Inductance Calculation and Layout Optimization for

Planar Spiral Inductors

Claudia Pacurar, Vasile Topa, Adina Racasan, Calin Munteanu
Department of Electrotechnics and Measurements
Technical University of Cluj-Napoca
26-28 Baritiu Street, 400027, Cluj-Napoca, Romania
Claudia.Pacurar@et.utcluj.ro, Vasile.Topa@et.utcluj.ro, Adina.Racasan@et.utcluj.ro, Calin.Munteanu@et.utcluj.ro

Abstract-The inductance calculation and the layout optimization extractions become more and more important for design,
for spiral inductors still are research topics of actuality and very optimization and design verification of the spiral inductors
interesting especially in radio frequency integrated circuits. Our
research work is fixed on the vast topics of this research area. In and their performances improvement.
this effect we create a software program dedicate to dc However things are the inductance calculation for spiral
inductance calculation and to layout optimization for spiral inductors, the spiral inductors optimization by finding the
inductors. We use a wide range of inductors in the application optimal layout for maximal inductance or for a given
made with our program; we compare our applications results inductance value, keeping a constant area for the inductor
with measurements results existing in the literature and with
three-dimensional commercial field solver results in order to spiral implementation in the integrated circuit are still needed
validate it. Our program is accurate enough, has a very friendly to improve the spiral inductors performances. To this aim we
interface, is very easy to use and its running time is very sort devise a software program that allow fast and accurate
compared with other similar programs. Since spiral inductors inductance calculation and spiral inductor layout
tolerance is generally on the order of several percent, a more optimization. The program name is CIBSOC (Spiral
accurate program is not needed in practice. The program is very
useful for the spiral inductor design because it calculates the Inductors Inductance Calculation and Layout Optimization)
inductance of spiral inductors with a very good accuracy and and it is composed of four modules. The first and the second
also for the spiral inductor optimization, because it optimize the one are devising to calculation the spiral inductor inductance,
spiral inductor layouts in terms of technological restrictions and the third and the fourth to optimize the spiral inductor
and/or in terms of the designers’ needs. layout. The first module of the program, CIPGC (Inductance
Calculation for Constant Geometrical Parameters) is designed
I. INTRODUCTION
for calculate the total inductance of the spiral inductors with
The passive components parameters extraction from radio square, hexagonal, octagonal and circular shape (Fig.1), for
frequency integrated circuits, to wit inductance, capacitance constant number of turns and constant projecting geometrical
and resistance extraction are research topics of great interest parameters. The second module, CIPGV (Inductance
and very provocative. This fact is motivated by the continual Calculation for Variable Geometrical Parameters) is designed
technological progress thanks to; it is now possible to for the inductance variation calculation for the same spiral
implement integrated circuit at microns dimensions. The inductor shapes in terms of the number of turn’s variation or
decrease of the integrated circuit dimensions at this extreme with the one of the geometrical parameters variation. The
level lead in an implicit way to the significant rise extraction
importance with accuracy of the inductance, capacitance and w w

resistance parameters form micrometric devices, opening di

di
multitudinous now research topics in this domain. Our
de
research work focus on the study, analysis, simulation, de
modeling, design and optimization of spiral inductors. We are
focus now to inductance calculation. The inductance value s
s
extraction was and is intense studied in the literature. At this
a) square b) hexagonal
moment exists many expressions and methods used for w
integrated circuits inductance extraction [1]-[6]. However, as s

is mentioned in the literature, there are limits in their

有些表达式没 application. For example, in some expressions not all the
有完全考虑螺 geometrical parameters that describe completely the spiral w di s
di

旋电感的几何 inductor are taken into account and so the typical errors are de
参数 larger or smaller, tolerate or not in applications. de

To design and to optimize the spiral inductors from

integrated circuits is first necessary to find the exact
inductance value. So the fast and accurate inductance c) octagonal d) circular
Fig. 1. Spiral inductors geometrical shapes used in CIBSOC program

978-1-4673-1653-8/12/$31.00 '2012 IEEE 225

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third module OCBS (Spiral Inductor Layout Optimization), where μ is the magnetic permeability, N is the number of
allow to optimize the layouts of the spiral inductors that have turns, w is the width of the turn, s is the distance between the
one of the shapes shown in Fig. 1, have a given inductance turns, the average diameter is d m = 0.5 ( de + di ) , de is the
value and that will be implemented in a fixed area in the exterior diameter, di is the interior diameter,
integrated circuit. The fourth module OCBSIM (Spiral d e = di + 2 Nw + 2( N − 1) s , d e = d m + N ( w + s ) − s , the fill ratio is
Inductor Layout Optimization for Maximal Inductance) allow ρ = ⎣⎡ N ( w + s ) − s ⎦⎤ / d m , ρ = ( d e − d i ) / ( d e + d i ) , and di ≥ s .
the layout optimization for square, hexagonal, octagonal and Studying the literature we found that these expressions are
circular spiral inductors, for the maximal inductance of the more simple and accurate, and we decide to use them in our
inductor and for a fixed area in which the inductor will be program to calculate the inductance and to optimize the
incorporated in the integrated circuits. layout for the square, hexagonal, octagonal and circular spiral
II. THEORETICAL CONSIDERATIONS inductors.
The geometrical parameters that appear in these
The program is based for the calculation part on the current expressions, the shape of the spiral inductor and number of
sheet method. Simple and accurate expressions for dc turns are defining completely the spiral inductors. Many
inductance of square, hexagonal, octagonal and circular spiral expressions that exist in literature take into account only
inductors can be obtained by approximating the spiral sides some of these parameters, so that is not a surprise that the
with equivalent symmetric current sheets with the same errors are 20% or even bigger in some cases 80%, which are
current densities [9]. For example, in the case of a spiral unacceptable in the inductors design and optimization [7]-[8].
inductor, as is show in Fig. 2, are obtain four identical current
III. APPLICATIONS IN CIBSOC PROGRAM
sheets: the current sheets from opposite sides are parallel one
each other, and the adjacent one are orthogonal one each We design a set of spiral inductors which we use in the
other. Using the symmetry and the fact that the orthogonal applications that we done with our program. We present in
current sheets have the mutual inductance zero, the this paper only the application in which we use the square
inductance calculation is reduce to the self inductance shape for spiral inductors, even if the program allows also the
evaluation of one current sheet and to mutual inductance calculation for any of the four types of geometrical shapes
evaluation of opposite current sheets. These self and mutual generally used for spiral inductors. We use CIBSOC program
inductances are determinate using the geometric mean effectively to calculate the total inductance for each of the
distance, the arithmetic mean distance and the arithmetic square spiral inductors that we design and for theirs layouts
mean square distance concepts [9], [10]. The expressions for optimization.
square, hexagonal, octagonal and circular spiral inductors
A. Applications in CIPGC Module of the Program
inductances calculation are [11]:
The CIPGC module of the CIBSOC program allows the
inductance calculation for the geometrical shapes of the spiral
μ N 2 d m 1.27 ⎡ ⎛ 2.07 ⎞ 2⎤ inductors that are show in Fig.1. The module has a drawing
Lsquare inductor = ⎢ln ⎜ ⎟ + 0.18ρ + 0.13ρ ⎥ , (1)
2 ⎣ ⎝ ρ ⎠ ⎦ algorithm that draw automatic the implemented spiral
inductor and an inductance calculation algorithm that
μ N 2 d m 1.09 ⎡ ⎛ 2.23 ⎞ 2⎤ compute the inductance of the implemented spiral inductors.
Lhexagonal inductor = ⎢ln ⎜ ⎟ + 0.17 ρ ⎥ , (2)
In module are used as input quantities to completely define
2 ⎣ ⎝ ρ ⎠ ⎦
the spiral inductors the following: the geometrical shape of
the spiral inductor, the number of turns, N; the turn width, w;
μ N 2 d m 1.07 ⎡ ⎛ 2.29 ⎞ 2⎤
Loctagonal inductor = ⎢ ln ⎜ ⎟ + 0.19 ρ ⎥ , (3) the distance between the turns, s; the exterior diameter, de; the
2 ⎣ ⎝ ρ ⎠ ⎦ interior diameter, di and the turn thickness, t as we can see in
the Fig. 3. The input quantities used in our applications have
μ N 2 d m ⎡ ⎛ 2.46 ⎞ 2⎤ the following numerical values: N ∈ [1,16] (turns)
Lcircular inductor = ⎢ ln ⎜ ⎟ + 0.2 ρ ⎥ . (4)
de=500(μm); s=5(μm) w=10(μm); and t=2(μm). All the
2 ⎣ ⎝ ρ ⎠ ⎦
dimensions used in the paper are in (μm) and the inductance
values in (nH). Also as input quantities is necessary to select
the type of material that is used for inductors’ spiral. The
w
program allows us to select the type of material from its
di library and also to add new materials. The material used for
dm the square spiral inductor that we design is the cooper, Cu,
de
because we note studying the literature that it is the most
nI usually used material for practical realization of the inductor
spiral. We detail for exemplification the square spiral
s
inductor with sixteen turns from the set of the eight square
r dm spiral inductors elaborate.
Fig. 2. The square spiral inductor and its equivalent current sheets.

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 number of turns, without the necessity of the independent
implementation of each inductor and without the need of
collecting the results in another program that allow also to
plot the results.
In Fig. 4 can be observed the details regarding the
implementation of the application in the module and also the
obtain results for the inductance variation versus the number
of turns. We consider the number of turn’s variation between
one and sixteen turns with the pass one. The other dimensions
of the analyzed inductors are those mentioned above.
In essence we calculate the inductance of sixteen square
spiral inductors in the same time directly, without being
necessary to analyze independent each one inductor to obtain
its inductance value and then to plot the inductance variation
Fig. 3. The square spiral inductor with sixteen turns implemented for in terms of the number of turns. The results appear in the
inductance calculation in the CIPGC module of the CIBSOC program. right side of the window as shown in Fig. 4a). This module
We have implemented these inductors in our program as is permits also the directly plot of the obtained values, plot that
show in Fig. 3. In Table I we present the results for all the is presented in Fig. 4b). The inductance varies direct
eight square spiral inductors obtained with CIBSOC program. proportional with the number of turns, i.e. if the number of
TABLE I turns increase also the inductance increase, normally knowing
INDUCTANCE CALCULATION RESULTS FOR THE EIGHT SQUARE SPIRAL that the number of turn predominate in almost all the
INDUCTORS DONE IN CIPGC MODULE OF CIBSOC PROGRAM inductance calculation expressions.
Number of turns, Inductance,
N (turns) LCIBSOC (nH) 2. Inductance Variation versus the Turn Width
1 1,80763 We analyze now the influence of turn width on the
2 5,58207 inductance value. For this analysis we implement the spiral
3 10,52683
4 16,14966
inductors in CIPGV module, varying for each one the turn
6 28,08099
8 39,34739
12 55,3146
16 60,06357
Is obvious the utility of this module of the program, his
simplicity referring to the implementation, calculation and
drawing facilities and to the short running times that the
program offers.
B. Applications in CIPGV Module of the Program
We study the inductance variation for the eight square
spiral inductors in terms of the number of turns variation or of
the one of the following geometrical parameters variation:
turn width, w, distance between the turns, s; exterior
diameter, de or turn thickness, t. In the module we can set the a) The results of inductance variation vs. number of turns, N;
variation of the parameter in terms of which we want the
inductance variation to be calculate and we can also to plot
the results. The module has an inductance variation
calculation algorithm and a graphical representation
algorithm that can be used to directly plot the obtained
results. We detail now the obtained results for the inductance
variation in terms of the number of turn variation, then in
terms of the turn width variation and then in terms of the
distance between the turns variation. The inductance variation
in terms of the turn thickness has no significant variation, so
it will be not presented in the paper.
1. Inductance Variation in terms of the Number of
Turns b) The plot of inductance variation vs. the number of turns
The application is implemented in the CIPGV module to Fig. 4. Inductance vs. number of turns obtained in CIPGV module.
obtain directly the inductance variation in terms of the

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width between 1(μm) and 10(μm) with pass one, in only one turn on the inductance value. For this study we implement
applications all the variations. only seven from the eight types of create spiral inductors in
The inductors have the other dimensions identical with CIPGV module, because the inductance of the square spiral
those presented above. We calculate the inductance variations inductor with one turn does not vary with the distance
in terms of the turn width for the square spiral inductor with between the turn. So we vary for each inductor the distance
one, two, three, four, five, six, eight, ten, twelve, fourteen and between the turn from 1(μm) to 5(μm) with the pass one. The
respectively sixteen turns. We synthesized in Fig. 5a) and b) other inductors dimensions for the analyzed square spiral
the inductance variation in terms of the turn width results and inductors remain the same.
in Fig. 5 c) and d) we present the plots of these results only We present only some of the applications for this variation
for the square spiral inductor with two respectively ten turns study with the afferent results and graphical representations,
for exemplification. i.e. the results for the four and respectively sixteen turns
Analyzing these plots we observe that the square spiral square spiral inductors in Fig. 6a), b) and respectively in Fig.
inductor inductance varies inverse proportional with the turn 6c) and d).
width, w. So if the turn width decreases the inductance In the train of this analysis we can conclude that the
increases. inductance varies also inverse proportional with the distance
between the turns, not only with the turn width. So the
3.
Inductance Variation in terms of the Distance
inductance is maximal when the distance between the turns in
between the Turns
We study now the influence of the distance between the minimal.
We can conclude at the end of these variations analyses
saying that the inductance is direct proportional with the
number of the turns and inverse proportional with all the
other geometrical parameters of the square spiral inductors.
All the applications made for these variations studies were

a) Results of L vs. s for 4 turns inductor; b) Results of L vs. s for 16 turns inductor;
a) Results of L vs. w for 2 turns inductor; b) Results of L vs. w for 10 turns inductor;

c) L vs. s variation for square spiral inductor with 4 turns;

c) L vs. w variation for square spiral inductor with 2 turns;

d) L vs. w variation for square spiral inductor with 10 turns; d) L vs. s variation for square spiral inductor with 16 turns;
Fig. 5. Inductances versus the width of the turn obtain in CIPGV module of Fig. 6. Inductances versus the distance between the turns obtain in CIPGV
CIBSOC program. module of CIBSOC program.

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done for the exterior diameter of the inductors equal with
500(μm).
We consider that the spiral has to have the same area
(imprint) on the integrated circuit in which it will be
incorporate.
We design the program to draw, to calculate and to
optimize only the effective spiral as component part of the
spiral inductor, not with via, underpass, oxide and substrate,
because we have analyzed those influence on the inductance
and we find that in direct current have no effects on its value.
So the program was design to analyze and to optimize the
square, hexagonal, octagonal, and circular spiral inductors in
direct current. In the program we impose the variations limits
for the number of turns and for the geometrical parameters. Fig. 7. Optimal solutions for square spiral inductor layout optimization for
These limits are: for the turn width, from 1-100(μm), with L=28,08099(nH) and de=500(μm).
pass 1; for the distance between the turns from 0.5-50(μm) to obtain only one optimal solution, depending of the
with pass 0.1; the exterior diameter from 1-800(μm) and for technological restrictions or the uses’ needs.
the number of turns from 1-32(turns) with pass 0.25. We
2. Optimal Layout for L=60,0635(nH) and de=500(μm)
chose these limits to cover a wide range of square spiral
Similar we want to find now the optimal solution for the
inductors.
square spiral inductor layout that has inductance value
C. Applications in OCBS Module 60,06357(nH) and the exterior diameter 500(μm). The
The OCBS module was design in order to optimize the program finds 114 optimal solutions, obvious between them
spiral inductor layout for a given value of the inductance and we retrieve that one for the square spiral inductor with sixteen
for a fixed area that the inductor need in the integrated circuit turns from which we start the application, marked in Fig. 8.
in which it will be used, i.e. and for a given exterior diameter. The user will apply the filters that he chose from those that
If the designer has a free area on an integrated circuit in the program offers, taking into account his needs and the
which must to integrate one or more inductors, and he has technological limitations.
some impose values for inductances, he can use this module
D. Applications in OCBSIM Module
of the program to find the optimal solutions with accuracy in
The OCBSIM module of CIBSOC software program let us
a very short running time. He has to know only the
to find directly the unique optimal solution for spiral inductor
geometrical shape of the inductor, the material used for spiral
layout. We present only the applications for square spiral
realization, the inductance value and the dimension of the
inductors to which we optimize theirs layouts for the maximal
exterior diameter.
inductance of each one.
We use the set of eight square spiral inductor in the
We can find the unique optimal solution of the spiral
applications, only that we want to find now the optimal
inductors for its maximal inductance and that use an area on
layouts for these inductors by imposing the value of
an integrated circuit described by a fixed exterior diameter. It
inductance to be one of those that we calculate in the first or
is known that the inductor has very good performances for the
in the second module, modules that are designed to calculate
maximal value of its inductance. For this reason we consider
the exact inductance values. We keep the exterior diameter
useful to find the inductor optimal layout dependent on its
constant, 500(μm). We find the optimal layout of those eight
maximal inductance value. So we consider some square spiral
square spiral inductors for the inductance values calculate in
the CIPGC module, values that are collected in Table I.
1. Optimal Layout for L= 28,08(nH) and de=500(μm)
We continue the applications for spiral inductors with
square geometrical shape and spiral made of cooper, Cu. We
choose the input error for the optimization 0.1(%). In Fig. 7
are presented all the details of the OCBS module. The
optimization algorithm of this program module displays all
the possible optimal solution in the right part of the window.
The total number of possible optimal solutions that the
program find are display in the bottom of the left side of the
window (Fig. 7). Thus and so are found all the possible
optimal layouts for the inductors of 28,08099(nH), that fill
perfectly the area described by the exterior diameter of the
Fig. 8. Optimal solutions for square spiral inductor layout optimization for
spiral 500(μm). The program offers filters that can be applied L=60,06357(nH) and de=500(μm)

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inductor of 10(μm); 50(μm); 100(μm); 200(μm); 300(μm); where Lm is the measured inductance, and LCIBSOC is the
500(μm); 700(μm); 800(μm) exterior diameters and we want calculated inductance with cu CIBSOC program.
to find the optimal solution for their layout so that each one to TABLE III
have maximal inductance value. Keeping the same INDUCTORS PARAMETERS. COMPARISON OF THE RESULTS.
geometrical shape for the spiral inductors and the same N de w s t Lm LCIBSOC εr
(turn) Ref.. (μm) (μm) (μm) (μm) (nH) (nH) (%)
material we found the square spiral inductor layout for the
2.75 [12] 279 18,3 1,9 0,9 3,1 3,009 2,93
exterior diameters write above. So, the obtain results are 7.50 [12] 166 3,2 1,9 0,9 12,4 11,987 3,33
detail presented only for the inductor with the exterior 9.50 [12] 153 1,8 1,9 0,9 18,2 17,842 1,96
diameter 10(μm) in Fig. 9 for exemplification, and the other 2.75 [12] 277 18,3 0,8 0,9 3,1 3,055 1,45
solutions are described in Table II. 5,00 [12] 171 5,4 1,9 0,9 6,10 5,859 3,95
3.75 [12] 321 16,5 1,9 0,9 6,1 6,026 1,21
TABLE II 3.00 [13] 300 19 4 0,9 3,3 3,4977 5,99
OPTIMAL LAYOUT FOR MAXIMAL INDUCTANCE 5.00 [13] 300 24 4 0,9 3,5 3,7736 7,81
Optimal Layout 9.00 [13] 230 6,5 5,5 0,9 9,7 9,6791 0,21
Exterior Maximal
Turn Distance Interior Number 8.00 [14] 226 6 6 0,9 9,00 9,143 1,58
diameter, inductance,
de(μm) Lmax (nH)
width, between turns, diameter, of turns, 16.00 [14] 300 5 4 0,9 34 36,544 7,48
w(μm) s (μm) di (μm) N(turns) 6.00 [13] 300 9 4 0,9 11,7 12,444 6,36
10 0,056 1 0,5 1 3,25 It is obvious the concordance between the results. The
50 6,006 1 0,5 3 16
100 47,056 1 0,5 6 31,75 program has high accuracy in calculation. The relative errors
200 244,565 1 0,5 105 32 are from 0,21 % to 7,81%.
300 502,7375 1 0,5 205 32
500 1109,9387 1 0,5 405 32
V. COMPARISON TO COMMERCIAL FIELD SOLVER
700 1792,185 1 0,5 605 32 To validate our CIBSOC program we considered very good
800 2153,348 1 0,5 705 32
the opportunity to use a commercial field solver dedicate to
All these studies and applications can be made with our
extraction of the parameters from different complex types of
program also for the other types of geometrical shapes used
integrated circuits. We implement in this program a wide
for spiral inductor. In this paper we treat only the spiral
range of square spiral inductors starting with the one we
inductors with square geometrical shape.
design for our applications. First we compare the results for
IV. COMPARISON TO MEASUREMENTS the inductance variation in terms of the number of turns. To
We calculate the inductance for some square spiral compare the obtained results we write them in Table IV.
inductors from the literature for which we have experimental TABLE IV
COMPARISON OF RESULTS. INDUCTANCE VS. NUMBER OF TURNS.
measurements to demonstrate the calculation accuracy of our
software program CIBSOC and implicit of its four modules Number of turns LCIBSOC (nH) L program (nH)
1 1,808 1,7309
designed to calculate the inductance and to optimize the 2 5,582 5,4228
layout of spiral inductors. 3 10,527 10,285
We calculate the inductances for two sets of square spiral 4 16,15 15,833
inductors detailed in Table III. So the results obtained with 6 28,081 27,637
8 39,347 38,794
our program are compared with the measurements results 12 55,315 54,704
from the literature. We find the relative errors, εr with the 16 60,064 59,646
formula:
L −L
ε r = m CIBSOC 100 (%) , (6)
Lm

Fig. 9. The layout optimization for the inductor with the exterior diameter Fig. 10. The comparison of the results obtained with CIBSOC program and
10(μm) for maximal inductance. with the commercial field solver.

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 The inductance variation in terms of the number of the 190,0 Inductance vs. the turn width - Field Solver

turns is plot in Fig. 10. We observe close agreement between 180,0

L 1 turn inductor
L 2 turns inductor
the results. With dotted line are plot the results obtained with
170,0
L 3 turns inductor
160,0

our CIBSOC program and with continuous line the results 150,0
L 4 turns inductor
L 6 turns inductor

obtained with the commercial field solver. To obtain the 140,0

L 8 turns inductor
L 12 turns inductor

results and the plots with our program was enough to

130,0 L 16 turns inductor
120,0

implement the inductors parameters in the programs modules

Inductance [nH]
110,0

and all were solved in the same time, very quickly. 100,0

90,0
But for the simulations of the same inductors in the 80,0

commercial field solver was necessary to implement 70,0

effectively each inductor and to solve them separately. Then 60,0

50,0
we copy the results in another program to plot theirs 40,0

variation. The implementation time, the running time, and the 30,0

postprocessor time are incomparable diminished with our 20,0

10,0
program. For the inductance variation in terms of the turn 0,0

width we present in detail only the results for the square 1 2 3 4 5 6

Turn width [um]
7 8 9 10

spiral inductor with one turn in Table V and the plots in Fig. 200,0
Inductance vs. the turn width - CIBSOC Program

11. We present the results comparison of all the analyzed 190,0 L bobina 1 spira
L bobina 2 spire

inductors in Fig. 12, i.e. those obtained with commercial field 180,0
170,0
L bobina 3 spire
L bobina 4 spire

solver and those obtained with our CIBSOC program. 160,0 L bobina 6 spire
L bobina 8 spire
150,0 L bobina 12 spire
TABLE V 140,0 L bobina 16 spire

THE RESULTS COMPARISON FOR THE INDUCTANCE VARIATION VS. THE TURN 130,0

Inductance [nH]
120,0
WIDTH FOR ONE TURN SQUARE SPIRAL INDUCTOR
110,0
t (μm) de (μm) s (μm) w (μm) LCIBSOC (nH) Lprogram (nH) 100,0
90,0
2 500 5 1 2,764 2,3235
80,0
2 500 5 2 2,482 2,1861 70,0
2 500 5 3 2,316 2,096 60,0
2 500 5 4 2,196 2,0212 50,0
40,0
2 500 5 5 2,103 1,9571 30,0
2 500 5 6 2,026 1,9021 20,0
2 500 5 7 1,961 1,8518 10,0

2 500 5 8 1,904 1,8076 0,0

1 2 3 4 5 6 7 8 9 10
2 500 5 9 1,853 1,7676 Turn width [um]

2 500 5 10 1,808 1,7309 Fig. 12. Results for inductance variation versus the turn width for all the
Similar we compare also de obtained results for the analyzed square spiral inductors.
2 500 2 10 31,815 31,375
distance between the turns variation case. We choose the
2 500 3 10 30,497 30,039
square spiral inductor with six turns for exemplification and 2 500 4 10 29,255 28,793
we detail these results in Table VI and we represent them 2 500 5 10 28,081 27,637
graphical in Fig. 13 for comparison. We consider that the distance between the turns varies form
TABLE VI 1-5(μm) with pass one to keep the exterior diameter constant
RESULTS COMPARISON FOR THE INDUCTANCE VARIATION VERSUS DISTANCE i.e. we want that the inductors to be implemented in the
BETWEEN THE TURNS FOR THE SQUARE SPIRAL INDUCTOR WITH SIX TURNS
integrated circuit in the same area.
t (μm) de (μm) s (μm) w (μm) LCIBSOC(nH) Lprogram(nH)
2 500 1 10 33,218 32,812

Inductance vs. the turn width for one turn square spiral inductor
3,0
L program [nH]
L CIBSOC [nH]

2,0
Inductance [nH]

1,0

0,0
1 2 3 4 5 6 7 8 9 10
Turn width [um]

Fig. 11. Results obtain with our program CIBSOC and with commercial Fig. 13. The results obtain with CIBSOC program and with field solver
field solver for the inductance variation versus the turn width program for inductance variation versus the distance between the turns

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 The errors are acceptable. The program is very simple and The program validation was done by comparison of the
the obtained results are accurate. We compare the inductance applications results obtained with our CIBSOC program with
variation versus the distance between the turns obtain with the measurements results taken form literature and with the
the commercial field solver and with CIBSOC program for all results obtained by modeling the square spiral inductors also
the studied square spiral inductors in Fig. 14. with a commercial field solver design especially for
Analyzing the graphical representations for these variations parameters extraction. Analyzing the results obtained on three
and for the others we can affirm without doubts that CIBSOC different ways we ascertain the results similitude, the small
program is very good, very simple and very efficient. The errors that prove the accuracy of our program.
program is well assembled and logic structured, each module We consider our program very useful for design and
is made for a kind of calculation type or optimization type. optimization of spiral inductors. It is easy to use; the running
The CIBSOC program results were compared with the times are small compared with other similar programs. We
measurements results form the literature and with simulations want to extent our program also for ac inductance calculation,
that we done with the commercial field solver, a program at high frequency.
special create to extract the dc and ac circuits’ parameters.
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文献12、13、14给出了
某些方形电感的测试值

Fig. 14 Plots of the results for inductance variation versus the distance
between turns for all the analyzed inductors.

232

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