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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)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "For the capacitor C we have
 the physical statement that it builds up its voltage as the plates ar
e becoming charged due to the current:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "V_C:=(1/C)*Int(i(s),s=0..t);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 144 "For the inductance we take into consideration the fac
t that as the current begins to flow it is building up a time-varying \+
magnetic field (flux " }{XPPEDIT 18 0 "Phi" "6#%$PhiG" }{TEXT -1 99 ")
. This time-varying flux according to Faraday's law induces a time-var
ying voltage proprtional to " }{XPPEDIT 18 0 "-d Phi/dt" "6#,$*(%\"dG
\"\"\"%$PhiGF&%#dtG!\"\"F)" }{TEXT -1 151 "  (the negative sign is imp
ortant, since a positive proportionality would lead to runaway solutio
ns). The magnetic flux is proportional to the current " }{TEXT 261 1 "
i" }{TEXT -1 1 "(" }{TEXT 303 1 "t" }{TEXT -1 111 "). For two coils th
e relationship between the voltage induced in coil 2 due to a current \+
in coil 1 is given as " }{XPPEDIT 18 0 "V[2] = - M  d i[1]/dt" "6#/&%
\"VG6#\"\"#,$**%\"MG\"\"\"%\"dGF+&%\"iG6#F+F+%#dtG!\"\"F1" }{TEXT -1 
181 " , where M is the mutual inductance. For a single coil the same p
henomenon occurs, since neighbouring turns of the coil experience thei
r mutual magnetic fluxes. The self-inductance " }{TEXT 304 1 "L" }
{TEXT -1 271 " - a property of the coil - serves as the proportionalit
y constant between current and self-induced  magnetic flux. An inducto
r serves as an energy-storing device, as it passing a current through \+
the coil requires to overcome a back-electromotoric force voltage give
n by " }{XPPEDIT 18 0 "V[b]=-L d i/dt" "6#/&%\"VG6#%\"bG,$**%\"LG\"\"
\"%\"dGF+%\"iGF+%#dtG!\"\"F/" }{TEXT -1 89 " . To overcome this back E
MF voltage we have to drive the solenoid with a forward voltage" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "V_L:=L*diff(i(t),t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 262 10 "RL circuit" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The differential equatio
n for the current follows from Kirchhoff's law for putting the two ele
ments in series and applying a voltage " }{TEXT 305 1 "U" }{TEXT -1 
31 " across the external terminals:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "KL_RL:=V_R+V_L=U;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 50 "The initial condition is that no current flows at " }{TEXT 263 
1 "t" }{TEXT -1 3 "=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "I
C:=i(0)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sol_RL:=dsolv
e(\{KL_RL,IC\},i(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "i_
RL:=factor(rhs(sol_RL));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Acros
s the resistor we have according to Ohm's law:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "U_R:=simplify(subs(i(t)=i_RL,V_R));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We can introduce the characteristi
c time constant for this circuit as " }{XPPEDIT 18 0 "tau=L/R" "6#/%$t
auG*&%\"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));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "For graphing we \+
now need to substitute only the scales for the axes, i.e., the time co
nstant 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_RL1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 75 "The voltage across the resistor is a measure of the curre
nt 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 b
uilding 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 \+
current 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);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 166 "Now we look at the solution as the voltage is suddenl
y disconnected and the circuit is shorted out at the terminals. We ass
ume that the steady-state current flows at " }}{PARA 0 "" 0 "" {TEXT 
323 1 "t" }{TEXT -1 43 "=5, where we terminated our previous graph." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "KL_RL0:=subs(U=0,KL_RL);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "IC0:=i(5)=U/R;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sol_RL0:=dsolve(\{KL_RL0,IC0
\},i(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We simplify the solu
tion by combining the exponentials:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "i_RL0:=combine(rhs(sol_RL0),exp);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 60 "The voltage drop across the resistor follows from
 Ohm's law:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "U_R0:=simpli
fy(subs(i(t)=i_RL0,V_R));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We e
xpress the latter using the time constant tau:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "U_R0:=simplify(subs(L=R*tau,U_R0));" }}}
{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);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "This
 graph shows the voltage across the resistor R (or the current " }
{TEXT 272 1 "i" }{TEXT -1 1 "(" }{TEXT 324 1 "t" }{TEXT -1 127 ")) tha
t would appear in the RL circuit as the terminals are connected to a s
ingle 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 asymptoti
c current value was reached). Note that for cases where the driving sq
uare-wave pulse has a shorter time constant than the RL circuit, one h
as to use the final current from the turn-on phase as a start value in
 the initial condition for the turn-off phase." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 319 "It is important to reali
ze that this is the result of the current continuing to flow in the sa
me direction as the external voltage is turned off (down-time of the s
quare-wave pulse). The voltage across the inductance L is the compleme
nt to the external voltage: for the turn-on part (red curve) it is the
 complement to " }{TEXT 274 1 "U" }{TEXT -1 128 ", for the turn-off pa
rt (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:=pl
ot(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=b
lue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(P_RL3,P_RL
4,title=\"voltage across L\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
181 "For the current to continue to flow in the same direction as the \+
down-time of the square-wave signal begins, the voltage across the coi
l has to jump to the opposite 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 bef
ore:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "KL_RC:=V_R+V_C=U;" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "This is an integral equation. \+
We are better off turning it into a differential equation by taking th
e derivative with respect to time:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "KL_RC:=diff(KL_RC,t);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 164 "In this form we have to supply an initial condition for \+
the current, and we have to consider different possibilities. 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 capac
itor is generated (the integral vanishes), and the current 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;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
83 "The solution to the equation for the current shows that it decreas
es exponentially:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol_RC
:=dsolve(\{KL_RC,IC1\},i(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "i_RC:=rhs(sol_RC);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The v
oltage across the resistor is found from Ohm's law as before:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "U_R:=simplify(subs(i(t)=i_RC
,V_R));" }}}{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));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We graph the result af
ter 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);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 379 "We can interpret this result as f
ollows: initially the uncharged capacitor poses no effective resistanc
e to the current; as the capacitor plates are charged up the effective
 resistance goes to infinity causing the voltage drop across R (which \+
is in series with C) to go to zero. For alternating currents (AC) capa
citors therefore act as frequency-dependent resistors (impedance)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The volta
ge across the charging capacitor is given by 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 abov
e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "Ag
ain, the time at which the voltage has fallen to half its value can be
 used to determine the time constant from a single reading." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "t_half:=solve(U_R=U/2,t);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 334 "Note that if we are interested \+
in an accurate determination of the time constant for either the RL or
 the RC circuit, we should record a time sequence of voltages across R
, and perform a fit to the data. An exponential fit is performed most \+
conveniently by taking the logarithm of the data and carrying out a li
near least squares fit." }}{PARA 0 "" 0 "" {TEXT -1 37 "This is carrie
d 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 n
ow disconnect the battery and short-circuit the open ends of the RC ci
rcuit, or alternatively consider a square-wave pulse connected at the \+
free terminals of the circuit for the moment where it switches from hi
gh to low. We expect the stored energy in C to be released and dissipa
ted in the resistance in analogy to the RL circuit case. There is, how
ever, a difference: the capacitor discharges by sending a current in t
he 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 the new situation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "KL_RC0:=lhs(KL_RC)=0;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "Assuming that the capacitor was fully cha
rged at " }{TEXT 329 1 "t" }{TEXT -1 38 "=5 we can set the initial cur
rent to -" }{TEXT 330 1 "U" }{TEXT -1 1 "/" }{TEXT 331 1 "R" }{TEXT 
-1 87 ", which corresponds to the full capacitor voltage discharging t
hrough the resistance R." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"IC0:=i(5)=-U/R;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sol_RC0
:=dsolve(\{KL_RC0,IC0\},i(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 
"We see that the current dies out exponentially with the characteristi
c time constant RC." }}{PARA 0 "" 0 "" {TEXT -1 28 "We combine the exp
onentials:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "i_RC0:=combin
e(rhs(sol_RC0),exp);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Ohm's law
 gives us the voltage drop across the resistor:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 37 "U_R0:=simplify(subs(i(t)=i_RC0,V_R));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "and we substitute the time constan
t 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));" }}}{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 "displ
ay(P_RC1,P_RC2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "The graph sh
ows that the current (and the voltage across the resistor) change dire
ctions at the switching times." }}{PARA 0 "" 0 "" {TEXT -1 165 "As wit
h the RL circuit, we can find the voltage across the interesting devic
e, i.e., the capacitor from Kirchhoff's law, using the fact that the t
erminal voltage is " }{TEXT 279 1 "U" }{TEXT -1 79 " for the first 5 t
ime 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,color=red):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P_RC4:=plot(0-subs(U=1,tau=1,U_R0),
t=5..10,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dis
play(P_RC3,P_RC4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 323 "Note that \+
we find the expected behaviour for the voltage at the capacitor: in th
e turn-on phase (red) it acquires charge on its plates, while during t
he turn-off phase it releases this charge. The sign of the voltage acr
oss the capacitor stays the same, but the current changes sign as one \+
goes from one phase to the other." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 271 "For the inductance L we had the opposite
 behaviour: the current kept its sign (the graph is identical to the v
oltage across the capacitor if the same time constant is chosen), but \+
the voltage across the inductance changed sign during the turn-off of \+
the external voltage." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 926 "This perfect symmetry between the two devices makes t
he LC circuit a very interesting object: the two distinct storage mech
anisms for electromagnetic energy (the capacitor builds up an electric
 field between its plates, while the solenoid builds up a magnetic fie
ld) allow for a perfect oscillating device. In an ideal setting (no re
sistance in the solenoid) an initial charge in the capacitor sets off \+
a current through L, thus building a magnetic field as it discharges. \+
Once it discharged, the inductance continues a current in the same dir
ection, thus charging the capacitor in the opposite sense. The analogy
 to a mechanical harmonic oscillator where a constant total energy osc
illates in form between kinetic and potential energy is perfect. In a \+
non-ideal world one cannot avoid dissipation, and thus we consider an \+
RLC circuit, which we show is governed by the differential equation fo
r the damped harmonic oscillator." }}{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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "KL_RLC:
=diff(KL_RLC,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The analogy o
f 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 inductance " 
}{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 " b
ecomes inverse capacitance 1/" }{TEXT 335 1 "C" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We have t
hree elements in series: the capacitor poses no resistance 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 be 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 differenti
al equation to solve, and thus we need a second condition. We can use \+
the known solution to the RL circuit, since the capacitance does not i
mpede 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_R
L,t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "IC:=i(0)=0,D(i)(
0)=i0prime;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sol_RLC:=dso
lve(\{KL_RLC,IC\},i(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "We re
cognize in the solution that the constant " }{TEXT 292 1 "R" }{TEXT 
-1 1 "/" }{TEXT 293 1 "L" }{TEXT -1 39 " is responsible for damping, a
nd 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 circ
uit is in the subcritical, critical, or overcritical damping regimes. \+
Note that for small values of " }{TEXT 296 1 "R" }{TEXT -1 195 " the d
iscriminant is negative, i.e., the 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 recognize that the natural circular fr
equency 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 shown 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)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "A
t t=0.5 microseconds we have te voltage (just to check that the curren
t is real-valued):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf
(subs(t=0.5*10^(-6),i_1));" }}}{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));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "We multiplied the solution by 100
 so that point-and-click interface can give us good read-outs from the
 graph. This is needed below to simulate what one does on an oscillosc
ope, i.e., to read off the attenuation of the signal from peak to peak
." }}{PARA 0 "" 0 "" {TEXT -1 71 "The reader should check the time con
stant against the parameter values:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 71 "According to the analogy we should have (in the case of small d
amping):" }}{PARA 0 "" 0 "" {TEXT -1 32 "harmonic oscillator: omega=sq
rt(" }{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)
));" }}}{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));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "This agrees with \+
the result found from the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 58 "For the attenuation constant we observe t
hat the solution " }{TEXT 19 7 "sol_RLC" }{TEXT -1 65 " given above ha
s 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:=eva
lf(subs(R=10,L=10^(-6),2*L/R));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
72 "Therefore the amplitude drops to half of the original value in the
 time:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(ln(2)*tau);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "It is, however easier in thi
s case to check whether within a period of oscillation (which happens \+
to agree with the attenuation time constant) the signal drops by a fac
tor of e (Euler's constant). Indeed it does, as the read-out lists:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "2.56/0.93=evalf(exp(1));" 
}}}{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 circuit (idealized \+
as the undamped oscillatory LC circuit) has many applications in analo
gue 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 the circu
it with a weak external signal. This is used for signal discrimination
: 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 that one i
s in the undercritical damping regime." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "102" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }