**1. Introduction**

30 Will-be-set-by-IN-TECH

56 Fourier Transform Applications

Gómez-Díaz, J. S., Gupta, S., Álvarez-Melcón, A. & Caloz, C. (2009c). Tunable talbot imaging

Gómez-Díaz, J. S., Gupta, S., Álvarez-Melcón, A. & Caloz, C. (2010). Effcient time-domain

*IEEE Transactions on Microwave Theory and Techniques* 57(12): 4010–4014. Gupta, S., Gómez-Díaz, J. S. & Caloz, C. (2009). Frequency resolved electrical gating

Horii, Y., Caloz, C. & Itoh, T. (2005). Super-compact multilayered left-handed transmission

Marques, R., Martín, F. & Sorolla, M. (eds) (2008). *Metamaterials with Negative Parameters:*

Oliner, A. A. & Jackson, D. R. (2007). Leaky-wave antennas, *in* J. L. Volakis (ed.), *Antenna*

Paul, C. R. (2007). *Analysis of Multiconductor Transmission Lines*, 2*nd* edition edn, Wiley-IEEE

Pipes, L. A. & Harvill, L. R. (1971). *Applied Mathematics for Engineers and Physicist*, 3rd edn,

Saleh, B. E. A. & Teich, M. C. (2007). *Fundamentals of Photonics*, 2nd edition edn,

Trebino, R. (ed.) (2002). *Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser*

Talbot, H. F. (1836). Facts relating to optical science no. IV, *Philos. Mag.* 9: 401–407.

*Theory, Design and Microwave Applications*, Wiley, Hoboken, NJ.

*Engineering Handbook*, 4 edn, McGraw-Hill, New York.

Pozar, D. (2005). *Microwave Engineering*, 3rd edn, John Wiley and Sons.

Oppenheim, A. V. (1996). *Signals and Systems*, Prentice Hall.

*of Applied Physics* 106: 084908–9.

*Microwave Conference*, Rome, Italy.

53(4).

Press.

McGraw Hill.

Wiley-Interscience.

*Pulses*, Springer.

distance using an array of beam-steered metamaterial leaky-wave antennas, *Journal*

análisis of highly-dispersive linear and non-linear metamaterial waveguide and antenna structures, *IET Microwaves, Antennas and Propagation* 4(10): 1617–1625. Gupta, S., Abielmona, S. & Caloz, C. (2009). Microwave analog real-time spectrum analyzer

(rtsa) based on the spatial-spectral decomposition property of leaky-wave structures,

principle for UWB signal characterization using leaky-wave structures, *39th European*

line and diplexer application, *IEEE Transactions on Microwave Theory and Techniques*

Atmospheric discharge is one of the most interesting and powerful natural phenomenon hiding from men its undiscovered features and secrets for centuries. Lightning discharges have been studied in many theoretical and experimental ways. However, application of Fourier transform has been introduced in this research only in recent few decades. It proved very useful in solving many lightning research problems.

There are two main groups of problems in lightning studies that involve using Fourier transform. One deals with determining how the energy is distributed over a continuous frequency spectrum for the quantity of interest. Channel-base currents, induced voltages and currents, electric and magnetic field components, so as integrals and derivatives of the same functions, distribute components over the entire frequency range in different ways. These are non-periodic functions scattering their energy throughout the frequency spectra.

Another important group of problems to be solved by Fourier transform deals with the calculation of lightning induced effects at different distances from the lightning discharge and risk assessment for buildings, various structures, people and property in an external impulse electromagnetic field. These calculations are based on experimental results for the measured lightning electromagnetic field (LEMF) and channel-base currents, which are used in different types of lightning stroke models including lossy ground effects. Lightning discharge channel is usually modeled by a thin vertical antenna at a lossy ground of known electrical parameters. Both ground and air are treated as linear, isotropic and homogeneous half-spaces. Even for such a simple approximation of the lightning channel - calculations can not be easily done in time domain, but transformation to frequency domain is used instead. Once the calculations are done in frequency domain, way back to time domain is made by Inverse Fourier transform applied to the obtained results.

In both groups of problems an impulse function which has the analytical derivative, integral and integral transformations is very useful. New functions proposed by the author for representing lightning currents are presented in this Chapter, Section 2. These can be also used in other high voltage technique calculations. The main problem for a user of the impulse functions already given in literature to approximate some quantity is the choice of parameters so to obtain desired waveshape characteristics or values adequate to

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 59

( ) [ ]

where *a* and *bi* are parameters, *ci* coefficients, *n* the chosen number of expressions in the decaying part, so that the total sum of *n* weighting coefficients *ci* is equal to unit, and *tm* is the rise-time to the maximum current value *Im*. For *n*=1, *c*1=1 and *b*1=*b*, NCBC function reduces to CBC function (Javor & Rancic, 2006) with four parameters (*Im*, *tm*, *a* and *b*). In the special case, for *n*=1, *a*=4 and *b*=0.0312596735, CBC function reduces to High-Voltage Pulse (HVP) function 1.2/50μs (Velickovic & Aleksic, 1986). Impulse duration time is defined as *ti*=*tk*-*ta'*, for *tk* the time in which the current decreased to half of its peak value (Fig. 1). The rising part of the function and its front rise-time are given in Fig. 2. The front rise-time is defined as *tc*=*tb'*-*tc'*, for *tb'* and *tc'* determined as the time values corresponding to the points B' and A', obtained from intersecting horizontal lines for the maximum *Im* and the zero function value (time axis) with the line drawn through the points A, for *i*(*t*)=0.3*Im* (Fig. 2) or for

NCBC function is an analytically prolonged mathematical function (but still continuous, so as its first derivative, whereas higher order derivatives are not continuous at the point of function maximum *Im*), the parameter *a* in the rising part can be chosen to approximate the front of the waveshape independently from parameters *bi* and weighting coefficients *ci* in the decaying part, which facilitates the approximation procedure. NCBC function belongs to C1 differentiability class. Parameters of NCBC function can be chosen so that it represents waveshape of the often used DEXP function with parameters given in (Bruce & Golde,

β=107s-1, has the decreasing time to half of the peak value of approximately 23μs. It can be

104s-1, and

1941). DEXP function *i*(*t*)=*Im*[exp(-α*t*)-exp(-β*t*)] for *Im*=11kA, *tm*=0.5826μs, α=3.

*I ctt b tt t t*

⎪ − ≤ ≤∞

*mi m i m m*

<sup>⎧</sup> − ≤ <sup>≤</sup> <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup>

( ) / exp (1 / ) , , *<sup>i</sup>*

/ exp (1 / ) , 0 ,

<sup>⎩</sup> <sup>∑</sup> (1)

*m m m m*

*I tt a tt t t*

( ) [ ]

*a*

*<sup>n</sup> <sup>b</sup>*

1

*i*

*i*(*t*)=0.1*Im* in some other definitions, and B, for *i*(*t*)=0.9*Im*.

Fig. 1. Normalized NCBC function

=

*i t*

experimentally measured. Parameters of the new channel-base current (NCBC) function (Javor, 2008; Javor & Rancic, 2011) are calculated according to IEC 62305 standard lightning currents (International Electrotechnical Commission, Technical Committee 81 [IEC, TC 81], 2006), and the procedure to choose function parameters is also explained in (Javor & Rancic, 2011). Functions to represent other typical lightning currents are proposed in (Javor, 2011b, 2011c), such as long stroke current (LSC), and two-rise function (TRF) as a multi-peaked current. These functions can be used to obtain the desired peak value, rise-time, decaying time to half of the peak value, current steepness, integral of the function (representing also impulse charge), or integral of the square of the function (representing also specific energy), etc. Fourier transform is obtained analytically and the results are presented in Section 3. Application of the Fourier transform in LEMF computation is shown in Section 4. Based on these results, some conclusions are given in Section 5.
