PHY132_L22 - Classical Physics II Classical Physics II...

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lassical Physics II Classical Physics II PHY132 Lecture 22 AC Circuits II Lecture 22 1 03/22/2010
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Series E LR Circuit Initially, switch S is open; at time t= 0 the switch is closed Kirchhoff’s Rule #2 in this circuit: S i ( t ) 0 di Li R dt  E R L E t d t 0 di Vi R L dt  0 1 Ld i R d t  () it Ldi L di 0 VRi t x t L 0 td 00 ) t LR VV t e  0 0 R 0 0 / R VR i 0 VR R x   0 0 ln R  0 1 t V e ) t di t Ve  Equivalently: the Voltage across the resistor: ( ) R V it R t R R   R i 0 L L d Vt t 0 1 t • the quantity L/R has units ·s/ = time • and is the “ Characteristic Time Constant of the LR circuit… V R ( t ) V 0 Lecture 22 2 t L/R 03/22/2010
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Energy and Power In the preceding circuit: E = V B = 100 V, R = 4 , L = 4 mH – Calculate current, the power stored in the S i ( t ) inductor, the power dissipated in the resistor, and the power delivered by the battery: • immediately after S is closed, at t= 0 : R L E t R 00 0 0; 0 A B t di VL i R i dt  (0 ) 0 k W R Pt  2 Li 0 ) 0 k W ; dU di t L i kW i V 0 () 1 LR V it e R •A t t = L/R (= 1.0 ms) : i 2 L U 0 L L t dt dt  0 0kW BB Pi 0 36.8V; L R L di V e dt  0 BLR BL VVV VVi R  63.2V 15.8A iR 2 (/ ) 999W RR PtLR iV iR U i Lecture 22 3 03/22/2010 2 1 ) 2 C UtLR L i 581 36.8V 15.8A W L L dU di PL i dt dt 1.58kW V
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Series LCR Circuit Quantative discussion: – Apply Kirchhoff #2 Note that i= –dq/dt : if q ( t ) de creases, i ( t )>0 , as sketched i ( t ) S a b R L E C 0 C di VL R i dt  damping term oscillating 2 2 0 qd q LR Cd t d t  2 1 0 dq q t dt C  Trial solution : –Then : 0 () cos d t qt qe t term ONLY correct for “underdamped ” case 2 dt q 0 sin d t t 2 2 t d q dt 0 2 sin d t q t 0 sin d t t 2 0 cos d t t  0 2 2 sin 2 d d t q q et q – Note the appearance of sin ω t and cos ω t terms; must cancel independently 2 eq’ns: dt d d d t q dd 2 n R R Lecture 22 4 cos : t 03/22/2010 d L R 0 sin : t 2 d  2 d 2 R R L 0 22 2 1 42 RR L CL L 2 2 2 1 4 R LC L 2 0 2 1 d 0 d 1 C 2 2 2 4 R L L    
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