Telfor Journal, Vol. 1, No. 2, 2009. 73
Modelling and Simulation of RF Multilayer
Inductors in LTCC Technology
of its inductance and Q factor. The verification of results
Abstract – This paper is aimed at presenting the models obtained on the basis of physical model has been
and characteristics of two types of inductors designed in performed by using the Ansoft HFSS electromagnetic
LTCC (Low Temperature Cofired Ceramic) technology. We simulator .
present the physical model of a 3D planar solenoid-type
inductor and of a serial planar solenoid-type inductor for the
RF (radio frequency) range. To verify the results obtained by
using these models, we have compared them with the results II. DERIVING THE PHYSICAL MODEL OF AN INDUCTOR
obtained by employing the Ansoft HFSS electromagnetic DESIGNED FOR LTCC TECHNOLOGY
simulator. Very good agreement has been recorded for the
The view of inductor structures designed for LTCC
effective inductance value, whereas the effective Q factor
value has shown a somewhat larger deviation than the technology is shown in Figure 1.
Keywords - Q factor, Physical model of inductor,
Inductance, Planar solenoid inductor.
A N integrated inductor is a component finding ever
wider applications in low-noise amplifiers, active
mixers, voltage-controlled oscillators, power converters,
miniature sensors, filters, etc. . The main features of an (a) 3D Inductor.
inductor are its inductance and Q factor.
Various techniques are used for fabricating a
component or circuit, depending on their characteristics
and dimensions to be achieved as well as on the place of
their intended use. Technologies used most commonly for
producing microstructures are monolithic technology ,
thick-film technology  and low temperature cofired
ceramic (LTCC) technology , . A designer of
microelectronic structures should be familiar with as many
technologies as possible to be capable of selecting a (b) Serial Inductor.
technology that will give the optimal characteristics of a Fig. 1. 3D View of Inductor Structures.
designed component. In addition to various technologies,
a designer’s knowledge should also include the electrical As may be seen from Figure 1, inductor structures have
characteristics describing an inductor. been designed using five dielectric stripe produced by
The aim of this paper is to derive a physical model of DuPont company – DuPont 951PT . DuPont’s silver
two types of planar solenoid inductors for operation in the pastes DuPont 6142D  were used for conducting lines.
radio frequency range. The simulation of inductor Inductors were designed so as to have the same
structures using electromagnetic simulators can give data conducting line widths (100 m), the same conducting line
on the effective values of inductance and Q factor as well thicknesses (9 m) and to occupy identical areas on a chip.
as on resonance frequency, but it cannot provide Via diameter was 100 m. Both inductors had the same
information on parameters such as series resistance, series dimensions (5.4 x 2.2 x 0.5)mm.
capacitance, parasitic capacitance to the substrate, etc. Knowledge of the physical model of inductor is very
This is why deriving the physical model of an inductor can important for its application, because it permits
show exactly how these values affect the effective values recognizing all the characteristic parameters affecting the
inductor. Precise inductor modeling is rather demanding
A. eli , Faculty of Technical Sciences in Novi Sad, Serbia (e-mail: because of numerous parasitic effects occurring at high
firstname.lastname@example.org). frequencies .
74 Telfor Journal, Vol. 1, No. 2, 2009.
Figure 2 presents the physical structure of 3D inductor obtained by using equation (6) and the total mutual
with the characteristic parameters describing the inductor inductance (LM2U) by summing the single mutual
at high frequencies (series inductance LS, series resistance inductance between these segments ,
RS, series capacitance CS, parasitic capacitance to the
substrate CK), and Figure 3 its equivalent -diagram. The
physical structure and equivalent -diagram of serial
inductor are identical to those of 3D inductor.
Fig. 4. Slant Segments at an Angle .
cos l2 l1
LM 2 2 l1 arth ( ) l 2 arth ( )
4 R1 R2 R1 R4
l2 l1 d
Fig. 2. Physical Structure of 3D inductor. arth ( ) arth ( ) ,
R3 R4 R2 R3 sin
where parameter is given by equation (7)
d 2 cos l1 l 2 sin 2 d 2 cos l1 sin 2
dR1 sin dR 2 sin
d 2 cos sin 2 d 2 cos l 2 sin 2
arctg arctg ,
dR 3 sin dR 4 sin
and auxiliary parameters R1, R2, R3, R4 are given by
Fig. 3. Equivalent -diagram of 3D inductor. 2
R1 d2 l1 2 l2 2 2 l1 l 2 cos ,
Series inductance LS is the sum of the selfinductances of 2 2 2 2
R2 d l1 2 l1 cos ,
inductor’s single segments (LO) and the mutual 2 2 2 2
R3 d 2 cos ,
inductances between the parallel and acute line segments
of inductor (LM1 + LM2), 2
R4 d2 2
l2 2 2 l2 cos .
LS LOU LM 1U LM 2U . (1) The series resistance of 3D inductor is given by equation
The selfinductance of one line segment is obtained
N l ll 2 l lK
using equation (2) and the total selfinductance (LOU) by RS
summing the selfinductances of all single line segments 2 wl tl 2 wl tl
N l tvia 2 N l tl
with the selfinductance introduced by vias (Lvia), equation ,
(3), 2 wvia / 2 w pad 2
O ll 2ll wl tl where N=8 is the total number of 3D inductor’s acute
LO ln 0,50049 , (2)
2 wl tl 3ll angles, l = 2.97*10-8 m is the specific resistance of
conducting line, lK is the length of connection segments
o r 1 tvia 2 2
Lvia tvia sinh tvia wvia wvia , (3) and wpad is the diameter of extracting at terminal of
conductive segment (pad). The parameter describing the
where ll, wl , tl are the lengths, widths and thicknesses of dependence of resistance on frequency is referred to as
line segments, respectively, wvia and tvia are via diameter skin depth and is given by equation (10).
and length, and r = 0.99998 is the relative permeability of
conducting line. . (10)
Mutual inductance between two parallel inductor
The series capacitance of 3D inductor is given by
segments is obtained by using equation (4) and the total
equation (11) and includes the capacitance between
mutual inductance (LM1U) by summing the single mutual
parallel segments, acute segments and the capacitance
inductance between parallel segments (9),
between extracting at terminal of conductive segment (via
O r l ll 2 G2 G pad)
LM 1 ll ln l 1 1 , (4)
2 G G ll 2 ll
N 2 ll tkor sin / 2 tl N 1 ll tl
CS 0 rk 0 rk
where parameter G is given by equation (5), tkor cos / 2 wl 2
tvia p/2 2
1 1 1 N w pad / 2 2 wvia / 2 2
ln G ln d
12 d / wl
60 d / wl
164 d / wl
, (5) 0 rk ,
and d is the distance between parallel segments. where rk = 7.8 is the relative permittivity of ceramic
Mutual inductance between the two segments of an substrate, and Tker = 100 m is the ceramic substrate
inductor that are positioned at an angle , Figure 4, is thickness. Parasitic capacitance to the substarte is
eli : Modelling and Simulation of RF Multilayer Inductors in LTCC Technology 75
calculated using equation (12) and exists between the III. COMPARISON OFRESULTS CALCULATED USING
inductor itself and the metal coating deposited from the INDUCTOR MODEL WITH SIMULATION RESULTS
lower side of the component. Figures 5 and 6 present a comparison of the results of
N N the effective values of the inductance and Q factor of 3D
wl l wl l
1 2 1 2 inductor obtained by calculation of the parameters of
CK o rk o rk
2 2Tker 2 3Tker inductor’s physical model and by simulation. The same
2 meshing was used for both inductor types in simulation.
1 2 wl l K 1 N w pad / 2
o rk o rk .
2 3Tker 2 2Tker
These equations correspond to the physical model of 3D
inductor. To calculate the values of the elements of
equivalent -diagram of serial inductor, we use the
following equations. The series resistance of serial
inductor is calculated on the basis of equation (13),
2 ll p
RS Ng Nd
2 wl tl 2 wl tl 2 wl tl
Ng Nd t t 2 Ng Nd
l via l l
2 wvia / 2 w pad / 2
where Ng = 8 is the total number of acute segments on the
upper conducting layer, Nd =8 is the total number of acute
segments on the lower conducting layer, lp is the end Fig. 5. Inductance of 3D Inductor.
segment connecting the segments of the upper and lower
conducting layer. As can be seen, in the case of 3D inductor the value of
The series capacitance of serial inductor is obtained on inductance obtained by calculation is slightly larger than
the basis of equation (14) and includes the capacitance the simulated value. The calculated inductance value of
between the parallel segments, acute segments, the 3D inductor at 1 GHz frequency is 6.53nH. The simulated
capacitance between extracting at terminal of conducting inductance value of 3D inductor at 1GHz frequency is
segments (via pad) and the capacitance between the 5.21nH. The inductance increases with the increasing
overlapping parts of the segments of the upper and lower frequency.
Ng Nd 2 ll t kor sin / 2 tl Ng Nd 2 ll t l
CS o rk o rk
t kor cos /2 2
t via 2
2 2 (14)
N g wl2 Ng Nd w pad / 2 wvia / 2
o rk o rk ,
where tkor is the distance between the two parallel
segments of serial inductor.
The parasitic capacitance to the substrate of serial
inductor is calculated on the basis of equation (15)
1 N d wl l 1 N g wl l 1 2wl l K
CK O rk o rk o rk
2 2Tker 2 3Tker 2 3Tker
1 wl l p 1 N g N d w pad / 2
o rk o rk .
2 3Tker 2 2Tker
The effective values of the inductances and Q factor of Fig. 6. Q Factor of 3D Inductor.
3D and serial inductors are calculated on the basis of
equations (16) and (17), The maximum calculated Q factor value of 3D inductor
w L 2 2
w L 2 2 was 91.33 and was obtained at 1.7GHz frequency. The
LS RS2 1 S
CS CK 1 S
maximum Q factor value obtained by simulation was
R S2 R S2
L 2 2
, (16) 87.42 and was obtained at 1.8GHz. It can be seen that the
w 2 w 2 L2S deviation is very small. This is due to the fact that many
LS R 1 S CS CK
RS R S2 parasitic effects are taken into account in the calculation.
wLS RS2 CS C K Figures 7 and 8 present a comparison of the results of
Q 1 w2 LS CS CK . the effective values of the inductance and Q factor of
RS LS (17)
serial inductor obtained by the calculation of the
parameters of inductor’s physical model and by
76 Telfor Journal, Vol. 1, No. 2, 2009.
This paper presents the physical models and
characteristics of two inductor types. The values of the
calculation and simulation for the effective inductance and
Q factor value are compared.
The inductance of serial inductor is larger than that of
3D inductor. This difference is introduced by the larger
number of segments of serial inductor, the larger number
of vias as well as the larger number of pads. The Q factor
of serial inductor is lower than the Q factor of 3D
inductor. This results from the higher values of the
elements of the equivalent -diagram obtained with the
serial inductor compared with the 3D inductor.
Briefly, if the inductance value is a decisive criterion
Fig. 7. Inductance of Serial Inductor. for the intended application of the inductor, it should be
realized as a serial one; on the other hand, if the Q factor
A good agreement between the calculated and simulated value is decisive, the inductor should be realized as a 3D
inductance value was also obtained in the case of serial inductor .
inductor. A slightly larger calculated inductance value was
obtained compared with the simulated one. The calculated ACKNOWLEDGEMENT
inductance value at 1GHz frequency is 10.13nH. The This paper forms part of a Master Degree Thesis and
simulated inductance value at 1GHz frequency is 8.8nH. has been prepared under the supervision of Ms Ljiljana
Živanov, PhD and Mr Goran Radosavljevi , MS, both
with the Faculty of Technical Sciences in Novi Sad.
Research and results reported in the paper have been
partially supported by Serbia's Ministry of Sceince and
Technology Development under Grant No 11023.
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