The antenna, which was the subject of optimization, can be designed to operate either in free space or in proximity of the human body. In Figure 5 the impedance matching of antenna, that was designed to operate in free space, is presented. It is expected to cover the frequency range from 2.4 GHz to 2.5 GHz with the maximum value for VSWR smaller than 1.3. This can be considered a wide bandwidth of impedance matching, which makes it less sensitive to impedance detuning. In the case of the antenna designed for the free space condition, the antenna impedance matching changes significantly, while placed very close to the human body. In Figure 5 the VSWR, obtained for free space and on-body case, is presented.

Figure 5 Impedance matching of antenna before optimization for free space and on body position (*x* = 8mm)

The optimization process aimed at finding a set of 6 antenna’s geometrical parameters for which the antenna would obtain the best impedance matching to 50 *Ω* (lowest value of VSWR) in the frequency range of 2.4 GHz – 2.5 GHz. To calculate the input impedance of the antenna the Remcom XFdtd code that utilizes finite difference time domain method was used [7]. The simulations were controlled by the script that was launched in XFdtd program. For the set of geometrical parameters given by the optimization procedure, the script generated the model of the antenna and launched the simulations of its input impedance vs. frequency. Based on the antenna impedance the objective function was calculated as the maximum value of voltage standing wave ratio - *VSWR* - within the frequency range of interest. Additionally, 3 different values of antenna distance to the body model (parameter *x* in Figure 4) equal to 2, 6 and 8 mm were considered and the greatest value of VSWR was taken as the objective function value. The design problem formulation can be cast as follows: given an initial solution (antenna prototype), find the minimizer with respect to design vector *g* = (*L*,*D*,*W*,*R*_{1},*R*_{2},*A*) of the following objective function *Ψ* (1) should be found:

$$\begin{array}{}{\displaystyle \mathit{\Psi}\phantom{\rule{thinmathspace}{0ex}}\left(\mathit{g}\right)\mathit{=}\phantom{\rule{thinmathspace}{0ex}}\underset{\mathit{x}\in {\mathit{\Omega}}_{\mathit{x}}}{sup}\underset{\mathit{f}\in \mathit{B}}{sup}\phantom{\rule{thinmathspace}{0ex}}\left|\mathit{V}\mathit{S}\mathit{W}\mathit{R}\mathit{(}\mathit{g}\mathit{,}\mathit{x}\mathit{,}\mathit{f}\mathit{)}\right|}\end{array}$$(1)

where:

*Ω*_{x} – the set of admissible x movements of antenna

*B* – the ISM f frequency band: 2.4 – 2.5 GHz

Therefore, (1) originates a double min-max problem: in principle, the solution to such class of problems might be non-smooth; therefore, derivative-free optimization algorithms, like evolutionary ones, is recommended. Coherently, the EStra optimization algorithm – an evolution strategy – was used. A version of the lowest order (*i*.*e*. a single parent generates a single offspring), which makes the search cost-effective, was chosen. A detailed analysis of the algorithm can be found *e*.*g*. in Ref. [2], while applications in electromagnetic simulations are presented *e*.*g*. in [8, 9].

The minimum value of objective function equal to *f*_{min} = 1.95 (that corresponds to the maximum value of VSWR) was achieved in 53 iterations. The values of the parameters found by the algorithm had following: *L* = 74.5 mm, *D* = 2.6 mm, *W* = 2.5 mm, *R*_{1} = 30.9 mm, *R*_{2} = 23.7 mm, *A* = 6.4 mm.

The antenna impedance matching simulations for the antenna geometry achieved after the optimization process are presented in Figure 6: for antenna distance to body equal to 2 mm – position 1, and 8 mm – position 3.

Figure 6 Impedance matching of antenna after optimization with EStra algorythm

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