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PHY108 - Chapter 31

PHY108 - Chapter 31 - Chapte 31 Ele r ctrom tic...

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Chapter 31: Electromagnetic Oscillations and Alternating Current o What is an LC circuit? o What is an LRC circuit? o Why do weuseAC power? o What happens to an LRC circuit when it is driven by and AC voltage? o What is thetime relationship of current and voltage for AC resistor circuit? o What is thetime relationship of current and voltage for AC capacitor circuit? o What is thetime relationship of current and voltage for AC inductor circuit? o What is thetime relationship of current and voltage for AC RLC circuit? o What is a transformer and why arethey useful?
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LC Circuits 2 2 2 2 2 ma 2 x Q dI Q d dQ L 0 L 0 C dt C dt dt d Q 1 Q dt LC 1 let LC d Q Q dt Looks like equation for simple harmonic motion Q(t)=Q cos( + ) ! t - - = - - = = - ϖ = ϖ φ = The charge on the capacitor oscillates, and similarly the current through inductor oscillates 90 degrees out of phase with the charge: m max ax dQ d I(t) Q(t)=Q cos( I(t)=-Q t+ ) dt dt sin ( t+ ) = ϖ ϖ = ϖ φ φ
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(33-18) Summary ( ) cos o q t Q t ϖ = sin o I Q t ϖ ϖ = - 2 2 2 cos 2 2 o C Q t q U C C ϖ = = 2 2 2 sin 2 = 2 o L Q t U L C I ϖ =
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. . t = 0 (33-19) 2 2 2 2 At 0 (0) (0) 0 is maximum 0 C L o C L q U C LI U t q Q I U U = = = = = = ( ) cos o q t Q t ϖ = sin o I Q t ϖ ϖ = -
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. . t = T/4 (33-20) 2 2 2 2 At 4 (0) 0 (0) 0 is maximum C L o C L q U C LI U T t q I Q U U ϖ = = = = = - = ( ) cos o q t Q t ϖ = sin o I Q t ϖ ϖ = -
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+q -q C L I I a b c d Why do we bother with LC-circuits? L(H) C(F) f(Hz) Range 10 -3 10 -6 5 × 10 3 audio 10 -4 10 -8 1.6 × 10 5 AM radio 10 -6 10 -12 1.6 × 10 8 Short wave radio (33-24) Important? You bet! 2 ( ) cos sin 1 1 2 2 o o q t Q t I Q t f LC LC ϖ ϖ ϖ ϖ ϖ π π = = - = = =
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RLC Circuits 2 2 2 2 Q dI RI L 0 C dt Q dQ d dQ R L 0 C dt dt dt d Q dQ Q L R 0 dt dt C Looks alot like the equation for damped harmonic motion d x dx m b kx 0!! dt dt - - - = - - - = + + = + + = Rt /2L max d 2 d Q(t) Q e cos( t) 1 R LC 2L - = ϖ ϖ = -
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The oscillation amplitude of the damped RCL circuit decays exponentially with time The reason is that some of the electric and magnetic energy is converted into heat by R and escapes in the surroundings 2 1 Solution: and ' 2 1 Note: When 0 ( then ) cos( ' ' ) t o R L L q t Q e t C R LC α α α ϖ ϖ ϖ ϕ - = = + - = = =
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Turbine generators produce Turbine generators produce AC AC The output is a time varying The output is a time varying Emf, voltage Emf, voltage Power loss in transmission due Power loss in transmission due to Joule heating in to Joule heating in transmission wires transmission wires Can reduce loss by tranmission Can reduce loss by tranmission at high voltages at high voltages
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Electric Generators By mechanically turning a loop relative to magnetic poles, change flux through loop and thus create a voltage source. The loop can be turned by for example using water flow. B B BAcos BAcos t d mf N NBA sin t dt peak emf =NBA Φ = θ = ϖ Φ ε = - = ϖ ϖ ϖ
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AC-generator An AC-generator consists of N wire loops (each of area A) rotating in a uniform magnetic field B with an angular velocity ϖ
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AC - generator θ= ϖ t 1 Magnetic flux through one loop cos BA θ Φ = B 1 Net magnetic flux cos N NBA t ϖ Φ = Φ = Induced emf sin B d NBA t dt ϖ ϖ Φ = - = E x max ma sin NB V t A V ϖ ϖ = E
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