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AC Circuit  Phasors Physics 102:  Lecture 13 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],I t V L V C V R L R C
Peak & RMS values  in AC Circuits (REVIEW) When asking about RMS or Maximum values relatively simple expressions V R, max  = I max R  V C, max  = I max X C   V L, max  = I max X L   L R C
Time Dependence  in AC Circuits ,[object Object],[object Object],[object Object],[object Object],[object Object],I t V L V C V R We solve this using  phasors V gen L R C
I  = I max sin(2  ft )  (   = 2  ft ) V L  = I max X L   sin(2  ft  +   )   V R  = I max R  sin(2  ft ) V C  = I max X C  sin(2  ft  –   ) Graphical representation of voltages  I max X L  I max R  I max X C L R C
Drawing Phasor Diagrams (4) Generator vector  (coming soon) V L,max ,[object Object],[object Object],V C,max ,[object Object],[object Object],V R,max ,[object Object],[object Object],V C (t) V R (t) V L (t) ,[object Object],[object Object]
Phasor Diagrams ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Voltage across  resistor  is always  in phase  with current!  Voltage across  capacitor  always  lags  current!  Voltage across  inductor  always  leads  current! Instantaneous Values: I max R I max R sin(2  ft) I max X L  cos(2  ft) -I max X C  cos(2  ft) I max X L I max X C
Phasor Diagram Practice ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],V L V C V R Inductor Leads Capacitor Lags R: It has largest vertical component Decreasing, spins counter clockwise Inductor, it has longest line. Example
Kirchhoff: generator voltage ,[object Object],V L,max -V C,max V L,max =I max X L V C,max =I max X C V R,max =I max R V gen,max =I max Z  V gen  (t) = V R  (t)   +V C  (t)   +V L  (t) Define  impedance  Z: V gen,max  ≡ I max  Z “ Impedance Triangle” “ phase angle”
Phase angle   I  = I max sin(2  ft ) V gen  = I max Z  sin(2  ft +   )    is positive in this particular case. 2  ft I max I max Z 2  ft +  
Drawing Phasor Diagrams V C V R V L ,[object Object],[object Object],V L,max ,[object Object],[object Object],V C,max ,[object Object],[object Object],V R,max ,[object Object],[object Object],[object Object],[object Object],V gen,max V gen
ACTS 13.1, 13.2, 13.3 When does V gen  = V R  ? When does V gen  = 0 ? The phase angle is: (1) positive  (2) negative  (3) zero? time 1 time 2 time 3 time 4
Problem Time! ,[object Object],I max  = 2.5/2.76 = .91 Amps I max  = V gen,max  /Z Example L R C
ACT: Voltage Phasor Diagram At this instant, the voltage across the generator is maximum. What is the voltage across the resistor at this instant? 1) V R  = I max R  2) V R  = I max R sin(  ) 3) V R  = I max R cos(  ) I max  X L I max  X C I max  R V gen,max 
Resonance and the Impedance Triangle R (X L -X C ) Z  X L  and X C  point opposite.  When adding, they tend to cancel! When X L  = X C  they completely cancel and Z = R.  This is resonance!   V max,gen  = I max  Z I max (X L -X C ) I max X L I max X C I max R V gen,max  L R C
Resonance R  is independent of f R X L  increases with f X L X C  decreases with f X C Z :  X L  and X C  subtract Z X C  =  1/(2  fC ) X L  =  2  fL Resonance:  X L  = X C f 0 Z is minimum at resonance frequency!
Resonance R  is independent of f X L  increases with f X C  decreases with f Z :  X L  and X C  subtract Z X C  =  1/(2  fC ) X L  =  2  fL Resonance:  X L  = X C Current I max  = V gen,max /Z f 0 Current is maximum at resonance frequency!
ACT: Resonance ,[object Object],[object Object],[object Object],[object Object],[object Object],L R C
Summary of Resonance ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],I max (X L -X C ) I max X L I max X C I max R V gen,max 
Power in AC circuits ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
AC Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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Lect13 handout

  • 1.
  • 2. Peak & RMS values in AC Circuits (REVIEW) When asking about RMS or Maximum values relatively simple expressions V R, max = I max R V C, max = I max X C V L, max = I max X L L R C
  • 3.
  • 4. I = I max sin(2  ft ) (  = 2  ft ) V L = I max X L sin(2  ft +  ) V R = I max R sin(2  ft ) V C = I max X C sin(2  ft –  ) Graphical representation of voltages  I max X L  I max R  I max X C L R C
  • 5.
  • 6.
  • 7.
  • 8.
  • 9. Phase angle  I = I max sin(2  ft ) V gen = I max Z sin(2  ft +  )  is positive in this particular case. 2  ft I max I max Z 2  ft + 
  • 10.
  • 11. ACTS 13.1, 13.2, 13.3 When does V gen = V R ? When does V gen = 0 ? The phase angle is: (1) positive (2) negative (3) zero? time 1 time 2 time 3 time 4
  • 12.
  • 13. ACT: Voltage Phasor Diagram At this instant, the voltage across the generator is maximum. What is the voltage across the resistor at this instant? 1) V R = I max R 2) V R = I max R sin(  ) 3) V R = I max R cos(  ) I max X L I max X C I max R V gen,max 
  • 14. Resonance and the Impedance Triangle R (X L -X C ) Z  X L and X C point opposite. When adding, they tend to cancel! When X L = X C they completely cancel and Z = R. This is resonance! V max,gen = I max Z I max (X L -X C ) I max X L I max X C I max R V gen,max  L R C
  • 15. Resonance R is independent of f R X L increases with f X L X C decreases with f X C Z : X L and X C subtract Z X C = 1/(2  fC ) X L = 2  fL Resonance: X L = X C f 0 Z is minimum at resonance frequency!
  • 16. Resonance R is independent of f X L increases with f X C decreases with f Z : X L and X C subtract Z X C = 1/(2  fC ) X L = 2  fL Resonance: X L = X C Current I max = V gen,max /Z f 0 Current is maximum at resonance frequency!
  • 17.
  • 18.
  • 19.
  • 20.

Editor's Notes

  1. 1
  2. 1
  3. Note: VR=IR VL=IXL VC=IXC
  4. Note the lagging and leading. Do demo with phasor board
  5. Have them go back and fill in (4)
  6. demo with RLC and oscilliscope: Note that XL > XC for f>f0 and vice versa.
  7. demo with RLC and oscilliscope