Lect20 - Physics 212 Lecture 20 AC Circuits Maximum...

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Unformatted text preview: Physics 212 Lecture 20 AC Circuits Maximum currents & voltages Phasors: A Simple Tool Physics 212 Lecture 20, Slide 1 Music Who is the Artist? A) B) C) D) E) Allman Brothers Jefferson Airplane CCR The Band ZZ Top WHY? Have you seen the movie? Best music movie ever?? 1978: Martin Scorcese 1978: Scorcese Clapton, Dylan. Dr. John, Joni Mitchell, Clapton, Van the Man, Neil Young, … ALSO: theme of the week Bands who have played with Bob Dylan Bands (in honor of his performance here Saturday !!) (in Physics 212 Lecture 20, Slide 2 Your Comments “This was confusing... I guess we can start This with why things are "in" and "out" of phase with each other?” with Think about what the voltage across Think each element is proportional to: Q, I, dI/dt dI/dt “Just more examples and problems about Just Absolutely Will Do ! Absolutely Will Do ! putting the whole thing together. It was easy to understand until we threw all three together Impedance Triangle fl Phases !! Impedance and started talking about XC and XL” and Rotation of Phasors fl Voltages at any time Rotation Phasors “Please discuss the impedance triangle.” “Star Trek Phasors. I am not sure how they Star Phasors am work after seeing this.“ work “I will be perfectly honest: I had the will prelecture playing in the background as I prelecture playing stayed up really late doing all the homework I'd let slide because I was studying for the Physics exam. So I would love it if you'd discuss everything, because I don't think this whole subliminal message thing is working out.” subliminal 05 We’ll see what we can do 35 30 25 20 15 10 5 0 Confused Confident Physics 212 Lecture 20, Slide 3 Resistors ε = Vmaxsin(ωt) R I = VR/R = Vmax/R sin(ωt) Amplitude = Vmax/R Amplitude Physics 212 Lecture 20, Slide 4 Capacitors Q = CV = CVmaxsin(ωt) I = dQ/dt dQ/dt ε = Vmaxsin(ωt) C I = VmaxωC cos(ωt) Amplitude = Vmax/XC Amplitude 90o where XC = 1/ωC is like the “resistance” is of the capacitor of XC depends on ω Physics 212 Lecture 20, Slide 5 Inductors dI/dt = VL = Vmaxsin(ωt) ε = Vmaxsin(ωt) L I = - Vmax/ωL cos(ωt) Amplitude = Vmax/XL Amplitude 90o where XL = ωL is like the “resistance” is of the inductor of XL depends on ω Physics 212 Lecture 20, Slide 6 RL ACT An RL circuit is driven by an AC generator as shown in the An figure. figure. BB XL = ωL L R As ω Ø 0, so does XL As ω Ø 0, resistance of circuit Ø R current gets bigger For what driving frequency ω of the generator will the For of current through the resistor be largest A) ω large A) B) Current through R doesn’t depend on ω C) ω small small Physics 212 Lecture 20, Slide 7 Summary R Imax = Vmax/R VR in phase with I in Because resistors are simple C Imax = Vmax/XC XC = 1/ωC L Imax = Vmax/XL XL = ωL VC 90o behind I behind Current comes first since it charges capacitor Like a wire at high ω VL 90o ahead of I ahead Opposite of capacitor Like a wire at low ω Like Physics 212 Lecture 20, Slide 8 Makes sense to write everything in Makes terms of I since this is the same since everywhere in a one-loop circuit: everywhere Phasors make this make simple to see simple Imax XL Vmax = Imax XC V 90o behind I behind Imax R C εmax L R Vmax = Imax XL V 90o ahead of I ahead Imax XC Vmax = Imax R V in phase with I in “Do you have any fancy-schmancy simulations for to show me?” Prelecture animation Always looks the same. Only the lengths will Only change change Physics 212 Lecture 20, Slide 9 Imax XC The voltages still add up The C εmax But now we are adding But vectors: Imax XL L R Imax R Imax XL εmax Imax R Imax R Imax XC Imax XC Imax XL Imax R Imax XC Imax XL εmax Physics 212 Lecture 20, Slide 10 10 Imax XC Making this simpler… C εmax Imax XL L Imax XL R Imax R Imax XL εmax Imax R Imax XC Imax R Imax XC Physics 212 Lecture 20, Slide 11 11 Imax XC Making this simpler… C εmax L Imax XL R Imax R Imax XL εmax = Imax Z max Imax R Imax(XL-XC) Imax R Imax XC Physics 212 Lecture 20, Slide 12 12 Imax XC Making this simpler… C εmax L Imax XL R Imax R εmax = Imax Z max Imax(XL-XC) Imax R Physics 212 Lecture 20, Slide 13 13 Imax XC Making this simpler… C εmax Imax XL R εmax = Imax Z max Imax R Imax(XL-XC) φ L Imax R (XL-XC) φ R Impedance Triangle X L − XC tan (φ ) = R Physics 212 Lecture 20, Slide 14 14 Imax XC Summary: C VCmax= Imax XC max εmax VLmax= Imax XL max L Imax XL R VRmax= Imax R max εmax = Imax Z max Imax R Imax = εmax / Z max Z = R + ( X L − XC ) 2 X L − XC tan (φ ) = R 2 (XL-XC) φ R Physics 212 Lecture 20, Slide 15 15 Example: RL Circuit Xc=0 Example: εmax L Imax XL R Imax R Imax XL εmax Imax R Physics 212 Lecture 20, Slide 16 16 Preflight 2 BB Draw Voltage Phasors Imax XL εmax Imax R A B C 60 50 40 30 20 10 0 Physics 212 Lecture 20, Slide 17 17 Preflight 4 BB Draw Voltage Phasors Imax XL εmax Imax R A B C 60 50 40 30 20 10 0 Physics 212 Lecture 20, Slide 18 18 Preflight 6 BB The CURRENT is THE CURRENT Imax XL φ A B C D εmax Imax R φ is the phase between generator and current 50 40 30 20 10 0 Physics 212 Lecture 20, Slide 19 19 Preflight 8 BB A B C What does the voltage phasor diagram look like when the current IXL is a maximum? IXL ε IR ε 50 40 IR 30 20 IXc 10 0 IXc Physics 212 Lecture 20, Slide 20 20 Preflight 10 IXL ε IR IR A B C IXc BB 40 IXc ε 30 20 10 IXL What does the voltage phasor diagram look like when the capacitor is fully charged? 50 0 Physics 212 Lecture 20, Slide 21 21 Preflight 12 IXL ε IR IR A B C IXc BB 40 IXc ε 30 20 10 IXL What does the voltage phasor diagram look like when the voltage across capacitor is at its positive maximum? 50 0 Physics 212 Lecture 20, Slide 22 22 Calculation C 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. V~ L R 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 Physics 212 Lecture 20, Slide 23 23 Calculation C 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. V~ L R 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 BB (D) Not enough information • This information is determined from the phase – Current leads voltage 45ο VL VL = ImaxXL VC = ImaxXC VR (phase of current) V VC IR V leads Physics 212 Lecture 20, Slide 24 24 Calculation C 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. V~ L R What is XL, the reactance of the inductor, at this frequency? What is Z, the total impedance of the circuit? (A) 70.7 kΩ (B) 50 kΩ (C) 35.4 kΩ BB (D) 21.1 kΩ Vmax 100V Z= = = 50k Ω I max 2mA Physics 212 Lecture 20, Slide 25 25 Calculation C 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. V~ R Z = 50kΩ What is XL, the reactance of the inductor, at this frequency? What is R? (A) 70.7 kΩ L sin(45)=.707 cos(45)=.707 (B) 50 kΩ (C) 35.4 kΩ (D) 21.1 kΩ • Determined from impedance triangle R 45ο Z=50kΩ (XC-XL) R cos(45) = Z BB R = Z cos(45o) = 50 kΩ x 0.707 = 35.4 kΩ Physics 212 Lecture 20, Slide 26 26 Calculation C 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. V~ R Z = 50kΩ What is XL, the reactance of the inductor, at this frequency? (A) 70.7 kΩ (B) 50 kΩ We start with the impedance triangle: R 45ο Z (C) 35.4 kΩ XC − X L = tan 45° = 1 R L R = 35.4kΩ (D) 21.1 kΩ XL = XC - R BB What is XC ? (XC-XL) VCmax = ImaxXC XL = 56.5 kΩ – 35.4 kΩ 113 XC = = 56.5kΩ 2 Physics 212 Lecture 20, Slide 27 27 ...
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This note was uploaded on 02/09/2012 for the course PHYSICS 212 taught by Professor Mestre during the Spring '11 term at University of Illinois at Urbana–Champaign.

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