1 W13D1 LC and Undriven RLC Circuits Today’s Reading Course Notes: Sections 11.7-11.9, 11.10, 11.13.6Announcements No Math Review April 29 Problem Set 9 due April 29 9 pm (just four problems) 23 Outline Simple Harmonic Oscillator LC Circuits Undriven RLC Circuits Transformers4 Mass on a Spring: Simple Harmonic Motion5 Demonstration Mass on a Spring: Simple Harmonic Motion Mass on a Spring (C 2) http://scripts.mit.edu/~tsg/www/demo.php?letnum=C%202&show=06 Mass on a Spring F = −kx = ma = md2xdt2md2xdt2+ kx = 0 x(t) = x0cos(ω0t +φ)(1) (2) (3) (4) What is Motion? ω0=km= Angular frequencySimple Harmonic Motion x0: Amplitude of Motion f: Phase (time offset)7 Simple Harmonic Motion Amplitude (x0) x(t) = x0cos(ω0t +φ) Period =1frequency → T =1fPeriod =2πangular frequency → T =2πω Phase Shift (φ) =−π2Concept Question: Simple Harmonic Oscillator Which of the following functions x(t) has a second derivative which is proportional to the negative of the function x(t) = Acos2πTt⎛⎝⎜⎞⎠⎟ x(t) = Ae− t /T x(t) = Aet /T x(t) =12at222?d xxdt∝ −1. 2. 3. 4.9 Consider two ways of writing a solution to the equation Show that: Group Problem: Phase and Amplitude C = Acos(φ) and D = − Asin(φ) A = C2+ D2 and tanφ= − D / C (a) x(t) = C cos(ω0t) + D sin(ω0t) d2xdt2+ω0x = 0 (b) x(t) = Acos(ω0t +φ)10 Mass on a Spring: Energy x(t) = x0cos(ω0t +φ)(1) Spring (2) Mass (3) Spring (4) Mass Energy has 2 parts: (Mass) Kinetic and (Spring) Potential K =12mdxdt⎛⎝⎜⎞⎠⎟2=12kx02sin2(ω0t +φ)Us=12kx2=12kx02cos2(ω0t +φ) dxdt= vx(t) = −ω0x0sin(ω0t +φ)Energy sloshes back and forth11 Analog: RLC Circuit Inductors are like masses (have inertia) Capacitors are like springs (store/release energy) Batteries supply external force (EMF) Charge on capacitor is like position, Current is like velocity – watch them resonate12 Analog: LC Circuit Mass doesn’t like to change velocity Kinetic energy associated with motion Inductor doesn’t like to have current change Energy associated with current F = ma = mdvdt= md2xdt2; K =12mv2 ε= − LdIdt= − Ld2qdt2; UB=12LI2 F →ε; v → I; m → L13 Analog: LC Circuit Spring doesn’t like to be compressed/extended Potential energy associated with compression Capacitor doesn’t like to be charged (+ or -) Energy associated with stored charge F = −kx; E =12kx2 ε=QC; UE=12Q2C F →ε; x → Q; v → I; m → L; k → C−114 LC Circuit 1. Set up the circuit above with capacitor, inductor, resistor, and battery. 2. Let the capacitor become fully charged. 3. Throw the switch from a to b. 4. What happens? 15 LC Circuit It undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass). Equivalently: trade-off between energy stored in electric field and energy stored in magnetic field.16 Energy stored in electric field Energy stored in magnetic field Energy stored in electric field Energy stored in magnetic field17 Concept Question: LC Circuit Consider the LC circuit at right. At the time shown the current has its maximum value. At this time: 1. the charge on the capacitor has its maximum value. 2. the magnetic field is zero. 3. the electric field has its maximum value. 4. the charge on the capacitor is zero.18 Concept Question: LC Circuit In the LC circuit at right the current is in the direction shown and the charges on the capacitor have the signs shown. At this time, 1. I is increasing and Q is increasing. 2. I is increasing and Q is decreasing. 3. I is decreasing and Q is increasing. 4. I is decreasing and Q is decreasing.19 LC Circuit: Simple Harmonic Oscillator d2Qdt2+1LCQ = 0 ⇒ Q(t) = Q0cos(ω0t +φ) ω0= 1 / LCCharge: Angular frequency: Amplitude of charge oscillation: Phase (time offset): QC− LdIdt= 0 ; I = −dQdtφSimple harmonic oscillator: Q020 LC Oscillations: Energy U = UE+ UB=Q22C+12LI2=Q022C UE=Q22C=Q022C⎛⎝⎜⎞⎠⎟cos2ω0t UB=12LI2=12LI02sin2ω0t =Q022C⎛⎝⎜⎞⎠⎟sin2ω0tTotal energy is conserved !! Notice relative phasesLC Circuit Oscillation SummaryGroup Problem: LC Circuit Consider the circuit shown in the figure below. Suppose the switch that has been connected to point a for a long time is suddenly thrown to b at t = 0. Find the following quantities: (a) the frequency of oscillation of the LC circuit. (b) the maximum charge that appears on the capacitor. (c) the maximum current in the inductor. (d) the total energy the circuit possesses at any time t. 2223 Adding Damping: RLC Circuits24 Demonstration Undriven RLC Circuits (Y 190)RLC Circuit: Energy Changes QCdQdt+ I2R + LIdIdt= 0 ⇒Include finite resistance: Multiply by ddtQ22C+12L I2⎡⎣⎢⎤⎦⎥= − I2R ddt(UE+ UB)= − I2R QC+ I R + LdIdt= 0 I =dQdt⇒Decrease in stored energy is equal to Joule heating in resistor26 Damped LC Oscillations Resistor dissipates energy and system rings down over time. Also, frequency decreases: Q(t) = Q0e− ( R / 2 L)tcos(ω't) ω' =ω02− (R / 2L)2 ω0= 1 / LC > R / 2L27 AC Circuits and Transformers28 Alternating-Current Circuit • sinusoidal voltage source V (t) = V0sinωt ω= 2πf : angular frequencyV0: voltage amplitude• direct current (dc) – current flows one way (battery) • alternating current (ac) – current oscillates29 Transformer εp= NpdΦdt;εs= NsdΦdtNs > Np: step-up transformer Ns < Np: step-down transformer Equal flux through each turn: εsεp=NsNp30 Demonstrations: 26-152 One Turn Secondary: Nail H10 26-152 Many Turn Secondary: Jacob’s Ladder H11 26-152 Variable Turns Around a Primary Coil H9 http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H 10&show=0 http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H 11&show=0 http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H 9&show=031 1. House=Left, Line=Right 2. Line=Left, House=Right 3. I don’t know Concept Question: Residential Transformer If the transformer in the can looks like the picture, how is it connected?32 Transmission of Electric Power Power loss can be greatly reduced if transmitted at high voltage33 Electrical Power Power is change in energy per unit time So power to move current through circuit elements: P =ddtU =ddtqΔV( )=dqdtΔVVIP Δ=34 Power - Resistor Moving across a resistor in the
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