i - 2) ), for N even 1 h h ( N + 1) / 2 M 1 = (2N + 4 4i) M a1 (,,h, ( 2 i - 2) ), for N o 1 62

7 Compact Form of Expressions for Inuctance Calculation of Meaner Inuctors N / 2 M 2 = (2N + 2 4i) M b (,, ( 2 i -1) ), for N even 1 ( N 1) / 2 M 2 = (2N + 2 4i) M b (,, ( 2 i -1) ), for N o 1 ( N 1) ) M 3 = 2 M b ( a, a, + a a N M 4 = 4 M a1( a,,b, i) 0 a N-1 i M 5 = ( 1) 2 c i 0 ( N - i) M ( h, ) h h M 6 = 2 M c ( b,( N + 1) ), for N even b b b M 6 = + 2 M a 3 ( b, b,( N + 1) ), for N o N i M 7 = ( 1) 4 M a2, 0 ( b, h i ) M tot = M1 + M 2 + M 3 + M 4 + M 5 + M 6 + M 7 L tot = Lselftot + M tot The monomial equation for the total inuctance L mon, of the meaner inuctor that gave the least relative error (maximally up to 12%) is Lmon = a h N w. (11) Thus, by this expression for the first time was presente the inuctance of the meaner inuctor in the monomial form, so that the optimisation of the inuctor can be one by proceure of the geometrical programming. Simplicity an relatively goo accuracy 63 b h b

8 G. Stojanovi}, M. Damjanovi} are the avantages of this expression, but on the other han the physical sense of the expression is being lost. Also, it is not possible to get information on mutual inuctance or of the self-inuctance of the meaner inuctor. 4 Results an Discussion In orer to emonstrate the valiity of results of the evelope software tool in this part we compare results that are obtaine by means of propose programs an experimental results taken from the literature. In the Table I are shown geometrical parameters an measure inuctances for three meaner inuctors as foun in the open literature [5]. Table II Dimensions of three types of meaner inuctors an their measure inuctances [5]. Name N w [µ m] [µ m] L meas [nh] In # In # In # In the Section 2 in great etails is escribe the proceure for calculation of the total inuctance of the meaner inuctor both for the even an o number the longest segments using the mathematical inuction, an by using the Greenhouse's metho [6]. Expressions that are summe in the Table I serve as the basis to write the program for calculation of the self-inuctance, positive or negative mutual inuctance an total inuctance for the meaner inuctors. The program is written in the program language Visual Basic 6.0, while the source coe of program has been completely evelope by our own knowlege acquire by previous scientific research an experience. The program has its graphical interface (Fig. 6), while the user gives the input ata an very + quickly gets the information about the self-inuctance ( L selftot ), positive ( M ) or negative ( M ) mutual inuctance an the total inuctance ( L tot ) of the meaner inuctors. Simulate results of inuctance for the three types of the meaners are given in the Table III. The program can also raw the layout of the simulate meaner, an gives the information on the use area of the inuctor. Table III Self-inuctance, positive, negative mutual inuctance an total inuctance for three types of meaner inuctors. Inuct. name L selftot [n H] + M [nh] M [nh] L tot [nh] In # In # In # In the Section 3 is explaine the proceure for obtaining the expression for calculation of the meaner inuctance in the monomial form an at the en of that section is 64

9 Compact Form of Expressions for Inuctance Calculation of Meaner Inuctors given the expression use in optimisation of meaner layout by use of proceure of geometrical programming (results of this proceure will be given in one of our future scientific papers). Comparison of the experimental results for the total inuctance, for three meaner inuctors given in the Table II, with results obtaine by propose new expressions (base on Greenhouse theory, Table I), or monomial expression (11) is given in the Table IV. Table IV Comparison of obtaine results for the inuctance of meaner inuctors. Inuct. name Measure inuctance [nh] This work (Table I) [nh] Monomial expression (11) [nh] In # In # In # As it is seen from the Table IV, the agreement of the results is very goo. It is necessary to mention again that in the scientific literature there is very small number of papers giving experimental ata for the meaner inuctors. However, on the other han our ealing with meaner inuctors an conformity that so far can be evient, gives even greater value to research in this fiel. It shoul be note that are no unphysical fitting factors in propose expressions epicte in Table I. Those expressions have the avantage that it inicates to the esigner how the relative contributions of self, positive, an negative mutual inuctance are relate to the geometrical parameters. Fig. 6 - One example of software tool interface for inuctance calculation of meaner inuctors. 65

10 G. Stojanovi}, M. Damjanovi} Fig. 7 - The total inuctance of meaner inuctors versus number of segments of the greatest length. How number of the longest segments N influence the total inuctance for tree type of teste meaner inuctors is epicte in Figure 7. Hence, In #1 has the wiest conuctor s segment the total inuctance rises greater with increase the number of the longest segments. When we change one of parameters, all another take the same values as in the Table II. The above figure emonstrates that all inuctance variations by varying geometry parameters are successfully preicte using propose software tool. 5 Conclusion Planar meaner inuctors of the simple layout are inevitable component in the contemporary RF integrate circuits. The calculation of the meaner's inuctance has been treate in the literature with very great importance. In this paper is presente a software tool, which in a fast an efficient manner to calculate inuctance of the meaner inuctors. There are given two ways for calculation of inuctance base on Greenhouse metho an the form of monomial expression. The accuracy of presente results is evaluate by comparison with earlier publishe experimental values of inuctance. The propose close form inuctance expressions is compatible with circuit simulators. 6 Acknowlegement This paper is part of the project IT B at the University of Novi Sa, Faculty of Technical Sciences, an was supporte by Ministry of Sciences an Environment Protection of Republic of Serbia. 66

11 Compact Form of Expressions for Inuctance Calculation of Meaner Inuctors 7 References [1] Y. Koutsoyannopoulos et al.: A generic CAD moel for arbitrary shape an multilayer integrate inuctors on silicon substrates, in Proc. ESSDERC, pp , [2] Snezana Jenei, Bart K., J. C. Nauwelaers, Stefan Decoutere: Physics-base closeform inuctance expression for compact moeling of integrate spiral inuctors, IEEE Journal of Soli-State Circuits, Vol. 37, No. 1, January 2002, pp [3] C. Harlaner, R. Sabelka, S. Selberherr: Efficient inuctance calculation in interconnect structures by applying the Monte Carlo metho, Microelectronics Journal, Vol. 34, No. 9, September 2003, pp [4] Z. Zhu, X. Xia, R. Streiter, G. Ruan, T. Otto, H. Wolf, T. Gessner: Close-form formulae for frequency-epenent 3-D interconnect inuctance, Microelectronic Engineering, Vol. 56, No. 3-4, August 2001, pp [5] Geral W. Dahlmann, Eric M. Yeatman: Microwave characteristics of meaner inuctors fabricate by 3D self-assembly, 8th IEEE International Symposium on High Performance Electron Devices for Microwave an Optoelectronic Applications, November 2000, pp [6] H. M. Greenhouse: Design of planar rectangular microelectronic inuctors, IEEE Trans. Parts Hybris, Packaging, Vol. PHP-10, 1974, pp [7] Freerick W. Grover: Inuctance calculations, working formulas an tables, Princeton, D. van Nostran company, inc., 1946, reprinte by Dover Publications, New York, [8] Sunerarajan S. Mohan, Maria el Mar Hershenson, Stephen P. Boy, Thomas H. Lee: Simple accurate expressions for planar spiral inuctances, IEEE Journal of Soli-State Circuits, Vol. 34, No. 10, October 1999, pp [9] Available: 67