Download Engineering Tools and Techniques-2006 2007 Exam-Electrical Engineering and more Exams Electrical Engineering in PDF only on Docsity!
THE MANCHESTER METROPOLITAN UNIVERSITY
FACULTY OF SCIENCE AND ENGINEERING
DEPARTMENT OF ENGINEERING AND TECHNOLOGY
SESSION 2006/
Examination for the BEng (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING YEAR/STAGE TWO
UNIT 64EE2102: ENGINEERING TOOLS AND TECHNIQUES
Thursday 26th^ April 2007
9:30 am to 12:30 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.
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.
25/07/2007 S
SECTION A
Answer ALL questions in this section.
This section carries 40% of the total mark of this examination.
A.1. (a) Determine
dy dx
given that y = x^2 ln(sin( x ))
(b) Evaluate the integral: y = ∫ 5 x e^5 xdx
[4]
A.2. Determine the partial derivatives
f x
f y
and
(^2) f
y x
of the function
f ( x , y ) = 2 x^4 + 2 xy + xy^2 + 5 y^2. [4]
A.3. Refer to the circuit of Figure QA.3, and write down, in matrix form[ Z ][ I ] = [ V ],
the three mesh equations in I 1 (^) , I 2 (^) , and I 3 , BUT DO NOT solve them.
[4]
Figure QA.
10 ∠ 10 0 j^^5 Ω
2 Ω −^ j^8 Ω
10 ∠ 90 o
I 1 I 2
3 Ω^2 Ω
I 3
− j 3 Ω
A.7. (a) Write down the two linear equations characterising a two-port network in terms of the h-parameter;
(b) The following test on a two-port network is made:
Port 2 short-circuited : V 1 = 100 mV , I 1 = 50 μ A , I 2 = 2 mA
Determine the parameters h 11 and h 21 ; and
(c) State what other test would be required to determine the parameters h 12 and h 22. [4]
A.8. Show, from first principles, that the Laplace transform of the function
f ( ) t = sin( ω t )is given by
F s ( ) (^) s 2 2
[4]
A.9. The characteristics of a non-linear circuit element can be modelled by
the function:
( i − v )
i = 3 v 2 v ∈[0,1]
(a) Find the linear Taylor polynomial approximation of this function around v =0.5; and
(b) Use it to obtain an approximation of and compare the approximation with the exact value of.
i (0.6) i (0.6 ) [4]
A.10. A numerical method was used to solve an initial value problem of the form y ′ = f ( x , y ). Three points of the discrete solution are: (0.1,2.2), (0.2,2.44), and (0.3,2.724). It is required to obtain an approximate continuous solution to the problem by fitting a quadratic polynomial to these points of the form 2 y = ao + a x 1 + a x 2
Write down, in matrix form, the system of equations -but DO NOT solve it- that must be solved in order to determine the coefficients of the above polynomial. [4]
____________________________End of Section A _____________________
B.3. The mesh equations of an electric network are given by the system of equations:
1 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]
B.4. (a) Show that the trigonometric Fourier series for the voltage signal
v ( ω t ) shown in Figure QB.4 is:
1
( ) sin(2 1) n (2^ 1)
A
v t n t n
∞
=
[12]
(b) Determine the ratio of the normalised average power contained in the third harmonic to the normalised average power in the fundamental component of the signal. [3]
Figure QB.
ω t
v ( ω t )
π
A
-A
25/07/2007 S
B.5. Consider the rectangular pulse shown in Figure QB.5, which is normally expressed as:
( ) ( )
t v t A
(a) Show that its Fourier transform is V ( f ) = A τ sinc( f τ), AND make a
sketch of V ( f ). [10]
(b) Explain what is meant by the terms Spectral Width and Reciprocal Spreading. [5]
Figure QB.
t
v t ( )
A
END
