{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "RC Response to a Single Sq uare Pulse" }}{PARA 0 "" 0 "" {TEXT -1 72 "Using Fourier Integrals one can work out the response to a single pulse." }}{PARA 0 "" 0 "" {TEXT -1 74 "This is straightforward. Once finds the Fourier transform for the driving " }}{PARA 0 "" 0 "" {TEXT -1 70 "pulse and then phase shifts the resulting frequency function by the RC" }}{PARA 0 "" 0 "" {TEXT -1 67 "response across the resistor and capacitor respectively. \+ Maple does" }}{PARA 0 "" 0 "" {TEXT -1 72 "not respond well to this ca lculation. A much simpler method is available" }}{PARA 0 "" 0 "" {TEXT -1 26 "using LaPlace transforms. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/ %)addtableG%(fourierG%+fouriercosG%+fouriersinG%'hankelG%(hilbertG%+in vfourierG%+invhilbertG%+invlaplaceG%*invmellinG%(laplaceG%'mellinG%*sa vetableG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "assume(t,real); assume(RC,real);assume(w,real);assume(a,real);assume(b,real);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "ft:=1/2*Heaviside(t+1)-1/2*H eaviside(-t-1)-1/2*Heaviside(t-1)+1/2*Heaviside(-t+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ftG,*-%*HeavisideG6#,&%#t|irG\"\"\"F+F+#F+\"\"# *&#F+F-F+-F'6#,&!\"\"F+F*F3F+F3*&#F+F-F+-F'6#,&F*F+F+F3F+F3*&F,F+-F'6# ,&F+F+F*F3F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Plot the drivin g function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(ft,t=-2. .2);" }}{PARA 13 "" 1 "" {GLPLOT2D 268 201 201 {PLOTDATA 2 "6%-%'CURVE SG6$7co7$$!\"#\"\"!$F*F*7$$!3MLLL$Q6G\">!#.\"F/F+7$$!3]LL37&)=@5F/F+7$$ !39+Dc^jz:5F/F+7$$!3zm;/\">//,\"F/F+7$$!3+]7y5\"3x+\"F/F+7$$!3VL3_I?,0 5F/F+7$$!3(oTg-&fJ-5F/F+7$$!3w++++()>'***!#=$\"\"\"F*7$$!3_++++Y0j&*Fi oFjo7$$!3E++++0\"*H\"*FioFjo7$$!35++++83&H)FioFjo7$$!3\\LLL3k(p`(FioFj o7$$!3Anmmmj^NmFioFjo7$$!3)zmmmYh=(eFioFjo7$$!3+,++v#\\N)\\FioFjo7$$!3 commmCC(>%FioFjo7$$!39*****\\FRXL$FioFjo7$$!3t*****\\#=/8DFioFjo7$$!3= mmm;a*el\"FioFjo7$$!3komm;Wn(o)!#>Fjo7$$!3IqLLL$eV(>!#?Fjo7$$\"3)Qjmm \"f`@')F`rFjo7$$\"3%z****\\nZ)H;FioFjo7$$\"3ckmm;$y*eCFioFjo7$$\"3f)** ****R^bJ$FioFjo7$$\"3'e*****\\5a`TFioFjo7$$\"3'o****\\7RV'\\FioFjo7$$ \"3Y'*****\\@fkeFioFjo7$$\"3_ILLL&4Nn'FioFjo7$$\"3A*******\\,s`(FioFjo 7$$\"3%[mm;zM)>$)FioFjo7$$\"3M*******pfa<*FioFjo7$$\"3Ckm;zy*zd*FioFjo 7$$\"39HLLeg`!)**FioFjo7$$\"3Cm\"H#3Mo+5F/F+7$$\"3c**\\i5KJ.5F/F+7$$\" 3*G$3-8I%f+\"F/F+7$$\"3@mmT:Gd35F/F+7$$\"3'GL3-UKQ,\"F/F+7$$\"3^***** \\-#4>5F/F+7$$\"3\"GL$eM7hH5F/F+7$$\"3Lmm;W/8S5F/F+7$$\"3;LLLj)o61\"F/ F+7$$\"3w****\\#G2A3\"F/F+7$$\"3Ymm\"H3XL7\"F/F+7$$\"3;LLL$)G[k6F/F+7$ $\"3#)****\\7yh]7F/F+7$$\"3xmmm')fdL8F/F+7$$\"3bmmm,FT=9F/F+7$$\"3FLL$ e#pa-:F/F+7$$\"3!*******Rv&)z:F/F+7$$\"3ILLLGUYo;F/F+7$$\"3_mmm1^rZF/F+7$$\"\"#F*F+-%'COLOURG6&%$RGB G$\"#5!\"\"F+F+-%+AXESLABELSG6$Q#t|ir6\"Q!6\"-%%VIEWG6$;F(F\\y%(DEFAUL TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Find the fourier transform of \+ the driving function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fw: =simplify(Re(fourier(ft,t,w)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# fwG,$*&-%$sinG6#%#w|irG\"\"\"F*!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Set RC for the ciruit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "RC:=0.2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RCG$\" \"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 " Maple cannot deal wi th the phase exp(I*arctan(1/(w*rc)). So write" }}{PARA 0 "" 0 "" {TEXT -1 29 "it in complex notation: a+I*b" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "a:=simplify(cos(arctan(1/(w*RC))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG*&-%$absG6#%#w|irG\"\"\"*$-%%sqrtG6#,&*$)F) \"\"#F*F*$\"#D\"\"!F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "b:=simplify(sin(arctan(1/(w*RC))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,$*&-%'signumG6#%#w|irG\"\"\"*$-%%sqrtG6#,&*$)F* \"\"#F+F+$\"#D\"\"!F+F+!\"\"$\"\"&F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 " Work on the frequency phase shift and amplitude this for the c apacitor." }}{PARA 0 "" 0 "" {TEXT -1 57 " In class we showed: Vc=-j /sqrt(1+(w*RC)**2)*exp(j*Phi)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "fw_all_c:=-j*simplify(fw*(a+I*b)/sqrt(1+w**2*RC**2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)fw_all_cG,$*&*(%\"jG\"\"\"-%$sinG6#%#w|ir GF),&-%$absGF,F)*&^#$\"\"&\"\"!F)-%'signumGF,F)F)F)F)*&F-F),&*$)F-\"\" #F)F)$\"#DF5F)F)!\"\"$!#5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 " M aple seem to be unable to deal with signum and the absolute value" }} {PARA 0 "" 0 "" {TEXT -1 37 "the introduced so reenter the formula" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "fw_all2_c:=-I*2*sin(w)/w*(w +I*5)/(1+w**2*RC**2)*RC;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*fw_all2 _cG*&*(^#$!\"%!\"\"\"\"\"-%$sinG6#%#w|irGF+,&F/F+^#\"\"&F+F+F+*&F/F+,& F+F+*&$\"\"%!\"#F+)F/\"\"#F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ftc:=invfourier(fw_all2_c,w,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ftcG,.*&-%$expG6#,&%#t|irG$!\"&\"\"!$\"\"&F.!\"\"\" \"\"-%*HeavisideG6#,&F+F2$F2F.F2F2$F1F.*&$\"+++++]!#5F2F3F2F2*&$\"++++ +]F " 0 "" {MPLTEXT 1 0 18 "plot(ftc,t=-2..2) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 268 201 201 {PLOTDATA 2 "6%-%'CURVESG6 $7bo7$$!\"#\"\"!$F*F*7$$!3MLLL$Q6G\">!#'***!#=$ \"3?i^e3\\%))*=!#?7$$!3E,++vE\"z))*FT$\"3ssU3\"y9-X&!#>7$$!3k+++]miz(* FT$\"3bS$G$*yIL/\"FT7$$!3-+++D1Mr'*FT$\"3$*HoL;vP::FT7$$!3_++++Y0j&*FT $\"31%[[gyXD'>FT7$$!3Q+++]D[Y$*FT$\"3/i:.lDU(y#FT7$$!3E++++0\"*H\"*FT$ \"30CtntHkFNFT7$$!3y++++K?@*)FT$\"3+=O,31,pTFT7$$!3=++++f\\7()FT$\"3?n ralH#ou%FT7$$!3f*******f)y.&)FT$\"3]Y:\\_zPn_FT7$$!35++++83&H)FT$\"3]# )RM%R\\jt&FT7$$!3!ommT&)Gg\"zFT$\"3[sR)zJkCZ'FT7$$!3\\LLL3k(p`(FT$\"3? k@n^(*[\"3(FT7$$!3M++](QYi3(FT$\"3;Pp_DFPqwFT7$$!3Anmmmj^NmFT$\"3d#GBE sL/9)FT7$$!3gnmm;*)o`iFT$\"3GtA*G:>OY)FT7$$!3)zmmmYh=(eFT$\"3Bvbv/ijI( )FT7$$!3+,++v#\\N)\\FT$\"3%*3(y/6ue=*FT7$$!3commmCC(>%FT$\"33Z#)\\=f_] %*FT7$$!39*****\\FRXL$FT$\"3Z>)))y*[/V'*FT7$$!3t*****\\#=/8DFT$\"3cA7y eRGj(*FT7$$!3=mmm;a*el\"FT$\"3!pG8!*>%zX)*FT7$$!3komm;Wn(o)Fgn$\"3sM\\ 3U`'f*)*FT7$$!3IqLLL$eV(>FW$\"3E*RZ*z?&>$**FT7$$\"3)Qjmm\"f`@')Fgn$\"3 [TYc8i@c**FT7$$\"3%z****\\nZ)H;FT$\"35*>2^#H(R]b\"***FT7$$\"3'o****\\7RV'\\FT$\"3#=O\"\\a'pV***FT7$$ \"3Y'*****\\@fkeFT$\"3#>$*ftQ5k***FT7$$\"3_ILLL&4Nn'FT$\"3]`TE:Xg(***F T7$$\"3A*******\\,s`(FT$\"3OeqI*eW%)***FT7$$\"3%[mm;zM)>$)FT$\"3mvp9%G [*)***FT7$$\"3M*******pfa<*FT$\"3)GKB5N9$****FT7$$\"3Ckm;zy*zd*FT$\"3[ _O1]$R%****FT7$$\"39HLLeg`!)**FT$\"3#f6D3cT&****FT7$$\"3^*****\\-#4>5F /$\"3C\\q?!RL\"*3*FT7$$\"3Lmm;W/8S5F/$\"3W=^Vyyf\"=)FT7$$\"3;LLLj)o61 \"F/$\"3#e=\")RMyYO(FT7$$\"3w****\\#G2A3\"F/$\"3DaWkDpKHmFT7$$\"36L$3F =wF5\"F/$\"3UB]koYU\")fFT7$$\"3Ymm\"H3XL7\"F/$\"3_2uvrO%oR&FT7$$\"3\") **\\7$)R\"R9\"F/$\"3]]h*\\P&Rp[FT7$$\"3;LLL$)G[k6F/$\"33!*QFVg\\$R%FT7 $$\"3\\mm\"zM]v?\"F/$\"3>_li6:MUNFT7$$\"3#)****\\7yh]7F/$\"3!)Rx3S;3cG FT7$$\"3GLLe**o4#H\"F/$\"35K&zeXK6K#FT7$$\"3xmmm')fdL8F/$\"3?2_!y?!Q') =FT7$$\"3bmmm,FT=9F/$\"3#[QJsllUB\"FT7$$\"3FLL$e#pa-:F/$\"3sn_,%[iU5)F gn7$$\"3!*******Rv&)z:F/$\"3'3%[wAF*f]&Fgn7$$\"3ILLLGUYo;F/$\"3u#460Q> ``$Fgn7$$\"3_mmm1^rZ%)oyBFgn7$$\"34++]sI@K=F/$\"37(GkR7B! f:Fgn7$$\"34++]2%)38>F/$\"3u4RxE1[S5Fgn7$$\"\"#F*$\"3M,]w'4Twt'FW-%'CO LOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q#t|ir6\"Q!6\"-%%VIEWG6$;F( Fg_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 " In class we showe d the response across the resistor was:" }}{PARA 0 "" 0 "" {TEXT -1 37 "Vr=w*RC/sqrt(1+w**2*RC**2)*exp(I*phi)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "fw_all_r:=w*RC*2*sin(w)/w*(w+I*5)/(1+w**2*RC**2)*RC ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)fw_all_rG,$*&*&-%$sinG6#%#w|ir G\"\"\",&F+F,^#\"\"&F,F,F,,&F,F,*&$\"\"%!\"#F,)F+\"\"#F,F,!\"\"$\"\")F 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ftr:=invfourier(fw_all _r,w,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ftrG,&*&-%$expG6#,&%#t| irG$!\"&\"\"!$\"\"&F.!\"\"\"\"\"-%*HeavisideG6#,&F+F2$F2F.F2F2F2*($F2F .F2-F(6#,&F+F,$F0F.F2F2-F46#,&F+F2$F2F.F1F2F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Plot the response across the resistor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(ftr,t=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 268 201 201 {PLOTDATA 2 "6%-%'CURVESG6$7ho7$$!\"#\"\"!$ F*F*7$$!3MLLL$Q6G\">!#.\"F/F+7$$!3]LL37&)=@5F/F+7$$!39+Dc^jz:5F/F+7$$!3 zm;/\">//,\"F/F+7$$!3+]7y5\"3x+\"F/F+7$$!3VL3_I?,05F/F+7$$!3(oTg-&fJ-5 F/F+7$$!3w++++()>'***!#=$\"3Q[T\"4b65)**Fio7$$!3k+++]miz(*Fio$\"3Wf;n5 #pm&*)Fio7$$!3_++++Y0j&*Fio$\"3$f^^R@au.)Fio7$$!3Q+++]D[Y$*Fio$\"3&zVo \\VxD@(Fio7$$!3E++++0\"*H\"*Fio$\"3'fnAj-dBZ'Fio7$$!3=++++f\\7()Fio$\" 3!G$GXMq<`_Fio7$$!35++++83&H)Fio$\"31=gl01ljUFio7$$!3!ommT&)Gg\"zFio$ \"32Gg,#oNv_$Fio7$$!3\\LLL3k(p`(Fio$\"3zNyK[-^=HFio7$$!3Anmmmj^NmFio$ \"39\"H@&*)e79)!#>7$$!3commmCC(>%Fio$\"3OLv,:3u%\\&Fcs7$$!39*****\\F RXL$Fio$\"3M0=6@5bpNFcs7$$!3t*****\\#=/8DFio$\"3!Hx(=7/;nBFcs7$$!3=mmm ;a*el\"Fio$\"3YPr')4!e?a\"Fcs7$$!3komm;Wn(o)Fcs$\"3t^1:zlMS5Fcs7$$!3Iq LLL$eV(>!#?$\"3'o,g_+#z/oF`u7$$\"3)Qjmm\"f`@')Fcs$\"3Y3e`V'y$yVF`u7$$ \"3%z****\\nZ)H;Fio$\"3!o1!G*[2F)HF`u7$$\"3ckmm;$y*eCFio$\"3sqguJ#e/(> F`u7$$\"3f)******R^bJ$Fio$\"3w+&\\e>**RG\"F`u7$$\"3'e*****\\5a`TFio$\" 3i%ej3Gg\\W)!#@7$$\"3'o****\\7RV'\\Fio$\"3^x(Q'3bMIcF\\w7$$\"3Y'***** \\@fkeFio$\"3)QMo+k7'*e$F\\w7$$\"3_ILLL&4Nn'Fio$\"3Y?r%et%[&R#F\\w7$$ \"3A*******\\,s`(Fio$\"3-77%Hp5ab\"F\\w7$$\"3%[mm;zM)>$)Fio$\"37aN-`er ^5F\\w7$$\"3M*******pfa<*Fio$\"3'Q!owm(*[co!#A7$$\"3Ckm;zy*zd*Fio$\"3H @M[j$*\\1cF[y7$$\"39HLLeg`!)**Fio$\"3O/a$)[5F/$!39[q?!RL\"*3*Fio7$$\"3\"GL$ eM7hH5F/$!3GW$)=*=NMi)Fio7$$\"3Lmm;W/8S5F/$!3a>^Vyyf\"=)Fio7$$\"3;LLLj )o61\"F/$!3#e=\")RMyYO(Fio7$$\"3w****\\#G2A3\"F/$!3ObWkDpKHmFio7$$\"3Y mm\"H3XL7\"F/$!3k3uvrO%oR&Fio7$$\"3;LLL$)G[k6F/$!3a*)QFVg\\$R%Fio7$$\" 3\\mm\"zM]v?\"F/$!3v_li6:MUNFio7$$\"3#)****\\7yh]7F/$!3!)Rx3S;3cGFio7$ $\"3xmmm')fdL8F/$!3#p?0y?!Q')=Fio7$$\"3bmmm,FT=9F/$!3S%QJsllUB\"Fio7$$ \"3FLL$e#pa-:F/$!3Lm_,%[iU5)Fcs7$$\"3!*******Rv&)z:F/$!3iM[wAF*f]&Fcs7 $$\"3ILLLGUYo;F/$!3#[460Q>``$Fcs7$$\"3_mmm1^rZ%)oyBFcs7$$ \"34++]sI@K=F/$!3C'GkR7B!f:Fcs7$$\"34++]2%)38>F/$!3*>\"RxE1[S5Fcs7$$\" \"#F*$!3\"\\'\\w'4Twt'F`u-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELS G6$Q#t|ir6\"Q!6\"-%%VIEWG6$;F(Fe`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Plot driving voltage and voltage response across resistor and capacitor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(\{ft ,ftr,ftc\},t=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 268 201 201 {PLOTDATA 2 "6'-%'CURVESG6$7bo7$$!\"#\"\"!$F*F*7$$!3MLLL$Q6G\">!#'***!#=$\"3?i^e3\\%))*=!#?7$$!3E,++vE\"z))*FT$\"3ssU 3\"y9-X&!#>7$$!3k+++]miz(*FT$\"3bS$G$*yIL/\"FT7$$!3-+++D1Mr'*FT$\"3$*H oL;vP::FT7$$!3_++++Y0j&*FT$\"31%[[gyXD'>FT7$$!3Q+++]D[Y$*FT$\"3/i:.lDU (y#FT7$$!3E++++0\"*H\"*FT$\"30CtntHkFNFT7$$!3y++++K?@*)FT$\"3+=O,31,pT FT7$$!3=++++f\\7()FT$\"3?nralH#ou%FT7$$!3f*******f)y.&)FT$\"3]Y:\\_zPn _FT7$$!35++++83&H)FT$\"3]#)RM%R\\jt&FT7$$!3!ommT&)Gg\"zFT$\"3[sR)zJkCZ 'FT7$$!3\\LLL3k(p`(FT$\"3?k@n^(*[\"3(FT7$$!3M++](QYi3(FT$\"3;Pp_DFPqwF T7$$!3Anmmmj^NmFT$\"3d#GBEsL/9)FT7$$!3gnmm;*)o`iFT$\"3GtA*G:>OY)FT7$$! 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