2 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 1 Introduction Well-defined representation of real electrostatic discharge (ESD) currents is needed in order to establish realistic requirements for ESD generators used in testing of the equipment and devices, as well as to provide and improve the repeatability of tests. Such representations should be able to approxi- mate the ESD currents waveshapes for various test levels, test set-ups and procedures, and also for various ESD conditions such as approach speeds, types of electrodes, relative arc length, humidity, etc. A mathematical func- tion is needed for computer simulation of ESD phenomena, for verification of test generators and for improving standard waveshape definition. Functions previously proposed in the literature for modelling of ESD currents, are mostly linear combinations of exponential functions, Gaussian functions, Heidler functions or other functions, for a short review see for example [1]. The Analytically Extended Function (AEF) was initially pro- posed in [2] and has been successfully applied to lightning discharge mod- elling [3–13] using nonlinear least-square curve fitting. In this paper we analyse the applicability of the AEF with p peaks to representation of ESD currents by interpolation of data points chosen ac- cording to a D-optimal design. This is illustrated through examples from two applications. The first application is modelling of an ESD commonly used in electrostatic discharge immunity testing, and the second modelling of lightning discharges. For the ESD immunity testing application we model the IEC Standard 61000-4-2 waveshape, [14,15] and an experimentally measured ESD current from [16]. For the lightning discharge application we model the IEC 61312-1 stan- dard waveshape [17, 18] and a more complex measured lightning discharge current from [19]. We also use the same method to approximate a measured derivative of a lightning discharge current derivative from [20]. In both applications the basic properties of the current (or current deriva- tive) are the same, these properties and how they are modelled with the AEF is discussed in the next section. 2 Modelling of ESD currents using the AEF Various mathematical expressions have been introduced in the literature that can be used for representation of the ESD currents, either the IEC 61000- 4-2 Standard one [15], or experimentally measured ones, e.g. [21]. These FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 32, No 1, March 2019, pp. 25 - 49 https://doi.org/10.2298/FUEE1901025L Karl Lundengård1, Milica Rančić1, Vesna Javor2, Sergei Silvestrov1 Received December 21, 2018 Corresponding author: Karl Lundengard Division of Applied Mathematics, UKK, Mälardalen University, Högskoleplan 1, Box 883, 721 23 Västerås, Sweden (E-mail: karl.lundengard@mdh.se) FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 28, No 4, December 2015, pp. 507 - 525 Doi: 10.2298/FUEE1504507S HORIZONTAL CURRENT BIPOLAR TRANSISTOR (HCBT) – A LOW-COST, HIGH-PERFORMANCE FLEXIBLE BICMOS TECHNOLOGY FOR RF COMMUNICATION APPLICATIONS Tomislav Suligoj1, Marko Koričić1, Josip Žilak1, Hidenori Mochizuki2, So-ichi Morita2, Katsumi Shinomura2, Hisaya Imai2 1University of Zagreb, Faculty of Electrical Engineering and computing, Department of Electronics, Micro- and Nano-electronics Laboratory, croatia 2Asahi Kasei Microdevices co. 5-4960, Nobeoka, Miyazaki, 882-0031, Japan Abstract. In an overview of Horizontal Current Bipolar Transistor (HCBT) technology, the state-of-the-art integrated silicon bipolar transistors are described which exhibit fT and fmax of 51 GHz and 61 GHz and fTBVCEO product of 173 GHzV that are among the highest-performance implanted-base, silicon bipolar transistors. HBCT is integrated with CMOS in a considerably lower-cost fabrication sequence as compared to standard vertical-current bipolar transistors with only 2 or 3 additional masks and fewer process steps. Due to its specific structure, the charge sharing effect can be employed to increase BVCEO without sacrificing fT and fmax. Moreover, the electric field can be engineered just by manipulating the lithography masks achieving the high-voltage HCBTs with breakdowns up to 36 V integrated in the same process flow with high-speed devices, i.e. at zero additional costs. Double-balanced active mixer circuit is designed and fabricated in HCBT technology. The maximum IIP3 of 17.7 dBm at mixer current of 9.2 mA and conversion gain of -5 dB are achieved. Key words: BiCMOS technology, Bipolar transistors, Horizontal Current Bipolar Transistor, Radio frequency integrated circuits, Mixer, High-voltage bipolar transistors. 1. iNtRoDUctioN in the highly competitive wireless communication markets, the RF circuits and systems are fabricated in the technologies that are very cost-sensitive. in order to minimize the fabrication costs, the sub-10 gHz applications can be processed by using the high-volume silicon technologies. it has been identified that the optimum solution might Received March 9, 2015 Corresponding author: tomislav Suligoj University of Zagreb, Faculty of Electrical Engineering and computing, Department of Electronics, Micro- and Nano-electronics Laboratory, croatia (e-mail: tom@zemris.fer.hr) ELECTROSTATIC DISCHARGE CURRENTS REPRESENTATION USING THE ANALYTICALLY EXTENDED FUNCTION WITH P PEAKS BY INTERPOLATION ON A D-OPTIMAL DESIGN 1Division of Applied Mathematics, UKK, Mälardalen University, Västerås, Sweden 2Department of Power Engineering, Faculty of Electronic Engineering, University of Niš, Niš, Serbia Abstract. In this paper the Analytically Extended Function (AEF) with p peaks is used for representation of the electrostatic discharge (ESD) currents and lightning discharge currents. The tting to data is achieved by interpolation of certain data points. In order to minimize unstable behaviour, the exponents of the AEF are chosen from a certain arithmetic sequence and the interpolated points are chosen according to a D-optimal design. The method is illustrated using several examples of currents taken from standards and measurements. Key words: Analytically extended function, electrostatic discharge (ESD) current, lightning discharge current, D-optimal design. 2 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 1 Introduction Well-defined representation of real electrostatic discharge (ESD) currents is needed in order to establish realistic requirements for ESD generators used in testing of the equipment and devices, as well as to provide and improve the repeatability of tests. Such representations should be able to approxi- mate the ESD currents waveshapes for various test levels, test set-ups and procedures, and also for various ESD conditions such as approach speeds, types of electrodes, relative arc length, humidity, etc. A mathematical func- tion is needed for computer simulation of ESD phenomena, for verification of test generators and for improving standard waveshape definition. Functions previously proposed in the literature for modelling of ESD currents, are mostly linear combinations of exponential functions, Gaussian functions, Heidler functions or other functions, for a short review see for example [1]. The Analytically Extended Function (AEF) was initially pro- posed in [2] and has been successfully applied to lightning discharge mod- elling [3–13] using nonlinear least-square curve fitting. In this paper we analyse the applicability of the AEF with p peaks to representation of ESD currents by interpolation of data points chosen ac- cording to a D-optimal design. This is illustrated through examples from two applications. The first application is modelling of an ESD commonly used in electrostatic discharge immunity testing, and the second modelling of lightning discharges. For the ESD immunity testing application we model the IEC Standard 61000-4-2 waveshape, [14,15] and an experimentally measured ESD current from [16]. For the lightning discharge application we model the IEC 61312-1 stan- dard waveshape [17, 18] and a more complex measured lightning discharge current from [19]. We also use the same method to approximate a measured derivative of a lightning discharge current derivative from [20]. In both applications the basic properties of the current (or current deriva- tive) are the same, these properties and how they are modelled with the AEF is discussed in the next section. 2 Modelling of ESD currents using the AEF Various mathematical expressions have been introduced in the literature that can be used for representation of the ESD currents, either the IEC 61000- 4-2 Standard one [15], or experimentally measured ones, e.g. [21]. These Electrostatic discharge currents representation using the analytically extended function...3 functions are to certain extent in accordance with the requirements given in Table 1. Furthermore, they have to satisfy the following: • the value of the ESD current and its first derivative must be equal to zero at the moment t = 0, since neither the transient current nor the radiated field generated by the ESD current can change abruptly at that moment. • the ESD current function must be time-integrable in order to allow numerical calculation of the ESD radiated fields. 2.1 The Analytically Extended Function (AEF) with p peaks A so-called analytically extended function (AEF) with p peaks has been proposed and applied by the authors to lightning discharge current modelling in [9–11]. Initial considerations on applying the function to ESD currents have also been made in [1,5]. The AEF consists of scaled and translated functions of the form x(β; t) =( te1−t )β that the authors have previously referred to as power-exponential functions [10]. Here we define the AEF with p peaks as i(t) = q−1∑ k=1 Imk + Imq nq∑ k=1 ηq,kxq,k(t), (1) for tmq−1 ≤ t ≤ tmq, 1 ≤ q ≤ p, and p∑ k=1 Imk np+1∑ k=1 ηp+1,kxp+1,k(t), (2) for tmp ≤ t. The current value of the first peak is denoted by Im1, the difference between each pair of subsequent peaks by Im2, Im3, . . . , Imp, and their cor- responding times by tm1, tm2, . . . , tmp. In each time interval q, with 1 ≤ q ≤ p + 1, the number of terms is given by nq, 0 < nq ∈ Z. Parameters ηq,k are such that ηq,k ∈ R for q = 1, 2, . . . , p + 1, k = 1, 2, . . . , nq and nq∑ k=1 ηq,k = 1. Furthermore xq,k(t), 1 ≤ q ≤ p + 1 is given by xq,k(t) =    x ( βq,k; t−tmq−1 tmq −tmq−1 ) , 1 ≤ q ≤ p, x ( βq,k; t tmq ) , q = p + 1. (3) 26 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 27 2 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 1 Introduction Well-defined representation of real electrostatic discharge (ESD) currents is needed in order to establish realistic requirements for ESD generators used in testing of the equipment and devices, as well as to provide and improve the repeatability of tests. Such representations should be able to approxi- mate the ESD currents waveshapes for various test levels, test set-ups and procedures, and also for various ESD conditions such as approach speeds, types of electrodes, relative arc length, humidity, etc. A mathematical func- tion is needed for computer simulation of ESD phenomena, for verification of test generators and for improving standard waveshape definition. Functions previously proposed in the literature for modelling of ESD currents, are mostly linear combinations of exponential functions, Gaussian functions, Heidler functions or other functions, for a short review see for example [1]. The Analytically Extended Function (AEF) was initially pro- posed in [2] and has been successfully applied to lightning discharge mod- elling [3–13] using nonlinear least-square curve fitting. In this paper we analyse the applicability of the AEF with p peaks to representation of ESD currents by interpolation of data points chosen ac- cording to a D-optimal design. This is illustrated through examples from two applications. The first application is modelling of an ESD commonly used in electrostatic discharge immunity testing, and the second modelling of lightning discharges. For the ESD immunity testing application we model the IEC Standard 61000-4-2 waveshape, [14,15] and an experimentally measured ESD current from [16]. For the lightning discharge application we model the IEC 61312-1 stan- dard waveshape [17, 18] and a more complex measured lightning discharge current from [19]. We also use the same method to approximate a measured derivative of a lightning discharge current derivative from [20]. In both applications the basic properties of the current (or current deriva- tive) are the same, these properties and how they are modelled with the AEF is discussed in the next section. 2 Modelling of ESD currents using the AEF Various mathematical expressions have been introduced in the literature that can be used for representation of the ESD currents, either the IEC 61000- 4-2 Standard one [15], or experimentally measured ones, e.g. [21]. These Electrostatic discharge currents representation using the analytically extended function...3 functions are to certain extent in accordance with the requirements given in Table 1. Furthermore, they have to satisfy the following: • the value of the ESD current and its first derivative must be equal to zero at the moment t = 0, since neither the transient current nor the radiated field generated by the ESD current can change abruptly at that moment. • the ESD current function must be time-integrable in order to allow numerical calculation of the ESD radiated fields. 2.1 The Analytically Extended Function (AEF) with p peaks A so-called analytically extended function (AEF) with p peaks has been proposed and applied by the authors to lightning discharge current modelling in [9–11]. Initial considerations on applying the function to ESD currents have also been made in [1,5]. The AEF consists of scaled and translated functions of the form x(β; t) =( te1−t )β that the authors have previously referred to as power-exponential functions [10]. Here we define the AEF with p peaks as i(t) = q−1∑ k=1 Imk + Imq nq∑ k=1 ηq,kxq,k(t), (1) for tmq−1 ≤ t ≤ tmq, 1 ≤ q ≤ p, and p∑ k=1 Imk np+1∑ k=1 ηp+1,kxp+1,k(t), (2) for tmp ≤ t. The current value of the first peak is denoted by Im1, the difference between each pair of subsequent peaks by Im2, Im3, . . . , Imp, and their cor- responding times by tm1, tm2, . . . , tmp. In each time interval q, with 1 ≤ q ≤ p + 1, the number of terms is given by nq, 0 < nq ∈ Z. Parameters ηq,k are such that ηq,k ∈ R for q = 1, 2, . . . , p + 1, k = 1, 2, . . . , nq and nq∑ k=1 ηq,k = 1. Furthermore xq,k(t), 1 ≤ q ≤ p + 1 is given by xq,k(t) =    x ( βq,k; t−tmq−1 tmq −tmq−1 ) , 1 ≤ q ≤ p, x ( βq,k; t tmq ) , q = p + 1. (3) Electrostatic discharge currents representation using the analytically extended function...3 functions are to certain extent in accordance with the requirements given in Table 1. Furthermore, they have to satisfy the following: • the value of the ESD current and its first derivative must be equal to zero at the moment t = 0, since neither the transient current nor the radiated field generated by the ESD current can change abruptly at that moment. • the ESD current function must be time-integrable in order to allow numerical calculation of the ESD radiated fields. 2.1 The Analytically Extended Function (AEF) with p peaks A so-called analytically extended function (AEF) with p peaks has been proposed and applied by the authors to lightning discharge current modelling in [9–11]. Initial considerations on applying the function to ESD currents have also been made in [1,5]. The AEF consists of scaled and translated functions of the form x(β; t) =( te1−t )β that the authors have previously referred to as power-exponential functions [10]. Here we define the AEF with p peaks as i(t) = q−1∑ k=1 Imk + Imq nq∑ k=1 ηq,kxq,k(t), (1) for tmq−1 ≤ t ≤ tmq, 1 ≤ q ≤ p, and p∑ k=1 Imk np+1∑ k=1 ηp+1,kxp+1,k(t), (2) for tmp ≤ t. The current value of the first peak is denoted by Im1, the difference between each pair of subsequent peaks by Im2, Im3, . . . , Imp, and their cor- responding times by tm1, tm2, . . . , tmp. In each time interval q, with 1 ≤ q ≤ p + 1, the number of terms is given by nq, 0 < nq ∈ Z. Parameters ηq,k are such that ηq,k ∈ R for q = 1, 2, . . . , p + 1, k = 1, 2, . . . , nq and nq∑ k=1 ηq,k = 1. Furthermore xq,k(t), 1 ≤ q ≤ p + 1 is given by xq,k(t) =    x ( βq,k; t−tmq−1 tmq −tmq−1 ) , 1 ≤ q ≤ p, x ( βq,k; t tmq ) , q = p + 1. (3) 4 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV Remark 1. When previously applying the AEF, see [9–11], all exponents (β-parameters) of the AEF were set to β2 + 1 in order to guarantee that the derivative of the AEF is continuous. Here this condition will be satisfied in a different manner. Since the AEF is a linear combination of elementary functions, its deriva- tive and integral can be found using standard methods. Explicit formulae are given in [11, Theorems 1-3]. Previously, the authors have fitted AEF functions to lightning discharge currents and ESD currents using the Marquardt least square method but have noticed that the obtained result varies greatly depending on how the waveforms are sampled. This is problematic, especially since the methodol- ogy becomes computationally demanding when applied to large amounts of data. Here we will try one way to minimize the data needed but still enough to get as good approximation as possible. The method examined here will be based on D-optimality of a regression model. A D-optimal design is found by choosing sample points such that the determinant of the Fischer information matrix of the model is maximized. For a standard linear regression model this is also equivalent, by the so- called Kiefer-Wolfowitz equivalence criterion, to G-optimality which means that the maximum of the prediction variance will be minimized. These are standard results in the theory of optimal design and a summary can be found for example in [22]. Minimizing the prediction variance will in our case mean maximizing the robustness of the model. This does not guarantee a good approximation but it will increase the chances of the method working well when working with limited precision and noisy data, and thus improve the chances of finding a good approximation when it is possible. 3 D-optimal approximation for exponents given by a class of arithmetic sequences It can be desirable to minimize the number of points used when construct- ing the approximation. One way of doing this is choosing the D-optimal sampling points. In this section we will only consider the case where in each interval the n exponents, β1, . . . , βn, are chosen according to βm = k + m − 1 c , m = 1, 2, . . . , n 26 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 27 4 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV Remark 1. When previously applying the AEF, see [9–11], all exponents (β-parameters) of the AEF were set to β2 + 1 in order to guarantee that the derivative of the AEF is continuous. Here this condition will be satisfied in a different manner. Since the AEF is a linear combination of elementary functions, its deriva- tive and integral can be found using standard methods. Explicit formulae are given in [11, Theorems 1-3]. Previously, the authors have fitted AEF functions to lightning discharge currents and ESD currents using the Marquardt least square method but have noticed that the obtained result varies greatly depending on how the waveforms are sampled. This is problematic, especially since the methodol- ogy becomes computationally demanding when applied to large amounts of data. Here we will try one way to minimize the data needed but still enough to get as good approximation as possible. The method examined here will be based on D-optimality of a regression model. A D-optimal design is found by choosing sample points such that the determinant of the Fischer information matrix of the model is maximized. For a standard linear regression model this is also equivalent, by the so- called Kiefer-Wolfowitz equivalence criterion, to G-optimality which means that the maximum of the prediction variance will be minimized. These are standard results in the theory of optimal design and a summary can be found for example in [22]. Minimizing the prediction variance will in our case mean maximizing the robustness of the model. This does not guarantee a good approximation but it will increase the chances of the method working well when working with limited precision and noisy data, and thus improve the chances of finding a good approximation when it is possible. 3 D-optimal approximation for exponents given by a class of arithmetic sequences It can be desirable to minimize the number of points used when construct- ing the approximation. One way of doing this is choosing the D-optimal sampling points. In this section we will only consider the case where in each interval the n exponents, β1, . . . , βn, are chosen according to βm = k + m − 1 c , m = 1, 2, . . . , n 28 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 29 4 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV Remark 1. When previously applying the AEF, see [9–11], all exponents (β-parameters) of the AEF were set to β2 + 1 in order to guarantee that the derivative of the AEF is continuous. Here this condition will be satisfied in a different manner. Since the AEF is a linear combination of elementary functions, its deriva- tive and integral can be found using standard methods. Explicit formulae are given in [11, Theorems 1-3]. Previously, the authors have fitted AEF functions to lightning discharge currents and ESD currents using the Marquardt least square method but have noticed that the obtained result varies greatly depending on how the waveforms are sampled. This is problematic, especially since the methodol- ogy becomes computationally demanding when applied to large amounts of data. Here we will try one way to minimize the data needed but still enough to get as good approximation as possible. The method examined here will be based on D-optimality of a regression model. A D-optimal design is found by choosing sample points such that the determinant of the Fischer information matrix of the model is maximized. For a standard linear regression model this is also equivalent, by the so- called Kiefer-Wolfowitz equivalence criterion, to G-optimality which means that the maximum of the prediction variance will be minimized. These are standard results in the theory of optimal design and a summary can be found for example in [22]. Minimizing the prediction variance will in our case mean maximizing the robustness of the model. This does not guarantee a good approximation but it will increase the chances of the method working well when working with limited precision and noisy data, and thus improve the chances of finding a good approximation when it is possible. 3 D-optimal approximation for exponents given by a class of arithmetic sequences It can be desirable to minimize the number of points used when construct- ing the approximation. One way of doing this is choosing the D-optimal sampling points. In this section we will only consider the case where in each interval the n exponents, β1, . . . , βn, are chosen according to βm = k + m − 1 c , m = 1, 2, . . . , n Electrostatic discharge currents representation using the analytically extended function...5 where k is a non-negative integer and c a positive real number. Note that in order to guarantee continuity of the AEF derivative the condition k > c has to be satisfied. In each interval we want an approximation of the form y(t) = n∑ i=1 ηit βieβi(1−t) and by setting z(t) = (te1−t) l c we obtain y(t) = n∑ i=1 ηiz(t) k+i−1. If we have n sample points, ti, i = 1, . . . , n, then the Fischer information matrix, M, of this system is M = U�U where U =   z(t1) k z(t2) k . . . z(tn) k z(t1) k+1 z(t2) k+1 . . . z(tn) k+1 ... ... ... ... z(t1) k+n−1 z(t2) k+n−1 . . . z(tn) k+n−1   . Thus if we want to maximize det(M) = det(U)2 it is sufficient to maximize or minimize the determinant det(U). Set z(ti) = (tie 1−ti) l c = xi then un(t1, . . . , tn) = det(U) = ( n∏ i=1 xki )   ∏ 1≤i 1) when xi = xmax · yi and xn = xmax, or some permutation thereof. Proof. This theorem follows from Theorem 1 combined with the fact that wn(k; x1, . . . , xn) is a homogeneous polynomial. Since wn(k; b·x1, . . . , c·xn) = bk+ n(n−1) 2 · wn(k; x1, . . . , xn) if (x1, . . . , xn) is an extreme point in [0, 1]n then (b · x1, . . . , b · xn) is an extreme point in [0, b]n. Thus by Theorem 1 the points given by xi = xmax · yi will maximize or minimize wn(k; x1, . . . , xn) on [0, xmax] n. Remark 3. It is in many cases possible to ensure the condition 1 < xmax · y1 without actually calculating the roots of P (2k−1,0) n−1 (1 − 2y). In the literature on orthogonal polynomials there are many expressions for upper and lower bounds of the roots of the Jacobi polynomials. For instance in [25] an upper bound on the largest root of a Jacobi polynomial is given and can be, in our case, rewritten as y1 > 1 − 3 4k2 + 2kn + n2 − k − 2n + 1 and thus 1 − 3 4k2 + 2kn + n2 − k − 2n + 1 > 1 xmax guarantees that 1 < xmax · y1. If a more precise condition is needed there are expressions that give tighter bounds of the largest root of the Jacobi polynomials, see [26]. 32 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 33 8 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 3.2 D-optimal interpolation on the decaying part Finding the D-optimal points for the decaying part is more difficult than it is for the rising part. Suppose we denote the largest value for time that can reasonably be used (for computational or experimental reasons) with tmax. This corresponds to some value xmax = (tmax exp(1 − tmax)) 1 c . Ideally we would want a corresponding theorem to Theorem 1 over [1, xmax] n instead of [0, 1]n. It is easy to see that if xi = 0 or xi = 1 for some 1 ≤ xi ≤ n − 1 then wn(k; x1, . . . , xn) = 0 and thus there must exist some local extreme point such that 0 < x1 < x2 < . . . < xn−1 < 1. This is no longer guaranteed when considering the volume [1, xmax] n instead. Therefore we will instead extend Theorem 1 to the volume [0, xmax] n and give an extra constraint on the parameter k that guarantees 1 < x1 < x2 < . . . < xn−1 < xn = xmax. Theorem 2. Let y1 < y2 < . . . < yn−1 be the roots of the Jacobi polynomial P (2k−1,0) n−1 (1−2y). If k is chosen such that 1 < xmax ·y1 then the determinant wn(k; x1, . . . , xn) given in Theorem 1 is maximized or minimized on the cube [1, xmax] n (where xmax > 1) when xi = xmax · yi and xn = xmax, or some permutation thereof. Proof. This theorem follows from Theorem 1 combined with the fact that wn(k; x1, . . . , xn) is a homogeneous polynomial. Since wn(k; b·x1, . . . , c·xn) = bk+ n(n−1) 2 · wn(k; x1, . . . , xn) if (x1, . . . , xn) is an extreme point in [0, 1]n then (b · x1, . . . , b · xn) is an extreme point in [0, b]n. Thus by Theorem 1 the points given by xi = xmax · yi will maximize or minimize wn(k; x1, . . . , xn) on [0, xmax] n. Remark 3. It is in many cases possible to ensure the condition 1 < xmax · y1 without actually calculating the roots of P (2k−1,0) n−1 (1 − 2y). In the literature on orthogonal polynomials there are many expressions for upper and lower bounds of the roots of the Jacobi polynomials. For instance in [25] an upper bound on the largest root of a Jacobi polynomial is given and can be, in our case, rewritten as y1 > 1 − 3 4k2 + 2kn + n2 − k − 2n + 1 and thus 1 − 3 4k2 + 2kn + n2 − k − 2n + 1 > 1 xmax guarantees that 1 < xmax · y1. If a more precise condition is needed there are expressions that give tighter bounds of the largest root of the Jacobi polynomials, see [26]. Electrostatic discharge currents representation using the analytically extended function...9 We can now find the D-optimal t-values using the lower branch of the Lambert W function as in equation (5), ti = −W−1(−e−1xci), where xi are the roots of the Jacobi polynomial given in Theorem 1. Since −1 ≤ W−1(x) < −∞ for −e−1 ≤ x ≤ 0 this will always give 1 ≤ ti < tmax = −W−1(−e−1xmax) so xmax is given by the highest feasible t. Remark 4. Note that here just like in the rising part tn = tp which means that we will interpolate to the final peak as well as p−1 points in the decaying part. 4 Examples of models from applications and experiments In this section some results of applying the described scheme to two different applications will be presented. The first application is modelling of ESD currents commonly used in electrostatic discharge immunity testing, and the second modelling of lightning discharge currents. The values of n and peak-times have been chosen manually, and k and c have been chosen by first fixing k and then numerically finding a c that gave a close approximation. For this purpose we used software for numerical computing [27], based on the interior reflective Newton method described in [28, 29]. This is then repeated for k = 1, . . . , 10 and the best fitting set of parameters is chosen. Note that this methodology uses all available data points to evaluate fitting but could probably be simplified further. For example, by using a simpler method for choosing c given k, only use a subset of available points to asses accuracy or, with sufficient experimentation find some suitable heuristic for choosing the appropriate value of k. Since the waveforms are given as data rather than explicit functions the D-optimal points have been calculated and then the closest available data points have been chosen. In these examples we did not require that the coefficients in the linear sums were positive. 4.1 Modelling of ESD currents 4.1.1 The IEC 61000-4-2 standard current waveshape ESD generators used in testing of the equipment and devices should be able to reproduce the same ESD current waveshape each time. This repeata- 32 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 33 10 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 61000-4-2 © IEC:1995+A1:1998 – 43 – +A2:2000 Values are given in table 2. Figure 3 – Typical waveform of the output current of the ESD generator tr Fig. 1: Illustration of the IEC 61000-4-2 Standard ESD current and its key parameters, [15]. bility feature is ensured if the design is carried out in compliance with the requirements defined in the IEC 61000-4-2 Standard, [15]. Among other relevant issues, the Standard includes graphical represen- tation of the typical ESD current, fig. 1, and also defines, for a given test level voltage, required values of ESD current’s key parameters. These are listed in Table 1 for the case of the contact discharge, where: • Ipeak is the ESD current initial peak; • tr is the rising time defined as the difference between time moments corresponding to 10% and 90% of the current peak Ipeak, fig. 1; • I30 and I60 are the ESD current values calculated for time periods of 30 and 60 ns, respectively, starting from the time point corresponding to 10% of Ipeak, fig. 1. 34 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 35 10 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 61000-4-2 © IEC:1995+A1:1998 – 43 – +A2:2000 Values are given in table 2. Figure 3 – Typical waveform of the output current of the ESD generator tr Fig. 1: Illustration of the IEC 61000-4-2 Standard ESD current and its key parameters, [15]. bility feature is ensured if the design is carried out in compliance with the requirements defined in the IEC 61000-4-2 Standard, [15]. Among other relevant issues, the Standard includes graphical represen- tation of the typical ESD current, fig. 1, and also defines, for a given test level voltage, required values of ESD current’s key parameters. These are listed in Table 1 for the case of the contact discharge, where: • Ipeak is the ESD current initial peak; • tr is the rising time defined as the difference between time moments corresponding to 10% and 90% of the current peak Ipeak, fig. 1; • I30 and I60 are the ESD current values calculated for time periods of 30 and 60 ns, respectively, starting from the time point corresponding to 10% of Ipeak, fig. 1. Electrostatic discharge currents representation using the analytically extended function...11 Table 1: IEC 61000-4-2 standard ESD current and its key parameters, [15]. Voltage [kV] Ipeak [A] tr [ns] I30 [A] I60 [A] 2 7.5 ± 15% 0.8 ± 25% 4.0 ± 30% 2.0 ± 30% 4 15.0 ± 15% 0.8 ± 25% 8.0 ± 30% 4.0 ± 30% 6 22.5 ± 15% 0.8 ± 25% 12.0 ± 30% 6.0 ± 30% 8 30.0 ± 15% 0.8 ± 25% 16.0 ± 30% 8.0 ± 30% In this section we present the results of fitting a 2-peak AEF to the Standard ESD current given in IEC 61000-4-2. Data points which are used in the optimization procedure are manually sampled from the graphically given Standard [15] current function, fig. 1. The peak currents and corresponding times are also extracted, and the results of D-optimal interpolation with two peaks are illustrated, see fig. 2. The parameters are listed in Table 3. In the illustrated examples a fairly good fit is found but typically areas with steeply rising and decaying parts are somewhat more difficult to fit with good accuracy than the other parts of the waveform. 4.1.2 3-peaked AEF representing measured current from ESD In this section we present the results of fitting a 3-peaked AEF to a waveform from experimental measurements from [16]. The result is also compared to a common type of function used for modelling ESD current, also from [16]. In figs. 3 and 4 the results of the interpolation of D-optimal points are shown together with the measured data, as well as a sum of two Heidler functions that was fitted to the experimental data in [16]. This function is given by i(t) = I1 ( t τ1 )nH 1 + ( t τ1 )nH e − t τ2 + I2 ( t τ3 )nH 1 + ( t τ3 )nH e − t τ4 , I1 = 31.365 A, I2 = 6.854 A, nH = 4.036, τ1 = 1.226 ns, τ2 = 1.359 ns, τ3 = 3.982 ns, τ4 = 28.817 ns. Note that this function does not reproduce the second local minimum but that all three AEF functions can reproduce all local minima and maxima 34 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 35 12 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV Table 2: IEC 61312-1 standard current waveshape and its key parame- ters, [17]. Protection level Parameter First stroke Subsequent stroke n 10 10 T 19.0 µs 0.454 µs τ 485 µs 143 µs η 0.930 0.993 I Ipeak 200 kA 50 kA II Ipeak 150 kA 37.5 kA III-IV Ipeak 100 kA 25 kA (to a modest degree of accuracy) when suitable values for the n, k and c parameters are chosen. In fig. 4 we can see that even small bumps in he rising part are successfully reproduced. 4.2 Modelling of lightning discharge currents 4.2.1 IEC 61312-1 standard current waveshape In this section we use the scheme to represent the IEC 61312-1 Standard current wave shape as it is described in [18]. Rather than being given graph- ically, as the IEC 61000-4-2 Standard current waveform, the shape is de- scribed using a Heidler function, i(t) = Ipeak η ( t T )n 1 + ( t T )n e− t τ (8) whose parameters are chosen according to Table 2. In figs. 5 and 6 the results of fitting an AEF by interpolating on a D-optimal design to the first stroke of a protection level I IEC 61312-1 Standard waveshape are shown. The parameters of the fitted AEF are given in Table 5. In this case the waveshape can be reproduced fairly well but gives a relatively complicated expression compared to (8). 36 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 37 12 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV Table 2: IEC 61312-1 standard current waveshape and its key parame- ters, [17]. Protection level Parameter First stroke Subsequent stroke n 10 10 T 19.0 µs 0.454 µs τ 485 µs 143 µs η 0.930 0.993 I Ipeak 200 kA 50 kA II Ipeak 150 kA 37.5 kA III-IV Ipeak 100 kA 25 kA (to a modest degree of accuracy) when suitable values for the n, k and c parameters are chosen. In fig. 4 we can see that even small bumps in he rising part are successfully reproduced. 4.2 Modelling of lightning discharge currents 4.2.1 IEC 61312-1 standard current waveshape In this section we use the scheme to represent the IEC 61312-1 Standard current wave shape as it is described in [18]. Rather than being given graph- ically, as the IEC 61000-4-2 Standard current waveform, the shape is de- scribed using a Heidler function, i(t) = Ipeak η ( t T )n 1 + ( t T )n e− t τ (8) whose parameters are chosen according to Table 2. In figs. 5 and 6 the results of fitting an AEF by interpolating on a D-optimal design to the first stroke of a protection level I IEC 61312-1 Standard waveshape are shown. The parameters of the fitted AEF are given in Table 5. In this case the waveshape can be reproduced fairly well but gives a relatively complicated expression compared to (8). Electrostatic discharge currents representation using the analytically extended function...13 4.2.2 Modelling a measured lightning discharge current In this section we fit an AEF function both with free parameters (as in [6]) and using interpolation on a D-optimal design, to data extracted from [20] that comes from measurements of a lightning strike on Mount Säntis in Switzerland [30]. We used a 13-peaked AEF and the results are shown in figs. 7a, 7c and 7e. Often the curves are similar enough that it can be hard to spot the differences so the residuals of the two models relative to the measured current is shown in figs. 7b, 7d and 7f. It can be seen that in most cases the AEF with free parameters gives a closer fit but the version interpolated on a D-optimal design is often comparable. Parameters for the D-optimal fitting can be found in Table 6. 4.2.3 Modelling the lightning discharge current derivative Here we present some results when attempting to reproduce the derivative of the waveshape of the lightning discharge current using the AEF interpolated on a D-optimal design. We also compare the result of this fitting scheme to the results in [13] where the parameters of the AEF are chosen freely and fitted using the Marquardt Least-Squares Method. The method for fitting an AEF described in this paper is applied to the modelling of lightning current derivative signals measured at the CN Tower [20]. The results of the fitting can be seen in fig. 8. From these figures it is clear that in this case of several peaks and few terms in each interval the two schemes for fitting the AEF are often similar in quality but sometimes the extra flexibility offered when letting all the exponents in the AEF be chosen individually can give a significantly better fit, an example of this can be seen in fig. 8. A possible explanation for this in this case is that in the scheme for D-optimal fitting you need many terms in order to have both small and large exponents. In fig. 9 we examine how well the different fitting schemes model the current when they are integrated. Here we can see that the free parameter version gives a considerably better matching to the numerically integrated measured values than the D-optimal fitting version. 36 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 37 14 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-8 0 2 4 6 8 i( t) 0 5 10 15 IEC 61000-4-2 2-peaked AEF Peaks Interpolated points Fig. 2: 2-peaked AEF representing the IEC 61000-4-2 Standard ESD current waveshape for 4kV. Parameters are given in Table 3. Table 3: Parameters’ values of AEF with 2 peaks representing the IEC 61000-4-2 standard waveshape. Local maxima and minima and corresponding times extracted from the IEC 61000-4-2, [15] Imax1 = 15 [A] Imin1 = 7.1484 [A] Imax2 = 9.0921 [A] tmax1 = 6.89 [ns] tmin1 = 12.85 [ns] tmax2 = 25.54 [ns] Parameters of interpolated AEF shown in fig. 2 Interval n k c 0 ≤ t ≤ tmax1 3 1 0.01385 tmax1 ≤ t ≤ tmax2 3 4 2.025 tmax2 < t 5 10 2.395 Electrostatic discharge currents representation using the analytically extended function...15 t [s] #10-8 0 1 2 3 4 5 6 7 8 9 i( t) 0 2 4 6 Measured data 3-peaked AEF Two Heidler function Peaks Interpolated points Fig. 3: 3-peaked AEF interpolated to D-optimal points chosen from mea- sured ESD current from [16, fig.3] compared with the sum of two Heidler functions suggested in [16]. Parameters are given in Table 4. Table 4: Parameters’ values of AEF with 3 peaks representing measured ESD. Local maxima and corresponding times extracted from [16, fig.3] Imax1 = 7.37 [A] Imax2 = 5.02 [A] Imax3 = 3.82 [A] tmax1 = 1.23 [ns] tmax2 = 6.39 [ns] tmax3 = 15.5 [ns] Parameters of interpolated AEF shown in fig. 3 Interval n k c 0 ≤ t ≤ tmax1 5 5 0.05750 tmax1 ≤ t ≤ tmax2 3 1 0.4920 tmax2 ≤ t ≤ tmax3 4 2 0.5967 tmax3 < t 6 1 1.019 38 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 39 14 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-8 0 2 4 6 8 i( t) 0 5 10 15 IEC 61000-4-2 2-peaked AEF Peaks Interpolated points Fig. 2: 2-peaked AEF representing the IEC 61000-4-2 Standard ESD current waveshape for 4kV. Parameters are given in Table 3. Table 3: Parameters’ values of AEF with 2 peaks representing the IEC 61000-4-2 standard waveshape. Local maxima and minima and corresponding times extracted from the IEC 61000-4-2, [15] Imax1 = 15 [A] Imin1 = 7.1484 [A] Imax2 = 9.0921 [A] tmax1 = 6.89 [ns] tmin1 = 12.85 [ns] tmax2 = 25.54 [ns] Parameters of interpolated AEF shown in fig. 2 Interval n k c 0 ≤ t ≤ tmax1 3 1 0.01385 tmax1 ≤ t ≤ tmax2 3 4 2.025 tmax2 < t 5 10 2.395 Electrostatic discharge currents representation using the analytically extended function...15 t [s] #10-8 0 1 2 3 4 5 6 7 8 9 i( t) 0 2 4 6 Measured data 3-peaked AEF Two Heidler function Peaks Interpolated points Fig. 3: 3-peaked AEF interpolated to D-optimal points chosen from mea- sured ESD current from [16, fig.3] compared with the sum of two Heidler functions suggested in [16]. Parameters are given in Table 4. Table 4: Parameters’ values of AEF with 3 peaks representing measured ESD. Local maxima and corresponding times extracted from [16, fig.3] Imax1 = 7.37 [A] Imax2 = 5.02 [A] Imax3 = 3.82 [A] tmax1 = 1.23 [ns] tmax2 = 6.39 [ns] tmax3 = 15.5 [ns] Parameters of interpolated AEF shown in fig. 3 Interval n k c 0 ≤ t ≤ tmax1 5 5 0.05750 tmax1 ≤ t ≤ tmax2 3 1 0.4920 tmax2 ≤ t ≤ tmax3 4 2 0.5967 tmax3 < t 6 1 1.019 38 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 39 16 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-9 0 2 4 i( t) 0 2 4 6 Measured data 3-peaked AEF Two Heidler function Peaks Interpolated points Fig. 4: Close-up of the rising part of a 3-peaked AEF interpolated to D- optimal points chosen from measured ESD current from [16, fig.3]. Parameters are given in Table 4. Table 5: Parameters’ values of AEF representing the IEC 61312-1 standard waveshape. Chosen peak time and current tmax = 28.14 [µ s] I = 92.54 [kA] Parameters of interpolated AEF shown in fig. 5 Interval n k c 0 ≤ t ≤ tmax 4 10 0.7565 tmax < t 5 1 41.82 40 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 41 16 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-9 0 2 4 i( t) 0 2 4 6 Measured data 3-peaked AEF Two Heidler function Peaks Interpolated points Fig. 4: Close-up of the rising part of a 3-peaked AEF interpolated to D- optimal points chosen from measured ESD current from [16, fig.3]. Parameters are given in Table 4. Table 5: Parameters’ values of AEF representing the IEC 61312-1 standard waveshape. Chosen peak time and current tmax = 28.14 [µ s] I = 92.54 [kA] Parameters of interpolated AEF shown in fig. 5 Interval n k c 0 ≤ t ≤ tmax 4 10 0.7565 tmax < t 5 1 41.82 Electrostatic discharge currents representation using the analytically extended function...17 t [s] #10-3 0 0.5 1 1.5 2 2.5 3 i( t) 0 20 40 60 80 IEC 61312-1 D-optimal AEF Peak Interpolated sample points Fig. 5: AEF with 1 peak fitted by interpolating D-optimal points sampled from the Heidler function describing the IEC 61312-1 waveshape given by (8). Parameters are given in Table 5. t [s] #10-5 0 1 2 3 i( t) 0 20 40 60 80 IEC 61312-1 D-optimal AEF Peak Interpolated points Fig. 6: Close-up of the rising part of the AEF with 1 peak fitted by interpo- lating D-optimal points samples from the Heidler function describing the IEC61312-1 waveshape given by (8). Parameters are given in Ta- ble 5. 40 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 41 18 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-4 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (a) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 888.1 µs. t [s] #10-4 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (b) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 888.1 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (c) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (d) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (e) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 5.116 µs. t [s] #10-6 0 2 4 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (f) Residuals when comparing the fitted function to the measured data from t = −0.3437 µs to t = 5.116 µs. Fig. 7: Comparison of two AEFs with 13 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [19]. One is fitted by interpolation on D-optimal points and the other is fitted with free parameters using the MLSM method. Parameters of the D-optimal version are given in Table 6. 18 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-4 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (a) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 888.1 µs. t [s] #10-4 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (b) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 888.1 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (c) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (d) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (e) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 5.116 µs. t [s] #10-6 0 2 4 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (f) Residuals when comparing the fitted function to the measured data from t = −0.3437 µs to t = 5.116 µs. Fig. 7: Comparison of two AEFs with 13 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [19]. One is fitted by interpolation on D-optimal points and the other is fitted with free parameters using the MLSM method. Parameters of the D-optimal version are given in Table 6. 18 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-4 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (a) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 888.1 µs. t [s] #10-4 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (b) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 888.1 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (c) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (d) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (e) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 5.116 µs. t [s] #10-6 0 2 4 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (f) Residuals when comparing the fitted function to the measured data from t = −0.3437 µs to t = 5.116 µs. Fig. 7: Comparison of two AEFs with 13 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [19]. One is fitted by interpolation on D-optimal points and the other is fitted with free parameters using the MLSM method. Parameters of the D-optimal version are given in Table 6. 42 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 43 18 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-4 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (a) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 888.1 µs. t [s] #10-4 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (b) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 888.1 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (c) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (d) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (e) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 5.116 µs. t [s] #10-6 0 2 4 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (f) Residuals when comparing the fitted function to the measured data from t = −0.3437 µs to t = 5.116 µs. Fig. 7: Comparison of two AEFs with 13 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [19]. One is fitted by interpolation on D-optimal points and the other is fitted with free parameters using the MLSM method. Parameters of the D-optimal version are given in Table 6. Electrostatic discharge currents representation using the analytically extended function...19 Table 6: Parameters’ values of AEF with 13 peaks representing measured data for a lightning discharge current from [30]. Local maxima and corresponding times extracted from [19, figs.6, 7 and 8] are denoted t and I and other parameters correspond to the fitted AEF shown in figs. 7a, 7c and 7e. Peak times and currents Parameters of fitted AEF t [µs] I [µs] Interval n k c t1 = 0.3998 I1 = 8.159 0 ≤ t ≤ t1 2 2 0.4773 t2 = 0.9468 I2 = 10.96 t1 ≤ t ≤ t2 2 10 2.148 t3 = 1.458 I3 = 11.14 t2 ≤ t ≤ t3 2 1 0.3964 t4 = 1.873 I4 = 10.26 t3 ≤ t ≤ t4 2 1 0.2210 t5 = 2.475 I5 = 10.07 t4 ≤ t ≤ t5 2 10 1.695 t6 = 2.904 I6 = 9.819 t5 ≤ t ≤ t6 2 1 0.4591 t7 = 3.533 I7 = 8.519 t6 ≤ t ≤ t7 2 1 0.3503 t8 = 3.985 I8 = 9.097 t7 ≤ t ≤ t8 2 10 3.716 t9 = 5.036 I9 = 8.485 t8 ≤ t ≤ t9 2 1 0.6963 t10 = 6.168 I10 = 8.310 t9 ≤ t ≤ t10 2 1 0.2954 t11 = 8.472 I11 = 8.413 t10 ≤ t ≤ t11 2 6 3.074 t12 = 20.48 I12 = 8.576 t11 ≤ t ≤ t12 2 1 0.2784 t13 = 137.5 I13 = 4.178 t12 ≤ t ≤ t13 2 1 0.6456 t13 < t 4 1 0.3559 18 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-4 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (a) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 888.1 µs. t [s] #10-4 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (b) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 888.1 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (c) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (d) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (e) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 5.116 µs. t [s] #10-6 0 2 4 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (f) Residuals when comparing the fitted function to the measured data from t = −0.3437 µs to t = 5.116 µs. Fig. 7: Comparison of two AEFs with 13 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [19]. One is fitted by interpolation on D-optimal points and the other is fitted with free parameters using the MLSM method. Parameters of the D-optimal version are given in Table 6. 18 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-4 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (a) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 888.1 µs. t [s] #10-4 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (b) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 888.1 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (c) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 6 8 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (d) Residuals when comparing the fitted function to the mea- sured data from t = −0.3437 µs to t = 9.280 µs. t [s] #10-6 0 2 4 i( t) 0 2 4 6 8 10 Measured data D-optimal AEF Free parameter AEF Peaks Interpolated sample points (e) Comparison of measured data and AEF functions from t = −0.3437 µs to t = 5.116 µs. t [s] #10-6 0 2 4 i( t) 0 0.2 0.4 0.6 0.8 1 1.2 Residual D-optimal AEF Residual D-optimal AEF (f) Residuals when comparing the fitted function to the measured data from t = −0.3437 µs to t = 5.116 µs. Fig. 7: Comparison of two AEFs with 13 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [19]. One is fitted by interpolation on D-optimal points and the other is fitted with free parameters using the MLSM method. Parameters of the D-optimal version are given in Table 6. 42 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 43 20 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-6 0 2 4 6 di /d t[ kA /s ] 0 10 20 30 Measured data D-optimal 12-peaked AEF Free parameter AEF Peaks Interpolated sample points Fig. 8: Comparison of two AEFs with 12 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [20]. One is fitted by interpolation on D-optimal points and one is fitted with free parameters using the MLSM method. Parameters are given in Table 7. t [s] #10-6 0 2 4 6 i [k A ] 0 5 10 Measured data D-optimal AEF Free parameter AEF Fig. 9: Comparison of results of integrating the results shown in fig. 8. 44 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 45 20 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV t [s] #10-6 0 2 4 6 di /d t[ kA /s ] 0 10 20 30 Measured data D-optimal 12-peaked AEF Free parameter AEF Peaks Interpolated sample points Fig. 8: Comparison of two AEFs with 12 peaks and 2 terms in each linear combination fitted to measured lightning discharge current derivative from [20]. One is fitted by interpolation on D-optimal points and one is fitted with free parameters using the MLSM method. Parameters are given in Table 7. t [s] #10-6 0 2 4 6 i [k A ] 0 5 10 Measured data D-optimal AEF Free parameter AEF Fig. 9: Comparison of results of integrating the results shown in fig. 8. Electrostatic discharge currents representation using the analytically extended function...21 Table 7: Parameters’ value of AEF with 12 peaks representing measured data for a lightning discharge current derivative from [20]. Chosen peak times are denoted t and I and other parameters correspond to the fitted AEF shown in fig. 8. Peak times and currents Parameters of fitted AEF t [µs] I [µs] Interval n k c t0 = −0.3437 I0 = 0 t0 ≤ t ≤ t1 2 10 0.06099 t1 = 0.9468 I1 = 36.65 t1 ≤ t ≤ t2 2 1 0.4506 t2 = 0.5001 I2 = −2.208 t2 ≤ t ≤ t3 3 1 0.04772 t3 = 0.9215 I3 = 6.89 t3 ≤ t ≤ t4 2 1 0.4502 t4 = 1.212 I4 = −7.322 t4 ≤ t ≤ t5 3 1 0.2590 t5 = 1.714 I5 = 3.402 t5 ≤ t ≤ t6 3 2 0.9067 t6 = 2.103 I6 = 1.319 t6 ≤ t ≤ t7 3 1 0.3333 t7 = 2.730 I7 = −1.844 t7 ≤ t ≤ t8 3 1 0.03732 t8 = 3.416 I8 = 16.08 t8 ≤ t ≤ t9 2 4 3.3793 t9 = 4.005 I9 = −5.787 t9 ≤ t ≤ t10 2 1 1.4912 t10 = 4.216 I10 = −0.1268 t10 ≤ t ≤ t11 2 2 0.09448 t11 = 4.875 I11 = 1.972 t11 ≤ t ≤ t12 2 6 2.288 t12 = 5.538 I12 = 1.683 t13 < t 3 1 0.001705 44 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 45 22 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 5 Conclusion In this work we examine a mathematical model for representation of various ESD currents or their derivative and apply it to some realistic cases, either taken from standards, see section 4.1.1 and 4.2.1, or measured data, see sections 4.1.2, 4.2.2 and 4.2.3. The model is based around the multi-peaked analytically extended func- tion (AEF), see section 2.1, has been proposed and successfully applied to lightning current modelling in [6,9–11]. It matches common requirements of ESD-type currents, such as stating that the function and its first derivative must be equal to zero at the starting time. Furthermore, the AEF function is time-integrable, [11], which is nec- essary for numerical calculation of radiated fields originating from the ESD current. We construct the model by restricting the exponents in the AEF to an arithmetic sequence and then interpolate points of the function we wish to approximate chosen according to a D-optimal design. This makes the modelling less flexible than the case where all exponents can be chosen freely but gives a scheme for fitting the function that scales better to many data points than the MLSM fitting scheme used in [6,9–11]. The resulting methodology can give fairly accurate results even with a modest number of interpolated points but strategies for choosing some of the involved parameters should be further investigated. The decaying part of the waveforms are consistently difficult to fit and if the models are used in a context where significant error propagation appears a more flexible approach can be desirable. Acknowledgments The authors would like to thank Dr. Pavlos Katsivelis from the High Volt- age Laboratory, School of Electrical and Computer Engineering, National Technical University of Athens, Greece, for providing measured ESD cur- rent data. References [1] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”Multi-Peaked Analyti- cally Extended Function Representing Electrostatic Discharge (ESD) Currents,” in AIP Conference Proceedings ICNPAA, La Rochelle, France, pp. 1–10, 2016. 46 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 47 22 K. LUNDENGÅRD, M. RANČIĆ, V. JAVOR AND S. SILVESTROV 5 Conclusion In this work we examine a mathematical model for representation of various ESD currents or their derivative and apply it to some realistic cases, either taken from standards, see section 4.1.1 and 4.2.1, or measured data, see sections 4.1.2, 4.2.2 and 4.2.3. The model is based around the multi-peaked analytically extended func- tion (AEF), see section 2.1, has been proposed and successfully applied to lightning current modelling in [6,9–11]. It matches common requirements of ESD-type currents, such as stating that the function and its first derivative must be equal to zero at the starting time. Furthermore, the AEF function is time-integrable, [11], which is nec- essary for numerical calculation of radiated fields originating from the ESD current. We construct the model by restricting the exponents in the AEF to an arithmetic sequence and then interpolate points of the function we wish to approximate chosen according to a D-optimal design. This makes the modelling less flexible than the case where all exponents can be chosen freely but gives a scheme for fitting the function that scales better to many data points than the MLSM fitting scheme used in [6,9–11]. The resulting methodology can give fairly accurate results even with a modest number of interpolated points but strategies for choosing some of the involved parameters should be further investigated. The decaying part of the waveforms are consistently difficult to fit and if the models are used in a context where significant error propagation appears a more flexible approach can be desirable. Acknowledgments The authors would like to thank Dr. Pavlos Katsivelis from the High Volt- age Laboratory, School of Electrical and Computer Engineering, National Technical University of Athens, Greece, for providing measured ESD cur- rent data. References [1] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”Multi-Peaked Analyti- cally Extended Function Representing Electrostatic Discharge (ESD) Currents,” in AIP Conference Proceedings ICNPAA, La Rochelle, France, pp. 1–10, 2016. Electrostatic discharge currents representation using the analytically extended function...23 [2] V. Javor, ”Multi-Peaked Functions for Representation of Lightning Channel- Base Currents,” 31st Int. Conference on Lightning Protection ICLP 2012, September 2-7, 2012, Proceedings of papers, Vienna, Austria, 2012. [3] V. Javor, ”New function for representing IEC 61000-4-2 standard electrostatic discharge current,” Facta Universitatis, Series: Electronics and Energetics, vol. 27(4), pp. 509–520, 2014. [4] V. Javor, K. Lundeng̊ard, M. Rančić, and S. Silvestrov, ”Measured electrostatic discharge currents modeling and simulation,” in Proc. of TELSIKS 2015, Nîs, Serbia, pp. 209–212, 2015. [5] V. Javor, ”New Function for Representing IEC 61000-4-2 Standard Electrostatic Discharge Current,” (invited paper), Facta Universitatis, Series Electronics and Energetics, Vol. 27(4), pp. 509–520, University of Ni, Serbia, 2014. [6] V. Javor, K. Lundeng̊ard, M. Rancic, S. Silvestrov, ”Analytical Representation of Measured Lightning Currents and Its Application to Electromagnetic Field Estimation,” IEEE Transactions on Electromagnetic Compatibility, vol. 60(5), pp. 1415–1426, 2018. [7] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”Application of the Marquardt least-squares method to the estimation of Pulse function parame- ters,” in AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, 2014, pp. 637–646. [8] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”Estimation of Pulse function parameters for approximating measured lightning currents using the Marquardt least-squares method,” in Conference Proceedings, EMC Europe, Gothenburg, Sweden, 2014, pp. 571–576. [9] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”An Examination of the multi-peaked analytically extended function for approximation of lightning channel-base currents,” in Proceedings of Full Papers, PES 2015, Nǐs, Serbia, Electronic, arXiv:1604.06517 [physics.comp-ph], 2015. [10] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”Application of the multi-peaked analytically extended function to representation of some measured lightning currents,” Serbian Journal of Electrical Engineering, vol. 13(2), pp. 1–11, 2016. [11] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”Estimation of pa- rameters for the multi-peaked AEF current functions,” Methodol. Comp. Appl. Probab., pp. 1–15, 2016. [12] K. Lundeng̊ard, M. Rančić, V. Javor, and S. Silvestrov, ”On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents” in: Sergei Silvestrov and Milica Rančić, editors, Engineer- ing Mathematics I: Electromagnetics, Fluid Mechanics, Material Physics and Financial Engineering, volume 178 of Springer Proceedings in Mathematics & Statistics. Springer International Publishing, 2016. 46 K. LUNDENgÅRD, M. RANcic, V. JAVoR S. SiLVEStRoV Electrostatic Discharge currents Representation Using the Analytically Extended Function... 47 24 K. LUNDENGÅRD, M. 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