p2s10_lec11_LC_oscillations_RC_RL

p2s10_lec11_LC_oscillations_RC_RL - q q = = Q Q cos( cos( t...

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Physics 2 2009 Lecture 11 RC and RL circuits LC circuit oscillations circuit equations energy transfer and storage RLC circuit oscillations
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P2F09 2 RC circuit review + - i ; 0 C C R C R C q dq V V iR R C dt q E V V E iR C dq q E dt RC R = - = - = - + + = - - = + = / Discharging: 0 1 ( ) t RC C dq q dt RC dq dt q RC V t Ee - + = = - =
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P2F09 3 RL circuit ; 0 0 L R R C E i R di V L V iR dt V V di L iR dt = = - = - + = + = Then switch to position b Initial condition: Start circuit with switch in “a” position so that current E/R flows through inductor. 0 R t L di R dt i L i i e - = =
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LC circuit oscillations Inductor energy – proportional to i 2 . Capacitor energy – proportional to q 2 . Last class, we saw how a mass on a spring oscillated. In PHYS 100 we also saw how one form of energy (kinetic) was exchanged for another (potential). In an inductor/capacitor circuit energy stored in the electric field of the capacitor can be alternated with energy stored in the magnetic field of the inductor.
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LC circuit Loop rule: 0 circuit di q V L dt C ∆ = - - = Substitute, dq i dt = 2 2 0 1 d q q dt LC + =
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Comparing equations 2 2 2 2 1 Mass on spring Inductor/Capacitor d x k x dt m d q q dt LC = - = - Solution to the LC circuit: Solution to the LC circuit:
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Unformatted text preview: q q = = Q Q cos( cos( t t + + ) ) and q is the charge on the capacitor. and q is the charge on the capacitor. where LC 1 = Energy in LC circuits 2 2 2 cos ( ) 2 2 E q Q U t C C = = + 2 2 2 1 1 2 2 sin ( ) B U Li LI t = = + constant B E U U U + = = 2 2 max max 2 2 E B Q LI U U C = = = Energy oscillation in LC circuit Energy vs time when the current is a maximum at t=0.-0.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Time in Periods Ratio of U E to U B U = U B + U E U B U E t = /8 t T = / 4 t T = 3 /8 t T = 5 / 2 t T = 4 3 2 1 6 5 /8 t T = 3 / 4 t T = 7 /8 t T = 7 8 1 2 3 4 5 6 7 8 Energy oscillation in an LC circuit 1 Writing the loop equation for emf: 2 2 1 d q dq L R q dt dt C + + = This is the equation for the motion of damped harmonic oscillator, with solution: ( 29 / ( ) cos ' t q t Qe t -= + 2 L R = 2 1 1 ' LC =-with: and RLC Circuit...
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p2s10_lec11_LC_oscillations_RC_RL - q q = = Q Q cos( cos( t...

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