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Lect13 handout

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

    • 1. AC Circuit Phasors Physics 102: Lecture 13
      • I = I max sin(2  ft)
      • V R = I max R sin(2  ft)
        • V R in phase with I
      • V C = I max X C sin(2  ft-  )
        • V C lags I
      • V L = I max X L sin(2  ft+  )
        • V L leads I
      I t V L V C V R L R C
    • 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. Time Dependence in AC Circuits
      • Write down Kirchoff’s Loop Equation:
      • V gen (t) = V L (t) + V R (t) + V C (t) at every instant of time
        • However …
        • V gen,max  V L,max +V R,max +V C,max
        • Maximum reached at different times for R, L, C
      I t V L V C V R We solve this using phasors V gen L R C
    • 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. Drawing Phasor Diagrams (4) Generator vector (coming soon) V L,max
      • (2) Inductor vector: upwards
        • Length given by V L,max (or X L )
      V C,max
      • (3) Capacitor vector: downwards
        • Length given by V C,max (or X C )
      V R,max
      • Resistor vector: to the right
        • Length given by V R,max (or R)
      V C (t) V R (t) V L (t)
      • (5) Rotate entire thing counter-clockwise
        • Vertical components give instantaneous voltage across R, C, L
    • 6. Phasor Diagrams
      • I = I max sin(2  ft)
      • V R = I max R sin(2  ft)
      • V C = I max X C sin(2  ft–  )
          • = –I max X C cos(2  ft)
      • V L = I max X L sin(2  ft +  )
          • = I max X L cos(2  ft)
      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
    • 7. Phasor Diagram Practice
      • Label the vectors that corresponds to the resistor, inductor and capacitor.
      • Which element has the largest voltage across it at the instant shown?
      • 1) R 2) C 3) L
      • Is the voltage across the inductor
      • 1) increasing or 2) decreasing?
      • Which element has the largest maximum voltage across it?
      • 1) R 2) C 3) L
      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
    • 8. Kirchhoff: generator voltage
      • Instantaneous voltage across generator (V gen ) must equal sum of voltage across all of the elements at all times:
      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”
    • 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. Drawing Phasor Diagrams V C V R V L
      • (5) Rotate entire thing counter-clockwise
        • Vertical components give instantaneous voltage across R, C, L
      V L,max
      • (2) Capacitor vector: Downwards
        • Length given by V C,max (or X C )
      V C,max
      • (3) Inductor vector: Upwards
        • Length given by V L,max (or X L )
      V R,max
      • Resistor vector: to the right
        • Length given by V R,max (or R)
      • (4) Generator vector: add first 3 vectors
        • Length given by V gen,max (or Z)
      V gen,max V gen
    • 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. Problem Time!
      • An AC circuit with R= 2  , C = 15 mF, and L = 30 mH is driven by a generator with voltage V(t)=2.5 sin(8  t) Volts. Calculate the maximum current in the circuit, and the phase angle.
      I max = 2.5/2.76 = .91 Amps I max = V gen,max /Z Example L R C
    • 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. ACT: Resonance
      • The AC circuit to the right is being driven at its resonance frequency. Compare the maximum voltage across the capacitor with the maximum voltage across the inductor.
      • V C,max > V L,max
      • V C,max = V L,max
      • V C,max < V L,max
      • Depends on R
      L R C
    • 18. Summary of Resonance
      • At resonance
        • Z is minimum (=R)
        • I max is maximum (=V gen,max /R)
        • V gen is in phase with I
        • X L = X C V L (t) = -V C (t)
      • At lower frequencies
        • X C > X L V gen lags I
      • At higher frequencies
        • X C < X L V gen lead I
      I max (X L -X C ) I max X L I max X C I max R V gen,max 
    • 19. Power in AC circuits
      • The voltage generator supplies power.
        • Only resistor dissipates power.
        • Capacitor and Inductor store and release energy.
      • P(t) = I(t)V R (t) oscillates so sometimes power loss is large, sometimes small.
      • Average power dissipated by resistor:
        • P = ½ I max V R,max
      • = ½ I max V gen,max cos(  )
      • = I rms V gen,rms cos(  )
    • 20. AC Summary
      • Resistors: V R,max =I max R
      • In phase with I
      • Capacitors: V C,max =I max X C X c = 1/(2  f C)
      • Lags I
      • Inductors: V L,max =I max X L X L = 2  f L
      • Leads I
      • Generator: V gen,max =I max Z Z = √R 2 +(X L -X C ) 2
      • Can lead or lag I tan(  ) = (X L -X C )/R
      • Power is only dissipated in resistor:
      • P = ½I max V gen,max cos(  )