Review of discharge theory and numerical research on intrinsically safe low voltage DC circuits
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摘要: 分析了本质安全电阻性、电感性和电容性电路的放电特性，指出电感性、电容性电路放电能量较大，且放电波形较复杂，而电阻性电路放电特性相对简单。总结了本质安全电感性电路和电容性电路放电机理及其放电数学模型研究现状，给出了放电电流线性衰减模型、放电电流抛物线模型、静态伏安特性模型、动态伏安特性模型、电弧电阻指数模型等的数学表达式。介绍了宏观及微观气体放电数值模拟方法的优缺点：宏观方法计算量小，但只能仿真气体放电的外部特性；微观方法计算量大，但能精确仿真放电过程中各粒子的运动特性。探究了直流放电电路和放电电弧的微观机理，得出了不同电极材料、不同电极间距及不同初始电气参数下，不同表面发射机理如热场致电子发射、场致发射等对气体放电的影响。针对研究现状，提出了本质安全直流电路放电研究需进一步解决的问题：① 宏观试验−数学模型表达式复杂、计算量大，部分模型适用范围单一，无法真正实现本质安全电路非爆炸评估。 ② 本质安全电路放电的研究大部分还是基于IEC火花放电装置，影响特定电路参数放电的研究。③ 放电电弧的数值仿真研究无法定量研究两电极接触分断过程的电弧形成机理。④ 针对电感性电路、电容性电路的电感、电容对放电电弧特性的影响，还没有更具说服力的研究成果。⑤ 目前对于电路断路电弧或短路火花如何引爆危险气体的理论研究较少。Abstract: The discharge characteristics of intrinsically safe resistive, inductive and capacitive circuits are analyzed. It is pointed out that inductive and capacitive circuits have larger discharge energy and more complex discharge waveforms. The resistive circuits have relatively simple discharge characteristics. The discharge mechanism and mathematical models of the intrinsically safe inductive and capacitive circuits are summarized. The mathematical expressions of different models, such as linear decay model of discharge current, parabolic model of discharge current, static volt-ampere characteristic model, dynamic volt-ampere characteristic model and arc resistance index model, are given. The advantages and disadvantages of the numerical simulation method of macroscopic and microscopic gas discharge are introduced. The macroscopic methods have a small amount of calculation, but it can only simulate the external characteristics of gas discharge. The microscopic methods have a large amount of calculation, but it can accurately simulate the motion characteristics of particles in the discharge process. The microcosmic mechanism of the DC discharge circuit and discharge arc is studied. Under different electrode materials, different electrode distances and different initial electrical parameters, the influence of different surface emission mechanisms such as thermal field emission and field emission on gas discharge is obtained. According to the current research situation, the problems that need to be solved in the discharge research of intrinsically safe DC circuits are put forward. ① The macro experiment-mathematical model has complex expression and a large amount of calculation. Some models have a single scope of application and cannot truly achieve the non-explosive evaluation of intrinsically safe circuits. ② Most of the research on discharge in intrinsically safe circuits is based on IEC spark discharge devices, which affects the discharge of specific circuit parameters. ③ The numerical simulation of discharge arc cannot quantitatively study the arc formation mechanism during the contact breaking process of two electrodes. ④ There are no more convincing research results on the influence of inductance and capacitance of the inductive circuit and capacitive circuit on the characteristics of discharge arc. ⑤ At present, there are few theoretical studies on how to ignite dangerous gases by electric arc due to circuit break arc or short circuit spark.
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图 7 高速摄像机拍摄的电阻性电路和电感性电路放电电弧
Figure 7. Discharge arc of resistive circuit and inductive circuit captured by a high-speed camera
图 8 钨丝表面晶须及放电光谱
Figure 8. Whisker and discharge spectrum on the surface of tungsten wire
图 11 激发温度和由电导率确定的温度在不同电流下的比较（50，60，100 mA点燃,电压30 V）
Figure 11. Comparison of excitation temperature and temperature determined by discharge conductivity at different current values (50，60，100 mA ignition, voltage 30 V)
图 12 激发温度与由电导率确定的温度比较（60 mA点燃，电压30 V）
Figure 12. Comparison of excitation temperature and temperature determined by conductivity (60 mA ignition, voltage 30 V)
表 1 电感性电路放电模型数学表达式
Table 1. Mathematical expression of inductive circuit discharge model
文献 模型表达式 各参数含义 文献[6] 静态伏安特性方程：
${V}_{{\rm{g}}}=K\left({l}_{0}+vt\right){I}_{{\rm{g}}}^{-\delta }+{V}_{{\rm{a}}}$
通过基尔霍夫电压方程变换并计算得$L\dfrac{ {\rm{d} } }{ {\rm{d} }t}{i}_{ {\rm{g} } }\left(t\right)+R{i}_{ {\rm{g} } }\left(t\right)+ \\ 8\;085\left({l}_{0}+25t\right){i}_{{\rm{g}}}^{-0.68}\left(t\right)+8.6={V}_{0}$${V}_{{\rm{g}}}$：放电电压
$ K $：常数系数
$ {l}_{0} $：初始放电间隙
$ v $：电极分离速度
$ t $：放电时间
${I}_{{\rm{g}}}$：放电电流
δ：常数系数
${V}_{{\rm{a}}}$：电弧阴极电位降
$ L $：电感
R：电路总电阻
${i}_{{\rm{g}}}$：电弧电流
$ {V}_{0} $：电源电压文献[11] 线性衰减模型：
放电电流：${i}_{{\rm{g}}}=I\left(1-\dfrac{t}{T}\right)$
放电电压：${u}_{{\rm{g}}}=\dfrac{E}{T}\left(t+\dfrac{L}{R}\right)$
放电时间：${T}=\dfrac{LI}{ {V}_{ {\rm{arc} }{\rm{min} } } }$${i}_{{\rm{g}}}$：电弧电流
$ I $：稳定电流值
${t}$：放电时间
$ T $：计算放电时间
${u}_{{\rm{g}}}$：放电电压
$ E $：电源电势
L：电感
$ R $：电路总电阻
$ {V}_{{\rm{arc}}{\rm{min}}} $：最小建弧电压文献[12] 动态伏安特性方程：${V}_{{\rm{g}}}={V}_{ {\rm{max} } }+\dfrac{ {V}_{ {\rm{max} } }-{V}_{ {\rm{min} } } }{I}{i}_{{\rm{g}}}$
电弧电阻表达式：${R}_{ {\rm{g} } }=\dfrac{ {V}_{ {\rm{min} } } }{I} \dfrac{E-\left({V}_{ {\rm{max} } }-{V}_{ {\rm{min} } }\right){ {\rm{exp} } }{\left(-\dfrac{t}{f} \right)} }{E-{V}_{ {\rm{max} } }+{V}_{ {\rm{min} } }{ {\rm{exp} } }{\left(-\dfrac{t}{f} \right)} }$
$ f=\dfrac{IL}{E-\left({V}_{{\rm{max}}}-{V}_{{\rm{min}}}\right)} $$V_{\rm{g}}$ ：放电电压
$V_{{{\rm{max}}}} $：放电终止电压
$V_{\rm{{\rm{min}}}} $：放电初始电压
I：稳定电流值
$i_{\rm{g}} $：电弧电流
$R_{\rm{g}} $：电弧电阻
E：电源电压
$ t $：放电时间
f：时间常数
L：电感文献[14] 放电终止电压：$ {V}_{{\rm{max}}}={V}_{{\rm{arc}}{\rm{min}}}+\left(39+13L\right)I $
电弧电流、电压表达式：${i}_{ {\rm{g} } }=I-\dfrac{ {V}_{ {\rm{arc} }{\rm{min} } } }{R-\left(39+13L\right)}\left[1-{ {\rm{exp} } }{ \left( \dfrac{R-\left(39+13L\right)}{L}t \right)}\right]$
${V}_{ {\rm{g} } }=\dfrac{ {V}_{ {\rm{arc} }{\rm{min} } } }{R- \left( 39+13L\right)}\left[ R-\left(39+13L\right) { {\rm{exp} } }{\left( -\dfrac{R-\left(39+13L\right)}{L}t \right) }\right]$
电弧电阻表达式：${r}_{{\rm{g}}}=\dfrac{ {V}_{ {\rm{arc} }{\rm{min} } }+\left(39+13L\right)I}{ {i}_{{\rm{g}}} }-\left(39+13L\right)$$ {V}_{{\rm{max}}} $：放电终止电压
$ {V}_{{\rm{arc}}{\rm{min}}} $：最小建弧电压
$ L $：电感
$ I $：稳定电流值
${i}_{{\rm{g}}}$：电弧电流
$ R $：电路总电阻
$ t $：放电时间
${V}_{{\rm{g}}}$：放电电压
${r}_{{\rm{g}}}$：电弧电阻文献[15] 电极分离速度表达式：${v}^{ {'} } \left(t\right) \approx 25+1\; 940\mathrm{sin}\left(\dfrac{ {\text{π} } }{730}t\right)$
电弧长度表达式：$l={l}_{0}+25\times {10}^{-6}t+0.45\left[1-\mathrm{cos}\left(\dfrac{{\text{π}} }{730}t\right)\right]$
动态伏安表达式：${V}_{ {\rm{g} } }=K{i}_{ {\rm{g} } }^{-\delta }\left\{ { {l}_{0}+25\times {10}^{-6}t+0.45\left[1-\mathrm{cos}\left(\dfrac{ {\text{π} } }{730}t\right)\right] } \right\}+{V}_{ {\rm{a} } }$
电弧电阻表达式： $ \begin{array}{l}{r}_{ {\rm{g} } }=\dfrac{ {V}_{ {\rm{g} } } }{ {i}_{ {\rm{g} } } }=\dfrac{9.45}{ {i}_{ {\rm{g} } } }+ 4 \; 500 \bigg\{ {l}_{0}+25\times {10}^{-6}t+ \\ \;\;\;\;\;\;\;\; 0.45\left[1-\mathrm{cos}\left(\dfrac{ {\text{π} } }{730}t\right)\right] \bigg\}{I}_{ {\rm{g} } }^{-1.6} \end{array}$$ {v}^{{'}} \left(t\right) $：电弧分离速度
$ t $：放电时间
$ l $：电弧长度
$ {l}_{0} $：初始放电间隙
${V}_{{\rm{g}}}$：放电电压
K：常数系数
${i}_{{\rm{g}}}$：电弧电流
$ \delta $：常数系数
${V}_{{\rm{a}}}$：电弧阴极电位降
rg：电弧电阻文献[18] 简单电感性电路放电时间表达式：$T={T}_{ {\rm{R} } }+{T}_{ {\rm{L} } }={d}_{ {\rm{R} }{\rm{max} } }{v}^{-{k}_{1} }+{d}_{ {\rm{L} }{\rm{max} } }{ {\rm{exp} } }{\left(-\dfrac{v}{ {k}_{2} } \right) }+{T}_{\infty }$
电弧等效时变电阻表达式：${R}_{ {\rm{arc} } }=\dfrac{v t}{a{t}^{2}+bt+c}$
利用莱文贝格−马夸特优化算法进行拟合，得到上式中参数值${T}_{{\rm{R}}}$：电阻电路放电时间
${T}_{{\rm{L}}}$：电感电弧放电时间
${d}_{{\rm{R}}{\rm{max} } }$：电阻电弧长度
${d}_{{\rm{L}}{\rm{max} } }$：电感电弧长度
$ v $：放电电弧电压
k1,k2：特征系数
$T_ {{\infty}}$：电弧放电时间稳
定值
a,b,c：极间电阻特征
参数
t：放电时间文献[20] 电感性电路简单放电数学模型：$v\left(i,l\right)={V}_{{\rm{f}}}+\alpha l\left(1+\dfrac{\beta }{ {i}^{n} }\right)$
恒定电弧长度拟合函数：$V\left(i\right)=13.16+\dfrac{3.077}{{i}^{0.296}} $
恒定放电电流拟合函数：$ V\left(l\right)=9.8+40.93l $
联合上式可得电感性电路简单模型数学表达式：$ v\left(i,l\right)=9.8+18.6l\left(1+\dfrac{0.914}{{i}^{0.296}}\right) $$ v $：放电电弧电压
$ i $：放电电弧电流
$ l $：放电电弧长度
${V}_{{\rm{f}}}$：阴极电压降
$ \alpha $，β，n：拟合参数
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表 2 电容性电路放电模型数学表达式
Table 2. Mathematical expression of capacitive circuit discharge model
文献 模型表达式 各参数含义 文献[22] 电容性电路有触点短路放电数学模型：
当0≤t≤2.35+1.02C时：$\left\{ \begin{array}{l}{u}_{\rm c}\left(t\right)={V}_{\rm i}{ {\rm{exp} } }{(t\alpha) }\\ {i}_{\rm c}\left(t\right)=-{V}_{\rm i}{C\alpha {\rm{exp} } }{(t\alpha )}\end{array}\right.$
当2.35+1.02C≤t≤4.04+7.35C时：$\left\{ \begin{array}{l}{u}_{\rm c}\left(t\right)={V}_{\rm H}\\ {i}_{\rm c}\left(t\right)=-{\alpha C{V}_{\rm H}{\rm{exp} } }{\left[\alpha \left(T-2.35-1.02\mathrm{C}\right)\right]}\end{array}\right.$
当t＞4.04+7.35C时：$\left\{ \begin{array}{l}{u}_{\rm c}\left(t\right)={V}_{\rm H}{ {\rm{exp} } }{(\beta) }\\ {i}_{\rm c}\left(t\right)=\dfrac{ {V}_{\rm H} }{ {R}_{\rm 3} }{ {\rm{exp} } }{(\beta) }\end{array}\right.$
$\alpha =\dfrac{{\rm{ln}}\left({V}_{\rm H}/{V}_{\rm i}\right)}{2.35+1.02C}，\beta =\dfrac{4.04+7.35C-t}{ {R}_{\rm 3}C}$$ t $：放电时间
$ C $：电路电容
$ {u}_{\rm c} $：电容电压
$ {i}_{\rm c} $：电容电流
$ {V}_{\rm i} $：初始电压
$ {V}_{\rm H} $：放电维持电压
$ T：\mathrm{计}\mathrm{算}\mathrm{放}\mathrm{电}\mathrm{时}\mathrm{间} $
R3：两电极闭合后放
电回路总电阻
$\alpha ， \beta $：拟合参数文献[23] 电容性电路指数模型：
放电电流：
${i}_{\rm g}=\dfrac{E-{u}_{\rm T} }{ {R}^{2}C\left[1-{ {\rm{exp} } }{\left(-\dfrac{1}{RC}T\right)}\right]}t{\; {\rm{exp} } }{\left(-\tfrac{1}{RC}t\right)}$
放电电压：${u}_{\rm g}=\dfrac{ {u}_{\rm T}-E{ {\rm{exp} } }{\left(-\dfrac{1}{RC}T \right)} }{1-{ {\rm{exp} } }{\left(-\dfrac{1}{RC}T \right)} }+\dfrac{ {E-u}_{\rm T} }{1-{ {\rm{exp} } }{\left(-\dfrac{1}{RC}T \right)} }{ {\rm{exp} } }{\left(-\dfrac{1}{RC}t \right)}$$ E $：电源电压
$ {u}_{\rm T} $：每次放电结束时
的放电电压
$ R $：电路总电阻
$ C $：电路电容
$ T：\mathrm{计}\mathrm{算}\mathrm{放}\mathrm{电}\mathrm{时}\mathrm{间} $
$ t $：放电时间文献[24] 截止型保护方式下电容性电路放电模型：
放电电压：$ {u}_{\rm g}={U}_{\rm h} $
放电电流：${i}_{\rm g}=\dfrac{E-{U}_{\rm h} }{R}{ {\rm{exp} } }{\left(-\dfrac{1}{RC}t \right)}$$ {U}_{\rm h}：\mathrm{最}\mathrm{小}\mathrm{建}\mathrm{弧}\mathrm{电}\mathrm{压} $
$ E $：电源电压
$ R $：电路总电阻
$ C $：电路电容
$ t $：放电时间
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表 3 气体放电数值仿真模型表达式
Table 3. Expressions of numerical simulation models of gas discharge
文献 模型表达式 各参数含义 文献[31] 空间电子运动：$m\dfrac{ {\rm{d} }{\boldsymbol{x}}}{ {\rm{d} }t}=qE$
电子发射电流密度：
${J}_{{\rm{TFE}}}=e{ \displaystyle \int }_{-\infty }^{\infty }N\left(W\right)D\left(W\right){\rm{d} }W$
其中：
$N\left(W\right)=\dfrac{4 {\text{π}} mkT}{ {h}^{3} }{\rm{ln}}\left[1+{\rm{exp} }\left(-\dfrac{\left(W-{E}_{{\rm{F}}}\right)}{kT}\right)\right]$
$D\left(W\right)= \\ \qquad {\rm{exp} }\left(-2{ \displaystyle \int }_{ {x}_{1} }^{ {x}_{2} }\sqrt{\dfrac{8{ {\text{π} } }^{2} }{ {h}^{2} }\left(\left|W\right|-\dfrac{ {e}^{2} }{16{\text{π} } { {\textit{ε}} }_{0}{\boldsymbol{x} } }-eE{\boldsymbol{x} }\right){\rm{d} }{\boldsymbol{x} } }\right)$m：电子质量
x：电子位移矢量
t：时间
q：电荷量
E：电场强度
e：电子电荷量
W：阴极发射材料
逸出功
k：玻尔兹曼常数
T：温度
h：普朗克常数
EF：阴极表面电场
$ {x}_{1}\mathrm{，}{x}_{2} $：电子位置
$ { {\textit{ε}} }_{0} $：真空介电常数文献[33] MHD模型基本方程组
质量守恒方程：$ \dfrac{\partial \rho }{\partial t}+\mathrm{d}\mathrm{i}\mathrm{v}\left(\rho v\right)=0 $
动量守恒方程：
$\begin{array}{l}\dfrac{\partial \rho { {\boldsymbol{v} } }_{i} }{\partial t}+\mathrm{d}\mathrm{i}\mathrm{v}\left(\rho v{ {\boldsymbol{v} } }_{i}\right)=-\dfrac{\partial p}{\partial {x}_{i} }+ \\ \qquad \displaystyle \sum _{k=1}^{2}\dfrac{\partial }{\partial {x}_{k} }\left[\eta \left(\dfrac{\partial { {\boldsymbol{v} } }_{i} }{\partial {x}_{k} }+\dfrac{\partial { {\boldsymbol{v} } }_{k} }{\partial { {\boldsymbol{v} } }_{i} }+{\left(J\times B\right)}_{i}\right)\right]\end{array}$
能量守恒方程：$\begin{array}{l}\dfrac{\partial \left(\rho H\right)}{\partial t}+\mathrm{d}\mathrm{i}\mathrm{v}\left(\rho {\boldsymbol{v}}H\right)-{\rm{div} }\left[\lambda {\rm{grad}}\left(T\right)\right]= \\ \qquad \dfrac{\partial p}{\partial t}-{S}_{ {\rm{R}}}+\dfrac{ {J}^{2} }{\sigma } \end{array}$
（4）电场方程：${E}_{{\rm{arc}}}\left(t\right)=I\left(t\right)/{ \displaystyle \int }_{ {S}_{{\rm{a}}} }^{}\sigma {\rm{d}}s$ρ：等离子体密度
t：放电时间
$ v $：电弧运动速度
${{\boldsymbol{v}}}_{i}\mathrm{、}{{\boldsymbol{v}}}_{k}$：不同方向上的速度矢量
${x}_{i}\mathrm{，}{x}_{k}$：笛卡尔坐标
$ p： $压力
$ \mathrm{\eta } $：黏度系数
$ J： $电流密度
$ B $：磁感应强度
$ H： $动态热焓
$ \lambda ： $热导率
$ T： $温度
$ {S}_{ {{\rm{R}}}}： $辐射冷却
$ \sigma ： $电导率
$ {E}_{\mathrm{a}\mathrm{r}\mathrm{c}} $：弧柱区电场
强度
$ I： $流过弧柱截面的电流
$ {s}_{\mathrm{a}} $：弧柱截面
$ s： $截面积文献[34-35] 分离瞬间动态热过程，能量守恒方程：${\rho }_{ {\rm{eq} } }{C}_{ {\rm{p} } }\dfrac{\partial T}{\partial t}+{\rho }_{ {\rm{eq} } }{C}_{ {\rm{p} } }{\boldsymbol{u} } \cdot \nabla T+\nabla \cdot \left(-{k}_{ {\rm{eq} } }\nabla T\right)=\dfrac{ {J}^{2} }{\sigma_{\rm{eq} } }$
粒子守恒方程：$\dfrac{\partial {n}_{ {\rm{a} } } }{\partial t}+\nabla \cdot {\varGamma }_{ {\rm{a} } }={R}_{ {\rm{a} } }$
电子能量守恒方程：
$\dfrac{\partial {n}_{{\rm{e}} {\text{ε} } } }{\partial t}+\nabla \cdot {\varGamma }_{{\rm{e}} {\text{ε} } }+E {\varGamma }_{ {\rm{e} } }={R}_{ {\rm{e} }{\rm{ {\text{ε} } } } }$
基于MS方程的重物质输运过程表征：$\mathrm{\rho }\dfrac{\partial }{\partial t}{w}_{k}+\rho \left({\boldsymbol{u} } \cdot \nabla \right){w}_{k}=\nabla \cdot {j}_{k}+{R}_{k}$${\rho }_{{\rm{eq}}}$：等效密度
${C}_{{\rm{p}}}$：等效恒压热容
$ T $：温度
$ t $：时间
$ \boldsymbol{u} $：液态金属速度矢量
$\nabla$：哈密顿算子
${k}_{{\rm{eq}}}$：等效导热系数
$ J $：电流密度
$ {\sigma }_{\rm{eq}} $：等效电导率
na：粒子a的数量
${\varGamma }_{{\rm{a}}}$：粒子a的通量
${R}_{{\rm{a}}}$：粒子a生成或去除
的源项
${n}_{{\rm{e {\text{ε}} }}}$：平均电子能
${\varGamma }_{{\rm{e {\text{ε}} }} }$：电子能通量
E：电场强度
${\varGamma }_{{\rm{e}}}$：电子通量
${R}_{{\rm{e } {\text{ε}} }}$：所有碰撞损耗能量之和
$ \rho $：混合物密度
$ {w}_{k} $：第k个物质质量
分数
$ {j}_{k} $：第k个物质扩散流通量
$ {R}_{k} $：第k个物质速率
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