Wireless Power Transfer to Medical Implants

John S. Ho and Ada S. Y. Poon johnho@stanford.edu, adapoon@stanford.edu

April 3, 2012 Computer Forum

I. Wireless Power Transfer!Wireless powering of medical implants enables removal of the bulky battery.

Researchers have traditionally considered large, inductively coupled coils in the near-field. In this work, we show that miniature receivers are possible with mid-field wireless powering.

cardiac defibrillator! cochlear implant! deep-brain

stimulator!

III. Multi-layer Model!

II. Power Transfer Efficiency!Evaluation of the efficiency η is complicated by the coupling between the transmitter and receiver.

Coupling is described by the two-port network 3

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Figure 2. Overall wireless power transfer system. This work focuses on the analysis and the optimization of the shaded region. (J1,M1) are the electric

and the magnetic current distributions on the external antenna structure while (J2,M2) are those on the implant antenna structure.

for a small receive dipole. Putting these together,

ηc =

���iωµAr α ·H1(rf ) + lr β ·E1(rf )���2

ω�z<−d1

Im �(r)��E1(r)

��2 dr. (10)

B. Expressions for the FieldsThe electromagnetic fields can be expressed in terms of

source through the Green’s functions:

H1(r) = iω�

�Ghm(r− r

�)M1(r�) dr� (11a)

E1(r) = −�

Gem(r− r�)M1(r

�) dr. (11b)

Taking the 2D Fourier transform with respect to (x, y) for a

given depth z yields

H1(kx, ky, z) = iω� Ghm(kx, ky, z)zM1z(kx, ky) (12a)

E1(kx, ky, z) = −Gem(kx, ky, z)zM1z(kx, ky). (12b)

In free-space, via the use of Weyl identity, the Green’s

functions are given by

Ghm,fs(kx, ky, z) =ie−ikzz

2kz

�I− kk

t

k2

�

Gem,fs(kx, ky, z) = −e−ikzz

2kzk× I

where kz =�

k2 − k2x − k2y , k =�kx ky −kz

�t, and

k is the wavenumber of free-space. In the multi-layered

medium, we need to include the reflection and the transmission

coefficients. From [?, Chapter 2], when z is in between −dn+1

and −dn, the Green’s functions can be written as

Ghm,n(kx, ky, z) =

i

2k1z

�I− k1k

t1

k21

�·An

�e−iknzz + RTE

n,n+1eiknz(z+2dn)

�

(13a)

Gem,n(kx, ky, z) =

− 1

2k1zk1 × I ·An

�e−iknzz + RTE

n,n+1eiknz(z+2dn)

�

(13b)

where knz =�

k2n − k2x − k2y , kn =�kx ky −knz

�t, and

kn is the wavenumber of the nth layer. The term RTEn,n+1

is the generalized reflection coefficient while An can be

interpreted as the generalized transmission coefficient. Their

expressions can be found in [?, Chapter 2]. Once M1z(kx, ky)is known, performing the inverse Fourier transform yields the

electromagnetic fields at any point in space.

IV. OPTIMAL TRANSMIT CURRENT DISTRIBUTION

We will now express the coupling efficiency in (??) in terms

of M1z(kx, ky). For a Fourier transform pair g(t) and G(ω),g(0) = 1

2π

�G(ω) dω. Therefore,

H1(0, 0,−zf ) =1

4π2

��H1(kx, ky,−zf ) dkxdky (14)

and hence,

H1(0, 0,−zf )

=iω�14π2

��Ghm,j(kx, ky,−zf )zM1z(kx, ky) dkxdky. (15)

Similarly,

E1(0, 0,−zf )

=− 1

4π2

��Gem,j(kx, ky,−zf )zM1z(kx, ky) dkxdky.

(16)

By Parseval’s theorem,�|g(t)|2 dt = 1

2π

�|G(ω)|2 dω.

Therefore,�

z<−d1

Im �(z)��E1(r)

��2 dr

=1

4π2

���

z<−d1

Im �(z)��E1(kx, ky, z)

��2 dkxdkydz (17a)

=1

4π2

�� ��

z<−d1

Im �(z)��Gem(kx, ky, z)z

��2 dz�

(17b)

·��M1z(kx, ky)

��2 dkxdky.

Defining

h(kx, ky) =1

4π2

�k21Arα

tGhm,j(kx, ky,−zf )z

+ lrβtGem,j(kx, ky,−zf )z

�

f(kx, ky) =

�ω

4π2

�

z<−d1

Im �(r)��Gem(kx, ky, z)z

��2 dz,

When the transmitter and receiver are loosely-coupled, the efficiency is given by

The left term is dependent only on the transmitter. For a point receiver, it can be expressed in terms of the electromagnetic fields as

3

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;23<=3%#)1$#23%4

>**)

?@=('&#'A($)*$+,-$)?B;%'&C@,CC,3&)(,&4

Figure 2. Overall wireless power transfer system. This work focuses on the analysis and the optimization of the shaded region. (J1,M1) are the electric

and the magnetic current distributions on the external antenna structure while (J2,M2) are those on the implant antenna structure.

for a small receive dipole. Putting these together,

ηc =

���iωµAr α ·H1(rf ) + lr β ·E1(rf )���2

ω�z<−d1

Im �(r)��E1(r)

��2 dr. (10)

B. Expressions for the FieldsThe electromagnetic fields can be expressed in terms of

source through the Green’s functions:

H1(r) = iω�

�Ghm(r− r

�)M1(r�) dr� (11a)

E1(r) = −�

Gem(r− r�)M1(r

�) dr. (11b)

Taking the 2D Fourier transform with respect to (x, y) for a

given depth z yields

H1(kx, ky, z) = iω� Ghm(kx, ky, z)zM1z(kx, ky) (12a)

E1(kx, ky, z) = −Gem(kx, ky, z)zM1z(kx, ky). (12b)

In free-space, via the use of Weyl identity, the Green’s

functions are given by

Ghm,fs(kx, ky, z) =ie−ikzz

2kz

�I− kk

t

k2

�

Gem,fs(kx, ky, z) = −e−ikzz

2kzk× I

where kz =�

k2 − k2x − k2y , k =�kx ky −kz

�t, and

k is the wavenumber of free-space. In the multi-layered

medium, we need to include the reflection and the transmission

coefficients. From [?, Chapter 2], when z is in between −dn+1

and −dn, the Green’s functions can be written as

Ghm,n(kx, ky, z) =

i

2k1z

�I− k1k

t1

k21

�·An

�e−iknzz + RTE

n,n+1eiknz(z+2dn)

�

(13a)

Gem,n(kx, ky, z) =

− 1

2k1zk1 × I ·An

�e−iknzz + RTE

n,n+1eiknz(z+2dn)

�

(13b)

where knz =�

k2n − k2x − k2y , kn =�kx ky −knz

�t, and

kn is the wavenumber of the nth layer. The term RTEn,n+1

is the generalized reflection coefficient while An can be

interpreted as the generalized transmission coefficient. Their

expressions can be found in [?, Chapter 2]. Once M1z(kx, ky)is known, performing the inverse Fourier transform yields the

electromagnetic fields at any point in space.

IV. OPTIMAL TRANSMIT CURRENT DISTRIBUTION

We will now express the coupling efficiency in (??) in terms

of M1z(kx, ky). For a Fourier transform pair g(t) and G(ω),g(0) = 1

2π

�G(ω) dω. Therefore,

H1(0, 0,−zf ) =1

4π2

��H1(kx, ky,−zf ) dkxdky (14)

and hence,

H1(0, 0,−zf )

=iω�14π2

��Ghm,j(kx, ky,−zf )zM1z(kx, ky) dkxdky. (15)

Similarly,

E1(0, 0,−zf )

=− 1

4π2

��Gem,j(kx, ky,−zf )zM1z(kx, ky) dkxdky.

(16)

By Parseval’s theorem,�|g(t)|2 dt = 1

2π

�|G(ω)|2 dω.

Therefore,�

z<−d1

Im �(z)��E1(r)

��2 dr

=1

4π2

���

z<−d1

Im �(z)��E1(kx, ky, z)

��2 dkxdkydz (17a)

=1

4π2

�� ��

z<−d1

Im �(z)��Gem(kx, ky, z)z

��2 dz�

(17b)

·��M1z(kx, ky)

��2 dkxdky.

Defining

h(kx, ky) =1

4π2

�k21Arα

tGhm,j(kx, ky,−zf )z

+ lrβtGem,j(kx, ky,−zf )z

�

f(kx, ky) =

�ω

4π2

�

z<−d1

Im �(r)��Gem(kx, ky, z)z

��2 dz,

III. Phantom Simulations!5

x

z y

1

2

...

j

...

rf

0

−d1

−d2

−dj−1

−dj

Fig. 5. The layered medium model for tissue consists of n stacked layerswhere each layer is assigned a dielectric permittivity �rj . The center of thetransmitter is positioned at the origin and the receiver is placed at depth zfin the layers.

the transfer efficiency, it is sufficient to optimize the currentdensity along a plane. For simplicity, the planar current sourceis modeled as an infinite array of small current loops, whichis represented as sheet of magnetic current density

M1(r) = M1z(x, y)δ(z)z (13)

where the transmitter and receiver are separated along the zcoordinate. Since the receiver is small at higher frequencies,the point source model described in Section III-B remainsvalid.

In order to more accurately model tissue, the multilayermodel shown in Figure 5 is used. The model consists of nstacked layers where the jth layer is assigned a dielectricpermittivity �rj . The case n = 2 is known as the half-space

model. A convenient way to describe a wave in each mediumis through its wavenumber kj = ω

√µ0�0�rj which gives the

spatial frequency of the wave.

C. Optimal Transmitter

Using the impedance definitions in Section III-C, the opti-mal transmitter is the current distribution M1 that maximizesthe coupling efficiency ηc in (11). To find the optimal transmitcurrent distribution, the fields are first expressed in terms ofM1. For a point source, the Green’s functions give the fieldresponse at any point in space. The fields due to a generaltransmit current distribution M1 are then given by the integralequations

H1(r) = iω�(r)

�Gm(r, r�)M1(r

�) dr� (14)

E1(r) = −�

Ge(r, r�)M1(r

�) dr�. (15)

The Green’s functions for homogeneous media can beextended to the multilayer model through a plane wave ex-

pansion. Through the Weyl identity, the Green’s functions canbe written the sum of plane waves whose propagation throughtissue interfaces can be described in terms of reflection andtransmission coefficients.

106

107

108

109

10!6

10!5

10!4

10!3

10!2

Frequency [Hz]

!c

OptimalUniform (2 cm)Point

Fig. 6. Coupling efficiency versus frequency for a point transmitter, 2 cm-wide uniform transmitter, and optimal transmitter in air-muscle half-space.The receive coil is at a depth of 5 cm facing the transmitter.

In [39], the optimal transmitter was analytically found bydecomposing the fields into plane waves through a 2D Fouriertransform in the x and y coordinates. On writing the couplingefficiency in terms of the decomposition, an upper boundwas established through the Cauchy-Schwarz inequality. Anexpression was then found for the Fourier transform of thecurrent distribution that achieves the bound. For the half-spacemodel, this was found to be

M1z(kx, ky) =Ar(k1z + k2z) Im k2zeik

∗2z(−zf+d1)

2π2ωeik1zd1(16)

where kx and ky are the Fourier transform dual variables andkjz satisfies

k2j = k2x + k2y + k2jz. (17)

The optimal current distribution can then be recovered by aninverse Fourier transform.

D. Results

We consider a small receive coil of area 4 mm2 placedat depth −5 cm in air-muscle half-space with d1 = 1 cm.For simplicity, the receive coil is assumed to be facingthe transmitter, although arbitrary orientations can be easilyaccounted for. Figure 6 shows the coupling efficiency of theoptimal transmitter, point source, and uniform source of width2 cm. Close to the optimal frequency, the optimal sourcedistribution provides about an order of magnitude higherefficiency compared to the point and uniform source. At lowfrequencies, the optimal transmitter begins to resemble a pointsource and the efficiency of the sources are comparable [39].

When the coil is facing the transmitter, the optimal currentdistribution is circularly symmetric. Figure 7 shows the radialmagnitude and phase of the optimal transmitter at 2 GHz. Animportant property of the distribution is that the magnitudedecays rapidly with radial distance ρ. This allows a finite-size

Using the expression for coupling efficiency (2), we obtain the optimal transmitter for an analytically tractable tissue model.

2

placed inside the patient’s body, as shown in Figure 1. Thebasis of energy transfer is Faraday’s law

V2 = iωµ0

�H1 · ds (1)

where ω is the frequency of the fields. Wireless poweringoccurs when the transmit coil produces a magnetic field H1

that induces emf V2 at the receive coil. As the magnetic fieldat the receiver varies with time, voltage proportional to therate of change is induced in the implant.

For a transmit current distribution, the magnetic field H1

can be found by solving Maxwell’s equations. Inductive cou-pling is based on the assumption that the fields are slowlychanging. For low-frequency fields, the displacement currentterm can be ignored; this is known as the quasi-static approxi-

mation. Ignoring tissue, the field H1 under this approximationis governed by the diffusion equation whose solution is givenby the Biot-Savart law. On substituting this into (1), the emfat the receive coil V2 will have the form

V2 = iωMI1 (2)

where the mutual inductance M is a real quantity dependentonly on the geometry of the coils. This can be even morecompactly written as V2 = Z21I1, where Z21 = iωM is themutual impedance between the coils.

Since same laws must apply from the receive to the transmitcoil as well, there will be an opposing emf V1 induced inthe transmit coil for a current I2 in the receive coil. Inaddition, there exists a self-inductance L that arises from acoil’s opposition to its own change in current. The total emfat the receiver is then

V2 = iω(MI1 − L2I2). (3)

It can be shown that M = k√L1L2 where the constant of

proportionality k < 1 is known as the coupling coefficient.Although the inductive coupling was described in air, tissuelosses are negligible at low frequencies and the results can bedirectly applied.

B. Two-port Network

A convenient way to summarize the various forms ofcoupling is the two-port network. In this model, the inducedemf in both coils are given by

�V1

V2

�=

�Z11 Z12

Z21 Z22

� �I1I2

�(4)

where Z12 = Z21, Z11 = −iωL1 and Z22 = −iωL2. Note thatthe impedances are all purely imaginary. This is a consequenceof the quasi-static approximation and indicates that the energyis stored in the fields around the coil rather than radiated. Inpractice, the impedances contain real parts R to account forconductivity losses in the coil and tissue. The current I2 at theimplant is also due to the induced emf at the receive coil V2

as given byI2 = −V2/ZL (5)

where ZL is the load impedance that depends on the circuitrythat the system is trying to power.

TABLE IFREQUENCY, RANGE, AND COIL SIZE OF SELECT IMPLANTS

Implant Carrier Range Transmitter ReceiverType (MHz) Coil Coil

Generic [8] 1-2 0.5 cm 1 cm 0.47 cmGeneric [9] 4 0.8 cmGeneric [10] 10 1.5 cm 3.5 cm 2.7 cmCortical visual [11] 5,10.33 1 cmRetinal [12] 1-10Retinal [13] 2.5 0.575 cm 1.34 cm 1.34 cmMuscular [14], [15] 2 9-20 cm 0.12 cmMuscular [16] 6.78Muscular [17] 2Bio-sensor [18] 27 9 cm 5 cmSensor [19] 0.133 < 1 mSensor [20] 13.56 10 cm 15 cm 6 cmSensor [21] 3.18 4 cm 4.8 cm 0.6 cm

Cp1

Cs1

L2

Cs2

RL

M

L1 Cp2

RC

Figure 2: Illustrates the tuning circuit configuration.

Table 1: Z1 and Z2 in different tuning circuit configurations.

Primary Secondary R1 X1 R2 X2

Untuned Untuned RC ωL1 RL ωL2

Untuned Series RC ωL1 RL ωL2 − 1ωC2

Untuned Shunt RC ωL1RL

1+ω2C22R

2L

ωL2 −ωC2R2

L1+ω2C2

2R2L

Series Series RC ωL1 − 1ωC1

RL ωL2 − 1ωC2

Series Shunt RC ωL1 − 1ωC1

RL1+ω2C2

2R2L

ωL2 −ωC2R2

L1+ω2C2

2R2L

Shunt Series RC

1+( 1ωL1

−ωC1)2R2C

( 1ωL1

−ωC1)R2C

1+( 1ωL1

−ωC1)2R2C

RL ωL2 − 1ωC2

Shunt Shunt RC

1+( 1ωL1

−ωC1)2R2C

( 1ωL1

−ωC1)R2C

1+( 1ωL1

−ωC1)2R2C

RL1+ω2C2

2R2L

ωL2 −ωC2R2

L1+ω2C2

2R2L

2

Fig. 2. Lumped circuit model for inductive coupling. The coils can be shunttuned by inserting capacitors Cp1 and Cp2 in parallel or series tuned withCs1 and Cs2.

For implantable devices, the current in the receive coil isusually very small compared to the transmit coil. As such, thecoils are loosely coupled and the induced emf at the transmitcan be approximated as due only to self-inductance. The power

transfer efficiency is then given by

η =Pr

Pt=

|Z21|2

R11

RL

|Z22 + ZL|2. (6)

In this expression, the load-dependent term on the right is thematching efficiency that depends on the relative impedance ofthe load and the antenna at the receiver. The term on the leftis the coupling efficiency and is dependent only on the coils.

The transmit power is usually limited to avoid interferencewith other devices, reduce the size of the transmitter, or theextend the device lifetime if an external battery is used [22].As such, the goal of most studies on wireless powering isto maximize the power transfer efficiency or minimize itssensitivity to variations in the link.

C. Link Optimization

Table I lists the frequency, range, and coil size of wirelesslypowered implantable devices described in literature. The listis not exhaustive, but is representative of the typical implantparameters. When the secondary coil is small or not in closeproximity to the primary coil, design techniques were proposedto maximize the efficiency such as the use of ferrite cores [32],impedance transformation with inductance tapping and voltagedoubler circuits [5], and efficient rectifying circuits [33].

In practice, the load impedance ZL can be changed byintroducing a matching network between the receive coil andthe load. A simple way to improve the matching efficiency is

The transmitter is abstracted by an infinite current sheet. This formalism avoids assumptions on the underlying transmitter structure.

6

0 10 200

0.2

0.4

0.6

0.8

1

! [cm]

Ma

gn

itud

e

!d=6.48 cm

0 10 20

!"

!"/2

0

"/2

"

! [cm]

Ph

ase

Fig. 7. Magnitude and phase of the circularly symmetric optimal transmitcurrent distribution at 2 GHz. The radius ρd = 6.48 cm contains 90% of theenergy in the current distribution.

x [cm]

z [c

m]

4 2 0 2 4

6

5

4

3

2

x [cm]4 2 0 2 4

6

5

4

3

2

z [c

m]

4 2 0 2 4

6

5

4

3

2

4 2 0 2 4

6

5

4

3

2

|emf|2 [V2]0 0.4 0.8

SAR [mW/cm3]0 0.4 0.8 1.2 1.6

Fig. 8. Received open voltage and SAR distribution in muscle at 2 GHzfor a uniform distribution of 2 cm width and the optimal transmitter for animplant at 5 cm depth. The transmit power has been normalized such the peakSAR is 1.6 mW/cm3.

transmitter to approach the optimal efficiency if the radius ρdcontains most of the energy in the optimal current distribution.

Figure 8 shows the received open voltage and SAR inmuscle for the uniform source and the optimal transmitter.In order to provide a convenient basis of comparison, thetransmit power is normalized such the peak SAR is equal tothe safety guideline. The fields due to the optimal transmitterexhibit focusing, the effect where the fields are redistributedsuch that they interfere constructively at the focal point anddestructively otherwise. This allows a greater amount of powerto be delivered to the focal point without violating the safetyguidelines.

y [cm]

z [c

m]

20 15 10 5 0 5 10 15

25

20

15

10

5

0

Fig. 9. Projection of an axial slab of the phantom. An implant is placed onthe surface of the heart at depth zf = −5 cm as indicated by the bright dot.The bright line at z = 0 cm shows the transmitter. The phantom is facing thepositive z direction and the contrast is from the dielectric permittivity of themodel.

V. PHANTOM STUDIES

A. Human Body Model

In this section, we consider wirelessly powering a cardiacimplant, as shown in Figure 9. The Zubal phantom [41] is usedto model the human body. The model designates anatomicalregions with an index number for each voxel, which aremapped to dielectric properties using the Debye relaxationmodel. The implant is placed 5 cm deep at near the surfaceof the heart.

Instead of an arbitrary current sheet, a 15×15 array of smallcoils is used to synthesize the transmitter. The coils are equallyspaced 8 mm apart such that the radius of the array extends7.9 cm at the corner, which approximates finite dimension ofthe optimal distribution as found in Section IV. The transmitteris placed 1 cm away from the chest surface facing the receivecoil. The magnitude and phase of each transmit coil can beadjusted to redistribute the fields inside the phantom. In orderto find the optimal weights, the fields due to each coil isseparately found. Using the formulation in Section IV, theweights maximizing the coupling efficiency can be computed.The fields are solved by the finite-difference time-domain(FDTD) method implemented with custom MATLAB code.

B. Phantom Results

The received voltage distribution along the y = 0 sagittalslice of the phantom is shown in Figure 10 at 20 MHz,200 MHz, and 20 GHz. Note that the transmit power hasbeen normalized such that the peak SAR is equal to the safetyguideline. Focusing can be clearly seen at 2 GHz whereas thefield distributions are relatively smooth at low frequencies.Figure 11 shows the SAR distribution along the same sagittalslice in the body. Note that the boundaries are different sinceSAR is computed over 1 cm cubes. Since the fields canbe redistributed by focusing at higher frequencies, the SARdistribution at 2 GHz is different from that at 20 and 200 MHz.

In practice, it is difficult to obtain a model of the bodyfrom which the optimal weights can be computed. The tissuegeometry can be instead approximated by a simpler model.

The optimal current distribution was derived by a plane wave decomposition and demonstrates a focusing effect.

5

x

z y

1

2

...

j

...

rf

0

−d1

−d2

−dj−1

−dj

Fig. 5. The layered medium model for tissue consists of n stacked layerswhere each layer is assigned a dielectric permittivity �rj . The center of thetransmitter is positioned at the origin and the receiver is placed at depth zfin the layers.

the transfer efficiency, it is sufficient to optimize the currentdensity along a plane. For simplicity, the planar current sourceis modeled as an infinite array of small current loops, whichis represented as sheet of magnetic current density

M1(r) = M1z(x, y)δ(z)z (13)

where the transmitter and receiver are separated along the zcoordinate. Since the receiver is small at higher frequencies,the point source model described in Section III-B remainsvalid.

In order to more accurately model tissue, the multilayermodel shown in Figure 5 is used. The model consists of nstacked layers where the jth layer is assigned a dielectricpermittivity �rj . The case n = 2 is known as the half-space

model. A convenient way to describe a wave in each mediumis through its wavenumber kj = ω

√µ0�0�rj which gives the

spatial frequency of the wave.

C. Optimal Transmitter

Using the impedance definitions in Section III-C, the opti-mal transmitter is the current distribution M1 that maximizesthe coupling efficiency ηc in (11). To find the optimal transmitcurrent distribution, the fields are first expressed in terms ofM1. For a point source, the Green’s functions give the fieldresponse at any point in space. The fields due to a generaltransmit current distribution M1 are then given by the integralequations

H1(r) = iω�(r)

�Gm(r, r�)M1(r

�) dr� (14)

E1(r) = −�

Ge(r, r�)M1(r

�) dr�. (15)

The Green’s functions for homogeneous media can beextended to the multilayer model through a plane wave ex-

pansion. Through the Weyl identity, the Green’s functions canbe written the sum of plane waves whose propagation throughtissue interfaces can be described in terms of reflection andtransmission coefficients.

106

107

108

109

10!6

10!5

10!4

10!3

10!2

Frequency [Hz]!

c

OptimalUniform (2 cm)Point

Fig. 6. Coupling efficiency versus frequency for a point transmitter, 2 cm-wide uniform transmitter, and optimal transmitter in air-muscle half-space.The receive coil is at a depth of 5 cm facing the transmitter.

In [39], the optimal transmitter was analytically found bydecomposing the fields into plane waves through a 2D Fouriertransform in the x and y coordinates. On writing the couplingefficiency in terms of the decomposition, an upper boundwas established through the Cauchy-Schwarz inequality. Anexpression was then found for the Fourier transform of thecurrent distribution that achieves the bound. For the half-spacemodel, this was found to be

M1z(kx, ky) =Ar(k1z + k2z) Im k2zeik

∗2z(−zf+d1)

2π2ωeik1zd1(16)

where kx and ky are the Fourier transform dual variables andkjz satisfies

k2j = k2x + k2y + k2jz. (17)

The optimal current distribution can then be recovered by aninverse Fourier transform.

D. Results

We consider a small receive coil of area 4 mm2 placedat depth −5 cm in air-muscle half-space with d1 = 1 cm.For simplicity, the receive coil is assumed to be facingthe transmitter, although arbitrary orientations can be easilyaccounted for. Figure 6 shows the coupling efficiency of theoptimal transmitter, point source, and uniform source of width2 cm. Close to the optimal frequency, the optimal sourcedistribution provides about an order of magnitude higherefficiency compared to the point and uniform source. At lowfrequencies, the optimal transmitter begins to resemble a pointsource and the efficiency of the sources are comparable [39].

When the coil is facing the transmitter, the optimal currentdistribution is circularly symmetric. Figure 7 shows the radialmagnitude and phase of the optimal transmitter at 2 GHz. Animportant property of the distribution is that the magnitudedecays rapidly with radial distance ρ. This allows a finite-size

For a receive coil facing the transmitter, the optimal current distribution is radially symmetric.

Fig 3. The field distribution and coupling efficiency for the optimal transmitter compared to an uniform source. The implant is located at z=-5 cm in muscle.

Fig 1. Two-port network coupling model.

Fig 2. Multilayer tissue model for transmitter analysis.

6

0 10 200

0.2

0.4

0.6

0.8

1

! [cm]

Magnitu

de

!d=6.48 cm

0 10 20

!"

!"/2

0

"/2

"

! [cm]

Phase

Fig. 7. Magnitude and phase of the circularly symmetric optimal transmitcurrent distribution at 2 GHz. The radius ρd = 6.48 cm contains 90% of theenergy in the current distribution.

x [cm]

z [c

m]

4 2 0 2 4

6

5

4

3

2

x [cm]4 2 0 2 4

6

5

4

3

2

z [c

m]

4 2 0 2 4

6

5

4

3

2

4 2 0 2 4

6

5

4

3

2

|emf|2 [V2]0 0.4 0.8

SAR [mW/cm3]0 0.4 0.8 1.2 1.6

Fig. 8. Received open voltage and SAR distribution in muscle at 2 GHzfor a uniform distribution of 2 cm width and the optimal transmitter for animplant at 5 cm depth. The transmit power has been normalized such the peakSAR is 1.6 mW/cm3.

transmitter to approach the optimal efficiency if the radius ρdcontains most of the energy in the optimal current distribution.

Figure 8 shows the received open voltage and SAR inmuscle for the uniform source and the optimal transmitter.In order to provide a convenient basis of comparison, thetransmit power is normalized such the peak SAR is equal tothe safety guideline. The fields due to the optimal transmitterexhibit focusing, the effect where the fields are redistributedsuch that they interfere constructively at the focal point anddestructively otherwise. This allows a greater amount of powerto be delivered to the focal point without violating the safetyguidelines.

y [cm]

z [c

m]

20 15 10 5 0 5 10 15

25

20

15

10

5

0

Fig. 9. Projection of an axial slab of the phantom. An implant is placed onthe surface of the heart at depth zf = −5 cm as indicated by the bright dot.The bright line at z = 0 cm shows the transmitter. The phantom is facing thepositive z direction and the contrast is from the dielectric permittivity of themodel.

V. PHANTOM STUDIES

A. Human Body Model

In this section, we consider wirelessly powering a cardiacimplant, as shown in Figure 9. The Zubal phantom [41] is usedto model the human body. The model designates anatomicalregions with an index number for each voxel, which aremapped to dielectric properties using the Debye relaxationmodel. The implant is placed 5 cm deep at near the surfaceof the heart.

Instead of an arbitrary current sheet, a 15×15 array of smallcoils is used to synthesize the transmitter. The coils are equallyspaced 8 mm apart such that the radius of the array extends7.9 cm at the corner, which approximates finite dimension ofthe optimal distribution as found in Section IV. The transmitteris placed 1 cm away from the chest surface facing the receivecoil. The magnitude and phase of each transmit coil can beadjusted to redistribute the fields inside the phantom. In orderto find the optimal weights, the fields due to each coil isseparately found. Using the formulation in Section IV, theweights maximizing the coupling efficiency can be computed.The fields are solved by the finite-difference time-domain(FDTD) method implemented with custom MATLAB code.

B. Phantom Results

The received voltage distribution along the y = 0 sagittalslice of the phantom is shown in Figure 10 at 20 MHz,200 MHz, and 20 GHz. Note that the transmit power hasbeen normalized such that the peak SAR is equal to the safetyguideline. Focusing can be clearly seen at 2 GHz whereas thefield distributions are relatively smooth at low frequencies.Figure 11 shows the SAR distribution along the same sagittalslice in the body. Note that the boundaries are different sinceSAR is computed over 1 cm cubes. Since the fields canbe redistributed by focusing at higher frequencies, the SARdistribution at 2 GHz is different from that at 20 and 200 MHz.

In practice, it is difficult to obtain a model of the bodyfrom which the optimal weights can be computed. The tissuegeometry can be instead approximated by a simpler model.

6

0 10 200

0.2

0.4

0.6

0.8

1

! [cm]

Magnitu

de

!d=6.48 cm

0 10 20

!"

!"/2

0

"/2

"

! [cm]

Phase

Fig. 7. Magnitude and phase of the circularly symmetric optimal transmitcurrent distribution at 2 GHz. The radius ρd = 6.48 cm contains 90% of theenergy in the current distribution.

x [cm]

z [c

m]

4 2 0 2 4

6

5

4

3

2

x [cm]4 2 0 2 4

6

5

4

3

2

z [c

m]

4 2 0 2 4

6

5

4

3

2

4 2 0 2 4

6

5

4

3

2

|emf|2 [V2]0 0.4 0.8

SAR [mW/cm3]0 0.4 0.8 1.2 1.6

Fig. 8. Received open voltage and SAR distribution in muscle at 2 GHzfor a uniform distribution of 2 cm width and the optimal transmitter for animplant at 5 cm depth. The transmit power has been normalized such the peakSAR is 1.6 mW/cm3.

transmitter to approach the optimal efficiency if the radius ρdcontains most of the energy in the optimal current distribution.

Figure 8 shows the received open voltage and SAR inmuscle for the uniform source and the optimal transmitter.In order to provide a convenient basis of comparison, thetransmit power is normalized such the peak SAR is equal tothe safety guideline. The fields due to the optimal transmitterexhibit focusing, the effect where the fields are redistributedsuch that they interfere constructively at the focal point anddestructively otherwise. This allows a greater amount of powerto be delivered to the focal point without violating the safetyguidelines.

y [cm]

z [c

m]

20 15 10 5 0 5 10 15

25

20

15

10

5

0

Fig. 9. Projection of an axial slab of the phantom. An implant is placed onthe surface of the heart at depth zf = −5 cm as indicated by the bright dot.The bright line at z = 0 cm shows the transmitter. The phantom is facing thepositive z direction and the contrast is from the dielectric permittivity of themodel.

V. PHANTOM STUDIES

A. Human Body Model

In this section, we consider wirelessly powering a cardiacimplant, as shown in Figure 9. The Zubal phantom [41] is usedto model the human body. The model designates anatomicalregions with an index number for each voxel, which aremapped to dielectric properties using the Debye relaxationmodel. The implant is placed 5 cm deep at near the surfaceof the heart.

Instead of an arbitrary current sheet, a 15×15 array of smallcoils is used to synthesize the transmitter. The coils are equallyspaced 8 mm apart such that the radius of the array extends7.9 cm at the corner, which approximates finite dimension ofthe optimal distribution as found in Section IV. The transmitteris placed 1 cm away from the chest surface facing the receivecoil. The magnitude and phase of each transmit coil can beadjusted to redistribute the fields inside the phantom. In orderto find the optimal weights, the fields due to each coil isseparately found. Using the formulation in Section IV, theweights maximizing the coupling efficiency can be computed.The fields are solved by the finite-difference time-domain(FDTD) method implemented with custom MATLAB code.

B. Phantom Results

The received voltage distribution along the y = 0 sagittalslice of the phantom is shown in Figure 10 at 20 MHz,200 MHz, and 20 GHz. Note that the transmit power hasbeen normalized such that the peak SAR is equal to the safetyguideline. Focusing can be clearly seen at 2 GHz whereas thefield distributions are relatively smooth at low frequencies.Figure 11 shows the SAR distribution along the same sagittalslice in the body. Note that the boundaries are different sinceSAR is computed over 1 cm cubes. Since the fields canbe redistributed by focusing at higher frequencies, the SARdistribution at 2 GHz is different from that at 20 and 200 MHz.

In practice, it is difficult to obtain a model of the bodyfrom which the optimal weights can be computed. The tissuegeometry can be instead approximated by a simpler model.

The distribution also decays rapidly with radial distance – the optimal distribution can be approximated by finite dimensions.

We consider powering a miniature cardiac implant in the mid-field on computational phantom.

heart

Fig 4. Computational phantom for studying a cardiac implant. The implant is located 5 cm from a transmitter 0.8 cm from the chest. A 15x15 array was used; the dimensions closely approximate the optimal current sheet.

7

Fig. 10. Received open voltage along the y = 0 sagittal slice of the phantomfor an optimal 15 × 15 transmit array operating at 20 MHz, 200 MHz, and20 GHz. The implant is located at depth zf = −5 cm.

TABLE IVMULTILAYER MODEL FOR THE CARDIAC IMPLANT

Layer j Type Width (cm) dj (cm)

1 Air 1.2 1.22 Skin 0.4 1.63 Fat 0.8 2.44 Muscle 0.8 3.25 Bone 1.6 4.86 Heart ∞ –

The optimal 15×15 weights were computed for the multilayerin Table IV approximating the region between the phantomchest and heart. Figure 12 shows delivered voltage and SARat the x = 0 axial slice of the phantom for the optimal weightsand the weights from the planar approximation. The fielddistributions due to the planar weights are highly similar tothose due to the optimal weights and results in comparablepower transfer efficiency. This suggests that the multilayermodel is sufficient for transmitter design at a given depth andfrequency of operation.

VI. CONCLUSION

We review the inductive coupling analysis of the wire-less link and design techniques to optimize the link. Theanalysis was extended to higher frequencies using a simplehomogeneous model of tissue and the optimal frequency forwireless powering was found to be in the low GHz range.At higher frequencies where the wavelength is comparable tothe distance of separation, the transmitter can be optimized toimprove power transfer efficiency. Using a multilayer modelof tissue, the optimal current distribution was analytically

Fig. 11. SAR along the y = 0 sagittal slice of the phantom for an optimal15 × 15 transmit array operating at 20 MHz, 200 MHz, and 20 GHz. Thetransmit power has been normalized such the peak SAR is 1.6 mW/cm3.

derived. The analysis was used to determine the optimalfrequency and design a transmitter for wirelessly poweringa cardiac implant embedded at a depth of 5 cm throughsimulations on a human body phantom.

Prototypes of implantable devices operating at higher fre-quencies have recently been demonstrated in literature [37][38]. The physical limits derived in the analysis at higherfrequencies show that there is considerable room for im-provement in developing wirelessly powered implants forapplications with demanding size and efficiency constraints.

REFERENCES

[1] A. S. Y. Poon, S. O’Driscoll, and T. H. Meng, “Optimal operatingfrequency in wireless power transmission for implantable devices,” inProc. Annual Intl. Conf. of the IEEE Engineering in Medicine andBiology (EMBC), Lyon, France, Aug. 2007, pp. 5673–5678.

[2] ——, “Optimal frequency for wireless power transmission over dis-persive tissue,” IEEE Trans. Antennas Propagat., vol. 58, no. 5, p.17391749, May 2010.

[3] J. C. Schuder, H. E. Stephenson, Jr., and J. F. Townsend, “High-levelelectromagnetic energy transfer through a closed chest wall,” IRE Intl.Conv. Rec, vol. 9, pp. 119–126, 1961.

[4] F. C. Flack, E. D. James, and D. M. Schlapp, “Mutual inductance ofair-cored coils: effect on design of radio-frequency coupled implants,”Med. Biol. Eng., vol. 9, no. 2, pp. 79–85, Mar. 1971.

[5] W. H. Ko, S. P. Liang, and C. D. Fung, “Design of radio-frequencypowered coils for implant instruments,” Med. Biol. Eng. Comp., vol. 15,pp. 634–640, 1977.

[6] D. C. Galbraith, “An implantable multichannel neural stimulator,” Ph.D.dissertation, Stanford University, Dec. 1984.

[7] W. J. Heetderks, “RF powering of millimeter- and submillimeter-sizedneural prosthetic implants,” IEEE Trans. Biomed. Eng., vol. 35, no. 5,pp. 323–327, May 1988.

[8] T. Akin, K. Najafi, and R. M. Bradley, “A wireless implantable mul-tichannel digital neural recording system for a micromachined sieveelectrode,” IEEE J. Solid-State Circuits, vol. 33, no. 1, pp. 109–118,Jan. 1998.

7

Fig. 10. Received open voltage along the y = 0 sagittal slice of the phantomfor an optimal 15 × 15 transmit array operating at 20 MHz, 200 MHz, and20 GHz. The implant is located at depth zf = −5 cm.

TABLE IVMULTILAYER MODEL FOR THE CARDIAC IMPLANT

Layer j Type Width (cm) dj (cm)

1 Air 1.2 1.22 Skin 0.4 1.63 Fat 0.8 2.44 Muscle 0.8 3.25 Bone 1.6 4.86 Heart ∞ –

The optimal 15×15 weights were computed for the multilayerin Table IV approximating the region between the phantomchest and heart. Figure 12 shows delivered voltage and SARat the x = 0 axial slice of the phantom for the optimal weightsand the weights from the planar approximation. The fielddistributions due to the planar weights are highly similar tothose due to the optimal weights and results in comparablepower transfer efficiency. This suggests that the multilayermodel is sufficient for transmitter design at a given depth andfrequency of operation.

VI. CONCLUSION

We review the inductive coupling analysis of the wire-less link and design techniques to optimize the link. Theanalysis was extended to higher frequencies using a simplehomogeneous model of tissue and the optimal frequency forwireless powering was found to be in the low GHz range.At higher frequencies where the wavelength is comparable tothe distance of separation, the transmitter can be optimized toimprove power transfer efficiency. Using a multilayer modelof tissue, the optimal current distribution was analytically

Fig. 11. SAR along the y = 0 sagittal slice of the phantom for an optimal15 × 15 transmit array operating at 20 MHz, 200 MHz, and 20 GHz. Thetransmit power has been normalized such the peak SAR is 1.6 mW/cm3.

derived. The analysis was used to determine the optimalfrequency and design a transmitter for wirelessly poweringa cardiac implant embedded at a depth of 5 cm throughsimulations on a human body phantom.

Prototypes of implantable devices operating at higher fre-quencies have recently been demonstrated in literature [37][38]. The physical limits derived in the analysis at higherfrequencies show that there is considerable room for im-provement in developing wirelessly powered implants forapplications with demanding size and efficiency constraints.

REFERENCES

[1] A. S. Y. Poon, S. O’Driscoll, and T. H. Meng, “Optimal operatingfrequency in wireless power transmission for implantable devices,” inProc. Annual Intl. Conf. of the IEEE Engineering in Medicine andBiology (EMBC), Lyon, France, Aug. 2007, pp. 5673–5678.

[2] ——, “Optimal frequency for wireless power transmission over dis-persive tissue,” IEEE Trans. Antennas Propagat., vol. 58, no. 5, p.17391749, May 2010.

[3] J. C. Schuder, H. E. Stephenson, Jr., and J. F. Townsend, “High-levelelectromagnetic energy transfer through a closed chest wall,” IRE Intl.Conv. Rec, vol. 9, pp. 119–126, 1961.

[4] F. C. Flack, E. D. James, and D. M. Schlapp, “Mutual inductance ofair-cored coils: effect on design of radio-frequency coupled implants,”Med. Biol. Eng., vol. 9, no. 2, pp. 79–85, Mar. 1971.

[5] W. H. Ko, S. P. Liang, and C. D. Fung, “Design of radio-frequencypowered coils for implant instruments,” Med. Biol. Eng. Comp., vol. 15,pp. 634–640, 1977.

[6] D. C. Galbraith, “An implantable multichannel neural stimulator,” Ph.D.dissertation, Stanford University, Dec. 1984.

[7] W. J. Heetderks, “RF powering of millimeter- and submillimeter-sizedneural prosthetic implants,” IEEE Trans. Biomed. Eng., vol. 35, no. 5,pp. 323–327, May 1988.

[8] T. Akin, K. Najafi, and R. M. Bradley, “A wireless implantable mul-tichannel digital neural recording system for a micromachined sieveelectrode,” IEEE J. Solid-State Circuits, vol. 33, no. 1, pp. 109–118,Jan. 1998.

(2)

(1)

Fig 3. Radial section of the optimal current distribution for an implant 5 cm in muscle.

implant

Under the same tissue heating constraints, the optimal frequency for wireless powering is in the low-GHz range, which corresponds to the mid-field.

Fig 5. Open-circuit voltage and tissue heating at 20 MHz, 200 MHz, and 2 GHz for an optimized 15x15 array of magnetic dipoles.

Mid-field wireless powering provides efficient energy transfer for miniature medical implants. Current work on designing the antenna equivalent to the optimal current distribution.

η :=Pr

Pt≈

��Z21

��2

R11

RL��Z22 + ZL

��2.

In this expression, the load-dependent term on the right is the

IV. Conclusion!