AC Circuit Phasors

1. L R C Physics 102:Lecture 13 AC Circuit Phasors

2. L R C Review: AC Circuit • 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

3. 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

4. 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

5. 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)

6. 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)

7. 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!

8. 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.

9. “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”

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

11. 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

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

13. 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

14. 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)

15. L ImaxXL R C Vgen,max Imax(XL-XC) f ImaxR ImaxXC Resonance and the Impedance Triangle Vgen,max = 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!

16. 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

17. 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

18. 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 At resonance XL = XC. Since everything has the same current we can write XL = XC XLImax = XCImax VL,max = VC,max Also VGen is in phase with current!

19. 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

20. 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)

21. 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)

22. See You Monday!