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Physics 212 Lecture 20

Physics 212 Lecture 20. AC Circuits Maximum currents & voltages Phasors: A Simple Tool. Music. Who is the Artist? Derek Trucks Band Robert Plant & Alison Kraus CCR Emmylou Harris & Gregg Allman Led Zeppelin. Theme of the week

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Physics 212 Lecture 20

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  1. Physics 212 Lecture 20 AC Circuits Maximum currents & voltages Phasors: A Simple Tool

  2. Music • Who is the Artist? • Derek Trucks Band • Robert Plant & Alison Kraus • CCR • Emmylou Harris & Gregg Allman • Led Zeppelin Theme of the week Bluegrass fiddlers (Mark O’Connor, Alison Kraus) doing something different !!

  3. Exam Tonight • Regular exam 7:00 • Conflict exam 5:15 in 100 MSEB • Covers material in Lectures 9 – 18 • Bring your ID: Rooms determined by discussion section (see link) • Final Exam • “Regular”: Fri. May 4, 1:30; “Conflict”: Fri. May 11, 8:00 • Sign up for the conflict in the gradebook if you wish

  4. Your Comments “For the first time, I can actually say that I don’t hate circuits!!!! What an accomplishment :)" Change today – driven circuit: Ea.c. “I thought the checkpoints were difficult! I understood the prelecture concept. Everything is pretty much tied to KVR. However I don’t completely get the correlation between maximum voltage or currents across inductors, resistors and capacitors.” “This stuff is very confusing. I like the phasor diagrams though.” “I don’t get how to read the phasordiagram” THE KEY TECHNIQUE: Draw the phasor diagram (and impedance triangle) for each problem! “You putting a lecture on the same day as an exam just killed a puppy.” “I could really use some AC to cool off my brain before this exam...” “Set phasers to maximum stun.” 05

  5. Resistors R Amplitude = Vmax/R e = Vmaxsin(wt) I = VR/R = Vmax/Rsin(wt)

  6. Capacitors C Amplitude = Vmax/XC 90o Q = CV =CVmaxsin(wt) I = dQ/dt e = Vmaxsin(wt) I =VmaxwCcos(wt) where XC = 1/wC: “Reactance” is like the “resistance”of the capacitor XC depends onw

  7. Inductors Amplitude = Vmax/XL L 90o L dI/dt= VL =Vmaxsin(wt) e = Vmaxsin(wt) I = -Vmax/wLcos(wt) where XL = wL: “Reactance” is like the “resistance”of the inductor XL depends onw

  8. CheckPoint 1a An RL circuit is driven by an AC generator as shown in the figure The voltages across the resistor and generator are A) Always out of phase B) Always in phaseC) Sometimes in and sometimes out of phase A B C “Resistor and inductor are out of phase, and generator is determined by vector sum, thus out of phase” “The phasors for resistor and generator are in phase.”“it depends on the voltage and resistance”

  9. CheckPoint 1a Imax XL emax Imax R An RL circuit is driven by an AC generator as shown in the figure Draw Voltage Phasors The voltages across the resistor and generator are A) Always out of phase B) Always in phaseC) Sometimes in and sometimes out of phase A B C

  10. CheckPoint 1b An RL circuit is driven by an AC generator as shown in the figure The voltages across the resistor and inductor are A) Always out of phase B) Always in phaseC) Sometimes in and sometimes out of phase “Voltage across resistor always lags the voltage across the inductor by 90 degrees.” “Their voltage is scaled by the current produced by the generator” “Dependent on time”

  11. Imax XL emax Imax R CheckPoint 1b An RL circuit is driven by an AC generator as shown in the figure Draw Voltage Phasors The voltages across the resistor and inductor are A) Always out of phase B) Always in phaseC) Sometimes in and sometimes out of phase

  12. CheckPoint 1c An RL circuit is driven by an AC generator as shown in the figure The phase difference between the CURRENT through the resistor and inductor A) Is always zero B) Is always 90oC) Depends on the value of L and RD) Depends on L, R and the generator voltage “Current = current, it’s the same current because it’s the same circuit.” “The phase difference will remain constant at 90 degrees.” “the phase is dependent on L and R” “current = v/R”

  13. CheckPoint 1c Imax XL emax f Imax R f is the phase between generator and current An RL circuit is driven by an AC generator as shown in the figure The CURRENT is THE CURRENT The phase difference between the CURRENT through the resistor and inductor A) Is always zero B) Is always 90oC) Depends on the value of L and RD) Depends on L, R and the generator voltage

  14. Review R C Imax =Vmax/XC VC 90o behind I XC = 1/wC “Reactance” L Imax =Vmax/XL VL 90o ahead of I XL = wL “Reactance” Imax =Vmax/R VR in phase with I Because resistors are simple Current comes first since itcharges capacitor Like a wire at highw Opposite of capacitor Like a wire at low w

  15. The Driven RLC Circuit Phasors make this simple to see C Imax XL L R Imax R Imax XC Makes sense to write everything in terms of I since this is the same everywhere in a one-loop circuit: Vmax =Imax XC V 90o behind I Vmax =Imax XL emax V 90o ahead of I Vmax =Imax R Always looks the same.Only the lengths will change V in phase with I

  16. Imax XC C Imax XL emax L R Imax R Imax XL Imax XL Imax R emax Imax XC Imax R Imax R Imax XL emax Imax XC Imax XC The voltages still add up Now we are adding three vectors:

  17. Imax XC C Imax XL emax L R Imax R Imax XL Imax R Imax XC Making this simpler… Imax XL emax Imax R Imax XC

  18. Imax XC C Imax XL emax L R Imax R Imax XL Imax R Imax XC Making this simpler… emax = Imax Z Imax(XL-XC) Imax R

  19. Imax XC C Imax XL emax L R Imax R emax = Imax Z Imax(XL-XC) Imax R Making this simpler…

  20. Imax XC C Imax XL emax L R Imax R f emax = Imax Z Imax(XL-XC) f Imax R Making this simpler… (XL-XC) R Impedance Triangle

  21. Imax XC C Imax XL emax L R Imax R Summary: VCmax= Imax XC VLmax= Imax XL VRmax= Imax R emax = Imax Z Imax =emax / Z (XL-XC) f R

  22. L Imax XL emax R Imax R emax Example: RL Circuit Xc=0 Imax XL Imax R

  23. CheckPoint 2a A driven RLC circuit is represented by the phasor diagram below. The vertical axis of the phasor diagram represents voltage. When the current through the circuit is maximum, what is the potential difference across the inductor? A) VL = 0 B) VL = VL,max/2 C) VL = VL,max “when current is max the Xl vector is zero” “There is a voltage drop across the resistor too.” “The current should be max when the voltage is also at max.”

  24. CheckPoint 2a e IXL What does the voltage phasor diagram look like when the current is a maximum? IR IXc e IR IXc IXL A driven RLC circuit is represented by the phasor diagram below. The vertical axis of the phasor diagram represents voltage. When the current through the circuit is maximum, what is the potential difference across the inductor? A) VL = 0 B) VL = VL,max/2 C) VL = VL,max

  25. CheckPoint 2b A driven RLC circuit is represented by the phasor diagram below. CheckPoint 2b A B C When the capacitor is fully charged, what is the magnitude of the voltage across the inductor? A) VL = 0 B) VL = VL,max/2 C) VL = VL,max “The voltage across the capacitor would be at a maximum, so the voltage across the inductor would be 0” “half voltage because there is a resistor and capacitor” “its negative but the magnitude is the same as its max”

  26. CheckPoint 2b e IXL IR What does the voltage phasor diagram look like when the capacitor is fully charged? IXc e IR IXc IXL A driven RLC circuit is represented by the phasor diagram below. When the capacitor is fully charged, what is the magnitude of the voltage across the inductor? A) VL = 0 B) VL = VL,max/2 C) VL = VL,max

  27. CheckPoint 2c A driven RLC circuit is represented by the phasor diagram below. CheckPoint 2c When the voltage across the capacitor is at its positive maximum, VC = +VC,max, what is the voltage across the inductor ? A) VL = 0 B) VL = VL,maxC) VL = -VL,max “all energy in capacitor” “The voltages have to be equal.” “The capacitor and inductor voltage phasors point in opposite directions by definition.”

  28. CheckPoint 2c e IXL IR What does the voltage phasor diagram look like when the voltage across capacitor is at its positive maximum? IXc e IR IXc IXL A driven RLC circuit is represented by the phasor diagram below. CheckPoint 2c When the voltage across the capacitor is at its positive maximum, VC = +VC,max, what is the voltage across the inductor ? A) VL = 0 B) VL = VL,maxC) VL = -VL,max

  29. Calculation C V ~ L R Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o L and R are unknown. What is XL, the reactance of the inductor, at this frequency? • Conceptual Analysis • The maximum voltage for each component is related to its reactance and to the maximum current. • The impedance triangle determines the relationship between the maximum voltages for the components • Strategic Analysis • Use Vmax and Imax to determine Z • Use impedance triangle to determine R • Use VCmax and impedance triangle to determine XL

  30. Calculation C V ~ L R IR 45o V VL VR (phase of current) V leads VC Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o L and R are unknown. What is XL, the reactance of the inductor, at this frequency? Compare XL and XC at this frequency: (A) XL < XC (B) XL = XC (C) XL > XC (D) Not enough information • This information is determined from the phase • Current leads voltage VL = ImaxXL VC = ImaxXC

  31. Calculation C V ~ L R Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o L and R are unknown. What is XL, the reactance of the inductor, at this frequency? What is Z, the total impedance of the circuit? (A) (B) (C) (D) 50 kW 35.4 kW 70.7 kW 21.1 kW

  32. Calculation C V ~ L R • Determined from impedance triangle R R = Z cos(45o) 45o (XC-XL) = 50 kW x 0.707 Z=50kW = 35.4 kW Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o L and R are unknown. What is XL, the reactance of the inductor, at this frequency? Z = 50kW sin(45)=.707 cos(45)=.707 What is R? (A) (B) (C) (D) 70.7 kW 35.4 kW 21.1 kW 50 kW

  33. Calculation C V ~ L R We start with the impedance triangle: XL = XC - R R 45o (XC-XL) Z XL = 56.5 kW – 35.4 kW Consider the harmonically driven series LCR circuit shown. Vmax = 100 V Imax = 2 mA VCmax = 113 V The current leads generator voltage by 45o L and R are unknown. What is XL, the reactance of the inductor, at this frequency? Z = 50kW R = 35.4kW (A) (B) (C) (D) 70.7 kW 35.4 kW 21.1 kW 50 kW What is XC ? VCmax = ImaxXC

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