**3.2.2 Discussions on the computed results**

By considering the set of eight magnetic dipoles presented in Fig. 4 which are excited by the same pulse current plotted in Fig. 5 yields the H-field component (*Hx*, *Hy* and *Hz*) mappings depicted in Fig. 6 at the arbitrary time *t*0 = 2 ns and in the horizontal plane parallel to (*Oxy*) referenced *z*0 = 6.5 mm above the radiating source.

Fig. 5. Transient plot of the considered excitation current *i*(*t*)

Fig. 6. Maps of H-field components detected at the height *z*0 = 6.5 mm above the radiating source: (a) *Hx*, (b) *Hy* and (c) *Hz* obtained from the direct calculation

10 Fourier Transform Applications

By considering the set of eight magnetic dipoles presented in Fig. 4 which are excited by the same pulse current plotted in Fig. 5 yields the H-field component (*Hx*, *Hy* and *Hz*) mappings depicted in Fig. 6 at the arbitrary time *t*0 = 2 ns and in the horizontal plane parallel to (*Oxy*)

Fig. 6. Maps of H-field components detected at the height *z*0 = 6.5 mm above the radiating

source: (a) *Hx*, (b) *Hy* and (c) *Hz* obtained from the direct calculation

**3.2.2 Discussions on the computed results** 

referenced *z*0 = 6.5 mm above the radiating source.

Fig. 5. Transient plot of the considered excitation current *i*(*t*)

This height was arbitrarily chosen in order to generate a significant NF effect in the considered frequency range. The dimensions of the mapping plane were set at *Lx* = 110 mm and *Ly* = 100 mm with space-resolution equal to *∆x = ∆y* = 2 mm. First, by using the harmonic expressions of the magnetic field components, the maps of the frequencydependent H-field are obtained from *fmin* = 0.05 GHz to *fmax* = 2.50 GHz with step *∆f* = 0.05 GHz. Fig. 7 represents the corresponding mappings of the H-field component magnitudes at *f*0 = 2 GHz.

Fig. 7. Maps of H-field components magnitude obtained at the frequency *f*0 = 2 GHz: (a) *Hx*, (b) *Hy* and (c) *Hz* directly calculated from expressions (A-8), (A-9) and (A-10)

After the Matlab program implementation of the algorithm indicated by the flow chart schematized in Fig. 2, the results shown in Figs. 8(a)-(d) are obtained via the combination of the frequency-dependent data of the H-field components associated to the frequencycoefficients of the excitation signal plotted in Fig. 5. One can see that the EM-maps presenting the same behaviors as those obtained via the direct calculations displayed in Fig. 6 were established. In addition, we compare also as illustrated in Fig. 9 the modulus of the H-fields from the method under study and from the 3D EM Field Simulator - CST (Computer Simulation Technology). Furthermore, as evidenced by Figs. 10(a)-(c), very good correlation between the profiles of the H-field components detected in the vertical cut-plane along *Oy* and localized at *x* = 23 mm was observed. To get further insight about the timedependent representation of the H-field components, curves showing the variations of *Hx*(*t*), *Hy*(*t*) and *Hz*(*t*) at the arbitrary point chosen of the mapping plane having coordinates (*x* = 19 mm, *y* = 35 mm) are plotted in Figs. 11(a)-(c).

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 13

Fig. 10. Comparisons of the H-field component profiles obtained from the time-frequency computation method proposed and the direct calculation, detected in the vertical cut-plane

*x* = 23 mm: (a) *Hx*, (b) *Hy* and (c) *Hz*

As results, once again, we can find that the H-field components from the frequency data fit very well the direct calculated ones. As aforementioned, due to the truncation effects, the Hx-component presents a slight divergence at the ending time of the signal. This is particularly due to the numerical noises at the very low value of the EM field as the case of the x-component which is absolutely twenty times less than the two other components.

In order to prove in more realistic way the relevance of the investigated method, one proposes to treat the radiation of concrete electronic devices in the next section.

Fig. 8. Maps of H-field components calculated from the time-frequency computation method proposed for *z*0 = 6.5 mm: (a) *Hx*, (b) *Hy*, (c) *Hz*

Fig. 9. Comparison of H-field maps modulus |*H*|(*t*0 = 2 ns) obtained from the direct formulae (in left) and from the time-frequency computation method proposed for *z*0 = 6.5 mm

12 Fourier Transform Applications

As results, once again, we can find that the H-field components from the frequency data fit very well the direct calculated ones. As aforementioned, due to the truncation effects, the Hx-component presents a slight divergence at the ending time of the signal. This is particularly due to the numerical noises at the very low value of the EM field as the case of the x-component which is absolutely twenty times less than the two other components.

In order to prove in more realistic way the relevance of the investigated method, one

Fig. 8. Maps of H-field components calculated from the time-frequency computation method

Fig. 9. Comparison of H-field maps modulus |*H*|(*t*0 = 2 ns) obtained from the direct formulae (in left) and from the time-frequency computation method proposed for

proposed for *z*0 = 6.5 mm: (a) *Hx*, (b) *Hy*, (c) *Hz*

*z*0 = 6.5 mm

proposes to treat the radiation of concrete electronic devices in the next section.

Fig. 10. Comparisons of the H-field component profiles obtained from the time-frequency computation method proposed and the direct calculation, detected in the vertical cut-plane *x* = 23 mm: (a) *Hx*, (b) *Hy* and (c) *Hz*

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 15

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

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

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

(a)

(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

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

view including the geometrical dimensions is represented in Fig. 12(a).

microwave circuit is pictured in Fig. 12(b).

dielectric substrate height *h* = 1.6 mm

**concept** 

Fig. 11. Comparisons of the H-field components temporal variation obtained from the timefrequency computation method proposed and the direct calculation, detected in the arbitrary point (*x* = 19 mm, *y* = 35 mm): (a) *Hx*, (b) *Hy* and (c) *Hz*

14 Fourier Transform Applications

Fig. 11. Comparisons of the H-field components temporal variation obtained from the timefrequency computation method proposed and the direct calculation, detected in the arbitrary

point (*x* = 19 mm, *y* = 35 mm): (a) *Hx*, (b) *Hy* and (c) *Hz*
