ELEC 201 Electric Circuit Analysis I Lecture 9(a) RLC Circuits: Introduction

Transcription

1 //6 All le courey of Dr. Gregory J. Mazzaro EE Elecrc rcu Analy I ecure 9(a) rcu: Inroucon THE ITADE, THE MIITAY OEGE OF SOUTH AOINA 7 Moulre Sree, harleon, S 949 V Sere rcu: Analog Dcoery _ 5 Ω pf eq 56µH 56µH 56µH 68µH eq nf nf nf nf nf pf 68 µh orange, whe rpe eq 5 Ω kω kω W 5 Ω 68 µh pf gn Sere rcu: Analog Dcoery Sere rcu: Analog Dcoery V _ 5 Ω pf 68 µh V _ 5 Ω pf 68 µh ke an or crcu, he olage ar a a ele alue an re/fall o a ele alue. Ung a quare wae o mmc a ep funcon a u V Unlke an or crcu, he olage oerhoo an rng before ele..e. For an or crcu, / oe no follow a mple exponenal re/fall. 4

2 //6 Sere rcu: PSpce All le courey of Dr. Gregory J. Mazzaro mulae meaure EE Elecrc rcu Analy I ecure 9(b) rcu: Equaon & General Soluon 5 THE ITADE, THE MIITAY OEGE OF SOUTH AOINA 7 Moulre Sree, harleon, S 949 Sere rcu: Equaon Sere rcu: Equaon V KV aroun he ere loop: 7 V τ V τ ( ) V ake / of he enre equaon V τ ( ) 8 rearrange erm & e by V τ α ω where α, ω

3 //6 Parallel rcu: Equaon Parallel rcu: Equaon K a he op noe: τ I τ ( ) rearrange erm & e by I τ I I τ ( ) 9 ake / of he enre equaon where α, ω α ω Sere & Parallel : Soluon Sere & Parallel : Soluon ere: α, ω ( ) α ( ) ω ( ) α, ω ( ) α ( ) ω ( ) x x α ω x x X e X e X ere: α, ω α, ω oluon o he general fferenal equaon : α α ω general fferenal equaon for () or () : α α ω x x α ω x x X e X e X ( ) x X X X x ( ) X X ere n your exbook x ( ) X α ampng coeffcen, mlar o a me conan, hgher α qucker ecay ω reonan frequency, expree n ra/, hgher ω faer ocllaon

4 //6 Soluon: Gue & heck Soluon: Gue & heck The propoe oluon o The propoe oluon o V h general form: V e V e V α ± α ω, V h general form: V e V e V α ± α ω, an he oluon afy he bounary (nal, fnal) conon? ( ) V V V V Vo V V V Three equaon, hree unknown. Ye. I he oluon al for all me? ( ) V e Ve V Ve Ve ( ) V e Ve V V e Ve ( ) ( ) V e V e V V e V e V 4 Soluon: Gue & heck Soluon: Gue & heck The propoe oluon o The propoe oluon o V h general form: V e V e V α ± α ω, V h general form: V e V e V α ± α ω, I he oluon al for all me? I he oluon al for all me? V e V e V e Ve Ve Ve Ve Ve Ye, f for all an, 4( )( ) b ± b 4ac ± a ± 4 ± α α ω, α, ω ± 5 6 4

5 //6 All le an conen courey of Dr. Gregory J. Mazzaro Soluon: Oer-ampe x X e X e X α ± α ω, EE Elecrc rcu Analy I ecure 9(c) rcu: lae of Soluon THE ITADE, THE MIITAY OEGE OF SOUTH AOINA 7 Moulre Sree, harleon, S 949 The oluon a o be oerampe f α > ω n whch cae he oluon c c ere: x X e X e X 8 > > c α α ω c α α ω Oerampe rcu Example Oerampe rcu Example Smulae he followng parallel crcu for a lea one full pule pero: I pule wh a pero of T 6 m, 6 µf,.6 mh, Ω IPUSE par, SOUE lbrary Smulae he followng parallel crcu for a lea one full pule pero: I pule wh a pero of T 6 m, 6 µf,.6 mh, Ω IPUSE par, SOUE lbrary For a parallel crcu, For a parallel crcu, α, ω α, ω ra α ( ) α > ω, oerampe ra ω 6 6 (.6 )( 6 ) 9 5

6 //6 Soluon: Uner-ampe Unerampe rcu Example x X e X e X α ± α ω The oluon a o be unerampe f α < ω ere:, < < Smulae he followng parallel crcu for a lea one full pule pero: I pule wh a pero of T 6 m, 6 µf,.6 mh, Ω For a parallel crcu, α, ω IPUSE par, SOUE lbrary n whch cae he oluon co( ω ) n ( ω ) x e X X X ω ω α ere n your exbook Unerampe rcu Example Soluon: rcally Dampe Smulae he followng parallel crcu for a lea one full pule pero: I pule wh a pero of T 6 m, 6 µf,.6 mh, Ω For a parallel crcu, α, ω IPUSE par, SOUE lbrary x X e X e X α ± α ω The oluon a o be crcally ampe f α ω, ere: ra α 6 6 ( ) ra ω 6 6 (.6 )( 6 ) α < ω, unerampe n whch cae he oluon x e X X X 4 ere n your exbook 6

7 //6 rcally Dampe rcu Example rcally Dampe rcu Example Smulae he followng parallel crcu for a lea one full pule pero: I pule wh a pero of T 6 m, 6 µf,.6 mh, 5 Ω IPUSE par, SOUE lbrary Smulae he followng parallel crcu for a lea one full pule pero: I pule wh a pero of T 6 m, 6 µf,.6 mh, 5 Ω IPUSE par, SOUE lbrary For a parallel crcu, For a parallel crcu, α, ω α, ω ra α ( ) α ω, crcally ampe ra ω 6 6 (.6 )( 6 ) 5 6 rcu: Poble epone rcu: Poble epone α, ω α, ω α > ω α ω α < ω OVEDAMPED ITIAY DAMPED UNDEDAMPED c c x X e X e X x e X X X X co ( ω ) x( ) e X X n ( ω ) α, ω α, ω α > ω α ω α < ω OVEDAMPED ITIAY DAMPED UNDEDAMPED c c x X e X e X x e X X X X co ( ω ) x( ) e X X n ( ω ) c α α ω c α α ω ω ω α 7 8 7