1 / 20

AC Circuit Phasors

L. R. C. Physics 102: Lecture 13. AC Circuit Phasors. I = I max sin(2 p ft) V R = I max R sin(2 p ft) V R in phase with I. I. V R. V C = I max X C sin(2 p ft- p/2 ) V C lags I. t. V L. V L = I max X L sin(2 p ft+ p/2 ) V L leads I. V C. L. R. C.

agilley
Télécharger la présentation

AC Circuit Phasors

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. L R C Physics 102:Lecture 13 AC Circuit Phasors • I = Imaxsin(2pft) • VR = ImaxR sin(2pft) • VR in phase with I I VR • VC = ImaxXC sin(2pft-p/2) • VC lags I t VL • VL = ImaxXL sin(2pft+p/2) • VL leads I VC

  2. L R C Peak & RMS values in AC Circuits (REVIEW) When asking about RMS or Maximum values relatively simple expressions VR,max = ImaxR VC,max = ImaxXC VL,max = ImaxXL

  3. L R C Time Dependence in AC Circuits Vgen Write down Kirchoff’s Loop Equation: Vgen(t) = VL(t) + VR(t) + VC(t)at every instant of time • However … • Vgen,maxVL,max+VR,max+VC,max • Maximum reached at different times for R, L, C I VR t VL VC We solve this using phasors

  4. L R C q+p/2 ImaxR ImaxXL q q-p/2 ImaxXC Graphical representation of voltages I = Imaxsin(2pft) (q = 2pft) VL = ImaxXLsin(2pft + p/2) VR = ImaxR sin(2pft) VC = ImaxXC sin(2pft – p/2)

  5. Resistor vector: to the right • Length given by VR,max (or R) VL,max VR,max • (2) Inductor vector: upwards • Length given by VL,max (or XL) VC,max • (3) Capacitor vector: downwards • Length given by VC,max (or XC) VL(t) VR(t) • (5) Rotate entire thing counter-clockwise • Vertical components give instantaneous voltage across R, C, L VC(t) Drawing Phasor Diagrams (4) Generator vector (coming soon)

  6. ImaxXL ImaxR ImaxXL cos(2pft) ImaxR sin(2pft) ImaxXC -ImaxXC cos(2pft) Phasor Diagrams Instantaneous Values: • I = Imaxsin(2pft) • VR = ImaxR sin(2pft) • VC = ImaxXC sin(2pft–p/2) • = –ImaxXC cos(2pft) • VL = ImaxXL sin(2pft + p/2) • = ImaxXL cos(2pft) Voltage across resistor is always in phase with current! Voltage across capacitor always lags current! Voltage across inductor always leads current!

  7. Example Phasor Diagram Practice Inductor Leads Capacitor Lags 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 VR VL R: It has largest vertical component VC Decreasing, spins counter clockwise Inductor, it has longest line.

  8. “phase angle” Kirchhoff: generator voltage • Instantaneousvoltage across generator (Vgen) must equal sum of voltage across all of the elements at all times: VL,max=ImaxXL Vgen (t) = VR (t)+VC (t)+VL (t) Vgen,max=ImaxZ VL,max-VC,max f VR,max=ImaxR VC,max=ImaxXC Define impedance Z: Vgen,max ≡ Imax Z “Impedance Triangle”

  9. ImaxZ Imax 2pft + f 2pft Phase angle f I = Imaxsin(2pft) Vgen = ImaxZ sin(2pft + f) f is positive in this particular case.

  10. Resistor vector: to the right • Length given by VR,max (or R) VL,max VR,max • (2) Capacitor vector: Downwards • Length given by VC,max (or XC) Vgen,max VC,max • (3) Inductor vector: Upwards • Length given by VL,max (or XL) • (4) Generator vector: add first 3 vectors • Length given by Vgen,max (or Z) Vgen Drawing Phasor Diagrams VL VR • (5) Rotate entire thing counter-clockwise • Vertical components give instantaneous voltage across R, C, L VC

  11. time 4 time 3 time 1 time 2 ACTS 13.1, 13.2, 13.3 When does Vgen = 0 ? When does Vgen = VR ? The phase angle is: (1) positive (2) negative (3) zero?

  12. L R C Example Problem Time! An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH is driven by a generator with voltage V(t)=2.5 sin(8pt) Volts. Calculate the maximum current in the circuit, and the phase angle. Imax = Vgen,max /Z Imax = 2.5/2.76 = .91 Amps

  13. Imax XL Vgen,max f Imax R Imax XC 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) VR = ImaxR 2) VR = ImaxR sin(f) 3) VR = ImaxR cos(f)

  14. L ImaxXL R C Vgen,max Imax(XL-XC) f ImaxR ImaxXC Resonance and the Impedance Triangle Vmax,gen = Imax Z Z (XL-XC) f R XL and XC point opposite. When adding, they tend to cancel! When XL = XC they completely cancel and Z = R. This is resonance!

  15. Z is minimum at resonance frequency! f0 Resonance R is independent of f XL increases with f XL = 2pfL XC decreases with f XC = 1/(2pfC) Z R Z: XL and XC subtract XC XL Resonance: XL = XC

  16. Current is maximum at resonance frequency! f0 Resonance R is independent of f XL increases with f XL = 2pfL XC decreases with f XC = 1/(2pfC) Z Imax = Vgen,max/Z Z: XL and XC subtract Current Resonance: XL = XC

  17. L R C 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. • VC,max > VL,max • VC,max = VL,max • VC,max < VL,max • Depends on R

  18. ImaxXL Vgen,max Imax(XL-XC) f ImaxR ImaxXC Summary of Resonance • At resonance • Z is minimum (=R) • Imax is maximum (=Vgen,max/R) • Vgen is in phase with I • XL = XC VL(t) = -VC(t) • At lower frequencies • XC > XL Vgen lags I • At higher frequencies • XC < XL Vgen lead I

  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)VR(t) oscillates so sometimes power loss is large, sometimes small. • Average power dissipated by resistor: P = ½ Imax VR,max = ½ Imax Vgen,max cos(f) = Irms Vgen,rms cos(f)

  20. AC Summary Resistors: VR,max=Imax R In phase with I Capacitors: VC,max =Imax XC Xc = 1/(2pf C) Lags I Inductors: VL,max=Imax XL XL = 2pf L Leads I Generator: Vgen,max=Imax Z Z = √R2 +(XL -XC)2 Can lead or lag I tan(f) = (XL-XC)/R Power is only dissipated in resistor: P = ½ImaxVgen,max cos(f)

More Related