Alternating current tutorial

3.1

Chapter 3
Alternating Currents
In the previous chapter we dealt with DC or direct current. Direct current is easily
visualised, since it flows in one direction only, so it behaves similarly to water flowing in
a pipe. However there is also another form of current called alternating current or AC.
In an alternating current the charge oscillates (i.e. moves backward and forward) instead
of flowing ever onwards. The flowing-water analogy applies to AC as well, but it may be
harder to visualise, because such oscillating water flows are not part of our everyday
experience.
AC is sometimes preferred over direct current, particularly for distribution of electrical
energy. It is also well suited for representing signals such as those carrying speech, music,
radio waves and for encoding of information. A typical AC signal is illustrated in Figure
3.1. The two forms of current, AC and DC, are not mutually exclusive, and it is possible to
combine the two. The two parts can simply be added together, as illustrated in Figure 3.2.
Similarly, it is possible to combine any number of alternating currents that vary
differently. This is illustrated in Figure 3.3.

Signal value

Vp-p

Time
Vp

Signal value

Figure 3.1 A typical AC signal.

Time

Figure 3.2 This signal is not pure AC, as it has a DC part.

ELEC166 Chap3.doc 02/01/06

3.2

Amplitude (V)

+5

0

-5
0

2

4

6

8

10

Time (msec.)
Figure 3.3 In this signal, two voltages varying at two different
frequencies were added together.
All the rules that we have learned so far in the context of direct currents still apply
to alternating currents, but only to the instantaneous values of currents and voltages.
So, for example, Kirchhoff's laws apply at each instant of time separately. Similarly,
if we pass an AC current through a resistor, we will observe across this resistor an AC
voltage whose instantaneous value obeys Ohm's Law. This voltage will alternate
its polarity1 as the current reverses direction, and is denoted as the AC voltage (see Figure
3.4). In addition to these familiar DC rules, alternating currents and voltages have a couple
of new features, not applicable to DC. Now we will turn to these more specific features of
simple alternating currents and voltages.

1

Polarity indicates whether a quantity is positive or negative.


Current (A)

Voltage (V)

3.3

10 V
Time
0
Resistance = 10 Ω
1A
0

Time

Figure 3.4 When the current flowing through a resistor alternates,
the voltage across this resistor alternates in the same fashion.
At all times, for this resistor, v(t) / i(t) = 10 .

3.1 Basic Concepts – Frequency, Amplitude, Phase
As mentioned earlier, an alternating voltage alternates its polarity (becomes positive and
negative in sequence). The number of these alternations per second is called the frequency
and is denoted by f . The unit of frequency is the hertz, abbreviated as Hz.
A common form of alternating voltage v(t) is expressed as a sinusoidal function of time
(see Figure 3.1)
v(t) = Vp sin (ω t) .
Here Vp is the peak value of the voltage, or its peak amplitude, measured in volts,

and ω is called the angular frequency and is expressed in units of radians per second.
The angular frequency ω is related to the frequency f through
ω=2πf .
Similarly, the same can be said about alternating currents, but of course the amplitude of
a current is measured in amperes.
It is sometimes important to know the peak-to-peak value Vp-p of a voltage (or current)
signal. This quantity is visualised in Figure 3.1 and, of course, it is equal to 2 Vp
(or equivalent for currents).
Now we turn our attention to the rms value of voltage or current, which is widely used.
(The abbreviation 'rms' means 'root-mean-square'.) We remind you here that, in the case of
DC, the electrical power dissipated in a resistor R varies with the square of the voltage V ,
according to the expression
P = V2 / R .


3.4

Of course, for DC voltages the value V stays constant. In the case of an AC voltage, the
same formula applies, but now V needs to be replaced by the actual voltage Vp sin (ω t),
so the instantaneous power dissipated in a resistor is not constant. This leads to
a somewhat complicated expression, of limited practical utility. In many practical
situations we are, however, concerned with the average power, where the average is taken
over time. The expression for the average power is put together in a way to imitate
the expression of power for DC voltages. This is possible if we define a voltage amplitude
Vrms which is related to the peak voltage Vp in the case of a sine wave by the formula
Vrms = Vp / 2 = 0⋅707 Vp .
Then the average power Paverage can easily be calculated through
Paverage = Vrms2 / R .
Example 3.1 The voltage at household electrical outlets in the USA is 110 volts rms
at a 60 Hz frequency. Calculate the peak-to-peak voltage Vp-p .
Solution
The peak-to-peak voltage Vp-p is the difference between the highest voltage and
the lowest voltage. In the case of a sinusoid, this difference is twice
the peak voltage Vp . We can find the peak voltage from the rms voltage by reversing
the operation given above.
Vp-p = 2 × Vp = 2 × 2 × Vrms
Vp-p = 2 × 1⋅414 × 110 = 311 V .

Notice that we retained in the final answer no more significant digits than in
the numbers with which we started.
Now we discuss the concept of phase. This will be done in two ways: mathematically
and graphically. Mathematically an AC signal is described as
v(t) = Vo sin (ω t + φ)

where φ is called the phase. Phase is measured in either degrees or radians
(radians in equations, degrees in practice). A choice of a particular phase is arbitrary
for one signal, but when we have more than one signal – say two, at the same frequency –
the difference between their phases is not arbitrary. The sequence of pictures below
(Figure 3.5) shows signals that differ in phase (with respect to the uppermost one)
by 45 degrees, 90 degrees, 135 degrees, 180 degrees etc. We can easily see that the waves
progressively move to the left as the phase difference increases. Phase differences
in between these values would give waveforms with intermediate shifts.


3.5

0°
45°
90°
135°
180°
225°
270°
315°
360°

Figure 3.5 The consecutive signals are progressively shifted in
phase by 45, 90, 135, ... , 360 degrees.
So the phase difference describes how much one signal is shifted in time with respect to
the other. To work out the phase difference, the two signals may have different
amplitudes, but they need to have the same frequency.

3.2 Measurement Techniques
Alternating currents and voltages can be measured in a similar way to DC. In earlier days
the ammeters for alternating currents were different from those for direct currents,
and likewise for voltages, so that to deal with all measurements several separate
instruments were needed. At present these functions are often combined in one
multimeter, which needs to be set appropriately to 'DC amperes' or 'AC amperes'
or 'DC volts' or 'AC volts'. It is very important to remember that the meter reading for AC
is the root-mean-square amplitude of voltage/current, which is less than the peak
amplitude.


3.6

Another useful instrument for measuring AC voltages which, in addition, allows us
to visualise other useful quantities, is called the oscilloscope, or cathode-ray oscilloscope
(CRO). It can be used to measure and display both AC and DC signals. It displays
on its screen a part of a rapidly changing waveform, so that measurements can be made
of the frequency, shape and size of the waveform. The CRO usually requires power
from the mains, but some models are battery-operated.
The voltage waveform on the CRO is shown through movement of a bright spot
on the screen. If this spot moves very fast and repeatedly, our eye perceives the track
as a continuous line. There are several controls on the CRO panel that allow us to control
the motion of the spot. If no signal is applied, then the spot moves uniformly from the left
to the right, so its position is proportional to time. We can make it move faster or slower
as indicated by the control called 'time/division'. If we set this control to be 1 millisecond
per division, the spot will be moving over one division on the screen in 1 msec.
When it reaches the right side of the screen, it rapidly goes back to the left. If we set 0⋅2
msec./division, the spot will speed up and move five times as fast (1/0⋅2 = 5).

100%

100%

90%

90%

10%

10%

0%

0%

Time scale = 1 msec. / division

Time scale = 0·2 msec. / division

Figure 3.6 A CRO screen shows a sinusoidal voltage signal.
Divisions, both major and minor, are marked across and down the screen.
The same signal is shown with two different time scales.
The applied AC voltage causes additional up-and-down movement of the spot, in step
with the voltage. For a sinusoidal voltage, the track will look like the sine wave shown in
Figure 3.6. We can make the image taller or shorter, by adjusting the control 'Y sensitivity'.
For example (Figure 3.7), a sensitivity of 1V/division mean that a signal with
an amplitude of 3⋅5 V and therefore with a peak-to-peak voltage of 7 V will be drawn
over 7 divisions. When we change the setting to 5V/division the same voltage waveform
will be only
7 V/(5 V/division) = 1⋅4 divisions high.


3.7

100%

100%

90%

90%

10%

10%

0%

0%

Y Sensitivity = 1 V / division

Y Sensitivity = 5 V / division

Figure 3.7 The same signal displayed with two different Y sensitivities.
A CRO allows us to make a simple measurement of frequency f . We simply look at
Figure 3.7 and measure how many divisions we have between two adjacent troughs
(or use zero crossings). In our case it is 2⋅5 divisions, and looking at our time/division
settings of 1 msec./division we compute that the time between the two troughs,
called the period, T , is 2⋅5 msec.
The frequency f is calculated as the inverse of the period
f = 1/T
and is equal to 400 Hz.

3.3 Voltage and Frequency Ranges – Examples
It is interesting to appreciate what is the approximate order of magnitude of voltages and
frequencies in some practical situations. In Australia, mains voltage (quoted as rms)
is 230 V and its frequency is 50 Hz; this voltage appears between the two upper pins
on your power point – the lower one is connected to earth. Actually, only one of the two
upper wires is at 230 V; the other is at near zero voltage2. Other countries, as you are
probably aware, have different power standards – and different power-point standards.
Actually, Europe is not far from us, with 230 V and 50 Hz, and appliances usually can be
used without alteration, but a power-point adaptor is still needed. In the USA the voltage
is 110 V / 60 Hz and, not only an adaptor, but also a transformer, is required for most
appliances. An attempt to plug in a USA appliance set to operate at 110 V to a 230V power
point is very likely to result in damage.
2

The latter is called the neutral, and is earthed near most houses in Australia
(multiple-earth neutral) as well as at the power source. There is a standard for
connection of the live and neutral to power points, but it may be fatal to assume that
it is always followed. 230 VAC is deadly, and power leads and their connections
should be regularly checked to be in good condition. A power-point tester is cheap
life insurance for a few dollars; newer installations have in-built safety features.


3.8

It is interesting to note that contact between different insulators through mechanical
friction can lead to extremely high voltages, in the order of tens of kilovolts – so high that
an electric discharge can easily be generated. This effect may cause severe problems
when sparks start to fly in unwanted areas, such as near petrol tanks.
Voltages are also found in the bodies of living organisms. Voltage signals transmitted
through nerves to our muscles make them contract. These voltages are very small,
in the order of millivolts. Recently researchers found a way to artificially generate
these signals and send them to the muscles of disabled people who are unable to walk.
The electronic circuitry allows some of them to (very partially) regain the ability to walk.
Examples of very weak signals include those generated by light from distant stars and
galaxies. Astronomers use very large telescopes to gather as much light as they can.
Electric signals produced from this weak light may be in the range of nV. Astronomers
have a way to accumulate these signals by collecting them for a very long period of time,
and so present-day astronomical images are usually a result of long-time accumulation.
For example the Deep Field Image of the most distant galaxies photographed by
the Hubble telescope required 276 exposures taken on 10 consecutive days.
Now we turn to frequency ranges. Electronics deals with limited frequency ranges,
up to say 200 GHz; beyond that conventional electronics circuits do not operate.
But, as we concentrate on electronics only, we will loosely divide up the electronic
frequency range into the audio range, video range, radio-frequency (RF) range and
microwave range. Audio range refers to the frequencies of sound that we can hear,
and in particular terms it means approximately 20 Hz to 20000 Hz. Electrical signals
in this range are easy to generate, send and receive. The early telephone devised by
Graham Bell used varying signals from this range. The frequency range of video signals
extends higher, roughly between 25 Hz and 5 MHz; the name 'video' arises because
TV pictures use this range. The radio-frequency range is roughly between 10 kHz and
several hundred GHz; this refers to the frequency of the carrier wave only. Actually
only some selected frequencies (or rather narrow regions called bands) are used for radio
transmission; these are allocated to radio stations by relevant authorities.
The carrier frequencies of AM (amplitude modulated) radio stations are separated by only
9 kHz. This means that the full 20kHz range in an audio broadcast cannot be used without
stations overlapping. (However, nearby stations are not allocated similar frequencies.)
FM (frequency modulated) radio allows the broadcasting of more of the audio frequency
range (up to 15 kHz) as well as signals in stereo.
AM radio stations broadcast at frequencies between 550 and 1650 kHz, FM radio between
88 and 108 MHz, and TV in three regions between 54 and 890 MHz. Other regions are
assigned to citizens'-band receivers, ships, aeroplanes, police, amateur-radio operators,
and satellite communication.


3.9

Microwaves are produced electronically, in small devices, ranging in size from a few
millimetres to a few metres. These devices are used in microwave ovens, radar, and the
long-distance transmission of telephone calls in some countries (USA). The microwave
range above 16 GHz is becoming increasingly important. Technologically, this range is
the most difficult, but is also very useful. Many important systems such as satellite
communication and radar (radio direction and ranging) are using 'carrier' signals in these
ranges – radar uses several GHz, and satellites use 12-14 GHz.
In addition to these earlier applications, electronic circuits that work at microwave
frequencies are becoming increasingly common. High speed is particularly important for
computer networks, which require data transfer at ever-increasing rates. The design of
such electronic chips needs to be much more refined, and also silicon may be replaced by
other chip materials such as gallium arsenide. Devices must also be made much smaller –
if a device is to operate at a frequency of say, 100 GHz (so the period is 10-11 sec.),
and the maximum speed of electrons is 105 m/sec., then the device must be much smaller
than 105 × 10-11 = 10-6 m = 1 µm. A human hair is 30 µm wide and, at present,
the smallest features in electronic devices have dimensions of about 0⋅1 µm.

3.4 Capacitors and Inductors
In AC circuits we often find two important classes of electronic components:
capacitors and inductors. A capacitor is basically two conducting plates separated by
an insulator, which may be a vacuum, air or a specialised material, as shown schematically
in Figure 3.8.

d

Area A

C

+Q

-Q

Figure 3.8 Symbol and schematic diagram for a capacitor.
A capacitor operates in the following way. Suppose that a DC voltage is applied to the
plates as shown. Since the two plates are separated by an insulator, essentially no electrons
can cross the gap, and a (steady, DC) current cannot pass; we shall see later that
time-varying currents can appear to cross the gap, despite electrons not passing between
the capacitor plates.


3.10

When the battery is connected, it repels electrons from conductors attached to its negative
terminal, and correspondingly attracts them from the conductors connected to the positive
terminal; these electrons spread out onto, or come from over, the area of each plate.
Note that, for a short time, electrons flow into the right-hand plate, and electrons flow out
of the left-hand plate. Thus current apparently flows through the capacitor, although
no electrons actually cross the insulating gap.
If we leave the battery connected for long enough, the voltage between the two plates will
be equal to the voltage on the battery terminals. Now, when the battery is removed,
the charge on the plates will remain in place, and the voltage across the capacitor will also
remain, at the previous value of the battery voltage. The capacitor is said to store charge,
much in the same way as a water tank stores water.
The charge, Q , on a capacitor is proportional to the voltage V between its plates, which is
expressed as:
Q=C×V
where the proportionality constant C is called the capacitance and is characteristic of
a given capacitor. The unit of capacitance is called the farad. One farad (F) is the
capacitance of a capacitor which carries a charge of 1 coulomb when the voltage is 1 volt.
1 farad is actually a very large capacitance; in electronics we often use microfarads (µF),
nanofarads (nF) and picofarads (pF), which are much smaller.
There is no general rule as to how a capacitor should look. Some look like metal cans,
some look like a rolled-up film, some look like a ceramic tablet. The capacitance value
is usually marked, along with the maximum voltage rating and perhaps tolerance.
Some capacitors are marked with colour bands – these allow us to read the capacitance
value. It should be noted that some capacitor designs have particular restrictions as to
which way round they should be connected. The + sign on one of the terminals indicates
that this terminal should be connected to the more positive voltage.
Earlier we discussed water models of resistors and batteries, as well as water models of
simple circuits. Capacitors also have their water models, although somewhat more
involved. The water analogue of a capacitor is a water-filled cylinder with a movable
piston (Figure 3.9). The piston is attached to one side of a cylinder with a spring.
In operation, no water passes around the piston. As water (charge) moves into the device
through one pipe, an equal amount must leave through the other. As water flows,
it displaces the piston and compresses or stretches the spring. Externally applied changes
of water pressure on one side are instantaneously passed through to the other side.

Figure 3.9 A water-filled cylinder with a movable piston
works in analogy to a capacitor.


3.11

It needs to be noted that, unlike resistors where the electrical power is converted to heat,
capacitors do not (ideally) waste or dissipate power; they store energy when charging and
return it when discharging.
Once we know that some currents (i.e. AC or varying currents) can flow through
a capacitor, we can ask if there is a measure of how hard it is for a given current to flow
through a capacitor. In other words is there a property of capacitors similar to resistance
in some way? The answer is yes but, as we are working with AC signals, this new
property called the capacitive reactance needs to be carefully defined. Capacitive
reactance, XC , is given by the ratio of peak voltage amplitude to peak current amplitude
XC = Vp / Ip .
So the definition looks a bit similar to the definition of resistance through Ohm's law
but not quite the same. The capacitive reactance is related to the capacitance of the
capacitor through
XC = 1 /ω C .
The ratio stays constant if the frequency is not changed. The reactance XC at any frequency

f can be calculated from the value of capacitance, C (using ω = 2π f ). In a way, reactance is
analogous to resistance in the sense that, for a given voltage, if the capacitive reactance is
doubled, then the current will drop by a factor of two. However, there is
a very important difference, namely that the voltage and the current in a circuit
with a capacitive reactance are not in phase. When an alternating voltage is applied to
a capacitor and alternating current flows, the voltage wave is not in step with the current
wave, but is one quarter of a wave later. We say that in a capacitor current leads voltage.
The phase shift between current and voltage is 90 degrees.
Because of phase shifts, capacitors and resistors in circuits are treated differently.
For example, rules to calculate the equivalent series and parallel connections of capacitors
are different from those for resistors.
3.4.1 Charging and Discharging a Capacitor
If a capacitor is connected in series with a resistor as in Figure 3.10 then the current
flowing in the circuit will keep on charging the capacitor until the capacitor approaches
the battery potential. The graph of the voltage across the capacitor is shown in Figure 3.10.
e−1
2
The time τ taken for the capacitor's voltage to reach about
(more precisely,
) of
3
e
the battery voltage is known as the time constant of the circuit. It is calculated as
τ =R×C .


3.12

Voltage (V)

V max

R

2/3 Vmax

C

Time
Time constant

Figure 3.10 When we start charging a capacitor, its voltage will only gradually
reach the target value, over a time close to the circuit time constant RC .
If we now take the same capacitor, charge it to the full voltage of our battery, and then
connect it as in Figure 3.11, charge will start to flow around the circuit and the voltage
across the capacitor will be progressively reduced. The time scale for the process is
once again given by τ (Figure 3.11). This process is called a capacitor discharge –
after some time there will be almost no charge on the capacitor plates and no voltage3.

Voltage (V)

Vmax

R

2/3 Vmax

C

Time
Time constant

Figure 3.11 When a capacitor is discharged, its voltage declines within the time
scale of the circuit time constant RC .
3.4.2 Inductors

Electric current passing through a wire creates a magnetic field around it. If many loops
of wire are wound together, the effect of the mutual magnetic field becomes significant.
Such a coil of wire is called an inductor. The commonly used symbol for an inductor is L
(Figure 3.12).

L

Figure 3.12 Symbol for an inductor .
3

After fully discharging the capacitor it will be safe to touch.


3.13

For constant currents the inductor simply behaves as a piece of wire, so that it will act as
a small resistance. But for varying currents new things start to happen. When the current i
varies, the two terminals of an inductor start to show a voltage difference, which is
proportional to the rate of current change4:
v = L × rate of current change = L × di(t) / dt .
The proportionality constant L is called the inductance of a given inductor. The unit of
inductance is called the henry (H). As with capacitors, the basic unit is rather large
and typical coils range from mH to nH. Inductors with very many loops of wire
have larger inductance, but for just a few loops the inductance will be small.
Current flow in an inductor creates a magnetic field inside the loops – and partly also
on the outside – and this magnetic field stores energy. The stored energy prevents
the current from changing too quickly. Conversely, when the current tries to stop,
energy stored in the magnetic field will try to maintain it, so it cannot decline too fast.
Thus, it is justified to model the behaviour of an inductor in a circuit as a turbine driving a
flywheel (Figure 3.13). The turbine is assumed to be a perfect, frictionless, bidirectional
device with no leaks. A height difference between the inlet and the outlet of the turbine
(or a pressure difference)5 is needed to cause the current to flow. When the current
is reduced, for example if the water levels in front of the turbine and behind the turbine
are suddenly made equal, the momentum of the flywheel will try to keep the water
current flowing. On the other hand, the flywheel will resist when the current is increased.

Flywheel

Turbine

Figure 3.13 A turbine driving a flywheel behaves like an inductor
in an electrical circuit.

4

di(t)/dt is the first derivative of current with respect to time.

5

In our analogy the corresponding quantity to water height difference (pressure) is
voltage.


3.14

As mentioned earlier, for constant currents an inductor behaves just like a resistor,
but only because the wire has some resistance. The fact that the wire is wound into a coil
makes it behave as more than just resistance for alternating currents. If the resistance of
the coil is very small, we find that the alternating voltage across the coil leads the current
by a quarter of a period. The phase shift between the voltage and current is 90 degrees
(in the opposite direction to that in a capacitor).
In addition to the ordinary resistance of their wire, inductors have another 'resistance-like'
property, effective only for AC currents, called the inductive reactance. The inductive
reactance XL is the ratio of the peak voltage amplitude to the peak current amplitude
Vp / Ip . The reactance remains constant for a fixed frequency, but its value is
frequency-dependent:
XL = ω × L .
The inductive reactance is related to the inductance of the coil, L .

3.5 Relative Phase of v and i in Circuits with L and C
In this section we will be discussing a circuit in which a capacitor C and an inductor L
are connected in series with a resistor R and an AC voltage source. Both capacitors and
inductors affect the phase shift between current and voltage waves for AC. Their effects,
however, are in opposite directions. For a capacitor the current waveform comes before
the voltage waveform. In an inductor the relationship is the opposite – the current
waveform lags the voltage waveform. In both cases the absolute value of the phase shift is
90 degrees.
A simple mnemonic helps to remember this – CiviL : For a capacitor C , current i leads
voltage v , while voltage v leads current i for an inductor L .
Suppose that a series circuit contains both capacitance and inductance, along with
the inevitable resistance (Figure 3.14). We will now discuss the response of this circuit
to alternating voltages. Of course, steady (DC) current cannot flow in this circuit, because
the capacitor forms an effective break in the circuit. However, if we pass an alternating
current through this circuit, the voltage across the capacitor will be 180 degrees out of
phase with the voltage across the inductor. This occurs because each of the two voltage
differences is 90 degrees out of phase with respect to the current, but one leads and
the other lags behind. The phase difference of 180 degrees means that the two voltage
differences may cancel each other out, as the two voltage differences depend on frequency,
because the reactances of a capacitor (1/ω C) and of an inductor (ω L) are frequency
dependent (in different ways).
At some frequency the two voltage differences will cancel out exactly, and the only
potential difference left is that across the resistor. In such conditions the current in
the circuit is at its maximum; this condition is called resonance. The resonance occurs
when the angular frequency satisfies ω 2 = 1/LC (because 1/ω C = ω L ).


3.15

L

C
R

Figure 3.14 A series resonant circuit. In this circuit the effects of capacitance
and inductance cancel each other at a particular frequency.
3.5.1 Tuned Circuits

The frequency selectivity of resonant circuits allows a radio to be tuned to one of a set of
transmitting stations. Tuning is usually undertaken by varying the capacitance of
an adjustable capacitor (or the inductance of an inductor). Resonant circuits are also
important for timing and for transmitting signals.


3.16

3.6 Filters
Filtering is an important action that is very commonly used in linear circuits and which
often has to be represented on block diagrams. Filtering means selecting one range of
frequencies from others, if a mixture of signals at different frequencies is present.
For example we may want to filter a mixed signal so that it contains only frequencies
detectable by the ear, through a loudspeaker or a microphone. We might like to filter
a mixed signal so that it contains only the higher frequencies, free from power-line
frequencies. We might also like to filter a mixed signal to remove the higher frequencies.
A filter is an electronic circuit that acts selectively on a range of frequencies, either to pass
or to reject that frequency range.
When a filter passes only the lower range of frequencies that is applied to its input,
we call it a low-pass filter. A typical graph of output amplitude plotted against
input frequency for such a filter is illustrated in Figure 3.15. One can also use a high-pass
filter with a gain-frequency response as in Figure 3.15 (middle), which rejects the lower
frequencies of a mixture and passes only the higher frequencies. Another basic type of
filter is the band-pass filter (Figure 3.15, right), which rejects both the highest and
the lowest frequencies, and passes only a range between these extremes. A radio tuner
carries out this type of band-pass action, so that we can receive only one station at a time.

High-pass

Band-pass

Output / Input

Low-pass

Frequency
Figure 3.15 Graphs of the ratio of output to input versus frequency for three
basic types of circuit.
One can also use a band-stop filter to reject a range of frequencies.
Simple filters can be built using an arrangement of capacitors, inductors and resistors.
Two examples of such designs are shown for illustrative purposes only in Figure 3.16.


3.17

Input

Output

vi

vo

Low-pass
Input

Output

vi

vo

High-pass
Figure 3.16 A simple low-pass filter (top) and a high-pass filter (bottom).

3.7 Decibels
Sometimes we have to talk about quantities that differ very much, by many orders of
magnitude. One can, of course express these quantities in a traditional way, but at some
point it really becomes cumbersome. Imagine that we are comparing people's earnings.
A poor part-time bartender is barely making $10,000 a year. His boss at the same time
makes $100,000 p.a. We can still count his zeros, so this number is not too impossible
to read. Let us turn now to the bartender's friends. His neighbour just won $1,000,000
in Lotto. At the same time the person he went to school with just closed a deal
worth $300,000,000 and personally made $10,000,000. Now the zeros are really getting
out of hand. Surely there is a better way expressing these long numbers?
The idea is to relate these numbers, N , to some reference number of dollars.
It is convenient to take the reference number Nref to be one dollar. Then we calculate
10 × log10(N/Nref ). This number is:
50 for the bartender,
60 for his boss, 70 for his neighbour,
80 for his school friend (who made $10,000,000).

The numbers still reflect who is richer and who is poorer, but they are much more
convenient. We use decibels for this new way of expressing various quantities.
A similar idea is used in electronics. Here the numbers can differ even more wildly.
We therefore define a similar logarithmic measure to express power, voltage, current
and gain/loss. Sound level can also be measured in decibels. In each of these cases
the definition is very similar to that above, but the reference value is different.


3.18

3.7.1 Decibel calculations for Power

To express a power P in decibels, we choose a reference power Pref and write
Power in decibels, dB = 10 log10[P/Pref ].
Example 3.1 Express the following quantities in dB:
a)
Express 50 milliwatts referred to 1 watt (dBW).
b) Express 50 milliwatts referred to 1 milliwatt (dBm).
c)
Given that log102 = 0·3010 express in dB, to one significant figure, a power
which is twice Pref .
d) Similarly for a power that is one-half Pref .
Answers
a) -13 dBW, b) 17 dBmW, c) 3 dB, d) -3 dB.

You should try to remember 3 dB and particularly -3 dB. This value is often used
in electronics. Later you may be asked, for example, 'what is the -3dB frequency for the
filter?'. You will then look for the frequency where the ratio of the filter output power
to the input power is 1/2.
3.7.2 Sound Level in Decibels

Audio engineers use a decibel measure for the amplitude of sound pressure.
The definition of sound pressure level (SPL) in decibels for a sound whose pressure is P is
Sound pressure level in dB
SPL = 20 log10[ P/Pref ] ,
where Pref is the pressure level corresponding to the average threshold of hearing
for humans, which is 0·0002 µbars or 0·0002 millionths of an atmosphere. For example
one microbar of sound pressure is about 74 dB SPL. This sound pressure corresponds
approximately to the sound produced by a loud conversation.
3.7.3 Gain and Loss in Decibels

Often the gain produced by an amplifier is specified by giving the ratio of the amplifier
output voltage (or current) to the input voltage (or current). When the gain is expressed
in decibels the input voltage (or current) is taken as the reference value of voltage
(or current) in the defining expression:
Voltage gain in dB = 20 log10[ Voutput/Vinput ]
Current gain in dB = 20 log10[ Ioutput/Iinput ] .
For example, if an amplifier gives one volt of output for a 2mV input (a ratio of 500) it has
a voltage gain of approximately 54 dB. Similar expressions are used for those circuits
whose outputs are smaller than their inputs, but the numbers turn out to be negative.
If we are measuring powers then we use the following expression:
Power gain in dB = 10 log10[ Poutput/Pinput ] .
Note that for voltages/currents we use the multiplier of 20 and for powers the multiplier
of 10. This is because power is proportional to the square of voltage/current.


3.19

3.8 Shorts and Fuses
If a single bulb is connected to a battery and then an additional wire is run directly
from one end to the other, the bulb goes out, the wire gets very hot and the battery
runs out very quickly, which tells us that most of the current flows through the wire.
Actually, a very small current is still passing through the bulb, but insufficient to make it
light up. The wire forms an alternative path with very little resistance, which is known as a
short circuit, or colloquially a 'short'. If a short develops in an instrument or appliance the
fuse is normally activated. Otherwise excessive current may flow, causing overheating and
possibly a fire. Fuses protect against this happening.
Current from the supply to the equipment flows through the fuse. The fuse is a piece of
wire which can carry a stated current. If the current rises above this value it will melt.
If the fuse melts ('blows') then there is an open circuit and no current can flow,
thus protecting the equipment by isolating it from the power supply. The fuse must be
able to carry slightly more than the normal operating current of the equipment to allow for
tolerances and small current surges. With some equipment there is a very large surge of
current for a short time at switch-on. If a fuse were fitted to withstand this large current
there would be no protection against faults which cause the current to rise slightly above
the normal value. Therefore special 'slow-blow' fuses may be fitted. These can withstand
many times the rated current for a very short time. If the surge lasts longer than that
the fuse will blow.
Always find out why a fuse blew before replacing it. Occasionally they grow tired and fail
just because of aging. If, however, the fuse case is black and silvery (fuse vaporised)
then it is likely that there is a dead short (very low resistance) somewhere; this may be
dangerous and the cause needs to be located. Instruments may have additional fuses
inside, and they might need replacing. If the instrument's fuses are insufficient protection,
then the household fuses and/or circuit breakers may be activated. Alternatively,
fuses in the power substation will protect the entire area.
The circuits in a home are wired in parallel so that electrical devices can be turned on
and off without affecting each other. As each new appliance is turned on, the electricity
provider supplies more and more current at the same nominal voltage. Each parallel
circuit, however, is wired in series with a fuse, so that a 'weak link' is deliberately put
in the circuit. If too many devices are plugged into the circuit, they will draw more current
than the wires can safely carry. For example, the wires might heat up at a weak spot
and start a fire. The fuse contains a short piece of wire that melts when the current exceeds
the rated value. This interrupts the current, shutting everything off. Nowadays, rather
than fuses, we tend to use circuit breakers, which serve the same purpose as fuses,
but only require resetting rather than replacement, and may also work faster.


3.20

3.9 Buying Electricity
When we buy electricity from a power company we are buying a certain number of units
called kilowatt-hours. They are calculated by taking the product of electric power
with time. They are measured by domestic electric meters, connected between the house
and the power company. The meters record how much energy has been used, very much
the same as the odometer in a car records the kilometres driven6.
The way to calculate how many kilowatt-hours were used is as follows. Let us calculate
how much will be paid for 5 hours use of an electric heater which is rated at 1000 W.
1000 W = 1 kW, 1 kW is multiplied by 5 (5 hours) and 5 kWh is obtained (50 cents
at 10 cents per kWh)7.

3.10 Power Generation
The overall scheme for power generation, transmission and distribution is shown in Figure
3.17. Electric power is generated by machines called generators that work
following the principle of the dynamo. They have a central shaft on which coils of wire
are mounted. These coils are supplied with direct current that produces a magnetic field
in the space around them. Surrounding the movable coils are stationary coils.
As the rotating coils move, the magnetic field rotates with them, crossing the wires of
the stationary coils. This produces a voltage and, consequently, a current that flows
through any circuit connected to the stationary coils.
The inner coils must be kept in motion for this process to work. This can be accomplished
in a variety of ways:
a)
Burning fossil fuels such as oil, coal, or natural gas. Generated heat allows water
in a boiler to turn into high-pressure steam which can make a turbine turn.
b) A nuclear reactor generates heat which can be used to produce steam as well.
c)
Water stored in a dam can be made to flow through turbines, making them turn.
Turbines can then be directly connected to a generator.
Each of the coils in a generator produces a voltage which is a sinusoidal function of
the angle of rotation of the generator shaft. For highest efficiency, usually
three independent stationary coils are used on such generators. As a result the voltages of
the three stationary coils have phases which are 120 degrees apart. The typical generator
output is 11-33 kilovolts.

6

On the other hand, the power that you require to run a particular appliance is very
much like the speed of your car, in that:
Power × Time = Energy
( kW × hr = kWhr)
Speed × Time = Distance
(km/hr × hr = km)

7

Off-peak electricity is considerably cheaper than daytime power.


3.21

11 kV

Power station

330 kV
Grid System
(long distance)

Transformer
Pylon
11 or 33 kV
(medium distance)
Substation

Industrial 415 V
Transformer
Domestic 230 V

Pole

Transformer

A (230 V)
Front of socket

N (0 V)
Earth

Figure 3.17 Power distribution system.
With increased environmental awareness, alternative methods of generating electricity are
being developed. These include:
a)
production by generators driven by wind.
b) production by generators run by geothermal sources.
c)
production by sunlight absorbed in solar cells made from semiconductors.
d) production using thermal energy generated from other industrial processes,
utilised further to power a turbine-generator system.
Power is sent from the power plant to the point of use through a special transmission
system. The first step involves a step-up transformer, where the high voltage (11-33 kV)
from the generator is made even higher, to anywhere between 110 kV and 500 kV.
The current is proportionally reduced. It can easily be shown that the amount of power
lost to heat is considerably reduced if the current is made smaller.
High efficiency in transporting power is obviously very important, and it was found that
it helps if power generated by three stationary coils in a generator is sent over three
conductors. This is why the transmission-line towers carry three of everything –
conductors, insulators and so on, and sometimes we see six of everything. The seventh
cable that we can sometimes see is connected to the towers without the use of insulators.
This is a cable that protects the rest of the lines from damage by lightning strikes.
The cables are typically about 2 cm in diameter and made up of 50 separate aluminium
strands, wound on a steel core for strength. They have no external insulation,
but are insulated from the towers by glass or ceramic insulators.


3.22

When the power reaches the city, the 'three-phase' voltage is stepped down
in transformers at a substation to anywhere from 4 kV to 21 kV, and carried on conductors
supported by insulators on poles along streets. Again there are usually three of
everything, one for each phase of voltage. The distribution cables are usually about 1 cm
in diameter and carry up to 800 amperes in normal operation. Near the point of use
a transformer (often mounted on a pole, or enclosed in a cabinet, for safety)
reduces the voltage to 230 V, with a corresponding stepping up of the current.
Homes are predominantly connected to a single phase. Only some (all-electric) homes
and most industrial sites are connected to the full three-phase power, which is necessary
to run high-power appliances, for example some electrical motors, ovens etc.
House electricity has some safety features. Incoming current is delivered through
one wire, called 'active', and, in the house wiring, this wire is of a certain colour dictated
by regulations. In Australia this colour is brown or red. The 'outgoing' current goes away
through another wire which in Australia is blue or black. This wire should not be at a very
high voltage, but touching it is not recommended anyway, because the circuit might be
incorrectly wired. Finally, there is always a third wire called 'earth' connected to the
ground, which is green or yellow and green. This wire is supposed to be at a zero
potential, and metal cases of your domestic appliances like the oven and the washing
machine are connected to it.
Most appliances are grounded, which means that their plug must have three pins.
The ground is usually connected to the case, which means that touching the case is
absolutely safe – unless the earth wire has deteriorated with time or snapped off
somewhere. So, regular inspections of appliances are essential for safety.

3.11 Dangers of Electricity
Electricity is dangerous because our bodies are electric machines. Our muscles contract
when an electric current is passing through them. This reaction is normally triggered by
a complex chain of chemical reactions but can also occur when an electric current is made
to flow through a muscle. This process is the origin of twitches. Uncontrolled muscle
spasms can become very dangerous when the muscle is part of the heart or breathing
system. Currents as small as 50 milliamperes can interrupt breathing (see Table 8.2
in the Appendix).
Often people feel that the danger is the voltage. Although high voltages are dangerous,
it is the current flowing through a body that is lethal. Ohm's law tells us that the current is
equal to the voltage divided by the resistance. If the conditions are such that the total
resistance is low, even a low voltage – a few tens of volts – can cause a dangerously high
current. The resistance between the skin of human hands and the body depends on
how moist they are. If our hands are dry we may have about 300 kΩ of resistance.


4.1

Chapter 4
Signal Concepts
4.1 Linear Systems
The large majority of people cannot explain what each component in a TV set does,
but most people can tune the channels to receive the TV stations and plug the set
into a power socket so they can watch it. The TV receiver is an example of a system
and it shows that we do not always need to know how specific systems work;
we may simply need to know what input we have to supply to the system and what
the system will generate at the output. (The TV set has two inputs and two outputs;
what are they?) Systems are often regarded as black boxes; this is because we do not
always need to understand the internal workings of a system, but only need to know
about what happens outside the black box, for example to its inputs and outputs.
A linear system is a black box in which an increase in the input signal brings about
a proportional increase in the output signal. A typical example is an amplifier, in which
the input signal might be a voltage and the output signal might also be a voltage.
There are other systems which we consider as linear, such as filters, equalisers and plenty
of others. Linearity is related to the ratio of the signal output to the signal input.
If this ratio stays constant, regardless of the value of the input signal, then the system is
linear. If it does not, then it is not linear. This is illustrated in Figure 4.1. Note that
the figure shows two different forms of nonlinearity.
A useful quantity associated with linear systems is called the gain. In an amplifier,
the voltage gain is the ratio of the output voltage to the input voltage. The increase
in the signal cannot happen without delivery of electric power from somewhere,
for example from the mains. The ratio of the output power to the input power is called
the power gain. We can also talk about current gain. Out of all three it is the power gain
which often has the most practical utility. Sometimes a circuit has a gain which is less than
unity; in this case we are talking about attenuation.
The amplifier voltage gain is calculated as
G (dB) = 20 log10(Vo /Vi ) .
G(dB) is said to be the voltage gain in decibels. Vo and Vi are the output and input
voltages, respectively. The current gain is calculated using the same formula.
The expression for power gain, as we recall, is
G (dB) = 10 log10(Po /Pi )
where Pi and Po are the input and output power, respectively.
The calculation of gain in dB requires the use of logarithms. Finally, we note that the type
of circuit can be deduced from its gain and its behaviour as a function of frequency.


4.2
14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14

1.0
0.8
0.6
0.2

Time

0.0

-0.2

Output

Input

0.4

-0.4
-0.6
-0.8
-1.0

Time

a)

b)
6

14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14

5

Time

Output

Output

4
3
2
1
0
-1
Time

c)

d)

12

10

10

8

8

6

6

4

4

2

Output

14

12

Output

14

Input

0
-1.0

-0.8

-0.6

-0.4

-0.2 -2 0.0

0.2

0.4

0.6

0.8

1.0

2

Input

0
-1.0

-0.8

-0.6

-0.4

-0.2 -2 0.0

-4

0.4

0.6

0.8

1.0

-4

-6

0.2

-6

-8

-8

-10

-10

e)

f)

Figure 4.1 Linear and nonlinear response. a) Input signal; b) Output signal for a linear
circuit; c) Output signal for a nonlinear circuit – the sine wave is distorted;
d) This output is also a nonlinear response to the input signal; e) The output vs input
dependence for a linear circuit; f) The output vs input dependence for the nonlinear
circuit with output as in c).


4.3

4.2 Frequency Domain and Time Domain
The characteristics of a sinusoidal voltage waveform can be defined by using
the frequency, phase and amplitude of the waveform. If we forget the phase
for the moment then we can plot the characteristics of a given waveform in a graph
by using the amplitude (voltage) and frequency as the axes on our graph. For example
let us examine Figure 4.3a. The single vertical line represents a pure sinusoidal wave
of frequency fo and amplitude Vo given by
v(t) = Vo sin ω t .
(ω = 2πfo )
The traditional picture of the same wave expresses the voltage as a function of time.
The two parts of this figure contain the same information, i.e. the amplitude Vo
and the frequency fo of the wave.
Certainly we are more familiar with the traditional representation, but in fact it does not
tell us anything more than the new representation (apart from the phase, which is often
ignored). The traditional picture is the representation of a wave in the time domain,
i.e. using time as an axis. The new representation is in the frequency domain, i.e. using
frequency as an axis. Sometimes we shall find it more useful to have representations
in the frequency domain than in the time domain.
It is instructive to examine the frequency-domain representations of some simple waves.
The first wave we will explore is a sinusoidal waveform with frequency fo , to which
we add another waveform of frequency 3fo . At every point in time the voltages in each
waveform can be simply added together. The resulting combined wave is shown in
Figure 4.2a. We can see that the rising part of the wave has been made steeper and that
the peak of the wave has been made flatter. If now a further frequency 5fo is added,
this continues to flatten the peak and make the rise and fall sharper (Figure 4.2b).
This sequence of diagrams is necessarily long and elaborate when presented in the time
domain, but the addition of extra waveforms at different frequencies can be represented
very easily in the frequency domain by simply adding a new vertical bar at each
appropriate frequency (Figure 4.2c). The basic waveform of frequency fo is called
the fundamental and the waveforms with frequencies equal to multiples of fo are called
harmonics1.

1

All these separate waveforms that make up the total signal are jointly called the
Fourier components, or the components of a Fourier series.


4.4

1.4
1.2
1.0
0.8

a)

Amplitude (V)

0.6
0.4
0.2

1kHz Sine
3kHz Sine
Sum 1,3

0.0
-0.2 0.0

0.5

1.0

1.5

2.0

-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
Time (msec.)

1.4
1.2
1.0
0.8

b)

Amplitude (V)

0.6
0.4
0.2

5kHz Sine
Sum 1,3,5

0.0
-0.2 0.0

0.5

1.0

1.5

2.0

-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
Time (msec.)

1.4
Vo = 4 / π

1.2

c)

Amplitude (V)

1.0

Fundamental

0.8
Harmonics

0.6

Vo / 3

0.4

Vo / 5

0.2
0.0
0

1

2

3

4

5

6

Frequency (kHz)

Figure 4.2 a,b) Signals at various frequencies are added together. c) Spectra.


4.5

a)
5

Single frequency

4

1.2

3
1.0

1

0.8

Amplitude (V)

Amplitude (V)

2

0
-1
-2
-3

0.4
0.2

-4

0.0

-5
-0.5

0.6

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0

0.5

1

2

3

4

5

Time (msec.)

6

7

8

9

10 11 12 13 14 15

Frequency (kHz)

b)
5

Two frequencies

4

1.2

3
1.0

1

Amplitude (V)

Amplitude (V)

2

0
-1
-2
-3

0.6
0.4
0.2

-4

0.0

-5
-0.5

0.8

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0

0.5

1

2

3

4

5

Time (msec.)

6

7

8

9

10 11 12 13 14 15

Frequency (kHz)

c)
5

Three frequencies

4

1.2

3
1.0

1

0.8

Amplitude (V)

Amplitude (V)

2

0
-1
-2
-3

0.4
0.2

-4

0.0

-5
-0.5

0.6

-0.3

-0.1

0.1

0.3

0.5

0

1

2

Time (msec.)

3

4

5

6

7

8

9

10 11 12 13 14 15

Frequency (kHz)

Figure 4.3(a-c) Frequency spectra of various signals. Time-varying
signals (time domain) are shown on the left and their frequency
spectra (frequency domain) are on the right.


4.6

d)
5

Five frequencies

4

1.2

3
1.0

1

0.8

Amplitude (V)

Amplitude (V)

2

0
-1
-2
-3

0.4
0.2

-4

0.0

-5
-0.5

0.6

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0

0.5

1

2

3

4

5

Time (msec.)

6

7

8

9

10 11 12 13 14 15

Frequency (kHz)

e)
5

Many frequencies

4

0.6
Amplitude density (V / kHz)

3
Amplitude (V)

2
1
0
-1
-2
-3
-4

0.4
0.3
0.2
0.1
0.0

-5
-0.5

0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0

1

2

Time (msec.)

3

4

5

6

7

8

9

10 11 12 13 14 15

Frequency (kHz)

Figure 4.3(d-e) Frequency spectra of various signals. Time-varying
signals (time domain) are shown on the left and their frequency
spectra (frequency domain) are on the right.
We summarise now the main ideas. If a signal has just one single frequency ω1 ,

then its frequency spectrum shows only one single peak at ω1 , as in Figure 4.3a.
If a signal is made up of two different frequencies, then its frequency spectrum has
two peaks as in Figure 4.3b. If a signal is made up of three frequencies, then we see
three peaks in the frequency spectrum as in Figure 4.3c. For five frequencies (Figure 4.3d)
the signal picture starts to look messy and we cannot really clearly distinguish by visual
inspection which frequencies are present. But there are electronic instruments called
spectrum analysers which can identify these frequencies, and they show five peaks
in the frequency spectrum. What if the signal contains very many frequencies, which are
all quite close together and all between say ωa and ωb ? Then the individual peaks
showing the frequencies are very close and cannot be seen individually. Instead,
the waveform and frequency spectrum may look something like that in Figure 4.3e.


4.7

We now continue our addition of various waveforms. If we continue to add higher and
higher odd harmonics with appropriate progressively decreasing amplitudes, as in Figure
4.2, then we will end up with an endless sum
v(t) = (4 Vo / π) × [sin(ω t) + sin(3ω t)/3 + sin(5ω t)/5 + sin(7ω t)/7 + ...]
where
ω = 2π fo .
Here, the term ω is called the fundamental frequency, and 3ω , 5ω , 7ω , ... are called
the third, fifth etc. harmonics.
1.2

First few of many frequencies

1.0

1.4

0.8

0.4

4 Vo / π

1.2
1.0

0.2

Time (msec.)

0.0
-0.2 0.0

0.2

0.4

0.6

-0.4
-0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Amplitude ( / Vo )

Amplitude (V)

0.6

0.8
0.6
4 V o / 3π

0.4

4 V o / 5π

0.2

-0.8
0.0

-1.0

0

-1.2

2

4

6

8

10

12

14

16

Frequency (kHz)

a)

b)

Figure 4.4 A square wave and its spectrum.
In the frequency domain this expression is represented by Figure 4.4. In the time domain
the result of adding all these waveforms together is a wave with increasingly steep
rise and fall, together with a flat crest and trough, i.e. a square wave (see Figure 4.4a).
The graph in Figure 4.4b is said to be the frequency spectrum of the corresponding graph
in Figure 4.4a. We emphasise once again that this frequency spectrum is just
a different way of presenting the same information (except phase). Neither is more correct
than the other; we are, however, more familiar with the presentation in the time domain.
We will see soon how useful the frequency-domain representation can be
in understanding the behaviour of signals. Here we have obtained the frequency spectrum
of a relatively simple square wave. In fact it is possible to obtain the frequency spectrum
of any signal in the time domain, regardless of its complexity – or simplicity.
Mathematics provides explicit procedures, which involve some integrals, to evaluate
the frequency spectra. In addition, many computer packages are capable of computing
a frequency spectrum with a single press of a button.
Figure 4.3 showed several signals and their frequency spectra.


4.8

5.0
4.5
4.0
Amplitude (V)

3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5

0.0

0.5

1.0

1.5

Time (msec.)

Figure 4.5 A waveform.
Exercise 4.1 We have seen that a square wave can be considered a sum of many sinusoidal
components. What additional component must be added to obtain the waveform
in Figure 4.5?
Answer
The only fundamental difference between the square wave in Figure 4.4
and the signal in Figure 4.5 is that in the latter a constant voltage of more than Vo
has been added to every point in the wave. Thus an extra component must be added
at zero frequency, f = 0 , i.e. a DC component (of 2⋅5 V). Although we have previously
implied a fundamental difference between AC and DC signals, when we come to
frequency spectra a DC signal is merely an additional component at zero frequency.
The fundamental frequency corresponds to the shortest time interval over which
the signal repeats itself.


4.9

3.5
VR

3.0

Amplitude (V)

2.5
VS

VO

2.0
1.5
1.0
0.5
0.0
-60

-40

-20

0

20

40

60

Time (msec.)

Figure 4.6 A more complex waveform.
Exercise 4.2 Consider now the waveform in Figure 4.6. We already know from the
previous exercise that there is a component at zero frequency. In this question
you are asked first to find the frequencies of the two (AC) components with the
lowest frequencies. You should find the first frequency very easily, but you may have
problems with the next frequency; note that the square-wave pulses are
of alternating height. Having found the two low-frequency components, can you
explain how the amplitude of these two components might be related to
the amplitudes VS and VR , and the starting level VO , in the diagram? Do not spend
too long on this part if you find it difficult.
Solution
If we ignore at first the difference in height of different pulses, the period of
the square wave becomes T2 = 20 msec. This gives a frequency f2 = 1/T2 = 50 Hz
(see Figure 4.7).


4.10

3.5
VR

3.0

VS

VO

Amplitude (V)

2.5
2.0
1.5
1.0
0.5
0.0
-60

-40

-20

0

20

40

60

Time (msec.)

Figure 4.7 This is how we work out the basic frequencies
in the complex rectangular waveform.
The next step is more difficult. This is where we try to take into account
the difference in sizes of the pulses. The variation of pulse size is sinusoidal
with a period T1 . The period T1 can be calculated to be 40 msec.; this gives a
frequency f1 = 1/T1 = 25 Hz .
The amplitude of the component at frequency f2 probably depends on the average
amplitude of the pulses, i.e. ( (VR-VO) + (VS-VO) ) /2 . The amplitude of
the component at frequency f1 depends on the difference between VR and VS ,
i.e. (VR – VS). You should see that this is consistent with our analysis of the square
wave, because the f1 component vanishes completely for a square wave,
i.e. for VR = VS .
In terms of the information, the f2 component gives the average amplitude of the
signal, and the f1 component gives the difference between the amplitudes of pulses.
(The two small pulse waveforms, not to scale, show how the signal may be obtained
as the sum of 10msec. pulses at 50 Hz and 10msec. pulses at 25 Hz.)
A complete Fourier analysis would also give other frequency components
and modify the above analysis, but these would just describe the rectangular shape
more accurately.


4.11
Flute

Clarinet

-25
-30
-40

Amplitude density

Amplitude density (dB)

-35
-45
-50
-55
-60
-65
-70
-75
-80
-85
0

1000

2000

3000

4000

0

1000

Frequency (Hz)

2000

3000

4000

Frequency (Hz)

Figure 4.8 Frequency spectrum for two different instruments.
We can use the idea of frequency spectra to help us understand why, if someone plays
the same note (say A4) on two different instruments, we can still tell which instrument is
playing. The frequency components for the two instruments are illustrated in Figure 4.8.
The point is that, although the fundamental frequency fo in both cases is the same –
for A4 it is 440 Hz – each instrument has a different mixture of harmonics, so the shapes of
the two frequency spectra are different.
(The flute curve – see http://newt.phys.unsw.edu.au/music/flute/modernB/A4.html –
is for a modern flute with a B foot using conventional fingering, and is somewhat
smoothed. The clarinet curve is unsourced and unscaled, but note the suppression of
even harmonics, quite unlike the flute curve.)
4.2.1 Bandwidth of a Signal
0 dB

Amplitude density

Amplitude density

-3 dB

Bandwidth

Bandwidth

Frequency

Frequency

a)

b)

Figure 4.9 Bandwidth of a signal.


4.12

The bandwidth of a signal is an important concept. It is easy to make it precise
for some specific signals, namely for those where the frequency spectrum resembles
a rectangle with rounded edges. The bandwidth is defined as the frequency width
between the two -3dB points, that is the points where the signal power falls to half of
its maximum value, as shown in Figure 4.9a. However, for many signals, the spectrum is
not as neat as that in Figure 4.9a. In this case (e.g. Figure 4.9b) the term 'bandwidth'
usually describes somewhat loosely the range of frequencies making up this signal.
1.0

Amplitude (V)

0.8

0.6

0.4

0.2

0.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time (msec.)

Figure 4.10 Another waveform.
Exercise 4.3 Consider the waveform given in Figure 4.10 (the absolute value of a 1kHz
sine wave) and answer the following questions about its frequency spectrum.
i)
Is there a DC (zero frequency) component in the spectrum?
ii) What is the lowest (AC) frequency component in the spectrum?


4.13

a)

Amplitude density

The sound of a whale

0

200

400

600

800

1000

800

1000

800

1000

Frequency (Hz)

b)

Amplitude density

Man-woman duet (A4; 440 Hz)

0

200

400

600
Frequency (Hz)

c)

Amplitude density

Voices of the rainforest

0

200

400

600
Frequency (Hz)

Figure 4.11 Frequency spectra of natural acoustic signals.
The general character of a signal can often be deduced from its frequency spectrum.
The spectrum in Figure 4.11a is that from a whale. We see a spectrum which is essentially
centred near 100 Hz. In the spectrum of Figure 4.11b a note sung in duet by a man
and a woman can be seen. The pair is not singing the same note, rather they are singing
in parts – the man's voice is at a lower frequency and the woman sings at a higher pitch
(A5). Finally the spectrum of Figure 4.11c is that of the voices of the rainforest.
A rich spectrum of frequencies shows a variety of birds all singing various melodies.


4.14

4.2.2 Sources of Electrical Signals
The signals discussed earlier were generated by natural sources, picked up
by a microphone and perhaps amplified to be visualised more easily. But in electronics
we are often concerned with electrical signals which are generated in instruments called
signal generators. These instruments will be treated by us as black boxes. We need to have
some idea of how it is possible to generate electrical signals at all, without getting into
their inner workings.
Electronic signals are principally generated in oscillators and transducers.
A very important action to do with this is called feedback2. Feedback means that
in our black box we take a portion of the output signal and apply it back to the input.
The feedback signal can be connected so that it either adds to the normal input or subtracts
from it. If the feedback signal is in phase and so is added to the input wave, then
it is called positive feedback, and its action is to increase the gain of the amplifier,
reduce its bandwidth and make the amplifier more sensitive to any changes, such as
a small change in the resistance value of the load resistor. If positive feedback is increased
beyond a certain point the result is oscillation – the amplifier will provide an output signal
without any input.
An oscillator is a circuit that is deliberately designed to generate signals from a steady
voltage supply. Any oscillator can be thought of as an amplifier with positive feedback
and some kind of circuit to determine the frequency. If this frequency-determining circuit
is a resonant circuit, the oscillator is a tuned oscillator and it will generate just one
frequency component, or nearly so. Other circuits can be used that will cause the oscillator
to generate a square or pulse waveform. It might be worth noting that oscillators can also
be designed so that their oscillation frequency can be varied when a steady voltage input
is varied, and this type of oscillator is called a voltage-controlled oscillator.
Transducers are another important class of electronic devices. A transducer converts
one form of energy to another. For electronics purposes important transducers are those
with an input or an output which is electrical, particularly if that input or output
takes the form of a wave. An example of a transducer is a microphone, which converts
sound waves into electrical signals. Other examples include various sensors, which
change their output in response to changes in their environment. A temperature sensor
will generate signals stimulated by temperature or temperature differences.
A tape-recorder head generates electrical signals on playback in response to varying
magnetic properties of the tape. Light-sensitive photocells and a fuel gauge in the fuel tank
– all can be regarded as transducers.

2

We will examine feedback in much more detail later on.


4.15

4.2.3 Bandwidth of a System
It is easiest to illustrate the concept of the system bandwidth by discussing
audio systems. Antiquated radio receivers do not reproduce voice very faithfully;
the voice is recognisable, but the music just does not sound right. Later models are better,
and we can enjoy music played by these, but nevertheless they are no match
for present-day hi-fi equipment. A major difference between these three examples
lies in their bandwidth, which got progressively greater as improved technologies
became available.

0 dB

Gain

Gain

-3 dB

∆f
fh

Frequency

Frequency

-3 dB

-3 dB
Gain

0 dB

Gain

0 dB

fl

∆f

fh

fl
Frequency

Frequency

Figure 4.12 Gain curves of various systems.
In a system the same value of the gain may not apply to all incoming
frequency components. This can be expressed by using a graph of gain versus frequency,
called the frequency response. Different systems can have different frequency responses
as shown in Figure 4.12.


4.16

Gain

1970 radio

Gain

Early radio

1

10

100

1000

10000

100000

1

10

Frequency (Hz)

100

1000

10000

100000

10000

100000

Frequency (Hz)
Spectrum of a signal

Gain

Amplitude density

Present radio

1

10

100

1000

10000

100000

1

10

Frequency (Hz)

100

1000

Frequency (Hz)

Figure 4.13 Audio bandwidth of various radio receivers and a signal.
The bandwidth can be easily visualised in a diagram of gain versus frequency.
Three curves, for an early radio, for a later model and for a modern hi-fi, are schematically
shown in Figure 4.13. The bandwidth essentially means the width (in Hz) of the curve
such as in Figure 4.13, but has a commonly used and more precise definition, as shown
in Figure 4.12. We determine the -3dB cutoff points fl and fh on the left and on the right
of the graph (the voltage gain drops to 1 / 2 of its maximum value at these points)
and measure the width in Hz between these points.

Sometimes the gain curve may be not exactly flat. This is generally undesirable in audio
systems, but the exact bandwidth is still measured in the same way.
The choice of a system with an appropriate bandwidth may be quite important.
Suppose that our acoustic signal has a very large bandwidth as in Figure 4.14a.
This particular signal has a very wide and almost flat spectrum between, say, 100 Hz and
10000 Hz. We are trying to use a microphone which converts sound waves into electrical
signals. The frequency response of this microphone is shown in Figure 4.14b
and it is flat, but it extends only between 300 Hz and 20000 Hz. After the signal is
picked up by the microphone the signal will have a spectrum as in Figure 4.14c.
The lowest signal frequency is now 300 Hz; lower ones are not picked up
by the microphone. The highest frequency remains at 10000 Hz. The microphone was
capable of picking up higher frequencies, but none were present.


4.17

0

a)

Voltage density (V/MHz; dB)

-5
-10
-15
Spectrum of input signal

-20
-25
-30
-35
-40
10

100

1000

10000

100000

Frequency (Hz)
40
35

b)

Gain (dB)

30

Gain curve for microphone

25
20
15
10
5
0
10

100

1000

10000

100000

Frequency (Hz)
40

c)

Voltage density (V/MHz; dB)

35
30
25

Spectrum of output signal

20
15
10
5
0
10

100

1000

10000

100000

Frequency (Hz)

Figure 4.14 The spectrum of the signal does not exactly match
the frequency response of the system. As a result, the spectrum of
the output signal will be changed.


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