THE EVALUATION of the electric field generated by a

Transcription

1 IEEE RANSACIONS ON ELECROMAGNEIC COMPAIBILIY, VOL. 49, NO. 3, AUGUS An Approximate ime-domain Formula for the Calculation of the Horizontal Electric Field from Lightning Célio Fonseca Barbosa and José Osvaldo Saldanha Paulino Abstract his paper presents an approximate formula to calculate the horizontal electric field from lightning, considering finitely conducting earth. he formula is represented by an analytical expression in the time domain, which is useful for the calculation of lightning-induced voltages on power and telecommunication lines, without the need of domain transformations. he paper also compares the results of the formula with the results obtained from frequency-domain techniques, namely the numerical calculation of Sommerfeld s integrals and the Cooray Rubinstein s formula. he comparison is favorable for a wide range of distances from the lightning channel and values of earth s conductivity. he horizontal electric field calculated by the formula is composed of two components of opposed polarities, one due to the return stroke charge and the other due to the return stroke current, resulting in an electric field with a bipolar wave shape. he charge component prevails in the region close to the lightning channel, while the current component prevails in the region far from it. Index erms Electromagnetic fields, finitely conducting earth, lightning, time-domain analysis. I. INRODUCION HE EVALUAION of the electric field generated by a lightning return stroke is necessary in order to compute the overvoltages induced in power and telecommunication lines. he vertical component of the electric field can be calculated analytically with relative facility considering the earth as a perfect conductor [1], and the resulting expression can be considered as representative of finitely conducting earth. However, the horizontal component of the electric field is strongly influenced by the earth s conductivity, which makes it difficult to assess. he exact solution for this problem has been formulated by Sommerfeld [2], but its implementation to the calculation of lightning fields leads to an inefficient procedure, as it requires the evaluation of Sommerfeld s integrals for several elementary dipoles at several frequencies. his calculation has been carried out by Zeddam and Degauque [3] for a set of representative parameters of lightning return strokes. For distances beyond a few kilometers from the lightning channel, the horizontal electric field can be evaluated by the wave tilt formula [4], [5]. Cooray [6] has shown that the surface impedance of the earth could be used to Manuscript received July 21, 2006; revised November 17, 2006 and March 11, C. F. Barbosa is with the Centro de Pesquisa e Desenvolvimento em elecommunicacoes (CPqD), Campinas, , Brazil ( grcelio@ cpqd.com.br). J. O. S. Paulino is with the Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil ( josvaldo@cpdee.ufmg.br). Digital Object Identifier /EMC compute the horizontal field at ground level, and that his formula could be applied for distances as short as 200 m. Rubinstein [7] complemented the work done by Cooray, and developed a formula to evaluate the horizontal electric field above the earth s surface, at close, intermediate, and long ranges. It is known as Cooray Rubinstein s formula, and is widely used nowadays to compute lightning-induced voltages in power and telecommunications lines. Cooray [8] proposed a modification to the original Cooray Rubinstein s formula that slightly enhances the negative peak of the horizontal electric field. More recently, Shoory et al. [9] developed a generalized equation from which the original Cooray Rubinstein s formula can be derived as a special case. From early Sommerfeld s work up to the Shoory s equation, most of the models have been derived in the frequency domain, which means that the results have to be converted to the time domain by means of transformations computed numerically. his inherent characteristic of the frequency-domain technique brings some drawbacks to the calculation process, such as the requirement of huge computational resources in terms of processing and data storage, as well as the introduction of errors due to the truncation of the infinite frequency spectrum of the lightning current, which is usually known as aliasing. he direct solution of Maxwell s equations in time domain can be carried out by numerical methods such as finite-difference time domain (FDD), as done by Baba and Rakov [10]. Although this approach does not require any approximate equation, it requires huge computational resources, and some errors may be introduced due to the segmentation of time and space. his paper presents the development of a time-domain formula for the calculation of the horizontal electric field generated by a lightning return stroke, so that the mentioned drawbacks of the frequency-domain approach and time-domain numerical methods can be avoided. In this development, the work carried out by Rusck [1] is revisited, and some of his time-domain equations are used. Section II describes the basic assumptions considered, and Section III develops the equation for the component of the horizontal electric field induced by the charge of the return stroke. Section IV develops the equation for the horizontal field induced at the earth s surface by a linearly rising magnetic field, while in Section V, this equation is used to compute the horizontal electric field induced at the earth s surface by the return stroke current. Section VI analyzes the total horizontal electric field generated by the return stroke at the line height, and Section VII compares the results of the time-domain formula that has been developed in the previous sections with the results from Sommerfeld s integrals carried out by Zeddam and /$ IEEE

2 594 IEEE RANSACIONS ON ELECROMAGNEIC COMPAIBILIY, VOL. 49, NO. 3, AUGUS 2007 Degauque [3] and with the results calculated by Rubinstein [7] based on the Cooray Rubinstein s formula. Section VIII analyzes the behavior of the developed formula for a region far away from the lightning channel, showing that it leads to a time domain equivalent of the wave tilt formula [5]. Section IX generalizes the application of the formula to an arbitrary current waveform, and Section X discusses the impact of the approximations considered in the development of the formula on its range of application. Finally, Section XI presents some conclusions. II. BASIC ASSUMPIONS CONSIDERED his paper considers a downward vertical electric field generated by a return stroke as positive. he horizontal electric field is radial, and its positive direction is considered as outward from the lightning channel. It is assumed that, at ground level, the vertical electric field and the horizontal magnetic field can be calculated to an acceptable accuracy using a perfect conducting earth model. his assumption has also been considered by the Cooray Rubinstein s formula, and some discussion on its range of validity can be found in [7]. he channel base current is considered as a trapezium, characterized by a peak value I 0 and a front time, as shown by (1), where t is the time from the initiation of the return stroke and u (x) is the Heaviside s function, which is 1 for x 0 and 0 otherwise. his waveform was selected because its mathematical simplicity facilitates the analytical manipulation, and an arbitrary waveform can be represented by the superposition of trapeziums. In (1), as well as in the other equations in this paper that are functions of time, it is implicit that they assume a zero value for negative values of t i = I 0 [t u (t )(t )]. (1) he return stroke is modeled by an upward traveling current with the waveform as described by (1). his model is known as transmission line (L) model, and has been proposed by Uman and McLain [11]. However, it shall be pointed out that the timedomain formula described in this paper could also be developed for other L models that consider a current decay with height, as the linear decay [modified transmission line model with linear current decay with height (MLL) model] proposed by Rakov and Dulzon [12] and the exponential decay [modified transmission line model with exponential current decay with height (MLE) model] proposed by Nucci [13]. he main reason for using the L model is that there are many published results of electric fields calculated using this model and frequency-domain techniques, so that it is possible to compare them with the results obtained using the formula presented in this paper. III. HORIZONAL ELECRIC FIELD COMPONEN INDUCED BY HE REURN SROKE CHARGE (E q ) Considering the earth as a perfect conductor, Rusck [1] calculated the scalar potential at a height h above the ground when the return stroke current is a step function. he result is reproduced in (2), where the electrostatic field existing prior to the Fig. 1. Geometry of the problem. (a) Origin of the orthogonal coordinate system is the point P at the earth s surface (plane xy). (b) Infinitesimal volume. return stroke was removed, because it has no inductive effect V = 60I { [ 0h u (t 0 ) (vt) 2 + ( 1 v 2 ) ] } 1/2 R r 2 v 0 r 1 0 R where v velocity of the return stroke; v R relative velocity of the return stroke (v R = v/c); c velocity of light (c = m/s); r 0 distance from the base of the lightning channel; 0 time to the arrival of the field wave ( 0 = r 0 /c). he horizontal field in this condition can be obtained by deriving V with respect to r 0. his leads to (3), where E S is the electrostatic field achieved after a sufficiently long period of time E q = E S u (t 0 ) 1 1 v R 2 [1 v 2R +(vt/r 0) 2] 3/2 (3) E S = 60I 0h v R r0 2. (4) he expression when the current is a trapezium instead of a step can be calculated by the Duhamel s integral [14], which is the convolution of the time derivative of (1) with (3), resulting in [ E q = E S u(t 0) t ( ) ] 2 1/2 vt 1 1 vr 2 + r 0 (2) [ [ u (t 0 ) (t ) 1 1 vr 2 ( ) ] 2 1/2 v (t ) +. (5) r 0 IV. HORIZONAL ELECRIC FIELD INDUCED A HE EARH S SURFACE he return stroke current produces a horizontal magnetic field H at the earth s surface that varies with time. Fig. 1(a) shows a line of force of the magnetic field and a coordinate system that has its origin in a point P at the earth s surface. he distance from this point to the channel base is r 0.hex-direction is radial from the channel and the z-direction is vertical. For the region

3 BARBOSA AND PAULINO: CALCULAION OF HE HORIZONAL ELECRIC FIELD 595 below the earth s surface, the following Maxwell s equations apply: H = σe + ε E (6) t E = µ H (7) t where σ, ε, and µ are the earth s conductivity, permittivity, and permeability, respectively. E and H are the electric field and magnetic field vectors, respectively. In order to solve (6) and (7) and to obtain the electric field at P, let us consider an infinitesimal volume in the earth defined by δx x δx, δy y δy, and δz z 0, as shown in Fig. 1(b). In this volume, H is in y-direction, so that H x = H z =0. he displacement currents in the air can be neglected, in comparison with those in the earth. his assumption is permissible because, as stated by Sunde [15] in similar analysis, the earth s permittivity is from 4 to 80 times that of free space. his leads to the conclusion that the vertical electric field E z below the earth s surface is negligible, and so is H y / x. herefore, (6) and (7) can be expressed as σe x + ε E x t E x z = H y z (8) = µ H y t. (9) Using the Laplace transform on (8) and (9) leads to (10). he solution of (10) is given by (11), where H y (z=0) is the magnetic field at the earth s surface H y sµ(σ + sε) = 2 H y z 2 (10) ( H y = H y (z=0) exp z ) sµ(σ + sε). (11) Substituting H from (11) into the frequency-domain equivalent of (8) leads to ( E x = H y (z=0) exp z ) sµ sµ(σ + sε). (12) σ + sε he minus sign means that E x points to the lightning channel, and it will be omitted in the following calculation. Considering that the magnetic field at the earth s surface rises linearly with time with a slope H R /, its frequency-domain equivalent is H R /(s 2 ). Substituting this expression into (12) and making z =0for the earth s surface leads to E x = H R µ s 3/2 1 σ + sε. (13) he time-domain equivalent of (13) can be derived by an asymptotic approximation. It is first considered that the conduction currents are preponderant over displacement currents, which means that σ sε. In this case, the time-domain equivalent of E x is given by E x = H R 4µ t. (14) πσ Fig. 2. Horizontal electric field and its asymptotic boundaries: (14) conduction currents only; (15) displacement currents only; (16) conduction and displacement currents (H R =2A/m, =1µs, µ = µ 0 =4π 10 7 H/m, σ = S/m, and ε =10ε 0 = F/m). Equation (14) is identical to the expression obtained by Rusck [1] for the same condition, i.e., neglecting displacement currents in the earth. Next, it is considered that the displacement currents are preponderant over the conduction currents, i.e., σ sε. In this case, the time-domain equivalent of E x is given by E x = H R µ t ε. (15) he general solution E x (t) shall be equivalent to (15) for small values of time, when the displacement currents are preponderant and equivalent to (14) for high values of time, when the conduction currents are preponderant. An observation of (14) and (15) reveals that E x (t) shall have the common factor (H R / ) µ and that its remaining term shall converge to (2/ πσ) t if t is large and to t/ ε if t is small. herefore, the remaining term shall be in the form t/( a + bt), where a and b are to be determined from the asymptotic equations (14) and (15). Letting t small leads to a = ε and letting t large leads to b = πσ/4, thus resulting in (16), which is a good approximation of the time-domain equivalent of (13) E x = H R t µ ε + tπσ/4. (16) Fig. 2 shows the horizontal electric field E x according to (14) (16). his figure shows that, except for very short times, the waveform of the electric field is determined by the conduction currents. he effect of displacement currents can be seen as providing a reduction on the electric field values, which is very significant for short times, but becomes progressively negligible as time goes on. Another relevant effect of the displacement currents is that they provide a finite time-derivative for the electric field at t =0, which is important for the calculation of induced voltages.

4 596 IEEE RANSACIONS ON ELECROMAGNEIC COMPAIBILIY, VOL. 49, NO. 3, AUGUS 2007 V. HORIZONAL ELECRIC FIELD COMPONEN INDUCED BY HE REURN SROKE CURREN (E i ) Once the horizontal electric field induced at the earth s surface by a linear rising magnetic field has been calculated, it is possible to compute the horizontal electric field induced by a return stroke current. Equation (16) is rewritten as (17), where the traveling time 0 taken for the wave to arrive to the point of interest and the minus sign have been introduced, the permeability of earth was considered equal to the permeability of free space (µ 0 ) and ε R = ε/ε 0 stands for the relative permittivity of earth E x = u (t 0 ) Z 0 ( HR ) G (t 0 ) (17) where ( µ0 Z 0 = G (τ) =τ ε R + πστ ) 1/2. ε 0 4ε 0 As shown by Barbosa et al. [16], the electric field induced by a trapezoidal magnetic field can be obtained from the equation for the linear rising magnetic field using superposition and the Heaviside s function, which gives E x = Z 0 ( HR ) [u (t 0 ) G (t 0 ) u (t 0 ) G (t 0 )]. (18) Making 0 in (18) leads to the electric field induced by a step magnetic field, as shown in (19), where G (τ)isdg(τ)/dt E x = u (t 0 ) Z 0 H R G (t 0 ). (19) Equation (19) allows the evaluation of the electric field induced in the earth s surface by its convolution with the time derivative of the magnetic field produced by the return stroke. he magnetic field produced at the earth s surface by a trapezoidal return stroke current, considering perfect conducting earth, has been calculated by Rusck [1], and is shown in ( ) 2 0 H = H S v R u (t 0) vt 1+ vr 2 1 ( ) 2 u (t 0 ) v (t ) 1+ vr 2 1 (20) where H S is the magnetostatic field achieved after a long period of time r 0 r 0 H S = I 0 2πr 0. (21) he time derivative of (20) is not suitable for an analytical convolution with (19). However, an investigation shows that (20) can be represented, with reasonable accuracy, by (22). his equation has been derived from the observation that the field rises almost linearly with time from t = 0 up to t = 0 +, and then, goes asymptotically toward the magnetostatic field H S following approximately a quadratic hyperbole function. Fig. 3 Fig. 3. Return stroke magnetic field from (20) and (22) for v R =0.5 and =1µs. shows a comparison between the magnetic field calculated by (20) and (22) for different distances from the lightning channel, where a good agreement between the curves can be seen. his agreement was also verified by repeating the test of (20) against (22) for other values of v R,, and r 0 { H = H S α u (t 0 ) (t 0) [ ( (t 0 ) α where α = 0 v R )( 1 u (t 0 ) H 2 )]} (t + H 0 ) 2 [ ( ) 2 ) 1+v 2 R +2( ] 1 0 H = 2 0 (1 α)(1+αv R / 0 ). v R (1 αv R ) 0 (22) Assuming that (22) represents the magnetic field produced by the return stroke, the electric field induced at the earth s surface can be obtained by the convolution between (19) and the time derivative of (22). An observation of (22) reveals that it is composed of the sum of three parcels. he first two represent a trapezium, so that their time derivative is a constant, and the convolution with (19) is straightforward. However, the convolution with the time derivative of the third parcel is far more complicated. he result is shown in (23), where G(τ) is derived from (17), and the subscript i has been introduced to indicate that this field component is induced by the return stroke current. It shall

5 BARBOSA AND PAULINO: CALCULAION OF HE HORIZONAL ELECRIC FIELD 597 final value of the horizontal electric field is the electrostatic field, as given in (4). It shall be observed that the final value is not zero because the current waveform considered remains in its peak value, so that the channel remains charged with a linear charge density equal to I 0 /v E H = E q + E i. (24) Fig. 4. otal horizontal electric field and its two induced components (I 0 = 10 ka, =1µs, r 0 = 300 m, v = 130 m/µs, h =6m, σ =10 3 S/m, and ε R = 10). be noted in (23) that for τ =0, the product G(τ)F (τ) =0: { u (t 0 ) G (t 0 ) u (t 0 ) E i where F (τ) = = Z 0H S α G (t 0 ) 2(τ + H ) + 3 H 4(τ + H ) 2 [ ( ) ]} F (t 0 ) α (23) H ln{( τ + H + τ)/( τ + H τ)} 8 τ(τ + H ) 2 τ + H. VI. RESULAN HORIZONAL ELECRIC FIELD Although the field component E i has been calculated at the earth s surface, it propagates above the earth with little disturbance up the usual heights of power and telecommunication lines. herefore, the induced horizontal electric field (E H )at height h is given by the superposition of the components induced by charge (E q ) and current (E i ), as shown in (24), where E q and E i are given by (5) and (23), respectively. As these components have opposed polarities, the resultant field will display a bipolar wave shape, with a first negative excursion determined by the current-induced component (E i ), followed by a positive excursion that asymptotically approaches the charge-induced component (E q ). For a region close to the lightning channel, the E q component is predominant, so that the negative excursion is very small. On the other hand, for a region far from the lightning channel, the E i component is predominant, so that the induced field remains negative for a longer period of time. Fig. 4 shows the horizontal electric field generated by a lightning return stroke, considering the waveform given by (1), where the contribution of the two induced components can be seen. he For the region far away from the lightning channel, the component induced by charge dies out, and the horizontal electric field is given exclusively by the component induced by current. As this component is not affected by the height, it comes out that the induced surges in lines far away from the lightning channel will be independent of the line height, as has been verified experimentally by Koga and Motomitsu [17]. Continuing to move away from the lightning channel, further simplifications can be observed in the expression for the electric field, as will be discussed in Section VIII, while the propagation effects are discussed in Section X. VII. COMPARISON WIH RESULS FROM FREQUENCY-DOMAIN ECHNIQUES Once a time-domain formula for the horizontal electric field induced by a lightning return stroke has been developed, it is interesting to compare the waveforms calculated by this formula with those calculated by frequency-domain techniques. he comparison is made with the solution of Sommerfeld s integrals carried out by Zeddam and Degauque [3] and with the Cooray Rubinstein s formula, as calculated by Rubinstein [7]. he conditions are the same as used by Rubinstein [7] in order to validate the Cooray Rubinstein s formula. Zeddam and Degauque [3] used the L return stroke model, the return stroke velocity v = m/s, and the relative permittivity of earth ε R =10. he field was calculated at h =6 m with the current represented by a double-exponential wave (25), where I 0 =10kA, a = s 1, and b =10 7 s 1 i = I 0 [ e at e bt]. (25) Fig. 5 shows the current waveform, along with its representation by trapeziums. If a high accuracy is not required, the double-exponential wave can be represented by a single trapezium. However, for a better representation of the current wave, three trapeziums are necessary. heir data are shown in able I, where the Delay column means the time delay of the current component with respect to the first one. he first trapezium composes the front, while the second and the third compose the tail of the wave. As the time-domain calculation has been performed with an electronic spreadsheet, the superposition of the trapeziums and their induced fields is straightforward. Fig. 6 shows the horizontal electric field calculated by (24) along with the values calculated by two frequency-domain techniques, namely the Cooray Rubinstein s formula calculated by Rubinstein [7] and the computation of Sommerfeld s integrals carried out by Zeddam and Degauque [3]. he calculations were made with σ =10 2 S/m and different distances from the lightning channel: 100, 500, and 1500 m. he time is considered from the instant that the inducing fields arrive at the point of

6 598 IEEE RANSACIONS ON ELECROMAGNEIC COMPAIBILIY, VOL. 49, NO. 3, AUGUS 2007 Fig. 5. Double-exponential wave and its representation by three trapeziums. ABLE I REPRESENAION OF A DOUBLE EXPONENIAL BY HREE RAPEZIUMS interest. Fig. 6 shows a very good agreement between the results obtained with the time-domain equation developed in this paper and the results obtained with the frequency-domain techniques. he relatively high value of the earth s conductivity used in the calculations reported in Fig. 6 is not adequate to observe the effect of the displacement currents, as in this case the conduction currents are strongly predominant. However, Rubinstein [7] has also carried out a calculation for an extremely low value of earth s conductivity, i.e., σ =10 4 S/m, at a distance r 0 = 500 m. In this calculation, Rubinstein used two versions of the Cooray Rubinstein s formula: one that neglects the displacement currents and other that considers the displacement currents in the earth. he results obtained are reproduced in Fig. 7, along with the results from (24) and from [3]. It can be seen in Fig. 7 that (24) agrees very well with the results obtained by Zeddam and Degauque and with the Cooray Rubinstein s formula that considers the displacement currents, while the Cooray Rubinstein s formula that neglects the displacement currents gives an initial peak value about three times higher than that of the other techniques. his result also shows that the effect of displacement currents is adequately represented in (17). VIII. ELECRICFIELD FAR FROM HE LIGHNING CHANNEL In Section VI, it was mentioned that, beyond some distance from the lightning channel, the horizontal electric field is given only by the current-induced component, as given by (23). Continuing to move away from the lightning channel, (22) shows that α v R for high values of r 0. It can also be seen that H becomes very high, which makes the function F (τ) in (23) to fade away. herefore, the equation for the horizontal electric Fig. 6. Comparison of (24) and frequency-domain techniques (σ =10 2 S/m, ε R =10,andh =6m). [(a) r 0 = 100 m. (b) r 0 = 500 m. (c) r 0 = 1500 m.] field becomes E H = Z 0H S v R {u (τ) G (τ) u (τ ) G (τ )} (26) where τ is the time from the arrival of the field at the point of interest. Considering the front of the wave, i.e., the time interval

7 BARBOSA AND PAULINO: CALCULAION OF HE HORIZONAL ELECRIC FIELD 599 Rubinstein and Uman [18] have calculated the horizontal magnetic field and the vertical electric field radiated by a return stroke, considering perfect conducting earth. hey used the L model [11] with a step current I 0 =30kA propagating with a velocity v R =1/3 and the distances r 0 =5km and r 0 = 200 km. Inserting these values in (28) gives the following peak values, which agree remarkably well with the values obtained by Rubinstein and Uman [18]: 1) H H =0.32 A/m and E V = 120 V/m for r 0 =5km; 2) H H = A/m and E V =3.0 V/m for r 0 = 200 km. Fig. 7. Comparison of (24) and frequency-domain techniques (σ =10 4 S/m, ε R =10,h =6m, and r 0 = 500 m). 0 τ, and substituting the expression for G(τ) in (26) leads to [ ( τ )] [ ] 1 E H = Z 0 H S v R. (27) εr + πστ/4ε 0 he first term in brackets in (27) is the vertical electric field E V at the earth s surface that is radiated by the return stroke, considering perfect conducting earth. he vertical electric field and the horizontal magnetic field are reproduced in (28). Equation (27) can be written in the form of (29), which expresses the ratio between the horizontal electric field developed in the earth s surface and the radiated vertical electric field at the earth s surface considering perfect conducting earth. his equation can be considered as a time domain equivalent of the wave tilt formula [5] for a radiated vertical electric field that rises linearly with time. For the sake of comparison, the wave tilt formula is reproduced in (30), where ω is the angular frequency. It shall be observed that the displacement currents in the air have been disregarded in the development of (29), so that it is valid for ε R 1, which is usually the case of earth. he same restriction should also apply to the wave tilt formula. In fact, making σ 0 in both (29) and (30) leads to E H /E V = ε 1/2 R. In this case, the earth behaves as a dielectric medium with refraction index equal to, and a vertically polarized wave that reaches the earth s surface with a grazing angle of incidence will be refracted into the earth with a propagation direction close to the vertical axis ε 1/2 R E V = I 0v R Z 0 2πr 0 τ [ ] E H (τ) E V (τ) = 1 εr + πστ/4ε 0 H H = E V Z 0 (28) (29) [ ] E H (ω) E V (ω) = 1. (30) εr + σ/jωε 0 IX. REPRESENAION OF AN ARBIRARY CURREN WAVEFORM It was mentioned in Section II that any arbitrary current waveform could be represented by the superposition of a set of trapeziums. In Section VII, the double-exponential waveform of Fig. 5 is represented by three trapeziums selected adequately, so that the horizontal electric field calculated by (24) was in good agreement with the results available in the literature [3], [7]. his section presents a more general method that can be used to represent any arbitrary waveform. he time period of interest is divided into n slots of length δt. A vector is then constructed with the current increments in each slot, according to δ i (j) =i (j) i (j 1), for j =1, 2, 3,...,n. (31) he outcome of (24) is computed for a unit trapezium (i.e., a trapezium of unit amplitude and front time = δt) for n time steps of length δt, resulting in a vector E HU (j). he electric field at time t = kδt(where k is an integer and 1 k n) is given by the superposition of the electric field produced by k unit trapeziums scaled by the current increments and considering the respective delays, as given by (32). his equation is a discrete convolution of E HU (t) with di/dt, because letting δt 0 leads to the Duhamel s integral [14] E H (k) = k E HU (j) δ i (k j +1). (32) j=1 In order to illustrate the application of (32), the channel base current described by Nucci [19] is selected. his waveform well represents the current waves of subsequent return strokes obtained from experiments, and it is characterized by a peak value equal to 12 ka occurring at about 0.8 µs and a maximum time derivative equal to 40 ka/ µs. Fig. 8 shows this waveform and its representation by multiple trapeziums (n = 100), in accordance with (32). As can be seen in Fig. 8, the two plots are indistinguishable. he horizontal electric field calculated by (32) is shown in Fig. 9, along with the curves calculated by Nucci [20], both using the same velocity of the return stroke (v = 190 m/ µs) and line height (h =8m). he calculation was carried out for two values of earth s conductivities (σ =10 3 S/m and σ = ) and two distances from the lightning channel (r 0 = 1000 m and r 0 = 250 m). he relative earth s permittivity for this plot is not clearly indicated in [20] and ε R =5was considered for the calculation by (32). In any case, it shall be pointed out that E H is not very sensitive to ε R, as doubling ε R in the conditions of

8 600 IEEE RANSACIONS ON ELECROMAGNEIC COMPAIBILIY, VOL. 49, NO. 3, AUGUS 2007 Fig. 8. Current from [19] and its representation by multiple trapeziums. Fig. 9(a) leads to a reduction of the negative peak of only 7%, and does not disturb the wave tail. he matching between the plots from (32) and [20] in Fig. 9(a) (r 0 = 1000 m) is remarkably good, especially when the different calculating procedures and return stroke models used are considered. While (32) was developed based on the L model, results from [20] were calculated using the MLE model, where the current decays exponentially with height. his result suggests that the horizontal electric field at line height is strongly influenced by the lower portions of the lightning channel, where the currents are similar for both models. Fig. 9(b) shows the same comparison for a point closer to the lightning channel (r 0 = 250 m). In this case, there is still a very good agreement between the curves up to 2 µs, but beyond this time, there is a mismatch between them: while the curves from (32) tend to bend toward lower values (i.e., display a negative time derivative), the curves from [20] tend to keep growing (i.e., display a positive time derivative). An analysis of the phenomenon seems to point in favor of the curves from (32), because as current fades away, the electric field should also fade away [see Fig. 6(a) and (b)]. X. DISCUSSION During the development of the analytical equations in this paper, some justified simplifications have been made, and the testing of their results against the data available in the literature showed a good agreement. his section discusses the implications of these simplifications on the range of validity of the developed formula. One key equation of the present formulation is (16) that, although may not be exact, is valid for the full range of earth s conductivity and permittivity. he approximation introduced in (22) does not have any practical implication, as it slightly disturbs the shape of the wave tail. In fact, the stronger approximation used in the present development is the consideration that the vertical electric field and the horizontal magnetic field calculated for perfect conducting earth is also valid for finite conducting earth. Fig. 9. Horizontal electric field from (32) and [20] (v = 190 m/ µs and h =8m). [(a) r 0 = 1000 m. (b) r 0 = 250 m.] Indeed, this is also the basic assumption behind the Cooray Rubinstein s formula, so that both formulations shall have the same range of application. Shoory et al. [9] have shown that the horizontal electric field is attenuated to about 67% after propagating 50 km, which indicates an attenuation constant in the order of m 1. If an error of 10% is acceptable, the Cooray Rubinstein s formula and (24) could be used up to 13 km (for the conditions considered in [9]). his indicates that both formulas are adequate to compute induced voltages in power and telecommunication lines, where only relatively highvoltage values are of interest. hey may also be used beyond this range if a higher error is acceptable or a correction factor is introduced. Another approximation used both by (24) and the Cooray Rubinstein s formula is to add the electric field at the earth s surface considering finitely conducting earth and the electric field at line height considering perfect conducting earth. his procedure has been analyzed in detail by Cooray and Scuka [21],

9 BARBOSA AND PAULINO: CALCULAION OF HE HORIZONAL ELECRIC FIELD 601 and they concluded that it is justified. With the model employed in the present paper, this procedure gets a physical meaning, as the field at line height is modeled by the superposition of two distinct components: one produced by the return stroke current and the other produced by the return stroke charge. If the earth s conductivity is set to be infinite, the former fades away and the latter remains unchanged. Finally, it is worth mentioning that one interesting feature of (24) and (32) is that they are very easy to be implemented into a computer code, because they are based on the direct evaluation of elementary functions. For instance, the plots of Fig. 9 were produced by a 43-line QBasic code, and its execution time in a PC Pentium III is negligible. XI. CONCLUSION he formula presented in this paper allows the calculation of the horizontal electric field from a lightning return stroke in the time domain, considering the effect of finitely conducting earth. Its inputs are the waveform and the velocity of the return stroke current, the distance from the lightning channel, the height from the soil, and the earth s characteristics (conductivity and permittivity). Its range of validity is equivalent to the Cooray Rubinstein s formula [7]. he results of testing this formula against frequency-domain techniques show that it is accurate for different distances from the lightning channel and different values of earth s conductivity. his formula is particularly useful for the calculation of lightning-induced overvoltages in power and telecommunication lines, where the electric field is needed at many points along the line. As it is based on the direct evaluation of elementary functions in time domain, the calculation process is straightforward, with minimum requirements for computational resources and negligible introduction of errors. REFERENCES [1] S. Rusck, Induced lightning overvoltages on power transmission lines with special reference to the overvoltage protection of low voltage networks Ph.D. dissertation, Royal Inst. echnol., Stockholm, Sweden, [2] A. Sommerfeld, Uber die Ausbreitung der Wellen in der drahtlosen elegraphie, Ann. Phys., vol. 28, no. 4, pp , [3] A. Zeddam and P. Degauque, Current and voltage induced on a telecommunication cable by a lightning stroke, in Lightning Electromagnetic, R. L. Gardner, Ed. Bristol, PA: Hemisphere, 1990, pp [4] E. M. hompson, P. J. Medelius, M. Rubinstein, M. A. Uman, J. Johnson, and J. W. Stone, Horizontal electric fields from lightning return strokes, J. Geophys. Res., vol. 93, pp , [5] K. A. Norton, Propagation of radio waves over the surface of the earth and in the upper atmosphere: Part II, Proc. IRE,vol.25,no.9,pp , Sept [6] V. Cooray, Horizontal fields generated by return strokes, Radio Sci., vol. 27, pp , [7] M. Rubinstein, An approximate formula for the calculation of the horizontal electric field from lightning at close, intermediate and long range, IEEE rans. Electromagn. Compat., vol. 38, no. 3, pp , Aug [8] V. Cooray, Some considerations on the Cooray Rubinstein formulation used in deriving the horizontal electric field of lightning return strokes over finitely conducting ground, IEEE rans. Electromagn. Compat., vol. 44, no. 4, pp , Nov [9] A. Shoory, R. Moini, S. H. H. Sadeghi, and V. A. Rakov, Analysis of lightning-radiated electromagnetic fields in the vicinity of lossy ground, IEEE rans. Electromagn. Compat., vol. 47, no. 1, pp , Feb [10] Y. Baba and V. A. Rakov, Voltages induced on an overhead wire by lightning strikes to a nearby tall grounded object, IEEE rans. Electromagn. Compat., vol. 48, no. 1, pp , Feb [11] M. A. Uman and D. K. McLain, Magnetic field of the lightning return stroke, J. Geophys. Res., vol. 74, pp , [12] V. A. Rakov and A. A. Dulzon, Calculated electromagnetic fields of lightning return stroke, ekh. Elektrodinam., vol. 1, pp , [13] C. A. Nucci, C. Mazzetti, F. Rachidi, and M. Ianoz, On lightning return stroke models for LEMP calculations, in Proc. 19th Int. Conf. Lightning Protection, Graz, Austria, 1988, pp [14] A. Greenwood, Electrical ransients in Power Systems Section 2.6: Duhamel s Integral Response of a Circuit to an Arbitrary Stimulus. Hoboken, NJ: Wiley, 1971, pp [15] E. D. Sunde, Earth Conduction Effects in ransmission Systems. New York: Dover, [16] C. F. Barbosa, F. E. Nallin, J. A. Rossi, J. Ribeiro, S. Person, and A. Zeddam, Lightning induced surges on aerial telecommunication lines with special reference to the effect of earth resistivity, in Proc. 8th Int. Symp. Lightning Protection, São Paulo, Brazil, 2005, pp [17] H. Koga and. Motomitsu, Lightning-induced surges in paired telephone subscriber cable in Japan, IEEE rans. Electromagn. Compat., vol. EMC-27, no. 3, pp , Aug [18] M. Rubinstein and M. Uman, Methods for calculating the electromagnetic fields from a known source distribution: Application to lightning, IEEE rans. Electromagn. Compat., vol. 31, no. 2, pp , May [19] C. A. Nucci, F. Rachidi, M. V. Ianoz, and C. Mazzetti, Lightning-induced voltages on overhead lines, IEEE rans. Electromagn. Compat.,vol.35, no. 1, pp , Feb [20] C. A. Nucci, Lightning-induced voltages on distribution systems: Influence of ground resistivity and system topology, in Proc. 8th Int. Symp. Lightning Protection, São Paulo, Brazil, 2005, pp [21] V. Cooray and V. Scuka, Lightning-induced overvoltages in power lines: Validity of various approximations made in overvoltage calculations, IEEE rans. Electromagn. Compat., vol. 40, no. 4, pp , Nov Célio Fonseca Barbosa was born in Nova Lima, Brazil, in He received the B.Sc. and M.Sc. degrees in electrical engineering from the Federal University of Minas Gerais, Belo Horizonte, Brazil, in 1983 and 1988, respectively. In 1984, he joined the nationwide Brazilian telecommunication operator (elebrás), where he was in charge of developing the operator s standards and laboratories on electrical protection, electromagnetic compatibility (EMC), and safety. In 1998, he joined the CPqD (foundation for R&D on telecommunication), Campinas, where he developed several research programs, including the operation of a test site with rocket-triggered lightning. His current research interests include the protection of telecommunication systems against electromagnetic disturbance and the propagation of communication signals on power lines (PL). Mr. Barbosa is a member of the International elecommunication Union- elecommunication (IU-) SG-5, where he is currently a Rapporteur on lightning protection of telecommunication systems. José Osvaldo Saldanha Paulino was born in Belo Horizonte, Brazil, in He received the B.Sc. and M.Sc. degrees in electrical engineering from the Federal University of Minas Gerais, Belo Horizonte, in 1979 and 1985, respectively, and the Dr. Sc. degree in electrical engineering from the State University of Campinas, Campinas, Brazil, in In 1980, he joined the Electrical Department, Federal University of Minas Gerais, as a Professor. His current research interests include high-voltage and electromagnetic compatibility.