**4. Application with the Transient NF emitted by a microstrip device proof-ofconcept**

To get further insight about the feasibility of the computation method under study, let us examine the transient EM-wave emitted by an example of more realistic microwave device. This latter was designed with the standard 3-D EM-tools HFSS for generating the frequencydependent data which is used for the determination of transient H-NF. Then, the simulation with CST microwave studio (MWS) was performed for the computation of the reference H-NF mappings in time-domain. As realistic and concrete demonstrator, a low-pass Tchebychev filter implemented in planar microstrip technology was designed. Its layout top view including the geometrical dimensions is represented in Fig. 12(a).

This device was printed on the FR4-epoxy substrate having relativity permittivity *εr* = 4.4, thickness *h* = 1.6 mm and etched Cu-metal thickness *t* = 35 µm. The cut cross section of this microwave circuit is pictured in Fig. 12(b).

Fig. 12. (a) Top view of the CST design of the under test low-pass microstrip filter (b) Crosssection cut of the under test microstrip filter with metallization thickness *t* = 35 µm and dielectric substrate height *h* = 1.6 mm

After simulations, we realize the results obtained and discussed in next subsections.

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 17

Similar to the previous section, these H-field components were mapped in the horizontal plane placed at the height *z*0 = 6 mm above the bottom surface of the considered filter and delimited by -38 mm < *x* < 38 mm and -24 mm < *y* < 24 mm with space-step *∆x* = *∆y* = 2 mm. By using the frequency-dependent data convoluted with the excitation test signal, we will

**4.2 Analysis of the results obtained from the transient field computation method** 

Fig. 14. Maps of frequency-dependent H-field components magnitude: (a) *Hx*, (b) *Hy* and (c)

*Hz* computed from HFSS at *f* = 1 GHz

present next that we can regenerate these transient H-field maps.

**proposed** 

#### **4.1 CST-computation results**

The low-pass filter under test was simulated with CST MWS in the time interval range delimited by *tmin* = 0 ns and *tmax* = 20 ns with step *∆t* = 0.2 ns. Therefore, the magnetic-field maps are presented in Fig. 13. These curves are recorded at the arbitrary instant time *t*0 = 2 ns. Note that this structure was excited by the input transient current plotted in Fig. 5. It results the graphs of H-field components displayed in Fig. 13.

Fig. 13. Maps of transient H-field components detected at *t* = 2 ns: (a) *Hx*, (b) *Hy* and (c) *Hz* computed from the commercial tool CST

16 Fourier Transform Applications

The low-pass filter under test was simulated with CST MWS in the time interval range delimited by *tmin* = 0 ns and *tmax* = 20 ns with step *∆t* = 0.2 ns. Therefore, the magnetic-field maps are presented in Fig. 13. These curves are recorded at the arbitrary instant time *t*0 = 2 ns. Note that this structure was excited by the input transient current plotted in Fig. 5. It

Fig. 13. Maps of transient H-field components detected at *t* = 2 ns: (a) *Hx*, (b) *Hy* and (c) *Hz*

computed from the commercial tool CST

results the graphs of H-field components displayed in Fig. 13.

**4.1 CST-computation results** 

Similar to the previous section, these H-field components were mapped in the horizontal plane placed at the height *z*0 = 6 mm above the bottom surface of the considered filter and delimited by -38 mm < *x* < 38 mm and -24 mm < *y* < 24 mm with space-step *∆x* = *∆y* = 2 mm. By using the frequency-dependent data convoluted with the excitation test signal, we will present next that we can regenerate these transient H-field maps.

## **4.2 Analysis of the results obtained from the transient field computation method proposed**

Fig. 14. Maps of frequency-dependent H-field components magnitude: (a) *Hx*, (b) *Hy* and (c) *Hz* computed from HFSS at *f* = 1 GHz

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 19

Despite the slight difference of Hx-maps, we observe that the maps are perfectly wellcorrelated to those introduced in Fig. 13 of subsection 4.1. For the further smart illustration of the results correlation, comparisons of H-field profiles calculated with the method proposed (grey curves) and those from CST (black curves) for *x* = -5 mm are plotted Fig. 16.

Fig. 16. Comparisons between *Oy*-profiles of the H-field components computed with CST

The imperfection of the results presented here are due to the numerical errors mainly caused by the solver and the meshing inaccuracies. Note that one evaluates relative errors of about 10 % for |*Hx*|, |*Hy*| and |*Hz*|. Despite the apparent difference between the results from the method under investigation and the commercial EM-tool CST-computations, once again, very good correlations between the profiles of the H-field components are realized by considering the data recorded in the vertical cut-plane equated by *x* = -5 mm as explained in

In nutshell, the computation results exposed in this paper reveal the effectiveness and the operability of the method developed for the case of elementary magnetic dipoles and also by

A computation method of transient NF EM-field radiated by electronic devices excited by a complex wave or ultra-short duration transient signal is stated in this chapter. In addition to the evanescent wave integration, the originality of the NF calculation method developed lies on the consideration of the radiation deeming the UWB structures which is literally from DC to microwave frequency ranges. It is based on the convolution of the frequency-dependent

considering the NF EM-radiation of realistic use case electronic devices.

software and those obtained from the proposed method

Fig. 16.

**5. Concluding remarks** 

After HFSS-simulations carried out in the frequency range starting from *fmin* = 0.05 GHz to *fmax* = 2.5 GHz with frequency-step *∆f* = 0.05 GHz, the maps of H-field component magnitudes displayed in Fig. 14 are recorded. These field components are mapped in the same horizontal plane as in the previous subsection by taking the height *z*0 = 6 mm. Among the series of the field maps obtained, we show here the data simulated at the frequency *f* = 1 GHz. According to the flow work depicted in Fig. 2, the frequency-dependent data *Hx*(*f*), *Hy*(*f*) and *Hz*(*f*) are employed for the determination of the time-dependent data *Hx*(*t*), *Hy*(*t*) and *Hz*(*t*) regarding the transient input current plotted earlier in Fig. 5. So that by application of the computation algorithm under investigation, the H-field maps calculated from Matlab are respectively, depicted in Figs. 15 at the instant time *t* = 2 ns.

Fig. 15. Maps of H-field components obtained from the proposed method and regarding the simulated frequency-dependent data from HFSS: (a) *Hx*, (b) *Hy* and (c) *Hz*

Despite the slight difference of Hx-maps, we observe that the maps are perfectly wellcorrelated to those introduced in Fig. 13 of subsection 4.1. For the further smart illustration of the results correlation, comparisons of H-field profiles calculated with the method proposed (grey curves) and those from CST (black curves) for *x* = -5 mm are plotted Fig. 16.

Fig. 16. Comparisons between *Oy*-profiles of the H-field components computed with CST software and those obtained from the proposed method

The imperfection of the results presented here are due to the numerical errors mainly caused by the solver and the meshing inaccuracies. Note that one evaluates relative errors of about 10 % for |*Hx*|, |*Hy*| and |*Hz*|. Despite the apparent difference between the results from the method under investigation and the commercial EM-tool CST-computations, once again, very good correlations between the profiles of the H-field components are realized by considering the data recorded in the vertical cut-plane equated by *x* = -5 mm as explained in Fig. 16.

In nutshell, the computation results exposed in this paper reveal the effectiveness and the operability of the method developed for the case of elementary magnetic dipoles and also by considering the NF EM-radiation of realistic use case electronic devices.
