**3.1 Description of the radiation source considered**

Fig. 4 displays the arbitrary placement representing the set of eight magnetic dipoles in the horizontal plane *Oxy* considered for the validation of the calculation method under study. To generate the various behaviours of EM-field graphs, the dipole axes are randomly oriented as follows: *M*1, *M*4, *M*5 and *M*8 along *Ox*-axis, *M*2 and *M*6 along *Oy*, and *M*3 and *M*<sup>7</sup> along *Oz*. These dipoles are in this case, flowed by an ultra-short duration transient current *i*(*t*) which is considered as a pulse signal. These elementary dipoles are supposed formed by wire circular loops having radius *a* = 0.5 mm.

Note that in the present study, the elementary dipoles are supposed ideal, thus, there is no coupling between each other. By reason of linearity, the total EM-field at any point *M*(*x*,*y*,*z*) is consequently the sum of each dipole contribution:

$$
\overline{H}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \sum\_{k=1}^{8} \overline{H}\_{M\_k}(\mathbf{x}, \mathbf{y}, \mathbf{z}) \,. \tag{5}
$$

After implementation of the computation algorithm introduced in Fig. 2 in Matlab program, we obtain the EM-calculation results presented in the next subsection.

Fig. 4. Configuration of the radiating source considered which is comprised of magnetic dipoles placed in the horizontal plane *xy*

#### **3.2 Validation results**

8 Fourier Transform Applications

H-field here denoted as *H*(*t*) according to the arbitrary form of the excitation and also knowing the frequency-dependent H-field data in the frequency range starting from the lowest value to the upper frequency limit equal to the inverse of the time-step *∆t* of the

This section presents the verification of the computation method established regarding the radiation of magnetic dipoles shown in Fig. 3 in time-domain. The results were realized by considering the frequency-dependent EM-NF-field directly calculated from predefined analytical formulas (Baum 1971 & 1976, Singaraju & Baum 1976, Balanis 2005). After the analytical description of the considered dipole source and also the mathematical definitions of the H-field expressions in frequency- and time-domains, we will explore the numerical

Fig. 3. Configuration of the elementary magnetic dipole formed by a circular loop with

In this figure, the magnetic dipole assumed as a circular wire having radius *a* is positioned at the origin of the system. Then the considered point *M* where the EM-field will be evaluated can be referred either in cartesian coordinate (*x*,*y*,*z*) or in spherical coordinate (*r*,*θ*,*φ*). By assuming that the magnetic loop depicted in Fig. 3 is fed by transient current denoted *i*(*t*), we have the H-field expressions recalled in appendix A (Baum 1971 & 1976,

Fig. 4 displays the arbitrary placement representing the set of eight magnetic dipoles in the horizontal plane *Oxy* considered for the validation of the calculation method under study. To generate the various behaviours of EM-field graphs, the dipole axes are randomly oriented as follows: *M*1, *M*4, *M*5 and *M*8 along *Ox*-axis, *M*2 and *M*6 along *Oy*, and *M*3 and *M*<sup>7</sup> along *Oz*. These dipoles are in this case, flowed by an ultra-short duration transient current *i*(*t*) which is considered as a pulse signal. These elementary dipoles are supposed formed by

radius *a* placed at the centre *O*(0,0,0) of the cartesian (*x*,*y*,*z*)-coordinate system

**3. Validation with the radiation of a set of magnetic dipoles** 

discrete data *x*(*tk*).

computation with Matlab programming.

Singaraju & Baum 1976).

**3.1 Description of the radiation source considered** 

wire circular loops having radius *a* = 0.5 mm.

In this subsection, comparison results between the transient EM-field maps radiated by the elementary dipoles displayed in Fig. 4 from the direct calculation and from the method proposed are presented.

#### **3.2.1 Description of the excitation signal**

In order to highlight the influence of the form and the transient variation of the disturbing currents in the electronic structure, the considered short-duration pulse excitation current *i*(*t*) is assumed as a bi-exponential signal analytically defined in appendix B. One points out that in order to take into account the truncation effect between *tmin* and *tmax*, the considered sampling data from *i*(*t*) should be multiplied by a specific time gate. So that accordingly, each component *I*( ) ω should be assumed as a sine cardinal. But here, this effect can be negligible if the assumed time step is well-accurate. The numerical application was made by taking the current amplitude *IM* = 1A and the time-constants *τ*1 = *τ*2/2 = 2 ns. So, from the analytical relation expressed in (B-4), we have *ω*95% ≈ 3.07 Grad.s-1. Fig. 5 displays the transient plot of this current excitation.

The time interval range of signal test was defined from *tmin* = 0 ns to *tmax* = 20 ns with step *∆t* = 0.2 ns. One can see that this baseband signal presents a frequency bandwidth *fmax* of about 2 GHz, where belongs more than 95-% of the spectrum signal energy. The data calculated *I fft i t* ( ) [ ( )] ω = generates the frequency-coefficient values of *i*(*t*) according to the relation expressed in (2) as described earlier in subsection 2.2.

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 11

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

Fig. 7. Maps of H-field components magnitude obtained at the frequency *f*0 = 2 GHz: (a) *Hx*,

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

(b) *Hy* and (c) *Hz* directly calculated from expressions (A-8), (A-9) and (A-10)

mm, *y* = 35 mm) are plotted in Figs. 11(a)-(c).

*f*0 = 2 GHz.
