**6. Appendix**

20 Fourier Transform Applications

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

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

It is interesting to point up that the NF computation method introduced in this chapter is

1. Simplicity of the EM-field maps determination for any waveform of transient excitation even with ultra-short duration which is very hard to simulate with most of commercial simulation tools. The method developed can be used for the determination of the NF maps in time-domain which is practically very difficult to measure in the realistic

2. It is flexible for various types of excitation signals which can be expressed analytically and also from the realistic use case of disturbing signal generally met in EMC area

3. It can be adapted also to different forms of electrical and electronic structures for lowand high-frequency applications. Globally speaking, it offers a possibility to work in

4. One can achieve significant EM-field measurement in very short time-duration with

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

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,

considered.

mathematical EM-formulae.

advantageous in terms of:

(Wiles 2003, Liu, K. 2011, Hubing 2011).

UWB from DC to unlimited upper frequency limit.

base band measured data in wide bandwidth.

computer for the achievement of high accurate results.

contexts.

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.

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

By definition, the magnetic dipole moment of the elementary circular loop shown earlier in Fig. 3 (see section 3) is written as:

$$
\vec{p}\_M(r,t) = p(t) \cdot \delta(r) \cdot \vec{u}\_{z,t} \tag{A-1}
$$

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

$$\vec{H} = \mathcal{H}\_r(\mathbf{r}, \theta, \phi, \mathbf{t})\vec{\boldsymbol{\mu}}\_r + \mathcal{H}\_\theta(\mathbf{r}, \theta, \phi, \mathbf{t})\vec{\boldsymbol{\mu}}\_\theta + \mathcal{H}\_\phi(\mathbf{r}, \theta, \phi, \mathbf{t})\vec{\boldsymbol{\mu}}\_\theta\,\tag{A-2}$$

are expressed as:

$$H\_r(r, \theta, \varphi, t) = \frac{\cos(\theta)}{2\pi r^2} \left[ \frac{p(\tau)}{r} + \frac{1}{v} \frac{\partial p(\tau)}{\partial t} \right] \tag{A-3}$$

$$H\_{\partial}(r,\theta,\varphi,t) = \frac{\sin(\theta)}{4\pi r^2} \left[ \frac{p(\tau)}{r} + \frac{1}{v} \frac{\partial p(\tau)}{\partial t} + \frac{r}{v^2} \frac{\partial^2 p(\tau)}{\partial t^2} \right] \tag{A-4}$$

$$H\_{\varphi}(r,\theta,\varphi,t) = 0\,,\tag{A-5}$$

where *v* is the wave-velocity, and *τ* is the time delayed variable which is defined as τ = −*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 harmonic current with amplitude *IM* denoted:

<sup>2</sup> ( ) *j ft <sup>j</sup> <sup>t</sup> If I e I e M M* − − π ω= = , (A-6)

are written as (Balanis 2005):

( ) 2 <sup>3</sup> ( , , , ) 1 cos( ) <sup>2</sup> *M jkr r I a Hr f jkr e r* θ ϕ θ<sup>−</sup> = + , (A-8)

$$H\_{\theta}(r,\theta,\varphi,f) = \frac{I\_M a^2 \sin(\theta)}{4r^3} (1 + jkr - k^2 r^2) e^{-jkr},\tag{A-9}$$

Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 23

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$$H\_{\varphi}(r,\theta,\varphi\rho,f) = 0 \,,\tag{A-10}$$

where *j* is the complex number −1 and the real *k fv* = 2 / π expresses the wave number at the considered frequency *f*. Then, through the classical relationship between the spherical and cartesian coordinate systems, one can determine easily the expressions of the components *Hx*, *Hy* and *Hz*.

#### **6.2 Appendix B: Spectrum analysis of bi-exponential signal**

A bi-exponential form signal with parameters *τ*1 and *τ*2 is analytically expressed as:

$$i(t) = I\_M \left( e^{-t/\tau\_1} - e^{-t/\tau\_2} \right) \,. \tag{\text{B-1}}$$

The analytical Fourier transform expression of this current is written as:

$$\underline{I}(oo) = I\_M(\frac{\tau\_1}{1 + jo\tau\_1} - \frac{\tau\_2}{1 + jo\tau\_2}) ,\tag{\text{B-2}}$$

with *ω* is the angular frequency. This yields the signal frequency spectrum formulation expressed as:

$$\mathbb{E}\left|\underline{I}(\boldsymbol{\alpha})\right| = I\_M \frac{\left|\boldsymbol{\tau}\_1 - \boldsymbol{\tau}\_2\right|}{\sqrt{(1 + \boldsymbol{\tau}\_1^2 \boldsymbol{\alpha}^2)(1 + \boldsymbol{\tau}\_2^2 \boldsymbol{\alpha}^2)}}.\tag{\mathbb{B}-3}$$

To achieve at least 95-% of excitation signal spectrum energy, the frequency-data should be recorded in baseband frequency range with angular frequency bandwidth equal to:

$$\rho\_{\text{g\\_\%}} = \frac{\sqrt{\sqrt{(\tau\_1^2 - \tau\_2^2)^2 + 1600\tau\_1^2 \tau\_2^2} - \tau\_1^2 - \tau\_2^2}}{\sqrt{2}\tau\_1\tau\_2}. \tag{B-4}$$

#### **7. Acknowledgment**

Acknowledgement is made to EU (European Union) and Upper Normandy region for the support of these researches. These works have been implemented within the frame of the "Time Domain Electromagnetic Characterisation and Simulation for EMC" (TECS) project No 4081 which is part-funded by the Upper Normandy Region and the ERDF via the Franco-British Interreg IVA programme.

#### **8. References**


http://www.agilent.com/find/eesof-emds

22 Fourier Transform Applications

the considered frequency *f*. Then, through the classical relationship between the spherical and cartesian coordinate systems, one can determine easily the expressions of the

> / / 1 2 () ( ) *t t it I e e <sup>M</sup>* − − τ

() ( ) 1 1 *I IM j j* τ

with *ω* is the angular frequency. This yields the signal frequency spectrum formulation

τ ω

To achieve at least 95-% of excitation signal spectrum energy, the frequency-data should be

<sup>−</sup> <sup>=</sup>

recorded in baseband frequency range with angular frequency bandwidth equal to:

τ

( ) 1600

<sup>=</sup> <sup>−</sup> + +

ωτ

π

 τ

1 2

1 2 22 22 1 2

τ τ

(1 )(1 )

2 22 22 2 2 1 2 12 1 2

 ττ

1 2

2

Acknowledgement is made to EU (European Union) and Upper Normandy region for the support of these researches. These works have been implemented within the frame of the "Time Domain Electromagnetic Characterisation and Simulation for EMC" (TECS) project No 4081 which is part-funded by the Upper Normandy Region and the ERDF via the

Adada, M. (2007). High-Frequency Simulation Technologies-Focused on Specific High-

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τ τ

+ +

 τω

> τ τ

− + −− <sup>=</sup> . (B-4)

 τ

> ωτ

= , (A-10)

= − . (B-1)

expresses the wave number at

, (B-2)

. (B-3)

ϕ θ ϕ

A bi-exponential form signal with parameters *τ*1 and *τ*2 is analytically expressed as:

(, , , ) 0 *Hr f*

components *Hx*, *Hy* and *Hz*.

expressed as:

**7. Acknowledgment** 

**8. References** 

17

Franco-British Interreg IVA programme.

[Online]. Available from:

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where *j* is the complex number −1 and the real *k fv* = 2 /

**6.2 Appendix B: Spectrum analysis of bi-exponential signal** 

1 2

95%

ω

The analytical Fourier transform expression of this current is written as:

ω

( )

ω

*I IM*

τ


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**0**

**2**

<sup>1</sup>*Spain* <sup>2</sup>*Canada*

**Impulse-Regime Analysis of Novel**

J. Sebastian Gomez-Diaz1, Alejandro Alvarez-Melcon1,

Shulabh Gupta2 and Christophe Caloz2

<sup>1</sup>*Universidad Politécnica de Cartagena* <sup>2</sup>*École Polytechnique de Montréal*

**Optically-Inspired Phenomena at Microwaves**

The ever increasing of needs for high data-rate wireless system is currently producing a shift from narrow-band radio towards ultra-wideband (UWB) radio operation [see Ghavami et al. (2007)]. Novel microwave tools, concepts, phenomena and direct applications must be developed to meet this demand. While the past decades have been focused on the "magnitude engineering" and filter design [see Pozar (2005)], there is a renewed interest in the "dispersion engineering". In the dispersion engineering approach, the phase is engineered to met various

In this context, the development of electromagnetic metamaterials over the last years [see Caloz & Itoh (2006) or Marques et al. (2008)], with their intrinsically dispersive nature and subsequent impulse-regime properties, may provide novel and original solutions (see Fig. 1). Metamaterials can easily be synthesized in planar technology under the form of composite right/left-handed (CRLH) transmission lines (TLs), using non-resonant [see Caloz & Itoh (2006)] or resonant [see Duran-Sindreu et al. (2009)] approaches. These structures have provided novel and exciting applications, such as multi-band components, diplexers, couplers, phase-shifters, power-dividers or antennas with enhanced features, to mention just a few [see Caloz (2009) or Eleftheriades (2009) for a recent review]. However, CRLH structures have mostly been analyzed in the harmonic regime to date, and therefore only a few impulse-regime components and systems have been proposed so far. An example of these applications is the tunable pulse delay line presented in in Abielmona et al. (2007).

In this chapter, we present recent advances based on Fourier transformation techniques to model dispersive UWB phenomena and far-field radiation from complex CRLH structures. Section 2 first employs inverse Fourier transforms to study pulse propagation along this type of medium. Then, a Fourier transform approach is applied to the current which flows along the CRLH line, accurately retrieving the time-domain far-field radiation of the structure [which behaves as a leaky-wave antenna, (LWA)]. The main advantages of the proposed techniques are the easy treatment of complex CRLH structures, a deep insight into the physics of the phenomena, and an accurate and a fast computation, which avoids the time-consuming

analysis required by completely numerical simulations.

specifications within a given frequency range, so as to process signals in real time.

**1. Introduction**

Yang, T.; Bayram, Y. & Volakis, J. L. (2010). Hybrid Analysis of Electromagnetic Interference Effects on Microwave Active Circuits Within Cavity Enclosures. *IEEE Trans. EMC*, (Aug. 2010), Vol.52, No.3, pp. 745-748, ISSN 0018-9375
