Time Varying CircuitsA look into the futureThe Final ExamSlide 4Slide 5We also showed thatAt t=0, the charged capacitor is now connected to the inductor. What would you expect to happen??The math …The Graph of that LR (no emf) circuit ..Slide 10Mass on a Spring ResultSlide 12LC CircuitThe Math Solution (R=0):New Feature of Circuits with L and CVariable Emf AppliedSinusoidal StuffSlide 18Different FrequenciesNote – Power is delivered to our homes as an oscillating source (AC)Producing AC GeneratorThe Real WorldSlide 23Slide 24The Flux:problems …Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Source Voltage:Average value of anything:Average ValueSo …Slide 38RMSUsually Written as:Example: What Is the RMS AVERAGE of the power delivered to the resistor in the circuit:PowerMore Power - DetailsResistive CircuitConsider this circuitSlide 46Alternating Current CircuitsPhase TermSlide 49Example: household voltageSlide 51Slide 52Slide 53Review: Resistors in AC CircuitsCapacitors in AC CircuitsI Leads V??? What the **(&@ does that mean??Slide 57Inductors in AC CircuitsSlide 59Phasor DiagramsSlide 61Slide 62Steady State Solution for AC Current (2)Slide 64Steady State Solution for AC Current (3)REMEMBER Phasor Diagrams?Reactance - Phasor Diagrams“Impedance” of an AC CircuitSlide 69Slide 70Impedance of an RLC CircuitPhasors for a Series RLC CircuitSlide 73Slide 74Slide 75Slide 76Slide 77Phase in an RLC CircuitSlide 79Slide 80Slide 81Slide 82Slide 83Slide 84Slide 85Slide 86Slide 87Slide 88Slide 89Slide 90Induction 1Time Varying CircuitsTime Varying Circuits20082008Induction 2A look into the futureA look into the futureWe have one more week after today (+ We have one more week after today (+ one day)one day)Time Varying Circuits Including ACTime Varying Circuits Including ACSome additional topics leading to wavesSome additional topics leading to wavesA bit of review if there is time.A bit of review if there is time.There will be one more Friday morning quiz.There will be one more Friday morning quiz.I hope to be able to return the exams on I hope to be able to return the exams on Monday at which time we will briefly review Monday at which time we will briefly review the solutions.the solutions.Induction Induction 33The Final ExamThe Final Exam8-10 Problems 8-10 Problems similar to (or similar to (or exactly) Web-exactly) Web-AssignmentsAssignmentsCovers the entire Covers the entire semester’s worksemester’s workMay contain some May contain some short answer short answer questions.questions.Induction 4Max Current Rate ofincrease = max emfVR=iR~currentInduction 5constant) (time )1(/RLeREiLRtWe Solved the loop equation.Induction 6We also showed that20202121EBcapacitorinductoruuInduction 7At t=0, the charged capacitor is now connected to the inductor. What would you expect to happen??Induction 8The math …For an RLC circuit with no driving potential (AC or DC source):2/122max2221)cos(:00LRLCwhereteQQSolutiondtQdLCQdtdQRdtdiLCQiRddLRtInduction 9The Graph of that LR (no emf) circuit ..LRte2LRte2IInduction 10Induction 11Mass on a Spring ResultEnergy will swap back and forth.Add frictionOscillation will slow downNot a perfect analogyInduction 12Induction 13LC CircuitHigh Q/CLowLowHighInduction 14The Math Solution (R=0):LCInduction 15New Feature of Circuits with L and CThese circuits produce oscillations in the currents and voltagesWithout a resistance, the oscillations would continue in an un-driven circuit.With resistance, the current would eventually die out.Induction 16Variable Emf AppliedemfSinusoidalDCInduction 17Sinusoidal Stuff )sin( tAemf“Angle”Phase AngleInduction 18Same FrequencywithPHASE SHIFTInduction 19Different FrequenciesInduction 20Note – Power is delivered to our homes as an oscillating source (AC)Induction 21Producing AC Generatorx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xInduction 22The Real WorldInduction 23AInduction 24Induction 25The Flux:tARemfitBAemftBAbulbsinsincos- ABInduction 26problems …Induction 2714. Calculate the resistance in an RL circuit in which L = 2.50 H and the current increases to 90.0% of its final value in 3.00 s.Induction 2818. In the circuit shown in Figure P32.17, let L = 7.00 H, R = 9.00 Ω, and ε = 120 V. What is the self-induced emf 0.200 s after the switch is closed?Induction 2932. At t = 0, an emf of 500 V is applied to a coil that has an inductance of 0.800 H and a resistance of 30.0 Ω. (a) Find the energy stored in the magnetic field when the current reaches half its maximum value. (b) After the emf is connected, how long does it take the current to reach this value?Induction 3016. Show that I = I0 e – t/τ is a solution of the differential equation where τ = L/R and I0 is the current at t = 0.Induction 3117. Consider the circuit in Figure P32.17, taking ε = 6.00 V, L = 8.00 mH, and R = 4.00 Ω. (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250 μs after the switch is closed. (c) What is the value of the final steady-state current? (d) How long does it take the current to reach 80.0% of its maximum value?Induction 3227. A 140-mH inductor and a 4.90-Ω resistor are connected with a switch to a 6.00-V battery as shown in Figure P32.27. (a) If the switch is thrown to the left (connecting the battery), how much time elapses before the current reaches 220 mA? (b) What is the current in the inductor 10.0 s after the switch is closed? (c) Now the switch is quickly thrown from a to b. How much time elapses before the current falls to 160 mA?Induction 3352. The switch in Figure P32.52 is connected to point a for a long time. After the switch is thrown to point b, what are (a) the frequency of oscillation of the LC circuit, (b) the maximum charge that appears