EE201 Lecture 18 P 1 Source free and step response Example Find iL t and vL t for t 0 if iL 0 10A and the switch closes at t 0 4s 0 4s vL 5 20 8H Step1 Compute RTH for t 0 4s RTH 5 Step 2 Evaluate iL t for t 0 4s 4 8H vL t R L t t0 iL t e TH iL t0 EE201 Lecture 18 P 2 Where t0 0 4 s But iL 0 4 is unknown Step 3 Evaluate iL t for 0 t 0 4 s RTH 20 20 8 t 0 iL t e iL 0 iL t 10 e 2 5t A At t 0 4 s iL 0 4 10e 1 3 68 A Step 4 Put solution for iL 0 4 into equation for iL t t 0 4 s 4 8 t t0 iL t e iL 0 4 iL t 3 68 e 0 5 t 0 4 A EE201 Lecture 18 P 3 Step 5 Compute vL t from Ohm s Law When 0 t 0 4 s vL t RTH iL t vL t 20 10 e 2 5t 200 e 2 5t V When t 0 4 s vL t RTH iL t vL t 4 3 68 e 0 5 t 0 4 vL t 14 72 e 0 5 t 0 4 V Unit step functions 1 t 0 u t 0 t 0 1 u t t0 0 t t0 t t0 Examples v t 10 u t V 10V if t 0 v t if t 0 0 iL t 3 68 e 0 5 t 0 4 A iL t 3 68 e 0 5 t 0 4 u t 0 4 A EE201 Lecture 18 P 4 Step Response Add a constant independent source to RL or RC circuit iC t iL t R L vS RL Circuit vL t R vS C vC t RC Circuit Step response considers circuits with constant current or voltage sources Derivation of differential equation for RL circuit iL t response vL t L d iL t dt From KVL vL t vs R iL t EE201 Lecture 18 d iL t dt R iL t L P 5 vS L 18 1 Derivation of differential equation for RC circuit vc t response ic t C d vc t dt From Ohm s Law ic t d vC t dt vs t vc t R vC t RC vS RC 18 2 EE201 Lecture 18 P 6 General Differential Eqn for 18 1 and 18 2 d x t 1 x t F 18 3 dt F a forcing function constant RL RC L RTH RTHC Solution form for general differential equation x t F x t0 F exp t t0 Interpretation of solution is obtained by examining each term see next page General strategy reduce complicated circuit attached to inductor or capacitor to its Thevenin equivalent attach inductor or capacitor to external terminals of network and use equations derived on Page 7 EE201 Lecture 18 P 7 RL Circuit Case iL t vs L L RTH iL t0 vs L L RTH exp RTH L t t0 lim t iL t vs L L RTH F final value iL t final iL initial iL final iL exp RTH L t t0 iL t iL iL t0 iL exp RTH L t t0 RC Circuit Case vc t Vs RTHC RTHC vc t0 vs RTHC RTHC exp t t0 RTHC lim t vc t vs RTHC RTHC F final value vc t final vc initial vc final vc exp t t0 RTHC vc t vc vc t0 vc exp t t0 RTHC
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