Mathematical method for air electrostatic discharge circuits calculation

Here, a new analytical method, state matrix transform method, is proposed for calculating the descriptions of current waveforms generated by commercial electrostatic discharge (ESD) generators. This method can be used to analyse and design lumped parameter ESD circuits. A relatively simple eight elements ESD circuit model is established and used to predict the current waveform of human-metal ESD events. To calibrate the ESD simulator, a mathematical expression of four exponentials curve for ESD current waveform is given. This careful theoretical study of ESD current waveform calculation makes an innovative contribution to ESD generator circuit simulation and design, and it provides a new mathematical description for ESD sensitivity test.


Introduction
With the development of miniaturisation and modularisation of the integrated circuits, the sensitive electronic components or systems face a great threat from electrostatic discharge (ESD) current, and ESD has become a major concern in the microelectronic and electronic industry [1][2][3].A typical ESD event is a fast transient electric charge transfer phenomenon, which occurs between bodies with different electrostatic potentials in proximity or through direct contact [4,5].In [6], that ESD event is a charge-driven phenomenon through an analysis of the effect of charge injection due to charged model ESD in capacitive microelectromechanical systems structures is proved.Directly or indirectly, ESD events can interfere with the device or equipment either by the current or by the associated transient electromagnetic fields [7].Therefore, it is significant to study the ESD current waveform and an accurate description of ESD current for ESD sensitivity test is necessary.
Under the influence of environmental factors such as temperature, humidity, the charging voltage, metal approach speed, the topological structure of human body and ESD events, electrode types, and human body own inner electrical parameters [1,8], ESD phenomenon has a poor repeatability [9] and may produce a fast electromagnetic pulses similar to travelling waves [10], accompanied by strong electromagnetic radiation.From this reason, the IEC 61000-4-2 standard [4] defines two kinds of discharge modes: contact discharge and air discharge, where the contact discharge mode is an idealised test method, but the air discharge mode is more closer to real events for establishing a common and reproducible basis for simulating the natural ESD phenomenon as well as possible.In addition, the standard presents the typical waveform of the contact ESD current, its four parameters including first peak current value I p , rise time t r , current value I 30 at 30 ns, and current value I 60 at 60 ns and ideal expression.Besides, the consists and values for commercial ESD test generators are also given in the standard.In view of this situation, researchers have proposed a good many ESD circuit implementation schemes [11][12][13].Moreover, some different algorithms have been used to fit the reference waveform described in the standard, such as BP neural network algorithm [2], genetic algorithm [14], the polynomial of pulse function [15], and the method of moments in time domain [16].Although the current expressions in [17] meet the standard requirements, there are still several shortcomings in current descriptions.That is, the resulting expressions will show diversity and cannot accurately reflect the reality of the ESD events if the typical current waveform just take mathematical fitting.
In [18], a dipole source model for predicting the ESD radiated fields is established, indicating that the near-fields are mainly depend on the ESD current value, and the time derivative of the current will largely determine the far-fields [19].In [20,21], it is shown that the ESD signal has a much wider frequency width, and may cause the destruction of the electronic devices through entering the coaxial cables connecting systems.The ESD current waveform usually may be simulated by commercially available ESD generators, and the current generating circuits of most commercial ESD simulators are lumped parameter circuit in order to ensure the repeatability of the test results.It is difficult to achieve the corresponding waveform using a specific circuit if the standard waveform curve is only analysed by data fitting.To achieve this aim, a new analytical method -state matrix transform method (SMTM) -suitable for general lumped parameter circuits is presented.According to the standard IEC 61000-4-2(2008), an eight elements ESD circuit model which is suitable to be implemented in commercial circuit simulators is given, and a new expression of ESD current waveform is provided by the way.The validation of the presented circuit and the corresponding current description are examined by comparing with the standard.
The goal of this paper is to establish a general calculation method for ESD current performed by ESD guns and to present an ESD circuit conform to the standard.The structure of this paper is as follows: Section 2 introduces the principle of a new analytical method -SMTM suitable for general lumped parameter circuits in detail.In Section 3, the topological structure and the related electrical parameters of an eight elements ESD circuit are given, and the eight elements ESD circuit is solved using SMTM, and a new mathematical expression of ESD current is yielded.The obtained current expressions are compared with the standard in Section 4. In Section 5, a final discussion is briefly carried out to summarise the main conclusions of this paper.

State matrix transform method
The current generated by a general linear lumped parameter circuit comprised m circuit inductances and n energy storage capacitors can be analysed using the state equations of the circuit given by where x is the state vector of the circuit, as follows: A (m+n)×(m+n) is the state matrix, as follows: and i m represents the current through the mth inductance, and u m+n is the voltage at the ends of the nth capacitor.
Let us assume that the initial conditions of the circuit are as follows: Solving differential equations ( 1) and ( 4), the details of the derivation are left to Appendix 8.1, we obtain the mathematical description for current waveforms produced by the commercial ESD generators, as follows: These represent the desired current waveform expressions.

Eight elements ESD circuit model
According to the anatomical structure model of human body, it is assumed that the ESD discharge circuit composed of the body and the earth is equivalent to a parallel path composed of a series of discharge resistance R 0 and loop inductance L 0 with a parasitic capacitor C 0 .We regard the fingertip and the metal as the positive and negative poles of the ESD source consisting of energycapacitor C, discharge resistor R, and inductance L, respectively.Therefore, an eight elements ESD circuit model shown in Fig. 1 is established, where K 0 represents discharge switch, and K is the charge switch.
The parameters of the eight elements ESD circuit are shown in Table 1.
The equivalent circuit diagram of the eight elements ESD circuit model is presented in Fig. 2.
The current simulated by the eight elements ESD circuit may be analysed using SMTM; the details of the derivation are left to Appendix 8.2.When the charge voltage is 4 kV, the current expression is obtained where The current waveform performed by the eight elements ESD circuit at 4 kV is presented in Fig. 2. Also, the current waveform parameters of the eight elements ESD circuit at 4 kV are given in Table 2.

Comparison
The International Electrotechnical Commission (IEC) 61000-4-2 standard [4] defines the typical waveform of ESD current, and gives four waveform parameters, namely first peak current value I p , rise time t r , current value I 30 at 30 ns, and current value I 60 at 60 ns.Fig. 3 shows the contrast between the typical ideal current waveform of the contact ESD described in IEC 61000-4-2 and the  current waveform of the eight elements ESD circuit at 4 kV.Table 3 gives the four main parameters of the ideal current waveform.
According to Fig. 3 and Tables 2 and 3, it is obvious that the current waveforms generated by the eight elements ESD circuit are close to the standard waveform and the four waveform parameters meet the standard requirements.

Conclusion
A new calculation method (SMTM) has been proposed.The method is an accurate way for solving general lumped parameter circuits, and provides a universal way for solving the problems about the first-order linear differential equations.The proposed methodology can calculate the analytical expression of the current and the voltage of the element in a circuit, and the PSPICE or fullwave simulation method cannot do it.The efficiency of SMTM has been confirmed by calculating the eight elements ESD circuit.
An eight elements ESD circuit model and a new ESD current expression have been provided.The eight elements ESD circuit is appropriate to be actualised in commercial ESD generators and can permit to reproduce the standard current wave shape specified by the IEC 61000-4-2 for the ESD immunity testing.The new mathematical current expression can be used to calibrate the ESD simulator.The efficiency and accuracy of the ESD circuit is validated by the satisfactory agreement between the results and the standard.

State matrix transform method
As mentioned in Section 2, for the easy of study, we let we apply a non-singular linear transformation to the state vector of the differential equations, as follows: where Substituting ( 8) into (1) yields V dy dt = AV y (10) which implies that where J (m+n)×(m+n) is the Jordan standard form of A (m+n)×(m+n) , as follows:  where and j p represents the algebraic multiplicity of the eigenvalue λ p of the matrix A (m+n)×(m+n) , q is the number of eigenvalues of matrix A (m+n)×(m+n) , the j p satisfy the equation as follows: If we let where We find that that is (see (18)) .We may now solve for y j p , and we find that Solving differential equations ( 18) and ( 19), we obtain a recurrence formula as follows: Solving the recurrence formula (20) yields (see equation below) which implies that (see equation below).Combining the results yields y = y j 1 , y j 2 , …, y j q T (15) For the non-singular transformation matrix V (m+n)×(m+n) , we can solve it by the following way.
If we let where We find that which implies that Combining the results yields Since that is Solving the linear equations (26) yields Combining the transformation relation (8) yields that is Thus, we obtain the mathematical description for current waveforms produced by the commercial ESD generators, as follows: These represent the desired current waveform expressions.

Eight elements ESD circuit model
The capacitor C 0 and C are assumed to have voltage u 0 and u, and the loop inductance L 0 and L are assumed to have current i 0 and i.

Fig. 3
Fig. 3 Comparison of different current waveforms at 4 kV

Table 2
Current waveform parameters of the eight elements ESD circuit at 4 kV Parameter Value High Volt., 2018, Vol. 3 Iss.3, pp.226-231 This is an open access article published by the IET and CEPRI under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/)