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Physics 42200 Waves & Oscillations Lecture 10 – French, Chapter 4 Spring 2013 Semester Matthew Jones Forced Oscillators and Resonance + + = • Natural oscillation frequency: = • Amplitude of steady-state oscillations: ⁄ = − cos = + • Phase difference: = tan − − 2 Resonance Phenomena • Change of variables: = • Natural oscillation frequency: = • = Amplitude of steady-state oscillations: = • Strong damping force Strong damping force = − / − Phase difference: = tan 2 + 1/ − = 1 1− / 2 1 large small Resonance Phenomena • Steady state amplitude: ( ⁄ ) Q=5 Q=4 Q=3 Q=2 Q=1 Peak is near ⁄ / ≈ 1. The peak occurs at exactly ⁄ = 1. Average Power • The rate at which the oscillator absorbs energy is: '(( ) = '(( ) /2 2 − 1 + 1 = 10 Full-Width-at-Half-Max: =5 ,-. = =3 =1 / = Examples 2 • Resonant RLC circuit: 0(1) / 3 ℰ cos 67 5 +7 6 1 ; 8+ : 7 9 6 The transformer does not play a role in the analysis of the circuit. It is just a convenient way to isolate the driving voltage source from the part of the circuit that oscillates. Resonant Circuit 67 1 ; 5 + 7 8 + : 7 6 = ℰ cos 6 9 • Differentiate once: 6 7 67 1 5 +8 + 7 = −ℰ sin 6 6 9 • Redefine what we man by “ = 0”: 6 7 67 1 5 +8 + 7 = ℰ cos 6 9 6 • Change of variables: 6 7 67 ℰ + + 7 = cos 6 6 5 Resonant Circuit 6 7 67 ℰ + + 7 = cos 6 6 5 • Amplitude of steady state current oscillations: / =ℰ 9 1 − + • Voltage across the capacitor: = =7 >? = 7 / 9 • Amplitude of voltage oscillations measured across C: / = =ℰ / 1 − + Actual Data Voltage • If we know that 5 = 235@-, can we estimate 8 and 9? 5 = 235@- Peak voltage is at AC DE = 290 -G C DE = 2B AC DE A = /2B [Mhz] Energy • Energy stored on a capacitor is H = 9= • The graph of the stored energy is proportional to the square of the voltage graph. • If we defined 2∆ as the FWHM on the graph of power vs frequency, then it will correspond to 1/ 2 of the peak voltage. Voltage Resonant Circuit 2∆A = 110 -G = 2B ∆A = 6.91 × 10L M 5 = 235@- But A= /2B Peak position is [Mhz] C DE = = 8/5 so we can find 8: 8 = 5 = 6.91 × 10L M = 162Ω O? − PQ 9 = 1.20nF 9= O 235@- Q XQ RSTUV W Q Lifetime of Oscillations • Amplitude of a damped harmonic oscillator: = Z P;/ cos • Maximum potential energy: 1 [= ∝ Z P; 2 • After time = 1/ , the energy is reduced by the factor 1/Z. • We call ] = 1/ the “lifetime” of the oscillator. Other Resonant Systems Other Resonant Systems Other Resonant Systems Wire Bond Resonance • Wire bonds in a magnetic field: ^ e _ Lorentz force is ` = a b 6ℓ × d The tiny wire is like a spring. A periodic current produces the driving force. Wire Bond Resonance Resonance in Nuclear Physics • A proton accelerated through a potential difference = gains kinetic energy f = Z=: = h Phys. Rev. 75, 246 (1949). Resonance in Nuclear Physics • In quantum mechanics, energy and frequency are proportional: H=i • A given energy corresponds to a driving force with frequency . • When a nucleus resonates at this frequency, the proton energy is easily absorbed. Nuclear Resonance “Lifetime” is defined in terms of the width of the resonance. Resonance • Resonances are the main way we observe fundamental particles.