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Self Inductance To Calculate: L = NΦ I 1. 2. 3. 4. Assume a current I is flowing in your device Calculate the B field due to that I Calculate the flux due to that B field Calculate the self inductance (divide out I) The Effect: Back EMF: ε dI ≡ −L dt Inductors hate change, like steady state They are the opposite of capacitors P25- 3 LR Circuit I (t ) = ε R − t /τ 1 − e ( ) 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. P25- 4 LR Circuit: AC Output Voltage Output Voltage Output (V) 3 2 1 Inductor (V) 0 3 Voltmeter across L 2 1 0 -1 -2 -3 Current I (A) 0.15 0.10 0.05 0.00 0.00 0.01 0.02 Time (s) 0.03 P25- 5 Non-Ideal Inductors Non-Ideal (Real) Inductor: Not only L but also some R = In direction of current: ε dI = − L − IR dt P25- 6 LR Circuit w/ Real Inductor Due to Resistance 1. Time constant from I or V 2. Check inductor resistance from V just before switch P25- 7 Mass on a Spring (1) (3) (2) What is Motion? d 2x F = − kx = ma = m 2 dt 2 d x m 2 + kx = 0 dt (4) Simple Harmonic Motion x(t ) = x0 cos(ω 0t + φ ) x0: Amplitude of Motion ω0 = φ: Phase (time offset) k = Angular frequency m P25- 10 Mass on a Spring: Energy (1) Spring (2) Mass x(t ) = x0 cos(ω 0t + φ ) (3) Spring (4) Mass x '(t ) = −ω 0 x0 sin(ω 0t + φ ) Energy has 2 parts: (Mass) Kinetic and (Spring) Potential 2 1 ⎛ dx ⎞ 1 2 2 K = m ⎜ ⎟ = kx0 sin (ω 0t + φ ) Energy 2 ⎝ dt ⎠ 2 sloshes back 1 2 1 2 2 and forth U s = kx = kx0 cos (ω 0t + φ ) 2 2 P25- 11 Simple Harmonic Motion Amplitude (x0) 1 Period (T ) = frequency (f ) 2π = angular frequency (ω ) x(t ) = x0 cos(ω0t − φ ) Phase Shift (ϕ ) = π 2 P25- 12 Analog: LC Circuit Mass doesn’t like to accelerate Kinetic energy associated with motion 2 dv d x 1 2 F = ma = m = m 2 ; E = mv 2 dt dt Inductor doesn’t like to have current change Energy associated with current 2 dI d q 1 2 ε = − L = − L 2 ; E = LI dt dt 2 P25- 14 Analog: LC Circuit Spring doesn’t like to be compressed/extended Potential energy associated with compression 1 2 F = − kx; E = kx 2 Capacitor doesn’t like to be charged (+ or -) Energy associated with stored charge 1 11 2 ε = q; E = q C 2C F → ε ; x → q; v → I ; m → L; k → C −1 P25- 15 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? P25- 16 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) P25- 17 LC Circuit Q dI dQ −L =0 ; I =− C dt dt 2 d Q 1 Q=0 + 2 dt LC Simple Harmonic Motion Q(t ) = Q0 cos(ω 0t + φ ) ω0 = Q0: Amplitude of Charge Oscillation φ: Phase (time offset) 1 LC P25- 19 LC Oscillations: Energy Notice relative phases ⎛ Q0 2 ⎞ 2 Q =⎜ UE = ⎟ cos ω0t 2C ⎝ 2C ⎠ 2 2 ⎛ ⎞ Q 1 2 1 U B = LI = LI 0 2 sin 2 ω0t = ⎜ 0 ⎟ sin 2 ω0t 2 2 ⎝ 2C ⎠ Q 2 1 2 Q0 2 U = UE +UB = + LI = 2C 2 2C Total energy is conserved !! P25- 20 Damped LC Oscillations Resistor dissipates energy and system rings down over time ⎛ R ⎞ Also, frequency decreases: ω ' = ω0 − ⎜ ⎟ ⎝ 2L ⎠ 2 2 P25- 22