Electronics: (Syllabus 6063)
Electronics: (Syllabus 6063)
Electronics: (Syllabus 6063)
CONTENTS
Page
INTRODUCTION 2
AIMS 2
ASSESSMENT OBJECTIVES 3
SCHEME OF ASSESSMENT 4
CONTENT STRUCTURE 5
SUBJECT CONTENT 5
SUMMARY OF KEY QUANTITIES, SYMBOLS AND UNITS 19
PRACTICAL GUIDELINES 20
GLOSSARY OF TERMS 21
SPECIAL NOTES 22
ASSESSMENT RUBRIC FOR PROJECT 22
GUIDELINES TO REPORT WRITING 29
DATA AND FORMULAE 30
INTRODUCTION
The O-Level electronics syllabus provides students with an understanding of the fundamental working of
electronic components and systems, as well as ideas of engineering design. The syllabus focuses on the
application of the knowledge of electronics components and circuit theories to design and build electronics
systems that can solve daily problems. The students will also develop testing and troubleshooting skills in the
realisation of an electronic system. Through these learning experiences, the subject should provide a broad-
base foundation for further studies in electronics engineering and related fields.
It is envisaged that teaching and learning programmes based on this syllabus would feature a range of
learning experiences designed to promote understanding of electronics and to develop values and attitudes
related to engineering. Teachers are encouraged to use a combination of appropriate strategies to effectively
engage students in hands-on and applied learning. It is expected that students will apply problem-solving
and engineering design skills, effectively communicate the intent of their design and appreciate the
contribution electronics makes to our modern living.
AIMS
These are not listed in order of priority.
1. develop attitudes relevant to engineering such as perseverance; curiosity; integrity; striving for
accuracy; open-mindedness; inventiveness; problem-solving (“can do” attitude); intellectual
thoroughness
2. develop abilities and skills related to the engineering design process such as
2.3 troubleshooting
4. develop an appreciation about the usefulness of electronics and its impact on modern society
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ASSESSMENT OBJECTIVES
A Knowledge with understanding
Candidates should be able to demonstrate knowledge and understanding of scientific facts, concepts,
theories and terminology in relation to:
1. electronic systems
4. digital electronics.
4. solve problems.
Candidates should be able to design, build and test electronic systems involving the following processes:
6. use appropriate test and measurement equipment to test and troubleshoot a prototype circuit
Project (Paper 2)
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SCHEME OF ASSESSMENT
Candidates are required to enter for Papers 1 and 2.
This project is carried out over a period of 32 hours in Year 2. It comprises two interrelated components.
The project report document shows evidence of the candidate’s activities and project implementation. The
report reflects the candidate’s understanding of the project specification, planning, design brief, analysis,
investigation, ideas generation, design proposal, development, building, testing and evaluation.
The report should articulate how information is obtained and used, and the basis on which decisions are
made in the development of design proposal. The candidate is advised to use flow diagrams to exemplify the
complex project stages and is encouraged to use pictures and graphical illustrations in the report. Due
recognition and acknowledgement should be accorded to information sources and person/s rendering help to
the project.
The report should include a description of the strengths and weaknesses of the design and how problems,
and issues surrounding the project were resolved.
The candidate is expected to demonstrate good quality work, appropriate use of electronic components and
constructional methods.
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CONTENT STRUCTURE
Section Topics
SUBJECT CONTENT
SECTION I: SYSTEMS
1. Electronic Systems
Electronic systems are designed to solve specific problems in many areas of our daily lives. A simple system
can consist of just an input, a process and an output1. However, complex systems can have multiple inputs,
processes and outputs. In addition, complex systems are usually made of subsystems with the output of a
subsystem becoming the input of another subsystem. Only when all subsystems are functioning properly will
the overall system be able to solve the intended problem. This understanding allows electronic engineers to
design, build, test and troubleshoot electronic systems in a logical and systematic manner.
Content
• Simple systems
• Electronic systems
• Electrical signals
Learning Outcomes
(b) use the symbols of common electrical and electronic components to represent an electrical/electronic
system
1
A component that has input, process and output can be considered to be a simple system, e.g. an IC chip. By this definition,
components such as wires and resistors are not considered a system.
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(d) identify the inputs, processes (amplification, logic, memory, decoding, timing and counting) and outputs
of an electronic system (e.g., an audio amplifier)
(e) describe a subsystem as a system that obtains input from, or provides input to, another subsystem
(g) state that an electronic signal is an electrical voltage or current that carries information
(h) recognise that electronic signals may be analogue or digital in nature, and differentiate between them.
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2. Current Electricity
Electronics builds on principles of electricity. It is necessary for learners of electronics to have a firm
understanding of the principles related to electric charges, current, voltage, resistance and power.
Content
Learning Outcomes
(a) recall the following base quantities and their SI units: mass (kg), length (m), time (s), electric current (A),
temperature (K)
(b) recall derived quantities related to electricity (e.g., electric charge, resistivity and frequency) and their SI
units
(d) use the following prefixes and their symbols to indicate decimal submultiples and multiples of the SI
units: pico (p), nano (n), micro (µ), milli (m), centi (c), kilo (k), mega (M), giga (G), tera (T)
(f) state that current is the rate of flow of charge and is measured in amperes (A)
(h) distinguish between electromotive force (e.m.f.) and potential difference (p.d.)
(i) state that both e.m.f. and p.d. are measured in volts (V).
(j) calculate the effective e.m.f. when several sources are connected in series and in parallel
(l) state and apply Ohm’s law to determine current, voltage, and resistance
(m) sketch and interpret the graphical linear relationship between current and voltage in a purely resistive
circuit
(n) describe the use of the heating effect of an electric current flowing through a conductor
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(p) recall the power equations P = VI, P = I2R and P =V2/R and apply the relationships P = VI and E = VIt to
solve problems involving resistive circuits
3. Resistors
Resistors are basic electrical components that are used to control the size of current flowing in different parts
of electrical circuits. They can be made from different materials and are usually available in standard values.
Content
• Resistivity
• Types of resistors
• Colour code
• Effective resistance
Learning Outcomes
(a) describe resistivity as the characteristic of a material that affects its electrical conductivity and apply the
formula R=pl/A to perform calculations
(b) describe the structures of various types of resistor (carbon and wire-wound) and select the appropriate
resistor for a particular circuit design
(c) use the resistor colour code to determine the ohmic value and tolerance of a resistor, and verify the
value by measurement
(d) select a suitable resistor from the E24 resistor series for a particular application
(e) determine the power rating of a resistor and explain the factors affecting it
(f) explain how changing the resistance in a circuit changes the current in the circuit
(g) recall and apply the formulae to calculate the effective resistance of resistors connected in series and in
parallel
4. Circuit Theories
A circuit consists of electrical paths that allow currents to flow. Components in an electronic system are
connected as circuits. Circuit theories can be used to determine the voltage and current at different parts of a
given circuit. Knowledge of circuit theories is also necessary to design, build and troubleshoot electronic
systems.
Content
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Learning Outcomes
(a) define the terms - circuit, load, source, open-circuit, short-circuit and overload
(e) identify a resistive voltage divider and apply the voltage-divider formula to solve related problems
(f) identify a resistive current divider and apply the current-divider formula to solve related problems
5. Alternating Currents
As opposed to direct currents (DC), which do not change direction, alternating currents (AC) change
direction in a regular manner. Many voltages and currents encountered in our daily lives are AC, e.g. the
mains supply and music signals. To describe these AC voltages and current, terms such as frequency,
period and peak voltage are commonly used.
Content
• Types of AC waveforms
Learning Outcomes
(a) distinguish between direct and alternating currents/voltages (in terms of whether there is a change of
direction)
(c) show understanding that alternating currents or voltages can be represented by waveforms
(d) recognise and sketch the common types of AC waveforms (sinusoidal, rectangular square and
triangular)
(e) determine the DC level, frequency, period, peak and peak-to-peak values of an alternating
current/voltage from its waveform
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6. Capacitors
Capacitors are components that store electrical charges. A capacitor is made of 2 conductors (usually plates)
separated by a dielectric material. It charges up when power is supplied to a circuit and discharges when the
power is turned off. Capacitors have many important uses in electronic systems such as smoothing voltages
and timing.
Content
• Capacitance
Learning Outcomes
(f) apply the relevant equations for capacitors connected in series and in parallel to solve related problems
(g) calculate the time constant in a simple resistor-capacitor (RC) circuit using τ = RC
(h) estimate the time for a capacitor to be charged to and discharged by 2/3 and 100% of the maximum
voltage.
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7. Semiconductor Diodes
A semiconductor is a material which has electrical conductivity between that of a conductor and an insulator.
There are two types of semiconductors: n-type and p-type. The simplest semiconductor component is the pn
junction diode and one of its important uses is the conversion of AC to DC in a process called rectification.
Light-emitting diodes and Zener diodes are special types of diodes. Unlike normal diodes, Zener diodes
allow current to flow in the opposite direction but only when the reverse voltage is large enough. Zener
diodes can be used to maintain a steady voltage when it is in reverse-biased thus making them useful to be
used in voltage regulators. Due to their low power consumption, LEDs are commonly used for lighting and
visual displays.
Content
• 7-segment display
Learning Outcomes
(a) state that there are two types of semiconductor: n-type and p-type
(b) describe the basic structure of the PN junction diode and explain how it is biased in the forward and
reverse directions
(f) describe and explain the use of diodes in half-wave and full-wave rectifiers
(g) interpret typical diode specifications (forward voltage, max current, max reverse voltage) using
datasheets
(h) state that LED is a special type of diode that emits light and infra-red
(i) describe the benefits of using LEDs for lighting as compared to incandescent bulbs
(j) explain why a resistor should be connected in series with an LED in a circuit and calculate its resistance
value
(k) state that a 7-segment display is made up of 7 LEDs which can be individually controlled
(l) describe the difference between the structure and operation of a common-anode and common-cathode
7-segment display
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A transducer is a device that converts a signal from one form of energy to another. As electronic systems
can only process electrical signals, input transducers play the important role of converting non-electrical
quantities (e.g. temperature) to electrical signals. Output transducers then convert the outcomes of the
process to non-electrical quantities. There are many different types of input and output transducer which are
differentiated by the type of conversion they perform.
Content
Learning Outcomes
(c) recall and apply the effect of changes in temperature on the resistance of a thermistor to practical
situations
(d) recall and apply the effect of changes in light intensity on the resistance of an LDR to practical situations
(f) describe the use of infrared diodes as transmitting and receiving devices
(g) describe the function of the following transducers: microphone, loudspeaker, buzzer, low voltage DC
motor and electromechanical relay.
A transistor is a 3-terminal semiconductor device. By applying a small current at one terminal, the current
flowing between the other two terminals can be controlled. This property allows the transistor to be used as
an amplifier or an electronic switch. Transistors are the basic building blocks of complex integrated circuits
and the modern-day computer processors consist of hundreds of millions of transistors packed into a small
integrated circuit. A basic type of transistor is the bipolar junction transistor. Before it can be used, a BJT
needs to be biased using an external circuit so that it will work in the correct operating region.
Content
• Operating regions
• BJT applications
Learning Outcomes
(a) describe the structures of the two types of bipolar junction transistor (BJT)
(b) describe the working principle of a BJT (a base current controls current between emitter and collector)
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(d) relate the operating regions to the different segments of an IC–VCE characteristic graph
(e) relate the operating regions to the use of a BJT as a switch and an amplifier
(f) explain how a BJT can be biased to operate as a switch and an amplifier
(g) identify common base (CB), common collector (CC) and common emitter (CE) transistor circuits
(h) apply the relationship between the current, voltage and power of a transistor to solve related problems
in common emitter circuits
(i) explain the function of coupling and bypass capacitors in transistor amplifier circuits
(k) explain the advantage of a Darlington pair over a single transistor in driving an output transducer
(l) interpret typical BJT specification (β, ICmax, VBE, VCE(sat)) using its datasheet.
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Digital electronics is based on analogue electronics. Most modern electronic devices such as the personal
computer and mobile phone, use digital electronics to tap its advantages in terms of cost, size, speed and
reliability. A key difference between analogue and digital electronics is the signals. While analogue signals
take on continuous values, digital signals take on two discrete levels. For this reason, almost every digital
system uses the binary number system. However, humans are more familiar with the decimal number
system. To help us deal with binary numbers, a third number system, hexadecimal, can be used.
Content
Learning Outcomes
(b) state that digital signals can be represented by two logic states: logic 1 (high voltage, usually 5 V); logic
0 (low voltage, usually 0 V)
(c) explain the use of ‘pull up’ and ‘pull-down’ resistors to provide the correct logic levels
(d) list the advantages and disadvantages of digital systems over analogue systems
(e) describe the need to convert between analogue and digital signals
(f) convert between binary, decimal and binary-coded decimal (BCD) systems
(g) describe the function of a BCD to 7-segment display module using a truth table.
The basic building blocks of digital electronics are logic gates. Each logic gate is a system on its own with
one or more inputs and one output. Logic gates are used to represent logical decisions which can be
presented in the form of truth tables, logic symbols and Boolean notation. Two of the logic gates, NAND and
NOR, are special gates called universal gates as they can be used to build all other types of logic gates.
Content
• Universal gates
Learning Outcomes
(a) describe the truth table as a way to show the output of a digital circuit for different combinations of
inputs
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(b) state that a logic gate is a device with one output and at least one input; the output is either logic 1 or 0
depending on the inputs
(c) draw symbols and construct truth tables for NOT, AND, OR, NAND and NOR gates
(d) use Boolean notation (‘—’, ‘■’ and ‘+’) to write the Boolean expression for NOT, AND, OR, NAND and
NOR gates
(e) state that NAND and NOR gates are universal gates
(f) show how NOT, AND and OR gates can be made using NAND or NOR gates
(g) describe basic characteristics (e.g. general structure, pin configuration, common notation) of a dual
in-line IC
(h) use datasheets to identify pin connections of common logic gate ICs.
Combinational logic circuits are built using basic logic gates to achieve more complex functions. There are
usually more inputs and outputs compared to a basic logic gate. Here, the combinational logic circuit can be
viewed as a larger system with the basic logic gates functioning as sub-systems where the outputs are
dependent on the inputs and how the logic gates are connected. A combinational logic circuit can also be
simplified to achieve the same function with fewer logic gates, helping to reduce cost and make the circuit
less prone to faults. Mathematical tools such as Boolean algebra and Karnaugh maps can help to perform
such simplifications in a systematic manner.
Content
Learning Outcomes
(a) use a truth table to describe the output of a digital system (up to three inputs)
(b) convert a truth table (up to three inputs) into a sum-of-product (SOP) Boolean expression
(c) simplify an SOP Boolean expression (up to three variables) using either Boolean algebra or a Karnaugh
map
(d) implement logic circuits using NOT, AND and OR gates given an SOP Boolean expression
(e) describe and explain the function of a given combinational logic circuit
(f) solve system problems using combinations of logic gates (up to three inputs).
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A key advantage of digital electronics is the ability to remember data. This ability enables digital circuits to
perform more complex operations such as counting. The most basic circuit with memory is the S-R latch
which can be used to store a single bit of data as either logic ‘1’ or ‘0’ allowing the latch to convert a
momentary occurrence into a constant output. Timing diagrams are frequently used to describe and analyse
circuits with memory. These diagrams show the logic state (‘1’ or ‘0’) of the system at any point in time and
also the time when a change in state occurs.
Content
• Debounced Switch
Learning Outcomes
(b) draw the symbolic representation of an S-R latch using NOR gates
(c) construct the truth table of an S-R latch and use the table to determine the output of the latch
(e) explain how an S-R latch is used to convert a momentary occurrence into a constant output
(f) explain how an S-R latch can be used to build a debounced switch.
Comparing voltages, timing and counting are important applications of electronics which can be carried out
by specific IC chips. Voltage comparator ICs, such as the LM311, are commonly used to compare the
voltage produced by an input transducer against a reference voltage. Depending on the result of the
comparison, the output will either be a high voltage (e.g. 5 V) or a low voltage (e.g. 0 V). This is an example
of analogue-to-digital conversion. The most widely used IC chip for timing applications is the 555 timer IC
that can be set up to operate as a monostable or astable multivibrator. The time needed to charge or
discharge the capacitor in the external circuit determines the duration and frequency of the multivibrator. The
74390 dual decade counter IC contains two 4-bit decade counters that can be used together to count up to
99, lending it to many useful applications that require counting.
Content
Learning Outcomes
(a) identify the pins of an LM311 voltage comparator IC from its specification sheet
(b) describe the operation and use of an LM311 voltage comparator IC (with single rail supply only)
(d) identify the pins of a 555 timer IC from its specification sheet
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(e) recognise whether a 555 timer IC is set up as a monostable or astable multivibrator from a given circuit
(students are not required to draw the set-up)
(f) use the formula T = 1.1RC to determine the time period of a 555 IC in monostable mode (formula will be
provided)
( R1 + 2R 2 ) C
(g) use the formula T = to determine the time period of a 555 IC in astable mode (formula will
1.44
be provided)
(i) identify the pins of a 74390 4-bit decade counter IC from its specification sheet
(k) show understanding of how the output of a 74390 IC can be shown on a 7-segment display
(l) show understanding of how two 4-bit decade counters in a 74390 IC can be connected to count to 99.
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The engineering design process covers the development of a product from problem definition to the design,
build and test of a prototype to the reporting of the development process. To ensure the successful
completion of a project, it is important to manage time and resources effectively. In electronics, this often
means the completion of an electronic product that can perform a specific function. In the design phase of
the project, computer simulation offers a cost- and time-effective means of checking the workability of a
design before building a prototype. Electronic engineers use test equipment to check if a prototype is
working as planned; if not, they will use the equipment to pinpoint the sources of problem in a process called
troubleshooting. The engineers also need to be able to document and communicate the processes, usually
in the form of a report.
Content
• Project management
• Project realisation
• Documentation of project
Learning Outcomes
(b) draw a Gantt chart for a project with known tasks, precedence, and duration
(c) create new processes, products or projects through the synthesis of ideas from a wide range of sources
by:
(i) using research methods including web search, textbooks, library resources, literature review, etc.;
(ii) specifying the requirements of an electronic product based on the problem definition; and
(iii) building a prototype circuit using a prototype board
(d) appraise the role of computer simulation in circuit design (advantages and limitations)
(f) use relevant test and measuring equipment (digital multimeter, function generator and oscilloscope) to
test and troubleshoot prototype circuits
(h) write a project report using information collated from the project work.
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PRACTICAL GUIDELINES
Applied subjects are, by their nature, application-based. It is therefore important that the candidates carry out
appropriate practical work to support and facilitate the learning of the electronics components, test
equipment and theories. A list of suggested practical work is provided below.
• use test equipment (DC power supply, function generator and oscilloscope)
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GLOSSARY OF TERMS
It is hoped that the glossary will prove helpful to candidates as a guide, although it is not exhaustive. The
glossary has been deliberately kept brief not only with respect to the number of terms included but also to
the descriptions of their meanings. Candidates should appreciate that the meaning of a term must depend in
part on its context. They should also note that the number of marks allocated for any part of a question is a
guide to the depth of treatment required for the answer.
1. Define (the term(s) ...) is intended literally. Only a formal statement or equivalent paraphrase, such as
the defining equation with symbols identified, being required.
2. Explain/What is meant by ... normally implies that a definition should be given, together with some
relevant comment on the significance or context of the term(s) concerned, especially where two or more
terms are included in the question. The amount of supplementary comment intended should be
interpreted in the light of the indicated mark value.
3. State implies a concise answer with little or no supporting argument, e.g. a numerical answer that can
be obtained ‘by inspection’.
4. List requires a number of points with no elaboration. Where a given number of points is specified, this
should not be exceeded.
5. Describe requires candidates to state in words (using diagrams where appropriate) the main points of
the topic. It is often used with reference either to particular phenomena or to particular experiments. In
the former instance, the term usually implies that the answer should include reference to (visual)
observations associated with the phenomena. The amount of description intended should be interpreted
in the light of the indicated mark value.
6. Discuss requires candidates to give a critical account of the points involved in the topic.
7. Predict or deduce implies that candidates are not expected to produce the required answer by recall but
by making a logical connection between other pieces of information. Such information may be wholly
given in the question or may depend on answers extracted in an earlier part of the question.
8. Suggest is used in two main contexts. It may either imply that there is no unique answer or that
candidates are expected to apply their general knowledge to a 'novel' situation, one that formally may
not be ‘in the syllabus’.
9. Calculate is used when a numerical answer is required. In general, working should be shown.
10. Measure implies that the quantity concerned can be directly obtained from a suitable measuring
instrument, e.g. length, using a rule, or angle, using a protractor.
11. Determine often implies that the quantity concerned cannot be measured directly but is obtained by
calculation, substituting measured or known values of other quantities into a standard formula.
12. Show is used when an algebraic deduction has to be made to prove a given equation. It is important
that the terms being used by candidates are stated explicitly.
13. Estimate implies a reasoned order of magnitude statement or calculation of the quantity concerned.
Candidates should make such simplifying assumptions as may be necessary about points of principle
and about the values of quantities not otherwise included in the question.
14. Sketch, when applied to graph work, implies that the shape and/or position of the curve need only be
qualitatively correct. However, candidates should be aware that, depending on the context, some
quantitative aspects may be looked for, e.g. passing through the origin, having an intercept, asymptote
or discontinuity at a particular value. On a sketch graph it is essential that candidates clearly indicate
what is being plotted on each axis.
Sketch, when applied to diagrams, implies that a simple, freehand drawing is acceptable: nevertheless,
care should be taken over proportions and the clear exposition of important details.
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SPECIAL NOTES
Calculators
An approved calculator may be used in all papers.
Research (5 marks)
Candidates should be able to conduct research for information needed to make informed decisions at
various stages of the design work. Information should be obtained from a range of sources that answer all
aspects of questions posed and allow relevant findings to be evaluated.
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Project Report
Project plan
• No project • Plan lacks details and • Fairly detailed work plan • Detailed and well
plan clarity that provides a general thought out plan that
• Plan is not an effective overview of the project provides a good
guide to implement the • Plan can be used as a overview of the project.
project guide to implement the • Plan can be used as an
project effective guide to
implement the project
Research
• No research • Minimal research was • Adequate research was • Extensive research was
work conducted conducted – some conducted – a range of
performed • No evaluation of the useful information was useful information was
findings obtained for further obtained for further
investigation and investigation and
generation of ideas generation of ideas
• Evaluation of some of • Evaluation of most of
the findings the findings
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• No circuit • Provide a partial list of • Provide a complete list • Provide a complete list
diagram the components used of the components of the components
drawn, or • Provide most of the used with some used with full technical
• No list of circuit diagrams with technical specifications specifications
components only some test points missing • Provide a complete set
and component values • Provide a complete set of organised, well-
labelled of circuit diagrams with structured circuit
• Describe some parts of most test points and diagrams with all test
the overall system component values points and component
labelled values labelled
• Describe most parts of • Describe clearly details
the overall system of the overall system
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Project evaluation
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Project Hardware
Functionality of project
Testing activities
• No testing • Some tests were • Testing done was • Testing done was
activities conducted adequate and mostly thorough, correct and
conducted correct – able to show if logically sequenced –
project’s performance able to show if project’s
met most specifications performance met all
specifications. Field test
was conducted and
documented
Measurement results
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Quality of project
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3. Research
This section should document the research work you have conducted and the evaluated findings.
8. Project Evaluation
This section should document your review the project planning and implementation based on the
learning points and challenges faced. The section should also include your assessment of the strengths
and weakness of the project, and possible improvements that can be made.
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1.0 1.1 1.2 1.3 1.5 1.6 1.8 2.0 2.2 2.4 2.7 3.0 3.3
3.6 3.9 4.3 4.7 5.1 5.6 6.2 6.8 7.5 8.2 9.1 and multiples of ten.
FORMULAE
Astable and monostable generators using 555 timers
( R1 + 2R 2 ) C
Astable mode frequency (t1 + t2) Period T =
1.44
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• 0 =1
Laws of Complementation • 1=0
• A=A
• A⋅0 = 0
Single variable • A ⋅1 = A
AND Laws
theorems • A⋅A = A
• A⋅A = 0
• A+0 = A
• A +1 = 1
OR Laws
• A+A = A
• A + A =1
• A+B = B+A
Commutative Laws
• A ⋅ B = B⋅A
• A + ( B + C ) = ( A + B) + C = A + B + C
Associative Laws
• A ⋅ ( B ⋅ C ) = ( A ⋅ B) ⋅ C = A ⋅ B ⋅ C
• A ⋅ ( B + C) = A ⋅ B + A ⋅ C
Distributive Laws
• ( A + B) ⋅ ( C + D ) = A ⋅ C + B ⋅ C + A ⋅ D + B ⋅ D
Multivariable
theorems
• A + A⋅B = A
• A ·(A + B) = A
Absorptive Laws • A + A ·B = A + B
• ( )
A⋅ A + B = A⋅B
• ( A + B) = A ⋅ B
DeMorgan’s Theorems
• ( A · B) = A + B
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