无限电阻网络等效电阻的计算

32220104JOURNALOFEEEVol.32No.2Apr.2010田社平,陈洪亮,张峰(上海交通大学电子信息学院,上海200240):20090702;:20090929:(1967),,,,,Email:[e-mail]:,,,,Matlab,:;;;Matlab:TM13:A:10080686(2010)02002903CalculationofEquivalentResistanceofInfiniteResistiveGridTIANSheping,CHENHongliang,ZHANGFeng(SchoolofElectricalandElectronicEng.,ShanghaiJiaoTongUniv.,Shanghai200240,China)Abstract:Infiniteresistivelatticesarecircuitswithspecialstructures.Itissuitableforstudentstounderstandknowledgeaboutcircuittheoremsandcircuitsymmetrythroughinfiniteresistivelattices.Solutionofequivalentresistancebetweentwoarbitrarynodesforinfiniteresistivelatticesisdiscussed.Byusing2dimensionalFouriertransform,theintegralexpressionforequivalentresistanceisdeduced.PrecisevalueofequivalentresistancecanbeobtainedthroughsymboliccomputationofMatlabsoftware.Thediscussionishelpfulfortheteachingofcircuit.Keywords:circuittheory;infiniteresistivelattice;Fouriertransform;Matlab,,,1[1,2],r,,(n,p)(n+1,p)1A,1,1A,,(n,p)40.25A,(n,p);,1A,(n,p)(n+1,p)0.25A,(n+1,p),(n,p)(n+1,p)(0.25+0.25)r=0.5r(V),0.5r(V)/1(A)=0.5r1(r),,[35],,,,,Matlab,1[6],(0,0)(m,k)Rmk(0,0)(m,k)I,(0,0),(0,0)(m,k)u00umk,(0,0)(m,k)Rmk=u00-umkI(1),1(n,p)Inp(),KCLInp=unp-u(n+1)pr+unp-u(n1)pr+unp-un(p+1)r+unp-un(p-1)r(2)rInp=4unp-u(n+1)p-u(n1)p-un(p+1)-un(p1)(3)unp,,unp=14(u(n+1)p+u(n1)p+un(p+1)+un(p1))(4)(3)rInp=4unp-4unp(5)unp=unp+rInp4(6)(6)F(x,y):[,][,]R,unp,unp=142!!F(x,y)ej(nx+py)dxdy(7),F(x,y)=∀n,punpej(nx+py)(8)(6)(8)F(x,y)=∀n,p(unp+rInp4)ej(nx+py)(9),(0,0)(m,k)I,∀n,prInp4ej(nx+py)=rI4(1-ej(mx+ky))(10)(4)(10)(9),F(x,y)=rI4(1-ej(mx+ky))+∀n,p[14(u(n+1)p+u(n1)p+un(p+1)+un(p1))]ej(nx+py)=rI4(1-ej(mx+ky))+14∀n,punp(ej[(n1)x+py]+ej[(n+1)x+py]+ej[nx+(p1)y]+ej[nx+(p+1)y])=rI4(1-ej(mx+ky))+14∀n,punpej(nx+py)(ejx+ejx+ejy+ejy)(11),∀n,pu(n+1)pej(nx+py)=∀n,punpej[(n-1)x+py]ejz=cosz+jsinz,F(x,y)=rI4(1-ej(mx+ky))+(cosx+cosy)2∀n,punpej(nx+py)(12)(8)(12)F(x,y)=rI4(1-ej(mx+ky))+(cosx+cosy)2F(x,y)(13)F(x,y)=rI(1-ej(mx+ky))4-2(cosx+cosy)(14)(7)u00umku00=r42!!I(1-ej(mx+ky))4-2(cosx+cosy)dxdy(15)umk=r42!!I(ej(mx+ky)-1)4-2(cosx+cosy)dxdy(16)u00-umk=r42!!I(2-ej(mx+ky)-ej(mx+ky))4-2(cosx+cosy)dxdy=r42!!I[1-cos(mx+ky)]2-(cosx+cosy)dxdy(17)(1)Rmk=u00-umkI=r42!!1-cos(mx+ky)2-(cosx+cosy)dxdy(18)30322Matlab(18),,[6](m,k)=(1,1)Rmk/r=/2,[3,5]Matlab(18)(m,k)0123456,Matlab,Rmk/r1:Rmk=Rkm,,R01=R10=r/2,1,[4],[5]11Rmk/rmk012345600122-4172-2440-36834012-188031042-4905215112212+4463-480-492664615-1402468-1569222-412+48312+436-23615972+223615336-36824353172-24463-412+43461512+24599835-826924105-1612440-368380-4926-2361512+24535210512+402110-12524554012-18803664615-140972+22361599835-812+40221112631523645-1261042-49052152468-15692336-368243526924105-161210-12524523645-121301634653,,[3][4]Matlab,[3][4][3][4],:[1].[M].:,2002[2],,.[M].:,2007[3]G.Venezian.Ontheresistancebetweentwopointsonagrid[J].Am.J.Phys.1994(62):10001004[4]D.AtkinsonandF.J.vanSteenwijk.infiniteresistivelattices[J].Am.J.Phys.1999(67):486492[5],,.[J].:,2001(14):7577(上接第19页薛文涛等文):[1],.[J].:(),2004,6(1):7275[2],,.[J].:,2004,30(3):1417[3],.()[M].:,2006[4].#∃[J].:,2005,9:2830[5],.:[M].:,1999[6].[J].:,2006,28(1):4850312,: