# Design Optimization of Wearable Multiband Antenna Using Evolutionary Algorithm Tuned with Dipole Benchmark Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Benchmark Problem

#### 2.2. Algorithm Comparison—Single Objective

#### 2.3. Pareto-Inspired Evolutionary Optimization: P-EStra Algorithm

_{e}; in turn. q is the rate of correction the standard deviation d must undergo during the optimization.

_{e}> p in order to increase the exploration capability of the algorithm. In contrast. ${d}_{k+1}=q\xb7{d}_{k}$ is set to force a smaller standard deviation of the Gaussian distribution in the next iteration; this happens when p

_{e}< p in order to increase the exploitation capability of the algorithm. It appears that an appropriate tuning of p and q values is mandatory for a good performance of the algorithm; the issue is discussed in the subsequent Section 2.4.

_{o}is smaller than the prescribed search tolerance.

_{o}, which contains the initial guess solution, and in turn, vector d is initialized as d

_{o}and the value of its elements is proportional to the admissible range of the corresponding design variables.

- c
_{0}. the hardware-dependent time necessary to run a single solution of the direct problem associated to the optimization problem; - n
_{i}. the number of convergence iterations for a prescribed search accuracy; - n
_{p}. the number of evolving solutions (in our case. n_{p}= 1); - n
_{f}. the number of objective functions.

#### 2.4. Algorithm Tuning—Bi-Objective Case

_{min}and G

_{max}that are the best values of those parameters identified over 108 runs. For VSWR

_{min}and the corresponding to this value of antenna gain: G ≠ G

_{max}we have calculated:

_{min}) was calculated for the solution which exhibits minimum value of VSWR, while RG(G

_{max}) was for solution with maximal Gain. It can be noted that the selection of p_ann and q_ann parameter influences computational cost of the algorithm, as well as the quality of the final solution.

#### 2.5. Case Study: Wearable Antenna Design

_{1}) and the relevant coordinates (x

_{n}.y

_{n}), as well as the radius (r

_{n}) of other circles. Figure 8 presents an antenna with two circles (n = 2), but we have also investigated the design with 3 circles.

_{o}= 271 mm, r

_{i}= 102 mm, ε

_{o}= 3.35, σ

_{o}= 0.36, ε

_{i}= 42.94, σ

_{i}= 2.03. The antenna was placed at half of the model height.

^{®}). Its thickness is only 25 μm and the thickness of metallization is 35 μm that is much smaller than the voxel size used for antenna simulation. The dielectric constant of base material given by the manufacturer is ε

_{r}= 3.4 and the dielectric loss tg(δ) = 0.005; however, in our previous research we have found that its effective value that can be used for numerical simulation with voxels that are thicker than the substrate (0.5 mm in our case) is ε

_{r}= 1.7 and the dielectric loss tg(δ) = 0.001. The dielectric properties and the dimensions (75 mm by 75 mm) of substrate materials remained unchanged in the optimization process. For the clarity of drawings, we have omitted the layer of this material in figures that are presenting the antenna.

- g: design vector (geometric parameters defining the multi-band antenna shape)
- Ω
_{g}: set of admissible values - B
_{1}: band at 1.8 MHz - B
_{2}: band at 2.4 GHz - B
_{3}: band at 3.5 GHz - B
_{4}: band at 5.8 GHz - VSWR: voltage standing wave ratio
- G
_{1}: maxim gain in x-z plane for band 1 - G
_{2}: maxim gain in x-z plane for band 2 - G
_{3}: maxim gain in x-z plane for band 3 - G
_{4}: maxim gain in x-z plane for band 4 - G: minimum gain of bands 1–4

_{0}within Ω

_{g}. the following f

_{1}objective is to be minimized:

_{2}objective is to be maximized (3):

_{1}.f

_{2}) evolve from the guess solution g

_{0}to convergence, treating f

_{1}and f

_{2}as individual objectives in mutual conflict.

## 3. Results of Wearable Antenna Optimization

#### 3.1. Wearable Antenna Optimization

_{1}. The ground rectangle can be integrated with a printed circuit board that contains electronic circuits. The dimensions of the ground plane (a, b parameters) were limited to 5 cm because we wanted to keep the antenna as small as possible, which is crucial for a wearable application. The maximum radius of the circle allowed in optimization procedure was equal to 25 mm.

_{1start}= 0.0228, a

_{start}= 0.033, b

_{start}= 0.033. For this starting point the VSWR component of the objective function was VSWR

_{start}= 3.64 and the Gain component was G

_{start}= 0.28 dBi. The constraints in the optimization process were geometry oriented, allowing only for sets of design variable values that preserved the assumed geometry of the antenna without self-intersections or overlapping sections. The optimization process required 56 iterations to satisfy the automatic stopping condition. The condition relies on the ratios of the standard deviation within the current iteration ${d}_{k}$ to the initial standard deviation iteration d

_{0k}for each k-th optimization variable. The deviation is normalized across all the variables. The process stops when $\underset{k}{sup}\left[\frac{{d}_{k}}{{d}_{0k}}\right]<s$. where s << 1 is a prescribed search tolerance. This corresponds to the situation when the current search region is sufficiently small for all variables. In this study, basing on experience gathered on optimization of benchmark problem, we assumed $s={10}^{-2}$. Final set of parameters was identified with 56 iterations r

_{1stop}= 0.02366. a

_{stop}= 0.02241. b

_{stop}= 0.04619 and the final objective function components were the following: G

_{stop}= 2.08. VSWR

_{stop}= 2.36. The final geometry of antenna and its radiation pattern is presented in Figure 10. Table 4 presents the parameters of optimized antenna: VSWR, gain G and realized gain (RG) for all 4 bands.

_{1start}. a

_{start}and b

_{start}were the same as final parameter*rs of single circle antenna. The set of initial parameters had the following values: r

_{1start}= 0.02366. r

_{2start}= 0.02. x

_{2start}= 0.025 y

_{2start}= 0. a

_{start}= 0.02241. b

_{start}= 0.04619 For this starting point the VSWR component of the objective function was VSWR

_{start}= 2.26 and the Gain component was G

_{start}= 1.33 dBi.

_{1stop}= 0.0209. r

_{2stop}= 0.016. x

_{2stop}= 0.0216. y

_{2stop}= 0.0225. a

_{stop}= 0.02388. b

_{stop}= 0.02217. The final values of objective function components were and G

_{stop}= 1.98. VSWR

_{stop}= 2.26. The final geometry of antenna and its radiation pattern is presented in Figure 11. Table 5 presents the parameters of optimized antenna: VSWR. gain G and realized gain (RG) for all 4 bands.

_{1start}= 0.0209, r

_{2start}= 0.016, x

_{2start}= 0.0216, y

_{2start}= 0.0225, r

_{3start}= 0.00279, x

_{3start}= 0.02793, y

_{3start}= −0.01242, a

_{start}= 0.02388, b

_{start}= 0.02217. For this starting point, the VSWR component of the objective function was VSWR

_{start}= 2.21 and the Gain component was G

_{start}= 2.0 dBi.

_{1stop}= 0.02195, r

_{2stop}= 0.01852, x

_{2stop}= 0.02552, y

_{2stop}= 0.01648, r

_{3stop}= 0.00266, x

_{3stop}= 0.02725, y

_{3stop}= −0.02574, a

_{stop}= 0.02384, b

_{stop}= 0.0226. The final values of objective function components were G

_{stop}= 2.08, VSWR

_{stop}= 1.76. The final geometry of the antenna and its radiation pattern is shown in Figure 13. Table 6 presents the parameters of optimized antenna: VSWR, gain G and realized gain (RG) for all four bands.

_{max}) and average values for the whole body (SAR

_{AVG}) averaged for 10 g of tissue are presented in Table 8 for all considered bands. The SAR values are below the limits for mobile phones that for Europe are 2 W/kg.

#### 3.2. Assessment of Antenna Prototype

^{®}material that is a polymer foil with one side of coper metallization. The thickness of the base material is only 25 μm and the thickness of metallization is 35 μm. This makes the antenna flexible and lightweight what is beneficial in wearable applications. The dielectric constant of base material given by the manufacturer is ε

_{r}= 3.4 and the dielectric loss tg(δ) = 0.005; however, in our previous research we have found that its effective value that can be used for numerical simulation with voxels that are thicker than the substrate (0.1 mm in our case) is ε

_{r}= 1.7 and the dielectric loss tg(δ) = 0.001 [20] Due to the small thickness of substrate and its flexibility we used laser printer to put the mask on the substrate. To make it solid, we printed 4 layers of mask and then etched the metallization. The prototype is presented in Figure 17. The impedance matching of the prototype antenna located on human subject was measured using a Rohde & Schwarz ZVB 14 vector network analyzer. To avoid reflections that may influence the results of measurements they were conducted in anechoic chamber. The antenna was attached to the torso of human subject. To control the distance between antenna and human body we have used the Styrofoam spacer that is presented in Figure 18. The prototype antenna was fed by a coaxial probe and the calibration plane was moved to the end of the coaxial cable.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Design parameter space (L.r) and objective function component space (VSWR. G) for dipole benchmark.

**Figure 5.**Set of 108 starting points regularly located in the parameter space (L.r) that are marked with “*” and objective function component—VSWR presented with contour lines for dipole benchmark.

**Figure 7.**Results of P–EStra optimization located on Pareto–front in objective space and in design space.

**Figure 10.**Single circle antenna optimized with Pareto Estra algorithm: (

**a**)—antenna geometry. (

**b**)—radiation pattern in x-z plane for on-body position.

**Figure 11.**Two circle antennae optimized with Pareto Estra algorithm: (

**a**)—antenna geometry. (

**b**)—radiation pattern in x-z plane for on-body position.

**Figure 13.**Three circle antenna optimized with P-EStra algorithm: (

**a**)—antenna geometry. (

**b**)—radiation pattern in x-z plane for on-body position.

**Figure 14.**Three circle optimized antenna radiation pattern: (

**a**)—antenna position on heterogenous model. (

**b**)—radiation pattern in horizontal plane.

**Figure 16.**The distribution of Specific Absorption Rate (SAR) parameter in the cross-section of human body model, at the position of antenna for 3.5 GHz.

**Figure 19.**The impedance matching of prototype antenna for on-body location for 10 mm distance from the body.

**Figure 20.**The impedance matching of prototype antenna for on-body location for 20 mm distance from the body.

**Figure 21.**The impedance matching of prototype antenna for on-body location for 30 mm distance from the body.

**Table 1.**Performance of the optimization in the case of VSWR minimization for 108 starting points (O.F.—Objective Function).

nr | Method | Minimum Value of O.F. | Average Value of O.F. | Maximum Value of O.F. | Minimum Number of O.F. Calls | Average Number of O.F. Calls | Maximum Number of O.F. Calls |
---|---|---|---|---|---|---|---|

1 | Nelder-Mead | 1.32352 | 2.10497 | 8.25074 | 13 | 38 | 200 |

2 | Interior point | 1.39020 | 4.65605 | 27.46539 | 3 | 34.96296 | 67 |

3 | Quasi-Newton (Matlab fminunc function) | 2.22250 | 12.94491 | 159.88125 | 3 | 24.7 | 53 |

4 | Powell | 1.32352 | 1.81385 | 7.84942 | 200 | 530.37 | 2083 |

5 | EStra | 1.32352 | 1.66089 | 2.58385 | 41 | 48.5 | 67 |

nr | Method | Minimum Value of O.F. | Average Value of O.F. | Maximum Value of O.F. | Minimum Number of O.F. Calls | Average Number of O.F. Calls | Maximum Number of O.F. Calls |
---|---|---|---|---|---|---|---|

1 | Nelder-Mead | −14.6865 | 1.71705 | 4.77681 | 15 | 56.80556 | 403 |

2 | Interior point | −9.3953 | 1.77494 | 4.73644 | 3 | 35.9722 | 68 |

3 | Quasi-Newton (Matlab fminunc) | −30.6766 | −2.902 | 4.75 | 3 | 25.018 | 53 |

4 | Powell | Inf | inf | 4.77681 | 198 | 2551.129 | 19,804 |

5 | EStra | 3.7473 | 4.5014 | 4.7768 | 41 | 58.2037 | 97 |

nr | p | q | VSWR_{min} | G_{max} | RG(VSWR_{min}) | RG(G_{max}) | Call_{max} | Call _{avg} |
---|---|---|---|---|---|---|---|---|

1 | 0.1 | 0.7 | 1.414632 | 4.591953 | 1.996183906 | −0.75110921 | 50 | 32.97777778 |

2 | 0.1 | 0.75 | 1.34811575 | 4.42096375 | 2.015335957 | −0.182593774 | 67 | 38.15555556 |

3 | 0.1 | 0.8 | 1.445974 | 4.77681375 | 1.981397027 | −2.279198103 | 73 | 44.55555556 |

4 | 0.1 | 0.85 | 1.34811575 | 4.58963775 | 2.015335957 | −0.678137816 | 73 | 56.55555556 |

5 | 0.1 | 0.9 | 1.39427975 | 4.771963 | 2.000326514 | −2.080904046 | 96 | 78.22222222 |

6 | 0.1 | 0.95 | 1.48944775 | 4.5113045 | 1.965376331 | −0.481571138 | 150 | 146.4 |

7 | 0.15 | 0.7 | 1.392596 | 4.44949325 | 1.963651401 | −0.162833517 | 60 | 32.57777778 |

8 | 0.15 | 0.75 | 1.38949375 | 4.66783225 | 1.983534412 | −1.249282283 | 61 | 38.73333333 |

9 | 0.15 | 0.8 | 1.34811575 | 4.47306625 | 2.015335957 | −0.230794849 | 69 | 43.75555556 |

10 | 0.15 | 0.85 | 1.32351925 | 4.77637975 | 2.021862788 | −2.182204644 | 73 | 55.97777778 |

11 | 0.15 | 0.9 | 1.39020125 | 4.66783225 | 1.997481276 | −1.249282283 | 96 | 78.75555556 |

12 | 0.15 | 0.95 | 1.32351925 | 4.44949325 | 2.021862788 | −0.162833517 | 150 | 146.1555556 |

13 | 0.2 | 0.7 | 1.5989625 | 4.66783225 | 1.909027684 | −1.249282283 | 44 | 30.44444444 |

14 | 0.2 | 0.75 | 1.48614675 | 4.54797475 | 1.94579842 | −0.708278582 | 55 | 35.75555556 |

15 | 0.2 | 0.8 | 1.33607375 | 4.499961 | 2.01051999 | −0.362681899 | 55 | 41.75555556 |

16 | 0.2 | 0.85 | 1.32351925 | 4.771963 | 2.021862788 | −2.080904046 | 71 | 53.93333333 |

17 | 0.2 | 0.9 | 1.33607375 | 4.5113045 | 2.01051999 | −0.481571138 | 82 | 76.17777778 |

18 | 0.2 | 0.95 | 1.33607375 | 4.66783225 | 2.01051999 | −1.249282283 | 150 | 145.2444444 |

19 | 0.25 | 0.7 | 1.39020125 | 4.77561775 | 1.997481276 | −2.367162409 | 34 | 30.08888889 |

20 | 0.25 | 0.75 | 1.32351925 | 4.5113045 | 2.021862788 | −0.481571138 | 59 | 36.15555556 |

21 | 0.25 | 0.8 | 1.32351925 | 4.581255 | 2.021862788 | −0.604848098 | 55 | 41.31111111 |

22 | 0.25 | 0.85 | 1.39020125 | 4.42096375 | 1.997481276 | −0.182593774 | 69 | 53.93333333 |

23 | 0.25 | 0.9 | 1.39020125 | 4.75530625 | 1.997481276 | −1.979800714 | 90 | 76.8 |

24 | 0.25 | 0.95 | 1.32351925 | 4.581255 | 2.021862788 | −0.604848098 | 150 | 145.2 |

25 | 0.3 | 0.7 | 1.59242375 | 4.522489 | 1.883580045 | −0.366358967 | 34 | 30.08888889 |

26 | 0.3 | 0.75 | 1.32351925 | 4.5070495 | 2.021862788 | −0.42351266 | 39 | 35.08888889 |

27 | 0.3 | 0.8 | 1.38949375 | 4.5070495 | 1.983534412 | −0.42351266 | 41 | 41 |

28 | 0.3 | 0.85 | 1.32351925 | 4.47306625 | 2.021862788 | −0.230794849 | 53 | 53 |

29 | 0.3 | 0.9 | 1.5184325 | 4.66783225 | 1.955894016 | −1.249282283 | 76 | 76 |

30 | 0.3 | 0.95 | 1.5272455 | 4.547011 | 1.937485067 | −0.44992074 | 145 | 145 |

nr | f [GHz] | G [dBi] | VSWR | RG [dBi] |
---|---|---|---|---|

1 | 1.8 | 2.08 | 1.49 | 1.91 |

2 | 2.5 | 3.7 | 1.59 | 3.47 |

3 | 3.6 | 4.1 | 1.65 | 3.83 |

4 | 5.8 | 2.66 | 2.36 | 1.96 |

nr | f [GHz] | G [dBi] | VSWR | RG [dBi] |
---|---|---|---|---|

1 | 1.8 | 1.98 | 1.42 | 1.85 |

2 | 2.5 | 3.74 | 2.09 | 3.16 |

3 | 3.6 | 4.13 | 1.99 | 3.62 |

4 | 5.8 | 3.08 | 2.26 | 2.37 |

nr | f [GHz] | G [dBi] | VSWR | RG [dBi] |
---|---|---|---|---|

1 | 1.8 | 2.08 | 1.18 | 2.05 |

2 | 2.5 | 3.75 | 1.76 | 3.41 |

3 | 3.6 | 3.73 | 1.69 | 3.43 |

4 | 5.8 | 3.04 | 1.73 | 2.72 |

nr | f [GHz] | Current Distribution on the Antenna Surface |
---|---|---|

1 | 1.8 | |

2 | 2.5 | |

3 | 3.6 | |

4 | 5.8 |

nr | f [GHz] | SAR_{max} [W/kg] | SAR_{AVG} [W/kg] |
---|---|---|---|

1 | 1.8 | 0.894 | 0.00094 |

2 | 2.5 | 0.335 | 0.00045 |

3 | 3.6 | 0.688 | 0.00041 |

4 | 5.8 | 0.032 | 0.00002 |

Frequency [GHz] | Simulated Gain [dBi] | Measured Gain [dBi] | Radiation Pattern |
---|---|---|---|

1.8 | 2.5 | 2 | |

2.5 | 6 | 5 | |

3.6 | 2.6 | 1.8 | |

5.8 | 5.9 | 4 |

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**MDPI and ACS Style**

Januszkiewicz, Ł.; Barba, P.D.; Kawecki, J.
Design Optimization of Wearable Multiband Antenna Using Evolutionary Algorithm Tuned with Dipole Benchmark Problem. *Electronics* **2021**, *10*, 2249.
https://doi.org/10.3390/electronics10182249

**AMA Style**

Januszkiewicz Ł, Barba PD, Kawecki J.
Design Optimization of Wearable Multiband Antenna Using Evolutionary Algorithm Tuned with Dipole Benchmark Problem. *Electronics*. 2021; 10(18):2249.
https://doi.org/10.3390/electronics10182249

**Chicago/Turabian Style**

Januszkiewicz, Łukasz, Paolo Di Barba, and Jarosław Kawecki.
2021. "Design Optimization of Wearable Multiband Antenna Using Evolutionary Algorithm Tuned with Dipole Benchmark Problem" *Electronics* 10, no. 18: 2249.
https://doi.org/10.3390/electronics10182249