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# Ch31 ac circuits_partial

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• 1. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Ch 31: Electromagnetic oscillations Exclude section 5 Nominal 2 lectures This chapters notes are incomplete Ch31-1/41 LC oscillations (qualitative to start) So far RC and RL circuits. Both had exponential decay or growth of i, q and V. Now LC combination, but without emf in circuit (to start). These circuits oscillate. Energy stored in electric field of capacitor: Energy stored in magnetic field of inductor: Convention: lc: q, i etc are instantaneous values uc: Q, I etc are the amplitudes (i.e. maxima) Ch31-2/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 1
• 2. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons LC oscillations (qualitative to start) Bar graphs represent energy stored in the E or B fields. Initially all energy stored in capacitor, and for first instant inductor looks like open circuit (opposing any current flow). Then +ve charge flows counterclockwise => UE decreases and energy stored in inductor’s field because current building. Ch31-3/41 LC oscillations (qualitative to start) Eventually capacitor loses all its stored energy/charge -it is now stored in magnetic field of inductor. -inductor looks like a piece of wire => maximum current flowing So why doesn’t current just stop flowing? Ans: inductor opposes any change in B => continues to drive current to maintain B and/or i but di/dt = 0. Uses stored energy Ch31-4/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 2
• 3. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons LC oscillations (qualitative to start) (d) Current decreases slowly as capacitor charges and field becomes stronger. Ch31-5/41 LC oscillations (qualitative to start) Eventually reach a stage where all energy is stored on capacitor again, and current momentarily stops flowing. i.e., back to original situation with charges reversed => cycle starts again but in reverse Ch31-6/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 3
• 4. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons LC oscillations (qualitative to start) Eventually reach state (a) again and cycles again with a frequency f and angular frequency ω = 2πf Ch31-7/41 LC oscillations (qualitative to start) Measure voltage across capacitor => shape in (a) above right. Could measure current by putting a small R in circuit and measure potential across it. Note that current is a maximum when there is no charge on capacitor and di/dt = 0 => inductor looks like a piece of wire. Ch31-8/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 4
• 5. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons LC oscillations (qualitative to start) In a real circuit there is resistance and instead of oscillating forever, amplitude of current and charge oscillations die away with an exponential nature (but we don’t cover in this course). Ch31-9/41 Question LC circuit, all charge is on capacitor at t= 0 and period of oscillation is T. When do the following reach their maximum (in terms of T)? options: T, T/2, T/4, other (a) Charge on capacitor (b) Voltage across capacitor with original polarity (c) energy stored in electric field (d) the current Ch31-10/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 5
• 6. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Question LC circuit, all charge is on capacitor at t= 0 and period of oscillation is T. When do the following reach their maximum (in terms of T)? options: T, T/2, T/4, other (a) Charge on capacitor Ans: starts at max - at T/2 is back to same max (reverse sign) (b) Voltage across capacitor with original polarity Ans: starts at max - at T is back to same max with same sign (c) energy stored in electric field Ans: starts at max - at T/2 is back to same max with same sign (d) the current Ans: starts at 0 - at T/4 is max when di/dt=0 in inductor Ch31-11/41 Sample problem 31-1 1.5 µF capacitor charged to 57 V then disconnected and 12 mH coil connected in series with C. What is maximum current in the coil? (R = 0 assumed). Key ideas: 1) current is max when all energy is in magnetic field and UB = Li2/2. 2) energy stored in electric field, UE=Q2/2C, is at its max for max Q 3) Ch31-12/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 6
• 7. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons The electrical-mechanical analogy We reviewed block-spring system because there is an analogy between that and LC circuit q, charge on capacitor corresponds to displacement x i corresponds to v 1/C corresponds to k L corresponds to m For block spring: By analogy here (proven shortly): Ch31-13/41 Block-spring details Derive equations for block-spring case At any instant, the total energy U of system is energy conservation says dU/dt = 0 substitute some defns: This is the differential eqn governing the block-spring system: Solution is Ch31-14/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 7
• 8. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Block-spring details (cont) where X is amplitude, ω angular frequency and φ is phase constant. Proof this is a solution: See approach p31-17, show it is a soln only if ω2=k/m Note proof says nothing re X or φ. Boundary conditions will establish them. Ch31-15/41 LC Oscillator Here we follow the same approach and consider total stored energy: Energy conservation: Equation governs LC circuit with no resistance. Note analogy to block-spring eqn Ch31-16/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 8
• 9. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons LC Oscillator (cont) Solution where Q is amplitude of charge variation, ω angular frequency and φ is phase constant. define Hence Ch31-17/41 LC Oscillator (cont) Prove this is a solution Substitute in eqn: cancel cos terms and we find So eqn above is a solution as long as ω has this value (which is what we had before by analogy) Ch31-18/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 9
• 10. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons LC Oscillator: energy oscillations Stored electrical energy Stored magnetic energy Total stored: Notes: 1) maximum of UE and UB are same 2) sum at any time is same constant 3) when one is a max, other is zero Ch31-19/41 AC generator Why AC p 836 - sign by convention Ch31-20/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 10
• 11. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Forced oscillations With emf get driven oscillations or forced oscillations No longer at natural angular frequency of symbol for generator with sinusoidal emf Ch31-21/41 Resistive load Loop rule from previous defn of R=V/i from previous comparison of 2=> applies to any resistance in any ac circuit Ch31-22/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 11
• 12. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Resistive load: phasors Phasors: Vectors that represent VR and IR across resistor at time t Properties: rotate counterclockwise with angular frequency ωd Lengths = amplitude of quantities, VR and IR Projection on y-axis = vR and iR at time t Rotation angle = phase of alternating quantity, i.e., Since in phase => rotate together Ch31-23/41 Sample problem 31-4 Checkpoint 3 If increase driving frequency in a resistive load circuit, do (a) the amplitude VR and (b) the amplitude IR increase, decrease or stay the same? Ans: Both stay the same Ch31-24/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 12
• 13. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Capacitive load Define capacitive reactance unit is ohm (C has units s/ohm -recall τ = RC) iC leads vC Physically makes sense since charge causes v to build on capacitor Ch31-25/41 Checkpoint 4 S=sin ωdt & others (a) Rank A, B, C in terms of value of φ, most +ve first. (b) Which curve corresponds to which phasor? Ch31-26/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 13
• 14. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Checkpoint 4 S=sin ωdt & others (a) Rank A, B, C in terms of value of φ, most +ve first. (b) Which curve corresponds to which phasor? Ans(a): C > B > A since the more -ve the more it leads and A leads Ans(b): 1 =A 2=B 3 =S 4=C since 1 leads the phasors Ch31-27/41 Sample problem 31-5 Ch31-28/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 14
• 15. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Inductive load Define inductive reactance unit is ohm Here i lags v. Also physical since as emf applied there is no current at first. NB The above sections are incomplete Ch31-29/41 Series RLC circuit (no longer on course) i same in all elements Use phasor analysis and phase relationships developed for individual cases. Define circuit’s impedance for driving frequency ωd as Steady state: ignores initial transients Ch31-30/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 15
• 16. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Series RLC circuit XL > XC => more inductive than capacitive => φ > 0 => i lags emf XL < XC => more capacitive than inductive => φ < 0 => i leads emf XL = XC => in resonance => φ = 0 => i and emf in phase Ch31-31/41 Series RLC circuit more capacitive more inductive in resonance Ch31-32/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 16
• 17. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Series RLC circuit resonance XL = XC => in resonance => = ω = natural frequency of circuit Imax is emf/R Ch31-33/41 Power in ac circuits In steady state, average total energy stored in capacitor and inductor is a constant (see 31-18 for demo for LC circuit). Net transfer of energy is from generator to resistor to thermal energy at time t is Need average over time where Irms is root-mean square (rms) value of the current i. Same old formula, just need Irms instead of i. Ch31-34/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 17
• 18. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Power in ac circuits ac instruments such as ammeters, voltmeters usually measure rms values. So 120 V power at home has Vrms = 120 and Vmax = 170V Since sqrt(2) in all cases, eqns look same: Ch31-35/41 Power in ac circuits power factor cos φ=cos-φ To maximize power => cosφ close to 1 => φ near 0, i.e., i and emf nearly in phase => cosφ = R/Z near 1 => R = Z => at resonance where XL = XC Say a circuit highly inductive (i.e., XL > XC => L > 1/C) To make it less so, add some capacitance in series => Ceq will decrease (1/Ceq = 1/Co + 1/Cnew will increase) => XL closer to XC => more power transferred Power transmission system have series connected capacitors Ch31-36/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 18
• 19. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Question (CP8) Sinusoidally driven RLC circuit. Current leads the emf. (a) Do we increase or decrease the capacitance to increase rate at which energy is supplied to the resistance? (b) Would this change make resonant angular frequency of circuit closer to the angular frequency of the emf or put it further away? Ch31-37/41 Question (CP8) Sinusoidally driven RLC circuit. Current leads the emf. (a) Do we increase or decrease the capacitance to increase rate at which energy is supplied to the resistance? (b) Would this change make resonant angular frequency of circuit closer to the angular frequency of the emf or put it further away? (a)Ans: current leads emf => circuit more capacitive => XL < XC => 1/C > L. To transfer more power we want (fixed)XL nearer XC => decrease XC => decrease 1/C => increase C (b) Ans: XL < XC => ωd < ω = 1/sqrt(LC), C increases => ω decreases and closer to resonance. Ch31-38/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 19
• 20. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Sample problem 31-8 Series RLC circuit driven with emfrms=120. V at fd = 60.0 Hz contains R= 200 Ω, inductance XL = 80.0 Ω, and capacitance with XC=150 Ω. (a) what are power factor cos φ and phase constant φ of circuit? two angles from calculated cos φ select correct by: circuit mostly capacitive => current leads emf => φ is < 0 (b) What is power dissipated in resistance? Pavg = Irms emfrmscos φ or Pavg = I2rmsR = (emfrms2/Z2)R Ch31-39/41 Sample problem 31-8 cont Series RLC circuit driven with emfrms=120. V at fd = 60.0 Hz contains R= 200 Ω, inductance XL = 80.0 Ω, and capacitance with XC=150 Ω. (c) what new capacitance would be needed to maximize power Resonance => XL = XC => Cnew = 1/(2pi fd XL) => ans Ch31-40/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 20
• 21. Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals ofPhysics, Copyright 2005 by Wiley and Sons Transformers We drop rms subscript here, but it is implied always Ch31-41/41Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 21