**2.1. Model of an infinite homogeneous lossy dielectric media**

The description of this model is based on the solution of the Helmholtz equation for a spherical wave which propagates in a homogeneous dissipative dielectric media. Due to this approach the absolute value of the electric field decreases proportionally to the reciprocal distance and shows an additional exponential attenuation with progressing distance compared to a lossless media. Therefore, the absolute value of the electric field *E* can be approximated by

$$E(d) \propto \frac{\mathcal{U}\_{\rm n}}{d} \mathbf{e}^{-\omega d},\tag{1}$$

where *d*ff denotes the approximated far field distance, *λ*<sup>i</sup> the wave length of a TEM wave in the dielectric media at the band edge frequency and *D*max the largest diameter of the antenna. Note that *D*max may be increased by passively excited components in the near field of the antenna which may contribute to the radiation, like the PCB of an implant or its encapsulation. Based on this far field model it is possible to deduce the equivalent gain of an in-body antenna. Doing this it seems logically consistent to refer the normalization constant *U*n to an isotropic

*<sup>G</sup>* <sup>=</sup> *<sup>S</sup>* (*<sup>d</sup>* <sup>≥</sup> *<sup>d</sup>*ff)

*<sup>S</sup>*iso,lossy <sup>=</sup> *<sup>P</sup>*rad

*<sup>η</sup>* (*d*ff) <sup>=</sup> <sup>1</sup>

*S*iso,lossy (*d* ≥ *d*ff)

<sup>4</sup>*πd*<sup>2</sup> *<sup>e</sup>*

 � *A*

In equation 4, *P*rad denotes the radiated power of the antenna. For an antenna whose losses are restricted to the surrounding medium *P*rad is equal to the power on the antenna *P*ant. Due to the dissipative nature of the tissue the antenna efficiency *η* decreases exponentially with

*P*ant

where *A* is the enclosure of the antenna at the distance *d*ff. In order to calculate the path loss between two in-body antennas, the receive properties of such antennas have to be characterized as well. Due to the fact that the equivalent tissue medium is source free, linear and isotropic the definition of the effective antenna area *A*eff [2] is also applicable for the in-body scenario. With respect to the definition of the gain in equation 3, the effective antenna

*A*eff = *G*

Note, that the theory given above is based on an intrinsic far field model. Aim of this theory is an approach to give an intuitive formulation for the antenna design and handy path loss estimations. It raises no claim to give a closed analytical solution of the given problem. Despite this fact the model enables even estimations of theoretic problematic scenarios, such as a totally immersed antenna which is not insulated from the surrounding tissue media. As shown in [4], a theoretical formulation of this problem would lead to an inexpressible formulation. Nevertheless, to find a description between source and far field it is suggested to use an antenna which is electrically insulated from the surrounding media. Moreover the applicability of the model depends on the specific in-body scenario. The following chapter

*λ*2 i

where *S* denotes the power density of the antenna and *S*iso,lossy denotes the power density of an isotropic source in a lossy medium at the same distance. Please note, due to the fact that the propagation medium itself contains losses and the directivity is by definition a lossless quantity, the definition of the directivity is not appropriate in this case. In order to achieve a constant normalization ratio versus the distance, the losses expressed by the exponential term of equation 1 have to be taken into account. Therefore, the absolute value of the lossy isotropic

, (3)

Antennas and Propagation for On-, Off - and In-Body Communications 155

<sup>−</sup>2*αd*. (4)

**S** · d**A**, (5)

<sup>4</sup>*<sup>π</sup>* . (6)

radiator. In this case the general definition of the gain *G* [2] is altered to

power density is given by

area yields

progressing far field distance and can be calculated by

addresses the quality of the proposed model.

where *d*, *α* and *U*n denote the distance to the antenna, the attenuation constant of the media and a normalization factor related to the equivalent source of the spherical wave. For radiating elements, other than point sources, the accuracy of this model depends on the spacial current distribution of the source and is therefore a function of the distance. Due to the analogy of this dependency to the free space scenario the standard far field criteria seems applicable with the usual phase restrictions [2]. Related to the design of antennas which operates within the UWB frequency range [3] the radiating component is electrically large at the upper band edge frequency. In this case the appropriate far field criterium leads to following formulation:

$$d\_{\rm ff} = \frac{2D\_{\rm max}^2}{\lambda\_{\rm i}},$$

where *d*ff denotes the approximated far field distance, *λ*<sup>i</sup> the wave length of a TEM wave in the dielectric media at the band edge frequency and *D*max the largest diameter of the antenna. Note that *D*max may be increased by passively excited components in the near field of the antenna which may contribute to the radiation, like the PCB of an implant or its encapsulation. Based on this far field model it is possible to deduce the equivalent gain of an in-body antenna. Doing this it seems logically consistent to refer the normalization constant *U*n to an isotropic radiator. In this case the general definition of the gain *G* [2] is altered to

2 Will-be-set-by-IN-TECH

**Figure 1.** Distribution of the electric field of an implant located within the human abdomen; Left: On-body scenario showing surface waves guided by the body curvature; Right: In-body scenario

From an electromagnetic point of view the human body consists of a large number of lossy dielectric materials of various combinations and spacial arrangements. In case of an in-body scenario the antenna is integrated into this complex dielectric structure and along an arbitrary propagation path various electromagnetic propagation effects occur. However, the most dominant effect is the attenuation of the propagating wave due to the lossy character of human tissues, see Figure 1. Furthermore, it has been shown that the average attenuation through the inhomogeneous tissue structure can be described for some scenarios by an equivalent homogeneous medium with appropriate properties [1]. In this case the analogy between the in-body and free space scenario enables an evident description of the related antenna far field. For this purpose the basic assumption of an isotropic point source is generalized for a lossy

The description of this model is based on the solution of the Helmholtz equation for a spherical wave which propagates in a homogeneous dissipative dielectric media. Due to this approach the absolute value of the electric field decreases proportionally to the reciprocal distance and shows an additional exponential attenuation with progressing distance compared to a lossless

media. Therefore, the absolute value of the electric field *E* can be approximated by

*<sup>E</sup>*(*d*) <sup>∝</sup> *<sup>U</sup>*<sup>n</sup>

where *d*, *α* and *U*n denote the distance to the antenna, the attenuation constant of the media and a normalization factor related to the equivalent source of the spherical wave. For radiating elements, other than point sources, the accuracy of this model depends on the spacial current distribution of the source and is therefore a function of the distance. Due to the analogy of this dependency to the free space scenario the standard far field criteria seems applicable with the usual phase restrictions [2]. Related to the design of antennas which operates within the UWB frequency range [3] the radiating component is electrically large at the upper band edge frequency. In this case the appropriate far field criterium leads to following formulation:

> *<sup>d</sup>*ff <sup>=</sup> <sup>2</sup>*D*max<sup>2</sup> *λ*i

*<sup>d</sup>* <sup>e</sup>−*αd*, (1)

, (2)

characterized by circular shaped attenuation within the body.

dielectric medium.

**2. Antenna de-embedding for in-body applications**

**2.1. Model of an infinite homogeneous lossy dielectric media**

$$G = \frac{S \left(d \ge d\_{\rm ff} \right)}{S\_{\rm iso, lossy} \left(d \ge d\_{\rm ff} \right)} \,\mathrm{}\,\mathrm{}$$

where *S* denotes the power density of the antenna and *S*iso,lossy denotes the power density of an isotropic source in a lossy medium at the same distance. Please note, due to the fact that the propagation medium itself contains losses and the directivity is by definition a lossless quantity, the definition of the directivity is not appropriate in this case. In order to achieve a constant normalization ratio versus the distance, the losses expressed by the exponential term of equation 1 have to be taken into account. Therefore, the absolute value of the lossy isotropic power density is given by

$$S\_{\rm iso,lossy} = \frac{P\_{\rm rad}}{4\pi d^2} e^{-2ad}.\tag{4}$$

In equation 4, *P*rad denotes the radiated power of the antenna. For an antenna whose losses are restricted to the surrounding medium *P*rad is equal to the power on the antenna *P*ant. Due to the dissipative nature of the tissue the antenna efficiency *η* decreases exponentially with progressing far field distance and can be calculated by

$$\eta \left( d\_{\rm ff} \right) = \frac{1}{P\_{\rm ant}} \iint\_{A} \mathbf{S} \cdot \mathbf{dA}\_{\prime} \tag{5}$$

where *A* is the enclosure of the antenna at the distance *d*ff. In order to calculate the path loss between two in-body antennas, the receive properties of such antennas have to be characterized as well. Due to the fact that the equivalent tissue medium is source free, linear and isotropic the definition of the effective antenna area *A*eff [2] is also applicable for the in-body scenario. With respect to the definition of the gain in equation 3, the effective antenna area yields

$$A\_{\rm eff} = G \frac{\lambda\_{\rm i}^2}{4\pi}.\tag{6}$$

Note, that the theory given above is based on an intrinsic far field model. Aim of this theory is an approach to give an intuitive formulation for the antenna design and handy path loss estimations. It raises no claim to give a closed analytical solution of the given problem. Despite this fact the model enables even estimations of theoretic problematic scenarios, such as a totally immersed antenna which is not insulated from the surrounding tissue media. As shown in [4], a theoretical formulation of this problem would lead to an inexpressible formulation. Nevertheless, to find a description between source and far field it is suggested to use an antenna which is electrically insulated from the surrounding media. Moreover the applicability of the model depends on the specific in-body scenario. The following chapter addresses the quality of the proposed model.

#### 4 Will-be-set-by-IN-TECH 156 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Antennas and Propagation for On-, Off- and In-Body Communications <sup>5</sup>

### *2.1.1. Validation of the in-body model on the example of an UWB teardrop antenna*

Using the example of [1] the validity of the proposed in-body far field approach has been discussed by the evaluation of UWB localization of deep brain implants. As result it has been shown that the attenuation of an electromagnetic pulse within the inner human brain structure can be modeled by a homogeneous tissue with dielectric properties equivalent to grey matter. Despite the complex structure of the focused brain region time domain analyses indicate a radial wave conservation within the inner brain structure. These investigations indicate an average propagation velocity in arbitrary directions of the head which emphasize to the applicability of the suggested model.

Table 1 shows the calculated values for the lower, upper and center frequency of the UWB frequency range investigated. The error rises with increasing frequency and is still within the

Δ*S*[%] 1.1 3.7 6.9 *G*(Θ = 90◦) 1.9 2.3 3.3 *<sup>η</sup>*eff(*d*ff) 3.8 · <sup>10</sup>−<sup>2</sup> 8.4 · <sup>10</sup>−<sup>5</sup> 1.5 · <sup>10</sup>−<sup>8</sup> *<sup>A</sup>*eff[*m*2] 2.75 · <sup>10</sup>−<sup>5</sup> 7.22 · <sup>10</sup>−<sup>6</sup> 4.74 · <sup>10</sup>−<sup>6</sup> **Table 1.** Calculated equivalent in-body antenna parameters of the UWB teardrop antenna at *d*ff = 3 cm. Table 1 also shows the gain calculated by the equations 3 and 4. As stated, additional derivations have shown that the value is nearly constant for distances greater than *d*ff. Nevertheless, the value is not constant within the observed frequency range and rises with increasing frequency due to the variation of the electrical length of the antenna. To characterize the losses within the near field of the antenna the efficiency has been calculated by equation 5 at the minimum far field distance *d*ff, see Table 1. As it might be expected, the efficiency decreases drastically with increased frequency due to the higher power consumption of the tissue medium. For greater distances the efficiency decreases exponentially with increasing distance. The effective antenna area, as shown in Table 1, is calculated using equation 6. The derived values of the gain and the effective antenna area enable the approximation of the path gain for arbitrary distances greater than the minimum far field distance. This ratio of transmitted to received power (path gain) is shown for the

As shown above the model of an homogeneous dissipative medium enables the definition of antenna parameters to establish an antenna de-embedding for in-body scenarios. Therefore, the consideration of the whole system is not necessary to achieve a path loss estimation.

Moreover this assumption enables a basis for a purposeful antenna development.

**Figure 3.** Calculated path gain of the UWB teardrop antenna.

*f* = 3.1 GHz *f* = 6.85 GHz *f* = 10.6 GHz

Antennas and Propagation for On-, Off - and In-Body Communications 157

typical range of the equivalent free space far field considerations.

related frequencies in Figure 3.

The following analysis summarizes the results of an in-body scenario within the human trunk. In this case the representative media parameters are set to homogeneous muscle tissue, calculated by [5], and the validity of the approach is shown on the example of an UWB teardrop antenna. The antenna has been designed for the ultra wideband frequency range from 3.1 GHz to 10.6 GHz which is specified by [3]. Within this frequency range the return loss of the immersed teardrop antenna has been optimized to be lower than 10 dB. The capsulation of the antenna has been designed for the center frequency of 6.85 GHz by a lossless dielectric cylindrical insulation. The related permittivity of the capsulation has been set to *ε*<sup>r</sup> = 49.9 to achieve an impedance matching between the insulation and the tissue. The validation of the model has been performed by numerical calculations with the FDTD simulation software EMPIRE XCcelTM [6].

**Figure 2.** Insulated teardrop antenna designed for the focused UWB frequency band.

According to equation 2, the minimum far field distance for the upper band edge frequency (*f*<sup>u</sup> = 10.6 GHz) is *d*ff = 3 cm. Here, due to the insulation, the maximum antenna size *D*max has been set to the maximal capsule dimension. The quality of the proposed far field model has been verified by the absolute difference Δ*S* between the calculated power density values *S*FDTD and the far field model *S*ff. For the angle Θ = 90◦ the difference Δ*S* has been calculated at the far field distance *d*ff by the following equation:

$$
\Delta \mathbf{S} = \frac{|\mathbf{S\_{FDTD}} - \mathbf{S\_{ff}}|}{\mathbf{S\_{FDTD}}}.\tag{7}
$$


4 Will-be-set-by-IN-TECH

Using the example of [1] the validity of the proposed in-body far field approach has been discussed by the evaluation of UWB localization of deep brain implants. As result it has been shown that the attenuation of an electromagnetic pulse within the inner human brain structure can be modeled by a homogeneous tissue with dielectric properties equivalent to grey matter. Despite the complex structure of the focused brain region time domain analyses indicate a radial wave conservation within the inner brain structure. These investigations indicate an average propagation velocity in arbitrary directions of the head which emphasize

The following analysis summarizes the results of an in-body scenario within the human trunk. In this case the representative media parameters are set to homogeneous muscle tissue, calculated by [5], and the validity of the approach is shown on the example of an UWB teardrop antenna. The antenna has been designed for the ultra wideband frequency range from 3.1 GHz to 10.6 GHz which is specified by [3]. Within this frequency range the return loss of the immersed teardrop antenna has been optimized to be lower than 10 dB. The capsulation of the antenna has been designed for the center frequency of 6.85 GHz by a lossless dielectric cylindrical insulation. The related permittivity of the capsulation has been set to *ε*<sup>r</sup> = 49.9 to achieve an impedance matching between the insulation and the tissue. The validation of the model has been performed by numerical calculations with the FDTD simulation software

*2.1.1. Validation of the in-body model on the example of an UWB teardrop antenna*

**Figure 2.** Insulated teardrop antenna designed for the focused UWB frequency band.

at the far field distance *d*ff by the following equation:

According to equation 2, the minimum far field distance for the upper band edge frequency (*f*<sup>u</sup> = 10.6 GHz) is *d*ff = 3 cm. Here, due to the insulation, the maximum antenna size *D*max has been set to the maximal capsule dimension. The quality of the proposed far field model has been verified by the absolute difference Δ*S* between the calculated power density values *S*FDTD and the far field model *S*ff. For the angle Θ = 90◦ the difference Δ*S* has been calculated

> <sup>Δ</sup>*<sup>S</sup>* <sup>=</sup> <sup>|</sup>*S*FDTD <sup>−</sup> *<sup>S</sup>*ff<sup>|</sup> *S*FDTD

. (7)

to the applicability of the suggested model.

EMPIRE XCcelTM [6].


**Table 1.** Calculated equivalent in-body antenna parameters of the UWB teardrop antenna at *d*ff = 3 cm.

Table 1 also shows the gain calculated by the equations 3 and 4. As stated, additional derivations have shown that the value is nearly constant for distances greater than *d*ff. Nevertheless, the value is not constant within the observed frequency range and rises with increasing frequency due to the variation of the electrical length of the antenna. To characterize the losses within the near field of the antenna the efficiency has been calculated by equation 5 at the minimum far field distance *d*ff, see Table 1. As it might be expected, the efficiency decreases drastically with increased frequency due to the higher power consumption of the tissue medium. For greater distances the efficiency decreases exponentially with increasing distance. The effective antenna area, as shown in Table 1, is calculated using equation 6. The derived values of the gain and the effective antenna area enable the approximation of the path gain for arbitrary distances greater than the minimum far field distance. This ratio of transmitted to received power (path gain) is shown for the related frequencies in Figure 3.

As shown above the model of an homogeneous dissipative medium enables the definition of antenna parameters to establish an antenna de-embedding for in-body scenarios. Therefore, the consideration of the whole system is not necessary to achieve a path loss estimation. Moreover this assumption enables a basis for a purposeful antenna development.

**Figure 3.** Calculated path gain of the UWB teardrop antenna.

#### 6 Will-be-set-by-IN-TECH 158 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Antennas and Propagation for On-, Off- and In-Body Communications <sup>7</sup>

### *2.1.2. Limitations of the in-body model*

As shown in [1], boundaries between high water content tissues (muscle tissue) and air filled regions (paranasal sinus, frontal sinus) or low water content tissues (fat tissue, bone tissue) lead to various electromagnetic interactions which reduce the accuracy of the approach presented above. The average attenuation of the electromagnetic field may still be characterized by the proposed model but especially with respect to time domain analysis the multi path behavior of these structures leads to insufficient results. Moreover, the specific anatomical location of the RF application may have a strong influence on the corresponding radiation characteristic of the antenna which cannot be adequately described by a homogeneous tissue model. Despite this fact the generality of the approach enables an extension of the theory by including anatomic realistic human models in the antenna design process to derive the resulting radiation characteristic. In this case the corresponding far field distance has to be enlarged related to the anatomical structure but the average description of the far field may still be given by the proposed homogeneous model.

electric field of any observation point *P* along a parallel surface path at the effective antenna

where the normalization value *U*n depends on the antenna excitation. The term *N* is a function of the current distribution of the source **i**v,h, the distance *d*, the frequency *f* and the complex tissue permittivity *ε*. The absolute value of the electric field is given by the superposition of

> 2*πd*ref *λ*

In equation 10, *λ* denotes the free space wave length, *R*v,h the plane-wave reflection coefficient of the ground, *F*v,h the surface-wave attenuation function and *dref* the reflected path at angle *ψ*. Both, the reflection and the attenuation functions depend on the current distribution of the antenna and are given by [9] for vertical and horizontal antenna orientations. The related far field solution is valid for sufficient great distances *d* and depends in first place on the mathematical description of the surface-wave attenuation function. If an adequate description can be assumed the minimal valid distance, referred by Norton in [8], has to be greater than one free space wave length. An additional limitation is the assumption of locally plane waves for the derivation of the reflection coefficient of the ground. Under the assumption that the surface acts as a perfect mirror the electrical antenna size may be enlarged by the mirror image.

*<sup>d</sup>*ff <sup>=</sup> <sup>2</sup>*D*<sup>2</sup>

where the modified maximum antenna dimension is denoted by *D*max,eff. For an adequate derivation of the far field distance the quantity *D*max,eff has to be chosen appropriate under the aspect that the enlargement of the antenna by the ground acts primarily in normal direction to the surface. For distances greater than the minimum far field distance, given by equation 11, the formulation of equation 10 enables the definition of the directivity for on-body scenarios. Analog to equation 3 the directivity *D* is defined by the normalization of the power density to

*<sup>D</sup>* <sup>=</sup> <sup>|</sup>*S*<sup>|</sup>


The related electromagnetic field is in general a superposition of TE-, TM- and TEM-wave components and therefore a function of the parameters given in equation 8. Despite this fact, the TEM-wave component contributes the most significant part to the power flux density. Even if a strong surface wave is excited, the resulting TM-wave component is comparatively

max,eff

+ �

<sup>1</sup> <sup>−</sup> *<sup>R</sup>*v,h�

**i**v,h, *d*, *h*, *f* ,*ε*

�

*E*(*d*) = *E*sp + *E*sw, (9)

Antennas and Propagation for On-, Off - and In-Body Communications 159

*F*v,hcos2 (*ψ*)*e<sup>j</sup>*

*<sup>λ</sup>* , (11)

. (12)

� �� � surface wave

2*πd*ref *λ*

⎤ ⎥ ⎥ ⎦

. (10)

, (8)

high *h* can be described by Norton's formulation *N* as follows:

the space wave *E*sp and surface wave *E*sw as follows:

*e<sup>j</sup>* <sup>2</sup>*π<sup>d</sup> λ* ���� direct wave

Analog to equation 2 the far field distance *d*ff is defined by:

where *E* is given by

the far field model:

*<sup>E</sup>*(*d*) <sup>∝</sup> *Un d*

⎡ ⎢ ⎢ ⎣ *<sup>E</sup>* <sup>∝</sup> *Un*

+ *R*v,hcos<sup>3</sup> (*ψ*)*e<sup>j</sup>*

� �� � ground-reflected wave

*<sup>d</sup> <sup>N</sup>* �
