1. 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

2. Peak & RMS values in AC Circuits (REVIEW)
When asking about RMS or Maximum values relatively simple expressions 5

3. Time Dependence in AC CircuitsL R C
Time Dependence in AC Circuits
Write down Kirchoff’s Loop Equation:
VG(t) = VL(t) + VR(t) + VC(t) at every instant of time I
However …
VG,max  VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C VR t VL VC 5

4. 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 41

5. a A reminder about sines and cosines q+p/2 q q-p/2y 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

6. 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

7. Phasor Diagrams: A Detailed Example
I = Imaxsin(2pft)
VR = VR,maxsin(2pft)
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 10

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. Drawing Phasor Diagrams
VR,max
Resistor vector: to the right
Length given by VR,max (or R)
VL,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)
(4) Generator vector (coming soon)
VC(t)
VR(t)
VL(t)
(5) Rotate entire thing counter-clockwise
Vertical components give instantaneous voltage across R, C, L
Note: VR=IR VL=IXL VC=IXC 15

15. 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! 17

16. 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
1) increasing or 2) 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. 21

17. 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)
Vgen,max=ImaxZ
Imax(XL-XC) f
Vgen,max = Imax Z
ImaxR=VR,max
ImaxXC=VC,max
“phase angle” 25

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

19. Drawing Phasor Diagrams
VR,max
Resistor vector: to the right
Length given by VR,max (or R)
VL,max
(2) Capacitor vector: Downwards
Length given by VC,max (or XC)
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,max
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
Rotates Counter Clockwise 27

20. 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 ?
time 2
When does Vgen = VR ?
time 3 30

21. 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 ?
time 2
When does Vgen = VR ?
time 3
The phase angle is: (1) positive (2) negative (3) zero?
negative
Look at time 1: Vgen is below VR 31

22. 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 Vgen,rms cos(f) 34

23. 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) 37

24. 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 41

25. 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 41

26. ACT/Preflight 13.1 Vgen(t) = VL(t)+VR(t)+VC(t) at all times!
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
40%
28%
32%
ImaxXL=VL,max
Vgen,max
Imax(XL-XC)
Rotates Counter Clockwise f
ImaxR
Vgen(t) = VL(t)+VR(t)+VC(t)
at all times!
Vrms ≠ VL,rms+VR,rms+VC,rms
ImaxXC = VC,max 43

27. 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 ) VR = ImaxR sin(f) 3) VR = ImaxR cos(f) 46

28. See You Monday!