May 2 2005 web.mit.edu/8.02x/wwwQuiz #4 preparations• Quiz 4: Wed, 5/4, 10AM, 26-100– 1 sheet with formulae etc– No books, calculators• Evening review: Tue, 5/3, 7PM, 26-100• Tutoring:– Angel Solis, Mon + Tue, 5/2, 5-7PM, Room 4-3XXMay 2 2005 web.mit.edu/8.02x/wwwReview for Quiz #4May 2 2005 web.mit.edu/8.02x/wwwTransformer Action• Transformer action~IAC = I0 sin(!t)BSecondaryPrimaryEMFp = - Np d"B/dtEMFs = - Ns d"B/dtNpNsEMFs /EMFp = Ns/NpFlux through single turnsameMay 2 2005 web.mit.edu/8.02x/wwwTransformer Action• Transformer action• Transformers allow change of amplitude for AC voltage – ratio of secondary to primary windings•Constructed such that "B identical for primary and secondary•EMFs /EMFp = Ns/NpMay 2 2005 web.mit.edu/8.02x/wwwWhat you need to know• Transformers• Basic principle • Transformer in HVPS• Relationship between I,V,P on primary/secondary side• Demos– Jacobs Ladder – Melting nailMay 2 2005 web.mit.edu/8.02x/wwwMutual Inductance~BN2Def.:M12 = N2 "B/I1N1I1"B ~ BB ~ I1May 2 2005 web.mit.edu/8.02x/wwwMutal Inductance•Coupling is symmetric: M12 = M21 = M• M depends only on Geometry and Material• Mutual inductance gives strength of coupling between two coils (conductors): •M relates EMF2 and #1 (or EMF1 and #2) • Units: [M] = V/(A/s) = V s /A = H (‘Henry’)EMF2 = - N2 d"B/dt = - M dI1/dtMay 2 2005 web.mit.edu/8.02x/wwwExample: Two SolenoidsN1N2Area AQ: How big is M = N2 "B/I1 ?A: M = µ0 N1N2 A/l Length lMay 2 2005 web.mit.edu/8.02x/wwwDemo: Two CoilsSpeakerRadio• Signal transmitted by varying B-Field• Coupling depends on Geometry (angle, distance)May 2 2005 web.mit.edu/8.02x/wwwSelf Inductance~IBCircuit sees flux generated by it selfDef.: L = N "B/ISelf-InductanceMay 2 2005 web.mit.edu/8.02x/wwwExample: Solenoid~IBQ: How big is L ?A: L = µ0 N2 A/LMay 2 2005 web.mit.edu/8.02x/wwwSelf Inductance• L is also measured in [H]• L connects induced EMF and variation in current: EMF = - L dI/dt• Remember Lenz’ Rule: Induced EMF will ‘act against’ change in current -> effective ‘inertia’• Delay between current and voltageMay 2 2005 web.mit.edu/8.02x/wwwWhat you need to know• Inductance• Mutual Inductance• Definition• Calculation for simple geometry• Self Inductance• Definition• Calculation for simple geometry• Direction of induced EMF (depends only on dI/dt)May 2 2005 web.mit.edu/8.02x/wwwRL CircuitsKirchoffs Rule: V0 + %ind = R I -> V0 = L dI/dt + R IQ: What is I(t)? L dI/dtR IVV0RL12May 2 2005 web.mit.edu/8.02x/wwwRL Circuitst$ = L/R63%I(t)V(t)=V0 exp(-t/$)t$ = L/R37%V(t)I(t)=V0/R [1-exp(-t/$)]May 2 2005 web.mit.edu/8.02x/wwwRL Circuits• Inductance leads to ‘delay’ in reaction of current to change of voltage V0• All practical circuits have some L and R– change in I never instantaneousMay 2 2005 web.mit.edu/8.02x/www‘Back EMF’V0RL• What happens if we move switch to position 2? 12May 2 2005 web.mit.edu/8.02x/wwwt$ = L/R63%I(t)t$ = L/R37%%(t)1 2$ = L/R#(t)=V0/R exp(-t/$)May 2 2005 web.mit.edu/8.02x/wwwRL circuit• L counteracts change in current both ways– Resists increase in I when connecting voltage source– Resists decrease in I when disconnecting voltage source– ‘Back EMF’• That’s what causes spark when switching off e.g. appliance, lightMay 2 2005 web.mit.edu/8.02x/www• Energy in Inductor– Start with Power P = V*I = L dI/dt I = dU/dt -> dU = L dI I -> U = 1/2 L I2• Where is the Energy stored?– Example: Solenoid (but true in general) U/Volume = 1/2 B2/µ0Energy Storage in InductorMay 2 2005 web.mit.edu/8.02x/wwwWhat you need to know• Inductors• I(t) in DC RL circuits• Energy storage in inductors• Practical useMay 2 2005 web.mit.edu/8.02x/wwwRLC circuits• Combine everything we know...• Resonance Phenomena in RLC circuits– Resonance Phenomena known from mechanics (and engineering)– Great practical importanceMay 2 2005 web.mit.edu/8.02x/wwwSummary of Circuit ComponentsRL~ VCV(t) = V0 cos(!t) VR = - IRVL = - L dI/dtVC = -Q/C = -1/C IdtMay 2 2005 web.mit.edu/8.02x/wwwR,L,C in AC circuit • AC circuit– I(t) = I0 sin(!t)– V(t) = V0 sin(!t + &)• Relationship between V and I can be characterized by two quantities–Impedance Z = V0/I0–Phase-shift & same !!May 2 2005 web.mit.edu/8.02x/wwwAC circuitI(t)=I0 sin(!t)V(t)=V0 sin(!t + &)2(/!I0V0&/!Impedance Z = V0/I0 Phase-shift & May 2 2005 web.mit.edu/8.02x/wwwFirst: Look at the componentsZ = R& = 0 V and I in phaseR~I(t)V = I RC~I(t)V = Q/C = 1/C IdtZ = 1/(!C) &= - (/2 V lags I by 90oL~I(t)V = L dI/dtZ = ! L &= (/2 I lags V by 90oMay 2 2005 web.mit.edu/8.02x/wwwRLC circuitRLC~V(t)May 2 2005 web.mit.edu/8.02x/wwwRLC circuitRLC~V(t)V – L dI/dt - IR - Q/C = 0L d2Q/dt2 = -1/C Q – R dQ/dt + V2nd order differential equationMay 2 2005 web.mit.edu/8.02x/wwwRLC circuitRLC~V(t)V – L dI/dt - IR - Q/C = 0L d2Q/dt2 = -1/C Q – R dQ/dt + VWaterSpringMass mFextm d2x/dt2 = -k x – f dx/dt + Fext‘Inertia’‘Spring’‘Friction’May 2 2005 web.mit.edu/8.02x/wwwResonanceI0&! = 1/(LC)1/2 '(/2 (/2 !!Like LLike CImax = V0/RHigh FrequencyLow FrequencyMay 2 2005 web.mit.edu/8.02x/wwwRLC circuitV0 sin(!t) = I0{[!L -1/(!C)] cos(!t – &) +R sin(!t – &)} Solution (requires two tricks):I0 = V0/([!L -1/(!C)]2 + R2)1/2 = V0/Z tan(&) = [!L -1/(!C)]/R -> For !L = 1/(!C), Z is minimal and & =0 i.e. !0 = 1/(LC)1/2 Resonance FrequencyMay 2 2005 web.mit.edu/8.02x/wwwResonance• Practical importance– ‘Tuning’ a radio or TV means adjusting the resonance frequency of a circuit to match the frequency of the carrier signalMay 2 2005 web.mit.edu/8.02x/wwwLC-Circuit• What happens if we open switch?LC– L dI/dt - Q/C = 0L d2Q/dt2 + Q/C = 0V0d2x/dt2 + !02 x = 0Harmonic Oscillator!May 2 2005 web.mit.edu/8.02x/wwwLC-CircuitLCSpring kMass m1/2 k x21/2 m v21/2 Q2/C1/2 L I2Potential Energy Kinetic EnergyOscillationEnergy in E-Field Energy in B-FieldOscillationMay 2 2005 web.mit.edu/8.02x/wwwLC-CircuitLCd2Q/dt2 + 1/(LC) Q = 0 !02 = 1/(LC) d2x/dt2 + k/m x = 0!02 = k/mSpring kMass mMay 2 2005 web.mit.edu/8.02x/wwwLC-Circuit• Total energy U(t) is conserved: Q(t) ~ cos(!t) dQ/dt ~ sin(!t) UL ~ (dQ/dt)2 ~ sin2 UC ~ Q(t)2 ~ cos2 cos2(!t) + sin2(!t) = 1LC1/2 Q2/C1/2 L I2Energy in E-Field Energy in B-FieldOscillationMay 2 2005 web.mit.edu/8.02x/www• In an LC circuit, we see oscillations:• Q: Can we get
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