**2. NCBC function**

Any mathematical function capable to approximate the impulse lightning channel-base current in electromagnetic field calculations can be also used to represent far lightning electromagnetic field as an external excitation inducing currents and voltages in objects and structures inside such a field. Such functions are necessary in lightning stroke modeling.

There is an overview of lightning return-stroke models in (Rakov & Uman, 1998, 2006) where they are classified into four classes according to the type of governing equations. An adequate return-stroke model would be the one that provides simultaneously an approximation to the experimentally measured channel-base current, to the lightning electric and magnetic field waveshapes and intensities at various distances, and to the observed return-stroke speeds. Functions implied in these models are based either on the double-exponential (DEXP) function (Bewley, 1929; Bruce & Golde, 1941) or Heidler's function (Heidler, 1985). DEXP first order derivative at *t*=0+ is too large, thus causing numerical problems in LEMF calculations. It also has physically non-realistic convex waveshape in the rising part. Heidler's function reproduces concave rising part and its derivative is equal to zero at *t*=0+. It is also used for representing lightning currents in the International standard IEC 62305 (IEC/TC 81, 2006). In order to obtain better agreement with experimental results (Berger et al., 1975) one function was proposed (Nucci et al., 1990) as a linear combination of the DEXP and Heidler's function. Another pulse function was also proposed in (Feizhou & Shange, 2004). All these functions need peak correction factors, and their parameters cannot be easily chosen according to the desired waveshape. However, NCBC function parameters in the rising part can be chosen independently from parameters in the decaying part. The exact rising time to the peak value can be chosen in advance, and the decaying time to half of the peak value can be selected in the approximation procedure. The maximum value of the function can be chosen without the peak correction factor. Maximum current steepness can be adjusted (Javor & Rancic, 2011) using analytical expression for the first derivative. Experimentally measured impulse charge and specific energy values (Berger et al., 1975; Anderson & Eriksson, 1980) can be achieved using analytically obtained integral of NCBC function and integral of the square of the function (Javor, 2011a).

NCBC function (Fig. 1) is given with the following expression:

58 Fourier Transform Applications

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

Any mathematical function capable to approximate the impulse lightning channel-base current in electromagnetic field calculations can be also used to represent far lightning electromagnetic field as an external excitation inducing currents and voltages in objects and structures inside such a field. Such functions are necessary in lightning stroke

There is an overview of lightning return-stroke models in (Rakov & Uman, 1998, 2006) where they are classified into four classes according to the type of governing equations. An adequate return-stroke model would be the one that provides simultaneously an approximation to the experimentally measured channel-base current, to the lightning electric and magnetic field waveshapes and intensities at various distances, and to the observed return-stroke speeds. Functions implied in these models are based either on the double-exponential (DEXP) function (Bewley, 1929; Bruce & Golde, 1941) or Heidler's function (Heidler, 1985). DEXP first order derivative at *t*=0+ is too large, thus causing numerical problems in LEMF calculations. It also has physically non-realistic convex waveshape in the rising part. Heidler's function reproduces concave rising part and its derivative is equal to zero at *t*=0+. It is also used for representing lightning currents in the International standard IEC 62305 (IEC/TC 81, 2006). In order to obtain better agreement with experimental results (Berger et al., 1975) one function was proposed (Nucci et al., 1990) as a linear combination of the DEXP and Heidler's function. Another pulse function was also proposed in (Feizhou & Shange, 2004). All these functions need peak correction factors, and their parameters cannot be easily chosen according to the desired waveshape. However, NCBC function parameters in the rising part can be chosen independently from parameters in the decaying part. The exact rising time to the peak value can be chosen in advance, and the decaying time to half of the peak value can be selected in the approximation procedure. The maximum value of the function can be chosen without the peak correction factor. Maximum current steepness can be adjusted (Javor & Rancic, 2011) using analytical expression for the first derivative. Experimentally measured impulse charge and specific energy values (Berger et al., 1975; Anderson & Eriksson, 1980) can be achieved using analytically obtained integral of NCBC function and integral of the square of the function

these results, some conclusions are given in Section 5.

NCBC function (Fig. 1) is given with the following expression:

**2. NCBC function** 

modeling.

(Javor, 2011a).

$$\mathbf{i}(t) = \begin{cases} I\_m \left( t \;/\; t\_m \right)^a \exp\left[ a \left( 1 - t \;/\; t\_m \right) \right], & 0 \le t \le t\_m, \\ I\_m \sum\_{i=1}^n c\_i \left( t \;/\; t\_m \right)^{b\_i} \exp\left[ b\_i \left( 1 - t \;/\; t\_m \right) \right], & t\_m \le t \le \infty, \end{cases} \tag{1}$$

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

Fig. 1. Normalized NCBC function

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, 1941). DEXP function *i*(*t*)=*Im*[exp(-α*t*)-exp(-β*t*)] for *Im*=11kA, *tm*=0.5826μs, α=3. 104s-1, and β=107s-1, has the decreasing time to half of the peak value of approximately 23μs. It can be

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 61

function. NCBC function is presented in Fig. 4 for different values of parameter *b*, for *tm*=1.9μs, but in a longer time period (300μs), as parameter *b* determines the decaying part of the function. If the rising time to the maximum value *tm* has values e.g. 0.5, 1, 2, 5, or 10μs, and other parameters are *a*=1.5 and *b*=0.01, the changes in the function waveshape are

Fig. 3. Normalized NCBC function in the first 1μs, for *b* = 0.03, *tm*=0.5826μs, and different

Fig. 4. Normalized CBC function for *a* = 1.5, *tm* =1.9μs, and different values of *b* as parameter

presented in Fig. 5.

values of *a* as parameter

approximated with NCBC function for *n*=1 and *Im*=11kA, *tm*=0.5826μs, *a*=0.5 and *b*=0.02, having also the decreasing time to half of the peak value of about 23μs. If using lightning stroke models and electromagnetic theory relations, lightning electric and magnetic field components above perfectly conducting ground in general have three terms, depending on the integral of the channel-base current function, on the function itself, and on the function derivative. Consequently, the DEXP function having just a convex rising part and great values of the first derivative at *t*=0+ makes numerical problems in LEMF calculations.

Fig. 2. The rising part of the normalized CBC function representing current *tc*/*ti*=1.2/50μs

NCBC function, for *a* > 1, satisfies the demand of having the first derivative equal to zero at *t*=0+ and the concave to convex rising part. The value of parameter *a* has to be chosen greater than 1 in order to obtain one saddle point in the rising part (Javor & Rancic, 2006). Heidler's function also has the first derivative equal to zero at *t*=0+, but it needs peak correction factor, and its rise-time to the maximum current value cannot be chosen in advance. Heidler's function doesn't have analytical integral and its Fourier transform cannot be obtained analytically, but just numerically (Heidler & Cvetic, 2002).

NCBC function can approximate also other channel-base currents used in lightning returnstroke modeling, as the one proposed in (Nucci et al., 1990). This function is used in many papers and approximates well experimental results for subsequent negative strokes. In order to represent the same current two terms are needed in the decaying part of NCBC function (*n*=2) and its parameters are calculated as *Im*=11kA, *tm*=0.472μs, *a*=1.1, *b*1=0.16, *c*1=0.34, *b*2=0.0047, and *c*2=0.66.

#### **2.1 Parameters of NCBC function**

The normalized NCBC function for different values of parameter *a*, in the first 1μs is presented in Fig. 3, for *tm*=0.5826μs. The parameter *a* determines the rising part of the 60 Fourier Transform Applications

approximated with NCBC function for *n*=1 and *Im*=11kA, *tm*=0.5826μs, *a*=0.5 and *b*=0.02, having also the decreasing time to half of the peak value of about 23μs. If using lightning stroke models and electromagnetic theory relations, lightning electric and magnetic field components above perfectly conducting ground in general have three terms, depending on the integral of the channel-base current function, on the function itself, and on the function derivative. Consequently, the DEXP function having just a convex rising part and great

values of the first derivative at *t*=0+ makes numerical problems in LEMF calculations.

Fig. 2. The rising part of the normalized CBC function representing current *tc*/*ti*=1.2/50μs

analytically, but just numerically (Heidler & Cvetic, 2002).

*c*1=0.34, *b*2=0.0047, and *c*2=0.66.

**2.1 Parameters of NCBC function** 

NCBC function, for *a* > 1, satisfies the demand of having the first derivative equal to zero at *t*=0+ and the concave to convex rising part. The value of parameter *a* has to be chosen greater than 1 in order to obtain one saddle point in the rising part (Javor & Rancic, 2006). Heidler's function also has the first derivative equal to zero at *t*=0+, but it needs peak correction factor, and its rise-time to the maximum current value cannot be chosen in advance. Heidler's function doesn't have analytical integral and its Fourier transform cannot be obtained

NCBC function can approximate also other channel-base currents used in lightning returnstroke modeling, as the one proposed in (Nucci et al., 1990). This function is used in many papers and approximates well experimental results for subsequent negative strokes. In order to represent the same current two terms are needed in the decaying part of NCBC function (*n*=2) and its parameters are calculated as *Im*=11kA, *tm*=0.472μs, *a*=1.1, *b*1=0.16,

The normalized NCBC function for different values of parameter *a*, in the first 1μs is presented in Fig. 3, for *tm*=0.5826μs. The parameter *a* determines the rising part of the function. NCBC function is presented in Fig. 4 for different values of parameter *b*, for *tm*=1.9μs, but in a longer time period (300μs), as parameter *b* determines the decaying part of the function. If the rising time to the maximum value *tm* has values e.g. 0.5, 1, 2, 5, or 10μs, and other parameters are *a*=1.5 and *b*=0.01, the changes in the function waveshape are presented in Fig. 5.

Fig. 3. Normalized NCBC function in the first 1μs, for *b* = 0.03, *tm*=0.5826μs, and different values of *a* as parameter

Fig. 4. Normalized CBC function for *a* = 1.5, *tm* =1.9μs, and different values of *b* as parameter

Fourier Transform Application in the Computation of Lightning Electromagnetic Field 63

Impulse charge is defined in IEC 62305 standard (IEC/TC 81, 2006) as the integral of the channel-base current function. For standard lightning currents impulse charge is calculated

Whether a function is given analytically, graphically or numerically, its Fourier transform can be obtained analytically, numerically or using some of the commercial programs. Faster or slower rising/decaying of the function and its waveshape at the end of the covered impulse

For NCBC function Fourier and Laplace transforms are calculated analytically. It should be noted that we obtain Fourier transform for any Fourier-transformable function which is zero

duration time determine the number *N* of needed points for Fast Fourier Transform.

The analytical expression for the unilateral Laplace transform of NCBC function is:

( ) ( ) 1 1

*m i i m <sup>a</sup> c b Is I t a a st b b st a st b st* + + <sup>=</sup>

exp( ) exp( ) ( ) ( 1, ) ( 1, ) *<sup>i</sup> <sup>n</sup> i i m m a b <sup>m</sup> i im*

*x*

 *a x t tt* ∞

<sup>=</sup> <sup>⎢</sup> γ+ + + Γ+ + <sup>⎥</sup> <sup>⎢</sup> + + <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

1

∑ (4)

+= − ∫ the incomplete Gamma functions,

⎡ ⎤

for *t*<0 if we substitute variable *s* with j2π*f* in its Laplace transform.

*<sup>a</sup>* γ+ = − *a x t tt* ∫ and ( 1, ) exp( )d *<sup>a</sup>*

Γ

Fourier transform of NCBC function is obtained from (4) for *s*= j2π*f*, so:

in (Javor, 2011a), so as specific energy as the integral of the square of NCBC function.

Fig. 6. The first derivative of CBC function in the first 1μs

**3. Fourier transform of NCBC function** 

for

0 ( 1, ) exp( )d *x*

as defined in (Abramowitz, Stegun, 1970).

Fig. 5. Normalized CBC function for *a* = 1.5, *b* = 0.01, and different values of *tm* as parameter

#### **2.2 Derivative of NCBC function**

NCBC function first order derivative is:

$$\frac{\det i(t)}{\mathbf{d}t} = \begin{cases} aI\_m \left(t \;/\; t\_m\right)^{a-1} \left(1 - t \;/\; t\_m\right) \exp\left[a(1 - t \;/\; t\_m)\right], & 0 \le t \le t\_{m'}\\ I\_m \sum\_{i=1}^n c\_i b\_i \left(t \;/\; t\_m\right)^{b\_i - 1} \left(1 - t \;/\; t\_m\right) \exp\left[b\_i \left(1 - t \;/\; t\_m\right)\right], & t\_m \le t \le m, \end{cases} \tag{2}$$

The first derivative of CBC function (NCBC for *n*=1) is presented in Fig. 6 up to 1μs, for *b* = 0.03 (but note that parameter *b* is irrelevant for the rising part), and different values of *a*, *tm* and *Im*.

#### **2.3 Integral of NCBC function**

Integral of NCBC function is calculated as:

$$\begin{cases} I\_m t\_m a^{-(a+1)} \exp(a) \gamma(a+1, at \mid t\_m), & 0 \le t \le t\_m \\\\ I\_0 \mathbf{t}(t) \mathbf{d} \, t = \left\{ I\_m t\_m \begin{bmatrix} a^{-(a+1)} \exp(a) \gamma(a+1, a) + \\\\ \end{bmatrix} \right. \\\\ \sum\_{i=1}^n c\_i b\_i^{-(b\_i+1)} \exp(b\_i) \left[ \gamma(b\_i+1, b\_i t \nmid t\_m) - \gamma(b\_i+1, b\_i) \right] \right\}, & t\_m \le t \le m \end{cases} (3)$$

62 Fourier Transform Applications

Fig. 5. Normalized CBC function for *a* = 1.5, *b* = 0.01, and different values of *tm* as parameter

( )( ) [ ]

⎪ − − ≤ ≤ ∞

*aI t t t t a t t t t*

*m ii m m i mm*

The first derivative of CBC function (NCBC for *n*=1) is presented in Fig. 6 up to 1μs, for *b* = 0.03 (but note that parameter *b* is irrelevant for the rising part), and different values of

exp( ) ( 1, / ), 0

*m m m m*

*I t a a a at t t t*

<sup>⎧</sup> γ + ≤≤ <sup>⎪</sup>

*mm m m m*

<sup>⎩</sup> <sup>∑</sup> (2)

[ ]

(3)

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

*cb b b bt t b b t t*

<sup>⎪</sup> ⎫⎪ <sup>⎪</sup> <sup>γ</sup> + −γ + ≤ ≤∞ <sup>⎬</sup> <sup>⎪</sup>

*i i i i im i i m*

⎩ ⎪⎭

( )( ) [ ]

/ 1 / exp (1 / ) , 0 , d() d / 1 / exp (1 / ) , , *<sup>i</sup>*

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

*t I cb t t t t b t t t t*

1

−

1

*a*

−

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

( 1)

*a*

− +

( 1)

<sup>⎪</sup> <sup>⎪</sup> =⎨ <sup>⎨</sup> γ+ +

( )d exp( ) ( 1, )

*a*

− +

*it t It a a a a*

( 1)

− +

1

*i*

=

∑

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

*m m*

<sup>⎪</sup> <sup>⎪</sup> <sup>⎪</sup> <sup>⎩</sup>

**2.2 Derivative of NCBC function** 

*i t*

**2.3 Integral of NCBC function** 

0

∫

*t*

*a*, *tm* and *Im*.

NCBC function first order derivative is:

1

*i*

Integral of NCBC function is calculated as:

⎪ <sup>⎪</sup> <sup>⎧</sup> <sup>⎪</sup>

=

Impulse charge is defined in IEC 62305 standard (IEC/TC 81, 2006) as the integral of the channel-base current function. For standard lightning currents impulse charge is calculated in (Javor, 2011a), so as specific energy as the integral of the square of NCBC function.

Fig. 6. The first derivative of CBC function in the first 1μs
