Design of Large Air-Gap Transformers
`for Wireless Power Supplies
`
`K. O'Brien, G. Scheible", H. Gueldner
`
`Dresden University of Technology
`Department of Electrical Engineering
`01062 Dresden, Germany
`
`*ABB Corporate Research
`Wallstadter Str. 59
`68526 Ladenburg, Germany
`
`Absrracr-This paper addresses the design of two-dimensional
`large air gap transformers for wireless power transmission. A
`theoretical analysis of the optimum shape and arrangement of
`the primary coils is presented. The magnetic field created by
`such a system is discussed and optimised. The effects of shielding
`of the secondary coils are discussed. The optimum coil
`arrangement for one example design is derived.
`
`I. INTRODUCTION
`
`Novel power supplies for use in applications such as
`robotics, automated production machines, and applications
`with high insulation requirements where wired energy
`transfer is not suitable have recently been proposed [I], [2],
`[3]. Some of these supplies use unconventional transformers
`with large air-gaps to supply energy to the load via magnetic
`fields over distances up to several meters and provide for the
`wireless supply of power to devices such as sensors,
`communication devices, or actuators. Figure 1 depicts the
`main transformer components being used. The primary coils
`define a system with the coils lying in several different
`planes. The secondary coil(s) are wound around a ferrite core
`and placed inside the box formed by the primary coil(s).
`For a reliable operation of the power supply the transformer
`must be designed and powered in such a way as to transfer
`the energy to secondary coils that may be shifting in position
`and may be magnetically shielded by metallic objects within
`the operating volume. The field created by the primary coils
`should be as uniform as possible over the greatest possible
`volume in order to provide adequate power to the secondary
`coils. Apart from the two dimensional system shown in
`Figure 1, which can be regarded as a practical compromise
`between performance and complexity, one and
`three
`dimensional systems can be conceived as being formed by
`one or several coils in each plane.'
`This paper explores various design options for the primary
`side of the transformer with respect to uniformity of power
`transfer, utilization of the power per volume unit, and ability
`to overcome the problems associated with shielding of the
`magnetic field. An example one-dimensional system using a
`current of 24A and a magnetic field intensity at the center of
`the system of 4A/m is described.
`' Patents pending
`
`Figure 1 - Power supply using a large air-gap transformer
`
`II. DESIGN OF SECONDARY COILS
`
`Energy is transferred from the primary to the secondary
`coil(s) via a magnetic field.
`If the magnetic field lines
`created by the primary coil(s) do not pass through the area
`enclosed by a receiving (secondary) coil(s), no voltage will
`be induced in those coils. In order to minimize the effect of
`the position, shielding, and orientation of the secondary coil
`on the transformer's energy transfer characteristics, three
`secondary coils are designed such that they form a three-
`dimensional orthogonal coil system. The coils are arranged
`at a 90" separation around a cube-shaped ferrite core as
`indicated in Figure 1.
`
`III. BASIC SHAPES OF PRIMARY COIL
`
`The size, shape, and uniformity of the field created by the
`primary coils depend significantly on the coil configuration.
`Square, rectangular, and circular coils are investigated.
`
`A, Magnetic Fielak due to Rectangular Coils
`
`Single rectangular coil: The intensity of the magnetic field
`created at an arbitrary point in space P when a current is
`applied to a straight current carrying wire of length 2L is
`described in the near field (2mf A << 1, where h is the
`wavelength of the current) by equation (I).
`
`0-7803-7754-0/03/$17.00 02003 IEEE
`
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`
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`Exhibit 1021
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`

`

`2L f-J2,y
`
`'P
`
`1%
`
`4 'P, Y
`
`X
`
`I
`
`X
`
`
`(b) (b)
`(a) (a)
`
`Figure 2 - (a) rectangular coil with field intensity described by (2) (b) Figure 2 - (a) rectangular coil with field intensity described by (2) (b)
`
`rectangular coil system with field intensity described by (3) rectangular coil system with field intensity described by (3)
`
`where I is the current in the wire, r is the perpendicular
`distance between the point P at which the field is being
`calculated and the current carrying wire, 41 is the angle
`between the imaginary line of length r connecting the field
`point with the wire and the x-axis.
`
`The magnetic field intensity at an arbitrary point in space
`due to a rectangular coil can be expressed as the vector sum
`of the fields created by each of the straight wires. A
`rectangular coil lying in the y-z plane (Figure 2a) where wires
`1 and 2 are parallel to the z-axis and wires 3 and 4 are parallel
`to the y-axis will produce
`
`z
`
`(b)
`(a)
`Figure 3 - (a) angles used to calculate field strength due to a single coil
`(xy-axis). (b) angles used to calculate field strength due to a single coil
`(xz-axis)
`where xi = 4
`, s = number of coils
`2n4m-
`On the axis of square coils, the equation describing the
`magnetic field simplifies greatly to
`
`As an example, Figure 4 shows the intensity of the
`magnetic field created by two square coils 3m x 3m in
`dimension carrying 24A. The surface shown covers all points
`where the value of H is greater than 4".
`Figure 5 shows a
`cross section of the field intensity at z=lS m.
`
`Figure 4 - Area of field strength greater than or equal to 4A/m created
`by two square coils carrying 24A
`
`Multiple Rectangular Coils: The intensity of the field at an
`arbitrary point created by a set of rectangular coils can be
`expressed as the vector sum of the fields created by each of
`the coils. Numbering the sides of the rectangular coils as in
`Figure 2(b),
`
`Figure 5 - Cross section (at z=1.5m) showng field intensity created by two
`square coils carrying 24A
`
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`

`

`B. Magnetic Field due to Circular Coils
`
`IV. OPTJMUh4 SHAPE OF PRIMARY COILS
`
`Fields can be created using circular, square, or rectangular
`coils. It is beneficial to examine some differences between
`the fields created by coils of each shape.
`Obtaining a configuration in which the greatest possible
`volume of stable (unchanging) magnetic field exists between
`two coils may be of interest for certain applications. For two
`equal coils of any shape and size, the optimum distance dH
`(also known as Helmholtz spacing) by which the two coils
`should be separated to achieve the greatest possible area of
`stable magnetic field can be determined by finding the
`distance at which the gradient of the magnetic field in the
`center of the coils is equal to zero [4], [SI, [6].
`
`Expanding (6) to find the magnetic field at the center of two
`circular coils (Figure 6b) lying in a plane perpendicular to the
`z-axis with equal radius al =a2 = a and a distance D=dH
`apart. The z-axis passes through the center of the coils and
`the origin of the z-axis is at the point midway between the
`coils.
`
`1
`
`where z is the distance from each coil to the point in question.
`z1 = z2 for a point in the center of the coils. Finding the
`gradient of the magnetic field
`
`Single circular coil: The equations describing the field
`created by a single circular coil are somewhat more complex.
`For one circular coil centered on the z axis (Figure 6a) the
`magnetic field intensity can be shown to be:
`
`where K ( k ) and E ( k ) are a complete elliptic integral of the
`first kind and second kind, respectively.
`
`On the axis of a circular loop, the equation simplifies to
`
`G =a,
`
`la2
`2(z2 + a2 TI2
`
`Multiple Circular Coils: The magnetic field created by two
`circular current carrying coils is the vector sum of the field
`components created by each single coil. The magnitude of
`the field created by two coils numbered 1 and 2 at an
`arbitrary point P is
`
`d
`
`Z
`
`Y
`
`Figure 6 - (a) circular coil in xy-plane, (b) two circular coils with equal
`radius
`
`lj_n
`
`It is now clear that
`
`= 0 when D = a. For circular
`
`ZzH
`coils dH is always equal to the radius. For square coils dH is
`always equal to 0.5445 times the length of one side [4], 151,
`[6]. For rectangular coils dH varies with the dimensions of the
`rectangle as shown in Figure 7.
`The total volume contained within the box created by two
`square coils at Helmholtz spacing is 1.3866 times larger than
`the total volume contained within the cylinder created by two
`circular coils also at Helmholtz spacing. However, this result
`does not provide adequate information about the volume of
`stable magnetic field created by these configurations.
`
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`Exhibit 1021
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`

`

`1.751
`
`I
`
`I
`
`1
`
`6
`
`7
`
`
`
`I
`I
`5
`4
`3
`2
`Length of one side (meters)
`Figure 7- Helmholtz distance vs. side length for rectangles of area=7,8,
`and 9m2
`In order to hrther investigate general differences between
`coil shapes, the volume of a sphere centered on the origin of
`the system (Figure 8) in which the magnitude of the magnetic
`field vector varies by up to a certain percentage om the
`value at the center of the system is plotted for circular and
`square coil configurations. The radii of the coils are set to a
`value of one (the "radii" of the square coils are assumed to be
`one-half the length of one side of one coil) [7].
`Figure 9 and Figure 12 show that while the volume is
`maximized at or near Helmholtz spacing in all cases, spacing
`becomes less critical as the percentage of allowable deviation
`is increased. These results are confirmed by [SI. Figure 10
`shows that for coils with the same radius at the same distance
`of separation, square coils have a larger volume of stable
`magnetic field for all investigated percentages of deviation
`from the center field values. Figure 13 shows the trade off
`between a larger total system volume and a larger volume of
`stable magnetic field. Although square coils provide the same
`volume of slable magnetic field with a smaller distance
`between them than do circular coils, circular coils provide a
`larger volume of stable magnetic field as a percentage of total
`volume taken up by the system. However, square coils can in
`practice be more easily assembled into a modular system [2].
`This is a clear advantage when developing scalable systems
`for industrial use.
`The most important criteria in choosing a coil configuration
`will involve finding the distance of separation at which the
`coils will produce the largest possible volume of specified
`minimum field strength within the system volume while
`maintaining field values outside the system volume which do
`not exceed minimum safety standards. An approximation of
`this volume can be found easily for square coils.
`For example two coils 3m x 3m each carrying 24A, the
`field strength parallel to the x-axis and through the center of
`the coils for distances of separation from 0.7 meters to 3.7
`meters at steps of 0.085 meters is shown in Figure 11 for field
`strengths greater than 4 A/m. All of the distances of
`separation considered lead to a field strength of 4A/m or
`more along the entire path. Figure 14 shows the field strength
`parallel to the z-axis at a distance of D/2 for distances of
`separation from 0.7 to 3.7 meters.
`
`0.25
`0
`
`1
`
`Figure 8- Sphere centered at the origin of a system of square coils
`In this direction the field strength does not remain at 4 N m
`or higher along the entire path for all distances of separation.
`Because of the symmetry of the field created by square
`coils, the field strength in the y-direction will be the same as
`that in the z-direction.
`Figure 15 shows a plot of the approximate volume in which
`the field strength is 4 N m or higher inside the cube volume
`for varying values of D. For 3m x 3m coils carrying %A, the
`distance of separation at which the maximum volume within
`the cube created by the coil pair has a field strength of greater
`than 4 A/m is approximately 2.8m.
`The optimum distance of separation for a coil system can be
`found most easily by first calculating Helmholtz spacing and
`then varying the distance slightly to achieve the maximum
`volume of desired field strength. Although the example above
`showed a one-dimensional system, this method is also useful
`when designing multi-dimensional systems.
`V. POWER TRANSFER
`
`One-dimensional systems create a pulsating magnetic field
`vector and do not provide constant power to the secondary.
`Two-dimensional systems with coil sets positioned 90" apart
`spatially (Figure 1) create a rotating field vector of constant
`length and provide constant power to the secondary when the
`current applied to the first set of coils is phase shifted by 90"
`from the current applied to the second set. Simplifymg
`equation (4) shows that the magnitude of the resulting vector
`is constant.
`= iixkl, sin on
`H = Cirkly COS ut
`
`The rotating field is also beneficial in that it mitigates the
`problem of shielding.
`
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`

`Figure 9 - Volume of specified field strength as a function of spacing
`for a pair of square coils
`
`Figure 12 - Volume of specified field strength as a function of spacing
`for a pair of circular coils
`
`Figure 10 - Maximum obtainable sphere volume vs. percent allowable
`deviation for circular and square coils
`
`Figure 13 - Percent of total system volume utilized vs. percent
`allowable deviation for circular and square coils
`
`14 r------r------T---IC----p------r---,
`
`3'
`-2
`
`-1
`
`I
`0
`
`Figure
`
`I
`I
`2
`1
`X value (m)
`11 - Field strength along the axis of a 3m x 3m square coil pair
`
`I
` 3
`
`I
`4
`
`5
`
`H (Nr
`
`Figure 14 - Field strength perpendicular to the axis of a 3m x 3m square
`coil pair
`
`2 value [m)
`
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`

`

`24
`
`22
`
`20
`
`18
`m3
`16
`
`14
`
`12
`
`/ '
`
`8 -
`
`I
`
`_ _
`
`R
`0.5
`
`1
`
`3
`2.5
`1.5
`2
`Coli Separation Distance D (m)
`Figure 15 - Approximate volume of magnetic field strength 4A/m or
`greater for a single coil pair carrying 24A
`
`3.5
`
`4
`
`VlI. CONCLUSIONS
`
`This paper presents the theory behind the optimum design
`for a large air-gap transformer for a wireless power
`transmission. The optimal coil shape and the optimum
`distance of separation between coils were found subject to
`the parameters of system volume, field volume, and usability.
`The rotating magnetic field vector was described, as were
`problems occurring due to shielding of the field.
`The optimum distance between coils is determined by first
`finding the Helmholtz distance dH. The actual distance
`between the coils is then altered slightly in order to achieve
`the maximum volume of specified field strength.
`The volume of a sphere centered on the origin of the system
`in which the magnitude of the magnetic field vector varies by
`up to a certain percentage from the value at the center of the
`system was found for square and circular coils. The spacing
`required to achieve the sphere of the greatest volume
`becomes less critical as the allowable percentage of allowable
`deviation is increased.
`Although square coils provide the same volume of stable
`magnetic field with a smaller distance between them than do
`circular coils, circular coils are more space efficient in that
`they provide a larger volume of stable magnetic field as a
`percentage of total volume taken up by the system. However
`square coils have the additional advantage of being modular.
`A rotating field vector created by two coil sets that are 90"
`out of phase both spatially and electrically maintains a
`constant magnitude over time and can provide constant
`power to the secondary coils. The rotating vector has the
`additional advantage of mitigating the problems of shielding.
`An extension of this theory by the addition of a third set of
`coils will further enhance the ability of the system to
`withstand the effects of shielding.
`
`REFERENCES
`
`G. Scheible, J. Schutz, C. Apneseth, "Load Adaptive Medium
`Frequency Resonant Power Supply," 28th Annual Conference of the
`IEEE Industrial Electronics Society, IECON 2002, vol. 1, pp. 282-
`287.
`J. Schutz, G. Scheible, C. Willmes, "Novel Wireless Power Supply
`for Wireless Communication Devices
`System
`in
`Industrial
`Automation Systems," 28th Annual Conference of the IEEE Industrial
`Electronics Society, IECON 2002, vol. 2, pp. 1358-1363.
`C. Fernhdez, 0. Garcia, R. Prieto, J.A. Cobos, J. Uceda, "Overview
`of Different Alternatives for the Contact-less Transmission of
`Energy," 28th Annual Conference of the IEEE Industrial Electronics
`Sociery, IECON 2002, vol. 2, pp. 1318-1323.
`F. R Crownfield, Jr., "Optimum Spacing of Coil Pairs," Rev. Sci.
`Instr., Vol. 35, 1964, pp. 240-241.
`M. G. Rudd, J. R. Craig, "Optimum Spacing of Square and Circular
`Coil Pairs," Rev. Sci. Instr., Vol. 39, 1968, pp. 1372-1 374.
`A. H. Firester, "Design of Square Helmholtz Coil Systems," Rev. Sci.
`Instr., Vol. 37, 1966, pp. 1264-1265.
`W.M. Frix, G.G. Karady, B.A. Venetz, "Comparison of calibration
`systems for magnetic field measurement equipment," IEEE Trans. on
`Power Delivery, Volume: 9 Issue: 1 ,Jan. 1994, pp. 100 -108.
`R. K. Cacak, J. R. Craig, "Magnetic Field Uniformity around Near-
`Helmholtz Coil Configurations, " Rev. Sci. Instr., Vol. 40, N. 1 1 , Nov.
`1969, pp. 1468-1470.
`
`VI. SHIELDING
`
`Partial or complete shielding can occur which may
`effectively prevent the secondary coils from receiving
`adequate power for operation. Shielding can occur when a
`material of sufficiently high permeability or conductivity is
`placed within the operating volume.
`Highly permeable materials ( p >>1) can provide a low-
`reluctance path for the magnetic flux and can effectively
`"guide"
`the flux away from the secondary coils. The
`effectiveness of such a material as a shield is inversely
`proportional to both its permeability and its cross-sectional
`area. The initial permeability (the permeability of a material
`at an AC flux density of 1mT) of all but high-cost materials
`designed for magnetic shielding is low enough at the
`expected frequency of operation of the system (120 kHz) to
`make this effect negligible in most cases.
`Conductive materials can cause shielding of the magnetic
`field when the eddy currents induced by the field create an
`opposing field strong enough to attenuate the field created by
`the primary coils. This effect is seen when metallic objects
`such as robot arms are placed within the operating area.
`The effects of both of these types of shielding are obviously
`dependent on the position of the objects causing the shielding
`relative to the orientation of the field created by the primary
`coils and the position of the secondary coil.
`two-dimensional
`the
`theory of a
`An extension of
`transformer to a three-dimensional transformer can be used to
`mitigate the shielding problem. Obviously the principle
`cannot be easily extended to a three-dimensional system if
`constant power is required on the secondary side. This is due
`to the fact that a phase shift of 120 degrees between the three
`coils sets will not produce a field vector of constant
`magnitude. Various altematives for implementing this three-
`dimensional transformer will be discussed in a future
`publication.
`
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