**5. Concluding remarks**

18 Fourier Transform Applications

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

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*

from Matlab are respectively, depicted in Figs. 15 at the instant time *t* = 2 ns.

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

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 21

Tröscher 2011, Kopp 2011) where the transient NF effects are susceptible to disturb the

This appendix contains two parts of theoretical parts concerning the transient NF radiated by the elementary magnetic dipoles (Baum 1971 & 1976, Singaraju & Baum 1976, Ravelo et al. 2011a & 2011b, Lui Y. et al. 2011a & 2011b) and the bi-exponential signal processing.

By definition, the magnetic dipole moment of the elementary circular loop shown earlier in

*pM*(,) () () *rt pt r u* = ⋅ ⋅ δ*z*

(, , ,) (, , ,) (, , ,) *<sup>r</sup> H H r tu H r tu H r tu* =+ + *<sup>r</sup>*

θ ϕ

2 cos( ) 1 () () (, , ,) <sup>2</sup> *<sup>r</sup> p p Hr t*

sin( ) 1 () () () (, , ,) <sup>4</sup> *pp p <sup>r</sup> Hr t*

where *v* is the wave-velocity, and *τ* is the time delayed variable which is defined as

 = −*trv* / . One underlines that the magnetic dipole is also an Hertzian dipole so that ∂ ∂= *irt r* ( , )/ 0 . In the frequency domain, the spherical coordinate of the H-field component formulas radiated by the magnetic dipole pictured in Fig. 3 which is supposed flowed by an

> <sup>2</sup> ( ) *j ft <sup>j</sup> <sup>t</sup> If I e I e M M* − − π

2 <sup>3</sup> ( , , , ) 1 cos( ) <sup>2</sup>

*I a Hr f jkr e r*

3 sin( ) (, , , ) (1 ) <sup>4</sup> *<sup>M</sup> jkr I a Hr f jkr k r e r*

θ

2

( )

*M jkr*

τ

π

θ

ϕ θ ϕ

π

θ

θ

1892), the H-field components in the spherical coordinate system:

θ ϕ

θ ϕ

θ

harmonic current with amplitude *IM* denoted:

are written as (Balanis 2005):

θ ϕ

(, , ,) 0 *Hr t*

*r*

θ ϕ

θ

θ ϕ

with *r* is the distance between the dipole centre and the point *M*(*r*,*θ*,*φ*) as shown in Fig. 3. By analogy with the definition of the time-variant vector established by Hertz in 1892 (Hertz

<sup>G</sup> <sup>G</sup> , (A-1)

θ ϕ

2

 τ

= = , (A-6)

<sup>−</sup> = + , (A-8)

= , (A-5)

ϕ

, (A-3)

, (A-4)

 ϕ

> τ

JJGG G G , (A-2)

θ

*r r vt*

2 2 2

τ

⎡ ∂ ∂ ⎤

ω

θ

2 2

<sup>−</sup> = +− , (A-9)

*r v r vt t*

= ++ <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ <sup>∂</sup> <sup>∂</sup> ⎥⎦

τ

<sup>⎡</sup> <sup>∂</sup> <sup>⎤</sup> <sup>=</sup> <sup>+</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>∂</sup> <sup>⎦</sup>

**6.1 Appendix A: Analytical study of the magnetic dipole radiation in time-domain** 

system functioning.

Fig. 3 (see section 3) is written as:

**6. Appendix** 

are expressed as:

τ

EM-wave data with a transient excitation pulse current. A methodological analysis was made by taking into account a complex waveform of the transient pulse signal exciting the radiation source structure considered. It was explained how the frequency-bandwidth of the frequency-dependent baseband EM-field must be chosen according to the excitation current considered.

In order to demonstrate the relevance of the method investigated, it was first, implemented into Matlab program and then, applied to the determination of the H-field radiated by a microwave circuit in UWB. As consequence, the feasibility of the method was verified with two types of structures. First, with the semi-analytical calculation implemented in Matlab by considering the frequency- and time-dependent expressions of the magnetic NF radiated by a set of magnetic dipoles, an excellent agreement with the results from the calculation method developed were found. Then, further more practical analysis was performed with the determination of transient H-NF from the frequency-dependent data computed with a standard commercial 3-D EM-tool. For this second test, the H-NF emitted by a low-pass planar microstrip filter was treated. For both cases, the excitation current injected to the structures was assumed as an ultra-short transient pulse having half-bandwidth lower than 5 ns which presents a baseband frequency spectrum with bandwidth of about 2.5 GHz from DC. With the examples of complex structures tested, very good agreement between the transient H-field component maps and profiles was realized from the method proposed and those directly calculated from the well-known standard tools and from classical mathematical EM-formulae.

It is interesting to point up that the NF computation method introduced in this chapter is advantageous in terms of:


However, its main drawback is the limitation in term of time step which depends on the frequency range of the initial frequency-data considered and also the necessity of powerful computer for the achievement of high accurate results.

In the next step of this work, we plane to extend this method to transpose in time-domain the modelling of EM-radiation with the optimized association of elementary dipoles (Vives-Gilabert et al. 2009, Fernández-López et al. 2009). Then, we are hopeful that the method developed in this chapter is very helpful for EMC/EMI investigations of modern electrical/electronic systems as the case of hybrid vehicle embedded circuits (Vye 2011, Tröscher 2011, Kopp 2011) where the transient NF effects are susceptible to disturb the system functioning.
