1. Lecture 6 (III): AC RESPONSE

2. OBJECTIVES
Explain the relationship between AC voltage and AC current in a resistor, capacitor and inductor.
Explain why a capacitor causes a phase shift between current and voltage (ICE).
Define capacitive reactance. Explain the relationship between capacitive reactance and frequency.
Explain why an inductor causes a phase shift between the voltage and current (ELI).
Define inductive reactance. Explain the relationship between inductive reactance and frequency.
Explain the effects of extremely high and low frequencies on capacitors and inductors.

3. AC RESISTOR

4. AC V AND I IN A RESISTOR
Ohm’s Law still applies even though the voltage source is AC.
The current is equal to the AC voltage across the resistor divided by the resistor value.
Note: There is no phase shift between V and I in a resistor.

5. AC V AND I IN A RESISTOR
vR(t)
PHASE SHIFT FOR R, =0

6. AC CAPACITOR

7. CURRENT THROUGH A CAPACITOR
The faster the voltage changes, the larger the current.

8. PHASE RELATIONSHIP
The phase relationship between “V” and “I” is established by looking at the flow of current through the capacitor vs. the voltage across the capacitor.

9. Note: Phase relationship of I and V in a capacitor
Graph vC(t) and iC(t)
Note: Phase relationship of I and V in a capacitor
vc(t)
90°
ic (t)

10. PHASE RELATIONSHIP
In the Capacitor (C), Voltage LAGS charging current by 90o or Charging Current (I) LEADS Voltage (E) by 90o
I. C. E.

11. CAPACITIVE REACTANCE
In resistor, the Ohm’s Law is V=IR, where R is the opposition to current.
We will define Capacitive Reactance, XC, as the opposition to current in a capacitor.

12. CAPACITIVE REACTANCE XC will have units of Ohms.
Note inverse proportionality to f and C.
Magnitude of XC

13. Ex.
Ex: f = 500 Hz, C = 50 µF, XC = ?

14. PHASE ANGLE FOR XC
Capacitive reactance also has a phase angle associated with it.
Phasors and ICE are used to find the angle

15. PHASE ANGLE FOR XC
If V is our reference wave:
I.C.E

16. The phase angle for Capacitive Reactance (XC) will always = -90°
XC may be expressed in POLAR or RECTANGULAR form.
ALWAYS take into account the phase angle between current and voltage when calculating XC or

17. AC INDUCTOR

18. VOLTAGE ACROSS AN INDUCTOR
Current must be changing in order to create the magnetic field and induce a changing voltage.
The Phase relationship between VL and IL (thus the reactance) is established by looking at the current through vs the voltage across the inductor.

19. Graph vL(t) and iL(t)
vL(t)
Note the phase relationship
90°
iL(t)

20. In the Inductor (L), Induced Voltage LEADS current by 90o or Current (I) LAGS Induced Voltage (E) by 90o.
E. L. I. V C I 90

21. INDUCTIVE REACTANCE
We will define Inductive Reactance, XL, as the opposition to current in an inductor.

22. INDUCTIVE REACTANCE XL will have units of Ohms (W).
Note direct proportionality to f and L.
Magnitude of XL

23. Ex.
f = 500 Hz, C = 500 mH, XL = ?

24. PHASE ANGLE FOR XL
If V is our reference wave:
E.L.I

25. The phase angle for Inductive Reactance (XL) will always = +90°
XL may be expressed in POLAR or RECTANGULAR form.
ALWAYS take into account the phase angle between current and voltage when calculating XL or

26. COMPARISON OF XL & XC
XL is directly proportional to frequency and inductance.
XC is inversely proportional to frequency and capacitance.

27. SUMMARY OF V-I RELATIONSHIPS
ELEMENT
TIME DOMAIN
FREQ DOMAIN

28. Extreme Frequency effects on Capacitors and Inductors
Using the reactance of an inductor and a capacitor you can show the effects of low and high frequencies on them.

29. Frequency effects At low freqs (f=0): At high freqs (f=∞):
an inductor acts like a short circuit.
a capacitor acts like an open circuit.
At high freqs (f=∞):
an inductor acts like an open circuit.
a capacitor acts like a short circuit.

30. Ex.
Represent the below circuit in freq domain;
CTH LITAR RLC

31. Solution =2 rad/s: Time domain Freq domain R = 2 Ω C = 0.25 F
XC = -j(1/C) = -j2 Ω
L = 1 H
XL = jL = j2 Ω
Vs = 5 cos 2t
Vs = 5<0

32. Solution

33. REVIEW QUIZ
- What is the keyword use to remember the relationships between AC voltage and AC current in a capacitor and inductor.
What is the equation for capacitive reactance? Inductive reactance?
T/F A capacitor at high frequencies acts like a short circuit.
T/F An inductor at low frequencies acts like an open circuit.
ELI and ICE
True
False

34. IMPEDANCE

35. IMPEDANCE The V-I relations for three passive elements;
The ratio of the phasor voltage to the phasor current:

36. From that, we obtain Ohm’s law in phasor form for any type of element as:
Where Z is a frequency dependent quantity known as IMPEDANCE, measured in ohms.

37. IMPEDANCE Impedance is a complex quantity:
R = Real part of Z = Resistance
X = Imaginary part of Z = Reactance

38. Impedance in polar form:
where;

39. IMPEDANCES SUMMARY ZR R+j0 ZL 0+jXL ZC 0-jXC Impedance Phasor form:
Rectangular form ZR
R+j0 ZL
0+jXL ZC
0-jXC

40. ADMITTANCE

41. ADMITTANCE The reciprocal of impedance. Symbol is Y
Measured in siemens (S)

42. ADMITTANCE Admittance is a complex quantity:
G = Real part of Y = Conductance
B = Imaginary part of Y = Susceptance

43. Z AND Y OF PASSIVE ELEMENTS
IMPEDANCE
ADMITTANCE

44. TOTAL IMPEDANCE FOR AC CIRCUITS
To compute total circuit impedance in AC circuits, use the same techniques as in DC. The only difference is that instead of using resistors, you now have to use complex impedance, Z.

45. Ex: SERIES CIRCUIT
R=20Ω
L = 0.2 mH
C = 0.25μF

46. Solution: Convert to Freq Domain

47. Circuit in Freq domain
R=20Ω
jL = j20 Ω
-j(1/C) = -j40 Ω

48. (a): Find Total Impedance

49. (b): Draw Impedance Trianglej q q XC XC ZT ZT
- j

50. (c): Find is, vR, vC, vL
RMS value for power calculation

51. (c): Find is, vR, vC, vL
Remember to convert to RMS values when converting from
sinusoid to phasors.

52. (d):Using Voltage Divider
Voltage divider still works, too.

53. TOTAL IMPEDANCE FOR PARALLEL CIRCUIT

54. As a conclusion, in parallel circuit, the impedance can be easily computed from the admittance:

55. Ex: Parallel Circuits

56. (a) Find total impedance
The circuit given is already in freq domain:

57. Circuit simplification

59. POSITIVE/NEGATIVE QUADRANT+j
-,+
+,+
SUGAR
ADD
180-j60
-, -
+,- TO
COFFEE -j

60. (b) Draw Impedance Trianglej q q XC ZT ZT
- j

61. (c) Find is, iC, iRL

62. (d) Using Current Divider

63. Ex: Find i and vc

64. Elements in Frequency domain
=4 rad/s
Time domain
Freq domain

65. Circuit in Freq Domain

66. (a) The current, i
The current in series circuit,
Total impedance is;

68. (b) The voltage, VC