Electricity and Magnetism-LC and LCR circuits

1P25Cllas 25: Outline Hour 1: Expt. 10: Part I: Measuring L LC Circuits Hour 2: Expt. 10: Part II: LRC Circuit 2P25Laas Time: Self Inductance 3 P25-Self Inductance L N I = Φ 1. Assume a current I is flowing in your device 2. Calculate the B field due to that I 3. Calculate the flux due to that B field 4. Calculate the self inductance (divide out I) dI L dt ε≡ − To Calculate: The Effect: Back EMF: Inductors hate change, like steady state They are the opposite of capacitors4 P25-LR 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. ( ) /( ) 1 t I t e R τ ε − = −5P25LL Circuit: AC Output Voltage 0.00 0.01 0.02 0.03 0.00 0.05 0.10 0.15 0 1 2 3 ) ) Current -3 -2 -1 0 1 2 3 VollI (ATime (sInductor (V)tmeter across L Output (V) Output Votage 6P25-Non-Ideal Inductors Non-Ideal (Real) Inductor: Not only L but also some R In direction of current: dIL IRdtε= −− = 7P25LL Circuit w/Real Inductor Due to Resistance 1. Time constant from I or V 2. Check inductor resistance from V just before switch 8P25Experrimen 10: Part I: Measure L, R STOP after you do Part I of Experiment 10 (through page E10-5) 9P25LL Circuits Mass on a Spring: Simple Harmonic Motion (Demonstration) 10 P25-Mass on a Spring 22 22 0 d x F kxmamdt d x m kx dt = − == + = 0 0 ( ) cos() x t xt ω φ = + (1) (2) (3) (4) What is Motion? 0 Angular frequency km ω= = Simple Harmonic Motion x0: Amplitude of Motion φ: Phase (time offset)11 P25-Mass on a Spring: Energy 0 0 ( ) cos() x t xt ω φ = + (1) Spring (2) Mass (3) Spring (4) Mass Energy has 2 parts: (Mass) Kinetic and (Spring) Potential 2 2 2 0 0 2 2 2 0 0 1 1 sin () 2 2 1 1 cos() 2 2 s dx K m kxt dt U kxkxt ω φ ω φ ⎛ ⎞ = = + ⎜ ⎟ ⎝ ⎠ = = + 0 0 0 '( )sin() x t xt ω ω φ = − + Energy sloshes back and forth12 P25-Simple Harmonic Motion Amplitude (x0) 0 0 ( ) cos() x t xt ω φ = − 1 Period () frequency () 2 angular frequency () T fπ ω == Phase Shift () 2π ϕ =13P25Electtroni Analog: LC Circuits 14 P25-Analog: LC Circuit Mass doesn’t like to accelerate Kinetic energy associated with motion Inductor doesn’t like to have current change Energy associated with current 2 2 2 1 ; 2 dv dx F ma mmEmv dt dt = = == 2 2 2 1 ; 2 dI dq L L E LI dt dt ε= − =−=15 P25-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 2 1 ; 2 F kx Ekx = − = 2 1 11 ; 2 q E q C C ε= = 1 ; ; ;; F x qvImLkC ε − → → →→→16P25LL Circuit resistor, and battery. 1. Set up the circuit above with capacitor, inductor, 2. Let the capacitor become fully charged. 3. Throw the switch from a to b 4. What happens? 17P25LL 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) 18P25PPR Questions: LC Circuit 19 P25-LC Circuit 0 ; Q dIdQ L I C dtdt − = =− 2 2 1 0 d Q Q dt LC + = 0 0 ( ) cos() Q t Qt ω φ = + 0 1LC ω = Q0: Amplitude of Charge Oscillation φ: Phase (time offset) Simple Harmonic Motion20 P25-LC Oscillations: Energy 2 2 2 0 1 2 2 2 E B Q Q U U ULI C C = + = += 2 2 2 0 0 cos 2 2 E Q Q U t C C ω ⎛ ⎞ = =⎜ ⎟ ⎝ ⎠ 2 2 2 22 0 0 0 0 1 1 sin sin 2 2 2 B Q U LILItt C ω ω ⎛ ⎞ = = =⎜ ⎟ ⎝ ⎠ Total energy is conserved !! Notice relative phases21P25Adddin Damping: RLC Circuits 22P25Dammpe LC Oscillations Resistor dissipates energy and system rings down over time Also, frequency decreases: 2 2 0 ' 2 R Lω ω ⎛ ⎞ = −⎜ ⎟⎝ ⎠ 23P25Experrimen 10: Part II: RLC Circuit Use Units 24P25PPR Questions: 2 Lab Questions