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REFERENCES
1.Mathematics for Electrical Technicians -Levels 4 and 5 by J.O.Bird and A.J.C. May
2.Basic Distribution and Significance Tables
3.Statistics for Engineers by A Greer, published by Stanley Thornes (Publishers)Ltd
4.Mathematics for Technicians-New Level III by A.Greer and G.W. Taylor
5.Mathematics for Technician Engineers Levels 4 and 5 by George E.Dyball
56
Tutorial 1
1. State the order of the following differential equations and say if they are linear
or non-linear.
(a)
d 4v
dv
4 4  2  v3  0
dt
dt
(c)
d 2v
 dv 
 2   v  4
2
dt
 dt 
(b)
2
(d)
d 4v
dv
4 4  2  v  t3
dt
dt
1
d2y
 2
 dy  2 dx
  x
 dx 
Answers (a) 4th order non-linear (b) 4th order linear (c) 2nd order non-linear
(d) 2nd order non-linear
2. In each case substitute the given solution into the given differential equation to
verify it is indeed a solution
(a) solution v  10sin(t ) , equation
(b) solution v  5e  kt , equation
d 2v
  2v
dt 2
dv
  kv
dt
(3) Given a general solution is i  10  ce 500t find the particular solutions
satisfying
(ii) i ' (0)  5000
(i) i (0)  0
Answers (a) i  10(1  e500t ) (b)
(4)
Solve differential equations of the form
i  10(1  e500t )
dy
 f ( x)
dx
Find the general solution of
(i)
(iii)
dy
 2x4
dx
dy
 5 x  sin( x)
dx
dy
 x  2 given that y =3 when
Find the particular solution of
dx
x=2
(ii)
57
Answers (i) y 
(5)
2 5
x C
5
(ii) y 
5 2
x2
x  cos( x)  C (iii) y  2 x   1
2
2
Solve differential equations of the form
dy
 f ( y)
dx
Find the general solution of
dy
 3  y2
dx
dy
 1  0 given y = 4 when x = 1/3
(iii) Find the particular solution of y
dx
(i)
dy
 3 2y
dx
Answers (i) y 
(iii)
(ii) y
1
1
ln(3  2 y )  C (ii)  ln(3  y 2 )  x  C
2
2
2 3
y  x5
3
58
Tutorial 1 (Continued)
Solve the differential equations given on pages 236 and 237 of Bird and May (Ref:1)
59
Tutorial 2
1. Solve the following first order differential equations by separation of variables:
(a) 20
di
 100i  0
dt
given i = 3 when t = 0
(b) 20
di
 100i  6
dt
given i = 0 when t = 0
(c) 100q  4
100q  4
dq
0
dt
dq
8
dt
given q = 0.1 when t = 0
given q = 0 when t = 0
2. Solve 1(b) and 1(d) by the method of Laplace Transforms using the special case
result that f (t )  0 when t = 0 (as it is in these cases)
L { f ' (t )}  s L { f (t )}
3. Solve 1(a) and 1(c) by the method of Laplace Transforms using the general
result
that
L { f ' (t )}  s L { f (t )}  f (0)
where
f (0) is the value of f(t) when t = 0
4. Referring to page 17 of the Course notes. A coil has L = 100 mH, R = 10 ohms
and the dc supply E = 10 V. (i) What is the time constant? How long does it take
the current to reach (ii) 50% (iii) 90% and (iv) 99% of its final value?
Answers:
1(a)
3e-5t
1(b)
0.06(1 - e-5t)
1(c)
0.1 e-25t
1(d)
0.08(1 - e-25t)
4 (i) L/R = 10 ms (ii) 6.9 ms (iii) 23 ms (iv) 46 ms (i.e. about 5 time
constants)
60
Tutorial 3
Find the Laplace Transform of the following:
e 2t ,
2t 2 ,
2t ,
2e3t
1.
2,
2.
1  et ,
3.
et  e2t , et  e3t , e 2t  e 4t .
4.
2sin(t ) , 3sin(2t ) , 4 cos(3t )
5.
et sin(2t ) , e3t sin(t ) , e2t sin(3t )
2(1  et ) ,
3(1  e2t ) .
Find the Inverse Laplace Transform of:
6.
1
,
2s  4
7.
1
1
1
,
,
(2s  1)( s  2) 2(2s  3)(3s  2) (2s  3)2
8.
1
,
s 1
1
,
s 2
9.
1
,
s (2 s 2  1)
1
1
,
2
( s  2)(3s  1) ( s  2) 2  32
10.
1
,
s  4s  7
1
1
,
2
2 s  8s  16 2 s  8s  8
2
2
1
,
3s  4
2
1
2s(2s  1)
1
2s  1
2
2
61
Tutorial 3
Answers
1
2 2 4
1
2
, 2, 3,
,
s s s s2 s3
2
1
2
6
,
,
s( s  1) s( s  1) s( s  2)
3
1
1
1
1
1
1

,

,

s 1 s  2 s 1 s  3 s  2 s  4
4
2
6
4s
, 2
, 2
s 1 s  4 s  9
5
2
1
3
,
,
2
2
( s  1)  4 ( s  3)  1 ( s  2) 2  9
6
e
2 t
2
,
e
4 t
3
3
t
2
,
(1  e )
2
t
7
(e 2  e  2 t ) (e
,
3
sin 2t
,
2
8
sin(t ),
9
1  cos
2 t
3
e
10
sin
3t
2
3t
) te 2
,
4
t
2
2
t
,
2
e 2 t
1
t
1
sin(
  ) where   tan 1 (
),
1 
4 3
13
3
2 3
e 2 t sin( 32t )
32
10
e 2 t sin( 3t ) e 2 t sin(2t ) te 2 t
,
,
4
2
3
62
2
Tutorial 4
1
(a)
Find the general solutions to the following differential equations by the
method of separation of variables:(i)
(ii)
di
 5i  0
dt
dv 2
v 0
dt
6
(b)
The charge q coulomb on a capacitor of capacitance C farad at a time t
after the start of discharge through a 1 k resistor is given by the differential
equation:
q
dq
 1000
 0.
C
dt
If the charge is 1 microcoulomb at time t = 0 and 0.2 microcoulomb at time
t = 1 millisecond determine
and
2
(a)
(i) the particular solution for q,
(ii) the capacitance C,
(iii) the current flowing at t = 1.5 millisecond.
Use the Tables of Laplace Transforms to determine:(i) L( 3e-2t) ,
(ii) L [2e-3tsin(4t)],
(iii) q given that L(q) = 4
+ 2
s (2s + 7)
s
and
(b)
(iv) the Inverse Transform of
1
.
2
(2s + 8s + 14)
The current, i, in a particular series R, L circuit supplied by a voltage
source ,v(t), is given by the equation :20
di
 100i  v(t )
dt
At the instant the voltage source is connected to the circuit the current is zero.
Use the method of Laplace Transforms to determine the particular solution
for i,
and
(i) if v(t) = 6
(ii) if v(t) = 6 - 3 sin (2t) .
63
Tutorial 4 Answers
1(a)
1(b)
2(a)
2(b)
(i)
i=Ae-0.833 t
(ii)
v  3(c  t 
6
1
3
0.001t
C
(i)
q  10 e
(ii)
C = 0.62 x 10-6 F
(iii)
i = - 0.144 mA
(i)
3
s2
(ii)
8
( s  3) 2  16
(iii)
0.571(1  e 3.5t )  2
(iv)
0.289e 2t sin( 1.732t )
(i)
0.06(1  e 5t )
0.06(1  e 5t )  0.3(
(ii)
where   tan 1
e 5t sin( 2t   )

)
29
2 29
2
5
64
TUTORIAL 5
65
TUTORIAL 5 continued
66
TUTORIAL 6
Partial Fractions
67
TUTORIAL 7
1
Voltage Transfer Function
Determine the Voltage Transfer Function for each of the following circuits:
(a)
(b)
2
For ‘circuit (a)’ of Question 1 obtain an expression for Vout as a function of
time given that Vin, applied at t=0, is (i) 6 volt (ii) (6  2e3t ) volt.
3
Repeat 2(i) for ‘circuit (b)’
4
Given the following VTFs find Vout as a function of time for Vin=3sin(2t)
(a) VTF =1/3
(b) VTF=1/(s+2)
Answers
1(a)
6.25
s  10
(b)
2(circuit (a)) (i) 3.75(1  e 10t )
3 (circuit (b)) (i)
104
s 2  100s  104
6  4 3e 50t sin( 50 3t   )
4(a) sin(2t)
3
3
(b) e 2t 
sin( 2t   ) where   tan 1 1
4
2 2
or
12.5 3t
( e  e 10t )
7
where   tan 1 3
3(ii) 3.75(1  e 10t ) 
3 2 t 3
3
e  sin( 2t ) - cos(2t)
4
4
4
68
TUTORIAL 8
De Moivre’s Theorem
69
TUTORIAL 9 Matrices (Reference 4 Chapter12)
70
TUTORIAL 9 Continued
71
TUTORIAL 10
Normal Distribution
A population of resistors which are normally distributed has a mean of 1 k and a
standard deviation of 20 .
1. Find the u values corresponding to the following resistances:
(i) 1.04 k
(ii) 1.015 k
(iii) 0.99 k
(iv) 0.97 k
(v) 0.985 k
(vi) 1.01 k

 Answers: (i) 2 (ii) 0.75 (iii) -0.5 (iv) -1.5 (v) -0.75 (vi) 0.5

Use Normal Distribution Tables (Table 3) to find the fraction (or percentage) of
resistors in this population having a resistance which is:(i) greater than 1.04 k
(ii) less than 1.015 k
(iii) greater than 0.99 k
(iv) less than 0.97 k
(v) between 0.985 k and 1.015 k
(vi) within 1% of 1 k
(vii) in the range “mean  1 standard deviation”
(viii) in the range “mean  2 standard deviations”
(ix) in the range “mean  3 standard deviations”

 Answers: (i) 0.02275 (2.275%) (ii) 0.7734 (iii) 0.6195 (iv) 0.0668 (v) 0.5468
(vi) 0.3830 (vii) 0.6826 (viii) 0.9545 (ix) 0.9973

 If the population size is 5000 resistors estimate how manyhave a resistance:(i) greater than 1.02 k
(ii) less than 1.05 k
(iii) between 0.995 k and 1.01 k
(iv) within 0.5% of 1 k
(v) greater than 0.98 k


Answers: (i) 794 (ii) 4969 (iii) 1451 (iv) 987 (v) 4207
72
TUTORIAL 11
Normal Distribution (Reference 3 Exercise 6)
Answers
Q1:
Q2:
Q3:
Q4:
Q5:
Q6:
Q11:
-1; -0.5; 0.25; 1.5
0.4192; 0.2967; 0.5694; 0.0451; 0.0526
0.00798; 0.1949; 0.0823; 0.015
0.02275; 0.0062; 0.7745
1771
115; 673
4.75%
73
TUTORIAL 12
Normal Distribution: Extracts from past end tests
74
TUTORIAL 13
1.
Vectors
Given the position vectors
r1  3i  2 j  k , r2  2i  4 j  3k and r3  i  2 j  2 k
find the vectors
(a) r3  r2
(b) r1  r2 + r3
(c) 2r1 - 3r2  5 r3
2.
Find the magnitude of the vectors shown in 1(a),(b) and (c) above.
3.
Find the unit vectors parallel to the vectors shown at 1(a),(b) and (c) above
and check that they are of unit magnitude.
Answers: 1(a)
(b)
(c)
-3i+6j+5k
4i –4j
5i-2j+k
2 (a)
(b)
(c)
8.37
5.66
5.48
3(a)
3
6
5
i
j
k = -0.36 i + 0.72 j + 0.60 k
70
70
70
(b)
(c)
0.70i – 0.70 j
0.91i – 0.37 j + 0.18 k
75
TUTORIAL 14
Dot and Cross Products
1. Evaluate each of the following.
(a) i . i
(b) i . k
(c) k . j
(d) j . ( 2i - 3j + k )
(e) ( 2i - j) . ( 3i + k)
2. If A = A1i +A2j + A3k, show that A =
3. Find the angle between
A . A  A12  A22  A23
A = 2i + 2 j - k and B = 6i - 3j + 2k
4. Evaluate each of the following:
(a) i  j
(b) j  k
(c) k  i
(d) k  j
(e) i  i
(f) j  j
(g) i  k
(h) 2 j  3k
(i) 3i  (-2k )
(i) 2 j  i - 3k
5. Given A = 2i - 3j - k and B = i + 4 j - 2k ,
Find
Answers.
(a) A  B
(b) B  A
(c) (A + B)  (A - B)
1(a) 1 (b) 0 (c) 0 (d) -3
(e) 6
3 79o
4(a) k
(f) 0
(b) i
(c) j
(g) -j (h) 6i
(d) -i
(e) 0
(i) 6j (j) -5k
5 (a) 10i + 3j + 11k
(b) -10i - 3j - 11k
(c) - 20i - 6j - 22k
76
Questions from past Phase Tests 1 and 2
1(a)
(b)
Find the general solutions to the following differential equations by the
method of separation of variables:(i)
2dq/dt + 10q = 0
(ii)
3dv/dt - 4v 0.5 = 0
The current, i ampere, through an inductor, L henry, and a resistor,
100 ohm, connected in series at a time, t second, after a voltage source for the
circuit has been short-circuited is given by the differential equation:
L di/dt + 100 i = 0 .
If the current is 0.5 ampere at time t=0 and 0.2 ampere at time t=1 millisecond
determine
(i) the particular solution for i ,
and
(ii) the inductance of the inductor.
2(a)
Use the Tables of Laplace Transforms to determine:(i) the Laplace Transform of 2e 3t ,
(ii) L ( 2e 4 t sin(3t ) ),
2
3s(2 s  3)
1
(iv) v, given that L (v) = 2
3s  12 s  39
(iii) the Inverse Transform of
and
(b)
Given that
L [f(t)] =
1
( s  1)( s  2)( s  4)
, find f(t) by using
(i) partial fractions and Laplace Transform Table A
(ii) Laplace Transform Table B
3
The voltage, v, across the capacitor in an L-C-R circuit at time t is given by
the differential equation:
d 2v
dv
 8  25v  4  3et
2
dt
dt
Solve this equation for v at any time t given that v=0 and dv/dt=0 both at t=0.
4
(a)
Find the Voltage Transfer Function for a network in which
77
di
 7i
dt
di
v in  4  10i  3q
dt
v out  2
(b)
The Voltage Transfer Function for a given network is 1/( s + 2)
If vin = 3 sin 4t , find vout
5
(a)
Express 1 + j4 in (i) polar and (ii) exponential form.
(b)
Find
z given that
z3 = 1 + j4
.
6
(c)
Find x given that
(a)
For the matrix A =
cosh x = 4
 3  1
4  5


find
(i) A , and (ii) A-1
(b) The currents, I1 and I2 in two branches of a network are related by:3 I1 - I2 = 2 + j7
and 4 I1 - 5 I2 = -1 + j24
Determine the values of I1 and I2 using the method of matrices.
(c)
Solve the equations
4x + 10y = 17
8x + y = 15
using the Gauss Seidel method of iteration.(Work to two places of decimals.)
7
The resistances of resistors produced by an automatic process are distributed
normally with a mean of 2.1 k and a standard deviation of 0.05 k .
Estimate (i) the fraction of resistors with a resistance less than 2.02 k,
(ii) the probability that one resistor selected at random has a
resistance greater than 2.2 k,
(iii) the number of resistors in a batch of 2000 that have a resistance
greater than 2.0 k.
Past End Test questions
78
Q1
(a)
Find the general solutions to the following differential equations by the
method of separation of variables:(i)
5
dv
dt
+ 2v = 0
(ii)
2.5
di
dt
- 0.8 i 3 = 0
[6 marks]
(b)
Use the Tables of Laplace Transforms to determine:(i)
( 3e-0.4t) ,
(ii)
[4 e-2t sin(4t)],
4
10

s(2 s  3) s
1
(iv) the Inverse Transform of
2
4s  8s  12
(iii) q given that (q) =
and
[10 marks]
(c)
An R-L series circuit is supplied with 10 V direct voltage which is switched on at
time t =0. The current is given by the differential equation:
L di/dt + 50 i = 10 .
If the current is zero at time t =0 and 0.1 ampere at time t =1 millisecond determine
(i) the particular solution for i ,
(ii) the inductance of the inductor,
and
(iii) the time for the current to reach 99% of its steady state value.
[9 marks]
79
Q2
(a)
An L-C-R series circuit is supplied with 10 V direct voltage which is switched on
at time t =0. The voltage, v, across the capacitor is given by the differential equation:
3
d 2v
dv
 15v  10 .
2  12
dt
dt
(i) Use Laplace Transforms to find the particular solution for v
dv
are zero when t = 0.
dt





given that both v and





marks]

(ii) Find the new Laplace Transform of v if ,instead, the capacitor is already charged to 1 volt
when the supply is switched on.
(You are not required to write the new particular solution)
(b)
[5 marks]
For a particular circuit the input voltage, vin (t), and output voltage, vout(t), are
given by the following differential equations:
v out (t )  v  8
dv
dt
vin (t )  v  24
dv
d 2v
2 2
dt
dt
Find the expression in the s-domain for the Voltage Transfer Function.
[5 marks]
(c)
The Voltage Transfer Function of a network is
2
.
s3
Find vout(t) if vin(t) = 6  2 sin(3t )
[5 marks]
80
Q3
(a)
The Characteristic Impedance, z0, of a network is given by :
z02 = 5 + j3
Calculate z0. Express your answer in polar, rectangular and exponential forms.
[5 marks]
(b)
Find, without using the hyperbolic functions on your calculator, the
real value of x when cosh x = 2.
[5 marks]
Q4
Inverting amplifiers are to be constructed which include two resistors, R1 and R2 .The resistors to be
used are each distributed normally with mean and standard deviation as shown below:
R1
Mean = 1 k
Standard deviation = 50 
R2
Mean = 10 k
Standard deviation = 200 
If there are 4000 of each resistor available estimate

(a) the fraction of R1 resistors with a resistance greater than 970 ,
(b) the probability of an R2 resistor being selected with a resistance greater
than 10.25 k
(c) the number of R1 resistors having a resistance between 940  and 1.1 k,
(d) the probability that the gain,
R2
, of the first amplifier constructed will be
R1
(i) more than 10 if R1 has been selected and is 970 
and

(ii) less than 10 if, instead, R2 has been selected and is 10.05 k









marks]

81
Q5
(a)
For the matrix shown find
(i) the value of the determinant
and
(ii) the inverse.
5 3 


 4  1
[4 marks]
(b)
The currents I1 and I2 in two branches of a network are given by :
5 I1 + 3 I2 = 13 + j
and
4 I1 - I2 = 7 - j6
Find the values of I1 and I2 , by the method of matrices.
[6 marks]
(c)
Use the Gauss-Seidel method of iteration to solve the equations:-
5V1 + V2 = 12.5
2V1 + 10V2 = 19.4
Show your working to 2 places of decimals.
[5 marks]
Q6
Given two position vectors, A  2i  4 j  5k and B  i  2 j  3k find
(a) the resultant of these vectors,
(b) a unit vector parallel to this resultant,

(c) the vector AB ,
(d) the angle between A and B .
[10 marks]
82