Microwaves & RF September
1999
Free EM Software Analyzes Spiral Inductor On
Silicon
One of the more difficulttomodel components, the spiral
inductor, can be flawlessly analyzed with this free and powerful
electromagnetic simulator.
By James C. Rautio President
Sonnet Software, Inc., 1020 Seventh North St., Suite 210,
Liverpool, NY 13088; (315) 4533096, FAX: (315) 4511694,
email: info@sonnetusa.com,
Internet: http://www.sonnetusa.com/
SPIRAL inductors appear simple but are difficult to analyze. With
the complexity of a conductive silicon (Si) substrate, the problem
challenges the most advanced modern analysis tools. Amazingly, a
freeofcharge electromagnetic (EM)analysis program called Sonnet Lite
can be used to provide a highaccuracy solution to this problem.
Sonnet Lite performs analysis of the threedimensional (3D) EM fields
around planar circuits. It is based on the industrystandard suite of EM
software tools from Sonnet Software (Liverpool, NY). Sonnet Lite can be
supplied on a compactdisc readonly memory (CDROM) from
Sonnet^{®} (see
phone number at the end of article) for only the cost of shipping, or
it can be downloaded free of charge from the company's website (also
listed at the end of the article) [the download requires approximately
14MB available harddisk memory]. This is not a demonstration program but
a fully operating (albeit scaleddown) version of Sonnet's em^{®}
program (see Microwaves & RF, August 1999, p. 152 for a review of Sonnet Lite). Sonnet Lite can handle all
smalltomediumsize problems using up to 16MB randomaccess memory (RAM)
for a full matrix solution, up to two metal levels (and three dielectric
layers), and as many as four ports. With care, a large number of problems
can be solved within these constraints. The spiral inductor on Si is one
of these problems.
The basic characteristic of Sonnet Lite (and its fullfledged
counterpart) that makes it ideal for analyzing a spiral inductor on Si is
the accuracy when conducting dielectrics are analyzed. The software is
based on Fast Fourier transform (FFT) analysis. Currently, there is no
accuracydegrading numerical integration. Also, all fields in each layer
are represented as a (large) sum of simple rectangular waveguide modes
(the sidewalls of the conducting box containing the circuit form the
rectangular waveguide). When loss is present, only the characteristic
impedance and velocity of propagation of each waveguide mode need to be
changed. Since these characteristics are known exactly for rectangular
waveguide, substrate conductivity is included with precisely the same
accuracy as seen in an equivalent lossless analysis. This is difficult to
perform in analyses based upon direct numerical integration of an
underlying Green's function. Thus, the only problem to overcome in
analyzing an inductor on Si using Sonnet Lite is to cast the problem into
a form that satisfies the analysis constraints.
Figure 1 shows the baseline inductor.
Fig 1. The baseline inductor has coplanar groundreturn lines for
low loss.
To reduce substrate loss, a coplanar inductor is being used. The wide
conductors in the figure represent the coplanar ground strips.
Groundreturn current flows in these strips. Thus, the electric field from
the signal line to ground need not pass through the entire Si substrate.
This results in lower loss.
The inductor's transmission lines are 8 µm wide with 8µm separation.
The ground strips are 16 µm from the edge of the inductor. The substrate
is 1000 µm thick with a relative dielectric constant of 12 and a
conductivity of 20 S/m. In addition, there is a 1mm layer of silicon
dioxide (SiO_{2}) with a relative dielectric constant of 4.0 on
top of the Si. Most of the inductor is on top of the SiO_{2}
layer. The connection from the center of the inductor out is below the
SiO_{2} on top of the Si. A metal loss of 0.04 /square (plus
appropriate skin effects) is included.
The task at hand is to simplify the model of the inductor without
affecting analysis accuracy. The Sonnet software meshes only the metal
surface and problem size increases rapidly with the number of subsections.
Thus, the number of subsections should be reduced as much as possible.
The first way this can be performed is by making the subsection size as
large as possible. With 8µmwide lines on 16µm centers, this is simple
enough using an 8µm cell size. Now, all 8µm lines are meshed one cell
wide. The impact of this subsection size on analysis error is
quantitatively evaluated later.
Another way that the subsection count can be reduced is by reducing the
metal area. Since the coplanar groundreturn lines are wide, it would be
helpful if they could be eliminated from the analysis. Since Sonnet Lite
places a perfectly conducting box sidewall at the edge of the substrate,
the coplanar groundreturn lines can be eliminated. This is performed by
removing each ground strip and substituting a box sidewall in its place.
Now, the ground current flows in the box sidewalls rather than in the
ground strips.
As a modification, the sidewalls are placed 24 µm from the inductor
(the ground strips were originally 16 µm away). Since this is a major
modification, it is important to analyze the resulting effects, and an
EManalysis program supports this comparison to the original circuit
conditions. Figure 3 shows a comparison of the two approaches to
this analysis, with reflection parameter S_{11} differing by
approximately 0.5 dB in the two approaches while forward transmission,
S_{21}, differs up to 2 dB, but only at the higher frequencies
where S_{21} is down to approximately 20 dB.
Click
to see enlarged image
Fig 3. The inductors of Figs. 1 and 2 were compared using Sonnet
Lite software. Note that the differences in S_{11} are
approximately 0.5 dB while the differences in S_{21} approach 2
dB, but only at higher frequencies.
If the differences are assumed to be small compared to the design
requirements, it is possible to adopt the second approach and proceed with
the evaluation of the inductor of Fig. 2.
Fig 2. The inductor of Fig. 1 was modified so that the box sidewalls
take the groundreturn current. Removing the ground strips results in a
faster analysis.
Using a typical Pentiumbased personal computer (PC) running at a
450MHz clock speed, the modified inductor of Fig. 2 can be analyzed in 1
s/frequency using only 1 MB of randomaccess memory (RAM). The original
coplanar inductor requires 2MB RAM and 2s/frequencyanalysis time. This
may seem like a small difference at this point, but it could prove to be
significant later. For those who are concerned with the 0.5dB difference
in analysis results, tradeoffs can be performed using the basic inductor
of Fig. 2. When the tradeoffs are completed, one final analysis can be
performed with all of the changes incorporated into the inductor of Fig. 1
if desired.
At this point, it is appropriate to determine the accuracy of the
analysis. While suppliers of EManalysis tools claim their programs to be
accurate, the analysis errors rather than analysis accuracy are generally
of interest to highfrequency engineers. It is also desirable to have a
quantitative value for the error. By comparing the estimated error with
the project requirements, it is possible to tell if one's trust in an
analysis approach is justified.
The error mechanisms for the full version of Sonnet have been
extensively investigated. In nearly all of the cases, the principal error
source is error that is due to the size of the cell. A unique
characteristic of this error source is the fact that the error can be cut
in half by shrinking the cell size in half (with a few wellunderstood
exceptions). Error that is due to cell size is easily evaluated. If the
cell size is cut in half and the circuit is then analyzed, then the error
is also cut in half.
The results of this type of a convergence analysis are shown in Fig.
4.
Click
to see enlarged image
Fig 4. By cutting the cell size in half (from 8 to 4 µm), it is
possible to determine error bounds. With at most 0.15dB difference
between the two results, model data should be within ±0.3 dB of the
correct answer.
The original cell size is 8 µm. The second analysis uses a cell size of
4 µm with an analysis time of 6 s/frequency. The difference between the
two curves is not easily seen, but is less than 0.15 dB nearly everywhere.
This means that the results obtained using the 8µm cell size should be
good to approximately ±0.3 dB.
If this level of accuracy (or error) is sufficient for the needs of a
particular analysis, then further modification is not necessary. But if
more accurate results are needed, then an approach known as the
"Richardson extrapolation" can be tried. In this technique, the total
error for each data point is first determined (i.e., double the difference
between the 8 and 4µm cellsize results), and then the error is
subtracted from the 8µm answer. A spreadsheet is useful for performing a
Richardson extrapolation. The results should now be accurate to
approximately ±0.05 dB.
To ensure that a Richardson extrapolation is indeed providing increased
levels of accuracy, a third analysis should be performed with a
onequartercell size. While this problem is slightly too large for the
memory constraints of Sonnet Lite, it was performed in the fullfeatured
version of Sonnet and confirmed expectations that the results are
converging.
There is one exception to the ±0.3dB error boundthe lowestfrequency
data point for the forward transmission (S_{21}) at 100 MHz
appears to be incorrect.
This can happen in any EM analysis when the cell size is small compared
to the wavelength. The 200MHz point is much better than the 100MHz
point, but still appears to have some error. Due to their smaller
wavelengths, frequencies that are above 200 MHz do not pose a problem. The
lowfrequency data are included here to show that no EM analysis can be
trusted completely. Some form of convergence test should be applied in
order to check the accuracy of the software.
Before attempting to reduce loss, it is helpful to perform an analysis
with loss that is completely removed. A comparison of this lossless
analysis with the baseline lossy inductor should provide an upper limit on
how much improvement remains for the lossy inductor. In the lossless
analysis, all losses (metal and substrate) were removed from the inductor
model of Fig. 2. Figure 5 compares this lossless analysis with the
original lossy results.
Click
to see enlarged image
Fig 5. Comparing the loss of the baseline inductor (from Fig. 2)
with the lossless version of the inductor shows that there is room for
approximately 7dB improvement at 40 GHz.
This comparison reveals approximately 7dB loss in S_{21} and
S_{11} at 40 GHz. Certainly, there is room for improvement. By
comparing Sparameter data for the lossless case, the baseline case, and
any case under consideration, the merits of each alternative can be
quickly determined. Note that there is no need to calculate inductor
quality factor (Q) which, for the complicated equivalent circuits possible
with planar spiral inductors, is neither uniquely defined nor simple to
calculate.
Reasoning that wider line widths should yield lower loss, modifications
to the baseline inductor of Fig. 2 begin by increasing the line width from
the original 8 µm to a width of 12 µm, with a 4µm cell size for analysis
(Fig. 6).
Fig 6. The line width was increased from 8 to 12 µm to evaluate the
effect on inductor loss.
Figure 7 shows the results of analyzing this modified inductor
model.
Click
to see enlarged image
The analysis now requires 11 s/frequency. Note that the loss has
generally increased. While the effect of conductor loss has undoubtedly
gone down, the effect of substrate conductivity has increased. The loss
drops unexpectedly around 40 GHz. The reason for this was not
investigated, but may be due to a resonance above 40 GHz.
Since it did not help to cut losses by increasing line widths, perhaps
it will help to decrease loss by decreasing the line widths from the
nominal 8 µm to a new value of 4 µm (Fig. 8).
Fig 8. Decreasing the line width from 8 to 4 µm results in this
geometry.
Figure 9 shows the analysis results (at 5 s/frequency). In these
results, it is apparent that the loss has decreased substantially. If
desired, this process of reducing the line width and analyzing the results
can continue until the optimum line width is found.
Click
to see enlarged image
9. Decreasing line width from 8 to 4 µm results in the desired
decrease in loss.
A spiral inductor on Si usually has a thin insulating layer (for
example, SiO_{2}) deposited underneath it. This reduces the effect
of substrate conductivity, especially at low frequency. To reduce the
loss, the insulating layer can be made thicker and/or formed with a
material having a lower dielectric constant. To investigate the effects of
changing the insulator thickness, it can be increased from its nominal
thickness of 1 µm to a new thickness of 2 µm, then analyzed (Fig.
10).
Click
to see enlarged image
Fig 10. By increasing the thickness of the insulating layer on top
of the Si substrate from 1 to 2 µm, it is possible to reduce the inductor
loss substantially.
Following this, the dielectric constant of the original 1µmthick
layer can be changed from 4.0 to 2.0 and then analyzed (Fig. 11).
Click
to see enlarged image
Fig 11. By decreasing the dielectric constant of the thin insulating
substrate on top of the Si substrate from 4.0 to 2.0, the inductor loss is
reduced substantially.
Both actions substantially reduce inductor loss at all frequencies.
Now, armed with the knowledge of the relative importance of each parameter
with respect to loss, and with knowledge of manufacturing and design
constraints, a designer has a variety of options for realizing the lowest
possible inductor losses.
To gain some further insight into this particular spiral inductor
design, the inductor of Fig. 2 was analyzed with a small cell size of 0.5
µm. As a result, each line of the inductor is now 16 cells wide. Due to
the size of this problem, the fullfeatured Sonnet software suite was used
in the analysis, rather than Sonnet Lite. The resulting current
distribution is shown in Fig. 12.
Fig 12. The current distribution for the baseline spiral inductor
was analyzed with a veryfine 0.5µm cell size. The disruption near the
underpass connection to port 2 is not a numerical artifact but a real
phenomenon.
Note that there is some disruption of the current distribution at the
location of the underpass connection to port 2. This is common in
underpass and overpass situations and is real rather than a numerical
artifact.
In Fig. 12, the color red represents a current density of approximately
1200 A/m. Port 1 is excited with a 1V source at 40 GHz connected in
series with a 50 resistor. Port 2 is terminated in an impedance of 50 . The
analysis requires more than 12,000 subsections and 305MB RAM. Since the
PC used in this investigation has only 256MB RAM, substantial memory
swapping increased the analysis time to approximately 8 h. Normally, with
adequate RAM for the problem, about one hour of analysis time would be
expected.
Although free of charge, Sonnet Lite software is quite effective for
solving a difficult and troublesome problemthe analysis of a spiral
inductor on a conductive Si substrate. Techniques were shown for modifying
the inductor in order to reduce inductor loss and characterizing the
analysis error of an EM investigation quantitatively. Richardson
extrapolation was shown to be an effective method for reducing analysis
error even further without resorting to the full Sonnet suite of programs.
Sonnet Software, Inc., 1020 Seventh North St., Suite 210, Liverpool, NY
13088; (315) 4533096, FAX: (315) 4511694, email:
info@sonnetusa.com, Internet: http://www.sonnetusa.com/
Acknowledgment
Sonnet Lite was made possible because of work funded by DARPA under the
MAFET program.
