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Transcript

RC-1 RC Circuits An RC circuit is a circuit with a resistor (R) and a capacitor (C). RC circuits are used to construct timers and filters. Example 1. Very simple RC circuit: a capacitor C, charged to an initial voltage V0 = Q0/C, attached to a resistor R with a switch. switch +Q0 C C R –Q0 Q V V0 Q0 C Close the switch at time t = 0 , so current I starts to flow. The charged capacitor is acting like a battery: it produces a voltage difference across the resistor which drives the current through the resistor: C At t = 0+, +Q I –Q R I Vacross C = Vacross R , VC = VR , dQ dt I0 V0 . R (– sign because Q is decreasing) Q Q dQ IR , R , C C dt dQ 1 Q dt RC RC = "time constant" = , has units of time is a differential equation of the form dx a x , where a is a constant. dt This equation says: (rate of charge of x) x exponential solution : x x0 exp a t d x 0 ea t dx a x 0 ea t a x . Check: dt dt a > 0 exponential growth , The solution to dQ 1 Q dt RC It works! a < 0 exponential decay is t t Q(t) Q0 exp C V0 exp RC Notice that at t = 0, the formula gives Q = Q0 . Last update: 4/30/2017 Dubson Phys1120 Notes, University of Colorado RC-2 In time , Q falls by a factor of exp(–1) = 1/e 0.37.. In time 2, Q falls by a factor of exp(–2) = (1/e)(1/e) 0.14.. Q approaches zero asymptotically, and so does V and I Q t t Q(t) Q 0 exp Q 0 exp RC Q0 Q0/e t I V=Q/C V dQ 0 exp( t / ) dt R t t Example 2: More complex RC circuit: Charging a capacitor with a battery. switch E C VC = Q / C Let's use symbol E for battery voltage (E short for emf) because there are so many other V's in this example. Before switch is closed, I = 0, Q = 0. R VR = I R Close switch at t = 0. Always true that E = VC + VR , by Kirchhoff's Voltage Law (Loop Law) The charge Q on the capacitor and the voltage VC = Q / C across the capacitor cannot change instantly, since it takes time for Q to build up, so .. At t = 0+ , Q = 0 , VC = 0, E = VC + VR = VR = I R I0 = E / R Although Q on the capacitor cannot change instantly, the current I = dQ/dt can change instantly. Last update: 4/30/2017 Dubson Phys1120 Notes, University of Colorado R2 RC-3 "Current through a capacitor" means dQ/dt . Even though there is no charge ever passing between the plates of the capacitor, there is a current + I going into one plate and the same current is coming out - I + of the other plate, so it is as if there is a current passing + through the capacitor. + C E E = VR VC Q C E = IR I R E = , I dQ dt dQ Q R dt C Qualitatively, as t , Q , VC = Q/C , VR , I = VR / R . After a long time , t >> = RC , the current decreases to zero: I = 0, VC = E , Q = C E Vc = Q / C Analytic solution: E VC (t) E [1 exp(t / RC) ] Q(t) E C [1 exp( t / RC) ] t Things to remember: Uncharged capacitor acts like a "short" ( a wire ) since VC = Q / C = 0. After a long time, when the capacitor is fully charged, it acts like an "open-circuit" ( a break the wire). We must have IC = 0 eventually, otherwise Q , VC . Last update: 4/30/2017 Dubson Phys1120 Notes, University of Colorado