1. Exam 2 in two weeks! Lecture material Discussion/HW material
Magnetism (Lect. 8) – AC circuits (Lect. 13)
Will cover this weeks material!
Discussion/HW material
Discussion 4 – 7
HW 4 – 7
Review session Sunday, March 13, 3pm
Will review HE2 from Fall ‘10

2. Physics 102: Lecture 12
AC Circuits L R C 1

3. Review: Self-Inductance
Recall inductor
Changing current
Changing Bsol field
Changing  through itself!
 proportional to I:
“Inductance”
Units: Henry (H)
Induced EMF (voltage)
Recall Faraday’s law:
Direction
Given by Lenz’s Law
Opposes change in current!
Energy stored:
U = ½ LI2

4. Mutual Inductance AC Generator Changing current in P
Primary
Coil
Secondary
AC Generator
Changing current in P
Changing B-field thru P
Changing B-field thru S
Changing  thru S
S proportional to IP:
“Mutual Inductance”
Induced EMF (voltage) in S
Recall Faraday’s law:
Demo 67

5. Review: Generators and EMF
Voltage across generator: 1
= w A B sin(q)
= w A B sin(wt)
= Vmax sin(wt) w • q v v r 2 x e
Vmax
-Vmax
Frequency = How fast its spinning
Amplitude = Maximum voltage t

6. AC Source Example V(t) = Vmax sin(t)=Vmax sin(2pf t)
Vmax = maximum voltage
f = frequency (cycles/second)
V(t) = 24 sin(8p t)
+24
-24
2pf t = 8pt
f = 4 Hz
T=(1/4)seconds/cycle
0.25
0.5
Ave V = 0
Ave V^2: Vmax/2
RMS: Root Mean Square Vrms=Vmax/√2

7. RMS? V(t) = Vmax sin(2pf t) RMS: Root Mean Square Vrms=Vmax/√2 +Vmax
Ave V = 0
Ave V^2: Vmax/2
square Root:
Vmax / √2
RMS: Root Mean Square Vrms=Vmax/√2

8. Preflight 12.1, 12.2 I(t) = 10 sin(377 t) Find Imax Find Irms L R C
Well… We know that the maximum value
sine is 1. So the maximum current is 10!
Imax = 10 A
78% correct
85% and 78% correct respectively
Just like Vrms=Vmax/√2 …
Irms=Imax/√2
=10/√2 A = 7.07 A
64% correct

9. Resistors in AC circuit
VR = I R always true – Ohm’s Law
VR,max = ImaxR R
Voltage across resistor is “IN PHASE” with current.
VR goes up and down at the same times as I does. I t VR
Frequency
Resistance (R)
Frequency does not affect Resistance!

10. Capacitors in AC circuit
VC = Q/C always true
VC,max = ImaxXC
Capacitive Reactance: XC = 1/(2pfC) C
Voltage across capacitor “LAGS” current.
VC goes up and down just after I does. I t
Frequency
Reactance (XC)
Frequency does affect Reactance!
Changing voltage tries to charge/discharge capacitor.
Low frequency limit: open circuit
High frequency limit: closed circuit (capacitor never has time to build up charge)
Lag: charge (voltage drop) lags behind current. t VC

11. Inductors in AC circuit
VL = +L(DI)/(Dt) always true
VL,max = ImaxXL
Inductive Reactance: XL = 2pfL L
Voltage across inductor “LEADS” current.
VL goes up and down just before I does. I t
Frequency
Reactance (XL)
Frequency does affect Reactance!
Low frequency: no voltage drop…nothing is changing
High frequency: large rate of change of current implies large emf (as per Faraday)
Voltage leads current. (rate is initially large, current then grows as rate drops) t VL

12. ACT/Preflight 12.4, 12.5 The capacitor can be ignored when…
(a) frequency is very large
(b) frequency is very small w XC
very large w gives very small XC
The inductor can be ignored when…
(a) frequency is very large
(b) frequency is very small
“can be ignored” means “behaves like a wire”…no voltage drop
67% and 66% got this right w XL
very small w gives very small XL

13. AC Circuit Voltages Example
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH has a current I(t) = 0.5 sin(8p t) amps. Calculate the maximum voltage across R, C, and L.
VR,max = Imax R L R C
= 0.5  2 = 1 Volt
VC,max = Imax XC
= 0.5  1/(8p0.015) = 1.33 Volts
VL,max = Imax XL
= 0.5  8p0.03 = 0.38 Volts

14. ACT: AC Circuit Voltages
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH has a current I(t) = 0.5 sin(8p t) amps. Calculate the maximum voltage across R, C, and L. L R C
Now the frequency is increased so I(t) = 0.5 sin(16p t).
Which element’s maximum voltage decreases?
1) VR,max
2) VC,max
3) VL,max
Stays same: R doesn’t depend on f
Decreases: XC = 1/(2pfC)
Increases: XL = 2pf L

15. Summary so far… I = Imaxsin(2pft) VR = ImaxR sin(2pft)L R C
I = Imaxsin(2pft)
VR = ImaxR sin(2pft)
VR in phase with I VR I
VC = ImaxXC sin(2pft–p/2)
VC lags I t VL VC
VL = ImaxXL sin(2pft+p/2)
VL leads I 1

16. Kirchhoff: generator voltage
Vgen
Write down Kirchhoff’s Loop Equation:
Vgen(t) = VL(t) + VR(t) + VC(t) at every instant of time I t VL VC VR
However …
Vgen,max  VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C
We solve this using phasors

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. AC circuit summary Kirchoff’s Loop Equation always holds true:
Vgen = VL + VR + VC
However, Vgen,max  VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C I VR L R C
VR in phase with I
VC lags I
VL leads I t VL VC
Phasors represent instantaneous voltages 1

27. See you next lecture.