AC Circuit Phasors Physics 102: Lecture 13 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 1

Peak & RMS values in AC Circuits (REVIEW) When asking about RMS or Maximum values relatively simple expressions

Time Dependence in AC Circuits L R C Time Dependence in AC Circuits Write down Kirchoff’s Loop Equation: VG + VL + VR + VC = 0 at every instant of time However … VG,max  VL,max+VR,max+VC,max Maximum reached at different times for R,L,C I VR t VL VC

Here is a problem that we will now learn how to solve: 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. Example L R C

a A reminder about sines and cosines q+p/2 q q-p/2 y q q+p/2 q-p/2 a Recall: y coordinates of endpoints are asin(q + p/2) asin(q) asin(q - p/2) x whole system rotates, theta=2pi*ft 1

Graphical representation of voltages q+p/2 ImaxXL I = Imaxsin(2pft) (q = 2pft) VL = ImaxXL sin(2pft + p/2) VR = ImaxR sin(2pft) VC = ImaxXC sin(2pft - p/2) L R C q ImaxR q-p/2 ImaxXC 1

Phasor Diagrams: A Detailed Example I = Imaxsin(p/6) VR = VR,maxsin(p/6) t = 1 f=1/12 2pft = p/6 VR,max VR,maxsin(p/6) p/6 Phasor Animation Length of vector = Vmax across that component Vertical component = instantaneous value of V

Phasor Diagrams I = Imaxsin(p/3) VR = VR,maxsin(p/3) t = 2 2pft = p/3 Length of vector = Vmax across that component Vertical component = instantaneous value of V

Phasor Diagrams I = Imaxsin(p/2) VR = VR,maxsin(p/2) t = 3 2pft = p/2 VR,maxsin(p/2)=V0 p/2 Length of vector = Vmax across that component Vertical component = instantaneous value of V

Phasor Diagrams I = Imaxsin(4p/6) VR = VR,maxsin(4p/6) t = 4 2pft = 4p/6 VR,max VR,maxsin(4p/6) 4p/6 Length of vector = Vmax across that component Vertical component = instantaneous value of V

Phasor Diagrams I = Imaxsin(p) VR = VR,maxsin(p) t = 6 2pft = p Length of vector = Vmax across that component Vertical component = instantaneous value of V

Phasor Diagrams I = Imaxsin(8p/6) VR = VR,maxsin(8p/6) t = 8 2pft = 8p/6 8p/6 VR,max VR,maxsin(8p/6) Length of vector = Vmax across that component Vertical component = instantaneous value of V

Phasor Diagrams I = Imaxsin(10p/6) VR = VR,maxsin(10p/6) t = 10 2pft = 10p/6 10p/6 VR,maxsin(10p/6) VR,max Length of vector = Vmax across that component Vertical component = instantaneous value of V

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

Phasor Diagrams I = Imaxsin(2pft) VR = ImaxR sin(2pft) Instantaneous Values: ImaxXL I = Imaxsin(2pft) VR = ImaxR sin(2pft) ImaxR ImaxXL cos(2pft) ImaxR sin(2pft) ImaxXC VC = ImaxXC sin(2pft-p/2) = -ImaxXC cos(2pft) -ImaxXC cos(2pft) VL = ImaxXL sin(2pft+ p/2) = ImaxXL cos(2pft) Note the lagging and leading. Do demo with phasor board Voltage across resistor is always in phase with current! Voltage across capacitor always lags current! Voltage across inductor always leads current!

Phasor Diagram Practice Example 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? Inductor Leads Capacitor Lags VR VL VC 21

Kirchhoff: Impedance Triangle Instantaneous voltage across generator (Vgen) must equal sum of voltage across all of the elements at all times: ImaxXL=VL,max Vgen (t) = VR (t) +VC (t) +VL (t) Vmax,gen=ImaxZ Imax(XL-XC) f Vgen,max = Imax Z ImaxR=VR,max ImaxXC=VC,max “phase angle”

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

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

ACTS 13.1, 13.2, 13.3 When does Vgen = 0 ? When does Vgen = VR ? time 1 time 2 time 3 time 4 Vgen VR When does Vgen = 0 ? When does Vgen = VR ? 30

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

Power P=IV The voltage generator supplies power. Resistor dissipates power. Capacitor and Inductor store and release energy. P = IV 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 Vrms cos(f)

AC Summary Resistors: VRmax=I R In phase with I Capacitors: VCmax =I XC Xc = 1/(2pf C) Lags I Inductors: VLmax=I XL XL = 2pf L Leads I Generator: Vgen,max=I Z Z= sqrt(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)

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. Example L R C

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. Example L R C Imax = Vgen,max /Z Imax = 2.5/2.76 = .91 Amps

ACT/Preflight 13.1 Vgen,max The statement that the voltage across the generator equals the sum of the voltages across the resistor, capacitor and inductor is true for: (1) instantaneous voltages only (2) rms voltages only (3) both rms and instantaneous ImaxXL=VL,max Vgen,max Imax(XL-XC) f ImaxR ImaxXC = VC,max

ACT: Voltage Phasor Diagram Imax XL Imax XC Imax R Vgen,max f 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)