2049_ch31BC - Topics LC Oscillations Conservation of energy...

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PHY2049: Chapter 31 1 Topics Î LC Oscillations ± Conservation of energy Î Damped oscillations in RLC circuits ± Energy loss Î AC current (Your responsibility--read the book) ± RMS quantities Î Forced oscillations ± Resistance, reactance, impedance ± Phase shift ± Resonance frequency ± Power Î Transformers (Your responsibility—read the book along with Chapter 30 lecture notes) ± Impedance matching
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PHY2049: Chapter 31 2 AC Circuits with RLC Components Î Enormous impact of AC circuits ± Power delivery ± Radio transmitters and receivers ± MRI, and NMR in general ± Tuners ± Filters ± Transformers Î Basic components ± R ± L ± C ± Driving emf Î Now we will study the basic principles
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PHY2049: Chapter 31 3 AC Circuits and Forced Oscillations Î RLC + “driving” EMF with angular frequency ω d Î General solution for current is sum of two terms sin md t ε εω = sin di q LR i t dt C ω ++= “Transient”: Falls exponentially & disappears “Steady state”: Constant amplitude Ignore /2 cos tR L ie t ±
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PHY2049: Chapter 31 4 Steady State Solution Î Assume steady state solution of form ± I m is current amplitude ± φ is phase by which current “lags” the driving EMF ± Must determine I m and φ Î Plug in solution: differentiate & integrate sin( ω d t- φ ) ( ) sin md iI t ω φ = () ( ) cos sin cos sin m d m d d m d d I I LI R t t t C ωτ φ ε −+ −− −= sin m di q LR i t dt C ++= sin t =− cos dm d di It dt ωφ cos m d d I qt Substitute
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PHY2049: Chapter 31 5 Steady State Solution (2) Î Expand sin & cos expressions Î Collect sin ω d t&cos ω d tterms±separate ly Î These equations can be solved for I m and φ (next slide) () 1/ cos sin 0 dd md d m m LC R IL C I R ω ωφ φ ε −− = −+ = sin sin cos cos sin cos cos cos sin sin d d tt t t φω −= + High school trig! cos ω d tterms sin ω d ( ) cos sin cos sin m d m d d m d d I I LI R t t t C ωτ φ
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PHY2049: Chapter 31 6 Steady State Solution (3) Î Solve for φ and I m to find i, which is I m sin( ω d t- φ ) Î Let’s obtain this solution using “phasors” to gain better understanding () 2 2 1 C ω L ω R I d d m m ε + = 1/ cos sin 0 dd md d m m LC R IL C I R ω ωφ φ ε −− = −+ = Same equations R C ω L ω φ d d 1 tan =
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PHY2049: Chapter 31 7 Understanding AC Circuits Using Phasors Î Phasor ± Voltage or current represented by “phasor” ± Phasor rotates counterclockwise with angular velocity ω d ± Length of phasor is amplitude of voltage (V) or current (I) ± y component is instantaneous value of voltage ( v ) or current ( i ) ± We will use the same symbol for amplitude and phasor ε m I m ω d t − φ () sin md iI t φ =− sin t εω = i Current “lags” voltage by φ d t
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PHY2049: Chapter 31 8 Î Voltage (potential difference) across R ± In phase with current i (phasor V R is on top of phasor I m Î Voltage (potential difference)
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This note was uploaded on 04/12/2008 for the course PHY 2049 taught by Professor Any during the Fall '08 term at University of Florida.

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2049_ch31BC - Topics LC Oscillations Conservation of energy...

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