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Файл:Basics of electronics. Study aid
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voltage
V
ab
(a)
V
0
ab
V
AB
voltage,v
ab
v
AB
v
ab
R.m.s. value
time
0
time
(b)
Fig. 2.9. Measures of voltage waveforms: (a) a d.c. voltage plus a sinusoid;
(b) a sinusoid voltage showing its r.m.s. value
The code used for letter symbols is:
A capital letter with a capital subscript refers to the d.c. quantity
(e.g. V
amplitude of the sinusoid (e.g. V
).
AB
A capital letter with a lower-case subscript usually refers to the
). This symbol is also used to refer to the
ab
r.m.s. value of any waveform.
A lower-case letter with a lower-case subscript refers to the
instantaneous value of the variable quantity (e.g. v
A lower-case letter with a capital subscript refers to the
instantaneous value of the total voltage (e.g. v
So, the total voltage can be written as
AB
vVvVV t
AB AB ab AB ab
).
).
ab
sin
.
Obviously a similar coding of subscripts is applied to the current, except that
normally only one subscript letter is needed to identify a current.
In general, it is important, when labelling a circuit with the voltage or
the current or when writing equations, to use the right symbol, especially
when phase changes occur in the circuit. In resistive circuits in which there
are no phase differences, it is quite possible to use either the symbol for the
amplitude or the symbol for the instantaneous value, because there is a oneto-one relationship between them. So, where phase differences do not occur
you can use either symbols for the amplitude or symbols for instantaneous
values. The ratio of amplitudes of the waveform in two parts of the circuit is
the same as the ratio of instantaneous values of the waveforms at those points
at any instant. But if there is a phase difference between the current and the
voltage, or between two voltages or between two currents, the ratio of
amplitudes is not, in general, the same as the ratio of instantaneous values, so
31
it is essential to use correct symbols. For example, if you look at fig. 2.6, you
will see that the amplitudes of these two waveforms are the same, but that the
ratio of v/i can be anything from minus infinity to plus infinity!
The fact that
V
is used to stand for the amplitude of a sinusoid as well
ab
as for its r.m.s. value can cause a confusion, so it is essential to be certain
which is meant. The symbol
V
is sometimes used for the amplitude if
ab m
there is danger of the ambiguity. Equally
V
is sometimes used for the
ab rms
r.m.s. voltage between A and B.
Although all this seems to be very complicated, it is usually obvious
from the context what is being referred to.
2.4. A.C. Components
2.4.1. Capacitors
Capacitors consist of two conducting films or surfaces separated by a
thin layer of the insulation often called the
dielectric, as shown in
fig. 2.10(a). The dielectric may be flexible, like polyethylene, in this case the
conductor/insulator sandwich can be rolled up so that the capacitor appears to
be cylindrical in shape, as in fig. 2.10(b). Otherwise the dielectric is a solid
like mica or ceramic, in this case the capacitor might be a multi-layer pack. In
integrated circuits, capacitors are usually formed by depositing a layer of
metallization on the top of the silicon dioxide (or silicon nitride) film on the
surface of silicon. So the silicon substrate is one of the conductors and the
dioxide, which is a very good insulator, is the dielectric.
Capacitors work in the following way. When a voltage is applied to
plates of a capacitor, the battery voltage is transferred to the plates of the
capacitor. This does not take place immediately. To create this potential
difference between the capacitor plates electrons have to be supplied to a
more negative plate and removed from a more positive one, and this takes a
little time depending on the current carrying the electrons. The electrons
supplied to one plate repel the electrons from the other one, leaving a net
positive charge of ionized atoms on it. Thus a current is required in both
halves of the circuit. Suppose that at a particular instant, v is the voltage that
has been built up between the plates and q is the magnitude of the charge
quantity that has been supplied. Either of electrons is supplied to one plate of
the capacitor or of the positive charge is left on the other plate. Then, the
capacitance C of the capacitor is defined as
q
C
.
v
32
The unit of the capacitance is the farad (symbol F). That is, if q is
measured in coulombs and the potential difference is measured in volts, the
capacitance will be in farads. Typical capacitors usually range in the value
from a few picofarads (pF) to a farad. Examples are given above.
Capacitances associated with integrated circuits are likely to be measured in
picofarads, or in fractions of a picofarad.
Metal plates
(foil)
(a) (b)
Fig. 2.10. Capacitors: (a) A parallel-plate capacitor; (b) a tubular capacitor made
from metal foil and a flexible dielectric such as polythene
Insulating
dielectric
Now, as it has been already indicated, supplying electrons to one plate
and repelling or displacing them from the other means that a current is
flowing towards one plate and away from the other. Indeed, these currents
continue to flow until the voltage across the capacitor becomes equal to the
battery voltage and the current falls to zero. But since the currents in the two
halves of the circuit are the same, the flow of electrons is just as if a current
was actually flowing round the circuit, despite the presence of the insulating
layer in the capacitor. The flow of electrons towards one plate and away from
the other one is indistinguishable in the rest of the circuit from the current
flowing round the circuit, except that the current flows only when the voltage
across the capacitor is changing.
If the voltage source in the circuit is an a.c. voltage instead of a d.c. one,
so that the voltage across the capacitor is changing continuously, a
corresponding, continuously changing a.c. current will apparently flow
through the capacitor. This apparent current is called the displacement
current. Though there may be a perfect insulator separating these two plates
of the capacitor it is helpful to think of this alternating current actually
flowing around the a.c. circuit as a whole.
Suppose a small change of the voltage Av is applied to a capacitor so
that a charge of q is supplied to one plate of the capacitor and removed from
the other. (If the change of charge on one plate is q, the change on the other
plate is –q). Then
33
.qC v
Dividing these changes of charge and voltage by a short interval of time
A
over which the changes occur gives an expression for the rate of change of
charge as a function of the rate of change of voltage. This provides an
alternative mathematical model of the capacitor. Thus, dividing by t gives
qv
tt
but the current is equal to the rate of change of the charge, so
C
,
qti
, and
the equation can be written as
v
iC
.
t
In the limit of very small changes in very short intervals of time,
tends to
dv dt
, the first derivative of v with respect to time, so
vt
dv
iC
.
dt
An alternative definition of the capacitance of a capacitor is therefore
the ratio of the instantaneous current flowing through the capacitor to the
rate of change of voltage across it. This is sometimes called the small-signal
capacitance to distinguish it from the ratio q/v.
The magnitude of the capacitance of a capacitor is proportional to the area
A of two conducting ‘plates’ and is inversely proportional to the distance of
separation, d, between the plates. That is, C is proportional to A/d. The constant
of proportionality is the
permittivity of the dielectric. See fig. 2.10(a) again.
Thus:
A
C
0.r
dd
The permittivity is usually written as the absolute permittivity
multiplied by the relative permittivity
is sometimes called the permittivity of free space. It is a physical constant of
0
so
r
the magnitude of 8.854 picofarads per metre (pF m
varies from 1, for free space and air, to the range 3–10 for most solid
r
0 r
. The absolute permittivity
–1
). The relative permittivity
0
insulators. Many commercially available capacitors are made from special
ceramics that have much higher relative permittivity value of 100 or more.
Capacitors in which the capacitance is independent of the magnitudes of
q or v are called linear capacitors. In some of them the capacitance varies
with the magnitude of the charge or voltage. This is true of ceramic
capacitors and of the capacitance associated with junction diodes. These are
therefore non-linear capacitors.
34
Circuits always contain stray capacitances between wires and
s
components, because most conductors are separated by the insulation in the
circuit. These stray capacitances provide the decrease in speed of response of
digital circuits or limiting the frequency range of analogue circuits. It is clear
from the equation
i C dv dt
. This equation indicates that the rate of
change of voltage across the capacitance is limited by the current, i, available
to charge it up and discharge it. So, the smaller the signal current, the slower
the possible rate of change of voltage across stray capacitances. To obtain
fast circuits it is necessary to have either large signal currents or small stray
capacitances or both.
Capacitive Reactance
Suppose a sinusoidal voltage is applied to a capacitor as shown in
fig. 2.11(a), so that
vt V t
sm
sin .
The current can be calculated with v
dv
iC
s
Differentiating Equation gives
tv
S
it CV t
sm
dt
From this result we can draw two conclusions.
v
i
S
V
S
m
C
V
S
0
i
S
I
m
substituted for v. Thus,
s
.
dv dt V t
sm
cos .
cos
, so
vS=Vm sin t
T
iS=CVm cos t
T
2
t
0
Fig. 2.11. Relationship between the current through a capacitor and the voltage
applied across it. In an ideal capacitor the voltage always lags behind the current
by /2 radians
35
90
o
t
Firstly, the phase of the sinusoidal voltage waveform across a capacitor
lags behind the sinusoidal current flowing through it by
/2 radians, as
shown in fig. 2.11. You can visualize why the voltage phase lags behind the
current phase as the current must flow ‘through’ the capacitor
bring about
the change in the voltage across it. Thus, voltage changes arise
in order to
when the current flow causing them starts to flow.
The
reactance of a capacitor, X
amplitudes, so the second conclusion is that
, is the ratio of voltage and current
C
the reactance of a capacitor is
1/C. That is,
V
X
C
m
CV C
1
.
m
This result is obtained by taking the ratio of voltage and current
amplitudes. Thus, the reactance of a capacitor
decreases with the increasing
frequency.
The reactance is measured in ohms, just like the resistance, but you should
remember that with the reactance, the voltage and the current are not in phase.
2.4.2. Inductors
An inductor is a coil of wire such as that illustrated in fig. 2.12. A
varying current flowing through the wire of the coil induces a voltage in the
same wire by the process of the electromagnetic induction, so that a voltagecurrent relationship is set up, which has nothing to do with the resistance of
the wire. The voltage between the ends of the coil is dependent on the
change
of the current flowing through it and on the number of turns of the
rate of
wire in the coil; it does not depend on the actual value of the current. These
properties depend on the principles of electromagnetism. The most important
can be summarized as follows.
Induced
voltage, e
Coil
(a) (b)
Fig. 2.12. The magnetic effect of an electric current: (a) the structure of an inductor;
(b) a toroidal core inductor
36
(1) An electric current creates a magnetic field similar to that produced
B
by magnets. This magnetic field encircles the wire carrying the current, as
shown in fig. 2.12(a). The rings drawn round the wire in this figure are
intended to indicate the existence of the field. The arrows show its direction
for the current shown. If you reverse the current you reverse the direction of
the field. The strength of the
magnetic field, H, can be calculated from
Ampere’s law, which states, as shown in fig. 2.12(a), that
I H circumference of the circular path
.
The field strength for a given current I is inversely proportional to the
distance from the wire.
When a current-carrying wire is coiled up as in fig. 2.12(a), the field is
concentrated within the coil. Each turn adds to the strength of the field.
(2) This magnetic field produces the
magnetic flux which follows the
direction of the field but the magnitude of the flux depends on the material
surrounding the wire. The flux per unit cross-section – called the
density,
B – depends on the permeability of the material through which
flux
the field passes; that is
Flux Density Permeability Magnetic Field Strength
or
.
H
The permeability is usually expressed as the product of the permeability
of free space,
different materials. The permeability of free space
, and the relative permeability
0
which is specific to
r
71
410 H
0
(henries
per metre). The relative permeability of most materials is close to 1; but there
is one class of materials,
permeability. For ferromagnetic materials
ferromagnetic materials, which have much greater
may be 1000 or more.
r
Ferromagnetic materials include iron, steel and various oxides of iron called
ferrites. Because of its high permeability, the flux produced in a ferromagnetic
material by a given magnetic field is much larger than the flux produced in air.
(3) A
changing magnetic flux induces an electrical voltage or e.m.f. in a
wire placed in this field. The magnitude of the e.m.f. can be calculated from
Faraday’s law, which states that
rate of the change of the flux linked with the wire.
the induced e.m.f. in a wire is equal to the
So, returning to fig. 2.12(a)
again, a current flowing in the coil will create a magnetic flux through the
coil as indicated by the arrows. If the current flowing through the coil is now
varied in the magnitude, the flux will vary correspondingly. As a result an
37
e.m.f. will be induced in the coil. This process is called the self-induction.
Applied to a coil of wire, Faraday’s law can be expressed as
eL
,
dt
di
where
e is the induced e.m.f. and L is the inductance of the coil.
The induced e.m.f. is sometimes called the
back e.m.f. because it
‘opposes’ the voltage applied to the coil.
The unit of the inductance is the
inductance of a coil is
L = 1 H if the e.m.f. of 1 V is induced in it when the
change of the current flowing through it is
henry (symbol H). Thus, the
–1
1di dt
A s
. The inductance of
a coil depends upon the number of turns in the coil and the nature of the
magnetic path. For a tightly wound coil, which diameter is much greater than
its length, L is proportional to the square of the number of turns.
The inductance L is greatly increased by giving the coil a ferromagnetic
core as illustrated in fig. 2.12(a). The greatest effect is achieved if the coil
and the core encircle each other as shown.
All conductors, even straight wires, possess some inductance. A 10 mm
length wire in air has the inductance of about 1 nH. The value is small, but it
is significant for amplifiers of television signals. Inductors used in electronic
circuits are usually in the range of millihenries or microhenries. The inductor
in fig. 2.12(a) is likely to be a few henries.
The permeability of a ferromagnetic core varies with the strength of the
applied magnetic field. The larger the field, the smaller the permeability. This
means that the inductance of a coil with a ferromagnetic core will depend
upon the magnitude of the current in the coil. In other words, coils with
ferromagnetic cores are usually non-linear. Air, however, has a constant
permeability, so air-cored coils are linear inductors.
Note that ferromagnetic materials used in coils are not the same as those
used to make permanent magnets. Permanent magnets hold their flux even if
the magnetic field is removed. But the flux in the materials used in coils
should ideally follow the variations of the field as precisely as possible.
Inductive reactance
Consider what happens when a sinusoidal current flows in an ideal
inductor in which the wire has zero resistance as shown in fig. 2.13(a).
By Faraday’s law an e.m.f. will be induced in the inductor which is
proportional to the rate of change of the current.
Note: The rate of the change of a sinusoid is again a sinusoid. It is of the
same frequency as the original but of different phase. In fact its phase is
always
/2 radians in advance of the original waveform.
38
If the current is given by
s
X
it I t
sin
m
, then, since
eLdidt
,
eLI t
cos .
m
So the amplitude of the sinusoidal e.m.f. induced in the coil is
as shown in fig. 2.13(b), and it has a phase lead of /2 rad.
i
i
I
m
i=I
L
V
S
0
e
(a)
V
m
0
(b)
T 2T
vS=e=LIm cos t
o
90
sin t
m
LI
,
m
t
t
Fig. 2.13. A sinusoidal voltage applied to an inductor, (a) the relationship between
the current and the voltage:
it is also true that
the current and the voltage across an inductor; the current always lags /2 rad behind
ev
. (b) Waveforms illustrating the phase relationship between
eLdidt
the voltage
. With no resistance in the circuit
The ratio of the e.m.f. amplitude to the current amplitude is called the
reactance of the coil, or sometimes the inductive reactance. The usual
L
XL. So
LI
m
I
m
.
L
symbol for the inductive reactance is
2.4.3. Transformers
Transformers consist of two inductors placed close together, as it is
indicated in fig. 2.14. They are placed rather close to each other. Thus, the
rate of the current change in one coil not only induces the voltage in itself by
the electromagnetic induction, but also induces the voltage in the other coil.
The magnetic flux produced by the current in the ‘primary coil’ – the one
connected to the source – affects both coils, and so a sinusoidal current in the
39
primary coil induces the sinusoidal voltage both in the primary and secondary
p
coils. Here we refer to coils as windings.
Fig. 2.14. Typical construction of a transformer
In fig. 2.14 the arrows indicate that the magnetic flux produced by the
current in the primary winding is linked with the secondary winding.
Figure 2.14 shows a construction of a transformer that might be used in a
mains power supply. As in the case with the inductor, the ferromagnetic core
increases the amount of the flux, and by encircling both windings it ensures
that any variations in the flux affect both windings equally.
The result is that the same e.m.f. is induced in each turn of the wire in
each winding, so the total e.m.f. induced in each winding is proportional to
its number of turns. If
N
and
are the numbers of turns in the primary
N
s
and secondary windings, as shown in fig. 2.15(a), and
and
e
1
e
2
are
instantaneous values of the back e.m.f. induced in them, then
N
e
1
eN
2
p
.
s
If we suppose that the coil resistances are negligibly small, the back
e.m.f. in the primary winding must be exactly equal to the applied voltage,
ev
then
1 s
If
larger voltage output than the input voltage, or a voltage gain of
If
. Thus,
NN
sp
NN
sp
N
v
s
eN
2
p
or
s
ve
s
NN
2
.
ps
, the transformer is called a step-up transformer, giving a
/
NN
sp
).
, the device is a step-down transformer with a voltage gain of
less than 1.
40
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