AN IMPROVED INDUCTANCE MODEL
by
Robert J Distinti
A thesis submitted in partial fulfillment of the
requirements for the degree of
Masters of ECE
Fairfield University
2006
Approved by _________________________________________________
Chairperson of Supervisory Committee
__________________________________________________
__________________________________________________
__________________________________________________
Program Authorized
to Offer Degree _______________________________________________
Date _________________________________________________________
FAIRFIELD UNIVERSITY
ABSTRACT
INDUCTANCE MODELING
USING NEW
ELECTROMAGNETISM
by Robert J Distinti
The inductive properties of circuits are modeled with a new
model
of
electromagnetic
physics
Electromagnetism (NE) [ne_intro].
known
as
New
The inductive properties
modeled include self inductance, mutual inductance, intrinsic
inductance and other effects coupled through a magnetic field.
Accurate modeling of high-speed circuit properties is critical to
developing reliable, efficient and cost-effective printed circuits
and integrated circuits.
Classical Electromagnetism (CE) performs well when modeling
the property of mutual inductance; however, it is unsuitable for
the other inductive properties mentioned above due to its
reliance of continuous mathematics (to be explained).
NE
provides an order of magnitude improvement in both precision
and computation time over CE due to the fact that it is a simple
“line of sight” model for which complex field abstractions are
unnecessary.
1.
2.
3.
4.
5.
6.
7.
Table of contents
Introduction ................................................................................... 5
Historical Review .......................................................................... 7
2.1. Self-inductance ........................ Error! Bookmark not defined.
2.2. Intrinsic inductance ................................................................. 18
New Electromagnetism ............................................................ 20
Application of New Electromagnetism ................................. 24
Validation .................................................................................... 28
Milestones .................................................................................... 29
Bibliography ................................................................................ 30
4
1. Introduction
As the density and frequency of modern integrated circuits and
printed circuits increase, the need for accurate modeling of
circuit properties becomes imperative.
The Inductance and
capacitance of an integrated circuit(IC) trace is governed by its
geometry, material as well as parasitic effects from other traces
and ground planes.
Any given trace could experience an
unwanted resonance or “roll-off” that, under certain conditions,
may lead to an intermittent failure. Capacitive and inductive
coupling between traces is another source of intermittent failure.
Accurate modeling of circuit properties prior to fabrication will
reduce the number of development “re-spins” required to get a
product
through
development
and
acceptance
testing.
Reduction of the number of development “re-spins” correlates to
faster time to market, lower cost and improved reliability.
Reducing “re-spins” further reduces the environmental impact
and cost overhead associated with short engineering production
runs.
Eliminating the uncertainty and time associated with computer
modeling of vast networks of signals allows computer tuning of
individual traces; thereby eliminating the wasteful “one-size fits
all,” “worst-case” termination methods. Utilizing this technique,
device designers can reduce power consumption by using
5
existing
geometries
more
efficiently.
Reducing
power
consumption reduces the energy cost to the user as well as
reducing the environmental impact associated with the use of
energy.
This research addresses a new, more accurate model of circuit
properties associated with magnetic fields.
include
self-inductance,
These properties
mutual-inductance
inductance.
6
and
intrinsic
2. Historical Review
In classical theory, the inductance of a trace is the combination
of the self-inductance, intrinsic inductance (internal inductance)
and the mutual inductive coupling between other traces (which
includes ground plane).
The properties of Mutual and Self-inductance are modeled using
Faraday’s law or derivatives of Faraday’s Law based on Vector
magnetic potentials. Of these derivatives, there are two from
which all others are derived: one is Neumann’s equation
[Lorraine] and the other is called Partial inductance [Ruehli].
Lastly, the property of intrinsic inductance [Hayt] is developed
using a conservation of energy approach.
Because classical theory performs well on mutual inductance,
this historical review focuses primarily on self-inductance and
intrinsic-inductance.
2.1. Faraday’s law
The most basic Self-inductance problem is the single turn loop
inductor as shown in Figure 1. Classical Theory teaches that the
self-inductance of this loop is determined by Faraday’s Law (1),
and the Biot-Savart flux model (3) [see Hayt].
7
W
I
S=perimeter
A=Area
Figure 1
1) emf   n
d
dt
2) emf   L
dI
dt
3) dB  K M
IdS  rˆ
r2
In the above equations,  is the number of flux lines contained in
(or linked by) the loop; n is the number of turns in the loop; emf is
the voltage induced in the loop as a function of flux change; I is
the current in the loop. L is the inductance in the loop; K M is

;
4
B is the flux density (Webbers per square meter); r is the distance
from a differential segment of wire to the point where the flux
density is being determined.
Substituting (2) into (1) and reducing (n=1) yields (4)
8
4) L 

I
To find  we integrate (3) over the area circumscribed by the
loop to arrive at:
5) ( I )   dBdA   K M
AS
AS
IdS  rˆ
dA
r2
This resolves to
6) (I )   thus L  
The above results are obviously useless; a typical reaction to this
result is to assume that the modeling of an inductor as a filament
is the reason why the inductance goes to infinity. This is wrong for
two reasons.
First, the diameter of the filament is not
parameterized; as such, there is no opportunity to “divide by a
zero diameter”.
Second, this filamentary model gives very
accurate answers for mutual inductance problems.
This model works well for mutual inductance because the
distance (r) from the primary to the area circumscribed by the
secondary does not go to zero. This occurs because the primary
and secondary have a non-zero distance between them.
9
2.2. The Neumann Equation
Another variation of Faraday’s Law is called the Neumann
equation (7) [Lorraine, Ruehli] which is derived from Faraday’s
Law using Vector Magnetic Potentials.
dP  dS
r
S P
7) M  K M  
The Neumann equation is expressed in terms of the mutual
inductive coupling between a primary and secondary loop as
shown in Figure 2.
P=primary
r=distance between
differential lengths
dP and dS.
dP
I
r
dS
W=radius
A=Area
S=secondary
D=distance
between loops
Figure 2 Parallel Loops
The Neumann equation (like Faraday’s Law) yields an undefined
answer when applied to self-inductance modeling (P and S
same loop). This is obvious since r would go to zero if d goes to
zero.
10
This does not seem to concern anyone since the Neumann
equation has become the basis for the predominant SelfInductance model called Partial Inductance [Ruehli].
2.3. Partial Inductance
Partial Inductance [Ruehli] divides a solid conductor into many
“current chunks” (da*dl), then uses the Neumann equation to
calculate
the
coupling
between
each
chunk
with
the
approximation that the current is a filament down the center of
the chunk (center of da) moving in parallel to dl. The result is the
“Partial Mutual Inductance” equation (8).
8) LPkm  K M
1
a K aM
cK cM

aK aM bK bM
dlK  dlM
daK daM
rKM
The specialized form of (8) for self-induction, known as the
“Partial Self Induction” model is given as (9).
9) LPij
1
 KM 2 2
T W
li l j
 
ai a j 0 0
dli  dl j
rij
dai da j
11
W=width of rectangular conductor
T=height
dli
dl j
dai
da j
L=length
rij
Figure 3
It should be plain to see that if the integral is resolved, then rij
goes to zero as the limit of da goes to zero yielding infinite
inductance, yet again.
There are dozens of papers that utilize the above models in some
form or another, for which reasonably good results are reported;
how can this be? The answer is actually quite surprising; these
papers report that the results were obtained from numerical
algorithms based on the models. This is ludicrous, because if the
continuous model does not converge, then neither should the
numerical equivalent. It is clear that if DeltaA (numerical
equivalent of da ) is chosen to be large, then a very small result is
obtained from the algorithm; consequently, if DeltaA is chosen
very small, a large output is obtained; therefore, because of this
non-convergent numerical model, values of DeltaA could be
12
selected such that the numerical output will match any desired
result.
Essentially, if the continuous models were well behaved, then the
numerical models developed from them will be well behaved.
In a well-behaved numerical model, reducing a “delta” value
would normally cause the numerical integration to converge
(less round-off error) to the value obtained from the continuous
model. Alternatively, a numerical algorithm developed from a
poorly behaved continuous model that converges on infinity will
“explode” as the size of the “delta” is reduced; therefore, any
reports of “good results” from these algorithms can not be taken
seriously.
2.4. A Fine Mesh
Another serious problem with the partial Inductance model is the
time spent resolving inductance meshes. Each current “chunk”
represents an inductor in a mesh were the mutual inductance
among the “chunks” is determined with the Partial Inductance
model. There are entire articles [(&&&) need reference here}
devoted to the problem of reducing these very large mutual
inductance matrices into a single inductance value using various
techniques.
Aside from the complexity, an assumption made by the Partial
Inductance model is that current only travels along the length of
the “chunks.” There is no provision to account for orthogonal
13
(radial) current displacement.
Such displacements do occur,
resulting in skin effects at high frequencies.
The New Induction model eliminates all of the complexity and
assumptions since all charge behavior is reduced to a “force per
charge” field over the volume of the conductor.
The
inductance is then found by finding the path of least work
between the terminals of the device under test.
Because a complete force matrix is produced, all charge motion
and behavior can be studied.
2.5. The Continuous Current distribution
Classical models do not work for self-inductance problems
because they are based upon the concept of a continuous
current distribution;
as
such,
the distance between
the
differential current elements limit to zero resulting in an undefined
answer in all cases. Conversely, excellent results are obtained in
the mutual inductance case simply because the primary and
secondary current distributions are separate; therefore, the
distance
between
the
current
elements
is
never
zero.
Furthermore, the approximation of a charge carrying conductor
as a continuous distribution is sufficiently accurate.
14
The solution to the problem is to realize that current is not
continuous, it is a corpuscular entity comprised of individual
charges;
therefore,
electromagnetic
models
which
are
compatible with corpuscular (or discrete) systems of charges are
required. A later section introduces New Electromagnetism (NE)
which is ideal for this task.
2.6. The closed path Fallacy
Another problem with classical theory is the concept that a
closed path is required to calculate inductance.
Partial
Inductance [Ruehli] supposedly deals with this problem by
theoretically closing loops at infinity; however, a simple thought
experiment can show that a closed loop should not be required
in order to calculate inductance.
In this thought experiment, we suppose that there are only two
charges inhabiting the universe; classical theory enables us to
deterministically calculate the electromagnetic interactions
based on charge position and velocity; but what about
acceleration, which is the case of induction (di/dt)? If one
charge accelerates, it will generate a changing magnetic field
that should affect the other charge.
According to classical
theory and Partial Inductance, the charge needs to be part of a
closed path in order to determine the direction and magnitude
of the force affecting it.
This is patently ridiculous since
(according to Partial Inductance) it is possible to choose any
15
arbitrary path; resulting in an arbitrary, non-deterministic, solution.
Therefore, the classical models are ambiguous. There must be a
fundamental point charge equation that deterministically
expresses the effect of charge acceleration between charges. I
propose the New Induction model found in a later section
wherein New Electromagnetism is introduced.
2.7. Another Valiant Attempt using Classical
Theory
Another valiant attempt at trying to solve self induction using
continuous current distributions is found in the text Classical
Electrodynamics by John David Jackson [Jackson].
The
derivation that starts on page 181 yields the following answer to
the self-inductance problem
  8a  7 
L   0 a ln    
  b  4
b
a
Figure 4
With careful analysis of the derivation, it is shown [ind_jackson]
that the derivation is not the self-inductance of a loop; rather, it is
the mutual inductance between two loops as show in Figure 5.
16
Red target filament
at periphery of toroid
a
b
Blue source
filament at center
of toroid
Figure 5
This is sensible because classical induction models are only valid
for mutual inductance cases.
2.8. Alarm Bells
The above derivation uses Vector Magnetic Potentials and
Green’s theorem to compute self induction. Vector Magnetic
Potential is just an abstraction of the Biot-Savart model and
Greens Theorem is simply a method whereby the total flux linked
by a closed loop can be found from a Vector Magnetic
Potential.
Therefore, these techniques are mathematically
identical to using the Biot-Savart model and Faraday’s law to
find self induction.
If it is not possible to develop a sensible (non-undefined) answer
to self induction using the basic models (Faraday’s Law, BiotSavart), then it is certainly not possible to develop a sensible
answer to the induction problem using abstractions of those
basic models (Green’s Theorem, Vector Magnetic Potentials). A
solution developed from abstracted techniques should have set
17
off alarm bells years ago; but it didn’t. Why? The above result
would be analogous to stating that a certain problem can not
be solved using Coulomb’s Law; however, it can be solved using
electric potentials.
This is ridiculous because electric potentials
are derived from Coulomb’s law and are thus limited to what
Coulomb’s law can solve; likewise, Vector Magnetic Potentials
are derived from Biot-Savart and therefore should not be able to
solve anything more than what is possible using Biot-Savart.
2.9. Intrinsic inductance
Intrinsic inductance is the inductance of the wire itself.
By
definition [Hayt], the inductance of an inductor is the sum of the
self-inductance (calculated from faraday’s law or a related
technique) plus the intrinsic inductance.
Since there is no way
to solve this intrinsic inductance property from classical field
equations, a “backdoor” conservation of energy technique is
often used [Hayt].
The resulting equation is

(H/m) which
8
shows intrinsic inductance to be a function of wire length and
not of thickness; however, a simple experiment teaches us that
this is not the case.
The following table shows the measured
inductance of two single turn loops of same diameter but slightly
different wire thickness.
According to classical theory, these
results should be the same; but they are not.
48 inch perimeter Measured
loops
inductance
18
22 AWG wire
26 AWG wire
nH
2055
2655
Furthermore, none of the self inductance techniques reviewed,
which profess “good results”, use this “intrinsic inductance” effect
as part of their solution (although Ruehli does mention it; it is not
used).
2.10. Wrap up of Historical Review
Over [n (&&&)TBF (you have to cite all the articles that you
reviewed)articles were reviewed in preparation of this work]; all
(except a few based empirical modeling) used one of the
above techniques in some fashion or another for modeling of
self-inductance, only those papers directly referenced are found
in the references section; the remainder are included by IEEE
article number and placed into groups after the bibliography.
19
3. New Electromagnetism
Before delving into the methodology used to determine
inductive properties; it is important to highlight the fundamental
field models that will be used in that endeavor.
As Stated in the previous section, needed are Electromagnetic
models suitable for discrete modeling. New Electromagnetism
provides such a set of models
The Equation
F 
2
K M QSQ T
r
F
Coulomb’s Model
K E QS QT rˆ
r
F
Name
2
v T  rˆ v S  v S  rˆ v S  ( v S  v T )rˆ 
 K M QS QT a S
New Magnetism
New Induction
r
Where: K M 

1
, KE 
4
4
These equations represent the Force affecting a “target” charge
due to the behavior and position of a “source” charge. The ‘S’
and ‘T’ subscripts in the above equations delineate properties
20
associated with the source and target charges.
The vector
r represents the vector from source to target in all equations.
QS
aS
r
vS
F
vT
QT
Figure 6: Two Body Diagram
The first model is Coulomb’s model which relates force to charge
position.
Other than notational differences, it is identical to
classical theory.
The second model is New Magnetism. New Magnetism relates
force to charge velocities. The first and third terms inside the
brackets are derived directly from F  QvxB and the Biot-Savart
model. The middle term is required to make the model match
well known experiments [secrets].
The third model is New Induction which was found through
computer search of experimental data. New Induction shows
that an accelerating source charge creates a disturbance that
21
expands spherically from the source which forces charges in the
opposite direction.
Both New Induction and New magnetism are based on a
spherical magnetic field model. The magnetic field of classical
theory, as described by the Biot-Savart model, is a “donut” or
torrid shaped field because it does not provide for a magnetic
field longitudinally to a moving charge (see Figure 7); therefore,
New Electromagnetism is not a reformation of classical theory; it
is a superset.
QT
r
F
vS aS
Biot-Savart
QS
New Magnetism
New Induction
Figure 7
New Electromagnetism is ideally suited for modeling Self
inductance and intrinsic-inductance simply because the models
reflect
the
electromagnetic
interactions
charges.
22
among
discrete
Furthermore, the spherical field models of New Electromagnetism
are a more complete expression of electromagnetic interactions
enabling accurate solutions to physical systems that are as yet
unexplained by classical theory ( homopolor paradox [nm], self
induction) etc.
Lastly, the spherical field shapes are vastly easier to process;
thereby, reducing processing time and code complexity.
23
4. Application of New Electromagnetism
Since New Electromagnetism enables us to model a problem as
a systems of discrete charges; that is in fact the approach that is
taken.
The first, and most important task, is to establish a model for the
relative spacing between charges. Since only mobile carriers
are of interest, we develop a relationship for mobile carrier
density from published material properties. The values in the
following table are taken from Hayt [Hayt] page 133 and
appendix C.2.
Material
Charge Mobility
m2
e
Vs
Copper
0.0032
Aluminum 0.0012
Silver
0.0056
Conductivity
1

m
Charge Density

Q

e m3
5.80x107
3.82x107
6.17x107
1.8125x1010
3.1833x1010
1.102x1010
From the above table it is apparent that copper has 18 billion
Coulombs of mobile carriers per cubic meter. This works out to
18 Coulombs of mobile carriers per cubic mm.
At first glance it might seem like an insurmountable task to model
conductor properties at the charge level; however, the task is
vastly simplified by use of a technique that shall be called
“seeding” for lack of a better term.
4.1. The Seeding Technique
The seeding technique begins with a single charge. Then the
inductive contributions from that charges nearest 10n neighbors
(this value is selectable) are calculated as shown in the following
diagram.
24
Radius R
Figure 8: The seed for a given metal
The neighbors are chosen to form a spherical template of radius
R. The number of neighbors is chosen such that R is sufficiently
large such that the charge in the remaining parts of the
conductor can be accounted for using continuous charge
techniques.
The seed value for each given metal only ever needs to be
computed once. The seed is then reused for all processing using
the seed growing method shown in the next section.
Seeds are needed for boundary conditions as well. Boundaries
include the sides of the conductors (Hemispherical seed) and
other boundary conditions which include those of dissimilar
metals such as solder or plating.
25
4.2. The Seed Growing Method
The seed is placed at a point in a metallic structure where the
inductive force is to be determined. Then spherical shells are
added to the system as shown
Cylindrical conductor (side view)
Outer shells
The seed
Figure 9: Seed growing (not to scale)
The thickness of the shells can become progressively larger
depending upon the inverse relationship of the effect being
modeled.
The purpose of the shell is to reduce the problem to a point
charge equation where the total charge affecting the charge at
the center of the seed is the charge density times the volume of
the shell intersecting the wire construct. The distance is a
function of the inner and outer radius of the shell.
The seeds are grown a various strategic points within the wire
and at boundaries in order to map out the patterns of force.
26
4.3. Two courses of Action
The development can go two different ways depending upon
how things work out.
The first assumes that the current distribution remains substantially
constant, as such the voltage and current through the wire is
essentially uniform. This case will enable full use of the seeding
method as is. The draw back is that this technique is only
applicable for systems where the highest frequency component
being modeled has a far larger wavelength than the dimensions
of the circuit. Thus this technique is only applicable to systems of
mid to low frequency [this may be enough to validate the thesis].
The second initiative allows for retarded time modeling of field
effects. This modeling requires complex computational data
structures which save representative current distributions and
voltages at the seed points such that constructive and
destructive interference can be accounted for when modeling
high frequency signals
27
5. Validation
Validation consists of achieving better than 0.5 % tolerance with
measured data. At this point I have not decided whether to use
published tables of measured values or measure my own. – this is
part of the timeline decision making process.
[ not 100% sure yet]
28
6. Milestones
[see Microsoft project application that accompanies this
document]
29
7. Bibliography
References to New Electromagnetism
[NE_intro] Distinti, Robert J., 2005.
New Electromagnetism
Introduction. www.distinti.com/docs/ne_intro.pdf
[NI]
Distinti,
Robert
J.,
www.distinti.com/docs/ni.pdf
1997.
New
Induction,
[NM] Distinti, Robert J., 2001.
New Magnetism, Area-46
Publishing, Fairfield, CT. www.distinti.com/docs/bk_000.pdf.
[Secrets] Distinti, Robert J., 2004. The Secrets
www.distinti.com/docs/the_secrets_of_qvxb.pdf
of
QvxB,
[ind_jackson] Distinti, Robert J., 2004. A classical Attempt at
Induction, www.distinti.com/docs/ind_jackson.pdf.
[ni_neumann] Distinti, Robert J., 2003. New Induction and the
Neumann Equation, www.distinti.com/docs/ni_neumann.pdf
References to general classical theory texts
[Jackson] Jackson, John David, 1999. Classical Electrodynamics,
Third Edition. John Wiley and Sons Inc., Hoboken, NJ. On page
181 is found a solution to a current ring using Vector Magnetic
Potentials.
[Hayt] Hayt, Wiiliam H., 1981. Engineering Electromagnetics,
Fourth Edition. McGraw-Hill Inc., New York.
[Lorraine] Lorrain, Paul, 1988. Electromagnetic Fields and waves,
Fourth Edition. W.H. Freeman and Company, New York.
References to papers proposing Inductive modeling techniques
30
[Ruehli] A. E. Ruehli, Inductance Calculations in a Complex
Integrated Circuit Environment, IBM Thomas J. Watson Research
Center, Yorktown Heights, New York.
[Schanen] Schanen, Jean-Luc et al., 1994. Modeling of Printed
Circuit Board Loop Inductance, IEEE Transactions on Magnetics,
Vol. 30, No. 5. [00312715#6]
References to useful techniques or measured data
[fft] Hu, Haitian et al., 2003. ,Fast On-Chip Inductance Simulation
Using a Precorrected-FFT Method, IEEE Transactions on Computer
Aided Design of Integrated Circuits and Systems, Vol. 22, No. 1.
[01158253#4]
Reviewed papers that are not referenced
[00622203#1] T.Nakazato et al., Inductance Computation of
Microscopic Superconducting Loop, IEEE Transactions on
Computer Aided Design of Integrated Circuits and Systems, Vol.
22, No. 1.
[00466178#2] Cherry, Paul C. et al., 1995. FDTD Analysis of High
Frequency Interconnect effects, IEEE Transactions on microwave
theory And Techniques, Vol. 43, No. 10.
[00783917#3] Zhang, Yongming, 1999. Modeling os a Long
Josphson Junction Coupled to SFQ Elements, IEEE Transactions
on Applied Superconductivity, Vol. 9, No. 2.
[00037205#5] Poltz, J. et al., 1989. Cross-Talk and Ringing on a
Multilayer PCB. IEEE National Symposium of Electromagnetic
Compatabillity, Denver, C). 23-25 May 1989.
[00346626#8] Griese, E. et al. (no year), Fast Simulation of
Reflection and CrossTalk Effects Using Pade Approximation, IEE
Electromagnetic Compatibility Conference, 5-7 September.
31
[00385661#9] Hubbing, Todd H. et al., 1994. Identifying and
Quantifying Printed Circuit Trace Inductance, University of
Missouri-Rolla.
[00473718#10] Leferink, Frank B. et al., 1993. Inductance of
Printed Circuit Ground Planes, IEEE (no other info given)
[00569518#11] Holloway, Christopher L., et al., 1998. Net and
Partial Inductance of a Microstrip Ground Plane, IEEE
transactions on Electromagnetic Compatibility, Vol. 40, No. 1.
[00678798#12] Swirbel, Tom, et al., 1998. Electrical Design and
Simulation of High Density Printed Circuit Boards. IEEE Electronics
Components and Technology Conference 1998.
[00892842#13] Tang et al., 2000. Characterization of Coreless
Printed Circuits Boards (PCB) transformers, IEEE Transactions On
Power Electronics, Vol. 15, No. 6.
[00974648#14 Leone, Marco, 2001. Design Expressions for the
Trace-to_Edge Common-Mode Inductance of a Printed Circuit
Board. IEEE Transactions on Electromagnetic Compatibility, Vol.
43, No. 4.
[00982839#15 Sohn, Young-Soo, et al. 2001. Empirical Equations
on Electrical Parameters of Coupled Microstrip Lines for Crosstalk
Estimation in Printed Circuit Board.
IEEE Transactions on
Advanced Packaging, Vol. 24, No. 4.
[01057897#16] Miller, Jason R, et al., 2002. Calculating Partial
Inductance of Vias for Printed Circuit Boards Modeling, IEEE
2002.
[01280307#17] Chen, Yang, et al., 2003. Analysis and
Measurement of Small Inductance of Loops and Vias on Printed
Circuit Board. IEEE 2003.
32
[01287848#18] Charyulu et al., 2003. Analysis and Measurement
of Crosstalk in Printed Circuit Board de to RF and Transient Pulses.
Proceedings of INCEMIC -2003.
[01296122#19] Lopez, Toni, et al., 2004. PCB Layout Inductance
Modeling Based on a Time Domain Measurement Approach.
IEEE 2004.
[00290725#20] Witulski, Aurther F., 1993. Modeling and Design of
Transformers and Coupled Inductors. IEEE 1993.
[00299741#21] Geiwski, W. et al., 1994. Hybrid Inductor Modeling
for Succesful Fileter Design. IEEE Transactions of Microwave
Theory and Techniques, Vol. 42, No. 7.
Followup references: H.M. Greenhouse Design of planar
rectangular microelectronic inductors IEEE trans. Parts, hybrids,
packaging, vol PHP-10 pp101-109 june 1974
33
INDEX
34
A
Aristotle,3
4