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Embedding magnetic layer in inductors is an attractive option for increasing inductance density, which is a critical issue for radiofrequency applications. In this work, a magnetostatic model for square spiral inductors incorporating a magnetic layer

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A New Analytical Model for Square Spiral Inductors Incorporating a Magnetic Layer
Cevdet Akyel
#1
, Slobodan Babic
#2
, Edvin Skaljo
#3
#
1
École Polytechnique
,
Département de Génie Électrique
, H3C 3A7, Montréal, Succ
.,
Centre-Ville, Québec, Canada
#
2
École Polytechnique
,
Département de Génie Physique
, H3C 3A7, Montréal, Succ
.,
Centre-ville, Québec, Canada
#
3
BH Telecom
, Sarajevo, Zmaja od Bosne
88, Bosnia and Herzegovina
cevdet.akyel@polymtl.ca
,
slobodan.babic@polymtl.ca, skaljo@bhtelecom.ba
Abstract
— Embedding magnetic layer in inductors is an attractive option for increasing inductance density, which is a critical issue for radiofrequency applications. In this work, a magnetostatic model for square spiral inductors incorporating a magnetic layer is developed. This analytical model provides a fast and accurate calculation of inductance for integrated inductors with embedded magnetic layers either for regular cases or a singular case. In the presented model we replaced square spiral inductors with by square rings and we shown that this replacement gives accurately alike results. The results of the presented approach have been confirmed by already published data
.
Index Terms
— Inductors, inductance, magnetic layer, square spiral, rings.
I. I
NTRODUCTION
An on-chip spiral inductor is an important component for radio frequency integrated circuits (RFICs) such as low-noise amplifiers, voltage-controlled oscillators, impedance matching networks, inductive coils, apparatus for bio-medical telemetry. Many works have been done on inductor design and modeling. Accuracy in the inductor model is an important part of RFIC design. An inductor typically has one or more "turns" that concentrate the magnetic field flux induced by current flowing through each turn of the conductor in an "inductive" area defined within the inductor turns. In some cases the aspect ratio of the turn can be large so that the turn forms an ellipse or a rectangle. There is a growing need for increased inductance density as the inductors occupy increasingly larger portion of the total circuit area. While the initial efforts were focused on the increased quality factor of inductors, there is a growing need for increased inductance density as the inductors occupy increasingly larger portion of the total circuit area. This is because the scaling, which progressively reduces the area occupied by active devices, does not quite shrink the area taken by passive devices, which is dominated by inductors. The most promising approach to increase the inductance density has been the incorporation of ferromagnetic material into the inductors as a part of their structure [1-10]. In [1] a magnetostatic model for square-
shaped spiral inductors above a magnetic layer is developed to calculate the inductance. This method, with some assumptions, has been obtained in the form of a double integral where the double numerical integration is required. In this paper we give the analytical solution of this integral which appears in the form of elementary analytical expressions. This expressions permit fast, easy and accurate calculation of inductance either for regular cases or a singular case. We confirmed the presented formula by results given in [1].
II. B
ASIC
E
XPRESSIONS
S.
Pinhas at al. treated a
basic configuration of a square spiral inductor above a magnetic layer, [1], (See Fig.1). The metallic square-spiral is assumed to have zero thickness, and to be located in free space. The configuration is shown in Fig. 1(a). The strip width of the square is denoted by
W
, its strip spacing by
S
0
, its numbers of turns
N
(
N
is an integer), and its outer side by
a
. The distance between the square spiral and the magnetic layer is denoted
D
0
. In order to facilitate the calculations, they used the following replacement and additional assumptions, as described below. They replaced the square spiral by concentric square rings with the same strip width
W
and strip spacing
S
0
. In this case the number of the concentric square rings is equal to the number of turns
N
of the square spiral. Also, the outer side of the outermost square ring is equal to the outer side ‘
a
’ of the square spiral [3-4]. They assumed that
a
= 2
N
(
W
+
S
0
). The configuration with this replacement is shown in Fig. 1(b), [1].They calculated the inductance by a magnetostatic model and they expected a good accuracy for sufficiently low frequencies. Also, they assumed that the total current in each square ring was the same for all concentric square rings and that the surface current density was uniform till ‘the diagonal’ of a corner of a square ring. When the current density is rotated by 90
0
it keeps its magnitude. The magnetic layer is infinite in the
x
and
y
directions, and the permeability of the magnetic layer is isotropic and infinite. They take the permeability of the half space
z <
0 to be isotropic and infinitive so that the magnetic-layer thickness does not appear in their calculations. The permeability of the half-space
z
> 0 remains the permeability of free space,
μ
0
. In the half-space
z
> 0 the vector potential
A
is calculated by the method of current image [11]. In this space the vector
A
is given by [1],
Proceedings of the Asia-Pacific Microwave Conference 2011978-0-85825-974-4 © 2011 Engineers Australia967
Fig.1. The configuration of a square spiral inductor above a magnetic layer (a) and the replacement of the square spiral by concentric square rings with the same strip width and strip spacing (b), [1].
001212
44
(1)
= +
SS
KK RR
AdSdS
µ µ π π
where
R
1
and
R
2
are the distances from current elements to the point where
A
is calculated,
dS
is a differential area-element,
S
1
is the
z
=
D
0
plane, and
S
2
is the
z
= -
D
0
plane.
K
is the srcinal surface-current density in the
z
=
D
0
plane. The self inductance is calculated by [1],
21
1
(2)
=
S
I
LAKdS
From (1) and (2) they obtained the total inductance of considering system,
0
(3)
= +
M
LLL
L
0
is the self inductance for free space (i.e., for the permeability
μ
0
for all values of
z
).
L
M
is the self inductance due to the magnetic half space
z <
0 with the infinite permeability. Using all previously mentioned assumptions to facilitate the calculations S.
Pinhas at al. obtained the following equation for
L
M
:
12123113
()()'0'20211()()()()'20()()
[
2(,,)'(,,)(4)
]
= =
= +
blbk N N M l k alak bk bl alak
dx
LdxFxxDdxdxFxxD
W
µ
π
where
1112223330
(1)(), ()(), () 222()(), (), ()()22()
alaak alWblalWak NN ak bkakWakWbkakW N aNWS
−= − = + = − += + = − = += +
''''20011''00
(,,)(,,)()()(,,)(,,)
xxxx FxxDpxxDFF pxxDpxxD
+ −= −
''2200
(,,)()4
xxDxxD
= − +
221
()ln(1)1
Fuuuuu
= + + − +
From (2) it is possible to find
L
0
as a limiting case of
L
M
with
D
0
approaching zero,
000
lim (5)
→
=
M D
LL
with
L
M
and
L
0
obtained from (2) and (3), the total inductance can be calculated from (1) for any length parameter values of the inductor configuration.
III. C
ALCULATION
M
ETHOD
Based on the previously given model, S. Pinhas and al. calculated numerically the inductance of inductors with various structures using the double integration. From the presented ring configuration it is obvious that an analytical solution of equation (2) exists. Even though it is very tedious work we think that the analytically solved problem will considerably facilitate the inductance calculation especially in the treatment of singular cases. This enormous calculation leads to relatively simple elementary functions. Thus, we give the final analytical solutions for
L
M
and
L
0
.
A. L
M
calculation (
D
0
0) Integrating (2), [12] the inductance
L
M
can be obtained in an analytical form:
[ ]
012211
(6)
2
= =
= +
NN Mnnlk
LSS W
µ π
where
4411121111121321222331323
(1)(,), (1)(,) (), (), () (), (), () (), (),
nnnnnnnnnnnn
SIlpSIlqlblpbkqbk lalpbkqbk lalpakqa
− −= =
= − = −= = == = == = =
3414243
() (), (), ()
k lblpakqak
= = =
220122022012202222220010220
()4()(,)sinh2()4()()12sinh6()48()()22(6)sinh32322
−−−
+ − −+= +− +− − +−+− +− − − +−+ ++
nnnnnnnnnnnnnnnnnnnnnnnnnnn
lsDlsls IlslsDlslsDlslsD DlslsD ssDl sD
968
222110022002233110002220002222212100002222220000
+
2(6)4()sinhsinh32222()284tantan332224222tan2tan224224(
− −− −− −
− − −− −+− ++ ++ +− + − +++ + + +
nnnnnnnnnnnnnnnnnnnnnnnnnn
llDsDlsls DlDlsD DDls D DlsDllsDslsD DlDs DlsDDlsD
2222200
;
)42 or 32
− − + +=
nnnnnnn
lsDlsD spq
B. L
0
calculation (
D
0
0) Finding the limiting case of (5) the inductance
L
0
can be obtained in an analytical form:
[ ]
0012211
(7)
2
= =
= +
NN nnlk
LPP W
µ π
where
2
441112111112132122331
(), (), () (), (), () (),
(1)(,), (1)(,)
− −= =
= = == = ==
= − = −
nnnnnnnnnnnn
lblpbkqbk lalpbkqbk lal
PRlpPRlq
3233414243
(), () (), (), ()
= == = =
pakqak lblpakqak
231222213311
()()(,)sinh[()]266()()()sinh23222sinhsinh; or 3232
−−− −
− −−= − − −− ++ − ++ +−+ =
nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn
lslslslssignlslslslslslsls slls spq sl
R
If
l
n
=
s
n
(singular case)
3311
2222(,)sinh[()]sinh[()]33
nnnnnnn
ls Rlssignlsigns
− −
= =
Thus, we obtained the analytical solution for the square spiral inductors incorporating a magnetic layer. Obtained expressions are suitable for fast and accurate calculations of the inductance for square spiral inductors with magnetic layer.
IV.
E
XAMPLES
The inductance of inductors with various structures was calculated analytically and compared by results obtained by the double numerical integration given in [1]. The total inductance
L
is calculated as a function of the number of turns (
N
) for various distances between the metallic-spiral plane and the magnetic layer (
D
0
), while keeping the strip width and strip spacing fixed at 10 and 5
μ
m, respectively, [1]. TABLE I
W
= 10
μ
m,
S
0
= 5
μ
m,
D
0
= 0.5
μ
m
N
L
M-NUMERICAL
(
n
H) [1]
L
M -ANALYTICAL
(
n
H)
This work
(6)
Discrepancy
(%) 1 0.02050152 0.02047538 0.128 2 0.12707460 0.12698409 0.071 3 0.39418411 0.39399100 0.049 4 0.89650103 0.89616710 0.037 5 1.70875897 1.70824599 0.030 6 2.90571993 2.90498969 0.025 7 4.56216125 4.56117551 0.022 8 6.75286945 6.75158999 0.019
TABLE II
W
= 10
μ
m,
S
0
= 5
μ
m,
D
0
= 5
μ
m
N
L
M-NUMERICAL
(
n
H) [1]
L
M -ANALYTICAL
(
n
H)
This work
(6)
Discrepancy
(%) 1 0.00582058 0.00582058 0.00 2 0.06121310 0.06121311 1.63E-05 3 0.23498410 0.23498411 4.26E-06 4 0.59986846 0.59986848 3.34E-06 5 1.22961422 1.22961426 3.25E-06 6 2.19838435 2.19838441 2.73E-06 7 3.58055303 3.58055310 1.96E-06 8 5.45061681 5.45061690 1.65E-06
TABLE III
W
= 10
μ
m,
S
0
= 5
μ
m,
D
0
= 25
μ
m
N
L
M-NUMERICAL
(
n
H) [1]
L
M -ANALYTICAL
(
n
H)
This work
(6)
Discrepancy
(%) 1 0.00024658 0.00024658 0.00 2 0.00742810 0.00742810 0.00 3 0.04763757 0.04763757 0.00 4 0.16540984 0.16540984 0.00 5 0.41639924 0.41639924 0.00 6 0.86256099 0.86256099 0.00 7 1.56958748 1.56958748 0.00 8 2.60552346 2.60552346 0.00
It is clear from Tables I-III that all results are in an excellent agreement and that equation (6) is correct analytical solution of equation (4) for calculating the inductance
L
M
. This formula can be also served to calculate the inductance
L
0
. The exact approach to calculate the inductance
L
0
is to find the limit of equation (5) when
D
0
0. It gives equation (7) from which
L
0
can be calculate easily. Obviously, the inductance
L
0
does not depend of the distance
D
0
and it increases with increasing
N
as expected. In Table IV we give the comparative results when equation (7) is used. All results are in an excellent agreement. Finally, the total inductance of considering system can be obtained by equation (3). We give results for one case in Table V comparing results obtained analytically (this approach) and numerically, [1].
969
All results are in an excellent agreement. Finally, the total inductance of considering system can be obtained by equation (3). We give results for one case in Table V comparing results obtained analytically (this approach) and numerically, [1]. TABLE IV
W
= 10
μ
m,
S
0
= 5
μ
m,
D
0
= 0
N
L
0
PURLY-ANALYTICAL
(
n
H)
Eq.
(7)
L
0
-NUMERICAL
(
n
H)
[1]
Discrepancy
(%) 1 0.02463516 0.02457728 0.24 2 0.14201468 0.14178982 0.16 3 0.42666932 0.42616607 0.12 4 0.95329019 0.95239659 0.09 5 1.79662099 1.79522472 0.08 6 3.03142983 3.02941835 0.07 7 4.73249813 4.72975878 0.06 8 6.97461534 6.97103535 0.05
TABLE V
W
= 10
μ
m,
S
0
= 5
μ
m,
D
0
= 0.5
μ
m
N
L
NUMERICAL
=
L
0
+
L
M
(
n
H) [1]
L
ANALYTICAL
=
L
0
+
L
M
(
n
H)
This work
(3)
Discrepancy
(%) 1 0.04507880 0.04511054 0.070 2 0.26886442 0.26899877 0.050 3 0.82035018 0.82066033 0.038 4 1.84889762 1.84945730 0.030 5 3.50398369 3.50486699 0.025 6 5.93513828 5.93641952 0.022 7 9.29192003 9.29367364 0.019 8 13.7239048 13.72620533 0.017
TABLE VI
W
= 10
μ
m,
S
0
= 5
μ
m,
N
= 5
D
0
L
NUMERICAL
=
L
0
+
L
M
(
n
H) [1]
L
ANALYTICAL
=
L
0
+
L
M
(
n
H)
This work
(1)
Discrepancy
(%) 0 3.59044944 3.59324199 0.078 0.1 3.58702959 3.57386264 0.368 1 3.42927395 3.43064312 0.040 10 2.70460125 2.70599752 0.052 100 1.82674690 1.82814317 0.076
From Table VI we confirm that the total inductance
L
increases with
N
as expected, while it decreases with increasing
D
0
, indicating the diminished effect of the magnetic material as it moves away from the spiral. The last calculation (Table VI) represents the total inductance
L
calculated for different distances
D
0
= 0; 0.1; 1, 10; 100
μ
m and the constant
N
= 5. Analytically obtained results are in an excellent agreement with those given in [1]. It was expected because we found analytical solutions of the total inductance
L
given in the form of a double integral, [1]
.
A
Matlab
implementation of previous formulas is available from the authors on the request.
V.
C
ONCLUSION
In this work, a new analytical model for square spiral inductors incorporating a magnetic layer has been developed, which enables the calculation of the inductance either for regular cases (
D
0
0) or a singular case (
D
0
= 0). This model can serve as an effective tool for a fast estimation of inductance for square spiral inductors with magnetic layer. All results obtained by this approach are in an excellent agreement with already published data.
A
CKNOWLEDGEMENT
This work was supported by Natural Science and Engineering Research Council of Canada (NSERC) under Grant RGPIN 4476-05 NSERC NIP 11963.
R
EFERENCES
[1] S. Pinhas, S. W. Hwang and J. S. Rieh,“A Magnetostatic Model for Square Spiral Inductors Incorporating a Magnetic Layer,”
IEEE Trans. Magn.
Vol. 44, No.8, pp. 2085-2087, August 2008. [2] B. Rejaei, J. L. Tauritz, and P. Snoeij, “A predictive model for Si-based circular spiral inductors,”
Proc. 1st Topical Meeting Silicon Monolithic Integrated Circuits RF Syst.
, Ann Arbor, MI, 1998, pp. 148–154. [3] M. Yamaguchi, M. Baba, and K. I. Arai, “Sandwich-type ferromagnetic RF integrated inductor, ”
IEEE Transactions on Microwave Theory and Techniques
, Vol. 49, no. 12, pp. 2331-2335, 2001. [4] B. Viala, A. S. Royet, R. Cuchet, M. Aid, P. Gaud, O. Valls, M. Ledieu, and O.Acher, “RF planar ferromagnetic inductors on silicon, ”
IEEE Transactions on
Magnetics
,Vol. 40, no. 4, pp. 1999-2001, 2004. [5] A. M. Crawford, D. Gardner, and S. X. Wang, “High-frequency microinductors with amorphous magnetic ground planes,”
IEEE Transactions on Magnetics
, Vol.38, no. 5, pp. 3168-3170, 2002. [6] W. A. Roshen and D. E. Turcotte, “Planar inductors on magnetic substrates,”
IEEE Transactions on Magnetics
, Vol. 24, no. 6, pp. 3213-3216, 1988. [7] W. A. Roshen, “Effect of finite thickness of magnetic substrate on planar inductors,”
IEEE Transactions on Magnetics
, Vol. 26, no. 1, pp. 270-275, 1990. [8] W. A. Roshen, “Analysis of planar sandwich inductors by current images,”
IEEE Transactions on Magnetics
, Vol. 26, no. 5, pp. 2880-2887, 1990. [9] W. G. Hurley and M. C. Duffy, “Calculation of self and mutual impedances in planar magnetic structures,”
IEEE Transactions on Magnetics
, Vol. 31, no. 4, pp. 2416-2422, 1995. [10] J. N. Burghartz and B. Rejaei, “On the design of RF spiral inductors on silicon,”
IEEE Trans. Electron Devices
, Vol. 50, no. 3, pp. 718–729, Mar. 2003. [11] W. R. Smythe
, Static and Dynamic Electricity
. New York: McGraw-Hill, 1950. [12] I. S. Gradshteyn and I. M. Rhyzik,
Tables of Integrals, Series and Products,
Dover, New York, 1972.
970

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