phy293_l07.page2 - 2 + 2 Drops to half of P max at ( 2- 2 )...

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Have already discussed that Q is an indication of how many oscillations occur before the Energy of a damped, un-driven, oscillator falls to E 0 /e Here we’ll see how Q is related to the resonant frequency and the width of the peak in the power resonance < P > = P max γ 2 ( ω 2 0 - ω 2 ) 2 2 + γ 2 Where P ( ω ) V ( ω ) 2 and we have P max = 1 2 ma 2 0 ω 3 0 Q Near the resonance defined by | ω - ω 0 | ± ω 0 we can re-write: ω 2 0 - ω 2 ω = ( ω 0 - ω )( ω 0 + ω ) ω ( ω 0 - ω )2 ω 0 ω 0 = 2( ω 0 - ω ) Note this will be good approximation if Q ² 1 In this limit ( | ω - ω 0 | ± ω 0 ) we can write: < P > = P max γ 2 / 4 ( ω 0 - ω ) 2 + γ 2 / 4 This is the Lorentzian function: L ( x ) = 1 / (1 + x 2 ) and is considerably simpler than the full expression for the power absorption Notice that the average power drops to 1 2 its maximum for ω = ³ γ/ 2 So the full width, at half-maximum, of this simplified < P ( ω ) > is γ = ω 0 /Q The higher the Q the narrower the power resonance will become 3. Width of the General Power Curve Lorentzian is a good approximation for Q ² 1 , but general curve also has width γ < P > = P max γ 2 ( ω 2 0 - ω 2 ) 2
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Unformatted text preview: 2 + 2 Drops to half of P max at ( 2- 2 ) 2 / 2 + 2 = 2 2 This is true when 2- 2 = This denes two frequencies: 1 , the half-power point above the resonance 2 , the half-power point below the resonance 2 1- 1- 2 = 0 [1] 2 2 + 2- 2 = 0 [1] Solve the to nd the width of the resonance at half-power [1]-[2] : 2 1- 2 2- ( 1 + 2 ) = 0 Factorise the difference of squares to get a term like ( 1 + 2 ) : ( 1- 2- )( 1 + 2 ) = 0 Since we dont have a physical interpretation for negative frequencies the second root is a false solution. The other root gives us 1- 2 = The difference in frequencies between the two mid-points is, again, This is true even at low Q ....
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This note was uploaded on 07/10/2011 for the course PHY 293 taught by Professor Pierresavaria during the Fall '07 term at University of Toronto- Toronto.

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