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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "LaPlace Example" }}{PARA 
0 "" 0 "" {TEXT -1 71 "LaPlace invented his own transforms to make app
lying Fourier Transforms" }}{PARA 0 "" 0 "" {TEXT -1 86 "to problems l
ike RC circuits driven by a square wave easily solved. Defining a comp
lex" }}{PARA 0 "" 0 "" {TEXT -1 77 "variable s=j*omega, resistor and c
apacitor filters reduce to poles and zeros." }}{PARA 0 "" 0 "" {TEXT 
-1 77 "Here is the same problem done previously with a truncated fouri
er series done" }}{PARA 0 "" 0 "" {TEXT -1 78 "here with LaPlace Trans
forms. Note that the previous problem solves the steady" }}{PARA 0 "" 
0 "" {TEXT -1 69 "state solution. Here we solve the transient and stea
dy state problem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(i
nttrans);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/%)addtableG%(fourierG%+
fouriercosG%+fouriersinG%'hankelG%(hilbertG%+invfourierG%+invhilbertG%
+invlaplaceG%*invmellinG%(laplaceG%'mellinG%*savetableG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "R:='R';C:='C';" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"RGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGF$" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "ft:=(Heaviside(t)-Heavisi
de(t-0.5)+Heaviside(t-1.0)-Heaviside(t-1.5)+Heaviside(t-2.0)-Heaviside
(t-2.5)+Heaviside(t-3.0)-Heaviside(t-3.5)-0.5)*2;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#ftG,4-%*HeavisideG6#%\"tG\"\"#*&F*\"\"\"-F'6#,&F)F,$
\"\"&!\"\"F2F,F2*&F*F,-F'6#,&F)F,$\"#5F2F2F,F,*&F*F,-F'6#,&F)F,$\"#:F2
F2F,F2*&F*F,-F'6#,&F)F,$\"#?F2F2F,F,*&F*F,-F'6#,&F)F,$\"#DF2F2F,F2*&F*
F,-F'6#,&F)F,$\"#IF2F2F,F,*&F*F,-F'6#,&F)F,$\"#NF2F2F,F2$F8F2F2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(ft,t=-1.0..4.0);" }}
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[$F[uFfn7$$\"3'[5:)=z7#\\$F[uFfn7$$\"3uD\"yv6*e&\\$F[uFfn7$$\"3gY6M;.0
*\\$F[uFfn7$$\"3ZnT5::^-NF[uF(7$$\"3%4Dc,Jcj^$F[uF(7$$\"3(RL3_5,-`$F[u
F(7$$\"3Y+DJ&p!*yb$F[uF(7$$\"3SnmT&G!e&e$F[uF(7$$\"3m+]P%37^j$F[uF(7$$
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IEWG6$;$!#5F)$\"#SF)%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "fs:=laplace(ft,t,s);" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%#fsG,2*&\"\"\"F'%\"sG!\"\"F'*&*&$\"\"#\"\"!F'-%$expG6#,$F($!+++++
]!#5F'F'F(F)F)*&*&$F-F.F'-F06#,$F($F)F.F'F'F(F)F'*&*&$F-F.F'-F06#,$F($
!+++++:!\"*F'F'F(F)F)*&*&F8F'-F06#,$F($!\"#F.F'F'F(F)F'*&*&$F-F.F'-F06
#,$F($!+++++DFEF'F'F(F)F)*&*&F8F'-F06#,$F($!\"$F.F'F'F(F)F'*&*&$F-F.F'
-F06#,$F($!+++++NFEF'F'F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "
 First write the complex filter across the resistor" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "frs:=R*C*s/(1+s*R*C);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$frsG*&*(%\"RG\"\"\"%\"CGF(%\"sGF(F(,&F(F(*(F*F(F'F
(F)F(F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The time solution
 is the product of the laplace transform for the square wave and the" 
}}{PARA 0 "" 0 "" {TEXT -1 55 "individual complex filter equations acr
oss the resistor" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "frt:=invlaplace(fs*frs,s,t);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$frtG*(%\"RG\"\"\"%\"CGF',2*&-%$expG
6#,$*&%\"tGF'*&F&F'F(F'!\"\"$F2\"\"!F'*&F&F'F(F'F2F'*&*($\"\"#F4F'-%*H
eavisideG6#,&F0F'$\"+++++]!#5F2F'-F,6#,$*&F=F'*&F&F'F(F'F2F3F'F'*&F&F'
F(F'F2F2*&*($F9F4F'-F;6#,&F0F'$F'F4F2F'-F,6#,$*&FLF'*&F&F'F(F'F2F3F'F'
*&F&F'F(F'F2F'*&*($F9F4F'-F;6#,&F0F'$\"+++++:!\"*F2F'-F,6#,$*&FYF'*&F&
F'F(F'F2F3F'F'*&F&F'F(F'F2F2*&*(FIF'-F;6#,&F0F'$F9F4F2F'-F,6#,$*&FaoF'
*&F&F'F(F'F2F3F'F'*&F&F'F(F'F2F'*&*($F9F4F'-F;6#,&F0F'$\"+++++DFfnF2F'
-F,6#,$*&F^pF'*&F&F'F(F'F2F3F'F'*&F&F'F(F'F2F2*&*(FIF'-F;6#,&F0F'$\"\"
$F4F2F'-F,6#,$*&F[qF'*&F&F'F(F'F2F3F'F'*&F&F'F(F'F2F'*&*($F9F4F'-F;6#,
&F0F'$\"+++++NFfnF2F'-F,6#,$*&FiqF'*&F&F'F(F'F2F3F'F'*&F&F'F(F'F2F2F'
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 " Next write the complex filte
r across the capacitor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f
cs:=1/(1+R*C*s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fcsG*&\"\"\"F&,
&F&F&*(%\"sGF&%\"RGF&%\"CGF&F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 85 "The time solution is the product of the laplace transform for t
he square wave and the" }}{PARA 0 "" 0 "" {TEXT -1 57 "individual comp
lex filter equations across the capacitor." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "fct:=invlaplace(fcs*fs,s,t);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$fctG,4*&,&$!\"#\"\"!\"\"\"*&$\"\"#F*F+-%$expG6#,$*&,
&%\"tGF+$\"+++++:!\"*!\"\"F+*&%\"RGF+%\"CGF+F9$F9F*F+F+F+-%*HeavisideG
6#F4F+F+*&,&F-F+*&$F.F*F+-F06#,$*&,&F5F+$F.F*F9F+*&F;F+F<F+F9F=F+F9F+-
F?6#FIF+F+*&,&F(F+*&F-F+-F06#,$*&,&F5F+$\"+++++DF8F9F+*&F;F+F<F+F9F=F+
F+F+-F?6#FUF+F+*&,&F-F+*&$F.F*F+-F06#,$*&,&F5F+$\"\"$F*F9F+*&F;F+F<F+F
9F=F+F9F+-F?6#F]oF+F+*&,&F(F+*&F-F+-F06#,$*&,&F5F+$\"+++++NF8F9F+*&F;F
+F<F+F9F=F+F+F+-F?6#FjoF+F+*&,&F(F+*&F-F+-F06#,$*&,&F5F+$\"+++++]!#5F9
F+*&F;F+F<F+F9F=F+F+F+-F?6#FgpF+F+*&,&F-F+*&$F.F*F+-F06#,$*&,&F5F+$F+F
*F9F+*&F;F+F<F+F9F=F+F9F+-F?6#FfqF+F+$F+F*F+*&$F+F*F+-F06#,$*&F5F+*&F;
F+F<F+F9F=F+F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Take the limit \+
w*R*C<<1. Voltage across R looks like differential" }}{PARA 0 "" 0 "" 
{TEXT -1 70 "Note this looks just like the Fourier Transform from the \+
previous case" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "R:=100.;C:
=evalf(0.002/Pi/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{ft,fc
t,frt\},t=0.0..2.0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG$\"$+\"
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG$\"+h))4$=$!#8" }}
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}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "But even more important, we no lo
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" {MPLTEXT 1 0 35 "f3t:=Heaviside(t)-Heaviside(t-0.1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$f3tG,&-%*HeavisideG6#%\"tG\"\"\"-F'6#,&F)F*$F*!
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