魏浩孫鳳舉呼義翔梁天學(xué)叢培天邱愛慈1)?
1)(西安交通大學(xué),電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室,西安 710049)2)(西北核技術(shù)研究所,強(qiáng)脈沖輻射環(huán)境模擬與效應(yīng)國(guó)家重點(diǎn)實(shí)驗(yàn)室,西安 710024)(2016年6月30日收到;2016年11月20日收到修改稿)
一種非軸對(duì)稱磁絕緣電子鞘層邊界的計(jì)算方法?
魏浩1)2)?孫鳳舉2)呼義翔2)梁天學(xué)2)叢培天2)邱愛慈1)2)?
1)(西安交通大學(xué),電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室,西安 710049)2)(西北核技術(shù)研究所,強(qiáng)脈沖輻射環(huán)境模擬與效應(yīng)國(guó)家重點(diǎn)實(shí)驗(yàn)室,西安 710024)(2016年6月30日收到;2016年11月20日收到修改稿)
長(zhǎng)距離磁絕緣傳輸線內(nèi)電極偏心、感應(yīng)腔注入電流非均勻分布引起電子鞘層邊界偏心等非對(duì)稱磁絕緣特性.電子鞘層邊界是研究非軸對(duì)稱磁絕緣特性的重要參數(shù).本文提出一種計(jì)算非軸對(duì)稱磁絕緣電子鞘層邊界的方法.通過(guò)引入角向非均勻分布的模數(shù),將經(jīng)典一維軸對(duì)稱Creedon穩(wěn)態(tài)磁絕緣理論推廣應(yīng)用于圓柱坐標(biāo)系下二維(r,θ)平面.建立了感應(yīng)電壓疊加器次級(jí)非軸對(duì)稱磁絕緣的二維Creedon物理模型,給出了非軸對(duì)稱磁絕緣電子鞘層邊界的數(shù)值計(jì)算方法和計(jì)算誤差.當(dāng)陰極角向磁場(chǎng)(陰極電流)角向分布滿足余弦函數(shù)時(shí),電子鞘層邊界接近高斯分布.陰極電流角向不均勻程度越大,電子鞘層邊界偏心程度越嚴(yán)重,計(jì)算誤差越大.
磁絕緣感應(yīng)電壓疊加器,注入電流非均勻分布,非軸對(duì)稱磁絕緣,電子鞘層邊界
磁絕緣傳輸線(magnetically insulated transmission lines,MITL)在超高功率脈沖傳輸和匯聚中具有重要應(yīng)用[1?4].大型脈沖功率裝置常采用數(shù)米至幾十米長(zhǎng)的MITL,長(zhǎng)距離MITL內(nèi)電極通常僅在首端與裝置連接固定,這種懸臂式支撐結(jié)構(gòu)易造成MITL內(nèi)電極偏心、電極間電/磁場(chǎng)非軸對(duì)稱分布,從而引起非軸對(duì)稱磁絕緣[4?7].此外,在磁絕緣感應(yīng)電壓疊加器(magnetically induction voltage adders,MIVA)中,由于感應(yīng)腔初級(jí)注入電流非均勻分布,也會(huì)導(dǎo)致MIVA次級(jí)磁絕緣呈現(xiàn)非軸對(duì)稱特性[7?11].
磁絕緣電子鞘層邊界是聯(lián)系MITL微觀特性(空間電子電荷密度、電荷質(zhì)心、漂移速率)和宏觀特性(線電壓、陰/陽(yáng)極電流、運(yùn)行阻抗)的關(guān)鍵參數(shù).電子鞘層邊界限定了磁絕緣空間電子的活動(dòng)區(qū)域,可視作MITL的虛陰極,電子鞘層邊界和陽(yáng)極構(gòu)成的傳輸線特征阻抗即是MITL的實(shí)際運(yùn)行阻抗[12].沿電子鞘層邊界對(duì)磁場(chǎng)強(qiáng)度進(jìn)行環(huán)路積分,可獲得MITL陽(yáng)極電流.在非軸對(duì)稱磁絕緣研究中,電子鞘層邊界是評(píng)估磁絕緣狀態(tài)和性能的重要物理量.
在通常的一維、軸對(duì)稱磁絕緣中,電子鞘層邊界是與MITL陰/陽(yáng)電極共圓心的同心圓 (rs≡常數(shù)).對(duì)于非軸對(duì)稱磁絕緣,電子鞘層邊界rs(θ)隨角向位置呈某種分布,即rs(θ)=f(θ).目前已有的磁絕緣模型(1975年Creedon層流理論[13],1979年Mendel[14]任意動(dòng)量理論,1990年Lawconnell和Neri[15]通用Laminar理論,2006年Ottinger和Schumer[16]基于粒子模擬修正的Mendel理論,以及2006年Stygar等[17]磁絕緣電子彈性碰撞理論)主要用來(lái)描述一維、軸對(duì)稱磁絕緣問(wèn)題.由現(xiàn)有模型很難直接獲得非軸對(duì)稱磁絕緣rs(θ)的解析解.
電子鞘層邊界rs(θ)也很難直接實(shí)驗(yàn)測(cè)量,目前主要采用粒子模擬(particle in cell,PIC)方法,通過(guò)跟蹤大量磁絕緣空間電子的軌跡來(lái)確定[16].確定非軸對(duì)稱磁絕緣的rs(θ)必須采用三維 PIC模型,這對(duì)PIC模擬軟件及代碼提出了較高要求.國(guó)外主要采用大型三維粒子模擬軟件LSP(large scale procedure),國(guó)內(nèi)電子科技大學(xué)研發(fā)了三維粒子模擬軟件CHIPIC3D,并用于“聚龍一號(hào)”裝置四層圓盤錐MITL和1 MV LTD裝置螺旋支撐桿MITL的三維粒子模擬[18?21].但是,大型脈沖功率裝置或器件的三維PIC計(jì)算耗時(shí)極長(zhǎng),模擬結(jié)果因受多種因素和假設(shè)條件的影響隨機(jī)性很大.因此,本文提出了一種確定MIVA次級(jí)非對(duì)稱磁絕緣電子鞘層邊界rs(θ)的方法.
圖1為MIVA次級(jí)MITL橫截面(垂直功率傳輸方向)示意圖.rc,ra分別為MITL陰、陽(yáng)極半徑,rs(θ)為待求解的磁絕緣電子鞘層邊界.MIVA感應(yīng)腔初級(jí)注入電流非均勻分布導(dǎo)致次級(jí)磁絕緣陰/陽(yáng)極電流、電/磁場(chǎng)角向非軸對(duì)稱分布[22,23].即使MITL陰/陽(yáng)電極同軸,仍然會(huì)導(dǎo)致rs(θ)偏心[24].
由磁絕緣Creedon層流理論[13],磁絕緣達(dá)到穩(wěn)態(tài)時(shí)空間電子區(qū)域(rc<r<rs(θ))和真空區(qū)域(rs(θ)< r < ra) 的相對(duì)論因子γi(r,θ)和γo(r,θ)分別滿足數(shù)學(xué)物理方程[25?28],
在陰極(r=rc)和陽(yáng)極 (r=ra)的邊界位置,γi(r,θ)和γo(r,θ)分別滿足如下邊界條件,
其中γ0=1+eV0/(mc2)為陽(yáng)極的相對(duì)論因子,e為電子電荷量,m為電子靜止質(zhì)量,V0為陰-陽(yáng)電極間線電壓.
在電子鞘層邊界rs(θ)處,滿足分界面銜接條件,
采取分離變量法[29]求解方程(1),考慮到γi(r,θ)和γo(r,θ)在[0,2π]內(nèi)關(guān)于θ= π 對(duì)稱,并代入邊界條件(2),得到相對(duì)論因子γi(r,θ)和γo(r,θ)的解析解分別為
其中n為非負(fù)整數(shù),表征陰/陽(yáng)極電流角向非均勻分布的模數(shù),若令n=0,(4)式即為經(jīng)典的一維、軸對(duì)稱磁絕緣Creedon層流模型[13,28];系數(shù)aj,bj(0≤j≤n,j為整數(shù))為待定常數(shù),其取值與陰/陽(yáng)極電流角向分布有關(guān).
當(dāng)給定MITL結(jié)構(gòu)參數(shù)(陰/陽(yáng)極半徑ra,rc)、線電壓V0和陰極電流角向分布Ic(θ)時(shí),聯(lián)合方程(3)和(4),可確定電子鞘層邊界rs(θ).但因方程(4)為超越方程,難以直接獲得rs(θ)的解析解,需要采用數(shù)值方法進(jìn)行求解.
3.1 確定系數(shù)aj(0≤j≤n)
由磁絕緣穩(wěn)態(tài)時(shí)空間電子徑向受力平衡[27],得到
(4)和(5)式聯(lián)合,得到在rc< r< rs(θ)區(qū)域Bθ(r,θ)滿足,
令(6)式中r=rc,得到陰極表面角向磁場(chǎng)Bθ(rc,θ),
采用正交三角函數(shù)系cos(jθ)將Bθ(rc,θ)在[0,2π]區(qū)間內(nèi)做Fourier級(jí)數(shù)展開,得系數(shù)aj,其中κ為常數(shù),κ=?erc/mc.
由(8)式可知,系數(shù)aj取決于陰極表面角向磁場(chǎng)Bθ(rc,θ)角向分布.實(shí)際中,Bθ(rc,θ)角向分布可由感應(yīng)電壓疊加器的電磁模擬或?qū)嶒?yàn)測(cè)量獲得.
3.2 確定系數(shù)bj(0≤j≤n)
由方程(3b)可以推導(dǎo)系數(shù)bj(0≤j≤n)滿足如下矩陣方程,
其中a=(a0,a1,...,an)T和b=(b0,b1,...,bn)T為n+1階向量.N(rs),M(rs)為(n+1)×(n+1)階矩陣,其元素分別為
(9)和(10)式的推導(dǎo)過(guò)程見附錄A.(9)和(10)式表明,當(dāng)系數(shù)aj確定后,系數(shù)bj僅與電子鞘層邊界rs(θ)有關(guān).因此,可依據(jù)MIVA次級(jí)磁絕緣電子鞘層邊界的物理實(shí)際和可能分布,假定rs(θ)滿足某種特定的函數(shù)分布f0(θ)(如高斯分布、余弦分布、二次拋物線分布等),f0(θ)中含有待定系數(shù).
3.3 確定f0(θ)的待定系數(shù)
采用最優(yōu)化問(wèn)題的全局搜索解法[30],確定函數(shù)f0(θ)中待定系數(shù).最優(yōu)化問(wèn)題可表述為
其中(11a)式為最優(yōu)化問(wèn)題的目標(biāo)函數(shù),表征當(dāng)rs(θ)為假定函數(shù)分布f0(θ)時(shí), γi(r, θ)和 γo(r, θ)偏離方程(3a)的程度,定義為計(jì)算誤差.搜索約束條件(11b)限定了電子鞘層邊界的取值范圍,即rc< f0(θ)<ra.
3.4 確定電子鞘層邊界rs(θ)
重復(fù)3.2–3.3節(jié)的步驟,比較多種假定函數(shù)分布(高斯分布、余弦分布、二次拋物線分布等)的計(jì)算誤差,計(jì)算誤差最小的分布視為電子鞘層邊界rs(θ).
給定MIVA次級(jí)MITL運(yùn)行參數(shù):陰極半徑rc=0.1 m,陽(yáng)極半徑ra=0.2 m,陰-陽(yáng)極線電壓V0=4 MV,γ0=9,磁絕緣運(yùn)行在最小電流工作點(diǎn).
在一維軸對(duì)稱情況(陰/陽(yáng)極電流角向均勻分布)下,由磁絕緣經(jīng)典Creedon層流理論,計(jì)算磁絕緣電子鞘層邊緣的相對(duì)論勢(shì)γs=2.0,電子鞘層邊界rs=0.119 m,陽(yáng)極電流Ia=132 kA,陰極電流Ic=65.5 kA,陰極表面角向磁場(chǎng)Bco=0.131 T.
二維、非軸對(duì)稱情況下,由于MIVA感應(yīng)腔初級(jí)注入電流角向的非均勻分布,靠近饋入點(diǎn)的角向位置(θ=0)Bθ(rc,θ)明顯偏大,遠(yuǎn)離饋入點(diǎn)的角向位置(θ= π)Bθ(rc,θ)偏小.假定角向磁場(chǎng)Bθ(rc,θ)滿足余弦分布,
其中Bco為一維軸對(duì)稱(電流角向均勻分布)時(shí)陰極角向磁場(chǎng),不均勻系數(shù)δ用于表征二維非軸對(duì)稱情況下Bθ(rc,θ)角向非均勻分布的程度.
假定MIVA次級(jí)MITL電子鞘層邊界rs(θ)滿足余弦分布、高斯分布和二次拋物線分布,其分布函數(shù)分別如(13)–(15)式所示,
由本文所述的數(shù)值計(jì)算方法,可以確定f1(θ),f2(θ)和f3(θ)中待定系數(shù).三種假定分布函數(shù)下電子鞘層邊界rs(θ)如圖2所示.余弦、高斯和二次拋物線三種分布函數(shù)下的計(jì)算誤差分別為2.7084,0.0682和9.6110.高斯分布的計(jì)算誤差最小,因此,當(dāng)陰極角向磁場(chǎng)Bθ(rc,θ)滿足余弦分布時(shí),MITL電子鞘層邊界rs(θ)最接近高斯分布.
改變Bθ(rc,θ)角向不均勻系數(shù)δ,電子鞘層邊界rs(θ)隨之變化.表1為幾種不均勻系數(shù)δ時(shí)三種分布函數(shù)的計(jì)算誤差.由表1可知,二次拋物線分布的計(jì)算誤差最大,高斯分布的計(jì)算誤差均最小.注入電流分布越均勻(角向磁場(chǎng)Bθ(rc,θ)不均勻系數(shù)δ越小),計(jì)算誤差越小.
當(dāng)Bθ(rc,θ)不均勻系數(shù)δ分別為25%,10% 和2.5%時(shí),采用高斯函數(shù)分布求解的電子鞘層邊界如圖3所示.電子鞘層邊界rs(θ)的最大值、最小值和電子鞘層不均勻系數(shù)ζ如表2所列,其中電子鞘層角向不均勻系數(shù)ζ定義為
其中max(rs(θ)),min(rs(θ))和mean(rs(θ)) 分別為rs(θ)在θ∈[0,2π]的最大值、最小值和平均值.(16)式定義的ζ物理含義為電子鞘層厚度的極差與平均厚度之比.
圖2 (網(wǎng)刊彩色)三種假定分布函數(shù)時(shí)電子鞘層邊界(Bθ(rc,θ)不均勻系數(shù)δ=25%)Fig.2.(color online)The electron sheath profi les under the three assumed dstribu tion functions as the asymmetric coffi cient of Bθ(rc,θ),δ,is abou t 25%.
表1 Bθ(rc,θ)不均勻系數(shù)δ對(duì)計(jì)算誤差的影響Tab le 1.The in fl uences of the asymmetric coeffi cient of Bθ(rc,θ),δ,on the calcu lation errors.
表2 Bθ(rc,θ)角向不均勻系數(shù)δ不同時(shí)電子鞘層邊界rs(θ)的特性Tab le 2.The parameters of the electron sheath profi les function as the asymmetric coeffi cients of Bθ(rc,θ).
由圖3和表2可知,Bθ(rc,θ)不均勻系數(shù)δ越大(注入電流角向不均勻程度越大),電子鞘層邊界rs(θ)偏心程度越嚴(yán)重.電子鞘層角向不均勻系數(shù)ζ與Bθ(rc,θ)角向不均勻系數(shù)δ數(shù)值相接近.當(dāng)Bθ(rc,θ)分布較均勻(δ=0.5%)時(shí),電子鞘層邊界rs(θ)≈0.119 m,此數(shù)值與一維軸對(duì)稱Creedon理論計(jì)算的結(jié)果相同.
圖3 不同Bθ(rc,θ)角向不均勻系數(shù)δ時(shí)的磁絕緣電子鞘層邊界 (a)δ=25%;(b)δ=10%;(c)δ=2.5%Fig.3.The electron sheath profi les under th ree different asymmetric coeffi cients of Bθ(rc, θ)for(a)δ=25%,(b)δ=10%,and(c)δ=2.5%.
在穩(wěn)態(tài)磁絕緣經(jīng)典、一維軸對(duì)稱Creedon理論基礎(chǔ)上,引入表征陰/陽(yáng)極電流角向非均勻分布的參數(shù)(角向非均勻分布模數(shù)n),建立了圓柱坐標(biāo)下MIVA次級(jí)非軸對(duì)稱磁絕緣的二維Creedon物理模型,給出了磁絕緣電子鞘層邊界rs(θ)的數(shù)值計(jì)算方法和計(jì)算誤差.當(dāng)給定磁絕緣線結(jié)構(gòu)參數(shù)(陰/陽(yáng)極半徑)、線電壓和陰極電流(陰極角向磁場(chǎng))角向分布時(shí),可計(jì)算磁絕緣電子鞘層邊界rs(θ).本文提出的磁絕緣電子鞘層邊界確定方法具有計(jì)算效率高、耗時(shí)短等優(yōu)點(diǎn),該方法已用于MIVA次級(jí)非軸對(duì)稱磁絕緣電子鞘層邊界的確定.結(jié)果表明,當(dāng)陰極角向磁場(chǎng)Bθ(rc,θ)角向分布滿足余弦函數(shù)時(shí),電子鞘層邊界rs(θ)呈高斯分布.研究了電流均勻性對(duì)磁絕緣電子鞘層邊界rs(θ)的影響,結(jié)果表明,隨著電流不均勻程度增加,電子鞘層偏心程度加大;電子鞘層rs(θ)的角向不均勻系數(shù)和陰極角向磁場(chǎng)Bθ(rc,θ)的不均勻系數(shù)數(shù)值相近.
感謝西安交通大學(xué)數(shù)學(xué)學(xué)院李東升教授、電氣工程學(xué)院馬西奎教授和葛曉宇博士對(duì)數(shù)值求解方法給予的良好建議.感謝西北核技術(shù)研究所付梅艷、尹佳輝、姜曉峰、曾江濤、張鵬飛、來(lái)定國(guó)、孫江等的討論.
附錄A 矩陣方程(9)的推導(dǎo)
分界面銜接條件(3)的物理意義是在分界面rs(θ)上,電勢(shì)和法向電場(chǎng)連續(xù).
當(dāng)電子鞘層偏心程度較輕(即(rs?rc)/rc? 1)時(shí),?/?n ~= ?/?r,(3b)式可簡(jiǎn)化為
將(4)式中γi(r,θ)和γo(r,θ)表達(dá)式代入(A1)式,得到
為了便于分析,將(A2)式等號(hào)兩邊均寫成級(jí)數(shù)求和形成,即
其中
由于方程(A3)對(duì)于[0,2π]內(nèi)任意θ均成立,因此,將θ離散化處理,在[0,2π]取(n+1)個(gè)離散點(diǎn),
對(duì)于任意一個(gè)θi均滿足方程(A3),可以得到,
將公式(A6)寫成矩陣方程的形式,即
其中,a=(a0,a1,...,an)T和b=(b0,b1,...,bn)T為n+1階向量;N(rs),M(rs)為(n+1)×(n+1)階矩陣,其元素分別為
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PACS:84.70.+p,07.85.Fv,41.20.–q,52.35.–gDOI:10.7498/aps.66.038402
Amethod tocalcu late the electron sheath profi le of the nonaxisymmetricalmagnetic insu lation?
Wei Hao1)2)?Sun Feng-Ju2)Hu Yi-Xiang2)Liang Tian-Xue2)Cong Pei-Tian2)Qiu Ai-Ci1)2)?
1)(State Key Laboratory of E lectrical Insu lation and Power Equipment,X i’an Jiaotong University,X i’an 710049,China)2)(State Key Laboratory of Intense Pu lsed Rad iation Simu lation and Effect,Northwest Institute of Nuclear Technology,Xi’an 710024,China)(Received 30 June 2016;revised manuscript received 20 November 2016)
The nonaxisymmetricalmagnetic insulation would occur due tothe disalignment of inner electrodes in longmagnetically insulated transmission lines,or the nonuniformdistributionsof injected currents in induction cavities ofmagnetically insulated induction voltage adders(MIVA).The electron sheath profi le is a very important parameter tocharacterize the nonaxisymmetricalmagnetic insulation.In the past,the three-dimensional particle in cell simu lation was usually used todetermine the electron sheath profi le,which is extremely time-consuming and ineffi cient.In this paper,a fast and effi cient calcu lation method is proposed.The classical one-dimensional Creedon theory of the magnetic insu lation equilibriumis generalized toa two-dimensional plane of(r,θ)via introducing a parameter defined as the azimuthalmode number.Two-dimensional Creedon is developed tomodel the asymmetric magnetic insulation of the MIVA.Provided the azimuthal distributions ofmagnetic fl ux density on the cathode,which is in proportion tothe cathode current,the two-dimensional Creedon model is numerically solved.Anumerical solution method tocalculate the electron sheath profi le is proposed,and then the calculation error is alsogiven.As the azimuthal distribution ofmagnetic flux density on the cathodemeets a cosine function,the profi le of the electron sheath is approximate tothe Gauss function.As the nonuniformportion of cathode current increases,the electron sheath becomesmore eccentric,and the calculation error is alsomuch larger.
magnetically insulated induction voltage adders,asymmetrical distribution of injected currents,nonaxisymmetricalmagnetic insulation,electron sheath profi le
10.7498/aps.66.038402
?國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):11505138,51577156)資助的課題.
?通信作者.E-mail:weihaoyy@nint.ac.cn
?通信作者.E-mail:qiuac@cae.cn
*Project supported by the National Natural Science Foundation of China(G rant Nos.11505138,51577156).
?Corresponding author.E-mail:weihaoyy@nint.ac.cn
?Corresponding au thor.E-mail:qiuac@cae.cn