Electricity and MagnetismRL CircuitsRL CircuitsRL Circuits‘Back EMF’RL CircuitsIn-Class Demo: Square Wave V0In-Class DemoRL as low-pass filterEnergy Storage in InductorRLC circuitsSummary of Circuit ComponentsR,L,C in AC circuitZ and fZ and fZ and f: Capacitance CZ and f: Capacitance CZ and f: Inductance LRLC circuitRLC circuitRLC circuitRLC circuitResonanceIn-Class Demo (on scope)Apr 26 2002Electricity and Magnetism•Reminder–RL Circuits– Energy storage in Inductor•Today–RLC circuits – Resonance in RLC AC circuitApr 26 2002RL CircuitsKirchoffs Rule: V0+ ξind= R I -> V0= L dI/dt + R IQ: What is I(t)? L dI/dtR IVV0L1R2Apr 26 2002RL CircuitsI(t)tτ = L/R63%I(t)=V0/R [1-exp(-t/τ)]ξ(t)ξ(t)=V0exp(-t/τ)tτ = L/R37%Apr 26 2002RL 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 instantaneousApr 26 2002‘Back EMF’V0RL• What happens if we move switch to position 2? 12Apr 26 2002tτ = L/R63%I(t)tτ = L/R37%2τ = L/RΙ(t)=V0/R exp(-t/τ)ξ(t)1Apr 26 2002RL Circuits• 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, lightApr 26 2002In-Class Demo: Square Wave V0RV(t)L~Apr 26 2002In-Class DemoVintttI(t)ξ(t)Apr 26 2002RL as low-pass filter• Again, like RC circuits, RL circuits act as low-pass filters• Sharp edges/high frequencies are removed– > In-Class Demo...• RC circuit: Energy gets stored in C when Voltage switched on, released when Voltage switched off• Energy storage in RL circuits?Apr 26 2002Energy Storage in Inductor•Energy in Inductor– Start with Power P = ξ I = L dI/dt I = dU/dt-> dU = L dI I-> U = ½ L I2• Where is the Energy stored?– Example: SolenoidU/Volume = ½ B2/µ0Apr 26 2002RLC circuits• Combine everything we know...• Resonance Phenomena in RLC circuits– Resonance Phenomena known from mechanics (and engineering)– Great practical importance–video...Apr 26 2002Summary of Circuit Components~VV(t)RVR = IRVL = L dI/dtLVC = 1/C IdtCApr 26 2002R,L,C in AC circuit •AC circuit– I(t) = I0sin(ωt)– V(t) = V0sin(ωt+ φ)• Relationship between V and I can be characterized by two quantities– Impedance Z = V0/I0– Phase-shift φsame ω!Apr 26 2002Z and φI(t)=I0 sin(ωt)V(t)=V0 sin(ωt+φ)2π/ωI0φ/ωV0Impedance Z = V0/I0Apr 26 2002Z and φ• First look at impedance and phase-shift for circuits containing only R,C or L• Then RLC circuit...Z and φ: Capacitance CR~I(t)V= I RImpedance Z = V/I = RPhase-shift φ = 0Apr 26 2002Apr 26 2002Z and φ: Capacitance CC~I(t)V = Q/C = 1/C IdtImpedance Z = 1/(ωC)Phase-shift φ = - π/2 V lags I by 90oApr 26 2002Z and φ: Inductance LL~I(t)V= L dI/dtImpedance Z = ω LPhase-shift φ = π/2 I lags V by 90oApr 26 2002RLC circuitLRC~V(t)V – L dI/dt - IR - Q/C = 0L d2Q/dt2= -1/C Q – R dQ/dt + V2ndorder differential equationApr 26 2002RLC circuitLRC~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’‘Drag’Apr 26 2002RLC circuit• Solve L d2Q/dt2= -1/C Q – R dQ/dt + V –for AC circuit:V = V0sin(ωt), I = I0sin(ωt–φ)•If I = I0sin(ωt–φ) then• Q(t) = - I0 / ω cos(ωt–φ) •dQ/dt= I0sin(ωt–φ) •d2Q/dt2=I0ω cos(ωt–φ)Apr 26 2002RLC circuitV0sin(ωt) = I0{[ωL -1/(ωC)] cos(ωt–φ) +R sin(ωt–φ)} Solution (requires two tricks):I0= V0/([ωL -1/(ωC)]2+ R2)1/2= V0/Ztan(φ) = [ωL-1/(ωC)]/R -> For ωL= 1/(ωC), Z is minimal and φ =0i.e. ω = 1/(LC)1/2Resonance FrequencyApr 26 2002ResonanceI0ω = (LC)1/2ωωφ−π/2π/2Like LLike CImax = V0/RLow Frequency High FrequencyApr 26 2002In-Class Demo (on scope)LRC~V(t)VR(t) ~
View Full Document
Unlocking...