Lesson 14 – Capacitors & Inductors

Learning Objectives Define capacitance and state its symbol and unit of

measurement. Predict the capacitance of a parallel plate capacitor. Analyze how a capacitor stores energy. Define inductance and state its symbol and unit of

measurement. Predict the inductance of a coil of wire. Analyze how an inductor stores energy.

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.

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

Capacitor Electrons are pulled from top plate (creating positive

charge) Electrons are deposited on bottom plate (creating

negative charge)

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

Definition of Capacitance Capacitance of the capacitor: is a

measure of the capacitor’s ability to store charge.

Unit is the farad (F). Capacitance of a capacitor (C):

Is one farad if it stores one coulomb of charge when the voltage across its terminals is one volt

( )Q

C FaradsV

Effect of Geometry Increased surface area means increased

Capacitance Larger plate will be able to hold more charge

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

Effect of Dielectric Substituting a dielectric material for the air gap will increase the

Capacitance

Dielectric Constant epsillon is calculated by using the relative dielectric constant and the absolute dielectric constant for air:

= r 0 (F/m)

where 0 = 8.854 x 10-12 F/m

Capacitance of Parallel-Plates Putting the factors together, capacitance of parallel-plate

capacitor is given by

where r is the relative dielectric constant0 = 8.854 x 10-12 F/m

A is aread is the distance between plates

(farads, F)o r

A AC

d d

A parallel-plate capacitor with a vacuum dielectric has dimensions of 1 cm x 1.5 cm and separation of 0.1 mm. What is its capacitance?

What would its capacitance be if the dielectric were mica?

Example Problem 1 (farads, F)o r

A AC

d d

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

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 (kV/mm)

V KdDielectric Strength (K)

Material kV/mm

Air 3

Ceramic (high Єr) 3

Mica 40

Myler 16

Oil 15

Polystyrene 24

Rubber 18

Teflon 60

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.

Capacitance and Steady State DC

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, while decreasing levels can be obtained by placing capacitors in series.

CAPACITORS IN SERIES AND IN PARALLEL

Capacitors in series are combined in the same manner as resistor sin parallel.

Capacitors in parallel are combined in the same manner as resistors in series.

1 2....

T nC C C C 1 2

1 1 1 1.....

T nC C C C

Power and work

The energy (or work) stored in a capacitor under steady-state conditions is given by

21

2w Cv

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.

Voltage is induced across coil when i is changing. When i is steady state, the voltage across coil returns to

zero.

Faraday’s Experiment #4: Self-induced voltage

Faraday’s Law

From these observations, he concluded:A voltage is induced in a circuit whenever the flux linking the circuit is changing and the magnitude of the voltage is proportional to the rate of change of the flux linkages.

demf

dt

EMF in an Inductor

Counter EMF Induced voltage tries to counter changes in current so it is

called counter emf or back voltage.

BEMF 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

INDUCTANCEInductor Construction

The level of inductance 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.

Induced emf 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.

die L

dt (volts, V)

Inductor calculations

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

Example Problem 2

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.

Inductance and Steady State DC

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 can be obtained by placing inductors in parallel.

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.

FIG. 11.56 Inductors in series.

FIG. 11.57 Inductors in parallel.

1 2...

T nL L L L

1 2

1 1 1 1....

T nL L L L

Inductor Energy Storage

2

0 0 0

1

2

t t idiW pdt L i dt L idi Li

dt (J)