Aims

The aims of the course are to:

• Instill fluency with the basic mathematical techniques which are needed as tools for engineers.
• Revise, and teach afresh where necessary, those parts of the A-level mathematics syllabuses which are necessary for the first two years of the engineering course, and to introduce those new mathematical techniques which are necessary for these courses.
• Place emphasis throughout upon the grasp of essentials and competency in manipulation.

Objectives

As specific objectives, by the end of the course students should be able to:

• Recognise the appropriate mathematical tools and techniques (from the following syllabus) with which to approach a wide variety of engineering problems.
• Specify a mathematical model of a problem.
• Carry out appropriate mathematical manipulations to solve the modelled problem.
• Interpret the significance of the mathematical result.

Content

Michaelmas term (24/16L)

The Michaelmas term course concerns revision and extension of concepts which most students will have met at school. It will be given in two versions, a standard course at a pace of three lectures per week and a fast course at a pace of two lectures per week. Both will cover the same syllabus and employ the same example sheets. The fast course is aimed primarily at those who have taken double mathematics at A level and who have good mathematical fluency, the standard course at those with less prior training. Examples papers will include exercises to encourage students to practice mathematical skills learnt in their previous studies.

Vectors (5/3L)

• Scalar and vector product.     
• Moment of a force and angular velocity vectors.
• Scalar and vector triple product.
• Examples of applications.
• Simple vector geometry, vector equations of lines and planes.
• Determinant of 3x3 matrices

Functions and Complex Numbers (7/5L)

• Definitions and simple properties of the hyperbolic functions.
• Statement of Taylor's theorem, examples including trigonometric and hyperbolic function, exp, ln.
• Simple ideas of series, approximations, limits, L'Hopital's rule.
• Asymptotic behaviour of functions for small and large argument.
• Revision of complex arithmetic and representation in the Argand diagram. Idea of a complex function.
• De Moivre's theorem, use of exp (iw t)

Introduction to Ordinary Differential Equations (ODE's) (5/3L)

• Linear equations of first order, integrating factor, separation of variables.
• Second order ODE’s: complementary functions, superposition and particular integrals.          
• Linear difference equations.
• Notions of a partial derivative.

Matrices (7/5L)

• Matrices as linear transformations: the range and the null space of a matrix.
• The inverse of a 3x3 matrix.
• Change from one orthogonal coordinate system to another, the rotation matrix.
• Symmetric, antisymmetric and orthogonal matrices.       
• Eigenvalues and eigenvectors for symmetric matrices.
• Special properties of symmetric matrices: orthogonality of eigenvectors, expansion of an arbitrary vector in eigenvectors.
• Examples, including small vibrations.

Lent Term (8L)

The course in the Lent and Easter terms introduces ideas which will be new to most students, but which find application across the whole range of engineering science.

Steps, impulses and linear system response (3L)

• Introduction to step and impulse functions. Step and impulse response of linear systems represented by ODE's.
• Use of convolution to obtain output given a general input.

Fourier series (4L)

• Fourier sine and cosine series. Full and half range, consideration of symmetries, convergence and discontinuities.
• Complex Fourier series. Physical interpretations, including effect of filtering a general periodic input.

Introduction to probability material in vacation programmed learning text (1L)

Easter vacation - Probability (Programmed learning text, equivalent to four lectures of material)

• Notion of probability. Conditional probability.
• Permutations and combinations.
• Mean,variance and standard deviation of probability distributions.
• Discrete and continuous distributions.
• The Normal distribution and experimental errors

Easter term (7L)

Functions of Several Variables (4L)

• Differentiation of functions of several variables.
• Chain rule, implicit differentiation.
• Introduction to definition of grad(f).
• Stationary values, unconstrained extrema.
• Taylor expansion of f(x,y).

Introduction to Laplace transforms (3L)

• Basic properties of Laplace transforms.
• Laplace transforms as a means of solving ODEs with initial conditions (using tables of transforms for inversion).

Aims

The aims of the course are to:

• Familiarise students with combinational and sequential digital logic circuits, and the analogue-digital interface,
• Familiarise students with the hardware and basic operation of microprocessors, memory and the associated electronic circuits which are required to build microprocessor-based systems.
• Teach the engineering relevance and application of digital and microprocessor-based systems, give students the ability to design simple systems of this kind, and understand microprocessor operation at the assembly-code level.

Objectives

As specific objectives, by the end of the course students should be able to:

• Know the nomenclature and the representation of basic gates and digital electronic components (including shift registers, counters, latches, RAM and ROM ICs)
• Understand Boolean algebra, and be able to convert verbal descriptions of requirements into Boolean notation
• Understand the need to simplify logic functions or rearrange them to use specific gate types; be able to use Boolean algebra and Karnaugh maps (for up to 4 variables) to achieve these tasks; be able to use "don't care states" in K-maps.
• Know about logic "families", the electronic circuit implementation of logic gates, and the resulting engineering issues (voltage thresholds, noise margin, finite rise time and delay)
• Know about Schmitt inputs; understand static hazards and be able to detect them using K-maps and correct them
• Be familiar with standard number codes for representing data (two's complement notation, sign+magnitude, one's complement, Gray code, ASCII); be able to convert between binary, hex, octal and decimal.
• Understand the operation of logic circuits for addition, negation and subtraction of binary integers
• Be familiar with examples of elementary VHDL and understand why it is useful
• Understand the distinction between combinational and sequential logic and the role of sequential logic; be familiar with unclocked and clocked S-R latches, D-type and JK flip-flops.
• Understand the operation and use of synchronous and asynchronous counters and shift registers.
• Understand state diagrams and their role in sequential circuit design; be able to convert a problem statement into a state diagram; be able to convert a state diagram into a circuit design based on JK flip-flops
• Understand unused states, and be able to guard against errors due to them. Be able to carry out the complete design process, from problem statement to circuit design.
• Understand the operation of weighted resistor and R-2R ladder DAC circuits
• Understand the operation of Full Adder and Ripple Carry Circuits
• Understand ROM and RAM memory circuits, the function of their control, address and data pins, and their use in digital (including microprocessor) systems
• Understand the use of tri-state outputs and busses.
• Understand and be able to design address decoders, including partial address decoders for simple systems.
• Be familiar with the system architecture of a typical PIC microprocessor system, including the ALU, memory, I/O;understand how it can be used in practical applications.
• Be familiar with the internal architecture of a typical PIC microprocessor (the PIC12F629/675) and its instruction set, and understand how instruction execution occurs.
• Understand the features of typical instruction sets,and be able to use the full instruction set (from the tables in the electrical data book).
• Be able to write simple programs in assembler mnemonics, including conditional branches, and calculate their execution times in clock cycles; know about the relationship of higher level languages to assembly level code.
• Understand (in outline only) stacks, subroutines and the hardware reset function.

Content

Digital Fundamentals and Combinational Logic

• Introduction, revision of simple logic gates, overview of logic circuit families. [1] Ch 3, [3] 392-399, [4] 12
• Circuits for inverters and basic logic gates in NMOS and CMOS. [3] 409-410, [5] Ch 2,
• Boolean algebra and its application to combinational logic. Karnaugh maps for function minimisation. [3] 436-446, [4] 39-60, [5] Ch 3,
• Gate delays, timing diagrams, hazards. [4] 391-398
• Introduction to VHDL.

Sequential Logic and its Applications

• Number codes, for example, hexadecimal, BCD, ASCII. 2's complement. [1] Ch 2, [2] Ch 3, [3] 430-435, [4] Ch 10, [6] Ch 3,
• RS and JK flip-flops, latches and simple counters.[3] 412-419, [5] Ch 4,
• Synchronous and asynchronous circuits, counters and shift registers. Serial communication. [3] 446-452, [6] Ch 5
• State diagrams and design methods for a sequencer. [4] Ch 4, [5] Ch 6
• D to A techniques. Weighted resistor and R-2R ladder networks. Schmitt trigger inputs. [3] 522-523
• Logic circuits for arithmetic functions.[3] 442, [4] 17

Introduction to Microprocessors

• Introduction to the architecture of a simple microprocessor. [1] Ch 1,5, [2] 1-9, [4] Ch 1 and Ch 5, [7] 10-13
• Memory circuits, RAM and ROM. Address decoding, definitions of read/write and chip select signals. [2] Ch 12, [3] 455-460, [4] Ch 6, [6] Ch 2
• PIC Microprocessor programming. Programme Development, Registers. [1] Ch 7, Ch 8
• Programming examples based on PIC12F629/675 instruction set. Addressing modes. Implementation using simple machine code. Assembly code and higher level languages. [1] Ch 8,9

REFERENCES

1) BATES, M. PIC MICROCONTROLLERS: AN INTRODUCTION TO MICROELECTRONICS
(2) DOWSING, R.D., WOODHAMS, F.W.D. & MARSHALL, I. COMPUTERS FROM LOGIC TO ARCHITECTURE
(3) FLOYD, T.L. DIGITAL FUNDAMENTALS
(4) GIBSON, J.R. ELECTRONIC LOGIC CIRCUITS
(5) SMITH, R.J. & DORF, R.C. CIRCUITS, DEVICES AND SYSTEMS
(6) TINDER, R.F. ENGINEERING DIGITAL DESIGN

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 12/02/2026 17:07

Aims

The aims of the course are to:

• Familiarise students with combinational and sequential digital logic circuits, and the analogue-digital interface,
• Familiarise students with the hardware and basic operation of microprocessors, memory and the associated electronic circuits which are required to build microprocessor-based systems.
• Teach the engineering relevance and application of digital and microprocessor-based systems, give students the ability to design simple systems of this kind, and understand microprocessor operation at the assembly-code level.

Objectives

As specific objectives, by the end of the course students should be able to:

• Know the nomenclature and the representation of basic gates and digital electronic components (including shift registers, counters, latches, RAM and ROM ICs)
• Understand Boolean algebra, and be able to convert verbal descriptions of requirements into Boolean notation
• Understand the need to simplify logic functions or rearrange them to use specific gate types; be able to use Boolean algebra and Karnaugh maps (for up to 4 variables) to achieve these tasks; be able to use "don't care states" in K-maps.
• Know about logic "families", the electronic circuit implementation of logic gates, and the resulting engineering issues (voltage thresholds, noise margin, finite rise time and delay)
• Know about Schmitt inputs; understand static hazards and be able to detect them using K-maps and correct them
• Be familiar with standard number codes for representing data (two's complement notation, sign+magnitude, one's complement, Gray code, ASCII); be able to convert between binary, hex, octal and decimal.
• Understand the operation of logic circuits for addition, negation and subtraction of binary integers
• Be familiar with examples of elementary VHDL and understand why it is useful
• Understand the distinction between combinational and sequential logic and the role of sequential logic; be familiar with unclocked and clocked S-R latches, D-type and JK flip-flops.
• Understand the operation and use of synchronous and asynchronous counters and shift registers.
• Understand state diagrams and their role in sequential circuit design; be able to convert a problem statement into a state diagram; be able to convert a state diagram into a circuit design based on JK flip-flops
• Understand unused states, and be able to guard against errors due to them. Be able to carry out the complete design process, from problem statement to circuit design.
• Understand the operation of weighted resistor and R-2R ladder DAC circuits
• Understand the operation of Full Adder and Ripple Carry Circuits
• Understand ROM and RAM memory circuits, the function of their control, address and data pins, and their use in digital (including microprocessor) systems
• Understand the use of tri-state outputs and busses.
• Understand and be able to design address decoders, including partial address decoders for simple systems.
• Be familiar with the system architecture of a typical PIC microprocessor system, including the ALU, memory, I/O;understand how it can be used in practical applications.
• Be familiar with the internal architecture of a typical PIC microprocessor (the PIC12F629/675) and its instruction set, and understand how instruction execution occurs.
• Understand the features of typical instruction sets,and be able to use the full instruction set (from the tables in the electrical data book).
• Be able to write simple programs in assembler mnemonics, including conditional branches, and calculate their execution times in clock cycles; know about the relationship of higher level languages to assembly level code.
• Understand (in outline only) stacks, subroutines and the hardware reset function.

Content

Digital Fundamentals and Combinational Logic

• Introduction, revision of simple logic gates, overview of logic circuit families. [1] Ch 3, [3] 392-399, [4] 12
• Circuits for inverters and basic logic gates in NMOS and CMOS. [3] 409-410, [5] Ch 2,
• Boolean algebra and its application to combinational logic. Karnaugh maps for function minimisation. [3] 436-446, [4] 39-60, [5] Ch 3,
• Gate delays, timing diagrams, hazards. [4] 391-398
• Introduction to VHDL.

Sequential Logic and its Applications

• Number codes, for example, hexadecimal, BCD, ASCII. 2's complement. [1] Ch 2, [2] Ch 3, [3] 430-435, [4] Ch 10, [6] Ch 3,
• RS and JK flip-flops, latches and simple counters.[3] 412-419, [5] Ch 4,
• Synchronous and asynchronous circuits, counters and shift registers. Serial communication. [3] 446-452, [6] Ch 5
• State diagrams and design methods for a sequencer. [4] Ch 4, [5] Ch 6
• D to A techniques. Weighted resistor and R-2R ladder networks. Schmitt trigger inputs. [3] 522-523
• Logic circuits for arithmetic functions.[3] 442, [4] 17

Introduction to Microprocessors

• Introduction to the architecture of a simple microprocessor. [1] Ch 1,5, [2] 1-9, [4] Ch 1 and Ch 5, [7] 10-13
• Memory circuits, RAM and ROM. Address decoding, definitions of read/write and chip select signals. [2] Ch 12, [3] 455-460, [4] Ch 6, [6] Ch 2
• PIC Microprocessor programming. Programme Development, Registers. [1] Ch 7, Ch 8
• Programming examples based on PIC12F629/675 instruction set. Addressing modes. Implementation using simple machine code. Assembly code and higher level languages. [1] Ch 8,9

REFERENCES

1) BATES, M. PIC MICROCONTROLLERS: AN INTRODUCTION TO MICROELECTRONICS
(2) DOWSING, R.D., WOODHAMS, F.W.D. & MARSHALL, I. COMPUTERS FROM LOGIC TO ARCHITECTURE
(3) FLOYD, T.L. DIGITAL FUNDAMENTALS
(4) GIBSON, J.R. ELECTRONIC LOGIC CIRCUITS
(5) SMITH, R.J. & DORF, R.C. CIRCUITS, DEVICES AND SYSTEMS
(6) TINDER, R.F. ENGINEERING DIGITAL DESIGN

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 14/12/2023 15:49

Aims

The aims of the course are to:

• Familiarise students with combinational and sequential digital logic circuits, and the analogue-digital interface,
• Familiarise students with the hardware and basic operation of microprocessors, memory and the associated electronic circuits which are required to build microprocessor-based systems.
• Teach the engineering relevance and application of digital and microprocessor-based systems, give students the ability to design simple systems of this kind, and understand microprocessor operation at the assembly-code level.

Objectives

As specific objectives, by the end of the course students should be able to:

• Know the nomenclature and the representation of basic gates and digital electronic components (including shift registers, counters, latches, RAM and ROM ICs)
• Understand Boolean algebra, and be able to convert verbal descriptions of requirements into Boolean notation
• Understand the need to simplify logic functions or rearrange them to use specific gate types; be able to use Boolean algebra and Karnaugh maps (for up to 4 variables) to achieve these tasks; be able to use "don't care states" in K-maps.
• Know about logic "families", the electronic circuit implementation of logic gates, and the resulting engineering issues (voltage thresholds, noise margin, finite rise time and delay)
• Know about Schmitt inputs; understand static hazards and be able to detect them using K-maps and correct them
• Be familiar with standard number codes for representing data (two's complement notation, sign+magnitude, one's complement, Gray code, ASCII); be able to convert between binary, hex, octal and decimal.
• Understand the operation of logic circuits for addition, negation and subtraction of binary integers
• Be familiar with examples of elementary VHDL and understand why it is useful
• Understand the distinction between combinational and sequential logic and the role of sequential logic; be familiar with unclocked and clocked S-R latches, D-type and JK flip-flops.
• Understand the operation and use of synchronous and asynchronous counters and shift registers.
• Understand state diagrams and their role in sequential circuit design; be able to convert a problem statement into a state diagram; be able to convert a state diagram into a circuit design based on JK flip-flops
• Understand unused states, and be able to guard against errors due to them. Be able to carry out the complete design process, from problem statement to circuit design.
• Understand the operation of weighted resistor and R-2R ladder DAC circuits
• Understand the operation of Full Adder and Ripple Carry Circuits
• Understand ROM and RAM memory circuits, the function of their control, address and data pins, and their use in digital (including microprocessor) systems
• Understand the use of tri-state outputs and busses.
• Understand and be able to design address decoders, including partial address decoders for simple systems.
• Be familiar with the system architecture of a typical PIC microprocessor system, including the ALU, memory, I/O;understand how it can be used in practical applications.
• Be familiar with the internal architecture of a typical PIC microprocessor (the PIC12F629/675) and its instruction set, and understand how instruction execution occurs.
• Understand the features of typical instruction sets,and be able to use the full instruction set (from the tables in the electrical data book).
• Be able to write simple programs in assembler mnemonics, including conditional branches, and calculate their execution times in clock cycles; know about the relationship of higher level languages to assembly level code.
• Understand (in outline only) stacks, subroutines and the hardware reset function.

Content

Digital Fundamentals and Combinational Logic

• Introduction, revision of simple logic gates, overview of logic circuit families. [1] Ch 3, [3] 392-399, [4] 12
• Circuits for inverters and basic logic gates in NMOS and CMOS. [3] 409-410, [5] Ch 2,
• Boolean algebra and its application to combinational logic. Karnaugh maps for function minimisation. [3] 436-446, [4] 39-60, [5] Ch 3,
• Gate delays, timing diagrams, hazards. [4] 391-398
• Introduction to VHDL.

Sequential Logic and its Applications

• Number codes, for example, hexadecimal, BCD, ASCII. 2's complement. [1] Ch 2, [2] Ch 3, [3] 430-435, [4] Ch 10, [6] Ch 3,
• RS and JK flip-flops, latches and simple counters.[3] 412-419, [5] Ch 4,
• Synchronous and asynchronous circuits, counters and shift registers. Serial communication. [3] 446-452, [6] Ch 5
• State diagrams and design methods for a sequencer. [4] Ch 4, [5] Ch 6
• D to A techniques. Weighted resistor and R-2R ladder networks. Schmitt trigger inputs. [3] 522-523
• Logic circuits for arithmetic functions.[3] 442, [4] 17

Introduction to Microprocessors

• Introduction to the architecture of a simple microprocessor. [1] Ch 1,5, [2] 1-9, [4] Ch 1 and Ch 5, [7] 10-13
• Memory circuits, RAM and ROM. Address decoding, definitions of read/write and chip select signals. [2] Ch 12, [3] 455-460, [4] Ch 6, [6] Ch 2
• PIC Microprocessor programming. Programme Development, Registers. [1] Ch 7, Ch 8
• Programming examples based on PIC12F629/675 instruction set. Addressing modes. Implementation using simple machine code. Assembly code and higher level languages. [1] Ch 8,9

REFERENCES

1) BATES, M. PIC MICROCONTROLLERS: AN INTRODUCTION TO MICROELECTRONICS
(2) DOWSING, R.D., WOODHAMS, F.W.D. & MARSHALL, I. COMPUTERS FROM LOGIC TO ARCHITECTURE
(3) FLOYD, T.L. DIGITAL FUNDAMENTALS
(4) GIBSON, J.R. ELECTRONIC LOGIC CIRCUITS
(5) SMITH, R.J. & DORF, R.C. CIRCUITS, DEVICES AND SYSTEMS
(6) TINDER, R.F. ENGINEERING DIGITAL DESIGN

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 07/02/2022 13:50