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Exercise:
An LC oscillator has LO inductance CO capacitance and RO resistance. abcliste abc Verify that the oscillation is only weakly damped. abc Calculate the number of oscillations it takes until the amplitude has decreased to ratioO of the initial value. abc What fraction of the initial energy is left at this moment? abcliste

Solution:
abcliste abc The damping ratio for this system is zeta fracdeltadelta_c fracdeltaomega_ fracfracRLfracsqrtLC zeF fracRtimessqrtfracCL zeS Since zell the damping is very low i.e. the angular frequency and the period are close to the values of the undamped oscillation. abc The period of the undamped oscillation is T TF The envelope of the damped oscillation is given by At A_ e^-t/tau A_ e^-delta t Solving for the time leads to t -fraclnAt/A_delta fraclnA_/At LR The number of oscillations during the time t is N fractT fraclnA_/At Lpi RsqrtLC NoscF fraclnratioinvPpitimesRtimessqrtfracLC Nosc approx resultNoscP abc The energy is proportional to the square of the amplitude so fracEtE_ eratF ratio^ erat resultepercP abcliste
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Exercise:
An LC oscillator has LO inductance CO capacitance and RO resistance. abcliste abc Verify that the oscillation is only weakly damped. abc Calculate the number of oscillations it takes until the amplitude has decreased to ratioO of the initial value. abc What fraction of the initial energy is left at this moment? abcliste

Solution:
abcliste abc The damping ratio for this system is zeta fracdeltadelta_c fracdeltaomega_ fracfracRLfracsqrtLC zeF fracRtimessqrtfracCL zeS Since zell the damping is very low i.e. the angular frequency and the period are close to the values of the undamped oscillation. abc The period of the undamped oscillation is T TF The envelope of the damped oscillation is given by At A_ e^-t/tau A_ e^-delta t Solving for the time leads to t -fraclnAt/A_delta fraclnA_/At LR The number of oscillations during the time t is N fractT fraclnA_/At Lpi RsqrtLC NoscF fraclnratioinvPpitimesRtimessqrtfracLC Nosc approx resultNoscP abc The energy is proportional to the square of the amplitude so fracEtE_ eratF ratio^ erat resultepercP abcliste
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Damped Oscillations
Tags
damping, energy, envelope, lc oscillator
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Difficulty
(2, default)
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0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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