一階與二階RLC 電路分析

1 RLC (((( ))))

2 ------------ dxxfetxetxettTHxtt)(1)()(000 = : 0)0(;xxfxdtdxTH=+=+ tTHefxdtdx1/*=+THtttfexedtdxe 11=+THttfexedtd 1= tt0dxxfetxetxttTHxttt)(1)()(000 += 0)0().

3 ()()(xxtftaxtdtdx=+=+ te /* "" , ,ot dxxfetxetxttTHxttt)(1)()(000 += 0)0(;xxfxdtdxTH=+=+ dxeftxetxttxtTHtt +=00)()(0 = xtxteeedxeeftxetxttxtTHtt +=00)()(0 ttxtTHtteeftxetx00)()(0 += += 00)()(0tttTHtteeeftxetx() 0)()(0ttTHTH eftxftx +=0tt 021;)(0tteKKtxtt += 021;)(0tteKKtytt += K_1 K_2 2%2%2%2% XXXX CRTH= +vSRSabC+vc_ THCCTH vvdtdvCR=+ 0)0(,==CSSvVv tSSCeVVtv =)(CRTH= tet 1% dtdvCCSSCRvv 0 : aKCL= +SSCcRvvdtdvC 1.)

4 2. KCL KVL.. 3. : ,,21KK()12 FACT: WHEN ALL INDEPENDENT SOURCES ARE CONSTANTFOR ANY VARIABLE, ( ), IN THE CIRCUIT THESOLUTION IS OF THE FORM( ),Ot tOy ty tKK et t =+>=+>=+>=+> y 001)0(yyfyadtdya=+=+ y feKKaeKatt= ++ 210210110afKfKa= =012010aaeKaat= = + teKdtdy =2 >+= 0,)(21teKKtyt 21)0(KKy+=+ 12)0(KyK += 1 00101afydtdyaafyadtdya=+ =+ 1K 2)0( .0),( SVvttv=> 0 t )( KCL>tv0)()(=+ tdtdvCRVtvS2/)0( SVv= 1 fydtdy=+ 2 )( , t 0 0,)( 121 Kv(t)teKKtvt >>+= SVvdtdv= =0 SVK= 1)( fKfydtdy==+1 3 1221)0()0(0 KvKKKvt = +== fvK =)0(22/2/)0(2 SSVKVv = =0,)2/()( :> = teVVtvRCtSS sVtvtdtdvRC=+)()(R/*)0();(0,)(21121+=+ =>+= xKKxKteKKtxt 0),( >tti 0tKVL> +Rv +Lv)(tiKVL)()(tdtdiLtRivvVLRS+=+=0)0()0( )0( 0)0(0=+ += = <iiiit 1 RVtitdtdiRLS=+)()(RL= 2 RVKiS== 1)( 3 21)0(KKi+=+ = RLtSeRVti1)( : )0().

5 (0,)(21121+=+ =>+= xKKxKteKKtxt 12( ),0ti tKK et =+> 0 t KCL >)()(tiRtvIS+=)(tv =)()(tdtdiLtv)()(titdtdiRLIS+= 1 2 SSIKIi= = 1)( 3210)0(KKi+==+ = RLtSeIti1)( : 0)0( :=+i RL= 12( ),0ti tKK et =+> 0 t>2)()(Rtvti= VvVkkkvC4)0(4)12(633)0(=+ =+= 0)()(||;0)()()(2121=+==++PPRtvtdtdvCRRRR tvtdtdvCRtv ==kkkRP26|| )10100)(102(63= == 1 20)(1== Kv 3 VKVKKv44)0(221= =+=+0],[4)( >= tVetvt0],[34)( >= tmAetit 0,)(21>+= teKKtit 0 t<VtvtdtdvOO6)()( +][3)()( )(4)(2 Atitdtdititdtdi=+=+])[(2)(VtitvO=0),( >ttvO KVL(t>0))(ti 0 t KVL>0)()()(311=+++ 10,)(21>+= teKKtvtO 2: K1 VvtvtdtdvOOO6)(6)()( =+1)(KvO= VK61= t<0 )0(+Ov t<0 )0().

6 (0,)(21121+=+ =>+= xKKxKteKKtxt ==12||2 THR044121= + I1 IKVLKVL][4421 VIVVOCTH= ==][41AI=][34)0()0(AiiL=+= 0,)(21>+= teKKtvtO ][38)0(34)0(VviO=+ =+0,353)( > = tetitab0],[3106)( > = tVetvtO (t<0))( tiL 3106382221= ==+KKKKLi0t<0),( >ttvO C1R2R 0 tKCL>0)()(0)(2121=++ =++ )10100)(106()(6321= =+= 1)(31)(422)(tvtvtvCCO=+= 20,)(21>+= teKKtvtC 01=K t<0 +)0(CvV)12(96=][88)0(221 VKKKvC= +==+ 30],[8)( >= tVetvtC0],[38)( >= tVetvtO )( tvc )0();(0,)(121121+=+ =>+= iKKvKteKKtvCtC 0),( 1>tti 0 tKVL> =+0)(1811tidtdiLL0)()(9111=+titdtdi)0(); (0,)(12111211+=+ =>+= iKKiKteKKtit 1s91= 201=K t<0 )0(1 i AVi11212)0(1= = 3][1)0()0(22111 AKKKii= +=+= 0],[][)( :9911>== tAeAetitt )(1ti +Lv CircuitwithresistancesandsourcesInductor orCapacitorabRepresentation of an arbitrarycircuit with one storage element +VTHRTHI nductororCapacitorab +VTHRTHabC+vc_Case across capacitor +VTHRTHabLiLCase through inductor a KCLciRi0=+RciidtdvCiCc=THTHCRRvvi =0= +THTHCCR vvdtdvCTHCCTH vvdtdvCR=+ KVL +Rv +LvTHLRvvv=+LTHRiRv=dtdiLvLL=THLTHLviRdt diL=+==+ THTHLLTHR vidtdiRLSCi 6 6 6 6H3V24 +)(tiO0=t0>t;(t)iFindO THTHOOTHR vidtdiRL=+ 0.)

7 (21>+= teKKtitO 6 6 6 6V24 +0>tThevenin for t>0at inductor terminalsab=THv0=THR))66(||6(6++ == 0; >=+ ++ tteKKeK0;)( >= teKtitO =01K : 6 6 6 6V24 +)0()0(+= OOii0<t K1=0 0;)( >= teKtitO 0+ 6322=K0;632)( >= tetitO 6 6 6 6H3V24 +)(tiO0<tCircuit for t<01i2i3i0)(6)(6621311= + +iiiii0)(6)(6243212= + + iiii0)(6)(62313= + iiii3)0(iiC=+mA632)(0i:C=+ ).0( +Oi1v80624661111= = ++vvvv6624)0(1viO+=+ +-0=tk6k6k6k6F 100V12)(tiO0t(t),iFindO> +Cv6kvi,0tCO=> v_c THCCTH vvdtdvCR=++-0>tk6k6k6k6V12)(tiOab +THvkkkRTH36||6== *100*10*363= = v=+CCCvdtdv += ++ tteKKeK61=K v_c(0+) +-0<tk6k6k6k6V12)(tiOcircuit in steady statebefore the switching +)0(CvVvC6)0(= VvC6)0(=+)0(21+=+CvKK0621= =KK >=0;6)(tVtvC0;16)(>==tmAkvtiCO : : : : KCLKCLKCLKCLRiLiCi0=+++ CLRS iiii)();()(1;)(00tdtdvCitidxxvLiRtviCLtt LR=+== SLttitdtdvCtidxxvLRv=+++ )()()(100 dtdiLvdtdvRdtvdCS=++122 : : : : KVL KVL KVL KVL +Rv +Cv0=+++ LCRS vvvv)();()(1.)

8 00tdtdiLvtvdxxiCvRivLCttCR=+== dtdvCidtdiRdtidLS=++22 SCttvtdtdiLtvdxxiCRi=+++ )()()(100 )( )( titv > <=000)(tIttiSSSiRLC RLC RLC RLC dtdiLvdtdvRdtvdCS=++1220;0)(>=ttdtdiS0122=++LvdtdvRdtvdC +Sv > <=000)(ttVtvSSRLC RLC RLC RLC dtdvCidtdiRdtidLS=++220;0)(>=ttdtdvS022=++CidtdiRdtidL )()()()(2122tftxatdtdxatdtxd=++ )()()( :cpcpxxtxtxtx+=0)()()(2122=++txatdtdxatd txdccc ffff((((tttt) ) ) ) )(2aAxAtfp= =AxadtxddtdxaAxpppp= == =22220 : )()()( 2txaAtxAtfc+== 0)(4)(2)(22=++txtdtdxtdtxd 0)(16)(8)(4 , ,22=++txtdtdxtdtxd 1 1 1 10)(4)(2)(22=++txtdtdxtdtxd2n n 20422=++ss 2= n 0)()()(2122=++txatdtdxatdtxd0)()(2)(222= ++txtdtdxtdtxdnn 21122222aaaaannn= == = n0222=++nnss 0)()(2)(222=++txtdtdxtdtxdnn 02 )(22=++=nnstssKetx ssss ststKesdtxdsKetdtdx222.

9 ( :== stnnnnKesstxtdtdxtdtxd)2()()(2)(22222 ++=++0222=++nnss )( 1 :1 > tstseKeKtx2121)(+=)( 1 :2 < dnnjsjs = =21()tAtAetxddt sincos)(21+= tjttjstdndneeee ==)( : tjteddtjd sincos =)( 1 :3 = ns =()tnetBBtx +=21)()022()02( :22=+=++nnnstssste tstseKeKtx2121)(+= =d *12 )(KKtx= []tjddeKtxjsKK)(1*12Re2)( + = ==2/)( 211jAAK+= 10)()(22222222 = == ++ nnnnnnnnsss = 0)(4)(4)(22=++txtdtdxtdtxd0442=++ss 0)2(04422=+ =++sss242= =nn 142= = n3) ( tstetBBtxetBBtx22121)()()()( +=+= 0)(16)(8)(422=++txtdtdxtdtxd0)(4)(2)(22= ++txtdtdxtdtxd 242= =nn = n2) (.)

10 12= = === ndn()()tAtAetxtAtAetxtddt3sin3cos)(sinco s)(2121+=+= 3103)1(4222jssss = =++=++ d FCHLRRLC2,2,1:== = 0122=++LvdtdvRdtvdC022=++CidtdiRdtidL042 10222222=++=++vdtdvdtvdvdtdvdtvd0163)41( 41222=++=++sss2141;21= == nn += tAtAetvtc43sin43cos)(214434112112= = = ndFFFCHLRRLC2,1, ,1;2== = 0222=++Cidtdidtid L:CCnn= == 22;1C= C= C= C= C= C= C= C= C= C= C= C= CCCC 41= C44 = Atxtdtdxtdtxdnn=++)()(2)(222 tstsneKeKAtx21212)(++= 212)0(KKAxn+= + 2211)0(KsKsdtdx+=+()tAtAeAtxddtnn sincos)(212++= 12)0(AAxn= + 21)0(AAdtdxdn + =+()tnnetBBAtx ++=212)(12)0(BAxn= + 21)0(BBdtdxn+ =+ FCHLR51,5,2== =VvAiCL4)0(,1)0(= = =+++tLdtdvCidxxvLRv00)0()(101122=++ ++ss ;1== ) ( = = )( += )0();0(++dtdvvVvvvCC4)0()0()0(==+=++=0t KCL AT0)0()0()0(=+++++dtdvCiRvLC5)5/1()1()5/ 1(24)0( = =+dtdv2; = =+KKKKKK0.