汽车电线电容和电感的数值计算.pdf

43Designresearch/设计研究汽车电器2 0 2 3年第8 期汽车电线电容和电感的数值计算周录尧，李海林，郭红，李国辉，胡瑞霞（一汽解放商用车开发院，吉林长春130 0 11）【摘要】汽车电线的电容和电感是汽车电线束电磁兼容分析计算的重要基础参数，本文通过整理计算公式，给出数值计算示例，方便查表并快速应用。【关键词】汽车电线；电容；电感；数值计算中图分类号：U463.62文献标志码：ANumerical Calculation of Capacitance and Inductance of Vehicle WireZHOU Lu-yao,LI Hai-lin,GUO Hong,LI Guo-hui,HU Rui-xia(FAW JIEFANG Co.,Ltd.,Commercial Vehicle Development Institute,Changchun 130011,China)Abstract】T h e c a p a c i t a n c e a n d i n d u c t a n c e o f a u t o m o b i l e w i r e a r e t h e i m p o r t a n t b a s i s o f e l e c t r o m a g n e t i ccompatibility analysis of automobile harness.In this paper,the numerical example is given by the calculation formulato facilitate the quick application of table lookup.Key words vehicle wire;capacitance;inductance;numerical calculation作者简介周录尧（19 8 8 一），男，工程师，从事商用车电气系统设计工作。本文关注汽车电线电气参数一一电容和电感的数值计算。根据文献资料的公式，对两平行直电线间的电容、单根长直电线的电感、矩形线圈电线的电感、单根长直电线与矩形电线回路的互感进行数值计算，为进一步深入研究汽车电线束电磁场提供电气参数计算数据支持。通过收集整理关于汽车电线电容和电感的计算公式，将公式推导过程的关键步骤详加阐述，方便读者识读理解并运用。公式固然重要，但定性分析的下一步应用往往需要进行定量数值计算，最终发挥作用的还是数据。本文利用Excel公式进行自动计算，改变已知变量，即可得到目标计算数值。1两平行直电线间的电容由电磁学教学参考知两平行长直电线间电容计算公式。1）一般情况：两条无穷长的平行直电线之间单位长度的电容公式计算如下，其中，导体半径分别为R,和R2，它们的轴线相距为d。C=arccosh(2）特殊情况1：半径相等的两条平行长直电线间单位长度的电容计算公式如下：C=arccosh(R收稿日期：2 0 2 2-12-18文章编号：10 0 3-8 6 39(2 0 2 3)0 8-0 0 43-0 6的电容计算如下。半径为R的一条长直电线，当其与地面平行，轴线到地面的距离为h时，与大地之间单位长度的电容公式如下：C=arcosh()4）电线电容的数值计算：真空电容率或真空介电常量1为8 08.8 5418 7 8 17 10-12 C/(Nm）。平行无穷长电线4k之间单位长度的电容计算数值见表1，汽车薄壁电线按ISO19642-3，导体半径R/mm、平均电线半径r/mm，d=r i+r 2+T，电线绝缘间距T/mm。2单根长直电线的电感2.1计算公式引用由直流输电线路导线电感的计算2 可知，长度为Q的电线导体的内电感计算公式为Li=Q；电线单位长度导体的内电感计算公式为：L=；长度为Q的电线导体的外电感计算公式为:L=-on2+-/2+,2T因此，单位长度的电线导体的外电感为：L=鼎ln2 元8 02TO+VP+0_VP+1。通常Qr，上式可简化为：L=(d2-Ri2-R2(1)2R1 R22 元8 0式中：R电线半径；d电线轴线距离。3）特殊情况2：平行于地面的电线与大地之间单位长度(3)8T8TrQQ别m20-11。可得长度为Q0的电线的总电感为：Lb-Lu+L.=2T0+/r2+Q2_/r2+Q2(2)08元2元2元8 0nQQ/设计研究/Design research44AutoelectricpartsNo.08,2023表1平行无穷长电线之间单位长度的电容计算数值规格0.35项目R2规格R10.350.450.50.550.750.651.00.751.50.902.51.104.01.401.7756.01.70102.25163.15203.45253.90505.25项目规格Rr/mm0.350.450.50.550.750.651.00.751.50.902.51.101.4254.01.406.01.70102.252.825163.15203.45253.90505.25项目规格Ri0.350.450.50.550.750.651.00.751.50.902.51.104.01.406.01.70102.25163.15203.45253.90505.250.50.450.550.650.75r/mm0.6516.14816.421 16.24916.36216.44416.06515.79315.67814.83915.17314.98814.63113.66335.1340.7516.74716.62516.77416.90116.56516.33316.24715.41315.79815.61215.246 14.23837.3120.9016.58316.77416.95816.70916.55516.5215.75416.19516.02115.667 14.66740.3471.0016.99417.21517.0116.899 16.895 16.15316.64116.47216.11715.10142.2431.1517.48617.34617.317.34116.64617.20217.04116.69215.666 44.9321.42517.32417.39317.51216.953 17.59417.46217.1516.17949.47317.5817.78517.3718.11918.02217.75116.84754.6982.07518.05417.74618.58718.51718.27917.42858.8032.82517.699 18.663 18.66418.52917.89867.9793.4019.81119.85119.7619.19274.2453.7019.91319.85419.36277.3124.2019.84719.4882.1685.8019.44896.068规格0.35R20.450.650.6512.01612.332 12.46412.66112.889 12.94413.0513.16212.9313.33713.28213.1312.61715.9470.7512.67312.82513.04113.29313.3713.501 13.63513.412 13.86313.8113.65713.12616.5840.9012.99913.23213.50713.61313.77613.93513.74314.23514.1914.04413.51817.4641.0013.48113.77513.90114.08914.26714.09114.62214.582 14.437 13.90518.011.1514.09614.25214.47514.68114.53315.11715.08414.94514.41118.78114.451 14.72514.97114.88515.53315.51715.39914.898 20.0781.77515.05815.35315.34216.07316.07815.98515.52821.5662.07515.688 15.73816.5416.56416.49316.07322.73615.92316.82116.88816.87816.59425.363.4017.87317.97318.00117.77927.1633.7018.08718.13717.96528.0484.2018.21918.132 29.4565.80一规格0.35R20.450.65T/mm0.657.103 7.2730.757.4527.5967.7297.9020.907.7467.8848.0658.2518.4868.6828.9199.3079.3881.008.0288.2151.158.4128.6168.8741.4258.8329.105 9.334 9.61510.07210.16910.28210.49511.221.7759.396 9.6439.94710.4410.54610.6710.90811.7322.0759.90410.22910.75410.86811.002 11.26112.1242.82510.58711.15611.28211.43311.73112.9763.4011.79511.93912.11212.461 13.5433.7012.08712.26712.632 13.8184.2012.45612.84414.255.8013.29315.470.750.650.900.50.750.550.650.750.900.50.750.550.650.750.907.417.5371.00.751.001.00.751.001.00.751.001.50.901.151.50.901.151.50.901.157.72.51.101.4251.7752.0752.825电容C/pF(T=0,h=1)2.54.01.101.401.4251.7752.0752.825电容 C/pF(T-=1,h=10)2.54.01.101.401.4251.775 2.075电容 C/pF(T=10,h=100)7.878.0818.2588.478.8178.8888.088.3048.4928.7179.0859.1629.2499.4099.9469.489.65110.2788.418.6548.859 9.1079.5139.597 9.6959.87610.489.099.3549.783 9.8739.97810.17210.7614.01.406.01.706.01.706.01.70102.25102.25102.252.825163.153.40163.153.40163.153.40203.453.70203.453.70203.453.70253.904.20253.904.2018.27533.5182550大地h3.905.254.205.808.979.11850大地h5.25/mm5.8050大地h5.25/mm5.80/mm9.745Designresearch/设计研究汽车电器2 0 2 3年第8 期表1（续）规格0.35项目R2规格Rt0.350.450.50.550.750.651.00.751.50.902.51.104.01.401.7756.01.70102.25163.15203.45253.90505.25项目规格Ri0.350.450.50.550.750.651.00.751.50.902.51.101.4254.01.406.01.70102.252.825163.15203.45253.90505.25故，单位长度电线的总电感为：L=L+L=+n2O+VP+Q2_ VP+Q1，通常Qr，单位长度电线的总QQ电感可简化表示为：L=8元2元2.2重点总结1）内电感：由L=可知，内电感数值为常数，不受8T电线导体半径的影响。2）外电感：被积函数Bp=4元Vx2-(Q-y)2是关于x和y的二元函数，磁链。可等效理解为求二重积分，即：Q一y中e=Bpdy dx:o4元xHol4元xMol2元，0.50.450.550.650.75r/mm0.654.5534.628 4.6924.7494.8224.9055.009 5.0955.2235.387 5.4325.4935.6416.9260.754.706 4.7724.8314.9074.9930.904.841.001.151.4252.0752.8253.403.704.205.80规格0.35R20.450.65r/mm0.653.3113.3513.3853.4153.453 3.4973.5513.5953.6613.7443.7663.7983.8755.3830.753.3923.4273.457 3.4973.5423.5973.6420.903.4633.4943.5341.003.5253.566 3.6131.153.6083.6563.7153.7633.7053.765 3.8153.889 3.9834.0094.0444.1325.8261.7753.8273.8793.9564.052 4.0794.1164.208 5.9632.0753.9324.0114.093 4.1974.2264.2654.3636.2753.404.306 4.3364.3784.4816.4093.704.3674.409 4.5146.4724.204.4524.559 6.5695.804.6726.829对于式（4），不定积分Vx2+Q2dx令x=Qtant,8元Q1+tant=sect,h=cost=Vx2+Q，C为常数，代人不定积分in20 1x2+Q2dx，可得：Vx2+QQsectQtantQsec?t dt=Qdcost=-Q(1-cos2 t)cos2t=-Q=-(1n+#)ah=-20J1Q一y-20-x2+y2d(1-h),d(1+h)1-h1+h1/x2+Q2+Q+C=/x2+Q-QIn+R+2+CInVx2+Q2-Qdydx/x2-(Q-y)2x2+x2+y2-/x2+(Q-y)dx0 x2+Q21dxX0.750.650.900.50.750.550.650.750.901.00.751.004.94.9625.0435.1345.2475.3425.1265.225.3175.4395.5415.6925.8885.9426.0156.1947.6755.5675.674 5.832 6.0386.095 6.1715.7855.956.1246.3526.414 6.499 6.7098.4746.5976.6656.756 6.9848.7196.733 6.8277.0598.8366.9237.1639.0177.4199.5121.01.50.750.901.001.15(4)1.50.901.154.9795.0685.1785.2715.4075.5835.632 5.6975.8577.2185.4825.6645.3375.4365.5812.54.01.101.401.425 1.775 2.0752.825电容C/pF(T=1000,h=10000)3.713.583.636 3.6823.7523.8383.8623.8953.9775.5583.673.7183.788 3.8773.9013.8363.9263.9524.114.1384.176dtsintcos?tdh(1-h2)h21dha-R1+h)+QdX将式(5)代人(4)式，得：Q+/x2+Q2中ex2+Q2-Qln2元2.51.101.4251.775 2.0752.825电容C/pF(T=100,5.15.196.01.704.01.406.01.70h=1000)5.3225.4935.5395.6035.7587.0525.7135.785.7695.826.1646.223 6.30310162.253.153.403.7953.8183.9354.0185.6173.9864.072 5.6984.276.0641(5)X(6)2102.25163.153.40203.453.70203.453.70253.904.205.9467.3185.896.0627.4556.367.9156.58.0942550大地h3.905.254.205.803.853.9350大地h5.25/mm5.80/mm5.459设计研究/Designresearch46AutoelectricpartsNo.08,2023由式(6)可知，外电感是电线长度和导体半径的函数。2.3单根长直电线电感的数值计算电线规格/mm0.35导体半径r/mm0.45电线长度Q长度的内电感Q/mmLio/h100.00051000.00510000.0520000.150000.25100000.5电线单位长度长度内电感Q/mmL:/h100.051000.0510000.0520000.0550000.05100000.053矩形线圈电线的外电感资料来源参考文献 8 ，本文通过公式推导，结果公式和参考文献 8 的论文结果公式不同，请读者注意。矩形线圈电线的外电感计算公式为：L=L1+L2+L3+L4L1dxdy4元Dxyx2+(b-y)2dxdy4Tx(b-y)D(a-x)2+y2dxdyL4元D(a-x)y真空磁导率o=410-Tm/A。单根长直电线电感的计算数值见表2。表2单根长电线的电感计算数据0.50.750.550.650.006180.00580.005480.005210.004880.004510.004090.003760.003290.002780.002650.002470.002090.107030.103030.09971 0.096870.093250.089280.084520.080690.07520.068640.066880.064520.058841.529971.489861.456471.427871.391431.351341.303171.264391.20844 1.141331.12321.098761.039583.337113.256863.190063.132843.059942.979722.883312.805712.69372.55929 2.52296 2.474012.355389.258949.058298.89125 8.748178.565888.365258.124157.930057.649867.31357 7.222667.100146.8031619.9040819.5027619.1686718.8824918.5178818.1165717.6343117.2460616.6855616.012815.8309215.585814.99157单位长度总电感L/uh0.617751.147311.6454 2.11682.554612.956093.314963.640543.919984.147744.362354.559834.718991.07026 2.050592.997713.916414.79894 5.641746.43697.193827.895778.53229.151039.7462210.28461.529972.969834.37635.754177.09568.396949.6501110.864512.0229413.1142714.1874715.2362316.225811.668563.246994.792026.308447.788419.2282710.6199311.9727813.2696314.4992715.7107516.8977518.025441.85179 3.613455.34177.041338.7045110.3275611.9023913.438414.9183716.3310817.7256119.0956420.406271.990413.890695.757567.595819.397611.1592612.8726914.547316.1658617.7171419.2502320.7588122.20797CQ21/x2+b2t2x2+x2L1dx41IJa1xra2 1dx(7)41Ja1x2+y(8)(9)(10)10.751.50.92.51.1长度Q的总电感Lo/h1241a1xdtdt(t-xt+xVx2+b1241.461.7102.252t2 x2Vx2+b1x2+b2x2+b1x2+b222+b1dt+dx(/2+b-41ra2dxx2+b2a1x2+bdln(t-x)-163.15212-x2dt=x2+b2+b203.45dt=253.9505.254TV(a-x)2+(b-y)2(a-x)(b-y)dxdyD下面将给出L的推导计算过程：L14元DD=(x,y)I 0i x2 a,0biyb2b利用换元法，设x2+y2=t，=V t 2-x 2代人式（12)，得到：Vx2+y2dxdy=Ca214XyJJD(11)x2+y2dxdyxy4元Ja1x1n(t+x)a2dx2JVx(12)x2+b12a21dxa1Xb1X2+b2+b2X(13)2+b2a1Xdxra2/x2+bi2a1Xdx+x2+b2x2+b22t-xdln2+b1t+xx2+b222u21nx2+bi+b1Xd24a1dxb22+b2b2dxXa2X2+bi?-b1a1(14)47Designresearch/设计研究汽车电器2 0 2 3年第8 期对式（14）中利用换元法和分部积分法进行计算，然后对其解得的部分多项式进行合并，推导过程如下：1）对于不定积分Xdx（换元法），令x=Qtant，Q1+tan=sec,，h=c o s=/x+，C为常数，代人式（14)，可得：x2+Q2dx-Qdcost-0(+N)ah-Q a-a+dh-1Q-sd-h+dth+Q1=1-h1+hQ/-In(1-h)+In(1+h)+Q=-1Qn0g+Q1+C-27x2+Q2-0Vx+Q-0 xIng+0+c2）对于不定积分（分部积分法)x2+b12b2b1x2+b1xnb1ha4元1b2lna1ai2+b22-a1ailnb2利用推导式（13)的方法，只要将式(9）、式(10）、式（11)中的-x、b-y 进行变量替换，将会得到如下结构类似的解析解：L2=4元2/az2+(b-b1)2-2/a12+(b-b1)2-2/az2+(b-b2)2+2/a,2+(b-b)2-(b-b)n g2+(0-b)+(b-b)+(-ai)2+b2-2/(a-a2)2+b2-Vx2+Q22/(a-ai)2+bi?+2/(a-a2)2+bi-b2ln(a-a,)2+b;2+bibiln(a-a)dtQtantdhdhhVx2+b22dx=xln(b2x2+b22-Vx2+bi2+C2 Jai2+b2 2 a2+bi2+a2?+b;2+b1+bilna2ai2+bi2+bia1a22+bi2-a2a2ln+ailnbiJ(a-a1)2+b2+b2(a-a.)(a-a2)2+b2+b2(a-a2)2+bi2+bi+b2lnbiln(a-a2)(a-a2)(a-a2)2+b2-(-a2)(a-a2)In b2(a-a)+b;2-(a-a1)(a-ai)n b1(15)L4=2/(a-ai)2+(b-b)2-2/(a-a2)2+(b-b,)2-2/(a-ai)2+(b-b2)2+2/(a-a2)2+(b-b2)2 矩形线圈示意图如图1所示，矩形线圈电线外电感的数值计算见表3。电线轴线(16)DXa2a222+b2a21b2ai?+bi2-b1a2(18)b2(a-a2)2+bi2-(a-a2)+(a-a2)Inb(a-al)(a-a.)(a-a2)(a-a2)(b-bh)(b-b.)(b-b2)(b-b2)bai4单根长直电线与矩形电线回路的互感1）单根长直电线与矩形电线回路的互感公式参数如下，详见图2、图3。电流I(19)a2a图1矩形线圈示意图电流bba2a1(b-b)a1(b-bi)(b-b2)C(17)图2长直电线与矩形回路业。b0c+aM(b-b2)1oaa/2al21图3矩形回路相对长直电线对称(20)In2元C设计研究/Designresearch48AutoelectricpartsNo.08,2023表3矩形线圈电感计算示例导体半电线规内电感外电感总电感序号a/mmb/mm径r/mm格/mmLid/uhLe/uhLe/uh1102100040003104100040000.75510610004000710810004000910101000400011101210004000131014100040002.25151016100040003.1517201810004000192040002010004000212022100040005.252320241000（上接第42 页）trackID贯穿整条链路；使用logstash来过滤有效的日志信息；使用elasticsearch来存储logstash过滤后的日志信息和skywalk-ing的链路数据；使用kibana来读取日志并展示统计信息页面。在存储和管理Android前端时的日志信息时，只要对日志的数据模型进行设计整合，即可对前后端的日志数据进行统一管理，降低后端服务资源消耗。此外，对于应用程序中的异步方法和多线程方法来说，日志信息的记录会呈现为乱序，不便于将同一调用链下的日志进行整合排查，而通过在前端请求中也加人trackID，打通前后端日志的调用链关系，可以实现更高效的全链路日志排查。8总结本文从控制能力、终端性能、数据安全性以及前后端整合的角度，论述了在Android平台下对于客户端日志采集系统的整体设计。在实际应用场景中，可以结合业务需求，对日志的配置文件和日志模型进行调整，也可以根据现有的后端日志系统进行适配性调整。2）单根长直电线与矩形电线回路的互感计算示例见表4。表4单根长直电线与矩形电线回路的互感计算示例序al6/c/R/1o/40000.650.6540000.7540000.90.940001.11.140001.41.440001.71.740002.2540003.1540003.453.453.93.940005.2540006.2540006.25MIE/uV1./uA0.750.4010.750.5147.25147.751.00.40148.2148.611.00.5144.38 144.881.50.40145.03 45.431.50.52.50.401 41.45 41.852.50.540.40137.0337.4340.560.40133.3533.7560.5100.40127.7528.15100.5122.32122.82160.40120.3820.78160.5115.45115.95200.40133.0433.44200.5250.40130.6331.03250.5500.40124.4524.85500.5700.40120.5320.93700.5101.68102.1850.6851.08140.72 141.22136.7137.2131.85132.35127.95 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