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 35 30 25 20 15 10 5 0 Confused Avg = 2.6 Confident Physics 212 Lecture 20, Slide 1 Music Who is the Artist? A) B) C) D) E) Professor Longhair Johnny Adams David Egan Dr. John Allen Toussaint classics classics The Theme of the week: New Orleans piano players New A great piano CD Physics 212 Lecture 20, Slide 2 Physics 212 Lecture 20 AC Circuits Maximum currents & voltages Phasors: A Simple Tool 35 30 25 20 15 10 5 0 Confused Avg = 2.6 Confident Physics 212 Lecture 20, Slide 3 Resistors ε = εmaxsin(ωt) R I = VR/R = Vmax/R sin(ωt) Amplitude = Vmax/R Amplitude Physics 212 Lecture 20, Slide 4 Capacitors Q = VC = Cεmaxsin(ωt) I = dQ/dt dQ/dt ε = εmaxsin(ωt) C I = VmaxωC cos(ωt) Amplitude = Vmax/XC Amplitude 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 = εmaxsin(ωt) ε = εmaxsin(ωt) L I = - Vmax/ωL cos(ωt) Amplitude = Vmax/XL Amplitude 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. L BB Imax = Vmax/XL XL = ωL R 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 V in phase with I in Because resistors are simple C Imax = Vmax/XC XC = 1/ωC L Imax = Vmax/XL XL = ωL V 90o behind I behind Current comes first since it charges capacitor Like a wire at high ω V 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 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~ What is XL, the reactance of the inductor, at this frequency? What is R? (A) 70.7 kΩ L R Z = 50kΩ sin(45)=.707 cos(45)=.707 (B) 50 kΩ (C) 35.4 kΩ (D) 21.1 kΩ • Determined from impedance triangle R 45ο Z= 50k Ω (XC-XL) R cos(45) = Z R = Z cos(45) BB = 50k Ω (.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 X L = XC − R BB What is XC ? (XC-XL) VC max = I max X C X L = 56.5kΩ − 35.4kΩ 113 XC = = 56.5kΩ 2 Physics 212 Lecture 20, Slide 27 27 ...
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This note was uploaded on 09/22/2011 for the course PHYSICS 212 taught by Professor Selig during the Fall '10 term at University of Illinois, Urbana Champaign.

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