{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 32 "RCL-vihtovirtapiiri: reso nanssi " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Olkoon tarkastelun kohteena tavallinen RLC-vaihtovirtapiiri. Piiri ss\344 on kolme komponenttia, " }{TEXT 258 1 "R" }{TEXT -1 15 " ohmin \+ vastus, " }{TEXT 259 1 "L" }{TEXT -1 23 " henryn induktanssi ja " } {TEXT 260 1 "C" }{TEXT -1 23 " faradin kapasitanssi.\n" }}{PARA 256 " " 0 "" {BITMAP 154 108 108 1 "?TMgcBB:nH>?NM^GF:C:;::JB :::::::::?cB?cB:::::;b:?cB?;>>?[\\::::Z:>:bB::::ZZN^ZN^\\bBJ;N:?cB:::: B:Z\\:;Z:::::?CBbBJ;>:<:::Z\\N^\\N>B:Z\\:?Z\\:::ZZBBD\\\\:?J;N^\\:::>> N>B:Z\\:?Z\\::::;b:N>JJ<:D<::JJD>N>B:Z\\:?Z\\:::::::::JJJKZ::D::JJDD<::>^\\JKZ:JKD\\\\:?Z\\::::BB?JK:<^\\bB?;;ZZZ\\:::;cBDLKZ:>:bBJ;bB::::B:;:DZ:NZ:N>;bBB:;JKbBJ ;bB::::B:;:DB:; Z\\bBJ;bB::::B:;:D:?BBbB<:?cB::JJDDB:;:D>N>B:Z\\:?Z\\:::::::::JJJKZ::DZ\\JKB:;:D<:bB>:D\\:JKbBN:D L;bB:::::::::BBbB<:bBN:DL;bB:::::::::J;bB<:bBN:DL;bB:::::::::J;bB<:bBN :DL;bB:::::::::J;bB<:bBN:DL;bB:::::::::J;bB<:bBN:DL;bB:::::::::J;bB<:b BN:DL;bB:::::::::;JKN:?cBZZJKD\\:>:bBN:DL;b B::::::::JJJK:?K;N^\\B:<:DZ:N>?J:D<;B:;bB;;?JKDL;bB::::::::Z:N >;BBbB>:;bB;Z\\N:DL;bB::::::::JJJK:>:;BB?Z\\N:?cB?Z\\:::::::::;;?C :?K:J:>:DLKDL;bB?Z\\:::::::::ZZZ\\J:>Z\\>>N:?cB?Z\\:::::::::>:?cB? J:D<;B:;bB;;?JKDL;bB:::::::::J;>Z\\J:>Z\\:N^\\N:D<:::::::::N:;bB>:;Z\\ N^\\Z\\N:D<:::::::::N:D\\::D?JKD\\:B:bBN:DL;bB::::::::Z:N>;BBN^\\B:;:D>N^:N >D\\:>:bBN:DL;bB::::::::JJJKZZJKD\\:>: bBN:DL;bB:::::::::;JKN:?cB:?Z\\::::J:DZ:N >D:::JJ<:D<<:bBJ;bB:::::::JJJKZ::D^\\JKZ::DB:Z\\:?Z\\::::BB?Z\\::::JJD^\\JK::::;;?;<:bBJ;bB:::::::JJJKZ::D>N>B:<:D>N>B:Z\\:?Z\\::: ::::::JJJKZ::D>N>B:Z\\:?Z\\::::J;> >DLK::::;;?;<:bBJ;bB::::N:;[\\N>:::JJJKZ::D>N>B:Z \\:?Z\\::::J;>>DLK::::;;?;<:bBJ;bB::::N:;[\\N>:::JJJKZ::D>N>B:Z\\:?Z\\::::J;>>DLK::::;;?;<:bBJ;bB::::N:;[\\N>:::JJJKZ::D >N>B:Z\\:?Z\\::::J;>>DLK::::;;?;<:bBJ;bB::::N:;[\\ N>:::JJJKZ::D>N>B:Z\\:;Z:::::J;>^:>Z:::::bBN>B:Z\\ ::::::?JJD\\\\N>:::::<:bB:::::J;>>DLK:::::B:Z\\::::::?JJbB?;::::Z::D<: ::::N:;[\\N>:::::<:bB:::::J;>>DLK:::::B:Z\\::::::?JJbB?;::::Z::D<::::: N:;[\\N>:::::<:bB:::::J;>>DLK:::::B:Z\\::::::?JJbB?;::::Z::D<:::::N:;[ \\N>:::::<:bB:::::J;>>DLK:::::B:Z\\::::::?JJbB?;::::Z:5:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Piir iin sy\366tettyyn j\344nnitteeseen kohdistuu kolme eri pudotusta, \n" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "E[L] = L*diff(I(t),t);" "6#/&%\"EG6 #%\"LG*&F'\"\"\"-%%diffG6$-%\"IG6#%\"tGF0F)" }{TEXT -1 15 " k\344 \344min yli," }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "E[R] = R*I(t);" "6#/& %\"EG6#%\"RG*&F'\"\"\"-%\"IG6#%\"tGF)" }{TEXT -1 23 " vastuksen yli , sek\344" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "E[C] = 1/C;" "6#/&%\"EG6 #%\"CG*&\"\"\"F)F'!\"\"" }{XPPEDIT 18 0 "int(I(t),t);" "6#-%$intG6$-% \"IG6#%\"tGF)" }{TEXT -1 24 " kondensaattorin yli." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "N\344iden summasta syn tyy piirin Kirchhoffin lain mukainen s\344hk\366motorinen voima E(" } {TEXT 256 1 "t" }{TEXT -1 65 "). Mik\344li piiriin sy\366tetty j\344nn ite on sinimuotoista vaihtovirtaa " }{XPPEDIT 18 0 "E(t) = E[0]*sin(om ega*t);" "6#/-%\"EG6#%\"tG*&&F%6#\"\"!\"\"\"-%$sinG6#*&%&omegaGF,F'F,F ," }{TEXT -1 25 ", saadaan piirille yht\344l\366" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "E(t) = L*diff(I(t),t) +R*I(t)+1/C;" "6#/-%\"EG6#%\"tG,(*&%\"LG\"\"\"-%%diffG6$-%\"IG6#F'F'F+ F+*&%\"RGF+-F06#F'F+F+*&F+F+%\"CG!\"\"F+" }{XPPEDIT 18 0 "int(I(t),t) \+ = E[0]*sin(omega*t);" "6#/-%$intG6$-%\"IG6#%\"tGF**&&%\"EG6#\"\"!\"\" \"-%$sinG6#*&%&omegaGF0F*F0F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Integraalitermist\344 p \344\344st\344\344n eroon derivoimalla yht\344l\366 ajan suhteen: \n" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "L*diff(I(t),`$`(t,2))+R*diff(I(t),t )+I(t)/C = E[0]*omega*cos(omega*t);" "6#/,(*&%\"LG\"\"\"-%%diffG6$-%\" IG6#%\"tG-%\"$G6$F.\"\"#F'F'*&%\"RGF'-F)6$-F,6#F.F.F'F'*&-F,6#F.F'%\"C G!\"\"F'*(&%\"EG6#\"\"!F'%&omegaGF'-%$cosG6#*&FCF'F.F'F'" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 117 "T\344m\344 on differentiaaliyht \344l\366 pakotetulle v\344r\344htelylle, jossa ulkoinen pakottava j \344nnite on sinimuotoinen amplitudina " }{XPPEDIT 18 0 "E[0];" "6#&% \"EG6#\"\"!" }{TEXT -1 8 ". \n\nJos " }{TEXT 261 1 "E" }{TEXT -1 1 "( " }{TEXT 262 1 "t" }{TEXT -1 112 ")=0, kyseess\344 vapaa v\344r\344hte lypiiri ja differentiaaliyht\344l\366 on homogeeninen. \n\nJos piiriss \344 ei ole vastusta, ts. " }{TEXT 263 1 "R" }{TEXT -1 150 " = 0, piir i on vaimentamaton. Vastuksen olemassaolo merkitsee, ett\344 piiriss \344 on vaimennnus. \n\nTarkastellaan differentialiyht\344l\366n ratka isuja vakioiden " }{TEXT 264 1 "L" }{TEXT -1 2 ", " }{TEXT 265 1 "R" } {TEXT -1 2 ", " }{TEXT 266 1 "C" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "E[0] ;" "6#&%\"EG6#\"\"!" }{TEXT -1 4 " ja " }{XPPEDIT 18 0 "omega;" "6#%&o megaG" }{TEXT -1 14 " eri arvoilla." }}{PARA 0 "" 0 "" {TEXT -1 73 "\n Aluksi h\344vitet\344\344n mahdollisista aiemmista laskuista j\344\344 neet muuttujat: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Virtaa kuvaava toisen kertaluvun yht\344l\366." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "yhtalo:= L*diff(i(t), t$2)+R*diff(i(t), t)+1/C*i(t)=omega*E[0]*cos(omega*t);" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 20 "Vaimentamaton tapaus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Vaimentamattoma ssa tapauksessa piiriss\344 ei ole vastusta ja yht\344l\366 on" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "yhtalo1:= subs(R=0, yhtalo);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Alkuehtona olkoon, ett\344 piir iss\344 ei tapahdu mit\344\344n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "alkuehto:= i(0)=0, D(i)(0)=0 ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "T\344m\344n ratkaisu on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "dsolve(\{yhtalo1, alkuehto\} , i(t)):\nratkaisu1:= subs(%, i(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Ratkaisu muodostuu kahdesta kos initermist\344: " }{XPPEDIT 18 0 "cos(t/sqrt(L*C));" "6#-%$cosG6#*&%\" tG\"\"\"-%%sqrtG6#*&%\"LGF(%\"CGF(!\"\"" }{TEXT -1 55 " on per\344isin homogeeniyht\344l\366n yleisest\344 ratkaisusta ja " }{XPPEDIT 18 0 " cos(t*omega);" "6#-%$cosG6#*&%\"tG\"\"\"%&omegaGF(" }{TEXT -1 238 " ep \344homogeeniyht\344l\366n yksitt\344isratkaisusta. Edellinen kuvaa pi irin sis\344ist\344 v\344r\344htely\344, j\344lkimm\344inen ulkoisen j \344nnitteen taajuudella tapahtuvaa v\344r\344htely\344. \n\nRatkaisu \+ t\344ss\344 muodossa ei kuitenkaan ole p\344tev\344, jos nimitt\344j \344 tulee nollaksi: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " restaajuus:= solve(-1+omega^2*L*C=0, omega);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Nollaksi tuloa vastaava a pakotteen taajuutta kutsutaan " }{TEXT 267 21 "resonanssitaajuudeksi " }{TEXT -1 170 ". T\344ll\366in piirin sis\344inen v\344r\344htelytaa juus ja pakotteen taajuus ovat samat. Kyseess\344 on tilanne, miss\344 ep\344homogeenisen yht\344l\366n yksitt\344isratkaisu saakin erilaise n muodon: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "yhtalo2:= subs(omega=sqrt(C*L)/(C*L), yhtalo1);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "dsolve(\{yhtalo2, alkuehto \}, i(t)):\nratkaisu2:= subs(%,i(t)):\nsimplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "T\344m\344 on t ulkittavissa v\344r\344htelytermiksi, jossa virran amplitudi " } {XPPEDIT 18 0 "E[0];" "6#&%\"EG6#\"\"!" }{TEXT 269 2 "t " }{TEXT -1 5 "/ (2 " }{TEXT 270 1 "L" }{TEXT -1 1 ")" }{TEXT 271 1 " " }{TEXT -1 25 "kasvaa rajatta muuttujan " }{TEXT 272 1 "t" }{TEXT -1 65 " mukana. K\344yt\344nn\366ss\344 t\344m\344 johtaisi piirin palamiseen. \n\nAr voilla " }{TEXT 273 9 "L = 0.1 H" }{TEXT -1 2 ", " }{TEXT 274 1 "C" } {TEXT -1 9 " = 0.001 " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 3 "F, \+ " }{XPPEDIT 18 0 "E[0];" "6#&%\"EG6#\"\"!" }{TEXT -1 56 " = 230 V saad aan resonanssitaajuudella seuraava kuvio: \n" }{TEXT 268 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "subs(\{L=0.1, C=0.001, E[0]= 230\}, ratkaisu2);\nplot(%, t = 0..1.5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Resonanssitaajuus on" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "subs(\{L=0.1, C=0.001\}, restaajuus[1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Jos pakotteen taajuu s on l\344hell\344 resonanssitaajuutta, saadaan erikoinen v\344r\344ht ely:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(\{L=0.1, C=0.001, E[0]=230, omega=90\}, ratkaisu 1);\nplot(%, t = 0..1.5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 357 "T\344ss\344 pakote aluksi vahvistaa piir iss\344 kulkevaa virtaa kuten resonanssitaajuuden tapauksessakin, mutt a koska taajuudet eiv\344t olekaan t\344sm\344lleen samat, v\344r\344h telyt siirtyv\344t v\344hitellen vastakkaisiin vaiheisiin, ja pakote a lkaa sammuttaa piirin virtaa. \n\nSyntynyt v\344r\344htely voidaan my \366s ymm\344rt\344\344 siniv\344r\344htelyksi, jolla on sinimuotoinen vaihteleva amplitudi: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "(cos(100*t)-cos(90*t))-(-2*sin(5*t)*sin(95*t)):\nsimplify(%);\n" } }}{EXCHG {PARA 3 "" 0 "" {TEXT -1 18 "Vaimennettu tapaus" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Yleisen vaimennetu n tapauksen ratkaisu on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "yhtalo;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 86 "dsolve(yhtalo, i(t)):\nsubs(%, i(t)):\nratkaisu:= c ollect(combine(%), \{_C1, _C2, exp\});\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Ratkaisu koostuu kahdesta termist\344:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sisainen:= select(has, ratkaisu, _C1)+select(has , ratkaisu, _C2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ulkoin en:= subs(\{_C1=0, _C2=0\}, ratkaisu);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Sis\344isen v\344r\344hte lyn termi esitt\344\344 vaimenevaa v\344r\344htely\344, sill\344 muutt ujan " }{TEXT 275 1 "t" }{TEXT -1 162 " kerroin kummassakin eksponenti ssa on joko negatiivinen reaaliluku tai kompleksiluku, jonka reaaliosa on negatiivinen. N\344m\344 termit siis kuolevat v\344hitellen pois. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "Ulko isen v\344r\344htelyn termi voidaan muokata seuraavasti. Pyrit\344\344 n saattamaan se muotoon, jossa on vain yksi sinifunktio kertoimena sop iva " }{TEXT 276 9 "amplitudi" }{TEXT -1 1 " " }{TEXT 277 1 "A" } {TEXT -1 24 " ja argumentissa sopiva " }{TEXT 278 11 "vaihesiirto" } {TEXT -1 1 " " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "uusimuoto:= expand(A*sin(omega*t+delta));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "Vaatimalla ett\344 \+ sini- ja kosinitermien kertoimet vanhassa ja uudessa esitysmuodossa ov at samat, saadaan ehdot, joista pyrit\344\344n ratkaisemaan " }{TEXT 279 1 "A" }{TEXT -1 4 " ja " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "ehdot:= zip((x, y)->x=y, map2(coeff, ulkoinen, \+ [sin(omega*t), cos(omega*t)]), map2(coeff, uusimuoto, [sin(omega*t), c os(omega*t)])):\nsimplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "solve(\{op(ehdot)\}, \{A, delta\}):\nratk:= allvalues(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Pa kotteen aiheuttama virta on siis sinimuotoinen amplitudina" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "amp litudi:= subs(ratk[1], A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 100 "Amplitudi riippuu pakotteen taajuudesta. Pyrit\344\344n m\344\344ritt\344m\344\344n maksimiamplitudi ja vastaa va taajuus:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "diff(amplitudi, omega):\nderivaatta:= simplify(%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "maksimikohta:= solve(de rivaatta=0, omega);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "subs (omega=sqrt(C*L)/(C*L), amplitudi);\nmaksimiamplitudi:= simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 "Maple on varovainen yhdistelless\344\344n neli\366juuria, ja aina kaan ohjelman versio 6 ei suostunut sievent\344m\344\344n lauseketta p idemm\344lle. Lauseke on kuitenkin yksinkertainen ja k\344sin sievent \344m\344ll\344 p\344\344st\344\344n tulokseen " }{XPPEDIT 18 0 "E[0]/ R;" "6#*&&%\"EG6#\"\"!\"\"\"%\"RG!\"\"" }{TEXT -1 49 ". Maksimiarvo sa adaan siis resonanssitaajuudella." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 92 "Piirret\344\344n pakotteen (ulkoisen j \344nnitteen) ja resonanssitaajuutta vastaavan virran kuvaajat:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "numulkoinen:= subs(\{L=0.1, C=0.0001, R=5, E[0]=230\}, ulkoinen) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "jannite:= subs(E[0]=23 0, E[0]*sin(omega*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "r es:= subs(\{L=0.1, C=0.0001, R=5, E[0]=230\},1/sqrt(L*C));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot(subs(omega=res, \{jannite, num ulkoinen\}),t=0..0.1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "Kuviossa on j\344nnite punaisella ja virt a vihre\344ll\344." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Vastaava kuvio, kun kyseess\344 ei ole resonanssitaajuus: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(subs(omega=280, \{jannite, numulkoinen\}), t=0.. 0.1);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "Teht\344vi\344" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 374 "Tutki, miten a lkuehtojen muuttaminen vaikuttaa ratkaisuun vaimentamattomassa tapauks essa. Miten ratkaisun luonne muuttuu, kun ollaan resonanssikohdan l \344hell\344? \n\nPiirr\344 kuvaaja, joka esitt\344\344 vaimennetun ta pauksen virtaa pakotteen taajuuden funktiona. Onko vaimennetun piirin \+ tapauksessa sellainen ratkaisu mahdollinen, jossa virran amplitudi raj atta kasvaa, ts. piiri palaa? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 257 24 "JP & SKK & MS 12.07.2001" }{TEXT -1 0 "" }}}}{MARK "0 2 2" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }