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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Fourier Example" }}{PARA 
0 "" 0 "" {TEXT -1 34 "Square Wave source into RC circuit" }}{PARA 0 "
" 0 "" {TEXT -1 49 "First write Fourier Series for Square Wave Source
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "V:=sum('4/Pi*sin((2*k+1
)*omega*t)/(2*k+1)','k'=0..40);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
\"VG,^p*&-%$sinG6#,$*&%#PiG\"\"\"%\"tGF-\"$Y\"F-F,!\"\"#\"\"%\"#t*&*&#
F2\"#rF--F(6#,$F+\"$U\"F-F-F,F0F-*&*&#F2\"#pF--F(6#,$F+\"$Q\"F-F-F,F0F
-*&*&#F2\"#nF--F(6#,$F+\"$M\"F-F-F,F0F-*&*&#F2\"#lF--F(6#,$F+\"$I\"F-F
-F,F0F-*&*&#F2\"#vF--F(6#,$F+\"$]\"F-F-F,F0F-*&*&#F2\"\"$F--F(6#,$F+\"
\"'F-F-F,F0F-*&*&F2F--F(6#,$F+\"\"#F-F-F,F0F-*&*&#F2\"\"&F--F(6#,$F+\"
#5F-F-F,F0F-*&*&#F2\"\"(F--F(6#,$F+\"#9F-F-F,F0F-*&*&#F2\"\"*F--F(6#,$
F+\"#=F-F-F,F0F-*&*&#F2\"#6F--F(6#,$F+\"#AF-F-F,F0F-*&*&#F2\"#8F--F(6#
,$F+\"#EF-F-F,F0F-*&*&#F2\"#:F--F(6#,$F+\"#IF-F-F,F0F-*&*&#F2\"#<F--F(
6#,$F+\"#MF-F-F,F0F-*&*&#F2\"#>F--F(6#,$F+\"#QF-F-F,F0F-*&*&#F2\"#@F--
F(6#,$F+\"#UF-F-F,F0F-*&*&#F2\"#BF--F(6#,$F+\"#YF-F-F,F0F-*&*&#F2\"#DF
--F(6#,$F+\"#]F-F-F,F0F-*&*&#F2\"#FF--F(6#,$F+\"#aF-F-F,F0F-*&*&#F2\"#
HF--F(6#,$F+\"#eF-F-F,F0F-*&*&#F2\"#JF--F(6#,$F+\"#iF-F-F,F0F-*&*&#F2
\"#LF--F(6#,$F+\"#mF-F-F,F0F-*&*&#F2\"#NF--F(6#,$F+\"#qF-F-F,F0F-*&*&#
F2\"#PF--F(6#,$F+\"#uF-F-F,F0F-*&*&#F2\"#RF--F(6#,$F+\"#yF-F-F,F0F-*&*
&#F2\"#TF--F(6#,$F+\"##)F-F-F,F0F-*&*&#F2\"#VF--F(6#,$F+\"#')F-F-F,F0F
-*&*&#F2\"#XF--F(6#,$F+\"#!*F-F-F,F0F-*&*&#F2\"#ZF--F(6#,$F+\"#%*F-F-F
,F0F-*&*&#F2\"#\\F--F(6#,$F+\"#)*F-F-F,F0F-*&*&#F2\"#^F--F(6#,$F+\"$-
\"F-F-F,F0F-*&*&#F2\"#`F--F(6#,$F+\"$1\"F-F-F,F0F-*&*&#F2\"#bF--F(6#,$
F+\"$5\"F-F-F,F0F-*&*&#F2\"#dF--F(6#,$F+\"$9\"F-F-F,F0F-*&*&#F2\"#fF--
F(6#,$F+\"$=\"F-F-F,F0F-*&*&#F2\"#hF--F(6#,$F+\"$A\"F-F-F,F0F-*&*&#F2
\"#jF--F(6#,$F+\"$E\"F-F-F,F0F-*&*&#F2\"#xF--F(6#,$F+\"$a\"F-F-F,F0F-*
&*&#F2\"#zF--F(6#,$F+\"$e\"F-F-F,F0F-*&*&#F2\"#\")F--F(6#,$F+\"$i\"F-F
-F,F0F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "To find Vr (voltage ac
ross resistor)" }}{PARA 0 "" 0 "" {TEXT -1 54 "Multiply terms by (2k+1
)*w*R*C/sqrt(1.+*(2*k+1)*w*R*C)" }}{PARA 0 "" 0 "" {TEXT -1 41 "Phase \+
shift terms by -arctan(-1/(w*R*C));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 118 "Vr:=sum('4./Pi*omega*R*C/sqrt(1+((2*k+1)*omega*R*C)*
*2)*sin((2*k+1)*omega*t-arctan(-1/(2*k+1)/omega/C/R))','k'=0..40);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#VrG,^p*&*(%\"RG\"\"\"%\"CGF)-%$sinG
6#,&%\"tG$\"+K$zqq$!\"(-%'arctanG6#,$*&F)F)*&F*F)F(F)!\"\"$\"+39a(p#!#
7F)F)F)*$-%%sqrtG6#,&$F)\"\"!F)*($\"+<PCu8!\"%F))F(\"\"#F))F*FIF)F)F)F
9$\"\")FC*&**FKF)F(F)F*F)-F,6#,&F/$\"+EcT\"e$F2-F46#,$F7$\"+$)>>#z#F<F
)F)F)*$-F?6#,&FBF)*($\"+)y`EG\"FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,
6#,&F/$\"+*4!Q*3&F2-F46#,$F7$\"+Se(['>F<F)F)F)*$-F?6#,&FBF)*($\"+!)*y,
f#FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+#op<q#F2-F46#,$F7$
\"+YxF,PF<F)F)F)*$-F?6#,&FBF)*($\"+<%f&*H(!\"&F)FHF)FJF)F)F)F9F)*&**FK
F)F(F)F*F)-F,6#,&F/$\"+*QLu#GF2-F46#,$F7$\"+7lxONF<F)F)F)*$-F?6#,&FBF)
*($\"+n&zV*zF`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+&4(4`HF2
-F46#,$F7$\"+$QviQ$F<F)F)F)*$-F?6#,&FBF)*($\"+^Cy?()F`qF)FHF)FJF)F)F)F
9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+,3wyIF2-F46#,$F7$\"+h+1[KF<F)F)F)*$-F?
6#,&FBF)*($\"+q!o(y%*F`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"
+xn'[l&!\")-F46#,$F7$\"+c#)Qo<!#6F)F)F)*$-F?6#,&FBF)*($\"+F=v(>$!\"'F)
FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+RQ]6pFau-F46#,$F7$\"+=J'o
W\"FguF)F)F)*$-F?6#,&FBF)*($\"+K&))ox%F_vF)FHF)FJF)F)F)F9F)*&**FKF)F(F
)F*F)-F,6#,&F/$\"++49o\")Fau-F46#,$F7$\"+$zoUA\"FguF)F)F)*$-F?6#,&FBF)
*($\"+xD&=n'F_vF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+izxC%*Fa
u-F46#,$F7$\"+aH.h5FguF)F)F)*$-F?6#,&FBF)*($\"+kRk#)))F_vF)FHF)FJF)F)F
)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+3`=$G'!\"*-F46#,$F7$\"+I%\\:f\"!#5F)
F)F)*$-F?6#,&FBF)*($\"+i<%y%RFauF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6
#,&F/$\"+#fb\\)=Fau-F46#,$F7$\"+pZ;0`FguF)F)F)*$-F?6#,&FBF)*($\"+&edIb
$F2F)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+aEfTJFau-F46#,$F7$\"
+h))4$=$FguF)F)F)*$-F?6#,&FBF)*($\"+/Wgp)*F2F)FHF)FJF)F)F)F9F)*&**FKF)
F(F)F*F)-F,6#,&F/$\"+;(H#)R%Fau-F46#,$F7$\"+W?ktAFguF)F)F)*$-F?6#,&FBF
)*($\"+jCWM>F_vF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+@E8X9F2-
F46#,$F7$\"+J,y>pF<F)F)F)*$-F?6#,&FBF)*($\"+#H3%)3#F`qF)FHF)FJF)F)F)F9
F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+wf5wDF2-F46#,$F7$\"+zy#=)QF<F)F)F)*$-F?
6#,&FBF)*($\"+,?KOmF`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+
\")*)Q7ZF2-F46#,$F7$\"+2f1A@F<F)F)F)*$-F?6#,&FBF)*($\"+\"*4m?AFGF)FHF)
FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+(o_!Q[F2-F46#,$F7$\"+7t%p1#F<
F)F)F)*$-F?6#,&FBF)*($\"+!Qv1M#FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,
6#,&F/$\"+$R;P'\\F2-F46#,$F7$\"+K&>Y,#F<F)F)F)*$-F?6#,&FBF)*($\"+V![QY
#FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+2XU/KF2-F46#,$F7$\"+
e^o?JF<F)F)F)*$-F?6#,&FBF)*($\"+UO$o-\"FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F
)F*F)-F,6#,&F/$\"+-:9o5F2-F46#,$F7$\"+va0i$*F<F)F)F)*$-F?6#,&FBF)*($\"
+pi#49\"F`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+4_!Q>\"F2-F4
6#,$F7$\"+\\fdw$)F<F)F)F)*$-F?6#,&FBF)*($\"+w3<D9F`qF)FHF)FJF)F)F)F9F)
*&**FKF)F(F)F*F)-F,6#,&F/$\"+:*o%>8F2-F46#,$F7$\"+7o!)yvF<F)F)F)*$-F?6
#,&FBF)*($\"+<#)*4u\"F`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"
+Fjzq:F2-F46#,$F7$\"+Ax>mjF<F)F)F)*$-F?6#,&FBF)*($\"+,6SnCF`qF)FHF)FJF
)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+L+Y'p\"F2-F46#,$F7$\"+@vi%*eF<F)
F)F)*$-F?6#,&FBF)*($\"+Wm(z(GF`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6
#,&F/$\"+RP7A=F2-F46#,$F7$\"+%[,\")[&F<F)F)F)*$-F?6#,&FBF)*($\"+A\\8?L
F`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+XuyZ>F2-F46#,$F7$\"+
A/.M^F<F)F)F)*$-F?6#,&FBF)*($\"+Lf(Qz$F`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F
)F*F)-F,6#,&F/$\"+_6Xt?F2-F46#,$F7$\"+iq(G#[F<F)F)F)*$-F?6#,&FBF)*($\"
+y'*>*H%F`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+e[6*>#F2-F46
#,$F7$\"+*3%GZXF<F)F)F)*$-F?6#,&FBF)*($\"+eh5O[F`qF)FHF)FJF)F)F)F9F)*&
**FKF)F(F)F*F)-F,6#,&F/$\"+k&yZK#F2-F46#,$F7$\"+Y\\[,VF<F)F)F)*$-F?6#,
&FBF)*($\"+s`f/aF`qF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+qAW]
CF2-F46#,$F7$\"+vf*33%F<F)F)F)*$-F?6#,&FBF)*($\"+>tm/gF`qF)FHF)FJF)F)F
)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+v_s'e%F2-F46#,$F7$\"+*p/-=#F<F)F)F)*
$-F?6#,&FBF)*($\"+v[!Q5#FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$
\"+cTt4UF2-F46#,$F7$\"+6pWvBF<F)F)F)*$-F?6#,&FBF)*($\"+nh=s<FGF)FHF)FJ
F)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+jyRNVF2-F46#,$F7$\"+yLf1BF<F)F)
F)*$-F?6#,&FBF)*($\"+jucz=FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F
/$\"+p:1hWF2-F46#,$F7$\"+;*=;C#F<F)F)F)*$-F?6#,&FBF)*($\"+Kq5!*>FGF)FH
F)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+QIuKQF2-F46#,$F7$\"+Fu44EF<
F)F)F)*$-F?6#,&FBF)*($\"+>>**o9FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,
6#,&F/$\"+WnSeRF2-F46#,$F7$\"+P*oi_#F<F)F)F)*$-F?6#,&FBF)*($\"+&R)*oc
\"FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+]/2%3%F2-F46#,$F7$
\"+&eP&[CF<F)F)F)*$-F?6#,&FBF)*($\"+WJ'zm\"FGF)FHF)FJF)F)F)F9F)*&**FKF
)F(F)F*F)-F,6#,&F/$\"+8#)3ILF2-F46#,$F7$\"+aM#H+$F<F)F)F)*$-F?6#,&FBF)
*($\"+^([*36FGF)FHF)FJF)F)F)F9F)*&**FKF)F(F)F*F)-F,6#,&F/$\"+>>vbMF2-F
46#,$F7$\"+Pis$*GF<F)F)F)*$-F?6#,&FBF)*($\"+L@A%>\"FGF)FHF)FJF)F)F)F9F
)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "To find Vc (voltage across c
apacitor)" }}{PARA 0 "" 0 "" {TEXT -1 43 "Multiply terms by 1/sqrt(1.+
*(2*k+1)*w*R*C)" }}{PARA 0 "" 0 "" {TEXT -1 46 "Phase shift terms by -
arctan(-1/(w*R*C))+Pi/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
122 "Vc:=sum('4./Pi/sqrt(1+((2*k+1)*omega*R*C)**2)*sin((2*k+1)*omega*t
-arctan(-1/(2*k+1)/omega/C/R)-Pi/2.)/(2*k+1)','k'=0..40);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%#VcG,^p*&-%$cosG6#,&%\"tG$\"+2XU/K!\"(-%'arct
anG6#,$*&\"\"\"F4*&%\"CGF4%\"RGF4!\"\"$\"+e^o?J!#7F4F4*$-%%sqrtG6#,&$F
4\"\"!F4*($\"+UO$o-\"!\"%F4)F7\"\"#F4)F6FHF4F4F4F8$!+E\"[l\\#!#6*&*&$
\"+2.-4FFLF4-F(6#,&F+$\"+&4(4`HF.-F06#,$F3$\"+$QviQ$F;F4F4F4*$-F>6#,&F
AF4*($\"+^Cy?()!\"&F4FGF4FIF4F4F4F8F8*&*&$\"+:?q:ZFLF4-F(6#,&F+$\"+L+Y
'p\"F.-F06#,$F3$\"+@vi%*eF;F4F4F4*$-F>6#,&FAF4*($\"+Wm(z(GF\\oF4FGF4FI
F4F4F4F8F8*&*&$\"+)=\"[!R%FLF4-F(6#,&F+$\"+RP7A=F.-F06#,$F3$\"+%[,\")[
&F;F4F4F4*$-F>6#,&FAF4*($\"+A\\8?LF\\oF4FGF4FIF4F4F4F8F8*&*&$\"+y\"eH4
&FLF4-F(6#,&F+$\"+Fjzq:F.-F06#,$F3$\"+Ax>mjF;F4F4F4*$-F>6#,&FAF4*($\"+
,6SnCF\\oF4FGF4FIF4F4F4F8F8*&*&$\"+\\;IeQFLF4-F(6#,&F+$\"+_6Xt?F.-F06#
,$F3$\"+iq(G#[F;F4F4F4*$-F>6#,&FAF4*($\"+y'*>*H%F\\oF4FGF4FIF4F4F4F8F8
*&*&$\"+ez=TMFLF4-F(6#,&F+$\"+k&yZK#F.-F06#,$F3$\"+Y\\[,VF;F4F4F4*$-F>
6#,&FAF4*($\"+s`f/aF\\oF4FGF4FIF4F4F4F8F8*&*&$\"+qs#yj$FLF4-F(6#,&F+$
\"+e[6*>#F.-F06#,$F3$\"+*3%GZXF;F4F4F4*$-F>6#,&FAF4*($\"+eh5O[F\\oF4FG
F4FIF4F4F4F8F8*&*&$\"+(e`PB#FLF4-F(6#,&F+$\"+EcT\"e$F.-F06#,$F3$\"+$)>
>#z#F;F4F4F4*$-F>6#,&FAF4*($\"+)y`EG\"FFF4FGF4FIF4F4F4F8F8*&*&$\"+QVA2
TFLF4-F(6#,&F+$\"+XuyZ>F.-F06#,$F3$\"+A/.M^F;F4F4F4*$-F>6#,&FAF4*($\"+
Lf(Qz$F\\oF4FGF4FIF4F4F4F8F8*&*&$\"+j(QBS#FLF4-F(6#,&F+$\"+8#)3ILF.-F0
6#,$F3$\"+aM#H+$F;F4F4F4*$-F>6#,&FAF4*($\"+^([*36FFF4FGF4FIF4F4F4F8F8*
&*&$\"+!*4)\\J#FLF4-F(6#,&F+$\"+>>vbMF.-F06#,$F3$\"+Pis$*GF;F4F4F4*$-F
>6#,&FAF4*($\"+L@A%>\"FFF4FGF4FIF4F4F4F8F8*&*&$\"+]![%)f#FLF4-F(6#,&F+
$\"+,3wyIF.-F06#,$F3$\"+h+1[KF;F4F4F4*$-F>6#,&FAF4*($\"+q!o(y%*F\\oF4F
GF4FIF4F4F4F8F8*&*&$\"+EJ.e@FLF4-F(6#,&F+$\"+K$zqq$F.-F06#,$F3$\"+39a(
p#F;F4F4F4*$-F>6#,&FAF4*($\"+<PCu8FFF4FGF4FIF4F4F4F8F8*&*&$\"+2T#e`&FL
F4-F(6#,&F+$\"+@E8X9F.-F06#,$F3$\"+J,y>pF;F4F4F4*$-F>6#,&FAF4*($\"+#H3
%)3#F\\oF4FGF4FIF4F4F4F8F8*&*&$\"+]a/jgFLF4-F(6#,&F+$\"+:*o%>8F.-F06#,
$F3$\"+7o!)yvF;F4F4F4*$-F>6#,&FAF4*($\"+<#)*4u\"F\\oF4FGF4FIF4F4F4F8F8
*&*&$\"+xVk*[(FLF4-F(6#,&F+$\"+-:9o5F.-F06#,$F3$\"+va0i$*F;F4F4F4*$-F>
6#,&FAF4*($\"+pi#49\"F\\oF4FGF4FIF4F4F4F8F8*&*&$\"+g2E,nFLF4-F(6#,&F+$
\"+4_!Q>\"F.-F06#,$F3$\"+\\fdw$)F;F4F4F4*$-F>6#,&FAF4*($\"+w3<D9F\\oF4
FGF4FIF4F4F4F8F8*&*&$\"+IOE)[)FLF4-F(6#,&F+$\"+izxC%*!\")-F06#,$F3$\"+
aH.h5FLF4F4F4*$-F>6#,&FAF4*($\"+kRk#)))!\"'F4FGF4FIF4F4F4F8F8*&*&$\"+0
1r99!#5F4-F(6#,&F+$\"+xn'[l&Ffdl-F06#,$F3$\"+c#)Qo<FLF4F4F4*$-F>6#,&FA
F4*($\"+F=v(>$FcelF4FGF4FIF4F4F4F8F8*&*&$\"+*3zka#FhelF4-F(6#,&F+$\"+a
EfTJFfdl-F06#,$F3$\"+h))4$=$FLF4F4F4*$-F>6#,&FAF4*($\"+/Wgp)*F.F4FGF4F
IF4F4F4F8F8*&*&$\"+9=8WUFhelF4-F(6#,&F+$\"+#fb\\)=Ffdl-F06#,$F3$\"+pZ;
0`FLF4F4F4*$-F>6#,&FAF4*($\"+&edIb$F.F4FGF4FIF4F4F4F8F8*&*&$\"+U.:%z*F
LF4-F(6#,&F+$\"++49o\")Ffdl-F06#,$F3$\"+$zoUA\"FLF4F4F4*$-F>6#,&FAF4*(
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