SUBSTRATE-DEPENDENT AIR-WOUND INDUCTOR MODEL IN
THE DC-4 GHZ RANGE
E. Benabe, H. Gordon and T. Weller
Electrical Engineering Department, University of South Florida
4202 E. Fowler Avenue, ENB 118, Tampa, FL 33620
benabe@eng.usf.edu, weller@eng.usf.edu, (813) 974-4851, (813) 974-2440
Abstract - This paper describes the development of an equivalent circuit model for air coil inductors in the
DC-4 GHz frequency range which includes substrate-dependent characteristics. The presented model is
applicable to microstrip-mounted components. Limitations of the model are discussed as well as
characteristics intrinsic to the inductors that greatly influence model extraction. A comparison of measured
data and simulated responses is presented for the model.
I. INTRODUCTION
Efforts in reducing fabrication costs and design development time typically rely heavily on the
availability of reliable, accurate component models. These models must take into consideration
characteristics intrinsic to the components as well as parasitic effects introduced into the system
due to properties of printed circuit boards or other circuitry surrounding the component. Air coil
inductors are widely used in RF/microwave circuit design, particularly when low loss
performance is critical, as in filters and some impedance matching circuits. Methods used to
predict the response of these elements include the use of scattering parameter measurements,
mathematical functions, and circuit parameter extraction-based models.
Measurement-based models provide an accurate representation of the device’s response but have
limited use because de-embedding of the component fixture or its surroundings is generally not
done. In addition, this approach requires a large amount of storage allocation. The majority of
equation-based models fail to take into consideration printed circuit board, parasitic or
frequency-related effects. In addition, the inherent complexity in deriving these formulas usually
compromises their accuracy and range of application. The use of equivalent circuit models, on
the other hand, provides physical insight of the component and its fixture, requires minimal
storage and memory allocation, and grants fast simulation time. Furthermore, model
development can be helped with the use of available equations that predict nominal element
values as a function of component geometry, printed circuit board substrate characteristics, and
frequency. Despite the extensive use of air coil inductors, there is minimal documentation
concentrating in equivalent circuit models for these devices.
This paper concentrates on the development of an equivalent circuit model for air coil inductors
(See Figure 1) using a linear circuit simulator. The model presented herein incorporates
theoretical equations to estimate the starting values of some of the high frequency parasitics and
substrate dependent components. Measured scattering parameter data taken in a series-thru (2port) configuration is used in the model extraction process. The objective is to extend the
frequency range of the models beyond the 2nd or 3rd harmonic of the fundamental frequency at
which the inductors are typically used; generally speaking, this places the target maximum
frequency above 3-4 GHz.
Figure 1. Air coil inductor (8 turns)
The measurement methodologies and characteristics of the inductor are discussed in the first
section. Then, the extraction procedure used for the substrate and frequency dependent model is
presented. One clear advantage of the presented model is its portability to different substrate
thickness and pad-stack1 configurations.
II. MEASUREMENT CONSIDERATIONS
The development of an accurate component model depends on reliable measurement data. A
Thru-Reflect-Line (TRL) calibration implemented using uniform microstrip lines was used in
this work. The reference planes were located at the outside edges of the fixture taper section that
connects to the inductor pad-stacks (See Figure 2) and the reference characteristic impedance
was 50. The inductors were measured using a Wiltron 360B network analyzer, a wafer-probe
station, and a personal computer with Wincal software.
Figure 2. Series-thru (2 port) fixture utilized to measure the inductors. Taper sections are
exaggerated for demonstration purposes.
1
pad-stack = microstrip geometry on which the components are mounted
Transmission response, S21 (dB)
The transmission response (magnitude) of a 11.03 nH inductor on 14, 31, and 62 mil FR-4
substrates is shown in Figure 3.
0
-5
-10
-15
-20
-25
Legend
-30
14 mil
-35
31 mil
-40
62 mil
-45
-50
400 1400 2400 3400 4400 5400 6400 7400
Frequency (MHz)
Figure 3. Transmission response (dB) of a 11.03nH inductor on 14, 31, and 62 mil FR-4 substrates.
A relationship between substrate thickness and the response of the inductor is evident from
Figure 3. This dependence will play a key role in the creation of a substrate-dependent model.
Another important factor in the development of the model is the radiation loss of the inductors.
The total loss is extracted from the measured data using the equation below.
LF 1 S112
S 212 Figure 4 shows the total loss of a 11.03nH inductor on the three substrates used in this study.
One point that becomes immediately clear is that the loss has radiation characteristics that
depend on the substrate used, as with the S-parameter measurements shown in Figure 3. This
particular inductor has a physical length of 1.83 mm, and thus the strong radiation peaks occur
when its length is approximately .40 o. Among the collection of inductors modeled in this
work, it was found that, on average, strong radiation is observed when the inductor length is
approximately .45 o. For some of the inductors, the radiation loss was severe as low as 2.0
GHz. This characteristic has implications in the equivalent model extraction process, since
simple R-L-C elements cannot accurately emulate radiation effects; an attempt to do so generally
results in degraded low frequency accuracy in the model. The loss shown in Figure 4 was
confirmed to be radiation-related by sweeping a waveguide horn antenna around several
inductors excited at only one port.
0.8
Legend
14 mil
31 mil
62 mil
Loss factor (mag)
0.65
0.5
0.35
0.2
0.05
-0.1
400
1400
2400
3400
4400
5400
6400
7400
8400
Frequency (MHz)
Figure 4. Radiation losses of the 11.03nH inductor on 14, 31, and 62 mil FR-4 substrates.
III. MODEL EXTRACTION AND RESULTS
To ensure a satisfactory physical representation of the inductor and appropriate substrate trends a
circuit emulating the physical mounting of the inductor on the fixture is created. The substrateand frequency-dependent model for the 11.03 nH inductor is shown in Figure 5.
C p2
L sbp
L sbp
C p1
C p1
L sbp
Cp
ESR
P1
Mtaper
MLIN
L sbp
Cp
ESR
ESL1
ESR
ESL1
ESL2 ESR
ESL2
ESL1
ESR
ESR
ESL1
M L I N Mtaper
MLEF
MLEF
ESR
ESR
ESR
C gs
ESR
C gs
Figure 5. Substrate- and frequency-dependent model for an air coil inductor.
In order to illustrate the approach used to attribute unique characteristics to separate sections of
the inductor, the model is given in its most detailed format. When finally implemented in a
circuit simulator, several neighboring elements can be combined to enhance computational
efficiency (e.g., separate ESR and ESL elements can be lumped together). Table 1 provides a
description of the elements used in the model.
P2
Parameter
Mtaper
MLIN
MLEF
ESR
Lsbp
Cp
ESL1
ESL2
Cgs
Cp1
Cp2.
Description
Models the stepping up or down from the 50 line to the
inductor pad-stack.
Represents the area in the inductor pad-stack where there are
no turns.
Corresponds to that space occupied by the inductor turn on
the pad and takes into consideration fringing effects.
Models coil conductor losses using an equation with DC and
frequency dependent terms.
Models turns not lying on the substrate using an equation that
includes frequency dependence.
Models a parasitic introduced by the capacitance between the
top portion of the turn and the inductor pad-stack.
Bottom portion of inductor turn lying on the substrate.
Top portion of inductor turn lying on the substrate.
Represents the substrate dependent capacitance to ground
formed between the inductor coils and the PCB ground plane.
Models the turn to turn capacitance.
Represents the end to end capacitance.
Substrate
dependent
Yes
Yes
Yes
No
No
No
Yes
Yes
Yes
No
No
Table 1. Description of parameters used in the substrate model shown in Figure 5.
The turn to turn modeling approach consists of breaking up the inductor turns that lay on the
substrate into two parts: top (ESL2) and bottom (ESL1). Table 2 provides a description of the
variables used in the equations to attribute substrate dependent characteristics to the model.
Variable
H (mm)
Lo_a (nH)
Lo_b and
Lobp_b
(nH/MHz)
f
hbi (i = 1, 2, 3)
e
L_Cgs (mm)
c (m/s)
Zo ()
W (mm)
H_sub (mm)
H_subf (mm)
r
Description
Distance from turn to ground plane.
Represents the DC inductance of the coil.
Terms that account for skin effects at high frequencies.
Frequency in MHz.
Second order function coefficients.
Effective dielectric constant.
Fitting factor added to determine the effective length of the inductor.
Speed of light.
Effective characteristic impedance.
Inductor’s turn outside width.
Substrate height.
Fitting factor used to calculate the effective distance from the inductor
to the ground plane.
Substrate dielectric constant.
Table 2. Description of parameters used to calculate substrate dependent terms.
Second order polynomial functions are used to predict the substrate dependent inductance term.
This function depends on the distance between the turn portion being simulated and the board
ground plane, H. The coefficients are interpolated internally within the simulator and a
distinction is drawn for the top and bottom portion of the turns. The coefficients Lo_a and Lo_b
are optimized within the simulator using initial estimates.
ESL1( H , f ) hb2 H 2 hb1 H hb3 Lo _ a Lo _ b * f ESL2 is calculated with a similar equation but using different coefficient and height values.
The capacitance to ground, Cgs, is calculated using a microstrip approximation [1].
C gs e
c * Zo
L _ Cgs
where
Zo 120
W
W
e * 1.393 .667 ln
1.444 H _ sub H _ subf
H _ sub H _ subf
and
e r 1 r 1
2
2
1
1 12
H _ sub H _ subf
W
A fitting factor, H_subf, is utilized to introduce an additional degree of freedom in the
calculation of the effective distance from the inductor to the ground plane. These factors (e.g.
H_sub and L_Cgs) can be attributed to tolerances in the fabrication of the board and nominal
dimensions provided of the inductor’s geometry. This scaling also helps to compensate for the
rounded nature of the coil since the formula was derived for flat conductors [1].
The inductance of the end turn that rests on the inductor pad stack, Lsbp is calculated using the
equation below. This equation assumes no substrate dependence due to the barrier presented by
the pad stack between the coil and the board ground plane. Any substrate dependent inductance
present in this turn will be “absorbed” by the MLEF element. The coefficients Lo_a and Lobp_b
are optimized within the simulator using initial estimates.
Lsbp ( f ) Lo _ a Lobp _ b * f
The end to end and turn to turn capacitors are estimated from equations [2, 3] and then optimized
in the simulation enforcing the following inequality Cp2 < Cp1.
The effective series resistance is calculated as the sum of a DC and AC resistance [4]. The AC
resistance is accounted for in the R_b coefficient.
ESR( f ) R _ a R _ b *
f
An average value of Cp is calculated by calculating the upper and lower limits of the capacitance
using microstrip and parallel plate approximations, respectively. The final value will be obtained
from optimizations that are bounded by the upper and lower limits.
Once starting values and equations are entered for the elements, models corresponding to each
substrate are optimized simultaneously, thus forcing non-substrate dependent elements to be the
same. A comparison to measured data for the 11.03 nH inductor is presented in Figures 6 and 7.
In order to avoid the frequency range in which radiation loss becomes significant, the model is
limited to 4 GHz.
0
S21 and S11 (dB)
-4
-8
Legend
Measured 14 mil
Measured 31 mil
Measured 62 mil
Model 14 mil
Model 31 mil
Model 62 mil
-12
-16
-20
400
900
1400
1900
2400
2900
3400
3900
Frequency (MHz)
Figure 6. S21 and S11 (dB) comparison of measured data and substrate dependent model responses
of 11.03nH inductor on 14, 31, and 62 mil substrates.
S21 (degrees)
-30
-60
Legend
Measured 14 mil
Measured 31 mil
Measured 62 mil
Model 14 mil
Model 31 mil
Model 62 mil
-90
-120
-150
400
900
1400
1900
2400
2900
3400
3900
Frequency (MHz)
Figure 7. Transmission response (degrees) comparison of measured data and substrate dependent
model responses of 11.03nH inductor on 14, 31, and 62 mil substrates.
IV. SUMMARY
During the initial stage of model development, several approaches were used to model the
substrate-dependence of the inductor elements in the equivalent circuit (Figure 5).
Unfortunately, a satisfactory method to represent the complex interaction of the inductor and the
substrate/ground plane using analytical equations was not found. The approach described herein
is to model the inductor elements using polynomial functions that depend on the substrate
thickness, as mentioned in the preceding section. This approach is valid in the general sense as
differences in the dielectric constant of the substrate should not affect the inductance.
Furthermore, the equations used to determine the value of capacitor elements do account for the
substrate thickness and dielectric constant.
We feel that the presented model should be of significant use in improving the accuracy of circuit
simulations that incorporate air coil inductors, and thereby improve the probability of first pass
success designs. The model has been applied to inductors with a wide range of dimensions and
inductance values, as listed in Table 3. Also, the results relating to the radiation characteristics
of these inductors should be of interest, particularly when one is concerned with performance at
higher order harmonic frequencies.
Parameter
Inductance (nH)
Number of turns
Length (mm)
Outside width (mm)
Inside height (mm)
Wire diameter (mm)
Minimum value
4.22
3
1.83
1.76
.89
.32
Maximum value
59.71
12
7.00
3.43
2.03
.64
Table 3. Range of inductance and physical dimensions of inductors simulated with the substrate
dependent model.
V. REFERENCES
1. David M. Pozar, Microwave Engineering, John Wiley & Sons, Inc., New York, 1998.
2. Grandi, G. et al., Stray Capacitances of Single-Layer Air-Core Inductors for High-Frequency
Applications, IEEE Industry Applications Society Meeting, 1996, p. 1384-1388.
3. Massarini, Antonio and Kazimierczuk, Marian K., Self-Capacitance of Inductors, IEEE
Transactions on Power Electronics, Vol.12, No.4, 1997, p. 671-76.
4. Lafferty, R. E., Capacitor Loss at Radio Frequencies, IEEE Transactions on Components,
Hybrids, and Manufacturing Technology, Vol. 15, No. 4, 1992, p.590-593.