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.
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without accepted special circumstances, will not be marked.
Extensions
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a guarantee that it will automatically be granted.
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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]

vx
v+