1. 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

2. 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

3. 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

4. Peak values in AC Circuits
reactance or impedance
voltage
(capacitive reactance)
(inductive reactance)
(impedance)
Physics 102: Lecture 13, Slide 4

5. 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

6. 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

7. 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

8. Drawing Phasor DiagramsVL
(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

9. 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

10. 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

11. 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

12. 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.

13. 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

14. 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”

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

16. Drawing Phasor DiagramsVL
(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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. Voltage Phasor Diagram
Rotates Counter Clockwise
Imax XL=VL,max
Vgen,max = ImaxZ f
ImaxR=VR,max
Imax XC=VC,max

27. Voltage Phasor Diagram
Imax XL
Imax XC
Imax R
Vgen,max f
Rotates Counter Clockwise

28. 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)

29. 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)

30. See you next time! Read Sections 21.6, 22.1, 4-5, 9
Physics 102: Lecture 13, Slide 30