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THE MANCHESTER METROPOLITAN UNIVERSITY
FACULTY OF SCIENCE AND ENGINEERING
DEPARTMENT OF ENGINEERING AND TECHNOLOGY
SESSION 2004/
Examination for the BEng (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING (FULL-TIME/PART TIME). YEAR/STAGE TWO
UNIT 64EE2102 : ENGINEERING TOOLS AND TECHNIQUES
Tuesday 26 April 2005
2.00 pm to 5.00 pm
Instructions to Candidates
This examination consists of TWO sections: SECTION A and SECTION B. Answer ALL questions in SECTION A and only FOUR questions from SECTION B. No programmable or graphical calculators may be used. No mathematical formula booklets are allowed.
S120 25/08/
SECTION A
Answer ALL 10 questions in this section. This section carries 40 % of this examination’s mark, and you are advised not to spend more than 60 minutes on it.
A.
[a] Determine
dy dx
given that y = 5ln( ) cos(2 ) x x
[b] Determine the integral: (^2) (1 2 )
x dx
∫ + x
[2]
A.
A periodic half-wave rectified sine wave is shown in Figure QA.2. Derive an expression for the average value of this waveform in terms of its maximum voltage value Vm.
v ( ω t )
Vm
ωt
Figure QA. [2]
A.
Refer to the circuit of Figure QA.3, and write down, in matrix form[ R ] [ I ] = [ V ], the
three mesh equations in I 1 (^) , I (^) 2 , and I (^) 3 , BUT do not solve them.
10 V
5 V
I 1 I^ 2 I 3
Figure QA.3 (^) [3]
A.
The non-linear device D, shown in the circuit of Figure QA.8(a), has the characteristic shown in Figure QA.8(b). Determine the voltage across, and the
current through, the device D. Make a sketch of the
i − v i − v characteristic, of the
device D, in your answer booklet to show your method of calculation.
[5]
i (^) D ( mA )
D^ ( V^ )
5 V
v (^) D V
i D
D
v
Figure QA.8(a) 0.5^1 1.5^2
Figure QA.8(b)
A. A voltage signal has the finite Fourier series representation:
v t ( ) = 4 + 10 cos(62.831 t + 30 ) o + 6sin(2 π 20 ) t −8cos 2π 40 t
[a]Plot the double-sided amplitude line spectrum of the signal; and [b]use Parseval’s theorem to determine the average power in the signal. [5]
A.
The step response of an R-L circuit is modelled by the initial value problem (IVP): ( ) ( ) ( ) (0) 0
di t Ri t L Vu t i dt
Where R is the resistance of the circuit in Ω , L is the inductance in H, is a unit
step function, and V is a dc forcing function in volts. Making use of Laplace transform, solve this IVP to determine the circuit current for.
u t ( )
i t ( ) t > 0 [6]
____________________________End of Section A _____________________
SECTION B
Answer only FOUR questions from this section. This section carries 60 % of this examination’s mark, and you are expected to complete it in 120 minutes.
Q UESTION B.1 [15 MARKS] In a displacement measuring system, a signal x, representing the displacement to be measured, is produced by a transducer such that 0 ≤ x ≤ 1. The signal is then amplified
by an amplifier whose output y is defined by the function f ( ) x :
10 ( ) , [0,1] 2
y f x x x
[a] Sketch a graph of the function f ( ), x x ∈ [0,1]. [1] [b] Select a suitable point as the centre of a linear Taylor’s polynomial that can approximate the function (^) f ( ) x over the interval x ∈ [0,1]. [1] [c] Determine the linear Taylor’s polynomial in [b] to three decimal places. [5] [d] Use the above approximating polynomial to calculate an estimate of and calculate the actual absolute error in this estimate.
y (0.65)
[2]
[e] Calculate an estimate of the upper error bound in approximating y (0.65). [4] [f] Compare this error bound with the actual error in the approximate value of y (0.65). [2]
Q UESTION B.2 [15 MARKS] The mesh equations of an electric network are given by the system of equations:
1 2 3 1 2 3 1 2 3
I I I A
I I I B
I I I C
[a] Write the system in matrix form; and [3] [b] use the Gaussian Elimination method to determine the unknown currents. No credit will be given for any other method of solution. [12]
Q UESTION B.5 [10 MARKS]
Consider the signal shown in Figure QB.5:
[a] Determine the amplitude line spectrum of the signal using the exponential Fourier series.
c nf ( (^) o )
[10]
[b] Sketch the amplitude spectrum c nf ( (^) o ) as a function of frequency assuming that
To = 4 τ and showing at least six components on either side of f = 0.
[5]
v t ( )
A
− T (^) o 2
− τ To
t
Figure QB.5:
S120 25/08/2005 END
