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 m d t ε ε ω = sin m d di q L Ri t dt C ε ω + + = “Transient”: Falls exponentially & disappears “Steady state”: Constant amplitude Ignore / 2 cos tR L i e t ω ±
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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 m d i I t ω φ = ( ) ( ) ( ) cos sin cos sin m m d d m d d m d d I I L I R t t t C ω ω τ φ ω φ ω φ ε ω ω + = sin m di q L Ri t dt C ε ω + + = ( ) sin m d i I t ω φ = ( ) cos d m d di I t dt ω ω φ = ( ) cos m d d I q t ω φ ω = − 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 t terms separately Î These equations can be solved for I m and φ (next slide) ( ) ( ) 1/ cos sin 0 1/ sin cos d d m d d m m L C R I L C I R ω ω φ φ ω ω φ φ ε = + = ( ) ( ) sin sin cos cos sin cos cos cos sin sin d d d d d d t t t t t t ω φ ω φ ω φ ω φ ω φ ω φ = = + High school trig! cos ω d t terms sin ω d t terms ( ) ( ) ( ) cos sin cos sin m m d d m d d m d d I I L I R t t t C ω ω τ φ ω φ ω φ ε ω ω + =
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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 1/ sin cos d d m d d m m L C R I L C I R ω ω φ φ ω ω φ 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 m d i I t ω
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