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

2. Review: AC Circuit I = Imaxsin(2pft) VR = ImaxR sin(2pft)L R C
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

3. 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. Time Dependence in AC CircuitsL 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. 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

6. 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

7. Phasor Diagrams Instantaneous Values: I = Imaxsin(2pft)
ImaxXL
I = Imaxsin(2pft)
VR = ImaxR sin(2pft)
ImaxR
ImaxXL cos(2pft)
ImaxR sin(2pft)
ImaxXC
-ImaxXC cos(2pft)
VC = ImaxXC sin(2pft–p/2)
= –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!

8. 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.

9. Kirchhoff: generator voltage
Instantaneous voltage 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
“phase angle”
VC,max=ImaxXC
Define impedance Z: Vgen,max ≡ Imax Z
“Impedance Triangle”

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

11. 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)
(4) Generator vector: add first 3 vectors
Length given by Vgen,max (or Z)
Vgen,max
VC,max
(3) Inductor vector: Upwards
Length given by VL,max (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

12. 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
The phase angle is: (1) positive (2) negative (3) zero?
Vgen is clockwise from VR

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

14. 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)

15. Resonance and the Impedance Triangle
Vgen,max = Imax Z
Imax(XL-XC)
ImaxXL
ImaxXC
ImaxR
Vgen,max f 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!
R is independent of f
XL increases with f
XL = 2pfL
Z is minimum at resonance frequency!
XC decreases with f
XC = 1/(2pfC) Z R
Z: XL and XC subtract
demo with RLC and oscilliscope: Note that XL > XC for f>f0 and vice versa. XC XL f0
Resonance: XL = XC

17. Current is maximum at resonance frequency!
R is independent of f
XL increases with f
Current is maximum at resonance frequency!
XL = 2pfL
XC decreases with f
XC = 1/(2pfC) Z
Imax = Vgen,max/Z
Z: XL and XC subtract
Current
demo with RLC and oscilliscope f0
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. Summary of Resonance At resonance At lower frequencies
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
Imax(XL-XC)
ImaxXL
ImaxXC
ImaxR
Vgen,max f

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!