IEA-2004 Engineering Analysis: Linear Systems

Semester 2 Examinations

IEA-2004 Engineering Analysis:

Linear Systems

Description

This exam is being set as an assignment. Youmay use all resources that have beenmade available to

you.

You will need to complete all questions in the paper.

Total number of marks: 50.

Issue date: Wednesday 29th April 2020, 09:00.

Due date: Friday 22th May 2020, 23:59.

Plagiarism Unfair Practice

Plagiarised work will be given amark of zero. Remember when you submit you agree to the standard

agreement:

This piece ofwork is a result ofmyownwork exceptwhere it is a group assignment forwhich approved

collaboration has been granted. Material from the work of others (from a book, a journal or theWeb)

used in this assignment has been acknowledged and quotations and paraphrasing suitably indicated.

I appreciate that to imply that such work is mine, could lead to a nil mark, failing the module or being

excluded from the University. I also testify that no substantial part of this work has been previously

submitted for assessment.

Late Submission

There is no late submission possible for this assessment. Any work received after the deadline,

without accepted special circumstances, will not be marked.

Extensions

Extensions will only be granted with the express recommendation of the student’s Personal Tutor.

Assessors do not need the full details of any special circumstances. Requesting an extensions is not

a guarantee that it will automatically be granted.

Submission Procedure

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draw graphs, diagrams, or write outmathematical equations, do so on paper and scan or take a photo

of the additional work and include it in your document.

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Semester 2 Examinations 1

IEA2004: Engineering Analysis: Linear Systems

1. The Laplace transform and its applications

a) Show from first principles that the Laplace transform of a one-sided sinusoidal signal is given by

the expression: ℒ{cos('()*(()} = --! + '!

with region of convergence Re{s} > 0. [5]

b) Use the first shift theorem: ℒ{/"#$0(()} = 1(- + 2)

where: 1(-) = 3 0(()/"%$4('

to find the inverse Laplace transform of: 5(-) = - + 4-! + 8- + 25

[5]

2. The Laplace transform and its applications

a) Use the Laplace transform to solve the linear constant coefficient differential equation: 4!:4(! + 74:4( + 10:(() = 40*(()

where H(t ) denotes the Heaviside step function, with initial conditions: :(0) = 8 :′(0) = 4:4(B$(' = −17

[10]

3. The Laplace transform and its applications – Extension question

This question will lead you through the derivation of a fourth-order low pass filter with a Butterworth

or “maximally flat” frequency response. Figure 3.1 shows a circuit diagram for a Sallen and Key low

pass filter, where the input to the filter is labelled vin and the output is labelled vout. Note that the

operational amplifier is connected as a unity gain buffer. This will be the building block from which

your filter will be assembled.

Figure 3.1 – Sallen and Key low pass filter

a) Derive an expression for the s-domain transfer function of the filter. [10]

The Butterworth, or “maximally flat” low pass frequency response is well-known in electronic

engineering. It is well-documented in the literature, and useful online sources are available. For

example: the Wikipedia entry https://en.wikipedia.org/wiki/Butterworth_filter in particular the

subsection

https://en.wikipedia.org/wiki/Butterworth_filter#Normalized_Butterworth_polynomials

describe the main mathematical features of the transfer function.

b) With reference to the web resources above, or any other relevant source, find the transfer

function of a fourth order Butterworth filter with corner frequency 5kHz. Most sources will

derive the pole locations for a normalized corner frequency of 1rad.s-1. You will need to find the

pole positions for that normalized filter, and then scale the frequency axis accordingly, to

change the corner frequency to 5kHz. You should also note that the denominator of the transfer

function must be factorized into two quadratic factors, each of which has a pair of complex

conjugate poles. Each of those factors will be implemented by a Sallen and Key circuit. [10]

c) Design a fourth order Butterworth low pass filter with corner frequency 5kHz by cascading two

Sallen and Key low pass filter sections, with their pole positions determined by your answer to

b) above. Keep your resistor values in the range 500W – 500kW. [10]