1. Lesson 11: Capacitors (Chapter 10) and Inductors (Chapter 11)

2. Learning Objectives
Define capacitance and state its symbol and unit of measurement.
Predict the capacity of a parallel plate capacitor.
Analyze how a capacitor stores charge and energy.
Explain and Analyze capacitor DC characteristics.
Define inductance and state its symbol and unit of measurement.
Predict the inductance of a coil of wire.
Explain Inductor DC characteristics.
Analyze how an inductor stores charge and energy.

3. Capacitors and Inductors
For resistive circuits, the voltage-current relationships are linear and algebraic.
Resistors can only dissipate energy; they cannot store energy and return it to a circuit at a later time.
This is not the case for capacitors and inductors. Capacitors and Inductors are dynamic elements.
The voltage-current relationships are non-linear and differential.
They are dynamic because they store energy.

4. Capacitor
A capacitor is passive element designed to store energy in its electric field. This energy can then be provided to a circuit at a later time.
Capacitors consist of two conductors (parallel plates) separated by an insulator (or dielectric).
Capacitors accumulate electric charge.
Conductive plates can become charged with opposite charges

5. Capacitor
Electrons are pulled from top plate (creating positive charge).
Electrons are deposited on bottom plate (creating negative charge).

6. Capacitor While charging a capacitor:
The voltage developed across the capacitor will increase as charge is deposited.
Current goes to zero once the voltage developed across the capacitor is equal to the source voltage.

7. Definition of Capacitance
Capacitance (C):of the capacitor: is a measure of the capacitor’s ability to store charge.
Unit is the farad (F).
Capacitance of a capacitor
Is one farad if it stores one coulomb of charge when the voltage across its terminals is one volt.

8. Effect of Geometry Increased surface area means increased Capacitance.
Larger plate will be able to hold more charge.

9. Effect of Geometry
Reduced separation distance means increased Capacitance.
As plates are moved closer together:
Force of attraction between opposite charges is greater.
Capacitance:
Inversely proportional to distance between plates.

10. Effect of Dielectric
Substituting a dielectric material for the air gap will increase the Capacitance.
Dielectric Constant  (epsilon) is calculated by using the relative dielectric constant and the absolute dielectric constant for air:
 = r  (F/m)
where 0 = x F/m and r is the relative permittivity or dielectric constant (see table 10.1 on pg 401 of text)

11. Capacitance of Parallel-Plates
Putting the factors together, capacitance of parallel-plate capacitor is given by where r is the relative dielectric constant 0 = x F/m
A is area d is the distance between plates

12. Example Problem 1
A capacitor consists of 2” x 2” plates separated by 1/32”. No dielectric is used.
Determine the capacitance.
Determine the charge on the plate if 48 V is applied across the plate.
Convert plate distance (d) to meters:
Now calculate Capacitance (C) in Farads (F):
Calculate area (A) in meters:
Finally calculate the charge (Q) in Coulombs (C):

13. Example Problem 2
A parallel-plate capacitor has dimensions of 1 cm x 1.5 cm and separation of 0.1 mm. What is its capacitance when the dielectric is mica?
Convert plate distance (d) to meters:
Now calculate Capacitance (C) in Farads (F):
Calculate area (A) in meters:

14. Dielectric Voltage Breakdown
High voltage will cause an electrical discharge between the parallel plates.
Above a critical voltage, the force on the electrons is so great that they become torn from their orbit within the dielectric.
This damages the dielectric material, leaving carbonized pinholes which short the plates together.

15. Dielectric Voltage Breakdown
The working voltage is the maximum operating voltage of a capacitor beyond which damage may occur.
This voltage can be calculated using the material’s dielectric strength (K) measured in kV/mm.
Dielectric Strength (K)
Material
kV/mm
Air 3
Ceramic (high Єr)
Mica 40
Myler 16
Oil 15
Polystyrene 24
Rubber 18
Teflon 60

16. Capacitance and Steady State DC
In steady state DC, the rate of change of voltage is zero, therefore the current through a capacitor is zero.
A capacitor looks like an open circuit with voltage vC in steady state DC.

17. Capacitors in Series and in Parallel
Capacitors, like resistors, can be placed in series and in parallel.
Increasing levels of capacitance can be obtained by placing capacitors in parallel.
Decreasing levels of capacitance can be obtained by placing capacitors in series.

18. Capacitors in Series and in Parallel
Capacitors in series are combined in the same manner as resistors in parallel.
Capacitors in parallel are combined in the same manner as resistors in series.
As an example - Solve for total charge and the voltage across each capacitor for the series circuit:

19. Example Problem 3
What is the total capacitance (CT) of the circuit shown?
Capacitors in parallel are combined in the same manner as resistors in series.
Capacitors in series are combined in the same manner as resistors in parallel.
Now, as you would with resistors in parallel, add Ceq to C1 (remember these are capacitors):

20. Example Problem 4
Assume that the circuit shown has been ‘on’ for a long time. Determine the voltage across and the charge on the capacitors?
Remember: A capacitor looks like an open circuit with voltage vC in steady state DC
To calculate the voltage, without knowing the current, use VDR:
Use KVL to solve for VC2:
VC2 = E – VC1 = 72V – 16 V = 56V
Now solve for the charges:
NOTE: R3 is effectively negated by the C2 (open therefore no current flow thus no voltage across R3

21. Power and Work
The energy (or work) stored in a capacitor under steady-state conditions is given by: 21

22. Example Problem 5
Using the same circuit from Example 4, determine the energy stored at each capacitor.
VC2 = E – VC1 = 72V – 16 V = 56V

23. Inductor
An inductor is a passive element designed to store energy in its magnetic field.
Inductors consist of a coil of wire, often wound around a core of high magnetic permeability.

24. Faraday’s Experiment #4: Self-induced voltage
Voltage is induced across coil when i is changing.
When i is steady state, the voltage across coil returns to zero.

25. Faraday’s Law
From these observations, Faraday concluded: A voltage is induced in a circuit whenever the flux linking the circuit is changing and the magnitude of the voltage (electro-magnetic force (emf) is proportional to the rate of change of the flux linkages over change in time.

26. EMF in an Inductor

27. Counter EMF
Induced voltage tries to counter changes in current so it is called counter emf or back voltage.
Back voltage ensures that any current changes in an inductor are gradual.
An inductor RESISTS the change of current in a circuit.
Acts like a short circuit when current is constant.

28. Inductance: Inductor Construction
The level of inductance (L) is dependent on the area within the coil, the length of the unit, to the number of turns of wire in the coil and the permeability of the core material.

29. Example Problem 6
For the inductor shown (the core is simply air), determine the inductance:
Convert diameter (d) to meters:
Calculate area (A) in meters:
Calculate length (l) in meters:

30. Inductor Calculations
Induced emf (e) in an inductor can be calculated:
L is the self-inductance of the coil. Units are Henry (H).
The inductance of a coil is one Henry if changing its current at one Ampere per second induces a potential difference of one Volt across the coil. 

31. Example Problem 7
The current in a 0.4 H inductor is changing at the rate of 200 A/sec. What is the voltage across it?

32. Inductance and Steady State DC
In steady state DC, the rate of change of current is zero, therefore the induced voltage across an inductor is also zero.
An inductor looks like a short circuit in steady state DC.

33. Inductors in Series and in Parallel
Inductors, like resistors and capacitors, can be placed in series or in parallel.
Increasing levels of inductance can be obtained by placing inductors in series, while
Decreasing levels of inductance can be obtained by placing inductors in parallel.

34. Inductors in Series and in Parallel
Inductors in series are combined in the same way as resistors in series.
Inductors in parallel are combined in the same way as resistors in parallel.
Inductors in Series
Inductors in Parallel

35. Example Problem 8 Determine I1 for steady state DC.
In a steady state, the inductor effectively looks like a short. Therefore, all current will flow through the inductor and NOT through R2.

36. Inductor Energy Storage

37. QUESTIONS?