Quasi-Static Derived Physically Expressive Circuit Model for Lossy Integrated RF Passives
ABSTRACT This paper presents a novel approach for deriving a physically meaningful circuit model for integrated RF lossy passives such as spiral inductors on a silicon substrate. The approach starts from a quasi-static partial element equivalent circuit (PEEC) model. The concept of complex inductance and capacitance is introduced to uniformly deal with the conductor and dielectric losses. Basic Y- Delta network transformation is used to ldquoabsorbrdquo the insignificant internal nodes of the original PEEC network and to reduce the order of the circuit model. The physically expressive circuit model given here can be very concise while preserving the major physical meanings and attributes of the original circuit layout. A low-temperature co-fired ceramic bandpass filter and two practical inductors fabricated using a 0.18-mum CMOS process are studied by the model to demonstrate the validity of this new approach. Furthermore, the stability condition of the model is also discussed.
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ABSTRACT: A new comprehensive wide-band compact-modeling methodology for on-chip spiral inductors is presented. The new modeling methodology creates an equivalent-circuit model consisting of frequency-independent circuit elements. A fast automated extraction procedure is developed for determining the circuit element values from two-port S-parameter measurement data. The methodology is extremely flexible in allowing for accurate modeling of general classes of spiral inductors on high- or low-resistivity substrate and for large spirals exhibiting distributed trends. The new modeling methodology is applied to general classes of spirals with various sizes and substrate parameters. The extracted models show excellent agreement with the measured data sets over the frequency range of 0.1-10 GHz.IEEE Transactions on Microwave Theory and Techniques 04/2004; · 2.23 Impact Factor
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ABSTRACT: A fast algorithm for computing the capacitance of a complicated three-dimensional geometry of ideal conductors in a uniform dielectric is described and its performance in the capacitance extractor FastCap is examined. The algorithm is an acceleration of the boundary-element technique for solving the integral equation associated with the multiconductor capacitance extraction problem. The authors present a generalized conjugate residual iterative algorithm with a multipole approximation to compute the iterates. This combination reduces the complexity so that accurate multiconductor capacitance calculations grow nearly as nm, where m is the number of conductors. Performance comparisons on integrated circuit bus crossing problems show that for problems with as few as 12 conductors the multipole accelerated boundary element method can be nearly 500 times faster than Gaussian-elimination-based algorithms, and five to ten times faster than the iterative method alone, depending on required accuracyIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 12/1991; · 1.09 Impact Factor
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ABSTRACT: In this paper, a closed-form integral representation for the eddy-current losses over a conductive substrate is presented. The results are applicable to monolithic inductors and transformers, especially when such structures are realized over an epitaxial CMOS substrate. The technique is verified against measured results from 100 MHz to 14 GHz for spiral inductorsIEEE Transactions on Microwave Theory and Techniques 02/2001; · 2.23 Impact Factor
1954IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 8, AUGUST 2008
Quasi-Static Derived Physically Expressive Circuit
Model for Lossy Integrated RF Passives
Hai Hu, Kai Yang, Ke-Li Wu, Senior Member, IEEE, and Wen-Yan Yin, Senior Member, IEEE
Abstract—This paper presents a novel approach for deriving
a physically meaningful circuit model for integrated RF lossy
passives such as spiral inductors on a silicon substrate. The
approach starts from a quasi-static partial element equivalent
circuit (PEEC) model. The concept of complex inductance and
capacitance is introduced to uniformly deal with the conductor
and dielectric losses. Basic
used to “absorb” the insignificant internal nodes of the original
PEEC network and to reduce the order of the circuit model. The
physically expressive circuit model given here can be very concise
while preserving the major physical meanings and attributes of
the original circuit layout. A low-temperature co-fired ceramic
bandpass filter and two practical inductors fabricated using a
0.18- m CMOS process are studied by the model to demonstrate
the validity of this new approach. Furthermore, the stability
condition of the model is also discussed.
? network transformation is
Index Terms—Equivalent circuits, low-temperature co-fired ce-
ramic (LTCC), microwave circuits, model-order reduction, RF in-
tegrated circuit (RFIC).
substrate-based system-in-package (SiP), are the key building
elements that construct a single-chip monolithic functional
system. New configurations and modeling techniques for the
passives in an RF integrated system have been drawing a great
deal of attention –. Needless to say, the characteristics
of integrated RF passive components, e.g., the
inductor, will decisively determine the overall performance
of the system. With the increase of the operating frequency,
how to represent the inevitable parasitic effects and the loss
properties of both substrate and conductors becomes one of
the major challenges to the designers. Even though the modern
full-wave electromagnetic simulation software packages are
more powerful than ever before, they still could not provide
good intuitive insights of those parasitic effects and a good
physically expressive circuit representation, not to mention the
NTEGRATED RF passive components, either in sil-
icon-based system-on-chip (SoC) or in ceramic/organic
value of an
Manuscript received October 4, 2007; revised May 6, 2008. First published
July 15, 2008; last published August 8, 2008 (projected). This work was sup-
Region under Grant 2150499.
H. Hu, K. Yang, and K.-L. Wu are with the Department of Electronic En-
gineering, The Chinese University of Hong Kong, Shatin, Hong Kong (e-mail:
firstname.lastname@example.org; email@example.com; firstname.lastname@example.org).
W.-Y. Yin is with the Center for Microwave and RF Technologies, Shanghai
Jiao Tong University, Shanghai 200240, China (e-mail: email@example.com).
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TMTT.2008.927307
long computation time for a large-scale system. In fact, a lot
of efforts have been devoted to the electromagnetic simulation
and model extraction methodologies for the integrated passives
in various of substrates. The studies worth mentioning include
 and , which intend to provide a simple computer-aided
design (CAD) tool for extracting either capacitance or induc-
tance values, respectively.
It is well understood that the performance of integrated
RF passives, such as the inductors integrated on a silicon
substrate, severely suffer from the losses associated with the
conductor and the substrate in the gigahertz frequency range.
Several previous studies have addressed these loss effects. In
, Niknejad and Meyer analyzed electrical substrate loss,
and presented a computationally efficient algorithm for de-
signing and optimizing spiral inductors . Cao et al. studied
the loss effect at the high-frequency range and proposed a
frequency-independent ladder RLC circuit to model the RF
spiral inductors in . Reference  classified the lossy
parasitics into different mechanisms, such as the skin effect and
eddy current, and suggested the use of finer mesh segments in
modeling the loss. In addition, Weisshaar et al. proposed the
idea of the “complex current image” in , which allows an
image current located at a complex distance to account for the
eddy-current effect. All these measures are effective to a certain
extent; they were attempting to use a simple and predefined
circuit to model the loss effects or to use a complicated RLC
network to represent an RF circuit layout. Nevertheless, there
are no efficient and generic methods that can derive a succinct
physically meaningful circuit model from the given layout of a
lossy RF integrated passive circuit.
The partial element equivalent circuit (PEEC)  technique
is based on the mixed-potential integral equation (MPIE). The
PEEC, which converts a circuit layout into a lumped RLC ele-
ment coupled network, has been widely used in the modeling
of integrated passives. For example,  proposed a scalable
model for a spiral inductor on a lossy silicon substrate using an
enhanced PEEC method. Unfortunately, the number of nodes
and the number of components in the generated network are
usually excessively large, which make the traditional SPICE-
like circuit solvers extremely slow in running and is prohibi-
tively difficult in revealing the physical insights. Therefore, re-
searchers have been searching for more effective means that
can reduce the circuit model order to accelerate circuit anal-
ysis – from the system transfer function point of view.
Frequently, one also needs to find an appropriate circuit repre-
sentation of a passive circuit from both physical intuition and
system response , . Recently, a systematic approach to
derive a quasi-static physically meaningful circuit model with a
specified order of accuracy for a lossless integrated passive was
0018-9480/$25.00 © 2008 IEEE
HU et al.: QUASI-STATIC DPEC MODEL FOR LOSSY INTEGRATED RF PASSIVES1955
Fig. 1. Substrate of a typical CMOS IC.
proposed . However, the quasi-static model is only for the
lossless case and cannot be directly applied to many practical
cases in which the losses are too significant to be ignored.
In this paper, the above derived physically expressive circuit
(DPEC) model is extended to the lossy integrated passive cir-
cuit case. This extension is based on a uniform treatment of
complex capacitances and inductances. The guideline of how
to ensure the passivity of the DPEC model is also discussed.
The case studies of a low-temperature co-fired ceramic (LTCC)
substrate are given to show the validity and effectiveness of the
extended DPEC model. It can be seen that, in all the examples,
the loss properties, such as
values versus frequency, are very
well preserved in the DPEC model. It is demonstrated that the
extended DPEC model for lossy circuits can effectively retain
the essences of a lossy integrated passive circuit.
A. Quasi-Static PEEC Circuit Model With Losses
The PEEC modeling starts by dividing a multilayer circuit
structure into a number of capacitive and inductive meshes. The
conductive loss is then incorporated in the complex numbered
self-inductors and the dielectric loss is included in the complex
numbered self capacitors.
The dielectric loss can be taken into account by including
the complex dielectric constant
Green’s functions that are associated to the substrate structure
under consideration. A brief description of PEEC modeling can
be found in .
to that used in the method of moment  can be employed in
the quasi-static PEEC modeling.
Since the quasi-static simplification is well known to be ac-
curate enough for modeling RF integrated circuit (RFIC) and
monolithic microwave integrated circuit (MMIC) devices, the
quasi-static Green’s functions are employed in this study. Due
to the lossy nature of silicon, the potential Green’s functions are
complex. The vertical structure of a typical silicon-based inte-
grated circuit (IC) is shown in Fig. 1. The spectral-domain po-
tential Green’s functions can be derived using transmission line
An important step during the derivation of the DPEC model
is to carefully convert all the mutual inductances between the
inductive mesh elements in a PEEC model into a directly con-
nected inductor network. A simple demonstrative example is
Fig. 2. Converting mutual inductances into self-inductances.
shown in Fig. 2. The mutual inductance
ductor pair of
The values for
be obtained from the values of
as they originate from the inductive meshes of a layout.
,, and are defined as inter-inductances, as they are
self-inductances. In general, inter-inductances are substantially
larger than the main inductances.
A point that deserves mentioning here is that the conductive
loss will only be associated with main inductors. Thus, no resis-
tance shouldbe associated to theinter-inductances afterthe mu-
tual inductance is transformed to inter-inductances. This point
is very important for preserving the passivity of a DPEC model.
between the self-in-
is equivalently transformed into four
are modified by
in Fig. 2 are called main inductances,
B. DPEC Model
In order to incorporate the conductive loss and dielectric loss
in a uniform manner, the concept of complex capacitance and
complexinductance is introduced. The impedance of a complex
inductor is expressed as
tained by multiplying the surface impedance with the integra-
tion of the basis function over an inductive cell. Similarly, for a
complex capacitor, one can define
is conductive loss of an inductive mesh, which is ob-
the Green’s functions. Since the lossy elements in (1) and (2)
are all frequency dependent, the resultant complex components
also depend on frequency. Having incorporated the above-de-
fined complex inductance and capacitance in a PEEC model, a
complexvaluednetwork representation of theoriginal lossy cir-
cuit layout can be obtained. Therefore, the next task is to reduce
the order of the excessively large PEEC network and to derive
a concise DPEC model.
The basic principle of the DPEC model is based on the
Gaussian elimination or the
-nodecircuit can be transformed to an
circuit. For readers’ convenience, the basic formula for the
DPEC model is repeated here.
accounts for the dielectric loss, which is included in
transformation . It says
1956IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 8, AUGUST 2008
The adjacent admittances in the
mation can be calculated by
circuit from the transfor-
is the total admittances connected to the node that has been re-
duced. Considering the generic case of
and are the admittances in thecircuit, and
and, which are in
Apparently, (4) cannot be restored to an admittance in the
form of an LC tank without a legitimate treatment. In order to
develop a recursive process, the third term in (4) needs to be
converted to a complex capacitance before removing the next
“removable” node such that
and are the equivalent inductance and capacitance of
The choice of the node that best satisfies the condition of
at each step of the process is critical for re-
taining the physical meanings of the circuit. That is to say, the
node that gives the smallest value of
step will be “absorbed.” The recursive iteration process is
repeated until the values of
are larger than a prescribed threshold value. Setting a threshold
value, say, 0.15, is to ensure a smooth and nearly monotonous
change of the component values in the frequency range of
interest. Obviously, the DPEC topology is determined at the
highest frequency of interest.
It should be mentioned that similar equations to (3)–(7) have
been given in . The differences are that here the inductance
and capacitance are complexnumbersand are frequencydepen-
dent in general.
at an iteration
for all the internal nodes
C. Passivity of the DPEC Mode for Lossy Circuits
Since the original PEEC model network is passive, the
passivity property of a DPEC model must be retained. In
circuit transformation, a positive imaginary
part of some complex capacitances
their values are substantially large, these positive imaginary
parts can be discarded without affecting the attributes of the
DPEC model. However, special attention needs to be paid to
the stability problem with inductance.
can be found. Since
Fig. 3. Vector diagram of ???? . The angles for reciprocals of the complex
inductances have been exaggerated in order to give a clearer sketch.
Assuming uniform meshes are used, a vector diagram can be
employed to illustrate how the passivity of a DPEC model can
be guaranteed if the inductive meshes are fine enough. As the
main complex inductance can be written in an exponential form
in (5) can be expressed by
, the reciprocal of the total inductance
The vector diagram for
adjacent inductances are considered. If
smallest and largest angles among
is shown in Fig. 3, where four
between. The new inductances
circuit transformation can be ex-
whose complex angle
can be rewritten as
and are the differences between
and , respectively. Thus, as long as
or , there will be no negative resistance
and, therefore, no instability problem.
A special case is that there is no difference between angles of
all main complex inductances. Thus, all main complex induc-
and thus, no instability problem should appear.
In ordinary cases, the inductive meshes need to be carefully
refined to avoid negative resistance. In section
, skin depth
HU et al.: QUASI-STATIC DPEC MODEL FOR LOSSY INTEGRATED RF PASSIVES1957
Fig. 4. Main inductance versus aspect ratio of a mesh.
Fig. 5. Cross-sectional view of the fabricated stacked circular inductors.
is assumed to be uniform, which makes the conductive loss
proportional to the aspect ratio
of a mesh, i.e.,
Fig. 4 shows some typical relations between the main induc-
as given in (13) for some illustrative mesh lengths. The main in-
ductance values are calculated using the formula given in .
From Fig. 4, the following can be observed.
• If is large, the inductance tends to be a linear function of
the mesh aspect ratio; hence, the angle differences
in (11) must approach zero.
• If issmall,theslopeofthecurveinFig.4becomessmaller
as the aspect ratio increases, i.e., the angles of the com-
plex inductances approach
, the resistance from the imaginary part of
in (10) is guaranteed to be larger than zero, and hence, a physi-
cally meaningful DPEC model can be guaranteed. This conclu-
sion can also be further verified by the examples discussed in
, which also guarantees
The first two examples presented in this paper are stacked
double circular spiral inductors fabricated using a 0.18- m
CMOS process. The cross-sectional view of the fabricated
inductor is shown in Fig. 5, where the vertical distance between
the double spirals is
strip thicknesses are
tively; the height of the metal layer 6 in the oxide substrate is
m andm with the relative permittivity
m. The upper and lower metal
m andm, respec-
Fig. 6. Geometry of the stacked single circular spiral inductor.
Fig. 7. (a) Capacitive meshes of the inductor shown in Fig. 6. (b) Segment of
the meshes. (c) PEEC model of the segment.
. The silicon substrate thickness is
with the relative permittivity
The first example is a stacked single circular spiral inductor.
Its geometry is shown in Fig. 6, where the inner radius
m, the trace width
The meshing scheme of the structure for the PEEC modeling
is shown in Fig. 7(a), where the trapezoid surface cells are for
capacitive meshes. The solid inductive cells are half-cell-length
shifted from the capacitive cells and are not shown in Fig. 7(a).
In Fig. 7(b),
,, andrepresent the capacitances to the
ground from the top, bottom, and side surfaces, respectively.
is shown in Fig. 7(c). All mutual inductance are suppressed in
Fig. 7(c) for the purpose of clarity.
Before a DPEC for the inductor was derived, the original
PEEC model of this RF inductor had 273 nodes and more than
4000 lumped components. The effective inductance and the
factor calculated by a DPEC model are compared with those
obtained from the measurement in Fig. 8. The threshold value
used in this example is 0.15. The result from the original PEEC
and the conductivity
m, and the trace spacing
1958IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 8, AUGUST 2008
Fig. 8. Characteristics of the stacked single circular spiral inductor by mea-
surement, PEEC and DPEC models. (a) Effective inductance. (b) ? factor.
Fig. 9. Geometry of a stacked double circular spiral inductor.
model is also superimposed as a reference. The definitions of
the effective inductance and the
factor in this study are
The second example is a stacked double circular spiral in-
ductor, as shown in Fig. 9. The original PEEC model for this in-
The effective inductances, as well as the
double circular RF inductor from measurement, the original
PEEC model, and a DPEC model are shown in Fig. 10.
The final DPEC model for the above two integrated RF in-
ductors can be a three-node network, as shown in Fig. 11. As
the circuits are derived at frequencies of interest, there is no ap-
proximation error in deriving the DPEC models.
factors of the
Fig. 10. Characteristics of the stacked double circular spiral inductor by mea-
surement, PEEC, and DPEC models. (a) Effective inductance. (b) ? factor.
Fig. 11. Obtained DPEC model of the RF inductors.
Although the quasi-static Green’s functions are used in the
PEEC modeling, the
given here are frequency dependent. Table I lists all the compo-
nent values at sampled frequencies of each example.
The third example in this study is an LTCC bandpass filter, as
2.36 mm0.55 mm, and consists of four metal layers in the
substrate with a 7.8 dielectric constant and 0.002 loss tangent.
The metal thickness is 0.012 mm and its conductivity is set to
10 S mto mimic a real situation if the uneven surface
is taken into account. The original PEEC model consists of total
is the average angular frequency of interest, a DPEC
model with only a few nodes can be achieved.
Although allthe loss-component valuesare frequencydepen-
GHz is used in the analysis and the
values in the DPEC models