Signals
Introduction


This unit uses voltage–time graphs to describe the characteristics of different types of signal. The supporting practical, Using an Oscilloscope, introduces the oscilloscope, a key instrument for measuring and displaying voltage–time graphs.

Figure 1.   Using an oscilloscope.
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About signals
In electronic circuits things happen. Voltage–time (Vt) graphs provide a useful method of describing the changes which take place.

Fig.2 shows the voltage–time graph that represents a direct
current
Current I is a flow of charged particles, usually electrons.
current
(
d.c.
In a d.c., or direct current, circuit, current always flows in the same direction.
d.c.
) signal.

Figure 2.   Voltage–time graph of a d.c. signal.
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This is a horizontal line a constant distance above the x-axis. In many circuits, fixed d.c. levels are maintained along power supply rails, or as reference levels with which other signals can be compared.

Compare Fig.2 with Fig.3, which shows the voltage–time graphs for several types of alternating current (
a.c.
In an a.c., or alternating current, circuit, current flows first in one direction, then in the other.
a.c.
) signals.

Figure 3.   Voltage–time graphs of a.c. signals.
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As you can see, the
voltage
Potential difference, or voltage V is a measure of the difference in energy between two points in a circuit. Charges gain energy in the battery and lose energy as they flow round the rest of the circuit.
voltage
levels change with time and alternate between positive values (above the x-axis) and negative values (below the x-axis).

Signals with repeated shapes are called waveforms and include sine waves, square waves, triangle waves, and sawtooth waves. A distinguishing feature of alternating waves is that equal areas are enclosed above and below the x-axis.

You can investigate some of these waveforms in the simulation of Fig.4.


Click on the figure below to interact with the model.

 Figure 4.  Waveforms



Move the probe to a new location to see each waveform.
Complete the following sentence.

  • In a voltage–time graph, voltage is plotted on the , or axis, with time on the , or axis.
  • Click here to mark the question
Sine waves
In electronics, sine waves are among the most useful of all signals in testing circuits and analysing system performance.

Fig.5 shows a sine wave in more detail.

Figure 5.   voltage–time graph of a sine wave.
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Sine waves are described in terms of:
Period – the time taken for each complete cycle.
Frequency – the number of cycles completed per second.
Amplitude – the displacement, or height of the wave.
     Phase – used to identify a particular point in the cycle.



You can find out more about each these by working through the rest of this section.


Period and frequency

The period is the time taken for one complete cycle of a repeating waveform. It is often thought of as the time interval between peaks, but can be measured between any two corresponding points in successive cycles.

The frequencyf is the number of cycles completed per second. The measurement unit for frequency is the hertz, Hz (1 Hz = 1 cycle per second).

If you know the period, the frequency of the waveform can be calculated from:



Conversely, the period is given by:



In rearrangeable format:



Signals you are likely to use vary in frequency from about 0.1 Hz, through values in kilohertz (kHz), thousands of cycles per second, to values in megahertz (MHz), millions of cycles per second.


Amplitude

In electronics, the amplitude, or height, of a sine wave is measured in three different ways:

The peak amplitudeVp is measured from the x-axis, 0 V, to the top of a peak, or to the bottom of a trough. In physics, 'amplitude' usually refers to peak amplitude.

The peak-to-peak amplitudeVpp is measured between the maximum positive and negative values. In practical terms, this is often the easier measurement to make. Its value is exactly twice Vp.

Although peak and peak-to-peak values are easily determined, it is sometimes more useful to know the root mean square (r.m.s.) amplitude Vrms of the wave, where:



and



In rearrangeable format:



Just why the r.m.s. amplitude is useful will be explained shortly.

Choose the most appropriate word for each of the following descriptions.
  • Time taken between peaks
    Number of cycles completed per second
    Maximum displacement from 0 V
  • Click here to mark the question
What units would be used in measuring each of the following quantities?
  • Amplitude
    Frequency
    Period
  • Click here to mark the question


Move the probe in Fig.6 to see the V–t graph of each signal.


Click on the figure below to interact with the model.

 Figure 6.  Three signal generators.


Which of the signals in Fig.6 has the highest frequency?
  • Click here to mark the question
Which of the signals in Fig.6 has the biggest amplitude?
  • Click here to mark the question


Phase

It is sometimes useful to divide a sine wave into degrees, °, as follows:

Figure 7.   Dividing a sine wave into degrees.
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Sine waves are generated by rotating electrical machines. A complete 360° turn of the voltage generator corresponds to one cycle of the sine wave. Therefore 180° corresponds to half a turn, 90° to a quarter turn, and so on. Using this method, any point on the sine wave graph can be identified by a particular number of degrees through the cycle.

If two sine waves have the same frequency and occur at the same time, they are said to be in phase:

Figure 8.   Sine waves and phase.
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On the other hand, if the two waves occur at different times, they are said to be out of phase. When this happens, the difference in phase can be measured in degrees, and is called the phase angle, θ. As you can see, the two waves in the second part of the diagram are a quarter cycle out of phase, so the phase angle θ = 90°.

Fig.9 shows another example of sine waves that are out of phase.

Figure 9.   Two sine waves.
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What is the phase angle θ between the waves in Fig.9?
  • Click here to mark the question
Understanding r.m.s.
What is r.m.s. amplitude and why is it important?
     The r.m.s. voltage is the d.c. voltage which will deliver the same average power as the a.c. signal.



To understand this, think about two lamps connected to alternative power supplies:

Figure 10.   Understanding r.m.s. amplitude.
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The brightness of the lamp connected to the a.c. supply looks constant, but the current flowing in the lamp is changing all the time and alternates in direction, flowing first one way and then the other. There is no current at the instant the a.c. signal crosses the x-axis. What you see is the average brightness produced by the a.c. signal.

The second lamp is powered by a d.c. supply and its brightness really is constant because the current flowing is always the same. It is obviously possible to adjust the voltage of the d.c. supply until the two lamps appear equally bright.

When this happens, the d.c. supply is providing the same average power as the a.c. supply. At this point, the d.c. voltage is equal to the Vrms value for the a.c. signal.
In the a.c.
circuit
A circuit is a closed conducting path.
circuit
, the current is …
  • Click here to mark the question
In the d.c. circuit, the current is …
  • Click here to mark the question


Nevertheless, the average power in the two circuits is the same.

A bit of mathematics is needed to explain why the equivalent d.c. voltage is called 'root mean square'. You don't need to know anything about this.

Expand the next section only if you are determined to find out:
Because the r.m.s. amplitude is the d.c. voltage that will deliver the same average power as the a.c. signal, you need to think about how to calculate the power delivered by the a.c. signal.
The unit Resistors introduced the following formula for calculating power:

Substituting or gives two additional formulae:
     and     

With a V–t graph, the formula provides the most convenient method for

calculating the power at any instant.
 
Figure 11.   Calculating average power.
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The upper graph in Fig.11 shows an a.c. signal with 10 V peak amplitude. This is the a.c. power supply lighting a lamp. Assume that the resistance of the lamp stays constant, R = 100 Ω.
The power developed is different at different times during each cycle. V is changing, and so P must also be changing.

For example, at t = 2 ms on the graph, V = 5.9 V and  = 0.35 W.

At t = 5 ms, V = 10 V, and  = 1.0 W.

While at t = 10 ms, V = 0, and P = 0.
Calculating a P value for every moment in the cycle gives the lower graph in the diagram. This shows how the power produced varies during each cycle.
The apparent brightness of the lamp depends on the average power. Look again at the power–time graph. You can see that the average power is equal to half the maximum power.
The average power Pav is given by:

Now think about the d.c. voltage Vd.c. that would deliver the same average power:

Combining these two formulae:

Rearranging:

Taking the square root of each side gives:



This is the same as the formula for Vrms as already defined.
It is now possible to explain that the r.m.s. value is the square root of the mean, or average, of the square of the voltages during a complete cycle of the a.c. waveform.
     Vrms is the single d.c. value which will deliver the same average power as the a.c. signal.

Waveform calculators
To help you with waveform calculations, you might like to use the Absorb Electronics calculators shown below:

Figure 12.   Frequency and period calculator.
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Use the calculator to answer the following questions:
The period of a sine wave is 20 ms, what is its frequency?

  • In kilohertz:   kHz

    In hertz:   Hz
  • Click here to mark the question
The frequency of a sine wave is 5 kHz, what is its period?

  • In milliseconds:   ms

    In microseconds:   μs
  • Click here to mark the question


To convert from peak amplitude to r.m.s. amplitude, or from r.m.s. to peak, use this calculator:

Figure 13.   Amplitude calculator.
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What is the r.m.s. voltage Vrms if the peak voltage Vp = 15 V? Give your answer correct to one decimal place.
  •  V
  • Click here to mark the question
What is the peak voltage Vp if the r.m.s. voltage Vrms = 12 V? Give your answer correct to one decimal place.
  •  V
  • Click here to mark the question
Listening to waves
You can understand what is meant by 'frequency' and 'amplitude' by comparing the sounds produced when different waves are played through a loudspeaker.

Not all frequencies are audible.
The frequency range of human hearing is usually quoted as from 20 Hz to 20 kHz. A good hi-fi system can accurately reproduce a slightly wider range of frequencies.
Your ears are particularly sensitive to sounds in the middle range, from about 100 Hz to 3 kHz, corresponding to the range of frequencies dominant in human speech.
     Telephone systems have a poor high frequency performance, but do work effectively in this middle range.



The pitch of a musical note is the same as its frequency. The intensity or loudness of a musical note is the same at its amplitude.

The animation below can show waveforms of different frequencies and amplitudes. Click on the buttons next to the graph to listen to the corresponding sounds:

Figure 14.   Listening to waves.
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Sine wave signals produce a 'pure' sounding tone. If the amplitude is increased, the sound is louder. If the frequency is increased, the pitch of the sound is higher. Comparing one note with another that is double the frequency, you should notice that they sound similar – the sounds are an octave apart.

Other shapes of signal generate sounds with the same fundamental frequency, but which can sound different. Switch to square wave signals in Fig.14. The square wave sound is harsher because the signal contains additional frequencies which are multiples of the fundamental frequency. These additional frequencies are called harmonics (or overtones). Sounds from different musical instruments are distinguished by their harmonic content:

Figure 15.   Piper.
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The distinctive sound of the bagpipes depends on the fundamental frequency of the note played and the harmonics produced. Additional pipes, called drones, generate sounds so that the overall effect is rich and complex. Some people like it.

You may be interested to know that an electronic device for checking the pitch of the drones is available.
Match the following words used to describe sounds with the corresponding word used to describe electronic waveforms.
  • Intensity
    Pitch
  • Click here to mark the question
Figure 16.   Four different waveforms.
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Which of the waveforms in Fig.16 has the highest frequency?
  • Click here to mark the question
Which of the waveforms in Fig.16 has the biggest amplitude?
  • Click here to mark the question
Making waves
Sine waves can be mixed with d.c. signals. or with other sine waves to produce new waveforms. Here is one example of a complex waveform:

Figure 17.   Complex waveform.
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'Complex' doesn't mean difficult to understand. A waveform like this can be thought of as consisting of a d.c. component with a superimposed a.c. component. It is quite easy to separate these two components using a capacitor, as is explained in the unit on Capacitors.

More dramatic results are obtained by mixing a sine wave of a particular frequency with exact multiples of the same frequency, in other words, by adding harmonics to the fundamental frequency. Click the button at lower left in the V–t graph below to find out what happens when a sine wave is mixed with its 3rd harmonic (3 times the fundamental frequency) at reduced amplitude:

Figure 18.   Adding sine waves.
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Keep clicking the button to see the effect of adding the 5th, 7th, and 9th harmonics. The waveform begins to look more and more like a square wave as more odd number harmonics are added in.

This surprising result illustrates a general principle first formulated by the French mathematician Joseph Fourier, namely that any repeating waveform can be built up from pure sine waves plus particular harmonics of the fundamental frequency. Square waves, triangular waves and sawtooth waves can all be produced in this way.

Other signals
This part of the unit outlines the other types of signal you are going to meet. Circuits which generate these signals are versatile building blocks and many practical examples are given elsewhere in Absorb Electronics.


Square waves

Like sine waves, square waves are described in terms of period, frequency, and amplitude:

Figure 19.   Square wave.
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Peak amplitude Vp and peak-to-peak amplitude Vpp are measured as you might expect. However, the r.m.s. amplitude Vrms of a square wave is bigger than that of a sine wave.

Remember that the r.m.s. amplitude is the d.c. voltage which will deliver the same average power as the signal.
If a square wave is connected across a lamp, the current flows first one way and then the other.
The current switches direction but its magnitude remains the same. In other words, the square wave delivers its maximum power throughout the cycle.
In this case, Vrms = Vp.
     If this is confusing, don't worry. The r.m.s. voltage of a square wave is not something you need to think about very often.



Although the square wave may change very rapidly from its minimum to maximum voltage, this change cannot be instantaneous. The rise time of the signal is defined as the time taken for the voltage to change from 10 to 90 per cent of its maximum value. Rise times are usually very short, with durations measured in microseconds, μs (1 μs = 10−6 s), or nanoseconds, ns (1 ns = 10−9 s).

Figure 20.   Square wave.
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From the Vt graph in Fig.20, estimate each of the following.

  • The peak amplitude of the waveform
     V (to one decimal place)

    The period of the waveform
     ms (to the nearest whole number)

    The frequency of the waveform
     Hz (   (to the nearest whole number))
  • Click here to mark the question


Pulse waveforms

Pulse waveforms look similar to square waves except that all the action takes place above the x-axis. At the beginning of a pulse, the voltage changes suddenly from a low level, close to the x-axis, to a high level, usually close to the power supply voltage:

Figure 21.   Pulse waveform.
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The low voltage level is often called logic 0, or just 0, while the high voltage is called logic 1, or just 1. Pulses are fundamental in
digital
In a digital circuit, information is represented by discrete voltage levels. A high voltage is called logic 1, or 1, while a low voltage is called logic 0, or 0.
digital
systems.

Sometimes the 'frequency' of a pulse waveform is called its repetition rate. This means the number of pulses per second, measured in hertz, Hz.

The length of time for which the pulse voltage is high is called the mark, while the length of time for which it is low is called the space. The mark and space do not need to be equal. The mark-space ratio is given by:



A mark-space ratio = 1.0 means that the high and low times are equal. A mark‑space ratio = 0.5 indicates that the high time is half as long as the low time.

Figure 22.   Pulse waveforms with different mark-space ratios.
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A mark-space ratio of 3.0 indicates that the mark period is three times as long as the space period. In general, a mark-space ratio greater than 1 indicates that the high time is longer.
Select the words which best complete each sentence.

  • In a pulse waveform, the high time is called the , while the low time is called the . The mark-space ratio is calculated as the divided by the .
  • Click here to mark the question


Another way of describing the same types of waveform uses the duty cycle, where:



When the duty cycle is less than 50 per cent, the high time is shorter than the low time, and so on.

A subsystem which produces a continuous series of pulses is called an astable.

As you will discover, it is useful to be able to change the frequency of the pulses to suit particular applications. Other pulse-producing subsystems include monostables and bistables.

Figure 23.   Four different pulse waveforms.
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Which of the waveforms in Fig.23 has the smallest value of mark-space ratio?
  • Click here to mark the question
Which two waveforms in Fig.23 have the same frequency (repetition rate)?
  • Click here to mark the question
Ramps

A voltage ramp is a steadily increasing or decreasing voltage, as shown in Fig.24 below.

Figure 24.   Ramp waveforms.
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The ramp rate is measured in volts per second, V/s. Such changes cannot continue indefinitely, but stop when the voltage reaches a saturation level, usually close to the power supply voltage.


Triangular and sawtooth waves

These waveforms consist of alternate positive-going and negative-going ramps. In a triangular wave, the rate of voltage change is equal during the two parts of the cycle, while in a sawtooth wave, the rates of change are unequal.

Sawtooth generator circuits are an essential building block in oscilloscope and television systems.
Figure 25.   Four different waveforms.
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Match each waveform in Fig.25 with the correct description.
  • A
    B
    C
    D
  • Click here to mark the question


Audio signals

Sound frequencies which can be detected by the human ear vary from around 20 Hz to 20 kHz. Audio signals usually consist of a mixture of different frequencies:

Figure 26.   Audio signal.
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Sometimes it is possible to see a dominant frequency in the voltage–time graph of an musical signal, but it is clear that other frequencies are present.


Noise

A noise signal consists of a mixture of frequencies with random amplitudes:

Figure 27.   Noise.
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Noise can originate in various ways. For example, heat energy increases the random motion of electrons and results in the generation of thermal noise in all components, although some components are 'noisier' than others.

Additional sources of noise include radio signals, which are detected and amplified by many circuits, not just radio receivers. Interference is caused by the switching of many appliances, and 'spikes' and 'glitches' are caused by rapid changes in current and voltage elsewhere in an electronic system.

Designers try to eliminate noise in most circuits, but special noise generators are used in electronic music synthesizers and for other musical effects.

Did you know …? Barcodes
Barcodes appear on all sorts of things including books, CDs, and almost everything you buy at the supermarket. Barcodes are useful because they are machine readable.
Figure 28.   Barcode.
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The checkout is connected to a computer network and reads the price of each item from a central database. This makes it easy for the supermarket to set and adjust prices and to collect information for stock control.

Figure 29.   Checkout.
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Barcode readers, scanners, and wands convert the printed pattern of the barcode into a voltage–time signal consisting of a series of pulses. All these devices shine light onto the barcode and detect variations in the intensity of reflected light. You can understand how this works by seeing what happens when you click the button in this animation:

Figure 30.   Scanning a barcode.
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LEDs inside the scanning wand provide the light source. The lens focuses the light from the LEDs into a beam. As the wand passes over a white area of the barcode, light is reflected back into the wand and is detected by a
photodiode
A photodiode is a sensor device which can be used to detect infrared or other wavelengths of light.
photodiode
. The photodiode responds much more quickly to changes in light intensity than a light-dependent
resistor
A resistor is an electronic component with a particular resistance values. Resistors limit current.
resistor
, or
LDR
A light-dependent resistor, or LDR, has a high resistance in the dark, and a low resistance in the light.
LDR
.

When the wand passes over a black bar, or stripe, much less light is reflected. These changes are converted into a pattern of pulses which exactly matches the pattern of bars in the barcode. The electronic circuit operates so that a high voltage, logic 1, corresponds to a bar.

The diagram below explains in more detail how an EAN 8 barcode, one of several common types, is constructed:

Figure 31.   Understanding a barcode.
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  • The narrowest element, either a bar or a space, is called a module.

  • At each end of the barcode, there is a quiet zone, consisting only of spaces.

  • Guard bars at the left and right of the barcode use a single module,
    bar–space–bar pattern to indicate where scanning should start and stop.

  • There is a middle zone separating the left and right sides of the barcode.

  • Within the left and right sections of the barcode, each number is represented by seven modules. For example, in the left part of the barcode, the number 5 is represented by space–bar–bar–space–space–space–bar, that is, 0 1 1 0 0 0 1. Zoom in on the diagram to see more detail.

  • Different codes are used to represent the same number if it appears in the right section of the barcode, instead of the left section. The coding system has been cleverly designed so that the barcode can be scanned in either direction.



Look out for applications of barcodes and think about why they are used.

Summary


Voltage–time, Vt, graphs are used to represent signals in electronic systems.

Sine waves are of fundamental importance. Essential features of sine waves and other repeating waveforms include (measurement units in brackets):

Period is the time taken for one complete cycle (s, ms, μs).

Frequency f is the number of cycles completed per second (Hz, kHz, MHz).
   or   

Peak amplitude Vp is the maximum displacement from 0 V (V, mV).

Peak-to-peak amplitude Vpp is the difference between the maximum positive and maximum negative values (V, mV).

Root mean square (r.m.s.) amplitude Vrms is the d.c. voltage which will deliver the same average power as the a.c. signal (V, mV).

For sine waves:



and



Phase angle θ is the angular difference between the peak times of two waves of the same frequency (°).

The pitch of a musical note is the same as its frequency. The amplitude of a musical note is the same as its loudness. Exact multiples of the fundamental frequency are called harmonics.

Complex waves are built up by adding d.c. signals or harmonics to the fundamental frequency. All repeating waveforms can by synthesized from sine waves.

Pulses change suddenly between low voltage (logic 0) and high voltage (logic 1) levels. The frequency of a repeating pulse waveform is sometimes called its repetition rate. The low time of the pulse is called the space, and the high time is called the mark. Mark-space ratio and duty cycle are used to describe the shape of pulse waveforms.

Ramp, triangular, sawtooth, audio, and noise signals can all be represented as voltage–time graphs.

Exercises
Figure 32.   Sine wave recorded from an electronic system.
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1. Answer the following questions from the graph in Fig.32.

  • What is the peak amplitude of the signal?
     V   (to the nearest whole number)

    What is the peak-to-peak amplitude of the signal?
     V   (to the nearest whole number)

    What is the r.m.s. amplitude of the signal?
     V   (to 1 decimal place)

    What is the period of the signal?
     ms   (to the nearest whole number)

    What is the frequency of the signal?
     Hz   (to the nearest whole number)
  • Click here to mark the question
Figure 33.   Pulse waveform.
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2. Answer the following questions from the graph in Fig.33.

  • What is the mark-space ratio of the signal?
       (to 2 decimal places)

    What is the duty cycle of the signal?
    %   (to the nearest whole number)

    What is the period of the signal?
     ms   (to the nearest whole number)

    What is the frequency or repetition rate of the signal?
     Hz   (to the nearest whole number)

    What is the logic 0 voltage level?
     V   (to 1 decimal place)

    What is the logic 1 voltage level?
     V   (to 1 decimal place)
  • Click here to mark the question
Well done!
Try again!