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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 34 "Signal generation by an R
C circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
50 "We consider the RC circuit with time constant tau=" }{TEXT 257 2 "
RC" }{TEXT -1 83 " driven by a time-dependent external source signal o
f triangular shape with period " }{TEXT 258 1 "T" }{TEXT -1 56 ". We w
ish to calculate the voltage across the capacitor." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "We begin with the proble
m for a single cycle of the source voltage, and then generalize to obt
ain the voltage for a finite number of cycles. We find the capacitor v
oltage for the regime of " }{TEXT 283 1 "T" }{TEXT -1 90 " < tau, and \+
study the transition from the transient to the steady-state capacitor \+
voltage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "We proceed as in the worksheet " }{TEXT 19 11 "RLCcirc.mw
s" }{TEXT -1 274 " to generate the time evolution of the current (or v
oltage across the resistor) using Kirchhoff's law and the relationship
s between voltage and current for the two elements. We need to modify \+
the rhs of Kirchhoff's loop rule to allow for a time-dependent electri
c potential." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "We begin by defining the voltage \+
drops across the resistor and the capacitor." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "V_R:=R*i(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$V_RG*&%\"RG\"\"\"-%\"iG6#%\"tGF'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "V_C:=(1/C)*Int(i(s),s=0..t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$V_CG*&-%$IntG6$-%\"iG6#%\"sG/F,;\"\"!%\"tG\"\"\"%\"C
G!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "According to Kirchhoff'
s rule we equate the sum of the drops across " }{TEXT 325 1 "R" }
{TEXT -1 5 " and " }{TEXT 324 1 "C" }{TEXT -1 46 " to the applied time
-dependent signal voltage " }{TEXT 323 1 "U" }{TEXT -1 1 "(" }{TEXT 
322 1 "t" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "KL_RC:=V_R+V_C=U(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&KL_RCG/
,&*&%\"RG\"\"\"-%\"iG6#%\"tGF)F)*&-%$IntG6$-F+6#%\"sG/F4;\"\"!F-\"\"\"
%\"CG!\"\"F)-%\"UGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "We differ
entiate the relationship to obtain a differential equation for the cur
rent " }{TEXT 327 1 "i" }{TEXT -1 1 "(" }{TEXT 326 1 "t" }{TEXT -1 2 "
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "KL_RC:=diff(KL_RC,t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&KL_RCG/,&*&%\"RG\"\"\"-%%diffG6
$-%\"iG6#%\"tGF0F)F)*&F-\"\"\"%\"CG!\"\"F)-F+6$-%\"UGF/F0" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "Note that in  the case of a constant sour
ce voltage " }{TEXT 284 2 "U " }{TEXT -1 188 "the value of the applied
 voltage disappeared from the equation due to the differentiation. How
ever,  it entered the solution through the initial condition. In the c
ase of a time-dependent " }{TEXT 288 1 "U" }{TEXT -1 1 "(" }{TEXT 287 
1 "t" }{TEXT -1 17 ") the derivative " }{TEXT 286 1 "U" }{TEXT -1 2 "'
(" }{TEXT 285 1 "t" }{TEXT -1 47 ") appears in the differentiated Kirc
hhoff rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 162 "We need to define the external voltage as a function of time. \+
We consider a triangle-shaped function. First we have a linear rise fr
om zero to a maximum value for" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{TEXT 328 2 "t " }{TEXT -1 2 "< " }{TEXT 329 1 "T" }{TEXT -1 53 "/2 . \+
We choose a slope of 1, i.e., the rise is 1 V/s:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 9 "U1:=t->t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#U1GR6#%\"tG6\"6$%)operatorG%&arrowGF(9$F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "KL_RC1:=simplify(subs(U(t)=U1(t),KL_RC));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'KL_RC1G/*&,&*(%\"RG\"\"\"-%%diffG6$
-%\"iG6#%\"tGF1F*%\"CGF*F*F.F*\"\"\"F2!\"\"F*" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 184 "Without simplification the rhs would not have been ev
aluated. This would have caused dsolve to fail in the lines below. The
 initial current vanishes, as there is no voltage applied at " }{TEXT 
330 1 "t" }{TEXT -1 90 "=0, and we assume that the capacitor is unchar
ged. Therefore, the initial condition reads:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "IC1:=i(0)=U1(0)/R;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$IC1G/-%\"iG6#\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "sol_c:=dsolve(\{KL_RC1,IC1\},i(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&sol_cG/-%\"iG6#%\"tG,&%\"CG\"\"\"*&-%$expG6#,$*&F)\"
\"\"*&F+\"\"\"%\"RG\"\"\"!\"\"!\"\"F,F+F,F9" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "i_RCc:=rhs(sol_c);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&i_RCcG,&%\"CG\"\"\"*&-%$expG6#,$*&%\"tG\"\"\"*&F&\"\"\"%\"RG
\"\"\"!\"\"!\"\"F'F&F'F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "We n
eed to know the voltage across the capacitor in order to be able to st
art the solution for the 'discharge' phase. By 'discharge' we mean the
 second half of the pulse during which the source voltage decreases. W
e obtain the capacitor voltage as the complement of " }{TEXT 260 1 "U
" }{TEXT -1 26 "_R to the battery voltage " }{TEXT 261 1 "U" }{TEXT 
-1 1 "(" }{TEXT 290 1 "t" }{TEXT -1 25 "). We also introduce tau=" }
{TEXT 259 2 "RC" }{TEXT -1 35 ", the time constant of the circuit." }}
{PARA 0 "" 0 "" {TEXT -1 75 "The voltage drop across the resistor duri
ng the 'charge' phase is given as:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "U_Rc:=subs(i(t)=i_RCc,V_R);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%U_RcG*&%\"RG\"\"\",&%\"CGF'*&-%$expG6#,$*&%\"tG\"\"
\"*&F)\"\"\"F&\"\"\"!\"\"!\"\"F'F)F'F6F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "U_Rc:=simplify(subs(R=tau/C,U_Rc));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%%U_RcG,$*&%$tauG\"\"\",&!\"\"F(-%$expG6#,$*&%\"tG
\"\"\"F'!\"\"F*F(F(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "The vol
tage across the capacitor is obtained as the complement between the vo
ltage applied to the circuit and the drop across the resistor:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "U_Cc:=U1(t)-U_Rc;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%U_CcG,&%\"tG\"\"\"*&%$tauGF',&!\"\"F'-%$e
xpG6#,$*&F&\"\"\"F)!\"\"F+F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
95 "This function describes how the voltage builds up across the capac
itor from a value of zero at " }{TEXT 262 1 "t" }{TEXT -1 98 "=0 up to
 asymptotic times. We explore its time behaviour for a linearly increa
sing source voltage " }{TEXT 281 1 "U" }{TEXT -1 1 "(" }{TEXT 280 1 "t
" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot([
U1(t),subs(tau=1,U_Cc)],t=0..10,color=[blue,red]);" }}{PARA 13 "" 1 "
" {GLPLOT2D 647 224 224 {PLOTDATA 2 "6&-%'CURVESG6$7S7$\"\"!F(7$$\"1nm
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$\"1+++D.&4]#F9FM7$$\"1+++vB_<FF9FP7$$\"1+++v'Hi#HF9FS7$$\"1nm\"z*ev:J
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&\\(F9F_r7$$\"1,+D1#=bq(F9Fbr7$$\"1LLL3s?6zF9Fer7$$\"1++DJXaE\")F9Fhr7
$$\"1ommm*RRL)F9F[s7$$\"1om;a<.Y&)F9F^s7$$\"1NLe9tOc()F9Fas7$$\"1,++]Q
k\\*)F9Fds7$$\"1NL$3dg6<*F9Fgs7$$\"1ommmxGp$*F9Fjs7$$\"1++D\"oK0e*F9F]
t7$$\"1,+v=5s#y*F9F`t7$$\"#5F(Fct-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-
F$6$7SF'7$F*$\"1m6(Gx')>@#!#<7$F.$\"1!oD+55aG(Fbu7$F1$\"1Xv$osvOe\"F,7
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>Ls9loF,7$FA$\"1$4sy&ROR&)F,7$FD$\"1R&R/MJ$H5F97$FG$\"1'3uGK1p@\"F97$F
J$\"1Qw@66,(Q\"F97$FM$\"1Z?D`t&He\"F97$FP$\"1b@sW0c$y\"F97$FS$\"1$GRtT
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%F97$F^p$\"1\"*HH1*p4A%F97$Fap$\"1^DwA\\%>T%F97$Fdp$\"1FJV>$)Q=YF97$Fg
p$\"1K-Q'>H=$[F97$Fjp$\"1'e7,nq2/&F97$F]q$\"1mHvAD.V_F97$F`q$\"1M0IUKq
naF97$Fcq$\"1r*z=%ykpcF97$Ffq$\"1kM82TK&)eF97$Fiq$\"1ZF$e]+33'F97$F\\r
$\"1Oc*4iWXH'F97$F_r$\"13D\"oz*o&\\'F97$Fbr$\"1i3_T&ofq'F97$Fer$\"1O*p
+#Qd6pF97$Fhr$\"1cE3>,%o7(F97$F[s$\"1q6P!>!=MtF97$F^s$\"1am#)ogAYvF97$
Fas$\"1i29qZ_cxF97$Fds$\"1lTPLOx\\zF97$Fgs$\"1t&ylck7<)F97$Fjs$\"1'*)*
yqIPp$)F97$F]t$\"1!)3ST<g!e)F97$F`t$\"1oB#oVxFy)F97$Fct$\"1i(H**RX++*F
9-Fft6&FhtFitF(F(-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(Fct%(DEFAULTG
" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 189 "We see that the capacitor voltage is non-linea
r for small times, but approaches a linear regime with a time delay as
 compared to the source voltage. We can explore the short-time behavio
ur:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "taylor(U_Cc,t=0,3);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG,$*&\"\"\"F'%$tauG!\"\"#\"
\"\"\"\"#\"\"#-%\"OG6#F+\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 
"The quadratic dependence on time, as well as the dependence of the co
efficient on the time constant of the RC circuit should be noted." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 17 "Now we introduce " }{TEXT 263 8 "T, i.e.
," }{TEXT -1 71 " the period of the triangle-wave signal that drives t
he RC circuit. At " }{TEXT 264 1 "t" }{TEXT -1 1 "=" }{TEXT 265 1 "T" 
}{TEXT -1 60 "/2 the wave stops growing, and the discharge regime begi
ns. " }}{PARA 0 "" 0 "" {TEXT -1 201 "The so-called duty cycle for the
 driving signal was chosen to be 50%. An interesting extension of this
 worksheet would consist of introducing the ratio of charge to dischar
ge regimes during the period " }{TEXT 337 1 "T" }{TEXT -1 41 " as a va
riable (e.g., charge regime: 0 < " }{TEXT 336 1 "t" }{TEXT -1 3 " < " 
}{TEXT 335 1 "T" }{TEXT -1 26 "/3, and discharge regime: " }{TEXT 334 
1 "T" }{TEXT -1 5 "/3 < " }{TEXT 333 1 "t" }{TEXT -1 3 " < " }{TEXT 
331 2 "T " }{TEXT -1 35 "corresponding to a 1/3 duty cycle)." }}{PARA 
0 "" 0 "" {TEXT -1 69 "We denote the voltage to which the capacitor wa
s charged at the time " }{TEXT 339 1 "t" }{TEXT -1 1 "=" }{TEXT 338 1 
"T" }{TEXT -1 9 "/2 as U0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "U_Th:=simplify(subs(t=T/2,U_Cc));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%U_ThG,(%\"TG#\"\"\"\"\"#%$tauG!\"\"*&F*F(-%$expG6#,$*&F&\"\"\"F*
!\"\"#F+F)F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "We formulate th
e solution for the discharge regime. We note that with the choice for \+
" }{TEXT 306 1 "U" }{TEXT 309 1 "1" }{TEXT -1 1 "(" }{TEXT 307 1 "t" }
{TEXT -1 2 ")=" }{TEXT 308 1 "t" }{TEXT -1 1 "," }{TEXT 332 1 " " }
{TEXT -1 39 "and a given choice of the period (e.g. " }{TEXT 310 1 "T
" }{TEXT -1 71 "=1) we imply a scale for the voltage axis, since the s
ource voltage at " }{TEXT 341 1 "t" }{TEXT -1 1 "=" }{TEXT 340 1 "T" }
{TEXT -1 36 "/2 will rise to some maximum value (" }{TEXT 312 1 "U" }
{TEXT -1 13 "_max=1/2 for " }{TEXT 311 1 "T" }{TEXT -1 28 "=1). The so
urce voltage for " }{TEXT 292 1 "T" }{TEXT -1 5 "/2 < " }{TEXT 291 1 "
t" }{TEXT -1 3 " < " }{TEXT 289 1 "T" }{TEXT -1 13 " is given as:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "U2:=t->T-t;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#U2GR6#%\"tG6\"6$%)operatorG%&arrowGF(,&%\"TG\"
\"\"9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Kirchhoff's \+
law is now formulated as follows:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "KL_RC2:=simplify(subs(U(t)=U2(t),KL_RC));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'KL_RC2G/*&,&*(%\"RG\"\"\"-%%diffG6$-%\"iG
6#%\"tGF1F*%\"CGF*F*F.F*\"\"\"F2!\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 102 "The initial condition on the current is given by the amo
unt that flows at the end of the charge cycle:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "IC2:=i(T/2)=simplify(subs(t=T/2,i_RCc));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IC2G/-%\"iG6#,$%\"TG#\"\"\"\"\"#,&%
\"CGF,*&-%$expG6#,$*&F*\"\"\"*&F/\"\"\"%\"RG\"\"\"!\"\"#!\"\"F-F,F/F,F
=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol_d:=dsolve(\{KL_RC2
,IC2\},i(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol_dG/-%\"iG6#%\"
tG,&%\"CG!\"\"*&*(-%$expG6#,$*&F)\"\"\"*&F+\"\"\"%\"RG\"\"\"!\"\"F,\"
\"\"F+F:,&!\"#F:-F06#,$*&%\"TGF4*&F+\"\"\"F7\"\"\"F9#F,\"\"#F:F:F4F=F9
F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We combine the exponentials
 in the solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "i_RCd:
=combine(subs(C=tau/R,rhs(sol_d)),exp);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&i_RCdG,&*&%$tauG\"\"\"%\"RG!\"\"!\"\"*&*(F'\"\"\",&!\"#F.-%$e
xpG6#,$*&%\"TGF(F'F*#F+\"\"#F.F.-F26#,&*&%\"tGF(F'F*F+F5#F.F8F.F(F)F*F
+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The voltage drop across the
 resistor is calculated in order to obtain the voltage across the capa
citor:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "U_Rd:=simplify(su
bs(i(t)=i_RCd,V_R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%U_RdG*&%$ta
uG\"\"\",(!\"\"F'-%$expG6#,$*&,&%\"tG!\"#%\"TGF'\"\"\"F&!\"\"#F'\"\"#F
6-F+6#,$*&F0F3F&F4F)F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"U_Cd:=simplify(U2(t)-U_Rd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%U_C
dG,,%\"TG\"\"\"%\"tG!\"\"%$tauGF'*&F*F'-%$expG6#,$*&,&F(!\"#F&F'\"\"\"
F*!\"\"#F'\"\"#F'F2*&F*F3-F-6#,$*&F(F3F*F4F)F'F'" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 77 "The voltage at t=T will be needed to continue the \+
solution in the next cycle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "U_T:=subs(t=T,U_Cd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$U_TG,(%
$tauG\"\"\"*&F&F'-%$expG6#,$*&%\"TG\"\"\"F&!\"\"#!\"\"\"\"#F'!\"#*&F&F
/-F*6#,$F-F2F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Now we have t
he voltage across the capacitor for the charge regime valid up until \+
" }{TEXT 269 1 "t" }{TEXT -1 1 "=" }{TEXT 268 1 "T" }{TEXT -1 62 "/2, \+
as well as for the discharge regime voltage valid between " }{TEXT 
267 1 "T" }{TEXT -1 7 "/2 and " }{TEXT 266 1 "T" }{TEXT -1 69 ". We ne
ed to piece them together, and to display them for one period " }
{TEXT 270 1 "T" }{TEXT -1 86 " as a function of  time with a choice of
 the parameter tau. The voltage scale for the " }{TEXT 271 1 "y" }
{TEXT -1 16 " axis is set by " }{TEXT 300 1 "U" }{TEXT 301 1 "1" }
{TEXT -1 1 "(" }{TEXT 305 1 "t" }{TEXT -1 6 ") and " }{TEXT 303 1 "U" 
}{TEXT 302 1 "2" }{TEXT -1 1 "(" }{TEXT 304 1 "t" }{TEXT -1 17 ") once
 we choose " }{TEXT 272 1 "T" }{TEXT -1 36 "=1 to set a scale for the \+
time axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 148 "We reassign the previously defined voltages across the capacit
or at the end of the charge and discharge cycles to refer specifically
 to the case of " }{TEXT 342 1 "T" }{TEXT -1 3 "=1:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "U_Cc:=simplify(subs(T=1,U_Cc));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%U_CcG,(%\"tG\"\"\"%$tauG!\"\"*&F(F'-%$exp
G6#,$*&F&\"\"\"F(!\"\"F)F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "U_Cd:=simplify(subs(T=1,U_Cd));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%U_CdG,,\"\"\"F&%\"tG!\"\"%$tauGF&*&F)F&-%$expG6#,$*&,&F'\"\"#F(F
&\"\"\"F)!\"\"#F(F1F&!\"#*&F)F2-F,6#,$*&F'F2F)F3F(F&F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 127 "It is of interest to observe the results
 of the charge&discharge cycles for the case where the time constant o
f the RC circuit " }{TEXT 19 3 "tau" }{TEXT -1 28 " is shorter than th
e period " }{TEXT 343 1 "T" }{TEXT -1 46 " of the driving signal. With
 the scale set by " }{TEXT 274 1 "T" }{TEXT -1 15 " we can choose " }
{TEXT 19 3 "tau" }{TEXT -1 4 " << " }{TEXT 273 1 "T" }{TEXT -1 4 " as \+
" }{TEXT 19 3 "tau" }{TEXT -1 5 " <<1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "P1c:=plot(subs(tau=1/10,U_Cc),t=0..1/2,color=red):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "P1d:=plot(subs(tau=1/10,U_
Cd),t=1/2..1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display
(P1c,P1d);" }}{PARA 13 "" 1 "" {GLPLOT2D 646 246 246 {PLOTDATA 2 "6&-%
'CURVESG6$7S7$\"\"!F(7$$\"1LLL3x&)*3\"!#<$\"1e]\"*[\"[*Gd!#>7$$\"1nm\"
H2P\"Q?F,$\"1R5QK2!G%>!#=7$$\"1LL$eRwX5$F,$\"1QC3C=!pN%F57$$\"1LL$3x%3
yTF,$\"1nVy.4oHwF57$$\"1mm\"z%4\\Y_F,$\"1FaTa27k6F,7$$\"1LLeR-/PiF,$\"
1f;f\")ef'f\"F,7$$\"1***\\il'pisF,$\"1sbOXpz*4#F,7$$\"1LLe*)>VB$)F,$\"
1n'R>J>Pn#F,7$$\"1***\\7`l2Q*F,$\"1N'*GMYc%H$F,7$$\"1nm;/j$o/\"!#;$\"1
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Ffn7$$\"1LLe*Gst!GFfn$\"19rH\"pNx'=Ffn7$$\"1+++DRW9HFfn$\"1`:#f5y'o>Ff
n7$$\"1***\\7j#>>IFfn$\"1f.5mK.o?Ffn7$$\"1+]i!RU07$Ffn$\"1(e/^hvY;#Ffn
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*QFfn7$$\"1+]iSj0xoFfn$\"1N*oF&\\*y\"QFfn7$$\"1mmm\"pW`(pFfn$\"1qlCrH;
[PFfn7$$\"1+]i!f#=$3(Ffn$\"1gvuA+foOFfn7$$\"1+](=xpe=(Ffn$\"1'H6eaF,f$
Ffn7$$\"1mm\"H28IH(Ffn$\"14'\\mgVd]$Ffn7$$\"1n;zpSS\"R(Ffn$\"1u]`?-@EM
Ffn7$$\"1LL3_?`(\\(Ffn$\"1]LLQnWQLFfn7$$\"1L$e*)>pxg(Ffn$\"1G$=Q%)H`C$
Ffn7$$\"1**\\Pf4t.xFfn$\"1*oHPq4G;$Ffn7$$\"1LLe*Gst!yFfn$\"1Q0RKwIsIFf
n7$$\"1+++DRW9zFfn$\"1$[$>SJXxHFfn7$$\"1***\\7j#>>!)Ffn$\"10$eY=bM)GFf
n7$$\"1+]i!RU07)Ffn$\"1pKuF&)[\"z#Ffn7$$\"1++v=S2L#)Ffn$\"17.\\H&>$)o#
Ffn7$$\"1mmm\"p)=M$)Ffn$\"1-Jg([kZf#Ffn7$$\"1****\\(=]@W)Ffn$\"1/y(*pP
2%\\#Ffn7$$\"1L$e*[$z*R&)Ffn$\"198&GH)=-CFfn7$$\"1++]iC$pk)Ffn$\"1*)zZ
U75,BFfn7$$\"1m;H2qcZ()Ffn$\"1BD*>xTa?#Ffn7$$\"1+]7.\"fF&))Ffn$\"1$4.[
nT\\5#Ffn7$$\"1mm;/Ogb*)Ffn$\"1/qX#>Ji+#Ffn7$$\"1+]ilAFj!*Ffn$\"1F#y;C
eC!>Ffn7$$\"1LLL$)*pp;*Ffn$\"1K(fc9O@!=Ffn7$$\"1LL3xe,t#*Ffn$\"1pb1*o)
>*p\"Ffn7$$\"1n;HdO=y$*Ffn$\"1wN1KY!of\"Ffn7$$\"1+++D>#[Z*Ffn$\"1AI9M.
Z-:Ffn7$$\"1mmT&G!e&e*Ffn$\"1Ry=mF4%R\"Ffn7$$\"1LLL$)Qk%o*Ffn$\"1_TC-j
%pH\"Ffn7$$\"1+]iSjE!z*Ffn$\"1XN*e;pJ>\"Ffn7$$\"1+]P40O\"*)*Ffn$\"1cXt
2xm$4\"Ffn7$$\"\"\"F($\"1DfJf]pl)*F,-Fhz6&FjzF(F(F[[l-%+AXESLABELSG6$Q
\"t6\"%!G-%%VIEWG6$;F(F`jl%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "We see \+
that the capacitor voltage nearly follows the source voltage with some
 rounding off occuring. The voltage to which the capacitor charges  at
 the end of the rise in the source voltage is:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 31 "evalf(subs(T=1,tau=1/10,U_Th));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+Zzt1S!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 53 "The value at the end of the cycle can be read off as:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf(subs(T=1,tau=1/10,U_T)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+f]pl)*!#6" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 80 "For comparable time constants of the RC circuit
 and the square wave signal, tau=" }{TEXT 313 2 "T " }{TEXT -1 10 "we \+
obtain:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P2c:=plot(subs(t
au=1,U_Cc),t=0..1/2,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 48 "P2d:=plot(subs(tau=1,U_Cd),t=1/2..1,color=blue):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(P2c,P2d);" }}{PARA 13 "" 1 
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L$eRwX5$F,$\"1!G!=)45(pZF57$$\"1LL$3x%3yTF,$\"1&f.\\=)*yg)F57$$\"1mm\"
z%4\\Y_F,$\"1C!z9\"p__8!#=7$$\"1LLeR-/PiF,$\"1&y()3y=_!>FE7$$\"1***\\i
l'pisF,$\"17\\FoMjuDFE7$$\"1LLe*)>VB$)F,$\"1;AN&zN)pLFE7$$\"1***\\7`l2
Q*F,$\"1#)\\TYC_lUFE7$$\"1nm;/j$o/\"!#;$\"1!Gr$eL.$H&FE7$$\"1LL3_>jU6F
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gn$\"1\"y0D)G*GD\"Fgn7$$\"1M$eR-/Pi&Fgn$\"1G2Roj;%G\"Fgn7$$\"1+]il'pis
&Fgn$\"1V`K:Z>:8Fgn7$$\"1L$e*)>VB$eFgn$\"1$)o)y\\]eM\"Fgn7$$\"1+]7`l2Q
fFgn$\"12'H@@s\\P\"Fgn7$$\"1nm;/j$o/'Fgn$\"1sb4!zZMS\"Fgn7$$\"1LL3_>jU
hFgn$\"1#>2;'*)HF9Fgn7$$\"1++]i^Z]iFgn$\"1=i#)z`y_9Fgn7$$\"1****\\(=h(
ejFgn$\"1ztH/v$pZ\"Fgn7$$\"1++]P[6jkFgn$\"1_AH\\F'))\\\"Fgn7$$\"1L$e*[
z(yb'Fgn$\"16!fq(oj<:Fgn7$$\"1mm;a/cqmFgn$\"1*)>.Z#o&Q:Fgn7$$\"1mmm;t,
mnFgn$\"1&\\X\\&y7b:Fgn7$$\"1+]iSj0xoFgn$\"1]fvqB0t:Fgn7$$\"1mmm\"pW`(
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gn$\"1I@wa9vP;Fgn7$$\"1LL3_?`(\\(Fgn$\"1Ka))z\"Ruk\"Fgn7$$\"1L$e*)>pxg
(Fgn$\"1h:2Ew?c;Fgn7$$\"1**\\Pf4t.xFgn$\"1Zi`*)yxi;Fgn7$$\"1LLe*Gst!yF
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$\"1!)QI#HEvn\"Fgn7$$\"1+]i!RU07)Fgn$\"1zt-dE2!o\"Fgn7$$\"1++v=S2L#)Fg
n$\"1?=<\"*Hn\"o\"Fgn7$$\"1mmm\"p)=M$)Fgn$\"1[&f->@?o\"Fgn7$$\"1****\\
(=]@W)Fgn$\"1\"3VYVm7o\"Fgn7$$\"1L$e*[$z*R&)Fgn$\"1z3?wzez;Fgn7$$\"1++
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.\"fF&))Fgn$\"11Wo!p&)zm\"Fgn7$$\"1mm;/Ogb*)Fgn$\"17\"))4_I@m\"Fgn7$$
\"1+]ilAFj!*Fgn$\"1nWO&eP\\l\"Fgn7$$\"1LLL$)*pp;*Fgn$\"11s:)e#*pk\"Fgn
7$$\"1LL3xe,t#*Fgn$\"1qfv^h%yj\"Fgn7$$\"1n;HdO=y$*Fgn$\"1Or=MfwF;Fgn7$
$\"1+++D>#[Z*Fgn$\"1Oz.')fi<;Fgn7$$\"1mmT&G!e&e*Fgn$\"1*>(\\g=)\\g\"Fg
n7$$\"1LLL$)Qk%o*Fgn$\"17n.')ov#f\"Fgn7$$\"1+]iSjE!z*Fgn$\"1JziA)z(y:F
gn7$$\"1+]P40O\"*)*Fgn$\"1gjk.+]k:Fgn7$$\"\"\"F($\"1`<Y<7=[:Fgn-Fiz6&F
[[lF(F(F\\[l-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(F^jl%(DEFAULTG" 1 
2 0 1 0 2 9 1 4 2 1.000000 43.000000 45.000000 0 }}}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 46 "The voltage to which the capacitor charges at " }
{TEXT 293 1 "T" }{TEXT -1 15 "/2 is given as:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "evalf(subs(T=1,tau=1,U_Th));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+(f1`1\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
39 "while at the end of the period we have:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 27 "evalf(subs(T=1,tau=1,U_T));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+A7=[:!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "We
 observe that as tau, " }{TEXT 295 4 "i.e." }{TEXT -1 90 ", the time c
onstant of the RC circuit approaches the time constant of the source v
oltage, " }{TEXT 296 1 "T" }{TEXT -1 324 " , the capacitor voltage doe
s not follow the source voltage. In fact, the capacitor charged during
 the 'discharge' phase to a higher voltage! Further below we will demo
nstrate that this is so only during the transient regime when the capa
citor is charged to a low voltage as compared to the maximum value (se
t by a scale of " }{TEXT 314 1 "U" }{TEXT -1 33 "_max=1/2  in  the sou
rce voltage " }{TEXT 298 1 "U" }{TEXT -1 1 "(" }{TEXT 297 1 "t" }
{TEXT -1 3 "))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 51 "Finally we go to the other extreme given by tau >> \+
" }{TEXT 299 1 "T" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "P3c:=plot(subs(tau=10,U_Cc),t=0..1/2,color=red):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "P3d:=plot(subs(tau=10,U_Cd),
t=1/2..1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "di
splay(P3c,P3d);" }}{PARA 13 "" 1 "" {GLPLOT2D 646 272 272 {PLOTDATA 2 
"6&-%'CURVESG6$7S7$\"\"!F(7$$\"1LLL3x&)*3\"!#<$\"1cR`s@zOf!#@7$$\"1nm
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3yTF,$\"1*G]y9`gr)F57$$\"1mm\"z%4\\Y_F,$\"1;$z3jzQP\"!#>7$$\"1LLeR-/Pi
F,$\"1mT2%4'*4%>FE7$$\"1***\\il'pisF,$\"1YE&G)\\'4j#FE7$$\"1LLe*)>VB$)
F,$\"1#>mmC&QaMFE7$$\"1***\\7`l2Q*F,$\"1(p$QS?@'Q%FE7$$\"1nm;/j$o/\"!#
;$\"1!)RE'Hh-Y&FE7$$\"1LL3_>jU6Fgn$\"1*46P$fC.lFE7$$\"1++]i^Z]7Fgn$\"1
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4np#*p]<F^p7$$\"1mmm\"pW`(>Fgn$\"10O>M-@Q>F^p7$$\"1+]i!f#=$3#Fgn$\"1p%
z#ee$[:#F^p7$$\"1+](=xpe=#Fgn$\"1CXzJ6qrBF^p7$$\"1nm\"H28IH#Fgn$\"1zu6
]](*3EF^p7$$\"1n;zpSS\"R#Fgn$\"1(pA1,\\n$GF^p7$$\"1LL3_?`(\\#Fgn$\"1@v
Uo)HI4$F^p7$$\"1L$e*)>pxg#Fgn$\"1ZN*=2l3P$F^p7$$\"1+]Pf4t.FFgn$\"1qIZI
3OAOF^p7$$\"1LLe*Gst!GFgn$\"1G:PK00/RF^p7$$\"1+++DRW9HFgn$\"1&zbt$=.1U
F^p7$$\"1***\\7j#>>IFgn$\"1Mr`1qB7XF^p7$$\"1+]i!RU07$Fgn$\"14s!3xR'=[F
^p7$$\"1++v=S2LKFgn$\"1V7CV=^q^F^p7$$\"1mmm\"p)=MLFgn$\"1>=Q0F9(\\&F^p
7$$\"1++](=]@W$Fgn$\"1!ohJh1o&eF^p7$$\"1L$e*[$z*RNFgn$\"1W\"=jYTC>'F^p
7$$\"1++]iC$pk$Fgn$\"1vLWg)[*plF^p7$$\"1m;H2qcZPFgn$\"1T<q(GD_$pF^p7$$
\"1+]7.\"fF&QFgn$\"1;k&GsruK(F^p7$$\"1mm;/OgbRFgn$\"1o?a#yd7s(F^p7$$\"
1+]ilAFjSFgn$\"1r>AA%3W9)F^p7$$\"1LLL$)*pp;%Fgn$\"1Dm&RlvCc)F^p7$$\"1L
L3xe,tUFgn$\"1`(zu-x1+*F^p7$$\"1n;HdO=yVFgn$\"1C,*yv\"*eW*F^p7$$\"1+++
D>#[Z%Fgn$\"1fc_1:Lk)*F^p7$$\"1nmT&G!e&e%Fgn$\"1xPY*H*[N5F,7$$\"1LLL$)
Qk%o%Fgn$\"1.)QtWe.3\"F,7$$\"1+]iSjE!z%Fgn$\"1zXJ&zH#H6F,7$$\"1+]P40O
\"*[Fgn$\"1<&o`%>+x6F,7$$\"1+++++++]Fgn$\"1$)Qr+XUH7F,-%'COLOURG6&%$RG
BG$\"*++++\"!\")F(F(-F$6$7SFdz7$$\"1LL$3x&)*3^Fgn$\"1Yv1b[&>G\"F,7$$\"
1n;H2P\"Q?&Fgn$\"1Wn#**z[mK\"F,7$$\"1MLeRwX5`Fgn$\"1ubDsuyv8F,7$$\"1LL
3x%3yT&Fgn$\"1RkRf40C9F,7$$\"1n;z%4\\Y_&Fgn$\"1nUGQ+*3Z\"F,7$$\"1M$eR-
/Pi&Fgn$\"1'[/n%>D8:F,7$$\"1+]il'pis&Fgn$\"1#HC(4w.c:F,7$$\"1L$e*)>VB$
eFgn$\"1bP%H\\M\"*f\"F,7$$\"1+]7`l2QfFgn$\"1x#QI`G4k\"F,7$$\"1nm;/j$o/
'Fgn$\"1F4(p(pq#o\"F,7$$\"1LL3_>jUhFgn$\"1j&y(\\&)[=<F,7$$\"1++]i^Z]iF
gn$\"1)\\e\"\\;jd<F,7$$\"1****\\(=h(ejFgn$\"14z)pYBdz\"F,7$$\"1++]P[6j
kFgn$\"1Ht_$3%GJ=F,7$$\"1L$e*[z(yb'Fgn$\"1wux69gi=F,7$$\"1mm;a/cqmFgn$
\"19:%y)Rj)*=F,7$$\"1mmm;t,mnFgn$\"1IQiTO8G>F,7$$\"1+]iSj0xoFgn$\"1B]U
#\\n7'>F,7$$\"1mmm\"pW`(pFgn$\"1F\"f')oP&*)>F,7$$\"1+]i!f#=$3(Fgn$\"1-
La%z6%>?F,7$$\"1+](=xpe=(Fgn$\"1+@.K([n/#F,7$$\"1mm\"H28IH(Fgn$\"15=<Y
&=T2#F,7$$\"1n;zpSS\"R(Fgn$\"11H#)Qh@)4#F,7$$\"1LL3_?`(\\(Fgn$\"1#ezUQ
(4B@F,7$$\"1L$e*)>pxg(Fgn$\"1g7gO@sZ@F,7$$\"1**\\Pf4t.xFgn$\"1!QpU5Z\"
o@F,7$$\"1LLe*Gst!yFgn$\"1Q+R!z]\"*=#F,7$$\"1+++DRW9zFgn$\"1+tZK$*p4AF
,7$$\"1***\\7j#>>!)Fgn$\"1?J>#)>nGAF,7$$\"1+]i!RU07)Fgn$\"1;dwAi'fC#F,
7$$\"1++v=S2L#)Fgn$\"1y,$yBXRE#F,7$$\"1mmm\"p)=M$)Fgn$\"1@'**>O.!zAF,7
$$\"1****\\(=]@W)Fgn$\"1F-[7j$RH#F,7$$\"1L$e*[$z*R&)Fgn$\"1x\"><wZkI#F
,7$$\"1++]iC$pk)Fgn$\"1&p3ga<!>BF,7$$\"1m;H2qcZ()Fgn$\"1W:xv%)yHBF,7$$
\"1+]7.\"fF&))Fgn$\"1xcL9O&*RBF,7$$\"1mm;/Ogb*)Fgn$\"16!>hM7)[BF,7$$\"
1+]ilAFj!*Fgn$\"1B[&z<WpN#F,7$$\"1LLL$)*pp;*Fgn$\"1\\$Q0\\sOO#F,7$$\"1
LL3xe,t#*Fgn$\"1!=WDTM%pBF,7$$\"1n;HdO=y$*Fgn$\"1t!)GLD.uBF,7$$\"1+++D
>#[Z*Fgn$\"1hJ+V)ysP#F,7$$\"1mmT&G!e&e*Fgn$\"1s;P(oZ)zBF,7$$\"1LLL$)Qk
%o*Fgn$\"1;p9KT5\"Q#F,7$$\"1+]iSjE!z*Fgn$\"1GBo!)>O\"Q#F,7$$\"1+]P40O
\"*)*Fgn$\"1OIFjTc!Q#F,7$$\"\"\"F($\"13<`M!p&yBF,-Fjz6&F\\[lF(F(F][l-%
+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(F_jl%(DEFAULTG" 1 2 0 1 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 179 "The answer is clearly not periodic after a single cycle, we wo
uld need to piece together several charge/discharge segments to see wh
ether a steady-state solution will be achieved." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 195 "The signal is st
rongly attenuated. The RC circuit acts as a low-pass filter: the volta
ge across the capacitor reflects the triangular-wave input to the circ
uit for low frequencies (time constant " }{TEXT 275 1 "T" }{TEXT -1 
233 " large compared to tau) with a rounding effect on the edges of th
e pulse. It suppresses throughput at high frequencies while distorting
 the shape considerably. The details of this behaviour are shown in th
e solution for many periods " }{TEXT 294 1 "T" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "T
he voltage to which the capacitor charges  at " }{TEXT 316 1 "T" }
{TEXT -1 7 "/2 and " }{TEXT 315 1 "T" }{TEXT -1 17 " respectively is:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(subs(T=1,tau=10,U
_Th));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")XUH7!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(subs(T=1,tau=10,U_T));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")!p&yB!\"*" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 4 "The " }{TEXT 282 19 "long-time behaviour" }{TEXT -1 25 
" of the capacitor voltage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 447 "The solution to a first-order equation depends o
n a single integration constant. It is the actual value of the current
 at the end of  the previous time segment that determines the further \+
time evolution. The solution to the problem is continuous and differen
tiable in this case as there are no discontinuities in the source volt
age (the derivative of the source voltage is discontinuous). Given tha
t the source voltage does not change abruptly at " }{TEXT 317 1 "t" }
{TEXT -1 1 "=" }{TEXT 318 1 "T" }{TEXT -1 194 "/2 we should assume as \+
the correct initial condition for the discharge cycle that the final v
alue of the current in the charge cycle represents the correct initial
 value for the discharge cycle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Now we are interested in piecing t
ogether a solution over several periods, e.g. " }{TEXT 19 7 "nmax=20" 
}{TEXT -1 9 " of them." }}{PARA 0 "" 0 "" {TEXT -1 97 "We follow a sim
ple strategy and let Maple do the work. We integrate the equation for \+
the current " }{TEXT 320 1 "i" }{TEXT -1 1 "(" }{TEXT 319 1 "t" }
{TEXT -1 183 ") as derived from the Kirchhoff rules for the charge and
 discharge regimes respectively, and calculate the voltages as before.
 We pick the case where the period of the driving signal " }{TEXT 344 
1 "T" }{TEXT -1 48 " equals the time constant tau of the RC circuit." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "i0:=0: T:=1: tau:=1; nmax
:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG\"\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%nmaxG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "for j from 1 to nmax do:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "tj:=(j-1)*T;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "so
lc:=dsolve(\{KL_RC1,i(tj)=i0\},i(t));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "U_Rc:=subs(i(t)=rhs(solc),V_R);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "U_Rc:=simplify(subs(R=tau/C,U_Rc));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "U_Cc:=U1(t-tj)-U_Rc;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "U_Ccj[j]:=U_Cc;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "P_c[j]:=pl
ot(U_Ccj[j],t=tj..tj+T/2,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 44 "i0:=evalf(subs(t=tj+T/2,R=tau/C,rhs(solc)));" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "sold:=dsolve(\{KL_RC2,i(tj+T/2)=i0\},i(t));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "U_Rd:=subs(i(t)=rhs(sold),V_R);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U_Rd:=simplify(subs(R=tau/C,U_Rd));
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "U_Cd:=U2(t-tj)-U_Rd;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "U_Cdj[j]:=U_Cd;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "P_d[j]:=plot(U_Cdj[j],t=tj+T/2..tj+T,color=blue):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "i0:=evalf(subs(t=tj+T,R=tau/C,rhs(s
old)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "print(`U_CTh: `,evalf(su
bs(t=tj+T/2,U_Ccj[j]),3),`  U_CT: `,evalf(subs(t=tj+T,U_Cdj[j]),3));" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6&%(U_CTh:~G$\"$2\"!\"$%)~~U_CT:~G$\"$a\"F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&%(U_CTh:~G$\"$,#!\"$%)~~U_CT:~G$\"$9#F&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$N#!\"$%)~~U_CT:~G$\"$L#F&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$\\#!\"$%)~~U_CT:~G$\"$T#F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$^#!\"$%)~~U_CT:~G$\"$X#F
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$d#!\"$%)~~U_CT:~G$
\"$W#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$`#!\"$%)~~U_C
T:~G$\"$X#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$_#!\"$%)
~~U_CT:~G$\"$Y#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$`#!
\"$%)~~U_CT:~G$\"$Z#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$
\"$c#!\"$%)~~U_CT:~G$\"$Y#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CT
h:~G$\"$a#!\"$%)~~U_CT:~G$\"$X#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%
(U_CTh:~G$\"$_#!\"$%)~~U_CT:~G$\"$X#F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6&%(U_CTh:~G$\"$d#!\"$%)~~U_CT:~G$\"$X#F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&%(U_CTh:~G$\"$b#!\"$%)~~U_CT:~G$\"$X#F&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$c#!\"$%)~~U_CT:~G$\"$W#F&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$d#!\"$%)~~U_CT:~G$\"$U#F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$e#!\"$%)~~U_CT:~G$\"$Z#F
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$c#!\"$%)~~U_CT:~G$
\"$Y#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$b#!\"$%)~~U_C
T:~G$\"$W#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%(U_CTh:~G$\"$b#!\"$%)
~~U_CT:~G$\"$Y#F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "We can displ
ay the sequences of graphs for the charge and discharge cycles:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(seq(P_c[j],j=1..nmax
),seq(P_d[j],j=1..nmax));" }}{PARA 13 "" 1 "" {GLPLOT2D 759 300 300 
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57$$\"1LL$3x%3yTF,$\"1&f.\\=)*yg)F57$$\"1mm\"z%4\\Y_F,$\"1C!z9\"p__8!#
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\"1LLe*)>VB$)F,$\"1;AN&zN)pLFE7$$\"1***\\7`l2Q*F,$\"1#)\\TYC_lUFE7$$\"
1nm;/j$o/\"!#;$\"1!Gr$eL.$H&FE7$$\"1LL3_>jU6Fgn$\"1MbctRM'G'FE7$$\"1++
]i^Z]7Fgn$\"1L0Q()o[-vFE7$$\"1++](=h(e8Fgn$\"1V'HHx')o#))FE7$$\"1++]P[
6j9Fgn$\"1Q?&px0+-\"F,7$$\"1L$e*[z(yb\"Fgn$\"1ut'e&e&G:\"F,7$$\"1nm;a/
cq;Fgn$\"1$Q-tCB3K\"F,7$$\"1nmm;t,m<Fgn$\"1J,.!QC:Z\"F,7$$\"1+]iSj0x=F
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$\"1gmPlWoE?F,7$$\"1+](=xpe=#Fgn$\"1&[RkSbSA#F,7$$\"1nm\"H28IH#Fgn$\"1
%454'Q-RCF,7$$\"1n;zpSS\"R#Fgn$\"1P-\"o&QZWEF,7$$\"1LL3_?`(\\#Fgn$\"1n
nh(f@Y(GF,7$$\"1L$e*)>pxg#Fgn$\"1*[`EIpH7$F,7$$\"1+]Pf4t.FFgn$\"1`r5hH
yYLF,7$$\"1LLe*Gst!GFgn$\"1te)*3!*R'f$F,7$$\"1+++DRW9HFgn$\"1%pmdR&ziQ
F,7$$\"1***\\7j#>>IFgn$\"1a!>pA-<8%F,7$$\"1+]i!RU07$Fgn$\"1\"Rjvj1'*R%
F,7$$\"1++v=S2LKFgn$\"1i6a_R(eq%F,7$$\"1mmm\"p)=MLFgn$\"1tG)eK*))))\\F
,7$$\"1++](=]@W$Fgn$\"1klV<I:*H&F,7$$\"1L$e*[$z*RNFgn$\"1KVwp^V(e&F,7$
$\"1++]iC$pk$Fgn$\"1WB%=s(G5fF,7$$\"1m;H2qcZPFgn$\"1XaS%o@8A'F,7$$\"1+
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lAFjSFgn$\"1Dr6F&R>C(F,7$$\"1LLL$)*pp;%Fgn$\"1Lz*pHj<f(F,7$$\"1LL3xe,t
UFgn$\"12A#=YRo&zF,7$$\"1n;HdO=yVFgn$\"1\"*)45DPhK)F,7$$\"1+++D>#[Z%Fg
n$\"1J7p$=z<n)F,7$$\"1nmT&G!e&e%Fgn$\"1]g8nBEv!*F,7$$\"1LLL$)Qk%o%Fgn$
\"1vq\"yY:FW*F,7$$\"1+]iSjE!z%Fgn$\"1#4]q<i7%)*F,7$$\"1+]P40O\"*[Fgn$
\"1oI@=&>H-\"Fgn7$$\"1+++++++]Fgn$\"1Nj7(f1`1\"Fgn-%'COLOURG6&%$RGBG$
\"*++++\"!\")F(F(-F$6$7S7$$\"\"\"F($\"1iLh;7=[:Fgn7$$\"1LL3x&)*3,\"!#:
$\"1?P5+;*>`\"Fgn7$$\"1n\"H2P\"Q?5Fj[l$\"1&)G[2&4!>:Fgn7$$\"1L$eRwX5.
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\"z%4\\Y_5Fj[l$\"1-05b_d#[\"Fgn7$$\"1LeR-/Pi5Fj[l$\"1.0*GMAOZ\"Fgn7$$
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$\"1+DJbw!Q4\"Fj[l$\"1S(yG**3AX\"Fgn7$$\"1nmTIOo/6Fj[l$\"1MMs%*oBZ9Fgn
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gt$\"1rSGu>UWEFgn7$$\"1LL$)*pp;z\"Fcgt$\"1(ovqEpii#Fgn7$$\"1L3xe,t#z\"
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6jk=Fcgt$\"1c94i]<#y#Fgn7$$\"1$e*[z(yb'=Fcgt$\"1\\*HDcX))y#Fgn7$$\"1n;
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?`(\\(=Fcgt$\"1WI&*f&QY!GFgn7$$\"1$e*)>pxg(=Fcgt$\"1/ZVD/s+GFgn7$$\"1]
Pf4t.x=Fcgt$\"1)yx:Fgjz#Fgn7$$\"1Le*Gst!y=Fcgt$\"1V43+)o1z#Fgn7$$\"1++
DRW9z=Fcgt$\"1f`ke%GPy#Fgn7$$\"1+DJE>>!)=Fcgt$\"1NXfem!fx#Fgn7$$\"1]i!
RU07)=Fcgt$\"1WO@'=xtw#Fgn7$$\"1+v=S2L#)=Fcgt$\"1pBj#\\5ov#Fgn7$$\"1nm
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$\"1<H2qcZ()=Fcgt$\"14cr6&eTp#Fgn7$$\"1]7.\"fF&))=Fcgt$\"1=Y3_3_yEFgn7
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$\"1LL$)*pp;*=Fcgt$\"1EmM%Hpii#Fgn7$$\"1L3xe,t#*=Fcgt$\"1a*z)**Gz1EFgn
7$$\"1<HdO=y$*=Fcgt$\"1E17\")ed'e#Fgn7$$\"1++D>#[Z*=Fcgt$\"1>&ysy9sc#F
gn7$$\"1nT&G!e&e*=Fcgt$\"1#p@C?6Ta#Fgn7$$\"1LL$)Qk%o*=Fcgt$\"1+:?/)GE_
#Fgn7$$\"1]iSjE!z*=Fcgt$\"1gvp\"*=)))\\#Fgn7$$\"1]P40O\"*)*=Fcgt$\"1=1
lOsMvCFgn7$Feb]l$\"1.([*4m=\\CFgnF]a_l-F$6$7S7$Fea^l$\"1%[WfO83b#Fgn7$
$\"1L$3x&)*3^>Fcgt$\"1)*>1V$pnd#Fgn7$$\"1<H2P\"Q?&>Fcgt$\"1\"o*\\EH;)f
#Fgn7$$\"1LeRwX5`>Fcgt$\"1<]&\\L74i#Fgn7$$\"1L3x%3yT&>Fcgt$\"1l\"3mKEC
k#Fgn7$$\"1<z%4\\Y_&>Fcgt$\"12#\\5'=ZiEFgn7$$\"1$eR-/Pi&>Fcgt$\"1T5n(p
^)zEFgn7$$\"1]il'pis&>Fcgt$\"190;p#Qmp#Fgn7$$\"1$e*)>VB$e>Fcgt$\"1-c)
\\.=Fr#Fgn7$$\"1]7`l2Qf>Fcgt$\"1u2[ONYFFFgn7$$\"1n;/j$o/'>Fcgt$\"1XnAa
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yOw#Fgn7$$\"1+](=h(ej>Fcgt$\"1\\//z<rtFFgn7$$\"1+]P[6jk>Fcgt$\"15H]t]<
#y#Fgn7$$\"1$e*[z(yb'>Fcgt$\"1wP$QdX))y#Fgn7$$\"1n;a/cqm>Fcgt$\"1$\\a$
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r3GFgn7$$\"1](=xpe=(>Fcgt$\"1b#e/'oH4GFgn7$$\"1n\"H28IH(>Fcgt$\"1[3WIk
x3GFgn7$$\"1<zpSS\"R(>Fcgt$\"1XtXotH2GFgn7$$\"1L3_?`(\\(>Fcgt$\"1\"3V-
dQY!GFgn7$$\"1$e*)>pxg(>Fcgt$\"1I>hN/s+GFgn7$$\"1]Pf4t.x>Fcgt$\"1<yl\"
Ggjz#Fgn7$$\"1Le*Gst!y>Fcgt$\"1Sq05)o1z#Fgn7$$\"1++DRW9z>Fcgt$\"16_^o%
GPy#Fgn7$$\"1+DJE>>!)>Fcgt$\"1S:Oom!fx#Fgn7$$\"1]i!RU07)>Fcgt$\"1f@)e>
xtw#Fgn7$$\"1+v=S2L#)>Fcgt$\"1&p#>-0\"ov#Fgn7$$\"1nm\"p)=M$)>Fcgt$\"1_
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1%zLU9IAs#Fgn7$$\"1+]iC$pk)>Fcgt$\"1y%)3TCB3FFgn7$$\"1<H2qcZ()>Fcgt$\"
1,lz?&eTp#Fgn7$$\"1]7.\"fF&))>Fcgt$\"1)[q5'3_yEFgn7$$\"1n;/Ogb*)>Fcgt$
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\"1Xlk*)ed'e#Fgn7$$\"1++D>#[Z*>Fcgt$\"1TCs&z9sc#Fgn7$$\"1nT&G!e&e*>Fcg
t$\"13Ex576WDFgn7$$\"1LL$)Qk%o*>Fcgt$\"1(4qC\")GE_#Fgn7$$\"1]iSjE!z*>F
cgt$\"1#Gz)**=)))\\#Fgn7$$\"1]P40O\"*)*>Fcgt$\"1W+vWsMvCFgn7$$\"#?F($
\"151'zh'=\\CFgnF]a_l-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(Fibam%(DEF
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{PARA 0 "" 0 "" {TEXT -1 276 "We observe that after a few cycles of th
e square-wave signal a steady-state regime is reached if tau=1. For la
rger tau it takes more cycles to reach the steady state. The steady-st
ate regime is reached as the oscillations become symmetric with respec
t to the midpoint voltage " }{TEXT 276 1 "U" }{TEXT -1 5 "=1/4." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "Various \+
properties of this solution can be exploited in practice: we mentioned
 the low-pass filtering property already; another property is the fact
 that under the right circumstances, i.e., for " }{TEXT 277 1 "T" }
{TEXT -1 159 " << tau, we can consider the circuit to be an approximat
e integrator. We learned that it turns a square-wave signal into an ap
proximately triangular shape (in " }{TEXT 19 14 "RLCcircuit.mws" }
{TEXT -1 324 "). This was found to be correct in the sense that an exp
onential function is approximated for small arguments by a straight li
ne. Here we explored the response to a triangular-shape source signal \+
and found that the response is a sequence of parabolic pieces that loo
k alomost like a sine function in the steady-state regime." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 311 "Thus, one can us
e this circuit in function generators to generate interesting shapes. \+
One can feed the near-sine wave signal coming from one RC circuit to d
rive a second one. To understand the response of the second RC circuit
 one then needs to solve a differential equation with a time-dependent
 driving force " }{TEXT 279 1 "U" }{TEXT -1 1 "(" }{TEXT 278 1 "t" }
{TEXT -1 128 "). This worksheet can be used for many further explorati
ons in this respect, as the user has control over the input voltages U
1(" }{TEXT 346 1 "t" }{TEXT -1 9 ") and U2(" }{TEXT 345 1 "t" }{TEXT 
-1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
68 "We explore the behaviour of the solution in the steady-state regim
e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "U_Ccj[20];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,(%\"tG\"\"\"$!#?\"\"!F%-%$expG6#,$F$$!\"\"F
($\"*Zf>A#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P1:=plot(U_
Ccj[20],t=19..19.5,color=green): display(P1);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 646 228 228 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"#>\"\"!$\"1%>o
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/P=S#F-7$$\"1LeRwX5.>F1$\"1@5\"[j(3zBF-7$$\"1L3x%3yT!>F1$\"1-2[VOddBF-
7$$\"1<z%4\\Y_!>F1$\"1IvN4\"GvL#F-7$$\"1$eR-/Pi!>F1$\"1&QFIF[,K#F-7$$
\"1]il'pis!>F1$\"13p$=qhLI#F-7$$\"1$e*)>VB$3>F1$\"1QyJO>G(G#F-7$$\"1]7
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@#F-7$$\"1n;a/cq;>F1$\"1)Gc_#pY/AF-7$$\"1nm;t,m<>F1$\"1@NgB)[)*>#F-7$$
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$\"1nT&G!e&e%>F1$\"1M4^p())eX#F-7$$\"1LL$)Qk%o%>F1$\"1+y+o6PxCF-7$$\"1
]iSjE!z%>F1$\"1FP!33=6]#F-7$$\"1]P40O\"*[>F1$\"1ms7OFlCDF-7$$\"1++++++
]>F1$\"1CL7jL\"3b#F--%'COLOURG6&%$RGBGF*$\"*++++\"!\")F*-%+AXESLABELSG
6$Q\"t6\"%!G-%%VIEWG6$;F($\"$&>!\"\"%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
214 "We calculate the precise location of the minimum to be able to co
mpare the solution with a sine shape. The sine needs to be phase shift
ed for comparison, and we can use the location of the minimum for this
 purpose." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "t_min:=solve(d
iff(U_Ccj[20],t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&t_minG$\"
+>q!>#>!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "We also need the \+
amplitude precisely. We assume (observe from the graph) that we have s
ymmetrical oscillations about U_avg=0.25, and obtain for the expected \+
amplitude:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=evalf(0.25
-subs(t=t_min,U_Ccj[20]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$
\")0)H4$!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "The capacitor vo
ltage is nearly periodic in the steady-state regime. We can compare it
 to a sine function with the expected period (2 Pi/" }{TEXT 321 1 "T" 
}{TEXT -1 57 "), the expected average voltage of 0.25 and an amplitude
 " }{TEXT 347 1 "A" }{TEXT -1 62 " and phase shift (related to t_min) \+
derived from the solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
86 "P2:=plot(0.25-A*sin((2*Pi/T)*(t-t_min)+Pi/2),t=19..19.5,color=maro
on): display(P1,P2);" }}{PARA 13 "" 1 "" {GLPLOT2D 648 234 234 
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\\y$\"1HPb7\\<eCF-7$Fay$\"1ybGedezCF-7$Ffy$\"1T&R=qA))\\#F-7$F[z$\"1-f
%f]O$>DF-7$F`z$\"1buy/B*)QDF-7$Fez$\"1A,'4kI(fDF--Fjz6&F\\[l$\")viobF_
[l$\")!\\DP\"F_[l$\")%yg>%F_[l-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F($
\"$&>!\"\"%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 660 "This justifies th
e use of a single RC integrator circuit as an approximate sine-wave ge
nerator. Better generators are based on the property that differential
 equations can be modelled by coupling several elements with feedback \+
(two integrators can model a second-order differential equation), ie.,
 true sine solutions can be obtained in electronics. The interesting m
essage from this worksheet is (apart from explaining the images genera
ted often on oscilloscopes and probably rarely understood) that piecin
g together exponential solutions can result in a nearly periodic respo
nse. This is the result of mathematics that leads to a limit cycle in \+
the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 135 "On modern digital oscilloscopes it is relatively straigh
tforward to record the transient (turn-on) behaviour by single-shot tr
iggering." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 \+
0 0" 34 }{VIEWOPTS 1 1 0 1 1 1803 }