1. Lecture 03: AC RESPONSE ( REACTANCE N IMPEDANCE )

2. AC RESISTOR 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.

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

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

5. 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.
In the Capacitor (C), Voltage LAGS charging current by 90o or Charging Current (I) LEADS Voltage (E) by 90o

6. 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)

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

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

9. PHASE ANGLE FOR XC
Capacitive reactance also has a phase angle associated with it.

10. PHASE ANGLE FOR XC
If V is our reference wave:

11. AC INDUCTOR
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

12. 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.

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

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

15. INDUCTIVE REACTANCE Magnitude of XL
We will define Inductive Reactance, XL, as the opposition to current in an inductor.
XL will have units of Ohms (W).
Note direct proportionality to f and L.
Magnitude of XL

16. Ex.
f = 500 Hz, L = 500 mH, XL = ?

17. PHASE ANGLE FOR XL or If V is our reference wave:
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

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

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

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

21. 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.

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

23. 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

24. Solution

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

26. 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.

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

28. Impedance in polar form:
where;

29. 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

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

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

32. Z AND Y OF PASSIVE ELEMENTS
IMPEDANCE
ADMITTANCE

33. 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.

34. TOTAL IMPEDANCE FOR PARALLEL CIRCUIT

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

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

37. Solution: Convert to Freq Domain

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

39. (a): Find Total Impedance

40. POSITIVE/NEGATIVE QUADRANT+j
-,+
+,+
SUGAR
ADD
20-j20
-, -
+,- TO
COFFEE -j

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

42. (c): Find is, vR, vC, vL

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

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