Free positioning for inductive wireless power system
Eberhard Waffenschmidt
Philips Research Europe
Eindhoven, The Netherlands
eberhard.waffenschmidt@philips.com
Abstract— Wireless power transmission suggest the freedom of
placement for power transmission. However, efficiency and
emitted magnetic fields limit the inductive power transfer to
close to a surface. But even there a lateral displacement of the
receiver coil to the transmitter coil may lead to a change of the
coupling factor and thus an unwanted variation of the power
transfer. Here, an algorithm to determine the turn distribution
to achieve homogeneous coupling between coils of different
diameter is described. As long as the coils overlap, the
variation of the coupling factor is very low. To achieve a lateral
displacement over an even larger area, an array of transmitter
coils can be used. The size of the receiver coil is selected such
that it always covers a complete transmitter coil. If only the
covered transmitter coil is activated by a suitable detection
circuit, the power transmission area can be arbitrary large
with homogeneous magnetic coupling.
I.
can be arbitrary large. This way, homogenous magnetic
coupling is achieved on top of one of the transmitter coils
and over the whole array.
II.
TURN DISTRIBUTION
The idea to find a suitable turn distribution is: First find a
current density distribution, which generates a homogeneous
field above the coil. Then find in a next step a turn
distribution, which narrows this current density distribution,
with equal current in each turn. In a third step, an
approximate analytical expression is derived. This is
illustrated in Figure 1.
H
Specified magnetic field
INTRODUCTION
Wireless power transmission suggest the freedom of
placement for power transmission. However, efficiency and
emitted magnetic fields limit the inductive power transfer to
close to a surface [1]. In inductive wireless power
transmission system a lateral displacement of the receiver
(Rx) coil to the transmitter (Tx) coil leads to a change of the
coupling factor and thus an unwanted variation of the power
transfer. For single coil systems literature describes solutions
with larger transmitter coils, which have a non-uniform
distribution of winding turns [2] [3]. Both sources use
empirical approaches, which lead to non-optimal designs.
Another approach is to use an array of transmitter coils,
where at least one transmitter coil is always covered by the
receiver, as e.g. published by the Wireless Power
Consortium [4]. But this approach may still lead to a
variation of the coupling, if the receiver’s position is varied
on top of one transmitter coil.
This paper aims in merging and improving these two
approaches. First, a procedure to calculate a winding turn
distribution to achieve a homogeneous magnetic coupling for
different sized coils is introduced. Contrary to existing single
coil solutions, this winding design is applied to the receiver
coil, which overlaps the transmitter coil. The transmitter
coils are arranged to an array, such that always one
transmitter is completely covered. If only the covered coil is
activated by a detection circuit, the power transmission area
1. step
Current density in a disk
2. step
Turn distribution
Figure 1: Illustration of the approach to calculate the turn density
A. Solving the inverse field problem
The task of finding a suitable current density distribution
relates to the task of solving the inverse magnetic field
problem. Here, a disk shaped coil with a limited outer radius
is used as example to derive and explain the algorithm.
H1
H2
H3
H4
a11 a12 a13 a14
J1
J2
J3
J4
HN
a1N
JN
Figure 2: Illustration of the inter-dependence of currents and magnetic
fields.
A discrete approach is used. The winding width is
divided into equally spaced current traces. Each current trace
contributes to the magnetic field at each magnetic field
position, as illustrated in Figure 2. The dependence of the
magnetic field on one of the current traces can be expressed
with a coefficient. The current values can be combined to a

vector J , the values for the magnetic field can be combined

to a vector H and the coefficients can be combined to a
matrix A. Then, Figure 2 can be expressed as equation
system:
(1)
H1
a11 a12 a13 ...
H2
a21 a22 a23 ...
H3 = a31 a32 a33 ...
:
: : :
HN
aN1 aN2 aN3 ...
H
=
a1N
a2N
a3N .
:
aNN
J1
J2
J3
:
JN
.
J
A
(2)
 1
A H
Thus, the unknown current distribution can be calculated
from the inverted coefficient matrix multiplied with the
vector of required magnetic field values.
Current distribution
0.6
2
Current
Magnetic field
0.4
0.3
1
0.2
0.1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Spec. magnetic field H / Ho
Trace current I / Io
0.5
An example for the resulting current distribution among
the traces is shown in Figure 3 (red curve). It also shows the
radial positions, at which the magnetic field is specified. The
figure is calculated for a planar coreless inductor. The radial
position is scaled to the outer radius Rout of the coil. It is
calculated for a number of magnetic field points and current
traces NI = 10. The magnetic field is specified at a height
above the coil of 5% of the coil diameter. To avoid
instabilities in the calculation, the magnetic field is not
specified until exactly the edge of the coil, but only to 90%
of the outer radius.
This method as described here is applicable to problems
with one-dimensional symmetry. Here, it will be applied to
circular inductors. In a similar manner, it can be applied to a
linear (or stadium shaped) inductor, if the round ends can be
neglected compared to the linear length. It’s even simpler,
because the magnetic field generated by a linear track is
simply calculated.
To obtain the current distribution, this equation is
inverted to:

J
coefficients can be calculated using Finite Element Method
(FEM) simulations.
0
1
Radial position r / Rout
Figure 3: Calculation of the current density distribution in the winding.
Red: Current density. Blue: Specified magnetic field.
To obtain one of the coefficients of matrix A, the
magnetic field at one position must be calculated from the
current in one of the traces for one arbitrary current value.
The coefficients of a circular coil in air can be calculated
from loops [5] [1] (see Appendix). If one side of the coil is
shielded with a soft-magnetic material, the algorithm
described in [6] can be used. For a general case, the
However, the general idea can also be applied to
problems, where there is no symmetry. This has not been
derived in detail in the course of this work, but I suggest
using finite sized current loops representing magnetic dipoles
distributed over the area. The resulting two-dimensional
current density distribution could then be converted into a
non-symmetric turn distribution in a similar manner as in the
following section. As a further modification of the algorithm
the number of magnetic field points could be overdetermined. Then the current vector is to be optimized by a
least square error fit. This may lead to a better stability of the
algorithm and might avoid possible oscillations between the
data points.
B. Distribution of turns
To match the current distribution to a turn distribution,
the following algorithm is used: At first, the current per turn
Iturn is calculated from the sum of all currents I0. To
determine the width of the track w, the current density times
a small x is summed up as well as the x to a width. If the
sum of all the small current equals the required current for
one turn Iturn, the necessary width of the track w is reached.
Then, the same current, summed up over the width w,
flows in the track of the distributed turn coil and in the traces
of the equally spaced structure. As a further approximation,
the resulting turns can be assumed as infinite thin with a
position in the centre of the planar track. A listing of the
algorithm will be presented in the final paper. Summarizing
the algorithm: Distribute the currents to the tracks with
variable width until one track has the right amount of
current.
The result is shown in Figure 4. It is calculated for the
previous example with a number of turns Nw = 10. The red
curve shows the current density distribution based on the
results shown in the previous Figure 3, but distributed over
the width of each current trace. The blue curve shows the
current density distribution for the distributed turns, where
each turn has a different width, but the current density adds
up to a constant track current. It is clearly seen that both
curves match as much as the discrete tracks allows.
Current density distribution
D. Resulting magnetic field
The resulting magnetic field is compared in Figure 6. The
gray curve shows the resulting magnetic field for the case of
equally spaced current traces with variable current with
NI = 10 current traces. The curve matches exactly the
specified points. The blue curve corresponds to the structure
with distributed turns with number of turns Nw = 10 turns.
Current variation
Trace position variation
NI  10
4
Nw  10
3
2
1
0
0
0.2
0.4
0.6
0.8
Radial position r / Rout
Figure 4: Calculation of the current density distribution in the winding.
Red: Constant trace width and distance. Blue: Variable trace width with
constant trace current.
C. Fit function
To ease the calculation of a turn distribution, a fit
function for a circular coil is derived. Figure 5 shows the turn
distribution as calculated in the previous section. N is the
number of turns and r(i) is the trace position of the turn with
index i. From the shape of this curve, an equation is guessed:
(3)
w
Here, the ripple is higher as for the current distribution,
because the track density is rather low at the inside of the
coil. The magnetic field for the fit function with w = 0.2 and
Nw = 10 is shown in red. The ripple is comparable to the
distribution calculated by solving with the inverse magnetic
problem.
Resulting magnetic field
1.5
Magnetic field H / [A/m]
Current density J / Jo
6
5
The parameter w is a fit parameter. It determines the
“bending” of the curve”. The term in the denominator scales
the curve to 1 at the outer turn. As Figure 5 shows w = 0.2
matches the optimal turn distribution for this particular case.
For a different height, the parameter w varies slightly.
1
0.5
0
i
1
N
1e
r( i)
1e
w
w
0.8
0.2
0.7
0.3
0.6
10
0.5
0.8
1
1.2
1
0.5
0
Specification
Nw  50
Variable current density z
H
Variable turn width
 0.1
Rout
Approx. turn width
0.5
0.4
0.3
1
0.2
0
0.2
0.4
0.6
0.8
1
1.2
Radial position r / Rout
0.1
0
0.6
Resulting magnetic field
w=
0.05 0.1
0.4
 0.1
1.5
Magnetic field H / [A/m]
Turn position r / Rout
0.9
0.2
zH
R out
Figure 6: Calculation of the resulting magnetic field for different
algorithms at height above the coil of zH = 0.1.Rout and NW = 10 turns.
Turn distribution
1
0
N w  10
Radial position r / Rout
1
Rout
Specification
Variable current density
Variable turn width
Approx. turn width
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Turn number i / N
Figure 5: Trace positions. Dots: Results from solving the inverse
magnetic field problem at a height of zH = 0.1.Rout. Lines: Fits with
different fit parameter w.
Figure 7: Calculation of the resulting magnetic field for different
algorithms at height above the coil of zH = 0.1.Rout and NW = 50 turns.
The ripple reduces, if the number of turns is increased as
Figure 7 shows. It was calculated for a winding distribution
with Nw = 50 turns. However, the equation system was
solved for only 10 current traces because of un-stabilities at a
higher number of traces. The figure clearly shows that the
“original” winding distribution has a ripple which is rarely
visible. The approximation function, however, has a slight
deviation in the centre of the winding, but is low with less
than 10%. A remaining ripple further reduces, if the finite
extension of a coupled coil is taken into account. This effect
becomes visible a the measurements in chapter 0.
Resistance R / Rref
100
Concluding, the fit function according to equation (3)
with a fit parameter of w = 0.2 gives sufficient good results
for a homogeneous magnetic field.
10
100
2  f L
0
0.1
wmin  0.2
0.3
0.5
10
,
where f is the operating frequency. To see, whether the
increase of the resistance R or of the inductance L is more
dominant, both and the quality factor Q are calculated for
exemplary structures with different turn distributions,
expressed by a variation of the fit parameter w. The results
are scaled to reference (index “ref”) values of an inductor
with the same copper thickness, but equally distributed turns.
The inductance values are calculated like in the previous
section and the resistance is calculated according to Ohm’s
law. The results turned out to be independent of the number
of turns in a wide range. To reduce discretization errors, the
calculations are performed for a “high” number of turns
N = 50.
The results are shown in Figure 8 as the red thick curves.
As expected, for low values of w, where the turns are
concentrated at the outer edge, the resistance, but also the
inductance increase. But overall, the resistance increase
exceeds the inductance increase such that the quality factor
decreases for inductors with distributed turns at the outer
edge. For the typical design case of w, = 0.2, the resistance
increases to about 10 times compared to the reference coil.
The inductance increases only by factor of 3. Therefore, the
quality factor decreases by 1/3 compared to the reference
coil.
1
0.1
1
10
Fit parameter w
b)
10
Quality factor Q / Q ref
R
1
Fit parameter w
a)
(4)
Q
10
1
0.1
Inductivity L / Lref
E. Quality factor of printed circuit board coils
If the coil with distributed turns is realized in printed
circuit board (PCB) technology, the width of the tracks is
usually adapted such that a maximum amount of the copper
layer is used. However, since the turns of an optimal
distribution are concentrated at the outer edge, these tracks
are significantly thinner than the average width. Therefore,
the coil with optimized turn distribution has a significant
higher resistance as a reference coil with an equal
distribution of the turns with the same number of turns.
However, the concentration of the turns at the outer edge
also increases the inductance compared to the reference coil.
Important for an application is the ratio of the inductance L
to its resistance R, expressed as the quality factor Q:
0
0.1
wmin  0.2
0.3
0.5
1
0
0.3
0.01
0.1
c)
0.1

wmin
0.2
0.1
0.5
1
10
Fit parameter w
Figure 8: Change of a) resistance, b) inductivity and c) quality factor of a
coil with distributed turns scaled to equally spaced turns (reference) as a
function of the fit parameter w.
Contrary, for a fit parameter  > 0.5 the quality factor
does no longer degrade significantly and is better than 90%
of the reference coil. However, a design with those fit
parameters lead to an inhomogeneous magnetic field
distribution.
To avoid the tracks becoming too thin, a minimum track
width wmin is introduced: no track must be smaller than this
value. To make the parameter independent of the particular
structure, the minimum track width wmin is related to the
track width of a reference structure with equal turn
distributions. To achieve a turn distribution satisfying this
criterion, the turns are first distributed from the outer edge to
the inside as close as possible. As soon as it is possible to
place a track on the optimal position (according to equation
(3) )without violating the width condition, the track is place
there. Thus, at the inside of the coil the turns are on the same
position as in the optimal distribution.
The effect of the introduction of a minimum track width
on resistance, inductance and quality factor is shown in
Figure 8. Especially for small values of w, where the turns
are concentrated at the outer edge, a significant improvement
with larger minimum track widths can be seen. Using a
minimum track width of wmin = 0.2 at an optimal fit
parameter of  = 0.2, the quality factor improves from 30%
of the reference value (red) to 60% (green) of the reference
value. A minimum width of wmin = 0.5 (blue) even
improves the quality factor to 90% of the reference value.
Concluding, the homogeneous magnetic field must be
paid with reduction of the quality factor, if the coil is
manufactured in PCB technology. However, the quality
factor reduction can be minimized by introducing a
minimum track width. For wire wound coils, this effect is
less obvious, because of the fixed wire diameter and wires
can overlap.
It should be mentioned that these investigations consider
only the DC (direct current) resistance. The AC (alternating
current) resistance is usually higher and dependent on the
track width. Therefore, for higher operating frequencies the
result may look different and are worth further
investigations.
III.
EXAMPLE AND MEASUREMENTS
Based on these considerations a transmitter and a
receiver coil are manufactured in wire-wound technology.
The purpose is to prove the calculations on the homogeneous
field distribution, not on quality factor. The geometric
dimensions of the coils are listed in Table 1.
2
1.5
Figure 10: Tx coil placed on Rx coils, both realized with litz wire.
1
0
0.5
0
-0.5
-1
0
Coupling measurements
0.1
w = 0.2
wmin  0.2
0.3
0.35
Nw = 10
0.3
0.5
0.2
Z = 0.1 .R
out
0.4
0.6
0.8
Radial position r / Rout
1
1.2
Figure 9: Magnetic field of distributed turns with minimum track width.
The influence on the magnetic field is shown in Figure 9.
As can be seen in the figure, the magnetic field in the centre
part of the coil is hardly affected by modifying the turn
distribution. Increasing the minimum track width only leads
to a less steep “edge” of the magnetic field at the outer edge
of the coil. As can be seen, a minimum track width of
wmin = 0.5 (blue) leads to a wide area of decay at the outer
edge, which is not desired. However, at a minimum track
width of wmin = 0.2 (green) the magnetic field shows hardly
a difference to the magnetic field of the optimal distribution
(red).
Coupling
Coupling factor
factor kk
Magnetic field H / [A/m]
2.5
0.25
0.2
0.15
z=0mm, meas.
0.1
z=5mm, meas.
max. overlap
0.05
0
0
5
10
15
20
Radial displacement r / mm
25
30
Figure 11: Measured coupling factor of the litz wire coils for a radial
displacement with vertical distance as parameter.
The turn distributions are calculated using the fit
function. Different to the exemplary simulations in the
previous chapter, the larger coil has only 2x8 turns, which
resulted from inductivity requirements of the intended
application (not further discussed in this paper).
The receiver is larger than the transmitter and selected
such that it always completely covers a transmitter coil in a
hexagonal array. Figure 10 shows the smaller transmitter coil
on top of the receiver coil. In the final system, the
arrangement would be reversed.
Figure 11 shows coupling measurement results at
different vertical distances z. The measurements are done
with an impedance analyzer HP4194 and manual placement.
The red curve is measured at a vertical distance of 5 mm
(= 0.1.Rout) and can be compared to the simulation shown in
Figure 6.
Clearly, a good lateral homogeneity is visible. The ripple
is lower despite the less turn positions in the previous
chapter, because the extension of the smaller coil averages
out the ripple. The coupling factor is very constant up to a
displacement of 28 mm, which marks the relevant range of
operation.
TABLE 1: GEOMETRIC PROPERTIES OF THE LITZ-WIRE COILS:
Rx coil
Design Name
Number of turns per layer
Number of serial layers
Wire type
Outer coil diameter / mm
Radial turn positions / mm
Tx coil
Design Name
Number of turns per layer
Number of serial layers
Wire type
Outer coil diameter / mm
Radial turn positions / mm
Hexagonal
Quadratic
Hex triple
Hex overlap
Hex triple overlap
Figure 12: Coil arrangement options: Red: Receiver, Green: Activated
cells, Blue: Non-active cells.
TABLE 2: TRANSMITTER COIL ARRANGEMENT OPTIONS
N_tpl
N_lay
d_out
r
N_tpl
N_lay
d_out
r
#1
8
2
180 x 0.03
100
23.4
#1
4
2
180 x 0.03
44
15.8
35.9
42.6
45 46.25
47.5 48.75
50
Diameter / m
#of cells per area /[1/m²]
# of layers per coil
System quality factor
Coupling factor
FOM Qk
Hex
Hex
Hex triple
Receiver Quadratic Hexagonal overlap triple
overlap
100E-3
41.4E-3
46.4E-3 63.4E-3 33.3E-3
50.0E-3
583
536
1149
1039
1848
2
3
3
1
3
1
144.3
152.8
103.1
123.0
106.0
0.201
0.237
0.374
0.262
0.426
29.1
36.3
38.6
32.3
45.2
Coupling factor conditions: 5mm distance, Tx 5 turns, Rx non-equal turn distance 10 turns
19.8
20.9
22
At the vertical distance of z = 0 the coupling variation is
remarkably larger, because this is not the distance for which
the system is designed. However, still a deviation of +/-10%
of the average coupling factor is achieved, which is still good
compared to conventionally designed coil combinations.
IV.
coupling between the transmitter cells is considered and an
effective coupling of the three combined coils to the receiver
coil is calculated. The Rx diameter is 10 cm, copper
thickness is 70 µm, frequency is 420 kHz.
SELECTION OF TRANSMITTER COIL ARRANGEMENT
In this chapter, the most suitable options for transmitter
arrays are analyzed and weighted. Figure 12 shows the most
reasonable arrangements of transmitter coils. For each
arrangement, the size of the transmitter coils is adapted such
that always at least one Tx coil is covered by the receiver.
The first row shows arrangements, where only one Tx cell
activated at the same time. In further arrangements, not one,
but three transmitter coils are activated simultaneously
(proposed by [4]). For comparison, the product of the quality
factor Q and the coupling factor k is calculated as a figure of
merit (FoM), which should be as large as possible. It is
assumed that all are made from the same copper thickness,
e.g. printed circuit board (PCB) coils. As a further decision
factor the required number of Tx cells per area is calculated.
Inductivity and resistance are calculated according to [7]
neglecting AC effects and the coupling calculations are
based on [1]. For the triple cell arrangements, the mutual
The calculation results are listed in Table 2. The
overlapping arrangements allow larger transmitter coils with
better coupling. However, the thickness can be only 1/3.
Therefore, the quality factor of the coils is lower. The triple
hex overlap arrangement achieves the best FOM because of
the good coupling of the triple cell to the receiver. However,
it requires by far the most number of cells. The least number
of transmitter cells are needed for the hexagonal
arrangement. It has a reasonable efficiency similar to the hex
overlap structure and the hex triple structure.
Concluding, the hexagonal arrangement in one layer is
the preferred solution for a low cost inductive power
transmission system with free positioning.
V.
TRANSMITTER ARRAY
As shown in the previous chapter, a regular hexagonal
arrangement is preferred. To give an impression of the
coupling homogeneity of a larger area consisting of 19
hexagonal arranged transmitter coils, its coupling is
simulated. The layout used for the simulation is shown in
Figure 13. The larger receiver coil (blue) is specified to
10 cm diameter and consists of 6 turns. The smaller
transmitter coils (red) are designed to 4.4 cm diameter and
have 4 turns. The turns at the outer edge may be difficult to
distinguish in the plot due to the printing resolution. In the
simulation, always only one transmitter coil is activated. For
this purpose, the first coil within a “detection radius” of 1.2
of the transmitter coil outer radius is used in the final plot.
0.1
Transmitter coils
Receiver coil
0.08
y-Position / m
0.06
0.04
The resulting coupling position dependence is shown
Figure 14 in a three dimensional plot. The z-axis of his plot
corresponds to the induced magnetic flux in the receiver coil
as a measure for the coupling. In plot a) showing the full
scale, no structure of a coupling variation is visible.
Therefore, the display range is limited to a range from 80%
to 100% of the maximum flux value in the bottom figure b).
Only by reducing the display range the slight variations of
the magnetic coupling become visible.
VI.
0.02
Free positioning of a wireless power receiver over a large
area can be achieved with a suitable winding design,
selecting a larger receiver coil which always covers one
transmitter and by using a hexagonal array of transmitter
coils.
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12 -0.08 -0.04
0
0.04
x-Position / m
SUMMARY
ACKNOWLEDGMENT
0.08
0.12
Figure 13: Coil layout for the simulation of a larger coil array.
Thanks to my colleague Reinhold Elferich, who triggered
the idea to the method, and to Michael Deckers, at that time
at Philips Research Aachen, for the design of related
hardware.
REFERENCES
[1]
2.74 µVs
2.19 µVs
1.50 µVs
[2]
[3]
0.82 µVs
[4]
a)
0 µVs
[5]
2.74 µVs
[6]
2.63 µVs
2.49 µVs
2.35 µVs
b)
2.19 µVs
Figure 14: 3D visualization of the coupling homogeneity simulation.
Detection radius is 27.6 mm from Tx cell centre, maximal one Tx cell is
activated. a) Linear scale from 0 to maximum. b) Reduced scale from 80%
to 100% of maximum!
[7]
Eberhard Waffenschmidt and Toine Staring, "Limitation of inductive
power transfer for consumer applications", 13th European Conference
on Power Electronics and Applications (EPE 2009), Barcelona,
Spain, 8.-10.Sept. 2009, paper #0607.
Xun Liu and S.Y.(Ron) Hui, "Optimal design of a hybrid winding
structure for planar contactless battery charging platform", IAS 2006.
Joaquin J. Casanova, Zhen Ning Low, Jenshan Lin and Ryan Tseng,
"Transmitting coil achieving uniform magnetic field distribution for
planar wireless power transfer system", Proceedings of IEEE Radio
and Wireless Symposium 2009, Jan. 18-22, 2009, p.530, paper
#TU4B-5.
System Description Wireless Power Transfer, Volume I: Low Power,
Part
1:
Interface
Definition.
Available
at
http://www.wirelesspowerconsortium.com/blog/2010/08/31/qispecification-availble-for-download/
Kathleen O’Brian, “Inductively coupled radio frequency power
transmission system for wireless systems and devices”, PhD Thesis,
TU Dresden, 5.12.2005, Shaker Verlag Aachen 2007, ISBN 978-38322-5775-0.
Eberhard Waffenschmidt, "Shielding properties of soft-magnetic
layers for planar inductors", 14th International Power Electronics and
Motion Control Conference - EPE-PEMC 2010, Ohrid, Republic of
Macedonia on September 6-8, 2010.
E. Waffenschmidt, B. Ackermann, "Size advantage of coreless
transformers in the MHz range", EPE 2001 (European Power
Electronic Conference), Graz, Austria, 27.- 29. Aug. 2001.
APPENDIX
A. Magnetic field of a coreless loop
For a single circular loop centered on the z-axis, the axial
magnetic field intensity Hz is (see [5], p. 34):
(5)

 a 2  r2  z2
1
Hz( r  z  a  I)  I

 E( k ( a  r  z) )  K ( k ( a  r  z) )

2
2

2
2
2  ( a  r)  z  ( a  r)  z

with
I = current in the loop
a = radius of the loop
r = radial position of the point
z = axial position of the point
K(k) is the Elliptic Integral of the first kind:
(6)

2

K( k )  


1
1   k  sin    
d
2
0
E(k) is the Elliptic Integral of the second kind
(7)

2

E( k )  

2
1   k  sin     d
0
And the auxiliary function k is defined as
(8)
k ( a  r  z)  2
ar
2
2
( a  r)  z