1. Inductance and Capacitance

2. Objectives
1. Find the current (voltage) for a capacitance or inductance given the voltage (current) as a function of time.
2. Compute the capacitance of a parallel-plate capacitor.
3. Compute the stored energy in a capacitance or inductance.
4. Describe typical physical construction of capacitors and inductors

3. Capacitors and Capacitance
Capacitance – the ability of a component to store energy in the form of an electrostatic charge
Capacitor – is a component designed to provide a specific measure of capacitance

4. Capacitors and Capacitance
Capacitor Construction
Plates
Dielectric

5. Capacitor Charge Electrostatic Charge Develops
Electrostatic Field Stores energy
Insert Figure 12.2

7. Capacitor Discharge
Insert Figure 12.3

8. Capacitors and Capacitance
Capacity – amount of charge that a capacitor can store per unit volt applied
where
C = the capacity (or capacitance) of the component, in coulombs per volt
Q = the total charge stored by the component
V = the voltage across the capacitor corresponding to the value of Q

10. Capacitance
Insert Figure 12.4

11. Capacitance Unit of Measure – farad (F) = 1 coulomb per volt (C/V)
Capacitor Ratings
Most capacitors rated in the picofarad (pF) to microfarad (F) range
Capacitors in the millifarad range are commonly rated in thousands of microfarads: 68 mF = 68,000 F
Tolerance
Usually fairly poor
Variable capacitors used where exact values required

12. Capacitors and Capacitance
Physical Characteristics of Capacitors
where
C = the capacity of the component, in farads
(8.85 X 10-12) = the permittivity of a vacuum, in farads per meter (F/m) or expressed as o
r = the relative permittivity of the dielectric
A = the area of either plate
d = the distance between the plates (i.e., the thickness of the dielectric)

13. Capacitance of the Parallel-Plate Capacitor

14. Capacitance
For DC
It acts as a voltage source

15. Voltage in terms of Current
, q(to) is the initial charge

16. Stored Energy

17. Series Capacitors Series Capacitors Where
CT = the total series capacitance
Cn = the highest-numbered capacitor in the string

18. Parallel Capacitors Connecting Capacitors in Parallel where
Cn = the highest-numbered capacitor in the parallel circuit

22. Inductance Unit of Measure – Henry (H)
Inductance is measured in volts per rate of change in current
When a change of 1A/s induces 1V across an inductor, the amount of inductance is said to be 1 H
Insert Figure 10.5

23. Inductance Induced Voltage where
vL = the instantaneous value of induced voltage
L = the inductance of the coil, measured in henries (H)
= the instantaneous rate of change in inductor current (in amperes per second)

25. Inductance
For DC
It acts as a short circuit

26. Current in terms of Voltage

27. Stored Energy

28. Inductance
Insert Figure 10.8

29. Connecting Inductors in Series
Series-Connected Coils
where
Ln = the highest-numbered inductor in the circuit

30. Characteristic of Capacitor and Inductor Under AC Excitation

31. Connecting Inductors in Parallel
Parallel-Connected Coils
where
Ln = the highest-numbered inductor in the circuit

32. Alternating Voltage and Current Characteristics
AC Coupling and DC Isolation: An Overview
DC Isolation – a capacitor prevents flow of charge once it reaches its capacity
Insert Figure 12.6

33. AC Coupling and DC Isolation
AC Coupling – DC offset is blocked
Insert Figure 12.7

34. Capacitor Current where
iC = the instantaneous value of capacitor current
C = the capacity of the component(s), in farads
= the instantaneous rate of change in capacitor voltage

35. Alternating Voltage and Current Characteristics
Sine-Wave Values of
reaches its maximum value when v = 0
Insert Figure 12.8

36. The Phase Relationship Between Capacitor Current and Voltage
Current leads voltage by 90°
Voltage lags current by 90°

38. Capacitive Reactance (XC)
Series and Parallel Values of XC
Insert Figure 12.18

39. Capacitive Reactance (XC)
Capacitor Resistance
Dielectric Resistance – generally assumed to be infinite
Effective Resistance – opposition to current, also called capacitive reactance (XC)
Insert Figure 12.15

40. Capacitive Reactance (XC)
Calculating the Value of XC

41. Capacitive Reactance (XC)
XC and Ohm’s Law
Example: Calculate the total current below
Insert Figure 12.17

42. The Phase Relationship Between Inductor Current and Voltage
Sine-Wave Values of
reaches its maximum value when i = 0
Insert Figure 10.9

43. The Phase Relationship Between Inductor Current and Voltage
Voltage leads current by 90°
Current lags voltage by 90°

44. Inductive Reactance (XL)
Inductor Opposes Current
Insert Figure 10.15

45. Inductive Reactance (XL)
Inductive Reactance (XL) – the opposition
(in ohms) that an inductor presents to a changing current
Calculating the Value of XL

46. Inductive Reactance (XL)
XL and Ohm’s Law
Example: Calculate the total current below

47. Capacitive Versus Inductive Phase Relationships
Voltage (E) in inductive (L) circuits leads current (I) by 90° (ELI)
Current (I) in capacitive (C ) circuits leads voltage (E) by 90° (ICE)

48. Alternating Voltage and Current Characteristics
Insert Figure 12.10

49. Euler’s identity In Euler expression, A cos t = Real (Ae j t )
Figure 4.23
In Euler expression,
A cos t = Real (Ae j t )

50. ( it is called the impedance of a capacitor)
( it is called the impedance of an inductor)

51. The impedance element
Figure 4.29

52. Impedances of R, L, and C in the complex plane
Figure 4.33

53. Figure 4.37

54. An AC circuit
Figure 4.41

55. AC equivalent circuits
Figure 4.44

56. Rules for impedance and admittance reduction
Figure 4.45