# Polynomial Chaos Expansion (PCE) Method

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## Design parameters effect on the antenna performance

The antenna has two resonant frequencies at 403 MHz and 915 MHz that correspond to data and power transmission respectively. For this antenna design, the following parameters are vital for determining the antenna’s resonant frequencies and these parameters are discussed below (it is worth pointing out that all the comments here are only suitable for the antenna in this thesis, it does not represent other ones):
• R1 and the first arc’s angle (θ1+ θ2+ θ4) determine the placement of the first resonant frequency at 403 MHz. As we increase the length of the first arc, the antenna resonates at a lower frequency than 403 MHz since the ‘resonating arm’ becomes longer.
• w1 affects the real part of the antenna’s impedance at the first resonant frequency. The resistance of the antenna at 403 MHz decreases as w1 decreases. However, due to the fabrication limit, w1 is restricted to be higher than 0.15 mm.
• t1 and t2 affect the impedance of the antenna at both frequencies. A thicker substrate may cause an increase of the antenna’s resistance at 403 MHz. However, since the Rogers company provides only the 0.64 mm thickness, the value 0.64 is used in simulations.
• R2, w2, and the inner arc’s angle (θ3+ θ4) affect the second resonant frequency in the same way as R1 and w1.
• The feeding point F affects the length of the current circulating on the radiating patch thus affects the antenna’s impedance at both resonant frequencies. The longer the current path, the higher the resistance.
• The via at W allows the appearance of the double resonant frequencies of the antenna.

### Propagation in tissue

Human tissue is an inhomogeneous material that contains many kinds of biological elements, such as: Bone, muscle, fat, skin, blood, and also some various organic materials. Among those elements, bone, muscle, fat, and skin are the most common and take the most volume percentage. Among those four kinds of tissue, bone and fat contain much less water than muscle and skin, and thus have less loss due to less free charge electron which could respond to an alternating electromagnetic (EM) field. The distribution of the time-harmonic EM field is mainly determined by its complex permittivity ε: 𝜀𝜀 = 𝜀𝜀0𝜀𝜀𝑟𝑟 where ε0 is the permittivity of the vacuum and εr is the complex relative permittivity, which could be defined as: 𝜀𝜀𝑟𝑟 = 𝜀𝜀𝑟𝑟 ′ − 𝑗𝑗.

#### Three-layer model

In order to find a compromise between the computational accuracy complexity and the representativeness of the environment, in this work, a three-layer model is chosen. The three layers are the most common tissues of the human body: bone, muscle, and skin. As fat has much less conductivity than muscle or skin and is less lossy, it is neglected in the purpose of model simplification. The radius of each layer is: Bone (25 mm), muscle (25 mm-47.5 mm), and skin (47.5 mm-50 mm). The implantation depth is calculated as the distance between the center of the radiating patch and the external surface of the skin layer. The length of the “arm” is set to the minimum value to ensure proper consideration of a “realistic” case and avoid heavy calculations. The details of the environmental model and the dielectric properties of the three kinds of human tissue are given in Figure 2.5 and Table 2.2. The contents in Table 2.2 are extracted from the CST database.

The radiation pattern describes the radiation of antenna and indicates the direction of the main lobe at far-field range. This helps to identify the direction in which it could receive more power and adjust the design. The radiation pattern is calculated in the center of the frequency band: 403 MHz and 915 MHz. Since the “human arm model” (the three-layer cylinder) is positioned along the Y-axis, the 2-D radiation patterns in xz plane for the first design are presented in Figure 2.13.
The antenna is implanted with 10 mm depth. At MedRadio band (403 MHz), the maximum gain in far-field is around -33.5 dBi and towards Z-axis (see Figure 2.7 for axis positioning). At the ISM band (915 MHz), the maximum gain is -33.65 dBi. It is worth pointing out that the antenna has better maximum gain at a deeper location: At 16 mm depth, its maximum gain increases to -31.6 dBi at 403 MHz and -33.1 dBi at 915 MHz since the radiation pattern has a narrower main lobe. For the advanced design, the radiation pattern at 10 mm implantation depth is shown in Figure 2.14. For the purpose to compare the radiation pattern between figure 2.13 and figure 2.14.,they are already in the same scale.

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Reflector’s geometry

Figure 3.5 shows the ideal structure of a reflector and how it works. According to the geometrical optics approaches, if high frequency EM wave could be represented as a distribution of vectors in a 2D plane, the reflector should be parabolic with a planar incident wave. But the external patch is not far or large enough to produce plane wave, so an elliptical reflector may be more appropriate. The ellipse has its two focal points at the embedded antenna’s and the external antenna’s positions. Its focal distance equals to the average of the sum of transmission distance and implantation depth. The geometry of the elliptical reflector is shown in Figure 3.12.

Reflector’s open angle

The open angle of the reflector is a key factor. It determines the amount of power that could be reflected and received by the embedded antenna. Here the transmission scenario is realized with a transmission distance of 600 mm (far-field) and an implantation depth of 14 mm. The reflector is made of aluminum. The relationship between the transmission efficiency enhancement and the reflector’s open angle at 915 MHz band (defined in Figure 3.12) is presented in Figure 3.15:  The S21 results are not monotonous with the increase of the open angle. There exists a local extremum at 30°. The enhancement at 30° is the best among all angles under 100°. The other extremum appears at 210° which is also the highest value. Therefore, in realistic cases, it is suggested that with a limited space which could not afford a reflector larger than 100°, a 30° reflector appears the best option. Last but not the least, we include all previous results in order to achieve the highest enhancement value. The transmission efficiency enhancement (increased value of S21) caused by the reflector at 915 MHz band is presented in the following Table 3.1.

Global introduction
General Introduction
Thesis structure
Chapter 1 State-of-art
1.1 Thesis context
1.2 State-of-art
1.2.1 Transmission methods comparison
1.2.1.1 Inductive charging
1.2.1.2 Strongly coupled magnetic resonance
1.2.1.3 Microwave power transfer
1.2.2 Antenna power transmission
1.2.2.1 Authorized frequency band
1.2.2.2 Antenna’s type, size, and operating frequency
1.2.2.3 Simulation environment
1.2.2.4 Experimental environment
1.2.3 Rectifying circuit
1.3 Thesis objective
1.4 Conclusion
Chapter 2 Antenna design and simulation
2.1 Introduction
2.1.1 Frequency range
2.2 Antenna Design
2.2.1 Preliminary design
2.2.2 Design parameters’ effect on the antenna performance
2.2.3 Simulation environment
2.2.3.1 Propagation in tissue
2.2.3.2 Single tissue (muscle) box
2.2.3.3 Three-layer model
2.3 Numerical results
2.3.1 Reflection coefficient and impedance
2.3.3 Specific Absorption Rate (SAR)
2.3.4 Surface current and polarization
2.4 CST configuration
2.4.1 Before simulation
2.4.2 Setup solver
2.4.3 Results observation
2.5 Conclusion
Chapter 3 Transmission link establishment and results
3.1 Introduction
3.2 FRIIS equation and transmission efficiency [74]
3.3 Power emitter standard
3.3.2 Industrial, Scientific, Medical (ISM) band
3.4 Transmission structure
3.4.1 External antenna choice
3.4.2 Reflector design
3.5 Numerical results
3.5.1 Antenna’s positioning
3.5.2 Transmission distance
3.5.2.1 Implantation depth
3.5.2.1 Transmission distance
3.5.3 Reflector issues
3.5.3.1 Reflector’s geometry
3.5.3.2 Reflector’s material
3.5.3.3 Reflector’s open angle
3.5.3 SAR issues
3.6 Sensitivity analysis
3.6.1 Polynomial Chaos Expansion (PCE) Method
3.6.2 Results and discussion
3.6.2.1 Leave-one-out (LOO) validation error
3.6.2.2 Sobol indices and sensitivity analysis
3.6.2.3 Result prediction
3.7 Conclusion
Chapter 4 Rectifying circuit design and simulations
4.1 Introduction
4.2 Circuit design
4.2.1 Rectifying circuit comparison
4.2.1.1 Circuit structure
4.2.1.2 Impedance matching
4.2.2 Combination design
4.2.3 Printed microstrip circuit design
4.3 Numerical results
4.3.1 Off-the-shelf components
4.3.2 Components number comparison
4.3.4 Diode comparison
4.3.5 Antenna-circuit combination analysis
4.4.1 Circuit microstrip circuit setup
4.5 Sensor analysis
4.6 Conclusion
Chapter 5 Measurements and experimental procedures
5.1 Introduction
5.2 Antenna measurements
5.2.1 Preparations
5.2.1.1 Vector network analyzer
5.2.1.2 RF coaxial cable
5.2.2 Antenna fabrication
5.2.3 Environmental setup and measurements
5.2.3.1 Biological environment
5.2.3.2 First measurement (Univ. Patras)
5.2.3.3 Second measurement (GeePs)
5.3 Transmission measurements
5.3.1 First transmission measurement (Univ. Patras)
5.3.2 Second transmission measurement (GeePs)
5.4 Circuit measurements
5.4.1 Circuit fabrication
5.4.2 Measurements
5.5 Conclusion
Conclusion and Perspectives
Conclusion
Perspectives
References

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