( ) ( ) ( ) ( ) ( ) 1 2. ELEC 201 Electric Circuit Analysis I Lecture 8(a) RL and RC Circuits: Single Switch 11/9/2017. Driven RL Circuit: Equation

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1 /9/7 Dren rcut: Equaton EE Electrc rcut Analyss I ecture 8(a) an rcuts: Sngle Swtch THE ITADE, THE MIITAY OEGE OF SOUTH AOINA All sles an content 7 Moultre courtesy Street, of harleston, Dr. Gregory S 949 J. Mazzaro s Intally, (a) the source s turne off, an (b) no energy s store, so (c) no current flows. s After the swtch changes, s V The nuctor has not yet ha tme to bul up current: A long tme after the swtch has been close,, ( ) V, t t just before t just after t (nuctor acts as a short crcut) Dren rcut: Equaton Dren rcut: Soluton ( ) V V t ( t) V Wrtng KV after the swtch changes, V V ( t) ( t) V V ( t ) ( t) homogeneous, frst-orer fferental equaton 3 Sole the fferental equaton by guess & check constant? snusoal? exponental ecay? exponental rse? t I e I I I ( ) I V I V 4

2 /9/7 Dren rcut: Soluton Dren rcut: Soluton V t ( t) V V t ( t) V V t V V t e ( e ) substtute nto the ff eq V V V V e e V ( e ) t tme constant of the crcut, a measure of how much tme s requre to charge/scharge 5 6 Example: Dren rcut Example: Dren rcut Plot the oltages s,, an for µs < t < µs. kω Plot the oltages s,, an for µs < t < µs. kω s V 5 mh ( t ) t I e V ( ) ( ) t kω 5 mh ( 5 ma)( ) I e I e I V kω 5 ma t e s 5 mh ( t ) t t e e 4 4t 5 5 ma kω V 7 8

3 /9/7 Example: Dren rcut Example: Dren rcut Plot the oltages s,, an for µs < t < µs. s kω V 5 mh s ( t) t < V t t < e V t t < t e V t ( t) 4t ( t) 4 s kω V 5 mh t t e 4t ( 5 mh) ( 5 ma)( ) t ( t) ( t) ( )( )( e ) e t 5 4 V t -e-6:e-7:e-6; _s ().* (t > ); _ (*exp(-4e4*t)).* (t > ); _ ( - *exp(-4e4*t)).* (t > ); plot(t/^-6,_s,'-',t/^-6,... _,'--',t/^-6,_,'-.') axs([-inf Inf - ]) legen('_s','_','_') ylabel('voltage (V)') xlabel('tme (\mus)') 9 Dren rcut: Equaton s s After the swtch changes, s V The capactor has not yet ha tme to bul up oltage: A long tme after the swtch has been close, Intally, (a) the source s turne off, an (b) no energy s store., ( ) V, t t just before t just after t (capactor acts as an open crcut) Dren rcut: Equaton Wrtng KV after the swtch changes, ( ) V V V V V ( t ) ( t ) homogeneous, frst-orer fferental equaton 3

4 /9/7 Dren rcut: Soluton Dren rcut: Soluton V ( t ) ( t ) ( ) V V ( t ) ( t ) ( ) V Sole the fferental equaton by guess & check t V e V V V ( ) V V V V 3 Substtute the exponental soluton nto the fferental equaton t t V ( V ) e ( V ) ( V ) e 4 t t V V e Dren rcut: Soluton rcut: PSpce V ( t ) ( t ) ( ) V Smulate ths crcut an plot the oltages an for < t < ms. t ms 4 kω V 5 nf ( t ) t V e tme constant of the crcut, a measure of how much tme s requre to charge/scharge Sw_tlose & Sw_tOpen, EVA lbrary 5 6 4

5 /9/7 Source-Free rcut Source-Free rcut t ( t) ( ) V energy store ntally no source rops to zero oer tme energy store ntally no source ( t ) ( t ) V ( ) rops to zero oer tme V e t V t V e t.368 V 7 8 Source-Free rcut: PSpce Source-Free rcut: PSpce Smulate ths crcut an plot the oltage an current for < t < µs. Assume that, at t, the oltage across the capactor s mv. 4 nf 6 kω Smulate ths crcut an plot the oltage an current for < t < µs. Assume that, at t, the oltage across the capactor s mv. 4 nf 6 kω IPINT part, SPEIA lbrary I ntal conon oltage for a capactor (n Volts) current for an nuctor (n Amps) 9 5

6 /9/7 Source-Free rcut: PSpce & rcut: General Soluton Smulate ths crcut an plot the oltage an current for < t < µs. Assume that, at t, the oltage across the capactor s mv. th 5 kω 4 nf µs 4 nf 6 kω The transent response (all oltages an currents) wthn a sngle-equalent-energy-storage-element crcut follow ths general form: crcut x t X e X X x X X x crcut th th th th t I e I t V e V t where th s the Theenn equalent resstance seen at the termnals of the energy storage element., General Soluton: Proceure () Ientfy the crcut as a transent or crcut. (e.g. not an crcut) () Assume that the crcut response ( or ) wll follow the general soluton: x t X e X 3 X x X X x (3) If necessary, etermne equalent nuctance ( eq ) or capactance ( eq ) by approprate seres/parallel combnatons. (4) Sole for the ntal nuctor current () or capactor oltage (). (5) Sole for the fnal nuctor current ( ) or capactor oltage ( ). (6) Determne the Theenn equalent resstance seen by the energy-storage element (nuctor or capactor), th. ompute / th or th. (7) Substtute results from steps 4, 5, 6 nto the general soluton (step ). (8) To sole for / that s not for the nuctor/capactor, use crcut analyss (KV, K, etc.) to sole for that partcular /. 6