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THE MANCHESTER METROPOLITAN UNIVERSITY
FACULTY OF SCIENCE AND ENGINEERING
DEPARTMENT OF ENGINEERING AND TECHNOLOGY
SESSION 2005/
Examination for the BEng (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING YEAR/STAGE TWO
UNIT 64EE2102 : ENGINEERING TOOLS AND TECHNIQUES
Tuesday 9 May 2006
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 THREE questions from SECTION B.
Show clearly the method of calculation used to arrive at final results.
No programmable or graphical calculators may be used.
No mathematical formula booklets are allowed.
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SECTION A
Answer ALL 12 questions in this section. This section carries 55% of this examination mark.
A.1.
(a) Determine
dy dx
given that y = xx
(b) Evaluate the integral: y = ∫ xe −^2 xdx
[4]
A.2.
Evaluate the partial derivatives f x
and
(^2) f
y x
for the function
f ( , x y ) = 5 x^3 − 4 xy + 2 xy^2 − 8 y at the point (1,1) [4] A.3.
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 don’t solve them.
Figure QA.
12 V
6 V
I 1^2
I
I 3
[4]
A.4.
Refer to the circuit of Figure QA.4. Determine value of the load impedance ZL which, when connected across the terminals a-b, would draw maximum power from the circuit. a
Figure QA.
V ∠ 0 0 12 Ω
4 Ω (5 + j 8)Ω
b
ZL
[4]
A.8.
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. Draw a sketch of the characteristic, of the device D, in your answer booklet to show your method of calculation.
i − v
i − v
iD ( mA )
vD ( V )
D
6 V
Figure QA.8(a) Figure QA.8(b) [4]
A.9.
A voltage signal has the finite Fourier series representation:
v t ( ) = 10 + 8 cos (314.16 t + 30 ) o +6sin(2 100 ) π t
(a) Sketch to scale 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.10.
(a) Write down the two linear equations characterising a two-port network in terms of the h-parameters; (b) The following test on a two-port network is made:
Port 2 short-circuited : V 1 = 25 mV , I 1 = 20 μ A , I 2 = 1 mA
determine the parameters h 11 and h 21 ; and (c) State what other test would be required to determine the parameters and.
h 12 h 22 [5]
A.11.
A two-port network when represented by its ABCD parameters has A = D = 0.8 ∠ 30 o ; B = 100 ∠ 60 o Ω ; and C = 5 10 x −^3 ∠ 90 oS. Determine: (a) The characteristic impedance of the network; and (b) the output current when the input voltage is and the load is a pure resistance of 14
120 ∠ 0 o^ V 1 Ω [6]
A.12.
A relay consists of an inductor whose resistance is R ( Ω) and whose inductance is L ( H ). The operation of the relay is modelled by the initial value problem (IVP): ( ) ( ) ( ) (0) 0
di t Ri t L Vu t i dt
Making use of Laplace transform, solve this IVP to determine an expression for the relay current i t ( ) for t > 0. [7]
____________________________End of Section A _____________________
Q UESTION B.2 [15 MARKS]
A second-order system is modelled by the second-order constant coefficients linear differential equation 2 2 3 2 2 5
d y dy y x dx dx
(a) Write down the auxiliary equation associated with the homogeneous differential equation; [1] (b) Solve the auxiliary equation and hence determine the complementary function yC ( ) x i.e. the general solution of the homogeneous differential equation; [2] (c) Find an appropriate particular integral yP ( ) x that satisfies the non- homogeneous differential equation; [3] (d) Write down the general solution that satisfies the non-homogeneous differential equation; [4] (e) Determine the particular solution of the non-homogenous differential equation given that the differential equation is subject to the initial conditions
(0) 0 , (0) 0
dy y dx
[5]
Q UESTION B.3 [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. [12]
Q UESTION B.4:[15 M ARKS]
(a) Show that the trigonometric Fourier series for the voltage signal shown in Figure QB.4 is:
1
( ) sin(2 1) (^2) n (2 1)
A A
v t n t
π n
∞
=
[10]
(b) Find the percentage third harmonic distortion; and [2] (c) Calculate the ratio of the normalised average power in the fundamental component of the signal to its total normalised average power. [3]
Figure QB.
ω t
v t ( )
π
A
Q UESTION B.5:[15 M ARKS]
Consider bandlimited finite energy signal m t ( )whose Fourier transform is M ( f )and whose spectral pattern is shown in Figure QB5. The signal is
multiplied by a sinusoid vc ( ) t = cos 2 π f tc , where fc >> fm.
(a) Determine, from first principles, the Fourier transform of the modulation
signal v t ( ) = m t ( ) cos 2 π f tc ; and
[10]
(b) Draw the amplitude spectrum of the modulation signal (^) v t ( )and comment on the effect of the modulation on the spectrum of m t ( ). M ( f )^ [5]
M (0)
Figure QB.
f − fm f m
END
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