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expressed either in degrees per second or radians per second. (Note that there
f
f
are 2 radians in 360°.) In practice, however, is always expressed in
radians per second.
Sine Wave
sin
4
radians2
A
y
Positive
Half
Amplitude
y=A
3
0
180
o
Negative
Half
T 2T
time
-A
(a)
First Period Second Period
y
A
y=A
2
3
cos
radians
4
0
time
(b)
-A
90
o
T 2T
Fig. 2.4. Sinusoidal waveforms: (a) a sine wave; (b) a cosine wave
The
frequency f of a sinusoid is the number of cycles of the wave per
second. The unit of frequency is the hertz or Hz, for short. (It is named after
Heinrich Hertz, 1857–1894, a German physicist who discovered radio waves
in 1885.) For example, one cycle of the waveform takes 20 milliseconds, so
the frequency of this sinusoid is
1 1000
ms
20 20
50
Hz
.
This is the frequency of the a.c. mains electricity power supply in
Europe. (In North America the frequency of the a.c. supply is 60 Hz.)
The frequency f is the number of cycles per second, and there are 2
radians per cycle, so the number of radians per second, which is , is
.
2
The waveforms in fig. 2.5 are not sinusoidal. They are
waveforms
. They possess a shape which is repeated over and over at a
periodic
constant repetition rate. The duration of the repeating pattern is called the
period of the waveform. The sinusoid is one example of a periodic
waveform. The waveform in fig. 2.5(a) might be produced by a steady note
played on the flute; that one in fig. 2.5(b) might be the voltage produced by
an ultrasound sensor; and the waveform in fig. 2.5(c) is the waveform applied
to the deflection plates of an oscilloscope.
21
V
0
(a)
V
0
(b)
V
0
(c)
Fig. 2.5. Examples of Periodic but Non-Sinusoidal Waveforms
T
T
T
T
2
2
time
T
2
time
time
T
Most electronic circuits are not built in order to handle periodic nonsinusoidal waveforms, such as those in fig. 2.5; they are designed and built to
deal with signals. So why is it important to discuss simple waveforms like
sine waves? One reason is that circuits that can handle waveforms such as
those in fig. 2.5 can usually handle signals as well; and furthermore, such
waveforms are, in effect, nothing but several different sinusoidal waveforms
added together. This is also true of other simple waveforms, but a sinusoidal
waveform of the voltage or current is unique in that when it is applied
continuously to a linear circuit the currents and voltages anywhere in the
circuit are always sinusoidal at that frequency. They will differ in other
respects in different parts of the circuit, but not in frequency. This makes the
analysis in terms of sinusoids much simpler than other forms of analysis.
There are three properties of sinusoids that are important in electronics.
The first two are the amplitude and the frequency. The third is the
phase. The
waveforms in figs. 2.4(a) and 2.4(b), for example, differ in the phase. This
concept of the phase is a little more difficult to grasp than the amplitude or
the frequency. To begin with, the phase and the phase difference are not quite
the same. First let’s consider the meaning of the phase.
0,v
The waveform in fig. 2.4(a) shows
when 0t
. Furthermore, the
value of v initially increases from zero (rather than decreases) as time
increases. This means that this waveform is sine waves with the zero phase.
The phase of a sinusoid depends on the value of v at a particular instant in
time, usually regarded as
.
0t
22
If the waveform of the voltage or the current versus t has a non-zero
t
0t
value at the moment
phase angle . Then the equation of the graph of such a voltage waveform
it can be described as a sine wave which has a
,
would be
a little before
lagging, or delayed with respect to a sine wave – that is
vt V t
If the waveform is in advance of, or leading, a sine wave – that is
sin
m
– the phase is said to be positive. If the waveform is
0t
.
0v a little after
0v
0t – the phase is said to be negative.
Figure 2.6 shows how the value of can be determined. In fig. 2.6 two
sinusoidal waveforms are plotted against . As you can see, the waveform
of the curve (a) lags behind a sine wave by the angle , which, in this case, is
6t
/6. (The voltage does not rise to 0 V until
6t
after a zero-phase sine wave would rise.) The peaks of the
waveform are to the right of the corresponding peaks of the sine wave.
A lagging phase is represented mathematically by a minus sign, so the
equation for the curve (a) is
V
0
/6
vt V t
sin 6
2
m
3
.
, that is a time
4
t, rad
(a)
V
/2
(b)
Fig. 2.6. Phase shift: (a) a sinusoid that lags in phase;
(b) a sinusoid that leads in phase
Similarly, graph 2.6(b) shows the sinusoid that is in advance of or
leading
(b) is
following. You can use the phase to refer to waveforms of different
frequencies but you must refer to a particular instant. Thus, it is possible to
sin t by the angle , where
vt V t
The difference between the terms phase and phase difference is the
sin 2
m
.
2
23
3
2
. So the equation of the curve
4
t, rad
say that waveforms can have the same phase at even though they have
t
t
different frequencies. But, if any instant other than
0
were regarded as the
0
reference instant, all the waveforms would have different phases.
The phase difference, on the other hand, refers to the comparison
between two sinusoids of the same frequency, like those of figs. 2.4(a) and
2.4(b). As they have the same frequency, the phase difference between these
two sinusoids is the same at all instants of time. Thus, there is a constant
2
phase difference of
behind the cosine wave by
sine wave by
2
between figs. 2.4(a) and 2.4(b). The sine wave lags
2
or, if you prefer, the cosine wave leads the
. There is no sense in speaking of the phase difference
between the waveforms of different frequencies.
To obtain the phase difference between two sinusoids, given the time
difference, you simply express the time difference as a fraction of the period of
one cycle and then multiply by the number of radians (or degrees) in one cycle:
Phase difference radians
time difference t
,2.
period of one cycle
Sinusoids of a different phase, but of the same frequency, arise in circuits
because certain components, such as capacitors and inductors, cause phase
differences between the current through them and the voltage across them.
It is possible to express a phase difference either as a lead of
equivalently as a lag of
2
since a waveform can be shifted in time in
or
either direction in order to make it coincide with another waveform of the same
frequency. In practice a smaller phase angle is required. So a sine wave is
usually said to lag behind a cosine wave by
2
rather than to lead it by
32
.
2.3. Voltage, R.M.S. and Power
2.3.1. Periodic Waveforms
Examples of simple periodic waveforms are sinusoids and squarewaves.
They have repeatedly recurring shapes. The interval over which the shape
repeats is called the
squarewaves, the rate of repetition is called the
period. In the case of non-sinusoids, such as
fundamental frequency.
Average Value
The average value of any periodic waveform can be found by taking n
equally spaced samples of the waveform over one period. Then
vvv v
Average value v
123
n
.
n
24
The average value of a waveform is denoted as v. This method gives an
approximate result because the number of samples is finite. To obtain the true
average of the waveform it is necessary to use a continuous averaging
process. This can be done by integrating the waveform in relation to time
over one period, and dividing by the period T:
T
1
vvtdt
T
0
.
This can be interpreted as: to find the average value v of a periodic
waveform, find the area between the waveform and the time axis over one
period, then divide by the period (the area below the time axis should be
taken as negative).
v=Vm sin t
V
(a)
-V
m
0
m
Period T=1/f
Sine Wave
t
V
Average
value
t
(b)
2V
V
0.5V
V
m
m
m
m
/
0
T=1/(2f)
2
v
2
2
0
(c)
T=1/(2f)
t
Fig. 2.7. Waveforms: (a) the sine wave; (b) the rectified sinusoid;
(c) the squared sinewave
Figure 2.7(b) shows a periodic waveform which is obtained by ‘rectifying’
the sinusoid in fig. 2.7(a) to obtain a periodic sequence of half sinusoids.
The rectified sinusoid has half the period
compared with the original. Its
fundamental frequency is twice the frequency of the sine wave.
To obtain the average value of the rectified sinusoid we can represent
one period of its waveform by a half-period of the original sinusoid. Thus,
for values of
25
sin
m
vt V
from 0 to , where is the angle
.
t
The average value is given by forming an integral over one period and
R
dividing by the period:
vV d
sin cos 1 1
m
0
12
VVV
mmm
0
Rectifying a sinusoidal waveform allows to measure its voltage.
Mean-Square Value of a Periodic Waveform
As it has been explained before, a d.c. voltage source of the magnitude
V driving the current I in the circuit of the resistance R, delivers to the circuit
a steady power of the value given by any of three inter-related formulae
,,PVI PV R PIR
22
The voltage may be changing with time but we can express the
instantaneous power which it delivers as where v is the instantaneous
2
voltage (or as vi or i
R). The average power Pav which it delivers, over a
period of time, will be the average value of
222
v average value of v v
The average value of
P
av
av
2
v
is usually written
2
Rv
2
over that time; so
Rv
.
RR
2
v and is referred to as the
‘mean-square voltage’. Since the terms ‘mean’ and ‘average’ are
interchangeable we can say that:
The mean-square voltage of a waveform is given by the mean (or
average) of the squared values of the waveform.
Consider the case of a sinusoidal waveform. Fig. 2.7(a) shows a
V
sinusoid voltage of the amplitude
, and fig. 2.7(c) is a graph of the voltage
m
squared. You will notice that when the sinusoid becomes negative, the sinesquared waveform is still positive. In fact, the sine-squared waveform repeats
itself every half-cycle of the original sinusoid. This can be shown
algebraically:
222
vV tV t
sin sin
mm
2
.
But, it is known that
2
sin cos2
1cos2 1 1
tt
.
t
222
The waveform
the frequency, with the amplitude
2
2
voltage
V
. This d.c. voltage is the mean value of
m
2
v
can be considered as an inverted cosine wave of twice
2
2
V
(and zero mean), added to a d.c.
m
2
v
. So, in this case, we
do not need to integrate to find the mean value. Provided we average over an
integral number of cycles:
26
2
R
R
R
P
V
2
m
v
for a sinusoid of amplitude
2
V
.
m
Then it follows that the average power delivered to a resistor is
2
V
m
P
av
for a sinusoid of amplitude
2
V
.
m
Strictly, this is the average power over a time interval which is an exact
number of half periods of the sinusoidal wave, but provided the time interval
is many periods long; it makes little difference if it is not an exact number of
half periods.
Similar analyses lead to three expressions for the average power
dissipated in a resistor. Thus, for a sinusoidal voltage of the amplitude V
and a corresponding sinusoidal current of the amplitude I
VI V I R
PPP
av av av
mm m m
,,
22 2
22
:
m
.
,
m
Root Mean Square Value
The square root of the mean-square value of any waveform (not just
sinusoidal) is called its ‘root mean square’ or r.m.s. value. Since the mean
square voltage is
2
, the r.m.s. voltage is
v
Vv . The average power in a
rms
2
resistor is
V
rms
R
2
.
2
v
P
av
It is important to use the correct sequence of operations when
calculating r.m.s. values, otherwise incorrect results will be obtained. Thus:
The r.m.s. value of a waveform is found by taking the square root of the
mean of the square of the waveform.
Mean-square current is defined in a similar way to the mean-square
voltage, so the average power is
av rms
2
IR
.
Thus, the average power dissipated in a resistor can be expressed as
PVI PV RP IR
av rms rms av rms av rms
,,.
22
In other words:
The equations for the average power (in a resistor) using r.m.s. voltages
and currents have the same form as the equations for the constant power
using d.c. voltages and currents.
27
R.m.s. values are consistent with Ohm’s relationship, that is, in a
resistor of the value R,
For sinusoids,
VIR
rms rms
Vv V
rms m
2
and they do not depend on the waveform.
2
VV
mm
2
0,707 .
2
When a numerical value is quoted for a sine wave voltage without
qualification, it is usually the r.m.s. value; thus a quoted a.c. mains voltage of
240 V is its r.m.s. value.
Note that mean-square and r.m.s. values are not restricted to sinusoids.
However, all waveforms of the voltage or current dissipate power in resistors,
so an r.m.s. value of the current or voltage can be stated for them.
2.3.2. Non-periodic Waveforms: Signals and Noise
The term ‘waveform’ is used in a fairly general way to refer to the timevarying voltage and current. The term ‘signal’ is used for waveforms that
transmit a message or information from one place to another. In order to
transfer information, a waveform must keep changing its shape and therefore,
must be non-periodic. There is an element of unpredictability in this.
Waveforms with this unpredictable behaviour are said to be
random, and all
signals in the real world vary in a non-periodic random mode.
When we begin to design an amplifier for audio-frequency signals, such
as speech or music, we need to know the voltage and frequency range of
these signals, to make sure that the amplifier will amplify them without any
distortion. The problem is how we measure or quantify the voltage and
frequency range of a randomly-varying signal.
There is another type of random waveforms, which is called the
noise. The
word ‘noise’ is used in electronics to describe a waveform which interferes with,
or corrupts, a signal. Thus the difference between a signal and noise is whether
we are interested in it or not and if we can interpret it. An example of the
interference is the reception by a radio receiver of another broadcast signal in
addition to the wanted one. Another example is a car ignition system radiating
electrical pulses which affect the radio or television reception.
Sources of the interference have one thing in common: in each case the
interference originates outside the electronic circuits that we are concerned
with. Another type of the noise is generated internally by all circuits. You can
sometimes hear it in an audio amplifier if you remove the signal source.
Then, if there is no audible external interference, you may hear a hissing
noise from the loudspeaker when you turn up the volume control. This is
sometimes called the
internal noise. In the specification and design of an
audio amplifier for example, the internally-generated noise has to be
28
considered. The amount of noise which is acceptable at the amplifier output
is first specified. Then the designer must calculate the noise that is likely to
be generated by the circuit, to see if the design meets the specification.
Fig. 2.8. Waveform of electrical noise
With such waveforms, the terms the amplitude, the frequency and the
phase have no clear meaning. Instead, the terms the r.m.s. voltage (or the
mean square voltage) and the power density spectrum are used.
The basic ideas in the preceding sections on the voltage and the power
of periodic waveforms are also applied to signals and noise. But the ideas
have to be modified, because of the non-periodic nature of signals and noise.
Average value
It is possible that a signal, such as that from a microphone, can exist
as voltage variations with the zero mean superimposed at a d.c. voltage level.
It is clear that in this case, the waveform’s average, or the mean value is the
value of the d.c. voltage.
Mean-square value
Just as with periodic waveforms, the instantaneous power delivered to a
resistor by a non-periodic waveform is , where v is the instantaneous
2
Rv
voltage of the waveform. This is true both for the signals and the noise. In the
same way, as before, the average power input by a non-periodic waveform is
2
vR
the average value of
:
22
P
av
vv
RR
av
.
Now, however, we are faced with non-periodic signal and noise
waveforms. There are two problems in finding the mean-square value of such
a waveform: the shape is not simple, and without a repetition period it is
difficult to know what interval to average over.
29
In one method of measuring the mean-square voltage of a waveform, an
analogue circuit, which has the output proportional to the square of its input
voltage, is used. Then the output is connected to an averaging meter, such as
a moving-coil voltmeter, which responds only to slowly changing
waveforms. Thus, if all the fluctuations of the waveform are faster than the
meter’s response, the meter responds only to the mean, or d.c. component,
and the mean-square voltage is displayed.
An alternative method uses a digital technique. This is the method used
in digital ‘true-r.m.s.’ meters. The input waveform is sampled, and each
sample is fed, in turn, to an analogue-to-digital converter. Then the digital
output values are squared by a digital circuit, and these squared digital values
are averaged. The digital output from the averaging process represents the
mean-square value of the input waveform. The final step is to take its square
root, and display this as the root mean square, or r.m.s., the value of the input.
In fact, both the average value and the r.m.s. value depend on the
number of samples and on the particular section of the waveform chosen.
In practice, it is conventionally assumed that the estimates of the
average and r.m.s. values of a random waveform can be progressively
improved by taking more samples over longer intervals. For most random
noise sources the average value, then, tends towards zero as the averaging
time is increased, while the r.m.s. value tends to a well-defined fixed value.
2.3.3. Symbols for Voltages and Currents
Several terms are used to describe the magnitudes of waveforms and d.c.
levels. Chapter 1 deals with the d.c. voltage and current. This chapter deals
with instantaneous values of waveforms, amplitudes or peak values of
sinusoids, voltages and currents which are the sum of both d.c. and a.c.
components, and R.m.s. values. Now it is necessary to clarify the distinction
between various measures of electrical quantities that are normally used.
Figure 2.9(a) shows a diagram of a voltage which begins as a d.c. voltage,
to which a sinusoid is added, so that the waveform can be thought of as the sum
of a d.c. and an a.c. voltage. The arrows on the diagram mark the principal
voltage values which can be referred to in connection with such a waveform.
Beside each arrow there is a letter symbol used to refer to each quantity.
The letter(s) used in the subscript identify the terminals between which the
voltage is measured. In this illustration the terminals referred to are called A and
B, so that V
When we consider transistors, for example, V
is the voltage of the terminal A with respect to the terminal B.
AB
would refer to the voltage of
BE
the base terminal relative to the emitter terminal. Different symbols
correspond to different kinds of the voltage.
30
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