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W12D1: RC and LR Circuits Reading Course Notes: Sections 7.7-7.8, 7.11.3, 11.4-11.6, 11.12.2, 11.13.4-11.13.5 1 Announcements Math Review Week 12 Tuesday 9pm-11 pm in 26-152 PS 9 due Week 13 Tuesday April 30 at 9 pm in boxes outside 32-082 or 26-152 2 Outline DC Circuits with Capacitors First Order Linear Differential Equations RC Circuits LR Circuits 3 DC Circuits with Capacitors 4 Sign Conventions - Capacitor Moving across a capacitor from the negatively to positively charged plate increases the electric potential DV = Vb - Va 5 Power - Capacitor Moving across a capacitor from the positive to negative plate decreases your potential. If current flows in that direction the capacitor absorbs power (stores charge) 2 Pabsorbed dQ Q d Q dU I V dt C dt 2C dt 6 RC Circuits 7 (Dis)Charging a Capacitor 1. When the direction of current flow is toward the positive plate of a capacitor, then dQ I=+ dt 2. When the direction of current flow is away from the positive plate of a capacitor, then dQ I=dt 8 Charging a Capacitor What happens when we close switch S at t = 0? 9 Charging a Capacitor Circulate clockwise Q å DVi = e - C - IR = 0 i dQ I=+ dt First order linear inhomogeneous differential equation dQ `1 =(Q - Ce ) dt RC 10 Energy Balance: Circuit Equation dQ I=+ dt Q e - - IR = 0 C dQ Multiplying by I = + dt 2 æ Q dQ d 1Q ö 2 2 eI = I R + = I R+ ç ÷ C dt dt è 2 C ø (power delivered by battery) = (power dissipated through resistor) + (power absorbed by the capacitor) 11 RC Circuit Charging: Solution ( dQ 1 =Q - Ce dt RC ) Solution to this equation when switch is closed at t = 0: dQ I(t) = + Þ I(t) = I0 e-t / t Q(t) = Ce (1- e ) dt t = RC : time constant (units: seconds) -t / t 12 Demonstration RC Time Constant Displayed with a Lightbulb (E10) http://tsgphysics.mit.edu/front/?page=demo.php&letnum=E%2010&show=0 13 Review Some Math: Exponential Decay 14 Math Review: Exponential Decay Consider function A where: A decays exponentially: dA 1 =- A dt t A(t) = A0 e -t t 15 Exponential Behavior Slightly modify diff. eq.: A “grows” to Af: dA 1 = - ( A - Af ) dt t A(t) = Af (1- e -t / t ) 16 Homework: Solve Differential Equation for Charging and Discharging RC Circuits 17 Concept Question: Current in RC Circuit 18 Concept Question: RC Circuit An uncharged capacitor is connected to a battery, resistor and switch. The switch is initially open but at t = 0 it is closed. A very long time after the switch is closed, the current in the circuit is 1. 2. 3. Nearly zero At a maximum and decreasing Nearly constant but non-zero 19 Concept Q. Answer: RC Circuit Answer: 1. After a long time the current is 0 Eventually the capacitor gets “completely charged” – the voltage increase provided by the battery is equal to the voltage drop across the capacitor. The voltage drop across the resistor at this point is 0 – no current is flowing. 20 Discharging A Capacitor At t = 0 charge on capacitor is Q0. What happens when we close switch S at t = 0? 21 Discharging a Capacitor Circulate clockwise Q å DVi = C - IR = 0 i dQ I=Þ dt First order linear differential equation dQ Q =dt RC 22 RC Circuit: Discharging dQ 1 -t / RC =Q Þ Q(t) = Qoe dt RC Solution to this equation when switch is closed at t = 0 with time constant t = RC Qo -t / t Qo -t / t dQ I=Þ I(t) = e = e dt t RC 23 Concept Questions: RC Circuit 24 Concept Question: RC Circuit Consider the circuit at right, with an initially uncharged capacitor and two identical resistors. At the instant the switch is closed: 1. I R = IC = 0 2. I R = e / 2R , I C = 0 3. I R = 0 , I C = e / R 4. I R = e / 2R , I C = e / R 25 Concept Question Answer: RC Circuit Answer: 3. I R = 0 IC = e R Initially there is no charge on the capacitor and hence no voltage drop across it – it looks like a short. Thus all current will flow through it rather than through the bottom resistor. So the circuit looks like: 26 Concept Q.: Current Thru Capacitor In the circuit at right the switch is closed at t = 0. At t = ∞ (long after) the current through the capacitor will be: 1. 2. 3. IC = 0 . IC = e R IC = e 2R 27 Con. Q. Ans.: Current Thru Capacitor Answer 1. IC = 0 After a long time the capacitor becomes “fully charged.” No more current flows into it. 28 Concept Q.: Current Thru Resistor In the circuit at right the switch is closed at t = 0. At t = ∞ (long after) the current through the lower resistor will be: 1. I R = 0 2. I R = e R 3. I = e 2R . R 29 Concept Q. Ans.: Current Thru Resistor Answer 3. Since the capacitor is “fullly charged” we can remove it from the circuit, and all that is left is the battery and two resistors. So the current is I R = e 2R . 30 Group Problem: RC Circuit For the circuit shown in the figure the currents through the two bottom branches as a function of time (switch closes at t = 0, opens at t = T>>RC). State the values of current (i) just after switch is closed at t = 0+ (ii) Just before switch is opened at t = T-, (iii) Just after switch is opened at t = T+ 31 Concept Q.: Open Switch in RC Circuit Now, after the switch has been closed for a very long time, it is opened. What happens to the current through the lower resistor? 1. 2. 3. 4. 5. 6. It stays the same Same magnitude, flips direction It is cut in half, same direction It is cut in half, flips direction It doubles, same direction It doubles, flips direction 32 Con. Q. Ans.: Open Switch in RC Circuit Answer: 1. It stays the same The capacitor has been charged to a potential of VC = e 2 so when it is responsible for pushing current through the lower resistor it pushes a current of I R = e 2R, in the same direction as before (its positive terminal is also on the left) 33 LR Circuits 34 Inductors in Circuits Inductor: Circuit element with self-inductance Ideally it has zero resistance Symbol: 35 Non-Static Fields dF B E × d s = ò dt E is no longer a static field 36 Kirchhoff’s Modified 2nd Rule dF B å D Vi = - ò E × d s = + d t i dF B Þ å D Vi =0 dt i If all inductance is ‘localized’ in inductors then our problems go away – we just have: dI å D Vi - L d t = 0 i 37 Ideal Inductor • BUT, EMF generated by an inductor is not a voltage drop across the inductor! ò dI e = -L dt E × ds = 0 ideal inductor Because resistance is 0, E must be 0! 38 Non-Ideal Inductors Non-Ideal (Real) Inductor: Not only L but also some R = In direction of current: e dI = - L - IR dt 39 Circuits: Applying Modified Kirchhoff’s (Really Just Faraday’s Law) 40 Sign Conventions - Inductor Moving across an inductor in the direction of current contributes dI L dt Moving across an inductor opposite the direction of current contributes e dI = +L dt 41 LR Circuit Circulate clockwise dI e - IR - L = 0 Þ dt First order linear inhomogeneous differential equation ö dI Ræ e =- çI - ÷ dt Lè Rø 42 RL Circuit ö dI Ræ e e -t / ( L / R) = - ç I - ÷ Þ I(t) = (1- e ) dt Lè Rø R Solution to this equation when switch is closed at t = 0: I(t) = e R (1- e -t / t ) L t = : time constant R (units: seconds) 43 RL Circuit t=0+: Current is trying to change. Inductor works as hard as it needs to to stop it t=∞: Current is steady. Inductor does nothing. 44 Group Problem: LR Circuit For the circuit shown in the figure the currents through the two bottom branches as a function of time (switch closes at t = 0, opens at t = T>>L/R). State the values of current (i) just after switch is closed at t = 0+ (ii) Just before switch is opened at t = T-, (iii) Just after switch is opened at t = T+ 45