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• 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 &gt; XC for f&gt;f0 and vice versa.
• demo with RLC and oscilliscope
• ### Lect13 handout

1. 1. AC Circuit Phasors Physics 102: Lecture 13 <ul><li>I = I max sin(2  ft) </li></ul><ul><li>V R = I max R sin(2  ft) </li></ul><ul><ul><li>V R in phase with I </li></ul></ul><ul><li>V C = I max X C sin(2  ft-  ) </li></ul><ul><ul><li>V C lags I </li></ul></ul><ul><li>V L = I max X L sin(2  ft+  ) </li></ul><ul><ul><li>V L leads I </li></ul></ul>I t V L V C V R L R C
2. 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. 3. Time Dependence in AC Circuits <ul><li>Write down Kirchoff’s Loop Equation: </li></ul><ul><li>V gen (t) = V L (t) + V R (t) + V C (t) at every instant of time </li></ul><ul><ul><li>However … </li></ul></ul><ul><ul><li>V gen,max  V L,max +V R,max +V C,max </li></ul></ul><ul><ul><li>Maximum reached at different times for R, L, C </li></ul></ul>I t V L V C V R We solve this using phasors V gen L R C
4. 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. 5. Drawing Phasor Diagrams (4) Generator vector (coming soon) V L,max <ul><li>(2) Inductor vector: upwards </li></ul><ul><ul><li>Length given by V L,max (or X L ) </li></ul></ul>V C,max <ul><li>(3) Capacitor vector: downwards </li></ul><ul><ul><li>Length given by V C,max (or X C ) </li></ul></ul>V R,max <ul><li>Resistor vector: to the right </li></ul><ul><ul><li>Length given by V R,max (or R) </li></ul></ul>V C (t) V R (t) V L (t) <ul><li>(5) Rotate entire thing counter-clockwise </li></ul><ul><ul><li>Vertical components give instantaneous voltage across R, C, L </li></ul></ul>
6. 6. Phasor Diagrams <ul><li>I = I max sin(2  ft) </li></ul><ul><li>V R = I max R sin(2  ft) </li></ul><ul><li>V C = I max X C sin(2  ft–  ) </li></ul><ul><ul><ul><li>= –I max X C cos(2  ft) </li></ul></ul></ul><ul><li>V L = I max X L sin(2  ft +  ) </li></ul><ul><ul><ul><li>= I max X L cos(2  ft) </li></ul></ul></ul>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. 7. Phasor Diagram Practice <ul><li>Label the vectors that corresponds to the resistor, inductor and capacitor. </li></ul><ul><li>Which element has the largest voltage across it at the instant shown? </li></ul><ul><li>1) R 2) C 3) L </li></ul><ul><li>Is the voltage across the inductor </li></ul><ul><li>1) increasing or 2) decreasing? </li></ul><ul><li>Which element has the largest maximum voltage across it? </li></ul><ul><li>1) R 2) C 3) L </li></ul>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. 8. Kirchhoff: generator voltage <ul><li>Instantaneous voltage across generator (V gen ) must equal sum of voltage across all of the elements at all times: </li></ul>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. 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. 10. Drawing Phasor Diagrams V C V R V L <ul><li>(5) Rotate entire thing counter-clockwise </li></ul><ul><ul><li>Vertical components give instantaneous voltage across R, C, L </li></ul></ul>V L,max <ul><li>(2) Capacitor vector: Downwards </li></ul><ul><ul><li>Length given by V C,max (or X C ) </li></ul></ul>V C,max <ul><li>(3) Inductor vector: Upwards </li></ul><ul><ul><li>Length given by V L,max (or X L ) </li></ul></ul>V R,max <ul><li>Resistor vector: to the right </li></ul><ul><ul><li>Length given by V R,max (or R) </li></ul></ul><ul><li>(4) Generator vector: add first 3 vectors </li></ul><ul><ul><li>Length given by V gen,max (or Z) </li></ul></ul>V gen,max V gen
11. 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. 12. Problem Time! <ul><li>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. </li></ul>I max = 2.5/2.76 = .91 Amps I max = V gen,max /Z Example L R C
13. 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. 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. 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. 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. 17. ACT: Resonance <ul><li>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. </li></ul><ul><li>V C,max > V L,max </li></ul><ul><li>V C,max = V L,max </li></ul><ul><li>V C,max < V L,max </li></ul><ul><li>Depends on R </li></ul>L R C
18. 18. Summary of Resonance <ul><li>At resonance </li></ul><ul><ul><li>Z is minimum (=R) </li></ul></ul><ul><ul><li>I max is maximum (=V gen,max /R) </li></ul></ul><ul><ul><li>V gen is in phase with I </li></ul></ul><ul><ul><li>X L = X C V L (t) = -V C (t) </li></ul></ul><ul><li>At lower frequencies </li></ul><ul><ul><li>X C > X L V gen lags I </li></ul></ul><ul><li>At higher frequencies </li></ul><ul><ul><li>X C < X L V gen lead I </li></ul></ul>I max (X L -X C ) I max X L I max X C I max R V gen,max 
19. 19. Power in AC circuits <ul><li>The voltage generator supplies power. </li></ul><ul><ul><li>Only resistor dissipates power. </li></ul></ul><ul><ul><li>Capacitor and Inductor store and release energy. </li></ul></ul><ul><li>P(t) = I(t)V R (t) oscillates so sometimes power loss is large, sometimes small. </li></ul><ul><li>Average power dissipated by resistor: </li></ul><ul><ul><li>P = ½ I max V R,max </li></ul></ul><ul><li> = ½ I max V gen,max cos(  ) </li></ul><ul><li> = I rms V gen,rms cos(  ) </li></ul>
20. 20. AC Summary <ul><li>Resistors: V R,max =I max R </li></ul><ul><li>In phase with I </li></ul><ul><li>Capacitors: V C,max =I max X C X c = 1/(2  f C) </li></ul><ul><li>Lags I </li></ul><ul><li>Inductors: V L,max =I max X L X L = 2  f L </li></ul><ul><li>Leads I </li></ul><ul><li>Generator: V gen,max =I max Z Z = √R 2 +(X L -X C ) 2 </li></ul><ul><li>Can lead or lag I tan(  ) = (X L -X C )/R </li></ul><ul><li>Power is only dissipated in resistor: </li></ul><ul><li>P = ½I max V gen,max cos(  ) </li></ul>