EE201 Lecture 15 P 1 Inductors A new circuit element i t B A A time varying current through the conductor induces a voltage vAB t between ends of wire vAB t L di t dt 1 Where the proportionality constant L is the inductance of the wire Symbol iL t L vl t EE201 Lecture 15 Units Henry H P 2 1H 1 Volt sec amp Inductors are energy storage elements Example Calculate vL t when iL t sin 20t 3 i L t Using Eq 1 vL t L diL t dt 0 5H vL t EE201 Lecture 15 P 3 d sin 20t 3 vL t 0 5 dt vL t 10 cos 20t 3 The integral relation is t 1 v d iL t L L Where iL 0 1 iL t L t0 t vL d vL d t0 Memory 2 EE201 Lecture 15 P 4 Memory initial current flowing through inductor before initial time of interest Example Find iout t given the current profile supplied by the input source shown iout t is t 0 5H vL1 0 25H 2vL1 is t Current supplied by current source 1 1 1 2 3 t s EE201 Lecture 15 P 5 Step1 Find voltage generated by dependent source dis t vL1 t 0 5 dt dis t 2 vL1 Slope of is t curve dt 2vL1 1 1 1 2 3 t sec EE201 Lecture 15 P 6 Step 2 Find iout t Using integral relationship iout t iout 0 1 L 0 0 t 2vL1 d Area under vL1 t curve iL t 2 4 6 8 1 2 3 4 t sec EE201 Lecture 15 P 7 Continuity Property of Inductors The current through an inductor is continuous even if the voltage across the inductor is discontinuous provided the voltage is not pulsed Discontinuous vL 1 vL 1 vL t iL t 1 1H vL t 1 2 3 t sec 1 t Continuous iL t iL t for all t iL t 1 vL d iL t L t0 1 2 EE201 Lecture 15 P 8 Power and Energy Instantaneous power absorbed by inductor pL t iL t vL t L iL t diL t dt Energy stored over an interval t0 t1 is WL t0 t1 t1 t p d 0 t1 diL WL t0 t1 L iL d d t0 iL t1 iLdiL WL t0 t1 L iL t0 EE201 Lecture 15 1 2 WL t0 t1 LiL t1 2 1 P 9 LiL2 t0 3 2 An instantaneous energy may be defined as 1 4 WL t L iL2 t 2 where t0 in Eq 3 is t0 and iL 0 Inductors in Series L1 iin t vin t v3 v2 v1 Leq vin v1 v2 v3 diin diin diin L2 L3 dt dt L1 dt diin L3 vin L1 L2 L3 dt Therefore Leq L1 L2 L3 L2 In series inductors add to form equivalent inductance EE201 Lecture 15 P 10 Inductors in Parallel iin t iL1 iL2 L1 L2 vin t Leq What is Leq L1 vin t L1 diL1 dt 1 L1 diL1 t dt vin t L2 vin t L2 diL2 dt 1 L2 diL2 t dt vin t EE201 Lecture 15 Equivalent circuit is iin t Leq vin t Where vin t Leq diin t dt 5 From KCL iin t iL1 t iL2 t diin t diL1 t diL2 t dt dt dt vin t vin t vin t L1 Leq L2 P 11 EE201 Lecture 15 1 1 L1 L2 1 Leq Leq 1 1 L1 P 12 1 L1 L2 L1 L2 L2 Inductors in parallel behave like resistors in parallel Current Division Formula two inductors in parallel iL 0 iin t iL1 t iL2 t EE201 Lecture 15 1 t vin d iL1 t L1 From Eq 5 t L L 1 1 2 iL1 t L1 L L 1 2 iL1 t L2 L1 L2 iin t P 13 diin d 1 L1 1 1 L2 L1 In general for n parallel inductors 1 Lj iin t iLj 1 n L n d iin t EE201 Lecture 15 P 14 Example Find the equivalent inductance 26mH 21mH 85mH Leq 15mH 18mH Leq 18mH 21mH 126mH 0 018 0 021 0 126 0 021 0 126 1 0 018 1 0 021 1 0 126 Leq 0 0356 H 36mH
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