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 we use AC power? o What happens to an LRC circuit when it is driven by and AC voltage? o What is the time relationship of current and voltage for AC resistor circuit? o What is the time relationship of current and voltage for AC capacitor o What is the time relationship of current and voltage for AC inductor o What is the time relationship of current and voltage for AC RLC circuit? o What is a transformer and why are they 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 = - 2 2 2 cos 2 2 o C Q t q U C C = = 2 2 2 sin 2 = 2 o L Q t U L C I =
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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 = -
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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 = sin o I Q t = -
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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 = = - = = =
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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 ϖ ϕ - = = + - = = =
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Turbine generators produce Turbine generators produce The output is a time varying The output is a time varying Power loss in transmission due Power loss in transmission due Can reduce loss by tranmission Can reduce loss by tranmission
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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 Φ = - = E x max ma sin NB V t A V = E
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Ε I R An AC generator has an emf Ε = V max sin ϖ t , V max = NBA ϖ We connect the generator to a resistor R. The current I is given by Ohm’s law.
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This note was uploaded on 10/10/2010 for the course PHY 108 taught by Professor Iashvili during the Spring '08 term at SUNY Buffalo.

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

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