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AC Circuit Phasors Physics 102: Lecture 13
Today’s lecture will cover Textbook Section 21.5 I = Imaxsin(2pft) VR = ImaxR sin(2pft) L R VC = ImaxXC sin(2pft-p/2) VL = ImaxXL sin(2pft+ p/2) C Physics 102: Lecture 13, Slide 1 1
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a A reminder about sines and cosines q+p/2 q q-p/2
Recall: y coordinates of endpoints are asin(q + p/2) asin(q) asin(q - p/2) Physics 102: Lecture 13, Slide 2 1
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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
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Peak values in AC Circuits
reactance or impedance voltage (capacitive reactance) (inductive reactance) (impedance) Physics 102: Lecture 13, Slide 4
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Phasor Diagrams I = Imaxsin(p/6) VR = VR,maxsin(p/6) f =1/12 t = 1
2pft = p/6 L R C VR,max VR,maxsin(p/6) p/6 Phasor Animation Length of vector = Vmax across that component Vertical component = instantaneous value of V Physics 102: Lecture 13, Slide 5
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Phasor Diagrams I = Imaxsin(p/3) VR = VR,maxsin(p/3) f =1/12 t = 2
2pft = p/3 Vmax VR,maxsin(p/3) p/3 Length of vector = Vmax across that component Vertical component = instantaneous value of V Physics 102: Lecture 13, Slide 6
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Phasor Diagrams I = Imaxsin(p/2) VR = VR,maxsin(p/2) t = 3 2pft = p/2
Vmax VR,maxsin(p/2)=V0 p/2 Length of vector = Vmax across that component Vertical component = instantaneous value of V Physics 102: Lecture 13, Slide 7
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Drawing Phasor Diagrams
VL (2) Inductor vector: upwards Length given by VL VR Resistor vector: to the right Length given by VR VC (3) Capacitor vector: downwards Length given by VC VL VR (4) Rotate entire thing counter-clockwise Vertical components give instantaneous voltage across R, C, L VC Physics 102: Lecture 13, Slide 8
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Phasor Diagrams I = Imaxsin(2pft) VR = ImaxR sin(2pft)
Instantaneous Values: ImaxXL ImaxR I = Imaxsin(2pft) VR = ImaxR sin(2pft) ImaxXL sin(2pft+p/2) ImaxR sin(2pft) ImaxXC ImaxXC sin(2pft-p/2) VC = ImaxXC sin(2pft-p/2) VL = ImaxXL sin(2pft+p/2) Voltage across resistor is always _______ with current! Voltage across capacitor always _______ current! Voltage across inductor always ________ current! Physics 102: Lecture 13, Slide 9
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Phasor Diagrams I = Imaxsin(2pft) VR = ImaxR sin(2pft)
Instantaneous Values: ImaxXL ImaxR I = Imaxsin(2pft) VR = ImaxR sin(2pft) ImaxXL sin(2pft+p/2) ImaxR sin(2pft) ImaxXC ImaxXC sin(2pft-p/2) VC = ImaxXC sin(2pft-p/2) VL = ImaxXL sin(2pft+p/2) Voltage across resistor is always in phase with current! Voltage across capacitor always lags current! Voltage across inductor always leads current! Physics 102: Lecture 13, Slide 10
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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 ) C ) L Is the voltage across the inductor increasing or decreasing? Which element has the largest maximum voltage across it? Physics 102: Lecture 13, Slide 11
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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 ) C ) L Is the voltage across the inductor increasing or decreasing? Which element has the largest maximum voltage across it? Inductor Leads Capacitor Lags VR VL R: It has largest vertical component VC Decreasing, spins counter clockwise Inductor, it has longest line.
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KVL: 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 Z = Physics 102: Lecture 13, Slide 13
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KVL: 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”
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Phase angle f I = Imaxsin(2pft) Vgen = ImaxR sin(2pft + f) 2pft + f
f is positive in this particular case.
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Drawing Phasor Diagrams
VL (2) Capacitor vector: downwards Length given by VC VR Resistor vector: to the right Length given by VR (4) Generator vector: add first 3 vectors Length given by Vgen Vgen VC (3) Inductor vector: upwards Length given by VL 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, gen VC Physics 102: Lecture 13, Slide 16
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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 ? Is the phase angle positive or negative? Physics 102: Lecture 13, Slide 17
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ACTS 13.1, 13.2, 13.3 f When does Vgen = 0 ? time 2
When does Vgen = VR ? time 3 Is the phase angle positive or negative? negative Look at time 1: Vgen is below VR Physics 102: Lecture 13, Slide 18
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Power P=IV The voltage generator supplies power.
Resistor dissipates power. Capacitor and Inductor store and release energy. P = IV so power loss is sometimes large, sometimes small. Average power dissipated by resistor: P = ½ Imax VR,max = ½ Imax Vgen,max cos(f) = Irms Vgen,rms cos(f) Physics 102: Lecture 13, Slide 19
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Power P=IV The voltage generator supplies power.
Resistor dissipates power. Capacitor and Inductor store and release energy. P = IV so power loss is sometimes large, sometimes small. Average power dissipated by resistor: P = ½ Imax VR,max = ½ Imax Vgen,max cos(f) = Irms Vgen,rms cos(f) Physics 102: Lecture 13, Slide 20
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AC Summary Resistors: VR=I R In phase with I
Capacitors: VCmax =Imax XC Xc = 1/(2pf C) Lags I Inductors: VLmax=Imax XL XL = 2pf L Leads I Generator: Vgenmax=Imax Z Z = R2 +(XL-XC)2 Can lead or lag I tan(f) = (XL-XC)/R Power is only dissipated in resistor: P = IrmsVrms cos(f) Physics 102: Lecture 13, Slide 21
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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 Z = Physics 102: Lecture 13, Slide 22
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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 Physics 102: Lecture 13, Slide 23
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Preflight 13.1 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 voltages ImaxXL=VL,max Vgen,max Imax(XL-XC) f ImaxR ImaxXC = VC,max
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Preflight 13.1 Vgen=VL+VR+VC at all times. Vrms does not! 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 voltages ImaxXL=VL,max Vgen,max Imax(XL-XC) Rotates Counter Clockwise f ImaxR Vgen=VL+VR+VC at all times. Vrms does not! ImaxXC = VC,max
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Voltage Phasor Diagram
Rotates Counter Clockwise Imax XL=VL,max Vgen,max = ImaxZ f ImaxR=VR,max Imax XC=VC,max
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Voltage Phasor Diagram
Imax XL Imax XC Imax R Vgen,max f Rotates Counter Clockwise
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ACT: Voltage Phasor Diagram
Imax XL Imax XC Imax R Vgen,max f Rotates Counter Clockwise At this instant, the voltage across the generator is maximum. What is the voltage across the resistor at this instant? 1) VR = ImaxR ) VR = ImaxR sin(f) 3) VR = ImaxR cos(f)
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ACT: Voltage Phasor Diagram
Imax XL Imax XC Imax R Vgen,max f Rotates Counter Clockwise At this instant, the voltage across the generator is maximum. What is the voltage across the resistor at this instant? 1) VR = ImaxR ) VR = ImaxR sin(f) 3) VR = ImaxR cos(f)
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See you next time! Read Sections 21.6, 22.1, 4-5, 9
Physics 102: Lecture 13, Slide 30
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