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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 301 12 "RLC Circuits" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 646 "Circuits that \+
involve resistors R, capacitors C, and inductances L can be thought of
 a network made up of elementary building blocks. We investigate the b
asic building blocks from first principles, namely the RL, RC, and RLC
 circuits. Often an idealization is made in that a pure inductance L i
s introduced, while in practice a solenoid always has some internal re
sistance associated with it. Thus, there is no need to discuss the pur
e LC circuit, since it emerges as the R=0 limit of the RLC circuit. Th
e RLC circuit is of great interest due to its oscillatory solutions th
at can be resonantly excited - a principle that any radio set relies o
n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 474 "Th
ese circuits contain very interesting physics. Before going into the d
etails, i.e., before setting up the equations we discuss first the phy
sical principles. In the RC circuit one studies how electricity is sto
red in a capacitor when a battery voltage is applied. If the circuit i
s then disconnected from the battery and the open terminals are shorte
d out, the capacitor discharges by sending a current through R that is
 in the opposite direction of the charging current. " }}{PARA 0 "" 0 "
" {TEXT -1 501 "The RL circuit is also an energy storing device: Farad
ay's law of induction as applied to the self-induction of a coil shows
 that the coil is opposing the free flow of current as the magnetic fi
eld is being built. Similarly, according to Lenz' rule the coil oppose
s the turning off of voltage, i.e., the energy stored in the magnetic \+
field is released by sending a current in the same direction as the or
iginal battery current after the battery is disconnected and the open \+
terminals are shorted out. " }}{PARA 0 "" 0 "" {TEXT -1 238 "Once thes
e different energy storage mechanisms are understood, it is not diffic
ult to comprehend the oscillatory properties of an LC circuit. Suppose
 the capacitor has some initial charge: it then discharges while build
ing up a magnetic (" }{TEXT 302 1 "B" }{TEXT -1 691 ") field in L; as \+
the capacitor has lost its charge, i.e., transferred its electrical en
ergy to the magnetic field in the coil, the magnetic field opposes the
 drop in voltage, i.e., sends a current to charge the capacitor in a s
ense that is opposite to its previous charge state, and then the cycle
 repeats. An oscillator emerges that is described by the same differen
tial equation as a harmonic oscillator. The addition of R in series to
 L and C results in a damped harmonic motion. The differential equatio
ns provide the mathematical detail for this physical reasoning and det
ermine how the physical characteristics of the components (R, C, and L
 respectively) determine the time constants." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "Our circuits are very simple: \+
we consider the two (or three) elements simply connected in series to \+
each other with a fixed DC voltage " }{TEXT 256 1 "U" }{TEXT -1 307 " \+
applied at the free terminals. When writing down the equations that go
vern the electric current in these circuits we can make use of Ohm's l
aw to relate the current in the circuit to the voltage across the resi
stor R. Note that all three building blocks are passive, i.e., they si
mply respond to a current " }{TEXT 260 1 "i" }{TEXT -1 1 "(" }{TEXT 
259 1 "t" }{TEXT -1 25 ") induced by the voltage " }{TEXT 258 1 "U" }
{TEXT -1 17 " applied at time " }{TEXT 257 1 "t" }{TEXT -1 3 "=0." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "We begin
 with the definition of the voltages across the three building blocks \+
as a function of current." }}{PARA 0 "" 0 "" {TEXT -1 24 "For R we hav
e Ohm's law:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "V_R:=R*i(t)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$V_RG*&%\"RG\"\"\"-%\"iG6#%\"tG
F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "For the capacitor C we hav
e the physical statement that it builds up its voltage as the plates a
re becoming charged due to the current:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "V_C:=(1/C)*Int(i(s),s=0..t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$V_CG*&-%$IntG6$-%\"iG6#%\"sG/F,;\"\"!%\"tG\"\"\"%\"C
G!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "For the inductance we \+
take into consideration the fact that as the current begins to flow it
 is building up a time-varying magnetic field (flux " }{XPPEDIT 18 0 "
Phi" "6#%$PhiG" }{TEXT -1 99 "). This time-varying flux according to F
araday's law induces a time-varying voltage proprtional to " }
{XPPEDIT 18 0 "-d Phi/dt" "6#,$*(%\"dG\"\"\"%$PhiGF&%#dtG!\"\"F)" }
{TEXT -1 151 "  (the negative sign is important, since a positive prop
ortionality would lead to runaway solutions). The magnetic flux is pro
portional to the current " }{TEXT 261 1 "i" }{TEXT -1 1 "(" }{TEXT 
303 1 "t" }{TEXT -1 111 "). For two coils the relationship between the
 voltage induced in coil 2 due to a current in coil 1 is given as " }
{XPPEDIT 18 0 "V[2] = - M  d i[1]/dt" "6#/&%\"VG6#\"\"#,$**%\"MG\"\"\"
%\"dGF+&%\"iG6#\"\"\"F+%#dtG!\"\"F2" }{TEXT -1 181 " , where M is the \+
mutual inductance. For a single coil the same phenomenon occurs, since
 neighbouring turns of the coil experience their mutual magnetic fluxe
s. The self-inductance " }{TEXT 304 1 "L" }{TEXT -1 271 " - a property
 of the coil - serves as the proportionality constant between current \+
and self-induced  magnetic flux. An inductor serves as an energy-stori
ng device, as it passing a current through the coil requires to overco
me a back-electromotoric force voltage given by " }{XPPEDIT 18 0 "V[b]
=-L d i/dt" "6#/&%\"VG6#%\"bG,$**%\"LG\"\"\"%\"dGF+%\"iGF+%#dtG!\"\"F/
" }{TEXT -1 89 " . To overcome this back EMF voltage we have to drive \+
the solenoid with a forward voltage" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "V_L:=L*diff(i(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$V_LG*&%\"LG\"\"\"-%%diffG6$-%\"iG6#%\"tGF.F'" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 262 10 "RL circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 133 "The differential equation for the curren
t follows from Kirchhoff's law for putting the two elements in series \+
and applying a voltage " }{TEXT 305 1 "U" }{TEXT -1 31 " across the ex
ternal terminals:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "KL_RL:
=V_R+V_L=U;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&KL_RLG/,&*&%\"RG\"\"
\"-%\"iG6#%\"tGF)F)*&%\"LGF)-%%diffG6$F*F-F)F)%\"UG" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 50 "The initial condition is that no current flows \+
at " }{TEXT 263 1 "t" }{TEXT -1 3 "=0:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "IC:=i(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG
/-%\"iG6#\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sol_RL:
=dsolve(\{KL_RL,IC\},i(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sol_
RLG/-%\"iG6#%\"tG,&*&%\"UG\"\"\"%\"RG!\"\"\"\"\"*&*&-%$expG6#,$*&*&F.F
0F)F0F-%\"LGF/!\"\"F0F,F0F-F.F/F:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "i_RL:=factor(rhs(sol_RL));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%i_RLG,$*&*&%\"UG\"\"\",&!\"\"F)-%$expG6#,$*&*&%\"RGF
)%\"tGF)\"\"\"%\"LG!\"\"F+F)F)F4F2F6F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 51 "Across the resistor we have according to Ohm's law:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U_R:=simplify(subs(i(t)=i_RL
,V_R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$U_RG,$*&%\"UG\"\"\",&!\"
\"F(-%$expG6#,$*&*&%\"RGF(%\"tGF(\"\"\"%\"LG!\"\"F*F(F(F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "We can introduce the characteristic time \+
constant for this circuit as " }{XPPEDIT 18 0 "tau=L/R" "6#/%$tauG*&%
\"LG\"\"\"%\"RG!\"\"" }{TEXT -1 30 " to simplify the expression to" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "U_R:=simplify(subs(L=R*tau,
U_R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$U_RG,$*&%\"UG\"\"\",&!\"
\"F(-%$expG6#,$*&%\"tG\"\"\"%$tauG!\"\"F*F(F(F*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 117 "For graphing we now need to substitute only the sca
les for the axes, i.e., the time constant and the applied voltage:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P_RL1:=plot(subs(tau=1,U=1,
U_R),t=0..5,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(P_R
L1);" }}{PARA 13 "" 1 "" {GLPLOT2D 647 224 224 {PLOTDATA 2 "6%-%'CURVE
SG6$7U7$\"\"!F(7$$\"1mmmT&)G\\a!#<$\"1H%)H!RvMI&F,7$$\"1LLL3x&)*3\"!#;
$\"1$=WoAoD.\"F27$$\"1+]i!R(*Rc\"F2$\"1bH6<u#yW\"F27$$\"1nm\"H2P\"Q?F2
$\"1j&y'**p&Q%=F27$$\"1LL$eRwX5$F2$\"1*3DM@'))oEF27$$\"1ML$3x%3yTF2$\"
1(*[X!o;^T$F27$$\"1nm\"z%4\\Y_F2$\"1R7]$>qB3%F27$$\"1MLeR-/PiF2$\"1t;*
zNW/k%F27$$\"1***\\il'pisF2$\"1GW)3r**G;&F27$$\"1MLe*)>VB$)F2$\"1oOkxE
r\\cF27$$\"1++DJbw!Q*F2$\"1l.'p*3?'3'F27$$\"1nm;/j$o/\"!#:$\"1$HL=,M&*
['F27$$\"1LL3_>jU6F_o$\"1!*)\\x>;-\"oF27$$\"1++]i^Z]7F_o$\"1&Hq828j8(F
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1L$e*[z(yb\"F_o$\"1xmje`<%*yF27$$\"1nm;a/cq;F_o$\"1*QSl3%e=\")F27$$\"1
nmm;t,m<F_o$\"1/lKhF()*G)F27$$\"1+]iSj0x=F_o$\"1K,#>f+'p%)F27$$\"1nmm
\"pW`(>F_o$\"1#*p#Q4lGh)F27$$\"1+]i!f#=$3#F_o$\"1*fC\\dnYv)F27$$\"1+](
=xpe=#F_o$\"14Y#y`+i())F27$$\"1nm\"H28IH#F_o$\"1Xvh@?Q!**)F27$$\"1n;zp
SS\"R#F_o$\"1k(fJ!)))\\3*F27$$\"1LL3_?`(\\#F_o$\"1wo)op@r<*F27$$\"1M$e
*)>pxg#F_o$\"1zj#>K7IE*F27$$\"1+]Pf4t.FF_o$\"1[#p7hZ/L*F27$$\"1MLe*Gst
!GF_o$\"1$>iG)fO'R*F27$$\"1+++DRW9HF_o$\"1rWy!>ewX*F27$$\"1++DJE>>IF_o
$\"1.k\\^Of6&*F27$$\"1+]i!RU07$F_o$\"1KT?bxme&*F27$$\"1++v=S2LKF_o$\"1
w%[*\\\"Rcg*F27$$\"1mmm\"p)=MLF_o$\"1\"QPl2lNk*F27$$\"1++](=]@W$F_o$\"
1#*Q)o&=/!o*F27$$\"1L$e*[$z*RNF_o$\"1)*=cP2')4(*F27$$\"1,+]iC$pk$F_o$
\"1-V[+-HR(*F27$$\"1m;H2qcZPF_o$\"1.Axf'\\Uw*F27$$\"1+]7.\"fF&QF_o$\"1
!ydi&*)y(y*F27$$\"1mm;/OgbRF_o$\"1w58A*G&3)*F27$$\"1+]ilAFjSF_o$\"1Sw3
?L2G)*F27$$\"1MLL$)*pp;%F_o$\"1>2l#[3]%)*F27$$\"1ML3xe,tUF_o$\"1e<k1Kg
g)*F27$$\"1n;HdO=yVF_o$\"1&*Rh.(=X()*F27$$\"1,++D>#[Z%F_o$\"1^2\"Q\\xg
))*F27$$\"1nmT&G!e&e%F_o$\"1.#=ep@!)*)*F27$$\"1MLL$)Qk%o%F_o$\"16ero(R
w!**F27$$\"1+]iSjE!z%F_o$\"1cOoiv*o\"**F27$$\"1,]P40O\"*[F_o$\"1,$H`.)
)[#**F27$$\"\"&F($\"1X\"4+`?E$**F2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(
-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(F\\[l%(DEFAULTG" 1 2 0 1 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 75 "The voltage across the resistor is a measure of the current in \+
the circuit " }{TEXT 264 1 "i" }{TEXT -1 1 "(" }{TEXT 265 1 "t" }
{TEXT -1 101 "). It shows that work is being done by the current in bu
ilding up a magnetic field over a time scale " }{XPPEDIT 18 0 "tau" "6
#%$tauG" }{TEXT -1 111 " (chosen to be 1 on the graph), and that the c
urrent reaches only asymptotically its maximum possible value of " }
{TEXT 306 1 "U" }{TEXT -1 1 "/" }{TEXT 307 1 "R" }{TEXT -1 31 " (assum
ing that the inductance " }{TEXT 308 1 "L" }{TEXT -1 73 " is ideal, i.
e., contributes no ohmic resistance, otherwise one replaces " }{TEXT 
314 1 "R" }{TEXT -1 4 " by " }{TEXT 313 1 "R" }{TEXT -1 1 "+" }{TEXT 
312 1 "R" }{TEXT -1 1 "_" }{TEXT 311 1 "L" }{TEXT -1 8 ", where " }
{TEXT 309 1 "R" }{TEXT -1 1 "_" }{TEXT 310 1 "L" }{TEXT -1 43 " is the
 ohmic resistance of the solenoid). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 77 "Note that the voltage across the soleno
id is given as the complement between " }{TEXT 315 1 "U" }{TEXT -1 1 "
_" }{TEXT 316 1 "R" }{TEXT -1 34 " and the external battery voltage " 
}{TEXT 317 1 "U" }{TEXT -1 191 ". It is falling as a function of time \+
allowing the interpretation that the solenoid acts as a big resistor a
s the magnetic field is building, but that its resistance goes to zero
 ideally (to " }{TEXT 318 1 "R" }{TEXT -1 1 "_" }{TEXT 319 1 "L" }
{TEXT -1 84 " realistically) for large times. Provide a simutaneous gr
aph of the voltages across " }{TEXT 321 1 "R" }{TEXT -1 5 " and " }
{TEXT 320 1 "L" }{TEXT -1 23 " as a function of time." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 157 "To determine the time
 constant from an experiment it is often advantageous to define the ti
me at which one-half of the asymptotic current (or voltage across " }
{TEXT 322 1 "R" }{TEXT -1 13 ") is reached:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 25 "t_half:=solve(U_R=U/2,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'t_halfG*&-%#lnG6#\"\"#\"\"\"%$tauGF*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 166 "Now we look at the solution as the volta
ge is suddenly disconnected and the circuit is shorted out at the term
inals. We assume that the steady-state current flows at " }}{PARA 0 "
" 0 "" {TEXT 323 1 "t" }{TEXT -1 43 "=5, where we terminated our previ
ous graph." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "KL_RL0:=subs(
U=0,KL_RL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'KL_RL0G/,&*&%\"RG\"
\"\"-%\"iG6#%\"tGF)F)*&%\"LGF)-%%diffG6$F*F-F)F)\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "IC0:=i(5)=U/R;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$IC0G/-%\"iG6#\"\"&*&%\"UG\"\"\"%\"RG!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sol_RL0:=dsolve(\{KL_RL0,IC0
\},i(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sol_RL0G/-%\"iG6#%\"tG
*&*&%\"UG\"\"\"-%$expG6#,$*&*&%\"RGF-F)F-\"\"\"%\"LG!\"\"!\"\"F-F5*&-F
/6#,$*&F4F5F6F7!\"&\"\"\"F4\"\"\"F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 55 "We simplify the solution by combining the exponentials:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "i_RL0:=combine(rhs(sol_RL0),
exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&i_RL0G*&*&%\"UG\"\"\"-%$ex
pG6#,&*&%\"RG\"\"\"%\"LG!\"\"\"\"&*&*&F.F(%\"tGF(F/F0F1!\"\"F(F/F.F1" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The voltage drop across the res
istor follows from Ohm's law:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "U_R0:=simplify(subs(i(t)=i_RL0,V_R));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%U_R0G*&%\"UG\"\"\"-%$expG6#,$*&*&%\"RGF',&!\"&F'%\"t
GF'F'\"\"\"%\"LG!\"\"!\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "W
e express the latter using the time constant tau:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "U_R0:=simplify(subs(L=R*tau,U_R0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%U_R0G*&%\"UG\"\"\"-%$expG6#,$*&,&!
\"&F'%\"tGF'\"\"\"%$tauG!\"\"!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "P_RL2:=plot(subs(tau=1,U=1,U_R0),t=5..10,color=blue):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(P_RL1,P_RL2);" 
}}{PARA 13 "" 1 "" {GLPLOT2D 775 217 217 {PLOTDATA 2 "6&-%'CURVESG6$7U
7$\"\"!F(7$$\"1mmmT&)G\\a!#<$\"1H%)H!RvMI&F,7$$\"1LLL3x&)*3\"!#;$\"1$=
WoAoD.\"F27$$\"1+]i!R(*Rc\"F2$\"1bH6<u#yW\"F27$$\"1nm\"H2P\"Q?F2$\"1j&
y'**p&Q%=F27$$\"1LL$eRwX5$F2$\"1*3DM@'))oEF27$$\"1ML$3x%3yTF2$\"1(*[X!
o;^T$F27$$\"1nm\"z%4\\Y_F2$\"1R7]$>qB3%F27$$\"1MLeR-/PiF2$\"1t;*zNW/k%
F27$$\"1***\\il'pisF2$\"1GW)3r**G;&F27$$\"1MLe*)>VB$)F2$\"1oOkxEr\\cF2
7$$\"1++DJbw!Q*F2$\"1l.'p*3?'3'F27$$\"1nm;/j$o/\"!#:$\"1$HL=,M&*['F27$
$\"1LL3_>jU6F_o$\"1!*)\\x>;-\"oF27$$\"1++]i^Z]7F_o$\"1&Hq828j8(F27$$\"
1++](=h(e8F_o$\"13HV92@IuF27$$\"1++]P[6j9F_o$\"1ke$4yf[o(F27$$\"1L$e*[
z(yb\"F_o$\"1xmje`<%*yF27$$\"1nm;a/cq;F_o$\"1*QSl3%e=\")F27$$\"1nmm;t,
m<F_o$\"1/lKhF()*G)F27$$\"1+]iSj0x=F_o$\"1K,#>f+'p%)F27$$\"1nmm\"pW`(>
F_o$\"1#*p#Q4lGh)F27$$\"1+]i!f#=$3#F_o$\"1*fC\\dnYv)F27$$\"1+](=xpe=#F
_o$\"14Y#y`+i())F27$$\"1nm\"H28IH#F_o$\"1Xvh@?Q!**)F27$$\"1n;zpSS\"R#F
_o$\"1k(fJ!)))\\3*F27$$\"1LL3_?`(\\#F_o$\"1wo)op@r<*F27$$\"1M$e*)>pxg#
F_o$\"1zj#>K7IE*F27$$\"1+]Pf4t.FF_o$\"1[#p7hZ/L*F27$$\"1MLe*Gst!GF_o$
\"1$>iG)fO'R*F27$$\"1+++DRW9HF_o$\"1rWy!>ewX*F27$$\"1++DJE>>IF_o$\"1.k
\\^Of6&*F27$$\"1+]i!RU07$F_o$\"1KT?bxme&*F27$$\"1++v=S2LKF_o$\"1w%[*\\
\"Rcg*F27$$\"1mmm\"p)=MLF_o$\"1\"QPl2lNk*F27$$\"1++](=]@W$F_o$\"1#*Q)o
&=/!o*F27$$\"1L$e*[$z*RNF_o$\"1)*=cP2')4(*F27$$\"1,+]iC$pk$F_o$\"1-V[+
-HR(*F27$$\"1m;H2qcZPF_o$\"1.Axf'\\Uw*F27$$\"1+]7.\"fF&QF_o$\"1!ydi&*)
y(y*F27$$\"1mm;/OgbRF_o$\"1w58A*G&3)*F27$$\"1+]ilAFjSF_o$\"1Sw3?L2G)*F
27$$\"1MLL$)*pp;%F_o$\"1>2l#[3]%)*F27$$\"1ML3xe,tUF_o$\"1e<k1Kgg)*F27$
$\"1n;HdO=yVF_o$\"1&*Rh.(=X()*F27$$\"1,++D>#[Z%F_o$\"1^2\"Q\\xg))*F27$
$\"1nmT&G!e&e%F_o$\"1.#=ep@!)*)*F27$$\"1MLL$)Qk%o%F_o$\"16ero(Rw!**F27
$$\"1+]iSjE!z%F_o$\"1cOoiv*o\"**F27$$\"1,]P40O\"*[F_o$\"1,$H`.))[#**F2
7$$\"\"&F($\"1X\"4+`?E$**F2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7U
7$F\\[l$\"\"\"F(7$$\"1nmT&)G\\a]F_o$\"1c,(4Y_'p%*F27$$\"1ML$3x&)*3^F_o
$\"19e:t<Vn*)F27$$\"1+D1R(*Rc^F_o$\"1Uq)Ges@b)F27$$\"1m;H2P\"Q?&F_o$\"
1T9K+I9c\")F27$$\"1LLeRwX5`F_o$\"16\\d'y86L(F27$$\"1ML3x%3yT&F_o$\"1.^
a>L)[e'F27$$\"1m;z%4\\Y_&F_o$\"1k()\\1)Hw\"fF27$$\"1L$eR-/Pi&F_o$\"1G$
3?kb&f`F27$$\"1+]il'pis&F_o$\"1tb6*G+r$[F27$$\"1L$e*)>VB$eF_o$\"1MjNAt
G]VF27$$\"1+]7`l2QfF_o$\"1O'RI5*z8RF27$$\"1nm;/j$o/'F_o$\"11n;))fY5NF2
7$$\"1ML3_>jUhF_o$\"14,D-Qy*=$F27$$\"1++]i^Z]iF_o$\"10(H'GpojGF27$$\"1
++](=h(ejF_o$\"1$4nbG*ypDF27$$\"1++]P[6jkF_o$\"1OT1>-9:BF27$$\"1M$e*[z
(yb'F_o$\"1BLOTY#e5#F27$$\"1nm;a/cqmF_o$\"15'fM\"fT\")=F27$$\"1nmm;t,m
nF_o$\"1&\\t'Qs75<F27$$\"1+]iSj0xoF_o$\"1p)z!3%*RI:F27$$\"1nmm\"pW`(pF
_o$\"13I<1\\8(Q\"F27$$\"1+]i!f#=$3(F_o$\"1+a2DCLX7F27$$\"1+](=xpe=(F_o
$\"1#Rv@Y*zB6F27$$\"1nm\"H28IH(F_o$\"1cCQyzh45F27$$\"1n;zpSS\"R(F_o$\"
1lBSo>6]\"*F,7$$\"1LL3_?`(\\(F_o$\"1Q78JIyG#)F,7$$\"1M$e*)>pxg(F_o$\"1
-it!yw)ptF,7$$\"1+]Pf4t.xF_o$\"19vI()Q_&p'F,7$$\"1MLe*Gst!yF_o$\"1s!y8
<Sj.'F,7$$\"1+++DRW9zF_o$\"1$Gb@4=MU&F,7$$\"1++DJE>>!)F_o$\"1rf.&[jS)[
F,7$$\"1,]i!RU07)F_o$\"1u'ezWALT%F,7$$\"1***\\(=S2L#)F_o$\"1V_^+&3O%RF
,7$$\"1mmm\"p)=M$)F_o$\"1'=EYB\\Vc$F,7$$\"1,+](=]@W)F_o$\"1w5;J9e*>$F,
7$$\"1L$e*[$z*R&)F_o$\"1;5QCER,HF,7$$\"1,+]iC$pk)F_o$\"1!)p:&*z42EF,7$
$\"1m;H2qcZ()F_o$\"1pzF-M]dBF,7$$\"1,]7.\"fF&))F_o$\"1,AUP/6A@F,7$$\"1
nm;/Ogb*)F_o$\"1T#*oy2r9>F,7$$\"1**\\ilAFj!*F_o$\"1.O7*zm#><F,7$$\"1NL
L$)*pp;*F_o$\"13G\\t^\"*\\:F,7$$\"1NL3xe,t#*F_o$\"1:CeLz'RR\"F,7$$\"1o
;HdO=y$*F_o$\"1^+'Q'H\"[D\"F,7$$\"1,++D>#[Z*F_o$\"1*[#*=1D#R6F,7$$\"1o
mT&G!e&e*F_o$\"1oz\"=/$y>5F,7$$\"1LLL$)Qk%o*F_o$\"1-*=%GJ-O#*!#=7$$\"1
**\\iSjE!z*F_o$\"1BVjJPC5$)Fhjl7$$\"1,]P40O\"*)*F_o$\"1k)pqY'>6vFhjl7$
$\"#5F($\"1na3**p%zt'Fhjl-Fa[l6&Fc[lF(F(Fd[l-%+AXESLABELSG6$Q\"t6\"%!G
-%%VIEWG6$;F(Fd[m%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "This graph shows t
he voltage across the resistor R (or the current " }{TEXT 272 1 "i" }
{TEXT -1 1 "(" }{TEXT 324 1 "t" }{TEXT -1 127 ")) that would appear in
 the RL circuit as the terminals are connected to a single cycle of a \+
square-wave pulse with turn-on at " }{TEXT 271 1 "t" }{TEXT -1 20 "=0,
 and turn-off at " }{TEXT 269 1 "t" }{TEXT -1 23 "=5 (we assumed that \+
at " }{TEXT 270 1 "t" }{TEXT -1 272 "=5 the asymptotic current value w
as reached). Note that for cases where the driving square-wave pulse h
as a shorter time constant than the RL circuit, one has to use the fin
al current from the turn-on phase as a start value in the initial cond
ition for the turn-off phase." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 319 "It is important to realize that this is \+
the result of the current continuing to flow in the same direction as \+
the external voltage is turned off (down-time of the square-wave pulse
). The voltage across the inductance L is the complement to the extern
al voltage: for the turn-on part (red curve) it is the complement to \+
" }{TEXT 274 1 "U" }{TEXT -1 128 ", for the turn-off part (blue curve)
 it is the complement to a zero voltage at the external terminals! We \+
remember that we used " }{TEXT 273 1 "U" }{TEXT -1 14 "=1, and graph:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "P_RL3:=plot(1-subs(tau=
1,U=1,U_R),t=0..5,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "P_RL4:=plot(0-subs(tau=1,U=1,U_R0),t=5..10,color=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(P_RL3,P_RL4,title=\"
voltage across L\");" }}{PARA 13 "" 1 "" {GLPLOT2D 568 300 300 
{PLOTDATA 2 "6'-%'CURVESG6$7U7$\"\"!$\"\"\"F(7$$\"1mmmT&)G\\a!#<$\"1d,
(4Y_'p%*!#;7$$\"1LLL3x&)*3\"F1$\"1<e:t<Vn*)F17$$\"1+]i!R(*Rc\"F1$\"1Xq
)Ges@b)F17$$\"1nm\"H2P\"Q?F1$\"1P9K+I9c\")F17$$\"1LL$eRwX5$F1$\"16\\d'
y86L(F17$$\"1ML$3x%3yTF1$\"1.^a>L)[e'F17$$\"1nm\"z%4\\Y_F1$\"1h()\\1)H
w\"fF17$$\"1MLeR-/PiF1$\"1F$3?kb&f`F17$$\"1***\\il'pisF1$\"1sb6*G+r$[F
17$$\"1MLe*)>VB$)F1$\"1LjNAtG]VF17$$\"1++DJbw!Q*F1$\"1N'RI5*z8RF17$$\"
1nm;/j$o/\"!#:$\"11n;))fY5NF17$$\"1LL3_>jU6Fao$\"15,D-Qy*=$F17$$\"1++]
i^Z]7Fao$\"10(H'GpojGF17$$\"1++](=h(e8Fao$\"1#4nbG*ypDF17$$\"1++]P[6j9
Fao$\"1OT1>-9:BF17$$\"1L$e*[z(yb\"Fao$\"1CLOTY#e5#F17$$\"1nm;a/cq;Fao$
\"16'fM\"fT\")=F17$$\"1nmm;t,m<Fao$\"1&\\t'Qs75<F17$$\"1+]iSj0x=Fao$\"
1o)z!3%*RI:F17$$\"1nmm\"pW`(>Fao$\"13I<1\\8(Q\"F17$$\"1+]i!f#=$3#Fao$
\"1+a2DCLX7F17$$\"1+](=xpe=#Fao$\"1\"Rv@Y*zB6F17$$\"1nm\"H28IH#Fao$\"1
bCQyzh45F17$$\"1n;zpSS\"R#Fao$\"1pBSo>6]\"*F.7$$\"1LL3_?`(\\#Fao$\"1O7
8JIyG#)F.7$$\"1M$e*)>pxg#Fao$\"1-it!yw)ptF.7$$\"1+]Pf4t.FFao$\"19vI()Q
_&p'F.7$$\"1MLe*Gst!GFao$\"1u!y8<Sj.'F.7$$\"1+++DRW9HFao$\"1&Gb@4=MU&F
.7$$\"1++DJE>>IFao$\"1rf.&[jS)[F.7$$\"1+]i!RU07$Fao$\"1y'ezWALT%F.7$$
\"1++v=S2LKFao$\"1S_^+&3O%RF.7$$\"1mmm\"p)=MLFao$\"1'=EYB\\Vc$F.7$$\"1
++](=]@W$Fao$\"1y5;J9e*>$F.7$$\"1L$e*[$z*RNFao$\"1;5QCER,HF.7$$\"1,+]i
C$pk$Fao$\"1!)p:&*z42EF.7$$\"1m;H2qcZPFao$\"1ozF-M]dBF.7$$\"1+]7.\"fF&
QFao$\"1-AUP/6A@F.7$$\"1mm;/OgbRFao$\"1U#*oy2r9>F.7$$\"1+]ilAFjSFao$\"
1,O7*zm#><F.7$$\"1MLL$)*pp;%Fao$\"14G\\t^\"*\\:F.7$$\"1ML3xe,tUFao$\"1
;CeLz'RR\"F.7$$\"1n;HdO=yVFao$\"1_+'Q'H\"[D\"F.7$$\"1,++D>#[Z%Fao$\"1*
[#*=1D#R6F.7$$\"1nmT&G!e&e%Fao$\"1pz\"=/$y>5F.7$$\"1MLL$)Qk%o%Fao$\"1&
*)=%GJ-O#*!#=7$$\"1+]iSjE!z%Fao$\"1;VjJPC5$)Fcz7$$\"1,]P40O\"*[Fao$\"1
r)pqY'>6vFcz7$$\"\"&F($\"1na3**p%zt'Fcz-%'COLOURG6&%$RGBG$\"*++++\"!\"
)F(F(-F$6$7U7$F_[l$!\"\"F(7$$\"1nmT&)G\\a]Fao$!1c,(4Y_'p%*F17$$\"1ML$3
x&)*3^Fao$!19e:t<Vn*)F17$$\"1+D1R(*Rc^Fao$!1Uq)Ges@b)F17$$\"1m;H2P\"Q?
&Fao$!1T9K+I9c\")F17$$\"1LLeRwX5`Fao$!16\\d'y86L(F17$$\"1ML3x%3yT&Fao$
!1.^a>L)[e'F17$$\"1m;z%4\\Y_&Fao$!1k()\\1)Hw\"fF17$$\"1L$eR-/Pi&Fao$!1
G$3?kb&f`F17$$\"1+]il'pis&Fao$!1tb6*G+r$[F17$$\"1L$e*)>VB$eFao$!1MjNAt
G]VF17$$\"1+]7`l2QfFao$!1O'RI5*z8RF17$$\"1nm;/j$o/'Fao$!11n;))fY5NF17$
$\"1ML3_>jUhFao$!14,D-Qy*=$F17$$\"1++]i^Z]iFao$!10(H'GpojGF17$$\"1++](
=h(ejFao$!1$4nbG*ypDF17$$\"1++]P[6jkFao$!1OT1>-9:BF17$$\"1M$e*[z(yb'Fa
o$!1BLOTY#e5#F17$$\"1nm;a/cqmFao$!15'fM\"fT\")=F17$$\"1nmm;t,mnFao$!1&
\\t'Qs75<F17$$\"1+]iSj0xoFao$!1p)z!3%*RI:F17$$\"1nmm\"pW`(pFao$!13I<1
\\8(Q\"F17$$\"1+]i!f#=$3(Fao$!1+a2DCLX7F17$$\"1+](=xpe=(Fao$!1#Rv@Y*zB
6F17$$\"1nm\"H28IH(Fao$!1cCQyzh45F17$$\"1n;zpSS\"R(Fao$!1lBSo>6]\"*F.7
$$\"1LL3_?`(\\(Fao$!1Q78JIyG#)F.7$$\"1M$e*)>pxg(Fao$!1-it!yw)ptF.7$$\"
1+]Pf4t.xFao$!19vI()Q_&p'F.7$$\"1MLe*Gst!yFao$!1s!y8<Sj.'F.7$$\"1+++DR
W9zFao$!1$Gb@4=MU&F.7$$\"1++DJE>>!)Fao$!1rf.&[jS)[F.7$$\"1,]i!RU07)Fao
$!1u'ezWALT%F.7$$\"1***\\(=S2L#)Fao$!1V_^+&3O%RF.7$$\"1mmm\"p)=M$)Fao$
!1'=EYB\\Vc$F.7$$\"1,+](=]@W)Fao$!1w5;J9e*>$F.7$$\"1L$e*[$z*R&)Fao$!1;
5QCER,HF.7$$\"1,+]iC$pk)Fao$!1!)p:&*z42EF.7$$\"1m;H2qcZ()Fao$!1pzF-M]d
BF.7$$\"1,]7.\"fF&))Fao$!1,AUP/6A@F.7$$\"1nm;/Ogb*)Fao$!1T#*oy2r9>F.7$
$\"1**\\ilAFj!*Fao$!1.O7*zm#><F.7$$\"1NLL$)*pp;*Fao$!13G\\t^\"*\\:F.7$
$\"1NL3xe,t#*Fao$!1:CeLz'RR\"F.7$$\"1o;HdO=y$*Fao$!1^+'Q'H\"[D\"F.7$$
\"1,++D>#[Z*Fao$!1*[#*=1D#R6F.7$$\"1omT&G!e&e*Fao$!1oz\"=/$y>5F.7$$\"1
LLL$)Qk%o*Fao$!1-*=%GJ-O#*Fcz7$$\"1**\\iSjE!z*Fao$!1BVjJPC5$)Fcz7$$\"1
,]P40O\"*)*Fao$!1k)pqY'>6vFcz7$$\"#5F($!1na3**p%zt'Fcz-Fd[l6&Ff[lF(F(F
g[l-%&TITLEG6#Q1voltage~across~L6\"-%+AXESLABELSG6$Q\"tF`\\m%!G-%%VIEW
G6$;F(Ff[m%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "For the current t
o continue to flow in the same direction as the down-time of the squar
e-wave signal begins, the voltage across the coil has to jump to the o
pposite sign, i.e., to -" }{TEXT 275 1 "U" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
266 10 "RC circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 35 "We apply Kirchhoff's law as before:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 17 "KL_RC:=V_R+V_C=U;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&KL_RCG/,&*&%\"RG\"\"\"-%\"iG6#%\"tGF)F)*&-%$IntG6$-F
+6#%\"sG/F4;\"\"!F-\"\"\"%\"CG!\"\"F)%\"UG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 134 "This is an integral equation. We are better off turning \+
it into a differential equation by taking the derivative with respect \+
to time:" }}}{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)\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 164 "In this form we have to supply an initia
l condition for the current, and we have to consider different possibi
lities. When the capacitor is uncharged and a voltage " }{TEXT 325 1 "
U" }{TEXT -1 15 " is applied at " }{TEXT 267 1 "t" }{TEXT -1 161 "=0, \+
one can see from the integral form of Kirchhoff's law that no voltage \+
across the capacitor is generated (the integral vanishes), and the cur
rent is given by " }{TEXT 268 1 "i" }{TEXT -1 4 "(0)=" }{TEXT 326 1 "U
" }{TEXT -1 1 "/" }{TEXT 327 1 "R" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 14 "IC1:=i(0)=U/R;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$IC1G/-%\"iG6#\"\"!*&%\"UG\"\"\"%\"RG!\"\"" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 83 "The solution to the equation for the current show
s that it decreases exponentially:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "sol_RC:=dsolve(\{KL_RC,IC1\},i(t));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%'sol_RCG/-%\"iG6#%\"tG*&*&%\"UG\"\"\"-%$expG6#,$*&
F)\"\"\"*&%\"CG\"\"\"%\"RG\"\"\"!\"\"!\"\"F-F3F7F9" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 18 "i_RC:=rhs(sol_RC);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%i_RCG*&*&%\"UG\"\"\"-%$expG6#,$*&%\"tG\"\"\"*&%\"CG
\"\"\"%\"RG\"\"\"!\"\"!\"\"F(F/F3F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 66 "The voltage across the resistor is found from Ohm's law as befo
re:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U_R:=simplify(subs(i
(t)=i_RC,V_R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$U_RG*&%\"UG\"\"
\"-%$expG6#,$*&%\"tG\"\"\"*&%\"CG\"\"\"%\"RG\"\"\"!\"\"!\"\"F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "This time tau=" }{TEXT 328 2 "RC" 
}{TEXT -1 36 " acts as a time constant and we have" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 33 "U_R:=simplify(subs(R=tau/C,U_R));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$U_RG*&%\"UG\"\"\"-%$expG6#,$*&%\"tG\"\"\"
%$tauG!\"\"!\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We graph th
e result after introducing scales for voltage and time:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P_RC1:=plot(subs(U=1,tau=1,U_R),t=0
..5,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(
P_RC1);" }}{PARA 13 "" 1 "" {GLPLOT2D 648 248 248 {PLOTDATA 2 "6%-%'CU
RVESG6$7U7$\"\"!$\"\"\"F(7$$\"1mmmT&)G\\a!#<$\"1d,(4Y_'p%*!#;7$$\"1LLL
3x&)*3\"F1$\"1<e:t<Vn*)F17$$\"1+]i!R(*Rc\"F1$\"1Xq)Ges@b)F17$$\"1nm\"H
2P\"Q?F1$\"1P9K+I9c\")F17$$\"1LL$eRwX5$F1$\"16\\d'y86L(F17$$\"1ML$3x%3
yTF1$\"1.^a>L)[e'F17$$\"1nm\"z%4\\Y_F1$\"1h()\\1)Hw\"fF17$$\"1MLeR-/Pi
F1$\"1F$3?kb&f`F17$$\"1***\\il'pisF1$\"1sb6*G+r$[F17$$\"1MLe*)>VB$)F1$
\"1LjNAtG]VF17$$\"1++DJbw!Q*F1$\"1N'RI5*z8RF17$$\"1nm;/j$o/\"!#:$\"11n
;))fY5NF17$$\"1LL3_>jU6Fao$\"15,D-Qy*=$F17$$\"1++]i^Z]7Fao$\"10(H'Gpoj
GF17$$\"1++](=h(e8Fao$\"1#4nbG*ypDF17$$\"1++]P[6j9Fao$\"1OT1>-9:BF17$$
\"1L$e*[z(yb\"Fao$\"1CLOTY#e5#F17$$\"1nm;a/cq;Fao$\"16'fM\"fT\")=F17$$
\"1nmm;t,m<Fao$\"1&\\t'Qs75<F17$$\"1+]iSj0x=Fao$\"1o)z!3%*RI:F17$$\"1n
mm\"pW`(>Fao$\"13I<1\\8(Q\"F17$$\"1+]i!f#=$3#Fao$\"1+a2DCLX7F17$$\"1+]
(=xpe=#Fao$\"1\"Rv@Y*zB6F17$$\"1nm\"H28IH#Fao$\"1bCQyzh45F17$$\"1n;zpS
S\"R#Fao$\"1pBSo>6]\"*F.7$$\"1LL3_?`(\\#Fao$\"1O78JIyG#)F.7$$\"1M$e*)>
pxg#Fao$\"1-it!yw)ptF.7$$\"1+]Pf4t.FFao$\"19vI()Q_&p'F.7$$\"1MLe*Gst!G
Fao$\"1u!y8<Sj.'F.7$$\"1+++DRW9HFao$\"1&Gb@4=MU&F.7$$\"1++DJE>>IFao$\"
1rf.&[jS)[F.7$$\"1+]i!RU07$Fao$\"1y'ezWALT%F.7$$\"1++v=S2LKFao$\"1S_^+
&3O%RF.7$$\"1mmm\"p)=MLFao$\"1'=EYB\\Vc$F.7$$\"1++](=]@W$Fao$\"1y5;J9e
*>$F.7$$\"1L$e*[$z*RNFao$\"1;5QCER,HF.7$$\"1,+]iC$pk$Fao$\"1!)p:&*z42E
F.7$$\"1m;H2qcZPFao$\"1ozF-M]dBF.7$$\"1+]7.\"fF&QFao$\"1-AUP/6A@F.7$$
\"1mm;/OgbRFao$\"1U#*oy2r9>F.7$$\"1+]ilAFjSFao$\"1,O7*zm#><F.7$$\"1MLL
$)*pp;%Fao$\"14G\\t^\"*\\:F.7$$\"1ML3xe,tUFao$\"1;CeLz'RR\"F.7$$\"1n;H
dO=yVFao$\"1_+'Q'H\"[D\"F.7$$\"1,++D>#[Z%Fao$\"1*[#*=1D#R6F.7$$\"1nmT&
G!e&e%Fao$\"1pz\"=/$y>5F.7$$\"1MLL$)Qk%o%Fao$\"1&*)=%GJ-O#*!#=7$$\"1+]
iSjE!z%Fao$\"1;VjJPC5$)Fcz7$$\"1,]P40O\"*[Fao$\"1r)pqY'>6vFcz7$$\"\"&F
($\"1na3**p%zt'Fcz-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$
Q\"t6\"%!G-%%VIEWG6$;F(F_[l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 379 "We can \+
interpret this result as follows: initially the uncharged capacitor po
ses no effective resistance to the current; as the capacitor plates ar
e charged up the effective resistance goes to infinity causing the vol
tage drop across R (which is in series with C) to go to zero. For alte
rnating currents (AC) capacitors therefore act as frequency-dependent \+
resistors (impedance)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 77 "The voltage across the charging capacitor is given b
y the complement between " }{TEXT 277 1 "U" }{TEXT -1 7 "_R and " }
{TEXT 278 1 "U" }{TEXT -1 94 ", i.e., it is a growing function of time
. we leave ot to the reader to graph it together with " }{TEXT 276 1 "
U" }{TEXT -1 15 "_R shown above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 131 "Again, the time at which the voltage has
 fallen to half its value can be used to determine the time constant f
rom a single reading." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "t_
half:=solve(U_R=U/2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'t_halfG*
&-%#lnG6#\"\"#\"\"\"%$tauGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 334 "
Note that if we are interested in an accurate determination of the tim
e constant for either the RL or the RC circuit, we should record a tim
e sequence of voltages across R, and perform a fit to the data. An exp
onential fit is performed most conveniently by taking the logarithm of
 the data and carrying out a linear least squares fit." }}{PARA 0 "" 
0 "" {TEXT -1 37 "This is carried out in the worksheet " }{TEXT 19 10 
"ExpFit.mws" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 494 "We can now disconnect the battery and short-ci
rcuit the open ends of the RC circuit, or alternatively consider a squ
are-wave pulse connected at the free terminals of the circuit for the \+
moment where it switches from high to low. We expect the stored energy
 in C to be released and dissipated in the resistance in analogy to th
e RL circuit case. There is, however, a difference: the capacitor disc
harges by sending a current in the direction opposite to the charging \+
current through the circuit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 132 "To find the corresponding solutions we need to
 modify the equations as well as the boundary conditions to reflect th
e new situation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "KL_RC0:
=lhs(KL_RC)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'KL_RC0G/,&*&%\"RG
\"\"\"-%%diffG6$-%\"iG6#%\"tGF0F)F)*&F-\"\"\"%\"CG!\"\"F)\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Assuming that the capacitor was fu
lly charged at " }{TEXT 329 1 "t" }{TEXT -1 38 "=5 we can set the init
ial current to -" }{TEXT 330 1 "U" }{TEXT -1 1 "/" }{TEXT 331 1 "R" }
{TEXT -1 87 ", which corresponds to the full capacitor voltage dischar
ging through the resistance R." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "IC0:=i(5)=-U/R;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IC0G/-%
\"iG6#\"\"&,$*&%\"UG\"\"\"%\"RG!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "sol_RC0:=dsolve(\{KL_RC0,IC0\},i(t));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(sol_RC0G/-%\"iG6#%\"tG,$*&*&%\"UG\"\"\"-%$exp
G6#,$*&F)\"\"\"*&%\"CG\"\"\"%\"RG\"\"\"!\"\"!\"\"F.F4*&-F06#,$*&F4F4*&
F6\"\"\"F8\"\"\"F:!\"&\"\"\"F8\"\"\"F:F;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 88 "We see that the current dies out exponentially with the c
haracteristic time constant RC." }}{PARA 0 "" 0 "" {TEXT -1 28 "We com
bine the exponentials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "i
_RC0:=combine(rhs(sol_RC0),exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&i_RC0G,$*&*&%\"UG\"\"\"-%$expG6#,&*&\"\"\"F/*&%\"CG\"\"\"%\"RG\"\"\"!
\"\"\"\"&*&%\"tGF/*&F1\"\"\"F3\"\"\"F5!\"\"F)F/F3F5F<" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 56 "Ohm's law gives us the voltage drop acros
s the resistor:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "U_R0:=si
mplify(subs(i(t)=i_RC0,V_R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%U_
R0G,$*&%\"UG\"\"\"-%$expG6#,$*&,&!\"&F(%\"tGF(\"\"\"*&%\"CG\"\"\"%\"RG
\"\"\"!\"\"!\"\"F(F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "and we su
bstitute the time constant tau = " }{TEXT 332 2 "RC" }{TEXT -1 1 ":" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U_R0:=simplify(subs(R=tau/
C,U_R0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%U_R0G,$*&%\"UG\"\"\"-%
$expG6#,$*&,&!\"&F(%\"tGF(\"\"\"%$tauG!\"\"!\"\"F(F4" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "P_RC2:=plot(subs(U=1,tau=1,U_R0),t=5..10,
color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(P_R
C1,P_RC2);" }}{PARA 13 "" 1 "" {GLPLOT2D 758 243 243 {PLOTDATA 2 "6&-%
'CURVESG6$7U7$\"\"!$\"\"\"F(7$$\"1mmmT&)G\\a!#<$\"1d,(4Y_'p%*!#;7$$\"1
LLL3x&)*3\"F1$\"1<e:t<Vn*)F17$$\"1+]i!R(*Rc\"F1$\"1Xq)Ges@b)F17$$\"1nm
\"H2P\"Q?F1$\"1P9K+I9c\")F17$$\"1LL$eRwX5$F1$\"16\\d'y86L(F17$$\"1ML$3
x%3yTF1$\"1.^a>L)[e'F17$$\"1nm\"z%4\\Y_F1$\"1h()\\1)Hw\"fF17$$\"1MLeR-
/PiF1$\"1F$3?kb&f`F17$$\"1***\\il'pisF1$\"1sb6*G+r$[F17$$\"1MLe*)>VB$)
F1$\"1LjNAtG]VF17$$\"1++DJbw!Q*F1$\"1N'RI5*z8RF17$$\"1nm;/j$o/\"!#:$\"
11n;))fY5NF17$$\"1LL3_>jU6Fao$\"15,D-Qy*=$F17$$\"1++]i^Z]7Fao$\"10(H'G
pojGF17$$\"1++](=h(e8Fao$\"1#4nbG*ypDF17$$\"1++]P[6j9Fao$\"1OT1>-9:BF1
7$$\"1L$e*[z(yb\"Fao$\"1CLOTY#e5#F17$$\"1nm;a/cq;Fao$\"16'fM\"fT\")=F1
7$$\"1nmm;t,m<Fao$\"1&\\t'Qs75<F17$$\"1+]iSj0x=Fao$\"1o)z!3%*RI:F17$$
\"1nmm\"pW`(>Fao$\"13I<1\\8(Q\"F17$$\"1+]i!f#=$3#Fao$\"1+a2DCLX7F17$$
\"1+](=xpe=#Fao$\"1\"Rv@Y*zB6F17$$\"1nm\"H28IH#Fao$\"1bCQyzh45F17$$\"1
n;zpSS\"R#Fao$\"1pBSo>6]\"*F.7$$\"1LL3_?`(\\#Fao$\"1O78JIyG#)F.7$$\"1M
$e*)>pxg#Fao$\"1-it!yw)ptF.7$$\"1+]Pf4t.FFao$\"19vI()Q_&p'F.7$$\"1MLe*
Gst!GFao$\"1u!y8<Sj.'F.7$$\"1+++DRW9HFao$\"1&Gb@4=MU&F.7$$\"1++DJE>>IF
ao$\"1rf.&[jS)[F.7$$\"1+]i!RU07$Fao$\"1y'ezWALT%F.7$$\"1++v=S2LKFao$\"
1S_^+&3O%RF.7$$\"1mmm\"p)=MLFao$\"1'=EYB\\Vc$F.7$$\"1++](=]@W$Fao$\"1y
5;J9e*>$F.7$$\"1L$e*[$z*RNFao$\"1;5QCER,HF.7$$\"1,+]iC$pk$Fao$\"1!)p:&
*z42EF.7$$\"1m;H2qcZPFao$\"1ozF-M]dBF.7$$\"1+]7.\"fF&QFao$\"1-AUP/6A@F
.7$$\"1mm;/OgbRFao$\"1U#*oy2r9>F.7$$\"1+]ilAFjSFao$\"1,O7*zm#><F.7$$\"
1MLL$)*pp;%Fao$\"14G\\t^\"*\\:F.7$$\"1ML3xe,tUFao$\"1;CeLz'RR\"F.7$$\"
1n;HdO=yVFao$\"1_+'Q'H\"[D\"F.7$$\"1,++D>#[Z%Fao$\"1*[#*=1D#R6F.7$$\"1
nmT&G!e&e%Fao$\"1pz\"=/$y>5F.7$$\"1MLL$)Qk%o%Fao$\"1&*)=%GJ-O#*!#=7$$
\"1+]iSjE!z%Fao$\"1;VjJPC5$)Fcz7$$\"1,]P40O\"*[Fao$\"1r)pqY'>6vFcz7$$
\"\"&F($\"1na3**p%zt'Fcz-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7U7$F
_[l$!\"\"F(7$$\"1nmT&)G\\a]Fao$!1c,(4Y_'p%*F17$$\"1ML$3x&)*3^Fao$!19e:
t<Vn*)F17$$\"1+D1R(*Rc^Fao$!1Uq)Ges@b)F17$$\"1m;H2P\"Q?&Fao$!1T9K+I9c
\")F17$$\"1LLeRwX5`Fao$!16\\d'y86L(F17$$\"1ML3x%3yT&Fao$!1.^a>L)[e'F17
$$\"1m;z%4\\Y_&Fao$!1k()\\1)Hw\"fF17$$\"1L$eR-/Pi&Fao$!1G$3?kb&f`F17$$
\"1+]il'pis&Fao$!1tb6*G+r$[F17$$\"1L$e*)>VB$eFao$!1MjNAtG]VF17$$\"1+]7
`l2QfFao$!1O'RI5*z8RF17$$\"1nm;/j$o/'Fao$!11n;))fY5NF17$$\"1ML3_>jUhFa
o$!14,D-Qy*=$F17$$\"1++]i^Z]iFao$!10(H'GpojGF17$$\"1++](=h(ejFao$!1$4n
bG*ypDF17$$\"1++]P[6jkFao$!1OT1>-9:BF17$$\"1M$e*[z(yb'Fao$!1BLOTY#e5#F
17$$\"1nm;a/cqmFao$!15'fM\"fT\")=F17$$\"1nmm;t,mnFao$!1&\\t'Qs75<F17$$
\"1+]iSj0xoFao$!1p)z!3%*RI:F17$$\"1nmm\"pW`(pFao$!13I<1\\8(Q\"F17$$\"1
+]i!f#=$3(Fao$!1+a2DCLX7F17$$\"1+](=xpe=(Fao$!1#Rv@Y*zB6F17$$\"1nm\"H2
8IH(Fao$!1cCQyzh45F17$$\"1n;zpSS\"R(Fao$!1lBSo>6]\"*F.7$$\"1LL3_?`(\\(
Fao$!1Q78JIyG#)F.7$$\"1M$e*)>pxg(Fao$!1-it!yw)ptF.7$$\"1+]Pf4t.xFao$!1
9vI()Q_&p'F.7$$\"1MLe*Gst!yFao$!1s!y8<Sj.'F.7$$\"1+++DRW9zFao$!1$Gb@4=
MU&F.7$$\"1++DJE>>!)Fao$!1rf.&[jS)[F.7$$\"1,]i!RU07)Fao$!1u'ezWALT%F.7
$$\"1***\\(=S2L#)Fao$!1V_^+&3O%RF.7$$\"1mmm\"p)=M$)Fao$!1'=EYB\\Vc$F.7
$$\"1,+](=]@W)Fao$!1w5;J9e*>$F.7$$\"1L$e*[$z*R&)Fao$!1;5QCER,HF.7$$\"1
,+]iC$pk)Fao$!1!)p:&*z42EF.7$$\"1m;H2qcZ()Fao$!1pzF-M]dBF.7$$\"1,]7.\"
fF&))Fao$!1,AUP/6A@F.7$$\"1nm;/Ogb*)Fao$!1T#*oy2r9>F.7$$\"1**\\ilAFj!*
Fao$!1.O7*zm#><F.7$$\"1NLL$)*pp;*Fao$!13G\\t^\"*\\:F.7$$\"1NL3xe,t#*Fa
o$!1:CeLz'RR\"F.7$$\"1o;HdO=y$*Fao$!1^+'Q'H\"[D\"F.7$$\"1,++D>#[Z*Fao$
!1*[#*=1D#R6F.7$$\"1omT&G!e&e*Fao$!1oz\"=/$y>5F.7$$\"1LLL$)Qk%o*Fao$!1
-*=%GJ-O#*Fcz7$$\"1**\\iSjE!z*Fao$!1BVjJPC5$)Fcz7$$\"1,]P40O\"*)*Fao$!
1k)pqY'>6vFcz7$$\"#5F($!1na3**p%zt'Fcz-Fd[l6&Ff[lF(F(Fg[l-%+AXESLABELS
G6$Q\"t6\"%!G-%%VIEWG6$;F(Ff[m%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
112 "The graph shows that the current (and the voltage across the resi
stor) change directions at the switching times." }}{PARA 0 "" 0 "" 
{TEXT -1 165 "As with the RL circuit, we can find the voltage across t
he interesting device, i.e., the capacitor from Kirchhoff's law, using
 the fact that the terminal voltage is " }{TEXT 279 1 "U" }{TEXT -1 
79 " for the first 5 time units and 0 for the second 5 time units, and
 that we set " }{TEXT 280 1 "U" }{TEXT -1 3 "=1." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 52 "P_RC3:=plot(1-subs(U=1,tau=1,U_R),t=0..5,colo
r=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P_RC4:=plot(0-su
bs(U=1,tau=1,U_R0),t=5..10,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "display(P_RC3,P_RC4);" }}{PARA 13 "" 1 "" {GLPLOT2D 
776 253 253 {PLOTDATA 2 "6&-%'CURVESG6$7U7$\"\"!F(7$$\"1mmmT&)G\\a!#<$
\"1H%)H!RvMI&F,7$$\"1LLL3x&)*3\"!#;$\"1$=WoAoD.\"F27$$\"1+]i!R(*Rc\"F2
$\"1bH6<u#yW\"F27$$\"1nm\"H2P\"Q?F2$\"1j&y'**p&Q%=F27$$\"1LL$eRwX5$F2$
\"1*3DM@'))oEF27$$\"1ML$3x%3yTF2$\"1(*[X!o;^T$F27$$\"1nm\"z%4\\Y_F2$\"
1R7]$>qB3%F27$$\"1MLeR-/PiF2$\"1t;*zNW/k%F27$$\"1***\\il'pisF2$\"1GW)3
r**G;&F27$$\"1MLe*)>VB$)F2$\"1oOkxEr\\cF27$$\"1++DJbw!Q*F2$\"1l.'p*3?'
3'F27$$\"1nm;/j$o/\"!#:$\"1$HL=,M&*['F27$$\"1LL3_>jU6F_o$\"1!*)\\x>;-
\"oF27$$\"1++]i^Z]7F_o$\"1&Hq828j8(F27$$\"1++](=h(e8F_o$\"13HV92@IuF27
$$\"1++]P[6j9F_o$\"1ke$4yf[o(F27$$\"1L$e*[z(yb\"F_o$\"1xmje`<%*yF27$$
\"1nm;a/cq;F_o$\"1*QSl3%e=\")F27$$\"1nmm;t,m<F_o$\"1/lKhF()*G)F27$$\"1
+]iSj0x=F_o$\"1K,#>f+'p%)F27$$\"1nmm\"pW`(>F_o$\"1#*p#Q4lGh)F27$$\"1+]
i!f#=$3#F_o$\"1*fC\\dnYv)F27$$\"1+](=xpe=#F_o$\"14Y#y`+i())F27$$\"1nm
\"H28IH#F_o$\"1Xvh@?Q!**)F27$$\"1n;zpSS\"R#F_o$\"1k(fJ!)))\\3*F27$$\"1
LL3_?`(\\#F_o$\"1wo)op@r<*F27$$\"1M$e*)>pxg#F_o$\"1zj#>K7IE*F27$$\"1+]
Pf4t.FF_o$\"1[#p7hZ/L*F27$$\"1MLe*Gst!GF_o$\"1$>iG)fO'R*F27$$\"1+++DRW
9HF_o$\"1rWy!>ewX*F27$$\"1++DJE>>IF_o$\"1.k\\^Of6&*F27$$\"1+]i!RU07$F_
o$\"1KT?bxme&*F27$$\"1++v=S2LKF_o$\"1w%[*\\\"Rcg*F27$$\"1mmm\"p)=MLF_o
$\"1\"QPl2lNk*F27$$\"1++](=]@W$F_o$\"1#*Q)o&=/!o*F27$$\"1L$e*[$z*RNF_o
$\"1)*=cP2')4(*F27$$\"1,+]iC$pk$F_o$\"1-V[+-HR(*F27$$\"1m;H2qcZPF_o$\"
1.Axf'\\Uw*F27$$\"1+]7.\"fF&QF_o$\"1!ydi&*)y(y*F27$$\"1mm;/OgbRF_o$\"1
w58A*G&3)*F27$$\"1+]ilAFjSF_o$\"1Sw3?L2G)*F27$$\"1MLL$)*pp;%F_o$\"1>2l
#[3]%)*F27$$\"1ML3xe,tUF_o$\"1e<k1Kgg)*F27$$\"1n;HdO=yVF_o$\"1&*Rh.(=X
()*F27$$\"1,++D>#[Z%F_o$\"1^2\"Q\\xg))*F27$$\"1nmT&G!e&e%F_o$\"1.#=ep@
!)*)*F27$$\"1MLL$)Qk%o%F_o$\"16ero(Rw!**F27$$\"1+]iSjE!z%F_o$\"1cOoiv*
o\"**F27$$\"1,]P40O\"*[F_o$\"1,$H`.))[#**F27$$\"\"&F($\"1X\"4+`?E$**F2
-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7U7$F\\[l$\"\"\"F(7$$\"1nmT&)
G\\a]F_o$\"1c,(4Y_'p%*F27$$\"1ML$3x&)*3^F_o$\"19e:t<Vn*)F27$$\"1+D1R(*
Rc^F_o$\"1Uq)Ges@b)F27$$\"1m;H2P\"Q?&F_o$\"1T9K+I9c\")F27$$\"1LLeRwX5`
F_o$\"16\\d'y86L(F27$$\"1ML3x%3yT&F_o$\"1.^a>L)[e'F27$$\"1m;z%4\\Y_&F_
o$\"1k()\\1)Hw\"fF27$$\"1L$eR-/Pi&F_o$\"1G$3?kb&f`F27$$\"1+]il'pis&F_o
$\"1tb6*G+r$[F27$$\"1L$e*)>VB$eF_o$\"1MjNAtG]VF27$$\"1+]7`l2QfF_o$\"1O
'RI5*z8RF27$$\"1nm;/j$o/'F_o$\"11n;))fY5NF27$$\"1ML3_>jUhF_o$\"14,D-Qy
*=$F27$$\"1++]i^Z]iF_o$\"10(H'GpojGF27$$\"1++](=h(ejF_o$\"1$4nbG*ypDF2
7$$\"1++]P[6jkF_o$\"1OT1>-9:BF27$$\"1M$e*[z(yb'F_o$\"1BLOTY#e5#F27$$\"
1nm;a/cqmF_o$\"15'fM\"fT\")=F27$$\"1nmm;t,mnF_o$\"1&\\t'Qs75<F27$$\"1+
]iSj0xoF_o$\"1p)z!3%*RI:F27$$\"1nmm\"pW`(pF_o$\"13I<1\\8(Q\"F27$$\"1+]
i!f#=$3(F_o$\"1+a2DCLX7F27$$\"1+](=xpe=(F_o$\"1#Rv@Y*zB6F27$$\"1nm\"H2
8IH(F_o$\"1cCQyzh45F27$$\"1n;zpSS\"R(F_o$\"1lBSo>6]\"*F,7$$\"1LL3_?`(
\\(F_o$\"1Q78JIyG#)F,7$$\"1M$e*)>pxg(F_o$\"1-it!yw)ptF,7$$\"1+]Pf4t.xF
_o$\"19vI()Q_&p'F,7$$\"1MLe*Gst!yF_o$\"1s!y8<Sj.'F,7$$\"1+++DRW9zF_o$
\"1$Gb@4=MU&F,7$$\"1++DJE>>!)F_o$\"1rf.&[jS)[F,7$$\"1,]i!RU07)F_o$\"1u
'ezWALT%F,7$$\"1***\\(=S2L#)F_o$\"1V_^+&3O%RF,7$$\"1mmm\"p)=M$)F_o$\"1
'=EYB\\Vc$F,7$$\"1,+](=]@W)F_o$\"1w5;J9e*>$F,7$$\"1L$e*[$z*R&)F_o$\"1;
5QCER,HF,7$$\"1,+]iC$pk)F_o$\"1!)p:&*z42EF,7$$\"1m;H2qcZ()F_o$\"1pzF-M
]dBF,7$$\"1,]7.\"fF&))F_o$\"1,AUP/6A@F,7$$\"1nm;/Ogb*)F_o$\"1T#*oy2r9>
F,7$$\"1**\\ilAFj!*F_o$\"1.O7*zm#><F,7$$\"1NLL$)*pp;*F_o$\"13G\\t^\"*
\\:F,7$$\"1NL3xe,t#*F_o$\"1:CeLz'RR\"F,7$$\"1o;HdO=y$*F_o$\"1^+'Q'H\"[
D\"F,7$$\"1,++D>#[Z*F_o$\"1*[#*=1D#R6F,7$$\"1omT&G!e&e*F_o$\"1oz\"=/$y
>5F,7$$\"1LLL$)Qk%o*F_o$\"1-*=%GJ-O#*!#=7$$\"1**\\iSjE!z*F_o$\"1BVjJPC
5$)Fhjl7$$\"1,]P40O\"*)*F_o$\"1k)pqY'>6vFhjl7$$\"#5F($\"1na3**p%zt'Fhj
l-Fa[l6&Fc[lF(F(Fd[l-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(Fd[m%(DEFAU
LTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 323 "Note that we find the expected behaviour
 for the voltage at the capacitor: in the turn-on phase (red) it acqui
res charge on its plates, while during the turn-off phase it releases \+
this charge. The sign of the voltage across the capacitor stays the sa
me, but the current changes sign as one goes from one phase to the oth
er." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 271 "F
or the inductance L we had the opposite behaviour: the current kept it
s sign (the graph is identical to the voltage across the capacitor if \+
the same time constant is chosen), but the voltage across the inductan
ce changed sign during the turn-off of the external voltage." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 926 "This perfect s
ymmetry between the two devices makes the LC circuit a very interestin
g object: the two distinct storage mechanisms for electromagnetic ener
gy (the capacitor builds up an electric field between its plates, whil
e the solenoid builds up a magnetic field) allow for a perfect oscilla
ting device. In an ideal setting (no resistance in the solenoid) an in
itial charge in the capacitor sets off a current through L, thus build
ing a magnetic field as it discharges. Once it discharged, the inducta
nce continues a current in the same direction, thus charging the capac
itor in the opposite sense. The analogy to a mechanical harmonic oscil
lator where a constant total energy oscillates in form between kinetic
 and potential energy is perfect. In a non-ideal world one cannot avoi
d dissipation, and thus we consider an RLC circuit, which we show is g
overned by the differential equation for the damped harmonic oscillato
r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
281 11 "RLC circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 42 "We begin again by stating Kirchhoff's law:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "KL_RLC:=V_R+V_L+V_C=U;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'KL_RLCG/,(*&%\"RG\"\"\"-%\"iG6#%\"tGF)F)*&%
\"LGF)-%%diffG6$F*F-F)F)*&-%$IntG6$-F+6#%\"sG/F9;\"\"!F-\"\"\"%\"CG!\"
\"F)%\"UG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "KL_RLC:=diff(K
L_RLC,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'KL_RLCG/,(*&%\"RG\"\"
\"-%%diffG6$-%\"iG6#%\"tGF0F)F)*&%\"LGF)-F+6$F--%\"$G6$F0\"\"#F)F)*&F-
\"\"\"%\"CG!\"\"F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The an
alogy of the damped harmonic oscillator goes as follows:" }}{PARA 0 "
" 0 "" {TEXT -1 15 "- the position " }{TEXT 286 1 "x" }{TEXT -1 1 "(" 
}{TEXT 287 1 "t" }{TEXT -1 18 ") becomes current " }{TEXT 288 1 "i" }
{TEXT -1 1 "(" }{TEXT 289 1 "t" }{TEXT -1 2 ");" }}{PARA 0 "" 0 "" 
{TEXT -1 11 "- the mass " }{TEXT 285 1 "m" }{TEXT -1 20 " becomes indu
ctance " }{TEXT 333 1 "L" }{TEXT -1 39 ", i.e., the inductance acts as
 inertia;" }}{PARA 0 "" 0 "" {TEXT -1 24 "- the friction constant " }
{TEXT 284 1 "b" }{TEXT -1 20 " becomes resistance " }{TEXT 334 1 "R" }
{TEXT -1 46 ", i.e. the resistance acts as a friction term;" }}{PARA 
0 "" 0 "" {TEXT -1 22 "- the spring constant " }{TEXT 283 1 "k" }
{TEXT -1 31 " becomes inverse capacitance 1/" }{TEXT 335 1 "C" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
71 "We have three elements in series: the capacitor poses no resistanc
e at " }{TEXT 282 1 "t" }{TEXT -1 166 "=0, since it is uncharged. The \+
solenoid, however, opposes the current, as the magnetic field has to b
e built up. Thus, the right initial condition for the current is " }
{TEXT 290 1 "i" }{TEXT -1 210 "(0)=0. However, we have a second-order \+
differential equation to solve, and thus we need a second condition. W
e can use the known solution to the RL circuit, since the capacitance \+
does not impede the current at " }{TEXT 291 1 "t" }{TEXT -1 3 "=0." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "i0prime:=simplify(subs(t=0,
diff(i_RL,t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(i0primeG*&%\"UG
\"\"\"%\"LG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "IC:=i(0
)=0,D(i)(0)=i0prime;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"i
G6#\"\"!F*/--%\"DG6#F(F)*&%\"UG\"\"\"%\"LG!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "sol_RLC:=dsolve(\{KL_RLC,IC\},i(t));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(sol_RLCG/-%\"iG6#%\"tG,&*&*(%\"UG\"\"\"%
\"CG\"\"\"-%$expG6#,$*&*&,&*&F/F0%\"RGF.F.*$-%%sqrtG6#*&F/F0,&*&)F9\"
\"#F0F/F0F.%\"LG!\"%F.F0F.F.F)F.F0*&F/\"\"\"FC\"\"\"!\"\"#!\"\"FBF.F0*
$-F<6#,&*&)F/FBF0FAF0F.*&F/F.FCF.FDF0FHFJ*&*(F-F0F/F0-F26#,$*&*&,&F8F.
F:FJF.F)F0F0*&F/\"\"\"FC\"\"\"FHFIF.F0*$-F<6#FNF0FHF." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "We recognize in the solution that the con
stant " }{TEXT 292 1 "R" }{TEXT -1 1 "/" }{TEXT 293 1 "L" }{TEXT -1 
39 " is responsible for damping, and that (" }{TEXT 295 2 "RC" }{TEXT 
-1 5 ")^2-4" }{TEXT 294 2 "LC" }{TEXT -1 156 " acts as a discriminant \+
that determines whether the circuit is in the subcritical, critical, o
r overcritical damping regimes. Note that for small values of " }
{TEXT 296 1 "R" }{TEXT -1 195 " the discriminant is negative, i.e., th
e solution involves exponentials with complex arguments, which signals
 oscillatory behaviour. If we ignore the friction (damping), i.e., go \+
to the limit of " }{TEXT 297 1 "R" }{TEXT -1 111 " going to zero we re
cognize that the natural circular frequency of oscillation is given as
 the inverse of sqrt(" }{TEXT 298 2 "LC" }{TEXT -1 3 ") ." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "We graph one case \+
of a solution, and refer the reader also to the oscillator solutions s
hown in " }{TEXT 19 11 "HOmovie.mws" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 77 "The parameters chosen are: U=1Volt, C=1 nanoFarad, R=10
 Ohms, L=1 microHenry:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "i
_1:=simplify(subs(U=1,C=10^(-9),R=10,L=10^(-6),rhs(sol_RLC)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$i_1G,$*(%\"IG\"\"\"-%%sqrtG6#\"#RF(
,&-%$expG6#,$*&,&\"\"\"F4*&F'F4F)F(F4F4%\"tGF4!(+++&F4-F/6#,$*&,&!\"\"
F4F5F4F4F6F(\"(+++&F=F4#F4\"$!R" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
89 "At t=0.5 microseconds we have te voltage (just to check that the c
urrent is real-valued):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "
evalf(subs(t=0.5*10^(-6),i_1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+Q?)e]#!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "An during the first
 microsecond the current looks as follows:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "plot(100*i_1,t=0..10^(-6));" }}{PARA 13 "" 1 "" 
{GLPLOT2D 692 248 248 {PLOTDATA 2 "6%-%'CURVESG6$7gr7$\"\"!F(7$$\"+qUk
CF!#=$\"+0j`%o#!#57$$\"+S&)G\\aF,$\"+m!osF&F/7$$\"+5G$R<)F,$\"+OoohxF/
7$$\"+3x&)*3\"!#<$\"+g?D75!\"*7$$\"+N@Ki8F=$\"+Q7eM7F@7$$\"+ilyM;F=$\"
+)p)*=W\"F@7$$\"+*)4D2>F=$\"+#f!4L;F@7$$\"+;arz@F=$\"+H5>2=F@7$$\"+*4b
Ql#F=$\"+942n?F@7$$\"+\"y%*z7$F=$\"+<jhpAF@7$$\"+kW8-OF=$\"+_M/8CF@7$$
\"+XTFwSF=$\"+8n%p\\#F@7$$\"+i!z&4UF=$\"+\\.\"*4DF@7$$\"+xR)GM%F=$\"+6
CF=DF@7$$\"+$*))=wWF=$\"+w!z?_#F@7$$\"+3Q\\4YF=$\"+YRQ@DF@7$$\"+RO5w[F
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p\"F@7$$\"+``T>yF=$\"+;:[&R\"F@7$$\"+S&phN)F=$\"+W3Uq5F@7$$\"+GEP!*))F
=$\"+B\"[,M(F/7$$\"+:ddC%*F=$\"+nWfZRF/7$$\"+.)y(e**F=$\"+h[z?i!#67$$
\"+*=)H\\5!#;$!+xEpZDF/7$$\"+ac#))4\"Fbt$!+?rkx_F/7$$\"+>JN[6Fbt$!+z*R
'\\xF/7$$\"+%e!)y>\"Fbt$!++p4<**F/7$$\"+[!3uC\"Fbt$!+QnHu6F@7$$\"+!pt*
\\8Fbt$!+vGFL9F@7$$\"+J$RDX\"Fbt$!+8P&\\_\"F@7$$\"+:x0z9Fbt$!+@R&>_\"F
@7$$\"+)4wb]\"Fbt$!+InZ3:F@7$$\"+#[%4K:Fbt$!+1)*)[[\"F@7$$\"+lGhe:Fbt$
!+88i^9F@7$$\"+K'\\;h\"Fbt$!+rr-e8F@7$$\"+)R'ok;Fbt$!+Z,:K7F@7$$\"+_(>
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$\"+dG\"\\)>Fbt$!+eNJ65F/7$$\"+LFHR?Fbt$\"+G2i](*F^t7$$\"+3En$4#Fbt$\"
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5b=RBFbt$\"+#>DzZ)F/7$$\"+:r5$R#Fbt$\"+&RVL)*)F/7$$\"+>z1?CFbt$\"+GljI
\"*F/7$$\"+@(GqW#Fbt$\"++(o&3#*F/7$$\"+C&*)RZ#Fbt$\"+(RE%=#*F/7$$\"+D.
&4]#Fbt$\"+(y\\>;*F/7$$\"+^jB4EFbt$\"+#)[j;$)F/7$$\"+vB_<FFbt$\"+fe\"G
k'F/7$$\"+Eg(=#GFbt$\"+B)pr\\%F/7$$\"+v'Hi#HFbt$\"+geI.@F/7$$\"+(y#*4-
$Fbt$!+uT;Ze!#77$$\"+(*ev:JFbt$!+)*p#)>?F/7$$\"+.%Q%GKFbt$!+xQF))QF/7$
$\"+347TLFbt$!+a$>c4&F/7$$\"+S$\\))Q$Fbt$!+x\")['Q&F/7$$\"+rxdOMFbt$!+
&\\0Ua&F/7$$\"+))>WgMFbt$!+[k'Qd&F/7$$\"+.iI%[$Fbt$!+(Qv9d&F/7$$\"+>/<
3NFbt$!+&f,y`&F/7$$\"+LY.KNFbt$!+(>sPZ&F/7$$\"+eO2VOFbt$!+pQd6[F/7$$\"
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]$*4<F/7$$\"+jB0pUFbt$\"+(>R.i#F/7$$\"+V&R<P%Fbt$\"+:Eg\"=$F/7$$\"+q`_
)R%Fbt$\"+a@)pE$F/7$$\"+&>6`U%Fbt$\"+:,*pK$F/7$$\"+@q4_WFbt$\"+uC(=O$F
/7$$\"+XG))yWFbt$\"+?l/sLF/7$$\"+r'oc]%Fbt$\"+8L4eLF/7$$\"+'\\aC`%Fbt$
\"+f#\\2K$F/7$$\"+@.CfXFbt$\"+5o*3E$F/7$$\"+Xh-'e%Fbt$\"+aVbzJF/7$$\"+
VrT%o%Fbt$\"+5`1;FF/7$$\"+R\"3Gy%Fbt$\"+degZ?F/7$$\"+.T1&*\\Fbt$\"+0Y2
;HF^t7$$\"+^7I0^Fbt$!+cepmdF^t7$$\"+(RQb@&Fbt$!+TFj)G\"F/7$$\"+e,]6`Fb
t$!+4&z)H<F/7$$\"+=>Y2aFbt$!+l$>+)>F/7$$\"+$e#GfaFbt$!+rWQL?F/7$$\"+[K
56bFbt$!+I,'4.#F/7$$\"+8R#Hc&Fbt$!+[qiv>F/7$$\"+yXu9cFbt$!+%*>]r=F/7$$
\"+9i\"=s&Fbt$!+%)\\0D:F/7$$\"+\\y))GeFbt$!+c2c\\5F/7$$\"+i_QQgFbt$\"+
u)RPG)!#87$$\"+@]tRhFbt$\"+Lo,-ZF^t7$$\"+!y%3TiFbt$\"+Q7f?%)F^t7$$\"+3
kh`jFbt$\"+B\\p;6F/7$$\"+O![hY'Fbt$\"+A8wH7F/7$$\"+4FEnlFbt$\"+e9F'>\"
F/7$$\"+#Qx$omFbt$\"+!Qh10\"F/7$$\"+y)Qjx'Fbt$\"+=as&*zF^t7$$\"+u.I%)o
Fbt$\"+O,$p'[F^t7$$\"+(pe*zqFbt$!+Aeev5F^t7$$\"+6=\"p=(Fbt$!+&R45$QF^t
7$$\"+C\\'QH(Fbt$!+GNG&*eF^t7$$\"+p%*\\%R(Fbt$!+ciGpqF^t7$$\"+8S8&\\(F
bt$!+Iz`buF^t7$$\"+5hK+wFbt$!++O()fqF^t7$$\"+0#=bq(Fbt$!+(z4!pfF^t7$$
\"+2s?6zFbt$!+H,FcDF^t7$$\"+IXaE\")Fbt$\"+?=c@8F^t7$$\"+l*RRL)Fbt$\"+s
I3dQF^t7$$\"+`<.Y&)Fbt$\"+z6]jWF^t7$$\"+8tOc()Fbt$\"+n`gFKF^t7$$\"+\\Q
k\\*)Fbt$\"+k`2z6F^t7$$\"+p0;r\"*Fbt$!+E-Ye6F^t7$$\"+lxGp$*Fbt$!+YTfeC
F^t7$$\"+!oK0e*Fbt$!++l([l#F^t7$$\"+<5s#y*Fbt$!+!z_w$=F^t7$$\"+++++5!#
:$!+cM<&4%F`^l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"%
!G-%%VIEWG6$;F(Fiam%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "We multiplied the
 solution by 100 so that point-and-click interface can give us good re
ad-outs from the graph. This is needed below to simulate what one does
 on an oscilloscope, i.e., to read off the attenuation of the signal f
rom peak to peak." }}{PARA 0 "" 0 "" {TEXT -1 71 "The reader should ch
eck the time constant against the parameter values:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 71 "According to the analogy we should have (in the
 case of small damping):" }}{PARA 0 "" 0 "" {TEXT -1 32 "harmonic osci
llator: omega=sqrt(" }{TEXT 336 1 "k" }{TEXT -1 1 "/" }{TEXT 337 1 "m
" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 36 "electronic oscillator
: omega=sqrt(1/" }{TEXT 338 2 "LC" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "We calculate for the undamped expected natural frequency:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "omega:=evalf(1/sqrt(10^
(-6)*10^(-9)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG$\"+gwFiJ!
\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The period of oscillation \+
is given as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "T:=evalf((2
*Pi/omega));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+`w\"p)>!#;" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "This agrees with the result fou
nd from the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 58 "For the attenuation constant we observe that the solution
 " }{TEXT 19 7 "sol_RLC" }{TEXT -1 65 " given above has an exponential
 damping with a time constant of 2" }{TEXT 342 1 "L" }{TEXT -1 1 "/" }
{TEXT 341 1 "R" }{TEXT -1 10 ", and not " }{TEXT 340 1 "L" }{TEXT -1 
1 "/" }{TEXT 339 1 "R" }{TEXT -1 32 " as observed for the RL circuit.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "tau:=evalf(subs(R=10,L=
10^(-6),2*L/R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG$\"+++++?!#
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Therefore the amplitude drop
s to half of the original value in the time:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "evalf(ln(2)*tau);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+hVH'Q\"!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "It is, h
owever easier in this case to check whether within a period of oscilla
tion (which happens to agree with the attenuation time constant) the s
ignal drops by a factor of e (Euler's constant). Indeed it does, as th
e read-out lists:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "2.56/0
.93=evalf(exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+s\")o_F!\"*
$\"+G=G=FF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "To the accuracy of
 the readout from the graph the two sides are equal." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "The RLC circui
t (idealized as the undamped oscillatory LC circuit) has many applicat
ions in analogue electronics (radio, TV, etc.). Depending on the value
 of " }{TEXT 299 2 "LC" }{TEXT -1 37 " (i.e., the natural frequency 1/
sqrt(" }{TEXT 300 2 "LC" }{TEXT -1 359 ")) one can resonantly excite t
he circuit with a weak external signal. This is used for signal discri
mination: an RLC oscillator picks out of the mixture of radiosignals (
received on the antenna) the one that comes in with the frequency that
 agrees with its natural frequency. This signal can then be amplified \+
without interference from the other radiosignals." }}{PARA 0 "" 0 "" 
{TEXT -1 103 "For the resonance oscillator to work one has to ensure t
hat one is in the undercritical damping regime." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 12 }{VIEWOPTS 1 1 0 1 1 
1803 }