CN103853696A - Impulse function representation method for lightning return stroke channel base current - Google Patents

Abstract

The invention provides an impulse function expression for describing lightning return stroke channel base current. An impulse function, which can be regarded as the modification of a double exponential function, can fully reflect standard-specified lightning current waveform characteristics and also overcomes the adverse factors of the base current models of the double exponential function and a Heidler function in lightning return stroke electromagnetic field calculation, not only is a first-order derivative continuous when t is equal to 0, but also the impulse function has an explicit time integral expression, and thereby a lightning return stroke electromagnetic field can be conveniently analytically calculated. In addition, in order to accurately obtain parameters which are used by the impulse function to represent the standard waveform of lightning current, a waveform nonlinear curve fitting method can be adopted.

Description

A kind of impulse function method for expressing of Fields of Lightning Return Stroke channel bottom electric current

Technical field

The present invention is applicable to the modeling of Fields of Lightning Return Stroke channel bottom electric current, describes lightning channel base current waveform with an impulse function, can be widely used in the theory calculating of Electromagnetic Fields of Lightning Return Stroke.

Background technology

Along with the fast development of microelectric technique and infotech, the integrated level of electronics and IT products is more and more higher, and its electromagnetic susceptibility is also more and more higher, and very weak electromagnetic field just may work the mischief.20 th century later, the harm that the Lightning Electromagnetic Pulse being produced by Fields of Lightning Return Stroke (LEMP) causes the electronic message unit that contains sensitive microelectronic component and system is more and more serious, and the loss causing is also increasing year by year.

The contingency occurring due to thunder and lightning is very strong, and very difficult actual measurement obtains in buildings or fights back parameatal Lightning Electromagnetic Fields distribution situation.In order to grasp the time space distribution of Electromagnetic Fields of Lightning Return Stroke, need to carry out modeling to Fields of Lightning Return Stroke process, and the source that Fields of Lightning Return Stroke channel bottom electric current produces as Electromagnetic Fields of Lightning Return Stroke is basis and the prerequisite that Electromagnetic Fields of Lightning Return Stroke is calculated.Calculate for simplifying, Fields of Lightning Return Stroke process can be simulated by simple antenna model, electrostatic field item in the Electromagnetic Fields of Lightning Return Stroke expression formula being obtained by this model is relevant with the time integral of return stroke current, and induction field item is relevant with return stroke current, and the derivative of radiation field Xiang Zeyu return stroke current is relevant.This just requires to fight back the preferably integrable function of analytical expression of channel bottom electric current, otherwise will carry out the very huge double integral of calculated amount in the time of numerical evaluation.Be commonly used to represent that, in the analytical function of Fields of Lightning Return Stroke channel bottom electric current, the expression formula of double-exponential function is

i(0，t)＝(I ₀/ω)[exp(-αt)-exp(-βt)] (1)

Wherein, I ₀be the maximal value of electric current, ω is the correction of peak value factor, and α, β are the time constants of determining electric current decline, rise time and maximum current steepness.This model-based originally can be reacted the situation of actual measurement base current, and is easy to differential and integration.But this function its derivative in the time of t=0 reaches maximal value, this is that explanation is obstructed for fighting back breakdown process physically; Secondly, derivative has point of discontinuity in the time of t=0, is not easy to the calculating of Electromagnetic Fields of Lightning Return Stroke.

The analytic expression of another common Heidler function is

$i_0\left(t\right)=({\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HSA00000817123800011.png"\; state="\{\{state\}\}">I</figure-callout>}}_0/\mathrm{\&eta;})[K_s^n/(1+K_s^n)]\exp (-t/{\mathrm{\&tau;}}_2)---\left(2\right)$

Wherein, K _s=t/ τ ₁, τ ₁, τ ₂be respectively Current rise, damping time constant, n is electric current steepness factor, η=exp[-(τ ₁/ τ ₂) (n τ ₂/ τ ₁) ^l/n] be the modifying factor of current maxima.Heidler function is by adjusting independently I ₀, τ ₁, τ ₂can change respectively current amplitude, maximum current derivative and charge transfer quantity, and be easy to prove that its single order electric current derivative is continuous in the time of t=0.Be integrable function not but the shortcoming of Heidler function is it, be not easy to directly carry out with it the analytical Calculation of Electromagnetic Fields of Lightning Return Stroke.For overcoming the shortcoming of double-exponential function and Heidler function, propose to represent with impulse function the method for lightning channel base current, and analytical expression using it as Fields of Lightning Return Stroke channel bottom electric current.

Summary of the invention

The object of this invention is to provide a kind of waveform character that can truly reflect lightning current, be again the Fields of Lightning Return Stroke channel bottom electric current analytical expression of integrable function, emphasis solves existing Fields of Lightning Return Stroke channel bottom current function not to be had continuous first order derivative or does not have explicit integration expression formula etc. to be not easy to carry out the problem of Electromagnetic Fields of Lightning Return Stroke calculating at t=0 place.

For addressing the above problem, the present invention adopts not only at t=0 place derivative continuously but also can represent Fields of Lightning Return Stroke base current waveform by long-pending impulse function, and its analytical expression is:

$i(0,t)=\frac{{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HSA00000817123800011.png"\; state="\{\{state\}\}">I</figure-callout>}}_0}{\mathrm{\&xi;}}{[1-\exp (-t/{\mathrm{\&tau;}}_1)]}^n\exp (-t{/\mathrm{\&tau;}}_2)---\left(3\right)$

Wherein, I ₀be peak point current, n > 1 is electric current steepness factor, τ ₁, τ ₂respectively the time constant that determines Current rise and decay,

for the correction of peak value factor.Under normal circumstances, electric current steepness factor is got n=2, down slope time constant τ ₁, τ ₂choose and can be similar in secundum legem the regulation of double-exponential function parameter alpha, β according to τ ₁≈ 1/ β, τ ₂≈ 1/ α determines.To more accurately obtain the parameter τ of impulse function ₁, τ ₂and n, can adopt the method for non-linear curve fitting to realize.

Represent lightning channel base current with impulse function, whole lightning current waveform features that not only can complete reflection standard regulation, and have the following advantages:

1) first order derivative of impulse function is continuous at t=0 place.This is consistent with actual physics situation.

2) impulse function has explicit integral expression.Solve the difficulty of calculating double integral in Electromagnetic Fields of Lightning Return Stroke expression formula, greatly reduced the amount of calculation of Lightning Electromagnetic Fields.

Brief description of the drawings

Below in conjunction with the drawings and specific embodiments, the present invention is further detailed explanation.

Fig. 1 is the channel bottom electric current that represents with impulse function, Heidler function and double-exponential function respectively and the comparison diagram of derivative waveform thereof.

Fig. 2 is the frequency spectrum of the current waveform that represents with impulse function, Heidler function and double-exponential function respectively in Fig. 1.

Embodiment

1) impulse function at t=0 place continuously and be the proof of integrable function

Analytical expression (3) the both sides differentiate of paired pulses function can obtain:

$\frac{\mathrm{di}(0,t)}{\mathrm{dt}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HSA00000817123800011.png"\; state="\{\{state\}\}">I</figure-callout>}}_0}{\mathrm{\&xi;}}{[1-\exp (-t/{\mathrm{\&tau;}}_1)]}^{n-1}\exp (-t/{\mathrm{\&tau;}}_2)[(\frac{n}{{\mathrm{\&tau;}}_1}+\frac{1}{{\mathrm{\&tau;}}_2})\exp (-t/{\mathrm{\&tau;}}_1)-\frac{1}{{\mathrm{\&tau;}}_2}]---\left(4\right)$

Easily prove, in the time of n > 1, the derivative of impulse function

be that derivative is continuous at t=0 place.

In order to ask i, (0, the t) integration to the time, by [1-exp (the t/ τ of first exponential term in formula (3) ₁)] ⁿexpansion obtains:

${[1-\exp (-t/{\mathrm{\&tau;}}_1)]}^n=\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}\frac{{(-1)}^{k}n!}{k!(n-k)!}\exp (-\mathrm{kt}/{\mathrm{\&tau;}}_1)---\left(5\right)$

Wherein, k unequal to k (k-1) 21.Therefore obtain:

$Q\left(t\right)=\underset{-\&infin;}{\overset{t}{\&Integral;}}i(0,\mathrm{\&tau;})\mathrm{d\&tau;}=\underset{0}{\overset{t}{\&Integral;}}i(0,\mathrm{\&tau;})\mathrm{d\&tau;}=\frac{{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HSA00000817123800011.png"\; state="\{\{state\}\}">I</figure-callout>}}_0}{\mathrm{\&xi;}}\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}\frac{{(-1)}^{k}n!}{k!(n-k)!}{\mathrm{\&tau;}}_k^{*}[1-\exp (-t/{\mathrm{\&tau;}}_k^{*})]---\left(6\right)$

Wherein,

hence one can see that, and this impulse function is an integrable function.

2) impulse function determination method for parameter

For the return stroke current waveform of standard, the parameter of general desirable impulse function is identical with the parameter of Heidler function, i.e. obtaining current steepness factor n=2, down slope time constant τ ₁, τ ₂all according in standard to the regulation of double-exponential function parameter alpha, β according to τ ₁≈ 1/ β, τ ₂≈ 1/ α determines.

Fig. 1 is the channel bottom electric current that represents with double-exponential function with impulse function, Heidler function respectively and the comparison diagram of derivative waveform thereof: when the parameter value of three kinds of functions is consistent, the channel bottom current waveform of the channel bottom current waveform representing with impulse function and double-exponential function and Heidler function representation is basically identical; And impulse function and Heidler function derivative waveform are substantially similar, are 0 at initial time, the derivative of double-exponential function reaches maximum in 0 moment.

Fig. 2 is the frequency spectrum of the current waveform that represents with impulse function, Heidler function and double-exponential function respectively in Fig. 1: the frequency distribution of the counterattack channel bottom current waveform of three kinds of function representations is basically identical, mainly concentrates on the frequency range of tens kilo hertzs.

For a certain specific return stroke current waveform, for more accurately determining the parameter τ of impulse function ₁, τ ₂and n, adopt the method for non-linear curve fitting to obtain corresponding undetermined parameter.The parameter that when table 1 is depicted as the exponential term n=2 of Heidler function, several conventional lightning current waveform Heidler functions and impulse function (underlining) represent and the error of the two waveform.

The comparison (n=2) of table 1Heidler function and impulse function parameter

The parameter matching degree of these two kinds of models is very good as can be seen from Table 1, and it is suitable as the analytical expression of Fields of Lightning Return Stroke channel bottom electric current that this explanation replaces Heidler function with impulse function.And, during with impulse function and the same waveform of Heidler function representation, τ ₂difference little, but τ ₁difference large (as I and V in table).Because the Section 1 of impulse function expansion is the dominant term of determining function decay just, the therefore τ of impulse function ₂the inevitable τ close to Heidler function ₂, and double-exponential function is had to α ≈ 1/ τ ₂.In fact impulse function double-exponential function in the time of n=1, general impulse function can be regarded as the correction of double-exponential function.Table 2 is according to IEC1312 standard, the parameter obtaining with impulse function matching when the exponential term of Heidler function is defined as to n=10.From table 2, can see, in the time of n=10, the error of Heidler function and impulse function is also smaller, is mainly that the variation range of exponential term n is larger.

The comparison (n=10) of table 2Heidler function and impulse function parameter

。

Claims (2)

1. be applied to an impulse function method for expressing of describing Fields of Lightning Return Stroke channel bottom electric current, it is characterized in that:

Impulse function of the present invention is as the analytical expression of Fields of Lightning Return Stroke channel bottom electric current, can be regarded as the correction of double-exponential function, compare with Heidler function with the double-exponential function of conventional description Fields of Lightning Return Stroke channel bottom electric current, have their advantage concurrently: its first order derivative is continuous at t=0, and it is integrable function, this not only makes the expression mode of lightning channel base current more meet the physical mechanism of Fields of Lightning Return Stroke, also solved the difficulty of double integral in Electromagnetic Fields of Lightning Return Stroke calculation expression simultaneously, the amount of calculation of Electromagnetic Fields of Lightning Return Stroke is reduced greatly, secondly, while representing standard thunder and lightning waveform with impulse function parameter choose also very convenient: parameter τ ₁and τ ₂choose and can be directly determine (that is:, τ by two exponential waveform parameter alpha and the β of national standard (GB50058-94) regulation ₁=1/ β, τ ₂=1/ α), parameter will obtain more accurately impulse function and represent reference waveform time, can adopt the method for waveform non-linear curve fitting to realize.

2. the modeling, the numerical evaluation of Electromagnetic Fields of Lightning Return Stroke etc. that the impulse function method for expressing of above-mentioned Fields of Lightning Return Stroke channel bottom electric current are applied to Fields of Lightning Return Stroke electric current, be also subject to the protection of this patent.

Images