1. Chapter 31
Alternating Current

2. Questions about AC Circuits
How do AC circuits work, compared with DC?
Advantages? Disadvantages?
Westinghouse vs. Edison?
What roles do inductors, capacitors, and resistors play in AC circuits?
How can we mathematically model AC circuits and the complex relationships of voltage and current through all components?

3. To use phasors to describe sinusoidally varying quantities
Goals for Chapter 31
To use phasors to describe sinusoidally varying quantities
To use reactance to describe voltage in a circuit
To analyze an L-R-C series circuit
To determine power in ac circuits
To see how an L-R-C circuit responds to frequency
To learn how transformers work

4. Introduction
How does a radio tune to a particular station?
Use a variable capacitor in concert with inductors and resistors!

5. Alternating currents
Voltage (supply) is a sinusoidal function of time V(t) = Vmaxcos wt

6. Alternating currents When Voltage is maximum, Current may not be!
Voltage (supply) is a sinusoidal function of time V(t) = Vmaxcos wt
Resulting current is ALSO a sinusoidal function in time i(t) = imax cos wt
But … phases of these are not necessarily the same through the circuit!
When Voltage is maximum, Current may not be!
V(t) = Vmaxcos (wt +/-f) but i(t) = imax cos (wt)
If f = 0 , Voltage and Current are described as “in phase”
If f  0 , Voltage and Current are described as “out of phase”

7. Alternating currents across a resistor…
How do Resistors affect an AC circuit?

8. Alternating current across a resistor…
Current and Voltage are in phase across resistors
VR(t) = Vmaxcos wt & iR (t) = imax cos wt

9. Alternating currents across a resistor…
Current and Voltage are in phase across resistors
VR(t) = Vmaxcos wt & iR (t) = imax cos wt

10. Alternating currents across a capacitor…
How do CAPACITORS affect an AC circuit?

11. Alternating currents across a capacitor…
Current and Voltage are out of phase across capacitors
VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt
Capacitors take time to reach maximum voltage
Voltage across capacitor LAGS behind current!

12. Alternating currents across a capacitor…
CAPACITORS
VOLTAGE lags CURRENT
CURRENT leads Voltage
I C E

13. Alternating currents across a capacitor…
Current and Voltage are out of phase across capacitors
VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt

14. Alternating currents across a capacitor…
Current and Voltage are out of phase across capacitors
VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt
Note current is max at time t = 0
But charge on capacitor is not yet built up to a maximum!
Charge on plates max AFTER current already decreasing (but still positive)

15. Alternating currents across a capacitor…
Current and Voltage are out of phase across capacitors
VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt
Note current is max at time t = 0
Voltage isn’t maximum until some time t = + f/w later!
Voltage E will “lag” current I across a capacitor C
Remember “I – C – E”

16. Alternating currents across a capacitor…
Current and Voltage are out of phase across capacitors
VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt

17. Alternating currents across a inductor…
How do INDUCTORS affect an AC circuit?

18. Alternating currents across an inductor…
Current &Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
Inductors “fight” current change, and push hardest in the opposite direction when current changes from – to + or + to -

19. Alternating currents across an inductor…
Current &Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
So voltage across the inductor will reach maximum BEFORE the current through it builds to max…

20. Alternating currents across an inductor…
INDUCTORS
CURRENT lags VOLTAGE
Voltage leads Current
E L I

21. Alternating currents across an inductor…
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt

22. Alternating currents across an inductor…
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
Consider cases: t = 0
Note current is max, and rate of change di/dt = 0
Voltage across inductor ONLY depends upon L di/dt!
So at that time, VL = 0!

23. Alternating currents
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
At t = 0, current max,
voltage across L = 0

24. Alternating currents across an inductor…
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
Consider cases: t >0
Note current is positive but decreasing, and rate of change di/dt <0
Voltage across inductor depends upon L di/dt!
Inductor reacts to decreasing current by continuing to provide EMF from a to b
So at that time, VL = Va - Vb <0!

25. At t > 0, current +, decreasing,
Alternating currents
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
At t > 0, current +, decreasing,
voltage across L <0

26. Alternating currents across an inductor…
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
Consider cases: t = ¼ of period…
Note current is 0 at some time wt = + /2
At that time, current is changing from + to – (large change in B field flux!)

27. Alternating currents
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
At wt = + /2 current 0, decreasing,
voltage across L max negative

28. Alternating currents across an inductor…
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
Note current is 0 and increasing at some time wt = 3/2
At that time, current is changing from - to + (large change in B field flux!)
Inductor reacts to this change, generating E to oppose this change
VL will be largest, positive (Va > Vb) pushing the other way!

29. Alternating currents
Current and Voltage are out of phase across inductors
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
At t, wt = +3/2 ,
current 0, increasing,
voltage across L max

30. How can we mathematically model AC circuits and the complex relationships of voltage and current, and power through all components?
Phasors!

31. How can we mathematically model AC circuits and the complex relationships of voltage and current through all components?
Phasors!

32. No, not PHASERS!

33. Phasors Graphical representation of current/voltage in AC circuits
Takes into account relative phases of different voltages
Example: current phasor graphs i (t) = imax cos wt

34. The “real” portion of a Phasor!
Projection of vector onto horizontal axis

35. The “real” portion of a Phasor!
Consider four different current phasors: IB IA w IC ID

36. The “real” portion of a Phasor!
Which phasor represents
Positive current becoming more positive?
Positive current decreasing to zero?
Negative current becoming more negative?
Negative current decreasing in magnitude? IA IB IC ID w

37. The “real” portion of a Phasor!
Which phasor represents
Positive current becoming more positive?
Positive current decreasing to zero?
Negative current becoming more negative?
Negative current decreasing in magnitude? IA IB IC ID w ID IA IB IC

38. Resistor in an ac circuit
VR = IR; VR in phase with I

39. Phasors for Voltage/Current across Resistor
VR(t) = Vmaxcos (wt) & iR (t) = imax cos wt

40. Capacitors in an ac circuit
VC(t) = Vmaxcos (wt-f) VC out of phase with I

41. Phasors for Voltage/Current across Capacitor
VC(t) = Vmaxcos (wt-f) & iC (t) = imax cos wt
I - C- E: Current Leads Voltage Across Capcitor

42. Capacitance in an ac circuit
The voltage amplitude across the capacitor is VC = IXC
Xc = “capacitive reactance” = 1/wC
Xc = DECREASES as angular frequency increases
WHY?

43. Inductors in AC circuits
VL(t) = Vmaxcos (wt+f)
VL out of phase with I

44. Phasors for Voltage/Current across Inductor
VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt
E-L-I: Voltage Leads Current Across Inductor

45. Inductor in an ac circuit
The voltage amplitude across the inductor is VL = IXL
XL = “inductive reactance” = wL
XL increases as frequency increases!
WHY?

46. Comparing ac circuit elements
Table 31.1 summarizes the characteristics of a resistor, an inductor, and a capacitor in an ac circuit.

47. Root-mean-square values

48. Current in a personal computer
Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug.
What is the:
AVERAGE current,
Average of the current squared,
Current amplitude?

49. Current in a personal computer
Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug.
What is the:
AVERAGE current?
0 amps!
Average over 1 period = 0!

50. Current in a personal computer
Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug.
What is the:
Average of current squared? = 7.3 Amps2

51. Current in a personal computer
Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug.
What is the:
Current amplitude?
Irms = .707 I
So I = 3.8 Amps

52. The L-R-C series circuit
Combine all three elements into simple series circuit
The voltage amplitude across an ac circuit is V = IZ
Overall effective resistance = Z (“impedance”)
Z = [R2 + (XL - Xc)2] ½

53. The L-R-C series circuit
Suppose inductive reactance > capacitive reactance?
XL > XC
Inductor is dominating
Current will be out of phase with supply voltage
“E – L – I “ reminds us that current will LAG voltage.

54. The L-R-C series circuit
Suppose inductive reactance > capacitive reactance?

55. The L-R-C series circuit
Suppose capacitive reactance > inductive reactance?
XC > XL
Capacitor is dominating
Current will be out of phase with supply voltage
“I – C – E ” reminds us that current will LEAD voltage.

56. The L-R-C series circuit
Suppose capacitive reactance > inductive reactance?

57. A resistor and a capacitor in an ac circuit
200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor VR = 1.20V cos (2500 rad/sec) x t
What is i(t)?
What is the reactance?
What is Vc(t)

58. A resistor and a capacitor in an ac circuit
200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t

59. A resistor and a capacitor in an ac circuit
200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t
Ohm’s Law applies (that’s why it is a LAW!  )
VR = IR so I = A cos (2500 rad/sec) x t Note current is in phase with the voltage across R!

60. A resistor and a capacitor in an ac circuit
200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t
Capacitive Reactance
XC = 1/wC = 1/(2500 rad/s) x 5.0 mF
= 80 W

61. A resistor and a capacitor in an ac circuit
200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t
Voltage across Capacitor
VC = I Xc = A x 80 W = 0.48 V
and
VC = I Xc = V cos (wt - p)

62. A useful application: the loudspeaker
The woofer (low tones) and the tweeter (high tones) are connected in parallel across the amplifier output.

63. An L-R-C series circuit
R = 300 Ohms
L = 60 mH
C = 0.50 mF
V = 50 V
w = 10,000 rad/sec
What are XL, Xc, Z, I, Phase angle f, and VR, Vc, VL?

64. An L-R-C series circuit
R = 300 Ohms
L = 60 mH
C = 0.50 mF
V = 50 V
w = 10,000 rad/sec
What are XL, Xc, Z, I, Phase angle f, and VR, Vc, VL?

65. Power in ac circuits Power = I x V Average Power = Irms Vrms cos f
Note that the net energy transfer over one cycle is zero for an inductor and a capacitor.

66. Resonance in ac circuits
At the resonance angular frequency 0, the inductive reactance equals the capacitive reactance and the current amplitude is greatest. (See Figure below.)

67. Tuning a radio
RMS voltage of 1.0V; what is resonance frequency? At that frequency what are XL and XC and Z?

68. Transformers
Power is supplied to the primary and delivered from the secondary.
Terminal voltages: V2/V1 = N2/N1.
Currents in primary and secondary: V1I1 = V2I2.

69. Real transformers
Real transformers always have some power losses