L13- First Order 1

# E 9 31812 r vc 0 vo ic vc vs vc vs c

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Unformatted text preview: r non‐homogeneous D.E. 9 3/18/12 R vc (0 + ) = vo ic vc Vs vc (! ) = Vs C ! = RC D.E. SoluCon: dvc 1 V + vc = s dt ! ! t ! 0+ t ! vc (t ) = Vs + ( vo ! Vs )! e &quot; ! ! ! t ! vc (t ) = ( final ) + ( initial ! final )! e &quot; ! t ! 0+ Note e = 2.71828 ExponenCal funcCon e! x Example: Natural Logarithm x=0 e!0 = 1 x =1 e!1 = 0.37 x=2 e!2 = 0.14 x=5 e!5 = 0.0067 &quot; 0 10 3/18/12 !! vc (t ) = vo ! e 1 &quot;! t # !! ! = RC vc (!Volt ) t (! s ) IniCal value Final value !! vc (0 ) = vo !! vc (! ) = 0 Note: signal reaches ﬁnal value for: t ! 5&quot; Example: t=0 5! 1! iL 20! 100V 10 mH 4! Switch was closed for long Cme Find : iL (t )!!!!! for !!!! t ! 0 + 11 3/18/12 Switch is closed !!!! t = 0 ! Before the acCon ! 1! a 5! iL 20! 100V 10 mH b 4! Steady state a 1! Vab = 100 20 / /5 = 4 ! 100V 4 = 80V 5 iL ( 0 ! ) = ! 80 = !1.6 A 5 b Switch is open !!!! t = 0 + Just aTer the acCon ! 5! 1! iL 100V 20! 10 mH != iL (0 ! ) = !1.6 A iL (0 + ) = !1.6 A != 4! L R 10 m = 2.5 ms 4 12 3/18/12 Switch is open !!!! t = ! ATer long Cme ! 5! 1! iL 20! 10 mH iL (0 + ) = !1.6 A 100V iL ( ! ) = 0 4! iL ( ! ) = 0 Summary t=0 5! 1! ! = 2.5 ms iL 20! 100V SoluCon: Note! 10 mH iL (t ) = !1.6 ! e ! t 2.5 m 4! !!! A ! t ! iL (t ) = ( final ) + ( initial ! final )! e &quot; ! 13 3/18/12 iL t = 5! t !1.6 A 14...
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