Understanding AC Response Relationship in Capacitors and Inductors

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. 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. AC V AND I IN A RESISTOR

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

6. AC CAPACITOR

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

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. 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 LAGScharging current by 90oor Charging Current (I) LEADSVoltage (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. If V is our reference wave: PHASE ANGLE FOR XC I.C.E

16. or • 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

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

20. V C 90 I C • In the Inductor (L), Induced Voltage LEADScurrent by 90o or Current (I) LAGSInducedVoltage (E) by 90o. • E. L. I.

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

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

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

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

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): • 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;

31. Solution • =2 rad/s:

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

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

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

46. Solution: Convert to Freq Domain

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

48. (a): Find Total Impedance

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

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