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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "This topic is a bit advanc
ed but fun to look at." }}{PARA 0 "" 0 "" {TEXT -1 81 "In the maple sq
uare wave response of an RC circuit solve using LaPlace Transforms" }}
{PARA 0 "" 0 "" {TEXT -1 74 "it was shown  RC circuits turned into pol
es and zeros. Active filters give" }}{PARA 0 "" 0 "" {TEXT -1 75 "us a
 very powerful way to shape incoming signals. A given OP AMP filter ca
n" }}{PARA 0 "" 0 "" {TEXT -1 81 "have a resistor and/or capacitor in \+
series into the inversing input, with another" }}{PARA 0 "" 0 "" 
{TEXT -1 77 "resistor and/or capacitor in series on the feedback resis
tor. If we have many" }}{PARA 0 "" 0 "" {TEXT -1 73 "such OP AMP filte
rs one gets a gain SUMi(Ri+jCi omega)/SUMk(Rk+jCk omega)" }}{PARA 0 "
" 0 "" {TEXT -1 65 "which interms of the complex laplace s=j omega can
 be written as " }}{PARA 0 "" 0 "" {TEXT -1 67 "SUMi (s+zi)/SUMj(s+pj)
 where zi are the zeros and pj are the poles." }}{PARA 0 "" 0 "" 
{TEXT -1 69 "Below are shown how this can be used to shape a short inc
oming square" }}{PARA 0 "" 0 "" {TEXT -1 11 "wave pulse." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7/%)addtableG%(fourierG%+fouriercosG%+fouriersinG%'ha
nkelG%(hilbertG%+invfourierG%+invhilbertG%+invlaplaceG%*invmellinG%(la
placeG%'mellinG%*savetableG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Ge
nerate a square wave pulse" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "ft:=(Heaviside(t)-Heaviside(t-1));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#ftG,&-%*HeavisideG6#%\"tG\"\"\"-F'6#,&F)F*F*!\"\"F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(ft,t=-1..2);" }}{PARA 
13 "" 1 "" {GLPLOT2D 268 201 201 {PLOTDATA 2 "6%-%'CURVESG6$7co7$$!\"
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Fep7$$\"3s**\\7G#\\31*F/Fep7$$\"3a****\\7TW)R*F/Fep7$$\"3g)*\\(=7;,b*F
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7$$\"3#****\\Pm,H.\"F[vF+7$$\"31++]7,Hl5F[vF+7$$\"3()**\\P4w)R7\"F[vF+
7$$\"3;++]x%f\")=\"F[vF+7$$\"3!)**\\P/-a[7F[vF+7$$\"3/+](=Yb;J\"F[vF+7
$$\"3')****\\i@Ot8F[vF+7$$\"3')**\\PfL'zV\"F[vF+7$$\"3>+++!*>=+:F[vF+7
$$\"3-++DE&4Qc\"F[vF+7$$\"3=+]P%>5pi\"F[vF+7$$\"39+++bJ*[o\"F[vF+7$$\"
33++Dr\"[8v\"F[vF+7$$\"3++++Ijy5=F[vF+7$$\"31+]P/)fT(=F[vF+7$$\"31+]i0
j\"[$>F[vF+7$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"#5F)F+F+-%+AXESLABELSG6$Q
\"t6\"Q!6\"-%%VIEWG6$;F(F]y%(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 40 " An Op Amp differentiator has gain s*RC." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fs:=laplace(ft,t,s);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#fsG,&*&\"\"\"F'%\"sG!\"\"F'*&-%$expG6#,$F(F)F
'F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "fsdiff:=s;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fsdiffG%\"sG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 81 "This leads to pure differentiation of the square wav
e with dirac delta functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "ftdiff:=invlaplace(fs*fsdiff,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'ftdiffG,&-%&DiracG6#%\"tG\"\"\"-F'6#,&F)F*F*!\"\"F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 " In a real situation one would not
 have a perfect square wave" }}{PARA 0 "" 0 "" {TEXT -1 66 "so the del
ta functions are a bit unrealistic. Lets smooth them out" }}{PARA 0 "
" 0 "" {TEXT -1 12 "with a pole." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "fsdiffreal:=s/(s+30.);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+fsdiffrealG*&%\"sG\"\"\",&F&F'$\"#I\"\"!F'!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ftdiffreal:=invlaplace(fs*fsdiffrea
l,s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+ftdiffrealG,&-%$expG6#,$
%\"tG$!#I\"\"!\"\"\"*($F.F-F.-%*HeavisideG6#,&F*F.$F.F-!\"\"F.-F'6#,&F
*F+$\"#IF-F.F.F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Below one can
 see a nice result. With differentiation we" }}{PARA 0 "" 0 "" {TEXT 
-1 51 "can choose between a rising edge and a falling edge" }}{PARA 0 
"" 0 "" {TEXT -1 24 "of a single square wave." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "plot(\{ftdiffreal,ft\},t=0..2);" }}{PARA 13 "" 
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*7$FdsF*7$FjsF*7$F`tF*7$FetF*7$F[uF*7$F`uF*7$FfuF*7$F[vF*7$FavF*7$FfvF
*7$F\\wF*7$FawF*7$FgwF*7$F\\xF*7$FbxF(7$FhxF(7$F]yF(7$FbyF(7$FgyF(7$F
\\zF(7$FazF(7$FfzF(7$F`[lF(7$Fj[lF(7$F_\\lF(7$Fd\\lF(7$F^]lF(7$Fh]lF(7
$Fb^lF(7$Fg^lF(7$F\\_lF(7$Fa_lF(7$Ff_lF(7$F[`lF(7$F``lF(7$Fe`lF(7$Fj`l
F(7$F_alF(7$FdalF(7$FialF(7$F^blF(7$FcblF(7$FhblF(7$F]clF(7$FbclF(7$Fg
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45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
40 "An OP AMP integrator has a gain 1/(s*RC)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "fsint:=1/s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&fsintG*&\"\"\"F&%\"sG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "t
his leads to pure integration of the square wave " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 32 "ftint:=invlaplace(fs*fsint,s,t);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&ftintG,&*&,&%\"tG!\"\"\"\"\"F*F*-%*Heavis
ideG6#,&F(F*F*F)F*F*F(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This \+
circuit is useful for measuring charge deposited from a detector:" }}
{PARA 0 "" 0 "" {TEXT -1 59 "photomultiplier, photodiode, proportional
 wire chamber, ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(
\{ft,ftint\},t=0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 268 201 201 
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DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "There of cours
e is always some resistance in series with the" }}{PARA 0 "" 0 "" 
{TEXT -1 60 "input capacitance no matter how small. Let's see the effe
ct." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fsintreal:=1/(s+0.1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*fsintrealG*&\"\"\"F&,&%\"sGF&$
F&!\"\"F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ftintreal:=i
nvlaplace(fs*fsintreal,s,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*fti
ntrealG,(*&,&$!#5\"\"!\"\"\"*&$\"#5F*F+-%$expG6#,&%\"tG$!+++++5F)$\"++
+++5F)F+F+F+F+-%*HeavisideG6#,&F3F+$F+F*!\"\"F+F+F-F+*&$F.F*F+-F06#,$F
3F4F+F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The circuit still inte
grates but the result drops off exponentially." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "plot(\{ft,ftintreal\},t=0...10);" }}{PARA 13 "
" 1 "" {GLPLOT2D 268 201 201 {PLOTDATA 2 "6&-%'CURVESG6$7]o7$$\"\"!F)F
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F-Ff`l7$$\"3#***\\7.K[V?F-Ff`l7$$\"3ALL$3FWYs#F-Ff`l7$$\"3%)***\\iSmp3
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FaqF(7$$\"3v\"zpB6Vf.\"FcqF(7$FgqF(7$F\\rF(7$FarF(7$FfrF(7$F[sF(7$F`sF
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(7$FgvF(7$F\\wF(7$FawF(7$FfwF(7$F[xF(7$F`xF(7$FexF(7$FjxF(7$F_yF(7$Fdy
F(7$FiyF(7$F^zF(7$FczF(7$FhzF(7$F][lF(7$Fb[lF(7$Fg[lF(7$F\\\\lF(7$Fa\\
lF(7$Ff\\lF(7$F[]lF(7$F`]lF(7$Fe]lF(7$Fj]lF(7$F_^lF(7$Fd^lF(7$Fi^lF(7$
F^_lF(7$Fc_lF(-Fh_l6&Fj_lF(F[`lF(-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6
$;F(Fc_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
66 "We often have a detector where a given amount of charge arrives in
" }}{PARA 0 "" 0 "" {TEXT -1 76 "a finite but short time period (propo
rtional drift chambers, ...). To build " }}{PARA 0 "" 0 "" {TEXT -1 
80 "an amplifier we would like to collect all of the charge and amplif
y it.The above" }}{PARA 0 "" 0 "" {TEXT -1 81 "circuit does this but w
e would also like the output to be in a nice simple shape " }}{PARA 0 
"" 0 "" {TEXT -1 84 "that we can analyse easily. Several OP AMP poles \+
will do this for us. Take the five " }}{PARA 0 "" 0 "" {TEXT -1 87 "po
le circuit below. This is accomplished with 5 OP AMPs where the integr
ating capacitor" }}{PARA 0 "" 0 "" {TEXT -1 44 "is in series with a re
sistor. One then gets:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f
smultipole:=1/(s+.1)**5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,fsmulti
poleG*&\"\"\"F&*$),&%\"sGF&$F&!\"\"F&\"\"&F&F," }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 76 " Lets look at the voltage delta function response of
 this multipole circuit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"ftmultipole:=invlaplace(fsmultipole,s,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%,ftmultipoleG,$*&)%\"tG\"\"%\"\"\"-%$expG6#,$F($!++++
+5!#5F*$\"+nmmmT!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The result
ing function is refered to in EE as a semi-Gaussian since it is close \+
" }}{PARA 0 "" 0 "" {TEXT -1 20 "to Gaussian in shape" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(\{ftmultipole\},t=0..100);" }}
{PARA 13 "" 1 "" {GLPLOT2D 268 201 201 {PLOTDATA 2 "6%-%'CURVESG6$7gn7
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"> " 0 "" {MPLTEXT 1 0 45 "ftmultipole2:=invlaplace(fs*fsmultipole,s,t
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-ftmultipole2G,0*&,.$!'++5\"\"
!\"\"\"*($\"+nmmmT!#5F+-%$expG6#,&%\"tG$!+++++5F/$\"+++++5F/F+F+)F4\"
\"%F+F+*($\"#:F*F+F0F+)F4\"\"$F+F+*($\"++++DX!\"(F+F0F+)F4\"\"#F+F+*($
\"+LLL[!*!\"'F+F0F+F4F+F+*&$\"++]P[!*!\"&F+F0F+F+F+-%*HeavisideG6#,&F4
F+$F+F*!\"\"F+F+$\"'++5F*F+*($\"+nmmmTF/F+F9F+-F16#,$F4F5F+FS*($\"+nmm
m;!\")F+F>F+FYF+FS*($\"$+&F*F+FDF+FYF+FS*($\"&++\"F*F+F4F+FYF+FS*&$FUF
*F+FYF+FS" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 " Once sees the delta
 function response and the square wave response are " }}{PARA 0 "" 0 "
" {TEXT -1 69 "nearly identical. The peak amplitude is obviously propo
rtional to the" }}{PARA 0 "" 0 "" {TEXT -1 46 "charge deposited in the
 finite amount of time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "
plot(\{ftmultipole2,ftmultipole\},t=0..100);" }}{PARA 13 "" 1 "" 
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