IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010
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Mutual Inductance Calculation Between Circular Filaments Arbitrarily
Positioned in Space: Alternative to Grover’s Formula
Slobodan Babic1 , Frédéric Sirois1 , Cevdet Akyel1 , and Claudio Girardi2
École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada
Texas Instruments France, 06271 Villeneuve-Loubet, France
In this paper, we present the full derivation of a new formula for calculating the mutual inductance between inclined circular filaments
arbitrarily positioned with respect to each other. Although such a formula was already proposed by Grover more than 50 years ago, the
formula presented here is slightly more general and simpler to use, i.e., it involves only a sequential evaluation of expressions and the
numerical resolution of a simple numerical integration. We derived the new formula using the method of vector potential, as opposed
to Grover’s approach, which was based on the Neumann formula. We validated the new formula through a series of examples, which
are presented here. Finally, we present the relationship between the two general formulas (i.e., Grover’s and our new one) explicitly
(Example 12).
Index Terms—Electromagnetic analysis, inductance, mutual coupling.
I. INTRODUCTION
HE mutual inductance calculation between coaxial circular filaments has been thoroughly treated by a number
of authors since the time of Maxwell, and an accuracy exceeding
anything required in practice is nowadays possible [1]–[7]. A
formula for two circles whose axes intersect was first given
by Maxwell [1]. Formulas for circular loops with parallel axes
have later been derived by Butterworth [2] and Snow [5]. Unfortunately, these formulas are slowly convergent and are applicable only within a restricted range of parameters. Using Butterworth’s formula [2], Grover developed a general method to
calculate the mutual inductance of inclined circular filaments in
any desired positions in the form of a single integral [3], [4].
Today, with the availability of powerful and general numerical methods, such as finite element method (FEM) and
boundary element method (BEM), it is possible to calculate
accurately and rapidly the mutual inductance of almost any
practical 3-D geometric arrangement of conductors. However,
in many circumstances, there is still an interest to address this
problem using analytic and semi-analytic methods, as they
considerably simplify the mathematical procedures, and often
lead to a significant reduction of the computational effort. For
instance, general computation techniques such as those presented in [8]–[12] have proved to be useful in application fields
as diversified as eddy-current tomography [13], [14], electronic
and printed circuit board design [15], [16], coreless printed
circuit board transformers [17], wireless battery chargers for
general purpose and biomedical applications [18], [19], force
and torque calculation [20]–[22], electromagnetic launchers
[23], plasma science [24], [25], superconducting magnetic
levitation [26], etc.
T
Manuscript received September 26, 2009; revised March 29, 2010; accepted March 30, 2010. Date of publication April 05, 2010; date of current
version August 20, 2010. Corresponding author: S. Babic (e-mail: slobodan.babic@polymtl.ca).
This paper has supplementary downloadable multimedia material available
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2010.2047651
Fig. 1. Filamentary circular coils with arbitrary lateral and angular misalignment (most general case).
In this paper, we use the approach of the magnetic vector potential to calculate the mutual inductance of two circular loops
arbitrarily positioned with respect to each other. We treated this
problem in its most general form, by considering plane equations for representing the positions of the loops. In the final form
of our equations, the mutual inductance is obtained by sequentially evaluating a few elementary expressions and solving numerically a single, well behaved integral. This new and general
mutual inductance formula was thoroughly validated against a
series of reference examples, and was shown to cover all cases
given by Grover, Snow, Kalantarov, and Dwight [3]–[7]. Its unlimited range validity therefore makes it an ideal choice in any
of the above-mentioned application fields.
II. BASIC EXPRESSIONS
Let’s consider two circular filaments as shown in Fig. 1,
where the center of the larger circle (primary coil) of radius
corresponds with the origin
of plane
, and
where the circle’s axis corresponds to the axis. The smaller
is located in an inclined
circle (secondary coil) of radius
plane whose general equation is given by
(1)
We also define the following points:
, the center of the secondary coil;
• Point
0018-9464/$26.00 © 2010 IEEE
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010
• Point
, an arbitrary point on the secondary
coil.
can
It is shown in Appendix A that coordinates of point
easily be expressed in terms of the plane parameters. For inand
stance, the points
6) The parametric coordinates of an arbitrary point
on the secondary coil is given by
(8)
(2)
and
will be used later
where
in our developments.
Let’s now define the basic mathematical expressions required
in the next section of the paper:
on the primary coil can
1) An arbitrary point
be expressed in terms of its parametric coordinates:
(3)
is the parameter. This is the well-known
where
parametric equation of circle in 3-D space.
7) The differential element of the secondary coil is given by
(9)
with
is the parameter.
where
2) The differential element along the primary coil’s path is
given by
(4)
3) The unit vector
normal to plane
is defined by
(5)
.
III. DERIVATION OF NEW FORMULA
The mutual inductance between the two inclined circular filaments defined above will be calculated by using the approach
of the magnetic vector potential. The magnetic vector potential
at point
produced by a circular current loop
carrying the current
(see Fig. 1), is given by
of radius
(10)
with
where
4) The unit vector
between points
and
in plane
is
(6)
with
(11)
and
a point on the secondary circle, e.g.,
or
, as defined in (2).
5) The unit vector tangential to the secondary loop is defined
as the cross product of and , i.e.,
(7)
with
Using Stokes’s theorem, the flux through the secondary circuit
due to a current in the primary circuit is
(12)
and
are respectively the cross section and the
where
perimeter of the secondary circle.
By definition, the mutual inductance between the secondary
and primary coils is given by
(13)
BABIC et al.: MUTUAL INDUCTANCE CALCULATION BETWEEN CIRCULAR FILAMENTS ARBITRARILY POSITIONED IN SPACE
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From (4), (9), (10), (11), (12), and (13), we obtain
(19)
(14)
where
is the complete elliptic function
is the complete
of the first kind and
elliptic function of the second kind.
After some algebraic manipulations, we obtain
The first integration is made with respect to , using the substitution
[27], and where
and
. Expressed in terms of
and , we have
(15)
(20)
where
The first integration gives
(21)
From (14) and (20), we obtain
(22)
Introducing some additional dimensionless variables, i.e.,
,
,
,
, and
, and performing the necessary transformations to remove the
and
terms, we obtain the
following form:
(23)
(16)
where
(17)
In order to solve the two distinct integrals appearing above,
we introduce the elliptic integrals of the first and second kind,
and
, defined as
respectively
Finally, using the explicit expressions of the unit vectors
and , defined in (5), (6), and (7), we obtain the new formula
for calculating the mutual inductance between inclined circular
filaments placed in any position (most general case) as a function of only the geometric parameters, i.e.,
and
;
1) the primary and secondary loops radii,
2) the parameters , , and defining the normal of the plane
containing the secondary coil;
3) the coordinates
defining the center of the secondary coil.
The new formula can be expressed as follows:
(24)
(18)
together with the following transformations [27], [28]:
where the following sequence of definitions must be used prior
to evaluate (24):
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010
Fig. 2. Geometric configurations and common notation used in examples 1 to 13 (Section IV of this paper). The three figures illustrated correspond to the following
cases: a) lateral misalignment only (
,
, axes y z and y
z coplanar); b) lateral and angular misalignment (
, axes y z and y
z
coplanar); c) arbitrary lateral and angular misalignment (no coplanar axes anymore). The sequence also illustrates graphically how to understand Grover’s notation
in [4], and therefore how to use correctly his formulas. We note that four parameters are sufficient to describe the arbitrary position of the secondary coil: , , and ( and define the relative position between the centers, is a “in-plane tilt angle” around axis x, and is a rotation angle around axis z , which allows
completing the secondary coil positioning). Because of the many angles involved, the set of parameters leading to a given coil arrangement is not unique.
=0 =0
0
0
(25)
and where the complete elliptic functions
and
are
defined in (18) and below (19). More details about the way we
obtained the above expressions are provided in Appendix B.
Equation (24) covers all possible cases except one, i.e., the
, following from an arbitrary algebraic
case
choice made in Appendix A. This corresponds to a secondary
coil with its normal aligned with the axis. In this case, one
to
and ,
should simply use alternative definitions for
, i.e.,
arising from the limit when
(26)
There remains only one very particular case for which numerwill be punctually equal to
ical problems can arise. Indeed,
0 whenever the integration variable corresponds to
and
, no matter the value of . In other words, if any
and
,
point of the secondary coil has coordinates
and if it happens that the numerical integration procedure falls
and an infinite value is returned
exactly on that point,
for the integrand. Attempts were made to find a general mathematical limit to take care of this problem analytically, but it is
a challenging task. A much simpler workaround here is to shift
the center of the secondary coil by a small value (
or
) and redo the computation. This approach
avoids the singularity and allows finding the mutual inductance
directly, in the sense of a numerical limit. It is interesting to
=0
0
0
note that all Grover’s formulas exhibit this singularity, although
Grover himself never reported this issue.
The new equation (24), together with the accompanying def, is
initions (25) and alternative formula (26) for the case
very simple to program. As long as the loops do not touch each
other, the integrand is never singular, and therefore any simple
numerical integration scheme can handle this integral numerically. The equation gives the same results if we chose the signs
above or below the or signs, since these signs correspond
respectively to the points
and
of the secondary circle,
given in (2). A Matlab implementation of this formula is available from the authors at http://ieeexplore.ieee.org.
IV. APPLICATION EXAMPLES
In this section, we show that the new formula (24) for the
mutual inductance calculation between two circular loops with
arbitrary orientations works indeed very well. Most examples
were taken from Grover’s book [4], although a few of them
are new examples. The latter were validated with the FastHenry
software [29]. All possible cases were tested, and none of them
failed.
In order to make easier the link with Grover’s book, we restate the reference problems in terms of a common notation (see
,
,
, and
Fig. 2), namely
. In all cases, the primary coil is located in plane
, with its center at the origin
.
Any numerical integration procedure can be used to solve
(24), although it is common practice to use an adaptive technique in order to make sure that a sufficient discretization is
used. It should also be noted that the loop with the larger radius
should be always taken as the primary coil in order to ensure
.
A. Example 1 (Example 62-Grover)
, with a
Given two circles of radius
between their centers, and an angle
distance
between the axis and the line joining their centers
[see Fig. 2(a)], we want to determine the mutual inductance between them. The two circles lie in planes parallel to each other.
BABIC et al.: MUTUAL INDUCTANCE CALCULATION BETWEEN CIRCULAR FILAMENTS ARBITRARILY POSITIONED IN SPACE
This case was solved in Grover’s book [4] with the help of
equation (159), for two circular filaments with parallel axes. The
result is
For this coil arrangement, we have the following plane equaor
, i.e.,
tion for the secondary coil, i.e.,
and
, with the center point
. Applying the new formula (24), the mutual inductance obtained is
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With their centers 3 in apart
cm , the mutual inductance now becomes positive (see explanation below)
For this coil arrangement, we have the plane equation
for the secondary coil, i.e.,
and
, with its center
at point
. Using the new formula
(24), the mutual inductance calculated is
For
cm, we find
B. Example 2 (Example 63-Grover)
For
cm, the mutual inductance is
Two circles of wire are located in planes parallel to each other.
.
Both have a diameter equal to 48 in
The horizontal and vertical distances between their centers are
in
cm and
in
respectively
cm [see Fig. 2(a)].
This case is also presented in [4] and solved with the help
of equation (159), for two circular filaments with parallel axes.
The result is
From these results, we see that both formulas give the expected
physical behavior, i.e., as the coils are moved closer together,
the mutual inductance is initially negative (coils far from each
other, weak coupling), then becomes zero, and finally becomes
positive when the major part of the area of the secondary loop
coincide with the primary loop.
D. Example 4 (Example 65-Grover)
As in the previous example, the plane equation is
,
and
, and the center point of the secondary
i.e.,
. Using the new formula
loop is
(24), the mutual inductance obtained is
The negative sign means that the electromotive force (emf)
induced in one coil by a change of current in the other coil is
in the opposite direction than the emf resulting from the same
change in current when the loops are arranged in a coaxial position (i.e., when
).
C. Example 3 (Example 64-Grover)
Two coplanar circles (i.e.,
) of 1 foot of diameter
cm are placed 1.5 feet apart
cm
[see Fig. 2(a)]. This case has been solved in [4] using equation
(161) for two circular filaments with parallel axes. The result is
If the circles are brought closer together
cm until
they are almost tangent (but both circles should not touch or
overlap each other, which is unphysical), the mutual inductance
becomes
In this example, we calculate the mutual inductance of circular filaments having parallel axes and unequal radii. We have
two circles with
cm and
cm, the distance becm and
[see Fig. 2(a)].
tween their centers is
This case has been solved in [4], using equation (162) for two
circular filaments with parallel axes. The result is
For this coil arrangement, the plane equation of the secondary
, i.e.,
and
, and the center of
coil is
.
the secondary coil is
Using the new formula (24), we find the following value for the
mutual inductance:
E. Example 5 (Example 66-Grover)
The two circles of the previous examples (
cm and
cm) are brought closer together, with the distance becm and
[see
tween their centers being now
Fig. 2(a)].
The solution presented in Grover’s book [4] and based on
equation (162) is
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For this coil arrangement, the plane equation of the secondary
, i.e.,
and
, and the center of
coil is
.
the secondary coil is
Using the new formula (24), we find the following value for the
mutual inductance:
IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010
. Using our new formula (24), the mutual inductance
obtained is
I. Example 9 (Example 71-Grover)
We observe a discrepancy of about 10% in this particular case.
This point is further explored in the next example.
F. Example 6 (Example 67-Grover)
In this example we use the same coil data as in the previous
example [referring to Fig. 2(a)], but this time we use Grover’s
general equation (163) for determining the mutual inductance
[4]. The result is
This result is identical to the one obtained with our new formula (24) in Example 5, and shows that some of Grover’s formulas do not give results as accurate as the most general one he
proposed (163). Therefore, Grover’s results should be used with
care.
cm,
cm have their
Two circles of radius
origin on the -axis, separated by a vertical distance of
cm. The
axes of the secondary coil are inclined at
with respect to the - plane [see Fig. 2(b)], i.e.,
.
This case has been solved in [4] with the help of equation
(170), which is the most general case when the center of the
secondary coil lies of the axis of the primary coil. The mutual
inductance obtained is
For this coil arrangement, the secondary coil can be described
,
and
exactly as in the previous example, i.e.,
, with the center point at
. Using our new
formula (24), we find
G. Example 7 (Example 69-Grover)
Let’s assume two circular filaments with
cm,
cm,
cm, and
[see Fig. 2(b)]. This is a case
of coaxial loops with an angular misalignment.
This problem has been solved in [4], using equation (168) for
two circular filaments whose axes intersect at the center of one
of the coils. The result is
For this coil arrangement, the secondary coil can be described
, or
,
by the plane equation
,
, and
, with the center
i.e.,
. Applying (24) of this work, we find a perfect
point at
agreement:
H. Example 8 (Example 70-Grover)
In this example, we assume two circular filaments with
in
,
,
,
and
(see Fig. 2(b)).
This case has been solved in [4], using equation (168) as in
the previous example. The result is
J. Example 10 (Example 73-Grover)
In this example, Grover calculated the mutual inductance of
inclined circular filaments whose axes - and
intersect,
but not at the center of either [see Fig. 2(b)]. The axes radii
cm and
cm, and axis
of two coils are
intersects with axis at a vertical distance
cm
from the origin . The distance between this intersection and
cm. The angle of
the center of the secondary coil is
inclination between coil axes is
. In terms of the notation
and
.
of Fig. 2(b), we have
Grover has solved this problem in addressed this problem
with his formula (174) [4], which provide a result in term of
a series containing Legendre polynomials. The result is
For this coil arrangement, the plane equation of the secondary
, i.e.,
,
coil is
,
, with the center point
.
Using the new formula (24), we obtain
K. Example 11
For this coil arrangement, the secondary coil can be described
by the plane equation
, or
, i.e.,
,
and
, with the center point at
Let’s consider two circular filaments with
cm and
cm. The primary coil lies in the plane
, and
is centered at
, as in all previous examples. The seccm, with its center located
ondary coil lies in the plane
BABIC et al.: MUTUAL INDUCTANCE CALCULATION BETWEEN CIRCULAR FILAMENTS ARBITRARILY POSITIONED IN SPACE
TABLE I
MUTUAL INDUCTANCE VALUES COMPUTED IN EXAMPLE 12, WITH VARIABLE ANGLE
at
cm. In terms of the notation of this paper (see
Fig. 2), we have
cm,
cm,
and
.
This problem was chosen to illustrate a case in which
, i.e.,
. This singular case is easily addressed using the
and
to
given in (26), together
alternate definitions for
with the general formula (24). In addition, since this singularity
occurs only because of an arbitrary choice in the mathematical
development of (24), it has to give the same results as Grover’s
formula (175) for
[4]. Explicitly, the mutual inductance
obtain with Grover’s formula is
L. Example 12
This example is based on the same geometric information as
in Example 10 (
cm,
cm,
cm,
cm,
), to which we add a variable rotation angle
, [as depicted in Fig. 2(c)], and according to the
notation used by Grover ( : latitude angle, : longitude angle).
Grover addressed this case with the help of its formula (179)
[4], which is also his most general formula. With the Grover’s
= 60
formula, one must take the center of the secondary coil at point
, where
and
. The same center point will be used with the new
formula (24) introduced in this work.
There remains to compute the equivalence between Grover’s
latitude and longitude angles and the , , and parameters
defining the secondary coil plane. It happens that this equivalence is that of a spherical to cartesian system of coordinates,
i.e.,
Using the new formula (24), we obtain
Thus, we obtained the correct result as expected, despite the singularity arising from
. Interestingly, if we compute the
mutual inductance by changing the position of the center of the
, we find
with our formula,
secondary coil to
whereas Grover’s formula (175) provides an indeterminate recm)
sult. Using a very small value for the (for instance
and applying again Grover’s formula for this numerical limit,
we find
, which confirms the validity of our formula.
AND
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(27)
Given the above equivalences, one can easily compute the
mutual inductance for various angles, and compare the results
obtained with formula (179) of Grover with those obtained with
formula (24) of this work. The results are shown in Table I.
Results obtained with the FastHenry software are also shown
in this table, in order to provide a third and independent means
of validation. From these results, we can confirm the validity of
the new formula (24) for this very general case.
M. Example 13
In this last example, we compare results obtained with our
formula and those obtained with the FastHenry software [29],
for seven arbitrary arrangements of coils described only by their
plane equations. Results are provided in Table II, and show very
good agreement. The slight discrepancies are not significant
and can be explained by the fact that the present method considers thin circular filaments of negligible cross section, whereas
FastHenry actually considers conductors of finite cross section,
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010
TABLE II
MUTUAL INDUCTANCE VALUES COMPUTED IN EXAMPLE 13, FOR ARBITRARY POSITIONED COILS
even if taken small. In addition, FastHenry discretizes curved
paths into many straight segments, which may also slightly affect the final result.
The equation of a circle of radius
and passing by point
centered at point
is
(A.31)
V. CONCLUSION
Eliminating the variable
In this paper, we derived a new formula for computing the mutual inductance between inclined circular filaments arbitrarily
positioned with respect to each other. In order to use this new
formula, whose final expression is given per (24), one needs
to provide the radius of the primary and secondary coils, the
position of the center of the secondary coil (the primary coil
is assumed to be centered at origin), and the plane equation in
which the secondary coil is located. With these parameters, the
problem is completely defined.
The new formula was verified against results available in literature (mainly Grover’s formulas), and double-checked with the
FastHenry software. In all cases, it proved to be successful, and
therefore we can claim that the new formula (24) is the most
general developed so far for this purpose. In particular, it replaces all Grover’s formulas at once. It is also more intuitive
to use than Grover’s formula (179), considered as its most general one, since using a plane equation (or a normal vector to this
plane) is more natural than specifying the set of latitude and
longitude angles that are difficult to figure out for an arbitrary
positioned coil.
APPENDIX A
DETERMINATION OF A POINT
THE SECONDARY COIL
BELONGING TO
and lies in plane ,
The secondary coil has a radius of
whose equation is given in (1). From this expression, is given
by
with the help of (A.30), we obtain
(A.32)
where
(A.33)
The solution of this algebraic equation is
(A.34)
where
(A.35)
This inequality gives the interval of definition for the variable
, i.e.,
(A.36)
After substituting back and by their corresponding expressions in terms of , and , we find the simple expression below:
(A.37)
(A.28)
where
(A.29)
An arbitrary point
necessarily satisfy (A.28), i.e.,
of the secondary coil must
(A.30)
in which
and
.
and
define the allowed interval for the
The values
. Any value taken in this interval allows defining a point that
is part of the secondary loop, using (A.30) and (A.34) allows to
and corresponding to a given . If we simply
determine
and
for , we define the following two points
choose
and
:
(A.38)
BABIC et al.: MUTUAL INDUCTANCE CALCULATION BETWEEN CIRCULAR FILAMENTS ARBITRARILY POSITIONED IN SPACE
Thus, the coordinates of the point
are exactly
determined.
APPENDIX B
DETERMINATION OF UNIT VECTORS , , AND OTHER
PARAMETERS APPEARING IN SECTION III
Remembering that
we easily obtain the following expressions for
:
(B.39)
and
:
(B.40)
Other expressions encountered in Section III can be much
simplified, i.e.,
(B.41)
In a similar way, we find
(B.42)
then
(B.43)
and
(B.44)
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Slobodan Babic received the Dipl. Ing. degree from the Faculty of Electrical
Engineering, University of Sarajevo, the M.Sc. degree from the Faculty of Electrical Engineering, University of Zagreb, Croatia, and the Ph.D. degree from the
Faculty of Electrical Engineering, University of Sarajevo, Bosnia and Herzegovina, in 1975, 1992, and 1980, respectively.
From 1975, he was with the Electrical Engineering Faculty of the University
of Sarajevo, where he held an Associate Professor position until 1994. Since
1997 he has lectured on the subjects of physics and electrical engineering at
École Polytechnique de Montréal, Montréal, QC, Canada. He is also research
associate. His major interests are in the mathematical modeling of stationary
and quasi-stationary fields, electromagnetic fields in machines, transformers,
computational electromagnetics, magnetic materials, and field theory. He has
authored or co-authored over 90 papers in major journals and conference proceedings. His research papers have been highly cited over 250 times.
Dr. Babic is a member of the International Compumag Society and a member
of the Editorial Board for the International Journal of Computer Aided Engineering and Technology. He is Editor-in-Chief for the WSEAS TRANSACTIONS
ON POWER SYSTEMS. He is a reviewer for IEEE TRANSACTIONS ON MAGNETICS,
International Journal of Computer Aided Engineering and Technology, Journal
of Electromagnetic Waves and Applications, and WSEAS TRANSACTIONS ON
POWER SYSTEMS.
Frédéric Sirois (S’96–M’05–SM’07) received the B.Eng. degree in electrical
engineering from Université de Sherbrooke, Sherbrooke, QC, Canada, in 1997,
IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010
and the Ph.D. degree from École Polytechnique de Montréal, Montréal, QC,
Canada, in 2003.
From 1998 to 2002, he was affiliated as a Ph.D. scholar with Hydro-Québec’s
Research Institute (IREQ). From 2003 to 2005, he was a research engineer at
IREQ. In 2005, he joined École Polytechnique de Montréal as an Assistant Professor, and was promoted to the rank of Associate professor in 2010. His main
research interests include modeling and design of electromagnetic and superconducting devices, integration studies of superconducting equipments in power
systems, and planning of power systems. He is also a regular reviewer for several international journals and conferences.
Cevdet Akyel (M’81) received the Sup. Ing. degree from the Technical University of Istanbul in 1971 and the M.Sc.A. and D.Sc.A. degrees from École Polytechnique de Montréal, Montréal, QC, Canada, in 1975 and 1980, respectively.
He had engineering positions in 1972 and 1976 at Northern Telecom of
Canada as a System Engineer in radio telecommunications. Since 1986, he
has been a Professor of Electrical Engineering at École Polytechnique de
Montréal, where he teaches electromagnetic theory and automated microwave
instrumentation. In 1991, he joined the Group of Poly-Grames involved in
space electronics and microwave advanced technologies at the same university.
His main research interests are the permittivity measurement with microwave
active cavity methods, the characterization of materials (conductive polymers,
superconducting ceramics, ferromagnetic materials, etc.), and high power
microwave measurement systems and applications.
Claudio Girardi (S’95–M’98) received the Laurea degree in telecommunications engineering from the Politecnico di Milano, Milan, Italy, in 1997, with a
thesis on the subject of high-speed/low-power amplifiers using pHEMTs.
In 1998, he joined the Optical Networks Division of Alcatel Italia, Vimercate, Italy, working on the hardware design of broadband optical communication equipment. From 2004 to 2005, he was with Esterel Technologies, Villeneuve-Loubet, France, in charge of the validation of the RF part of a wireless system-on-a-chip (SoC). In 2005 he joined Texas Instruments, VilleneuveLoubet, France, working on the integration of a GSM/EDGE RF front end in
a SoC and on the evaluation of the coexistence aspects between the integrated
RF functions and the digital baseband. He has also been involved in signal and
power integrity analysis for systems in package. His main interests include highspeed circuit design, power integrity, signal integrity, interconnections modeling, and electromagnetic compatibility for high-speed digital circuits.