Calculations of near-field emissions in frequency-domain into time- dependent data with arbitrary wave form transient perturbations

This paper is devoted on the application of the computational method for calculating the transient electromagnetic (EM) near-field (NF) radiated by electronic structures from the frequency-dependent data for the arbitrary wave form perturbations i(t). The method proposed is based on the fast Fourier transform (FFT). The different steps illustrating the principle of the method is described. It is composed of three successive steps: the synchronization of the input excitation spectrum I(f) and the given frequency data H0(f), the convolution of the two inputs data and then, the determination of the time-domain emissions H(t). The feasibility of the method is verified with standard EM 3D simulations. In addition to this method, an extraction technique of the time-dependent z-transversal EM NF component Xz(t) from the frequency-dependent x- and y- longitudinal components Hx(f) and Hy(f) is also presented. This technique is based on the conjugation of the plane wave spectrum (PWS) transform and FFT. The feasibility of the method is verified with a set of dipole radiations. The method introduced in this paper is particularly useful for the investigation of time-domain emissions for EMC applications by considering transient EM interferences (EMIs).


Introduction
With t he unintentional e lectromagnetic interferences (EMIs), the design engineers needs to take into account the electromagnetic c ompatibility ( EMC) models during t he electronic s ystems manufacture p rocess [1][2][3]. The mo st disturbing E MC ef fects ca used b y t he el ectrical/electronic system i ntegration can b e due t o t he E M n ear-field ( NF) radiations a nd t he c ouplings b etween t he different circuits as t he el ectrical cab les and el ectronic eq uipments [4][5]. Therefore, NF emission models and scanning measurement techniques were proposed [6][7][8][9]. Nevertheless, large amount of t he NF investigation were performed in f requencydomain. H owever, the t ransient perturbations a re susceptible to degrade the mixed electronic s ystems as digital a nd r adio f requencies ( RFs) [ 10] an d i ntegrated systems [11]. It has been found that the EMC engineering should include the transient EM-NF emissions especially in time-domain [10][11][12][13][14][15][16][17][18][19][20]. Currently, this topic attracts many o f electronic engineers an d r esearchers. With t he i ncrease o f integration d ensity a nd t he o perating r ate, EM NF analysis is necessary for the RF/digital electronic boards [1][2][3]. Undesired transient effects can be created by different perturbations a s t he no n-linearity o f electronic devices during their c ommutations [ 10]. These transient E M-field emissions need to be canceled out for the reliability. For this reason, E M transient a nalysis is required. A s r eported i n [21], analog/mixed (AM) electronic designers use regularly software t ools s uch a s S PICE, while t hose working on RF/microwave engineering focus i n f requency-domain simulation tools based on the S-parameters. In practice, one needs t he fusion of both a pproaches a s AM engineers a re required to make further analysis on the critical components by using EM simulation tools. This constitutes an improvement t echnique in t he EMC area. I n t his o ptic, t he EM emission modeling by t he mixed co mponents becomes one o f t he cr ucial s teps b efore t he i mplementation. Therefore, the issues both in frequency-and time-domains should be forecasted.
Basically, the transient EM-field computation was initially determined with e lementary EM dipole radiations [22][23][24][25][26][27]. As r eported i n [7][8], an y el ectronic ci rcuit NF r adiations can b e r eproduced with arrays of e lementary d ipoles. Moreover, time-domain N F radiation was al so co nducted with excitation b ased o n t he ar bitrary wave f orm signals [28]. T his computational approach is advantageous for the modeling of E M N F r adiated by c omplex electrical/electronic s tructures which cannot be modeled with most of standard tools [29][30]. This computation method r emains co mplicated when c onsidering electronic devices operating w ith UWB a nd base band s ignal. So, more r ecently, E M c omputational method based on t he plane wave spectrum ( PWS) t heory was proposed [31][32][33]. This method is based on the exploitation of the fundamental plane wave's properties and FFT, then, transposed in timedomain. I t a llows simplifying considerably t he reconstruction of the EM NF radiations ( including the evanescent waves) as t he cal culation o f l ongitudinal component ( along z -axis) from t ransversal c omponents (along x -and y -axes) [31][32], the NF/NF t ransform [ 33] and al so ex traction o f t he el ectric N F co mponents ( E x , E y and E z ) from 2D data H x and H y [34]. In the continuation of these works, a s a special is sue of [ 35], the g eneralized methodology of a time-frequency E M N F computation is presented in this paper.
The pa per i s mainly di vided i n three s ections. S ection 2 is focused on the application of the routine algorithm of EM NF time-frequency method proposed in [28] [35]. Section 3 introduces a time-domain EM c omputation method based on the PWS transform for extracting EM NF component X z from X x and X y with a rbitrary excitations. S ection 4 is the conclusion.

Calculation method of time-dependent nearfield maps with transient perturbations from frequency-dependent data
This s ection d escribes th e time-frequency computation methodology presented in th is. T he basic theoretical approach an d the r outine a lgorithm are d etailed. In t his paper, w e will e xtend t he FFT a nd I FFT i nstructions to reconstitute th e t ime-dependent m agnetic NF m aps H x,y,z (t) radiated b y an el ectronic device f rom t he frequency components H x,y,z (f) for any excitation undesired currents or voltages for the EMC applications as proposed in [28][35].

Theoretical approach of the time-frequency computation method understudy
Let us c onsider th e ti me-dependent pl ot of t he a rbitrary signal x(t) presented in Figure 1. This signal is supposed as the ex citation o f t he electronic s tructure under consideration. As indicated in this figure, the sampled data corresponding t o t he signal under t est is s upposed a nd discretized f rom t he starting time t min to t he s top ti me t max with ti me step ∆t. It means t hat the number of t imedependent samples is equal to: with int(α) generates the lowest integer number greater than the real number α. By definition, we can determine mathematically the frequency-dependent s pectrum o f i(t) as a co mplex n umber denoted as: In t his e xpression, t he variable k f represents th e s ampling of t he frequency variable. T hese f requencies can b e ex tracted f rom t he sampling time parameters by the following expression: In order to operate with the excitation signal, the frequency spectrum magnitude n eeds t o b e n ormalized as a co mplex coefficient. F or th is r eason, 0 I is as sumed as a h armonic component sinusoidal c urrent n ecessary for g enerating t he electric or magnetic field spectrum ) ( 0 f H , the harmonics of the input cu rrent can b e n ormalized with t he following complex coefficients: This normalization is illustrated by spectrum representation shown in Figure 2   For the base band applications, it is interesting to note t hat the s tarting f requency f min must b e eq ual t o t he frequency step Δf. I n t his s cope, the s pectrum value can b e extrapolated linearly to generate the DC-component of the excitation signal. According to the signal processing theory, the D C-component of the ultra-short tr ansient s ignal is negligible at very low frequency band. So, the extrapolation operating will not change the calculation results. The upper frequency f max must b elong in the frequency bandwidth containing higher t han 95% s pectrum e nergy o f the excitation signal.
Once, the frequency spectrum coefficients are defined, the time-dependent N F d ata co rresponding t o t he t ransient current signal c an b e c arried o ut b y c onvoluting the frequency co efficients c k and t he frequency-domain f ield data. The r outine p rocess will b e p resented i n the ne xt subsection.

Computational process of the proposed method
The c omputation method developed in t his p aper can be summarized in t wo s teps. After ex tracting t he frequency spectrum co efficient k c from t he tr ansient e xcitation s ignal i(t) as explained in the previous subsection, we will focus on the co nvolution b etween f requency s pectrum co efficients and the frequency EM field data.
Let u s d enote To r econstitute th e ti me-domain r esults, t he i maginary p art of t he d ata ( ) t z y x H , , , 0 is n ot n ecessary. T herefore, t he desired ti me-domain r esults ar e o btained with t he expression: where t he f unction Real(α) r epresents t he r eal p art o f t he complex num ber α. T he r outine pr ocess of t he pr oposed computation method is presented in Figure 4 [35]. This work flow is pe rformed with di fferent ope rations i n or der t o provide th e ti me-domain E M N F r adiated b y t he d evice under test with the arbitrary transient excitation signal i(t).
To v alidate t he i nvestigated m ethod, a M atlab program h as been i mplemented acco rding t o t he routine algorithm described in Figure 4.

Illustration results
In t his s ubsection, a comparison b etween t he transient EMfield radiated by a concrete microstrip device described from the 3 D s oftware s imulation and t hose o btained f rom the proposed method is realized.

Description of the Assumed Excitation Signal
In o rder t o hi ghlight t he i nfluence o f t he f orm a nd t he transient variation of the disturbing currents in the electronic structure, t he co nsidered s hort-duration p ulse excitation current i(t) is assumed as a Gaussian signal modulating 1.25 GHz sine carrier, defined by the analytical formula: ,

Description of the device under test
The microstrip circular resonator shown i n F igure 6 was designed and considered as the d evice under test i n order to validate the method under i nvestigation. T he r esonator i s based on a substrate with relative permittivity 10 = ε . It is fed by the via hole port with the transient current presented above. The top view of the resonator is shown by Figure 6. To validate the method proposed in this paper, comparisons of different r esults were made b etween t he C ST M icrowave simulations and the computation method proposed.

Transient EM-Field Determined by CST MWS simulation
By c onsidering th e circular r esonator p resented i n Figure 6 , excited b y th e p ulse c urrent p lotted in F igure 5 y ields t he electric and magnetic field co mponents mappings depicted in Figure 7 and Figure 8 at the arbitrary time t 0 = 7.611 ns and in the horizontal plan parallel to (Oxy) referenced by z 0 = 2 mm. The dimensions of the mappings were set at L x = 56 m m and L y = 56 mm with resolutions respectively, equal to ∆x = 1 mm and ∆y = 1 mm.
x, mm y, mm

Computed Results from the Proposed Method
First, b y analyzing t he f requency-domain r esults a chieved by CST Microwave Studio, one obtains the cartographies of the frequency-dependent electric and magnetic field from f min = 1 GHz to f max = 1.5 GHz step ∆f = 0.01 GHz.
After the program execution of the algorithm indicated by the flow chart described by Figure 4, one gets the results shown in Figure 9 and Figure 10 via the combination of the frequencydependent d ata o f t he el ectric o r m agnetic f ield components associated to the frequency coefficient of the excitation signal. One can see that one establishes the cartographies having the same behaviors as those generated via the direct calculations displayed in Figure 7 and Figure 8. Furthermore, as illustrated by Figure 11 and Figure 12, a very good correlation be tween t he profiles along Ox ar Oy of t he EM field components detected in the vertical plane placed at x = 22 mm or y = 30.3 mm was observed. In addition t o t his c omputation method, we pr opose f urther method e nabling t o e xtract t he t hird c omponent ( along zdirection) of EM fields in time-domain knowing the t wo first components (along x-and y-directions) in the next section.

Extraction method of the transverse component X z (t) from X x (f) and X y (f) with ultra-short duration transient perturbations
To reduce the order of complexity and the processing time of measurement, we pr opose a method t o ex tract t he t ime EM transversal component X z (t) from t he known the longitudinal components X x (t) and X y (t). To do this, we use the Plane Wave Spectrum ( PWS) method associated with th e r adiation o f electric dipoles in the time domain as i ntroduced r ecently in [31][32].

The b asic ap proach o f E M f ield c haracterization in t ime domain i s ex tracted f rom t he f requency d ata co mbined via
FFT. First, the P WS t heory which was in itially introduced in [36][37][38] w ill be a pplied t o t he obt ained da ta i n frequency domain. F inally, t he f requency d ata will b e transposed in to time domain with IFFT.

Principle of the time-frequency method of the zcomponent calculation from x-/y-components of the EM NF
By d efinition, th e P WS m ethod [31][32][33][36-38] is a b asic method de dicated to th e d ecomposition o f a ny E M-field plotted in 2D as a sum of plane waves propagating in different space directions. One denotes: the wave v ector i n t he r ectangular co ordinate s ystem ( Oxyz) with unit vectors, x u , y u and z u . The modulus of this wave vector, what is also known as the wave number, is given by : where ) ( f λ is th e wavelength at t he operating f requency f. , , ( π . (11) Similar to the 2D Fourier transform, the inverse PWS (IPWS) of EM field is given by the following equation: The ho rizontal X-Y plane is with di mensions L x × L y . I t is discretized with the steps Δx and Δy, r espectively, so that the discrete indexes, n x and n y are: In this case, the horizontal co mponents o f t he wave vector k x and k y vary respectively between: and with the numerical step: From e quation (10), one ca n d etermine t he co rresponding vertical component [31][32][33][ [36][37][38]: To avoid the unexpected case, the following r elation must b e respected.
And also, at the boundary condition, the field components and their P WS c omponents, According to the plane wave properties, wave vectors k and P must be perpendicular to each other: Start X x (x,y,z 0 ,t) X y (x,y,z 0 ,t) x min <x<x max step ∆x y min <x<y max step ∆y Orthogonality property X.k=0 Figure 13: Routine algorithm illustrating the computation method of X z from X x and X y by using the PWS transform.
. F igure 1 1 s ummarizes t he r outine algorithm of the method proposed.
As a co nclusion, al l t he p rocedure ab ove means t hat z X P , obviously X z , ca n b e e xtracted f rom x X P and y X P which can be calculated from the IPWS equation expressed in (12) if the 2D da ta X x and X y are gi ven. In summary, with t he pr oposed method, the EM NF measurement processes can be simplified.

Application results
To validate the computation method proposed in this paper, a set of elementary electric dipoles with arbitrarily chosen configuration is placed in the X-Y plane as displayed in Figure  14. It is considered as a r adiating source defined by analytical equations pr oposed in [39][40][41]. All th e e lectric d ipoles a re simultaneously excited b y t he s ame time va rying c urrent I(t). Figure 15 displays th e c urrent e xcitation. The f requency spectrum of I(t) plotted in F igure 16 presents a maximum frequency of about 5 GHz. So, th e minimum wave le ngth is λ min = c/f max = 0. 6 m. According to the wave propagation theory, the NF zone is up to about λ min /10 = 6 cm above the dipoles plane.  First, w e w ill calculate the tr ansient e lectric fields with formulae expressed in [39][40]. T he results are shown in Figure 17, the plots plane is at the height z = 10 mm and at the time t 0 = 50 ns. The profile of the Electric field along the line equated b y x = 0 mm i s shown b y F igure 18. The t hree transient el ectric field co mponents ar e o btained at t he t hree different point in the plane z = 10 mm, shown as Figure 19.
Second, the vertical electric field component E z (t) extracted by the P WS m ethod w ill be co mpared w ith the own E z (t) calculated d irectly. T his co mparison i s s hown b y Figure 19. The c omparison of the E z distribution at t = 5 0 n s a cross t he horizontal plane at z = 10 mm, is shown in Figure 20. One can s ee a good ag reement in t he zone w ith hi gher f ield strengths with almost the same distribution. Figure 17: Calculated e lectric field c omponents E x , E y and E z and the to tal magnitude |E| for the d ipoles in Figure 14 at the horizontal plane at the height z = 10 mm at the instant time t = 50 ns.   We can also find some errors in t he zone with lower field strengths. H owever, t hese er rors ar e r elatively s mall. T hey can be visualised in Figure 20. We can see that the relative errors are very small at the randomly chosen points.
In order to verify the relevance of the proposed method, we also s imulate th e tr ansient r adiation o f th e s et o f d ipoles in Figure 14 with commercial 3D EM modelling software, CST Microwave S tudio™. Figure    : Simulated el ectric f ield co mponents E x , E y and E z and th e to tal magnitude |E| for t he di poles in Figure 14 at th e horizontal plane at the height z = 10 mm at the instant time t = 50 ns. Figure 23: Simulated electric field components E x , E y and E z for the dipoles in Figure 21 along the line z = 10 mm and x = 0 and at the instant time t = 50 ns.
Through these figures, we can see that the simulations and calculation r esults present a v ery g ood co rrelation. T he almost same field distributions are found. However, we can find some d ifferences in the magnitude, when we compare the simulation and calculation results. These differences can be co nsidered as t he p roblem o f t he mesh s izes of t he considered EM simulator.

Conclusions
The methodology of time-frequency EM NF computation is successfully de veloped i n t his pa per. The method pr oposed consists mainly i n c onvoluting t he E M N F o btained f rom frequency calculation, simulations or measurements in wide frequency b and a nd a ny t ransient a rbitrary wave f orm perturbations.
In the first part of the paper, theoretical approach illustrating the r outine a lgorithm o f t he method i s e stablished. Then, application by comparing simulations of a microwave device with a standard co mmercial t ool an d s emi-analytical calculations r un i n M atlab p rogramming e nvironment was made t o ve rify t he va lidity of t he method. T o do t his, a transient current with pulse wave f orm presenting s ome n s time-duration was co nsidered. A s ex cepted a g ood correlation with r esults f rom a nalytical c alculation w as found.
In t he s econd pa rt of the pa per, a t ransposition of a frequency method based on the PWS spectrum is presented.
The f low ch art summarizing the co mputation o f E M wave component X z from X x and X y in 2 D i s e xplained. Then, application with the radiation of set of EM dipoles is presented. O nce ag ain, as e xpected a v ery good co rrelation with ti me-domain r esults f rom a 3 D E M s imulation commercial tool is performed.
The approach introduced in this paper can be very useful for time domain EM near field modeling and characterization in EMC a pplication. The m ethod es tablished i s cu rrently extended for the modeling of EM NF emissions based on the set of elementary dipoles [41] based on the frequency models developed in [7][8].