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  ( Farads)
V
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
A
A
C  or
d
d
where
(farads, F)
r is the relative dielectric constant
0 = 8.854 x 10-12 F/m
A is area
d is the distance between plates
Example Problem 1
C 
A
A
or
d
d
(farads, F)
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?
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)
Dielectric Strength (K)
V  Kd
Material
kV/mm
Air
3
Ceramic (high Єr)
3
Mica
40
Myler
16
Oil
15
Polystyrene
24
Rubber
18
Teflon
60
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.
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.
1
1
1
1


 ..... 
C
T
C C
1
2
C
n
Capacitors in parallel are
combined in the same
manner as resistors in
series.
C  C C
T
1
2
 ....  C n
Power and work

The energy (or work) stored in a capacitor under
steady-state conditions is given by
1 2
w  Cv
2
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.
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.
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.
d
emf 
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
INDUCTANCE
Inductor 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.
Inductor calculations

Induced emf in an inductor can be calculated:
di
eL
dt



(volts, V)
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.
Example Problem 2
The current in a 0.4 H inductor is changing at the rate of
200 A/sec. What is the voltage across it?
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.
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
FIG. 11.56 Inductors in
series.
L


T
 L1  L 2  ...  L n
FIG. 11.57 Inductors
in parallel.
1
L

T
1

1
L L
1
 .... 
2
1
L
n
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.
Inductor Energy Storage
i
di
1 2
W   pdt  L  i dt L  idi  Li
0
0 dt
0
2
t
t
(J)