{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "List \+ Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Subitem" 14 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 270 0 }} {SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 42 "Unit 21 -- Applica tion: Electrical Circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu /~meade/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "Industrial Mathematics I nstitute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 258 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" " " }}{PARA 259 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "ht tp://www.sc.edu/" "" }}{PARA 260 "" 0 "" {TEXT -1 19 "Columbia, SC 292 08\n" }}{PARA 262 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://ww w.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 263 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 38 "Copyright \251 2001 b y Douglas B. Meade" }}{PARA 265 "" 0 "" {TEXT -1 19 "All rights reserv ed" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 67 "-------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline of Unit 21" } }{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "21.A" 1 "" "21.A" }{TEXT -1 17 " Electric Circuit" }}{PARA 270 "" 0 "" {HYPERLNK 17 "21.A-1" 1 "" "21 .A-1" }{TEXT -1 32 " Model for a General RLC Circuit" }}{PARA 270 "" 0 "" {HYPERLNK 17 "21.A-2" 1 "" "21.A-2" }{TEXT -1 37 " Solution for t he General RLC Circuit" }}{PARA 270 "" 0 "" {HYPERLNK 17 "21.A-3" 1 " " "21.A-3" }{TEXT -1 28 " Special Case #1: LC Circuit" }}{PARA 270 "" 0 "" {HYPERLNK 17 "21.A-4" 1 "" "21.A-4" }{TEXT -1 28 " Special Case # 2: RC Circuit" }}{PARA 270 "" 0 "" {HYPERLNK 17 "21.A-5" 1 "" "21.A-5 " }{TEXT -1 28 " Special Case #3: RL Circuit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initia lization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 0 {PARA 3 "" 0 "21.A" {TEXT -1 23 "21.A Electrical Circuit" }} {SECT 0 {PARA 4 "" 0 "21.A-1" {TEXT -1 38 "21.A-1 Model for a General \+ RLC Circuit" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Consider an RLC ser ies circuit with resistance " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 19 " (ohm), inductance " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 26 " ( henry), and capacitance " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 40 " \+ (farad). Denote the electric charge by " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 89 " (coulomb). The current in the circuit is the instantane ous rate of change of the charge:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "charge := i(t) = diff( q(t), t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "The unit for current is ampere. One of Kirchof f's Laws states that the sum of the instantaneous voltage drops (chang es in potential) around a closed circuit must be zero. That is," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "KirchoffLaw := E[R] + E[L] + E[C] - E[emf] = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "E[R]" "6#&%\"EG6#%\"RG" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "E[L]" "6#&%\"EG6#%\"LG" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "E[C]" "6#&%\"EG6#%\"CG" }{TEXT -1 87 " are the voltage \+ drops across the resistor, inductor, and capacitor, respectively, and \+ " }{XPPEDIT 18 0 "E[emf]" "6#&%\"EG6#%$emfG" }{TEXT -1 65 " is the vol tage drop produced by an attached electromotive force." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "According to Ohm's \+ Law, the voltage drop across a resistor is proportional to the current with constant of proportionality " }{XPPEDIT 18 0 "R" "6#%\"RG" } {TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "E[R] := R*i(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Faraday's La w states that the voltage drop across the inductor is proportional to \+ the instantaneous rate of chage of the current, with " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 33 " as the proportionality constant:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E[L] := L*diff(i(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "The voltage drop across a capacitor is proportional to the electric charge on the capacitor, \+ with constant of proportionality " }{XPPEDIT 18 0 "1/C" "6#*&\"\"\"F$% \"CG!\"\"" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "E[C] := 1/C * q(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "21.A-2" {TEXT -1 43 "21.A-2 Solution for the General RLC Circuit" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "When the modeling assumptions for the po tential drop across each component in the circuit are inserted into Ki rchoff's Law, the resulting ODE is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "KirchoffLaw;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "To put this into the form of a second-order ODE for the charge " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "odeQ := eval( KirchoffLaw, \{charge,E[emf]=e(t)\} ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The corresponding initial conditions are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "icQ := q(0)=q0, D(q)(0)=i0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The solution to this generic \+ IVP is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "solQ := dsolve( \{ odeQ, icQ \}, q(t) );" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 0:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The \+ current can be found by differentiation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solI := eval( charge , solQ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "The same expressions for the charge and \+ current can be derived as the solution to the first-order system of OD Es formed by Kirchoff's Law and the constitutive relation between curr ent and charge. The appropriate first-order IVP is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sysQI := od eQ, charge;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "icQI := q(0) =q0, i(0)=i0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "with solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "solQI := ds olve( \{ sysQI, icQI \}, \{ q(t), i(t) \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "That the two forms of the solutions are the same is a little difficult to veri fy with Maple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "simplify( eval( q(t), solQ ) - eval( q(t), sol QI ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "collect( simplify( eval( i(t), solI ) - eval( i(t), solQI ) ), \{R,C,L\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "As partial v erification of the equivalence of the solutions, note that the charge \+ obtained from the first-order system satisfies the appropriate second- order ODE:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "odetest( op(select( has, solQI, q(t) )), odeQ );" } }}}{SECT 0 {PARA 4 "" 0 "21.A-3" {TEXT -1 34 "21.A-3 Special Case #1: \+ LC Circuit" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "If the circuit does \+ not have a resistor, " }{XPPEDIT 18 0 "R=0" "6#/%\"RG\"\"!" }{TEXT -1 34 ", then the solutions \"simplify\" to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "LCsolQI := simplify( eval( solQI, R=0 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 " 0" "6#\"\"!" }{TEXT -1 273 " imply that the exponentials in this probl em all involve imaginary exponents, the real-valued solutions will inv olve sine and cosine. In particular, there is no transient solution fo r an LC circuit. Given that the second-order ODE for the charge in an \+ LC circuit reduces to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eval( odeQ, R=0 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "which is equivalent to an undamped oscillator, this is not surp rising." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "21.A-4" {TEXT -1 34 "21.A- 4 Special Case #2: RC Circuit" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "I f the circuit does not have an inductor, " }{XPPEDIT 18 0 "L=0" "6#/% \"LG\"\"!" }{TEXT -1 65 ", it is not possible to obtain the solutions \+ by simply inserting " }{XPPEDIT 18 0 "L=0" "6#/%\"LG\"\"!" }{TEXT -1 45 " into the general solution to the RLC circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify( e val( solQI, L=0 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The problem is that when " } {XPPEDIT 18 0 "L=0" "6#/%\"LG\"\"!" }{TEXT -1 53 " the ODE for the cha rge is no longer of second order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "RCodeQ := eval( odeQ, L=0 ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "RCsolQ := expand( dsolve( \{ RCodeQ, q(0)=q0 \}, q(t) ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "RCsolI := i(t) = expand( diff( rhs(RCsolQ), t ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Here, because " }{XPPEDIT 18 0 "C*R" "6#*&%\"CG\"\"\"%\"RGF%" } {TEXT -1 3 " > " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT -1 31 ", the deno minators all tend to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT -1 4 " as " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " -> " } {XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 349 ". This means t hat the initial charge in the system has no bearing on the long-time b ehavior of the solution. If the exponentially growing term in the inte grand for the particular solution survives in a form that exactly canc els the exponential growth in the denominator, then the steady-state s olution for the charge will be nontrivial. For example," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 57 "collect( value( eval( \{ RCsolQ, RCsolI \}, \+ e=1 ) ), exp );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "21.A-5" {TEXT -1 34 "21.A-5 Special Case #3: RL \+ Circuit" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "If the circuit does not have a capacitor, " }{XPPEDIT 18 0 "1/C = 0" "6#/*&\"\"\"F%%\"CG!\"\" \"\"!" }{TEXT -1 120 ", it is possible to obtain the charge on the cir cuit by simply taking a limit of the general solution to the RLC circu it" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "RLsolQ := simplify( map( limit, solQ, C=infinity ), s ymbolic );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "This substitution works because the ODE \+ for the charge is still of second-order, with two distinct eigenvalues :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "RLodeQ := limit( odeQ, C=infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Note that while the formal substitution " }{XPPEDIT 18 0 "C=inf inity" "6#/%\"CG%)infinityG" }{TEXT -1 20 " yields the same ODE" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eval( odeQ, C=infinity );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "this substitution cann ot be used directly on the solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eval( solQ, C=infinity ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "but the charge in an LR circuit can be obtained as \+ the limit of the generic solution for an RLC circuit as " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 4 " -> " }{XPPEDIT 18 0 "infinity" "6#%)i nfinityG" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "simplify( map( limit, solQ, C=infin ity ), symbolic );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The corresponding current is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "RLsolI := i(t) = expand( diff( rhs(RLsolQ), t ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "Careful inspection of the expression for the charge in the cir cuit verifies that this solution is obtained from a second-order ODE w ith eigenvalues " }{XPPEDIT 18 0 "lambda=0" "6#/%'lambdaG\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "lambda=-R/L" "6#/%'lambdaG,$*&%\"RG \"\"\"%\"LG!\"\"F*" }{TEXT -1 147 ". The presence of a zero eigenvalue implies that the initial charge and current in the circuit impact the solution for all time. Moreover, because " }{XPPEDIT 18 0 "R/L" "6#*& %\"RG\"\"\"%\"LG!\"\"" }{TEXT -1 143 " > 0, only the portion of the ge neral solution to the homogeneous term corresponding to the negative e xponential is guaranteed to be transient." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Powertool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }