Bird J. Electrical Circuit Theory and Technology
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Assignment 14
This assignment covers the material contained in chapters 42
to 45.
The marks for each question are shown in brackets at the end of
each question.
1 A filter section is to have a characteristic impedance at zero freque-
ncy of 720 and a cut-off frequency of 2MHz. To meet these
requirements, design (a) a low-pass T section filter, and (b) a low-
pass section filter. (8)
2 A filter is required to pass all frequencies above 50kHz and to have a
nominal impedance of 620 . Design (a) a high-pass T section filter,
and (b) a high-pass section filter to meet these requirements.
(8)
3 Design (a) a suitable ‘m-derived’ T section, and (b) a suitable ‘m-
derived’ section having a cut-off frequency of 50 kHz, a nominal
impedance of 600 and a frequency of infinite attenuation 30kHz.
(14)
4 Two coils, A and B, are magnetically coupled; coil A has 400 turns
and a self inductance of 20mH and coil B has 250 turns and a self
inductance of 50mH. When a current of 10 A is reversed in coil A,
the change of flux in coil B is 2mWb. Determine (a) the mutual
inductance between the coils, and (b) the coefficient of coupling.
(4)
5 Two mutually coupled coils P and Q are connected in series to a
200V d.c. supply. Coil P has an inductance of 0.8 H and resistance
2; coil Q has an inductance of 1.2H and a resistance of 5.
Determine the mutual inductance between the coils if, at a certain
instant after the circuit is connected, the current is 5A and increasing
at a rate of 7.5 A/s. (5)
6 For the coupled circuit shown in Figure A14.1, calculate the values
of currents I
P
and I
S
.(9)
Figure A14.1
7 A 4km transmission line has a characteristic impedance of
600
6
30
°
. At a particular frequency, the attenuation coefficient
of the line is 0.4Np/km and the phase-shift coefficient is 0.20rad/km.
Calculate (a) the magnitude and phase of the voltage at the receiving
end if the sending end voltage is 5.0
6
0
°
V, and (b) the magnitude
and phase of the receiving end current. (5)
8 The primary constants of a transmission line at a frequency of 5kHz
are: resistance, R D 20/loop km, inductance, L D 3mH/loop km,
Assignment 14 959
capacitance, C D 50nF/km, and conductance, G D 0.4 mS/km. De-
termine for the line (a) the characteristic impedance, (b) the propa-
gation coefficient, (c) the attenuation coefficient, (d) the phase-shift
coefficient, (e) the wavelength on the line, and (f) the speed of trans-
mission of signal. (13)
9 A loss-free transmission line has a characteristic impedance of
600
6
0
°
and is connected to an aerial of impedance 250 C
j200. Determine (a) the magnitude of the ratio of the reflected to
the incident voltage wave, and (b) the incident voltage if the reflected
voltage is 10
6
60
°
.(5)
10 A low loss transmission line has a mismatched load such that the
reflection coefficient at the termination is 0.5
6
150
°
. The character-
istic impedance of the line is 200. Determine (a) the standing wave
ratio, (b) the load impedance, and (c) the incident current flowing if
the reflected current is 15 mA. (11)
11 Determine, in the s-domain, the impedance of (a) an inductance of
1mH, a resistance of 100 and a capacitance of 2
µF connected
in series, and (b) a 50 resistance in parallel with an inductance
of 5 mH. (6)
12 A sinusoidal voltage 9 sin 2t volts is applied to a series R–L circuit.
Assuming that at time t D 0, current i D 0 and that resistance, R D
6 and inductance, L D 1.5 H, determine using Laplace transforms
an expression for the current flowing in the circuit. (12)
Main formulae for part 3
advanced circuit theory and
technology
Complex numbers:
z D a C jb D rcos C j sin D r
6
,
where j
2
D1 Modulus, r DjzjD
a
2
C b
2
Argument, D argz D arctan
b
a
Addition: a C jb C c C jb D a C c C jb C d
Subtraction: a C jb c C jd D a c C jb d
Complex equations: If a C jb D c C jd, then a D c and b D d
If z
1
D r
1
6
1
and z
2
D r
2
6
2
then
Multiplication: z
1
z
2
D r
1
r
2
6
1
C
2
and Division:
z
1
z
2
D
r
1
r
2
6
1
2
De Moivre’s theorem: [r
6
]
n
D r
n
6
n D r
n
cosn C j sinn
General:
Z D
V
I
D R C jX
L
X
C
DjZj
6
,
where jZjD
[R
2
C X
L
X
C
2
]and D arctan
X
L
X
C
R
X
L
D 2fL X
C
D
1
2fC
Y D
I
V
D
1
Z
D G C jB
Series: Z
T
D Z
1
C Z
2
C Z
3
ÐÐÐ
Parallel:
1
Z
T
D
1
Z
1
C
1
Z
2
C
1
Z
3
CÐÐÐ
P D VI cos or P D I
2
R
RSD VI Q D VI sin
Power factor D cos D
R
Z
If V D a C jb and I D c C jd then P D ac C bd
Q D bc ad S D VI
Ł
D P C jQ
R–L–C series circuit:
f
r
D
1
2
p
LC
Q D
ω
r
L
R
D
1
ω
r
CR
D
1
R
L
C
D
V
L
V
D
V
C
V
D
f
r
f
2
f
1
f
r
D
p
f
1
f
2
Main formulae for part 3 advanced circuit theory and technology 961
LR–C network:
f
r
D
1
2
1
LC
R
2
L
2
R
D
D
L
CR
Q D
I
C
I
r
D
ω
r
L
R
LR–CR network:
f
r
D
1
2
p
LC
R
2
L
L/C
R
2
C
L/C
Determinants:
ab
cd
D ad bc
abc
def
ghj
D a
ef
hj
b
df
gj
C c
de
gh
Delta-star:
Z
1
D
Z
A
Z
B
Z
A
C Z
B
C Z
C
etc
Star-delta:
Z
A
D
Z
1
Z
2
C Z
2
Z
3
C Z
3
Z
1
Z
2
etc
Impedance matching:
jzjD
N
1
N
2
2
j
Z
L
j
Complex waveforms:
I D
I
2
0
C
I
2
1m
C I
2
2m
C ...
2
i
AV
D
1
0
idωt form factor D
r.m.s
mean
P D V
0
I
0
C V
1
I
1
cos
1
C V
2
I
2
cos
2
C ... or P D I
2
R
power factor D
P
VI
Harmonic resonance: nωL D
1
nωC
Harmonic analysis:
a
0
³
1
p
p
kD1
y
k
a
n
³
2
p
p
kD1
y
k
cosnx
k
b
n
³
2
p
p
kD1
y
k
sinnx
k
Hysteresis and Eddy
current:
Hysteresis loss/cycle D A˛ˇ J/m
3
or hysteresis loss D k
h
vfB
m
n
W
Eddy current loss/cycle D k
e
B
m
2
f
2
t
3
W
962 Electrical Circuit Theory and Technology
Dielectric loss:
Series representation: tan υ D R
S
ωC
S
D 1/Q
Parallel representation: tanυ D
1
R
p
ωC
p
Loss angle υ D 90
°
Power factor D cos ³ tanυ
Dielectric power loss D V
2
ωC tanυ
Field theory:
Coaxial cable: C D
2ε
0
ε
r
ln
b
a
F/m E D
V
r ln
b
a
V/m
L D
µ
0
µ
r
2
1
4
C ln
b
a
H/m
Twin line: C D
ε
0
ε
r
ln
D
a
F/m L D
µ
0
µ
r
1
4
C ln
D
a
H/m
Energy stored: in a capacitor, W D
1
2
CV
2
J; in an inductor W D
1
2
LI
2
J
in electric field per unit volume, ω
f
D
1
2
DE D
1
2
ε
0
ε
r
E
2
D
D
2
2ε
0
ε
r
J/m
3
in a non-magnetic medium, ω
f
D
1
2
BH D
1
2
µ
0
H
2
D
B
2
2µ
0
J/m
3
Attenuators:
Logarithmic ratios: in decibels D 10 lg
P
2
P
1
D 20 lg
V
2
V
1
D 20 lg
I
2
I
1
in nepers D
1
2
ln
P
2
P
1
D ln
V
2
V
1
D ln
I
2
I
1
Symmetrical T-attenuator: R
0
D
R
2
1
C 2R
1
R
2
D
p
R
OC
R
SC
R
1
D R
0
N 1
N C 1
R
2
D R
0
2N
N
2
1
Symmetrical -attenuator: R
0
D
R
1
R
2
2
R
1
C 2R
2
D
p
R
OC
R
SC
R
1
D R
0
N
2
1
2N
R
2
D R
0
N C 1
N 1
L-section attenuator: R
1
D
p
[R
OA
R
OA
R
OB
]
R
2
D
R
OA
R
2
OB
R
OA
R
OB
Main formulae for part 3 advanced circuit theory and technology 963
Filter networks
Low-pass T or : f
C
D
1
p
LC
R
0
D
L
C
C D
1
R
0
f
C
L D
R
0
f
C
Z
0T
D R
0
1
ω
ω
C
2
Z
0
D
R
0
1
ω
ω
C
2
High-pass T or : f
C
D
1
4
p
LC
R
0
D
L
C
C D
1
4R
0
f
C
L D
R
0
4f
C
Z
0T
D R
0
1
ω
C
ω
2
Z
0
D
R
0
1
ω
C
ω
2
Low and high-pass:
Z
0T
Z
0
D Z
1
Z
2
D R
2
0
I
1
I
2
D
I
2
I
3
D
I
3
I
4
D e
5
D e
˛Cjˇ
D e
˛
6
ˇ
Phase angle ˇ D ω
p
LC
time delay D
p
LC
‘m-derived filter sections:
Low-pass m D
1
f
C
f
1
2
High pass m D
1
f
1
f
C
2
Magnetically coupled
circuits
E
2
DM
dI
1
dt
DšjωMI
1
M D N
2
d
2
dI
1
D N
1
d
1
dI
2
D k
p
L
1
L
2
D
L
A
L
B
4
964 Electrical Circuit Theory and Technology
Transmission lines:
Phase delay ˇ D ω
p
LC wavelength 7 D
2
ˇ
velocity of propagation u D f7 D
ω
ˇ
I
R
D I
S
e
n5
D I
S
e
n˛
6
nˇ
V
R
D V
S
e
n5
D V
S
e
n˛
6
nˇ
Z
0
D
p
Z
OC
Z
SC
D
R C jωL
G C jωC
5 D
p
[R C jωLG C jωC]
Reflection coefficient, 9 D
I
r
I
i
D
Z
O
Z
R
Z
O
C Z
R
D
V
r
V
i
Standing-wave ratio, s D
I
max
I
min
D
I
i
C I
r
I
i
I
r
D
1 Cj9j
1 j9j
P
r
P
t
D
s 1
s C 1
2
Transients:
C R circuit t D CR
Charging:
v
C
D V1 e
t/CR
v
r
D Ve
t/CR
i D Ie
t/CR
Dischargin: v
c
D v
R
D Ve
t/CR
i D Ie
t/CR
L R circuit t D
L
R
Current growth:
v
L
D Ve
Rt/L
v
R
D V1 e
Rt/L
i D I1 e
Rt/L
Current decay:
v
L
D v
R
D Ve
Rt/L
i D Ie
Rt/L
Part4GeneralReference
Standard electrical
quantities—their symbols
and units
QUANTITY UNIT
QUANTITY SYMBOL UNIT SYMBOL
Admittance Y siemen S
Angular frequency ω radians per second rad/s
Area A square metres m
2
Attenuation coefficient
(or constant)
˛ neper per metre Np/m
Capacitance C farad F
Charge Q coulomb C
Charge density coulomb per square
metre
C/m
2
Conductance G siemen S
Current I ampere A
Current density J ampere per square
metre
A/m
2
Efficiency per-unit or per cent p.u. or %
Electric field strength E volt per metre V/m
Electric flux coulomb C
Electric flux density D coulomb per square
metre
C/m
2
Electromotive force E volt V
Energy W joule J
Field strength, electric E volt per metre V/m
Field strength, magnetic H ampere per metre A/m
Flux, electric coulomb C
Flux, magnetic weber Wb
Flux density, electric D coulomb per square
metre
C/m
2
Flux density, magnetic B tesla T
Force F newton N
Frequency f hertz Hz
Frequency, angular ω radians per second rad/s
Frequency, rotational n revolutions per
second
rev/s
Impedance Z ohm
Inductance, self L henry H
968 Electrical Circuit Theory and Technology
QUANTITY UNIT
QUANTITY SYMBOL UNIT SYMBOL
Inductance, mutual M henry H
length l metre m
Loss angle υ radian or degrees rad or
°
Magnetic field strength H ampere per metre A/m
Magnetic flux weber Wb
Magnetic flux density B tesla T
Magnetic flux linkage weber Wb
Magnetising force H ampere per metre A/m
Magnetomotive force F
m
ampere A
Mutual inductance M henry H
Number of phases m
–
–
Number of pole-pairs p
–
–
Number of turns (of a
winding)
N
–
–
Period, Periodic time T second s
Permeability, absolute henry per metre H/m
Permeability of free
space
0
henry per metre H/m
Permeability, relative
r
–
–
Permeance weber per ampere Wb/A or
or per henry /H
Permittivity, absolute ε farad per metre F/m
Permittivity of free
space
ε
0
farad per metre F/m
Permittivity, relative ε
r
–
–
Phase-change
coefficient
ˇ radian per metre rad/m
Potential, Potential
difference
V volt V
Power, active P watt W
Power, apparent S volt ampere VA
Power, reactive Q volt ampere reactive var
Propagation coefficient
(or constant)
(
–
–
Quality factor,
magnification
Q
–
–
Quantity of electricity Q coulomb C
Reactance X ohm
Reflection coefficient *
–
–
Relative permeability
r
–
–
Relative permittivity ε
r
–
–
Reluctance R
m
ampere per weber or A/Wb or
per henry /H
Resistance R ohm