Bird J. Electrical Circuit Theory and Technology
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748 Electrical Circuit Theory and Technology
From Section (40.6), inductance per metre due to internal linkages
D
1
4
0
r
2
henry/metre
Thus inductance per metre due to internal linkages of two conductors
D 2
1
4
0
r
2
D
0
r
4
henry/metre
Therefore, at low frequency, total inductance per metre of the two
conductors
D
0
r
4
C
0
r
ln
D
a
i.e.,
L =
m
0
m
r
p
1
4
Y ln
D
a
henry=metre
40.23
(This is often referred to as the ‘loop inductance’).
In most practical lines the relative permeability,
r
D 1.
Problem 19. A single-phase power line comprises two conductors
each with a radius 8.0 mm and spaced 1.2 m apart in air. Deter-
mine the inductance of the line per metre length ignoring internal
linkages. Assume the relative permeability,
r
D 1.
From equation (40.22), inductance
D
0
r
ln
D
a
D
4 ð 10
7
1
ln
1.2
8.0 ð 10
3
D 4 ð10
7
ln150
D 20.0
× 10
−7
H=m or 2.0 mH=m
Problem 20. Determine (a) the loop inductance, and (b) the
capacitance of a 1 km length of single-phase twin line having
conductors of diameter 10 mm and spaced 800 mm apart in air.
(a) From equation (40.23), total inductance per loop metre
D
0
r
1
4
C ln
D
a
D
4 ð 10
7
1
1
4
C ln
800
10/2
Field theory 749
D 4 ð 10
7
0.25 C ln160
D 21.3 ð 10
7
H/m
Hence loop inductance o
fa1kmlength of line
D 21.3 ð 10
7
H/m ð 10
3
m
D 21.3
× 10
−4
H or 2.13 mH
(b) From equation (40.14), capacitance per metre length
D
ε
0
ε
r
lnD/a
D
8.85 ð 10
12
1
ln800/5
D 5.478 ð 10
12
F/m
Hence capacitance o
fa1kmlength of line
D 5.478 ð 10
12
F/m ð 10
3
m
D 5.478 nF
Problem 21. The total loop inductance of an isolated twin power
line is 2.185
µH/m. The diameter of each conductor is 12 mm.
Determine the distance between their centres.
From equation (40.23),
total loop inductance D
0
r
1
4
C ln
D
a
Hence
2.185 ð 10
6
D
4 ð 10
7
1
1
4
C ln
D
6
where D is the distance between centres in millimetres.
2.185 ð 10
6
4 ð 10
7
D
0.25 C ln
D
6
ln
D
6
D 5.4625 0.25 D 5.2125
D
6
D e
5.2125
from which, distance D D 6e
5.2125
D 1100 mm or 1.10 m
750 Electrical Circuit Theory and Technology
Further problems on the inductance of an isolated twin line may be found
in Section 40.9, problems 28 to 32, page 756.
40.8 Energy stored in an
electromagnetic field
Magnetic energy in a nonmagnetic medium
For a nonmagnetic medium the relative permeability,
r
D 1and
B D
0
H.
Thus the magnetic field strength H is proportional to the flux density B
and a graph of B against H is a straight line, as shown in Figure 40.21.
It was shown in Section 38.3 that, when the flux density is increased by
an amount dB due to an increase dH in the magnetic field strength, then
energy supplied to the magnetic circuit D area of shaded strip
(in joules per cubic metre)
Thus, for a maximum flux density OY in Figure 40.21,
total energy stored in the magnetic field D area of triangle OYX
D
1
2
ð base ðheight
D
1
2
OZOY
Figure 40.21
If OY D B teslas and OZ D H ampere/metre, then the total energy stored
in a non-magnetic medium,
!
f
=
1
2
HB joules=metre
3
40.24
Since B D
0
H for a non-magnetic medium, the energy stored,
ω
f
D
1
2
H
0
H
i.e.,
!
f
=
1
2
m
0
H
2
joules=metre
3
40.25
Alternatively, H D B/
0
, thus the energy stored,
ω
f
D
1
2
HB D
1
2
B
0
B
i.e.,
!
f
=
B
2
2m
0
joules=metre
3
40.26
Magnetic energy stored in an inductor
Establishing a magnetic field requires energy to be expended. However,
once the field is established, the only energy expended is that supplied to
maintain the flow of current in opposition to the circuit resistance, i.e.,
the I
2
R loss, which is dissipated as heat.
Field theory 751
Figure 40.22
For an inductive circuit containing resistance R and inductance L (see
Figure 40.22) the applied voltage V at any instant is given by V D
v
R
C v
L
i.e., V D iR C L
di
dt
Multiplying throughout by current i gives the power equation:
Vi D i
2
R C Li
di
dt
Multiplying throughout by time dt seconds gives the energy equation:
Vi dt D i
2
Rdt C Li di
Vi dt is the energy supplied by the source in time dt, i
2
R dt is the energy
dissipated in the resistance and Lidi is the energy supplied in establishing
the magnetic field or the energy absorbed by the magnetic field in time
dt seconds.
Hence the total energy stored in the field when the current increases
from 0 to I amperes is given by
energy stored, W
f
D
I
0
Lidi D L
i
2
2
I
0
i.e., total energy stored,
W
f
=
1
2
LI
2
joules
40.27
From Section 40.5, inductance L D N)/I, hence
total energy stored D
1
2
N)
I
I
2
D
1
2
N)I joules
Also H D NI/l, from which, N D Hl/I,and) D BA. Thus the total
energy stored,
W
f
D
1
2
N)I D
1
2
Hl
I
BAI
D
1
2
HBlA joules
or ω
f
D
1
2
HB joules/metre
3
since lA is the volume of the magnetic field. This latter expression has
already been derived in equation (40.24).
Summarizing, the energy stored in a nonmagnetic medium,
!
f
=
1
2
BH
=
1
2
m
0
H
2
=
B
2
2m
0
joules=metre
3
and the energy stored in an inductor,
W
f
=
1
2
LI
2
joules
752 Electrical Circuit Theory and Technology
Problem 22. Calculate the value of the energy stored when a
current of 50 mA is flowing in a coil of inductance 200 mH. What
value of current would double the energy stored?
From equation (40.27), energy stored in inductor,
W
f
D
1
2
LI
2
D
1
2
200 ð 10
3
50 ð 10
3
2
D 2.5 × 10
−4
J or 0.25 mJ or 250 mJ
If the energy stored is doubled, then 22.5 ð 10
4
D
1
2
200 ð 10
3
I
2
from which
current I D
42.5 ð 10
4
200 ð 10
3
D 70.71 mA
Problem 23. The airgap of a moving coil instrument is 2.0 mm
long and has a cross-sectional area of 500 mm
2
. If the flux density
is 50 mT, determine the total energy stored in the magnetic field
of the airgap.
From equation (40.26), energy stored,
ω
f
D
B
2
2
0
D
50 ð 10
3
2
24 ð 10
7
D 9.95 ð 10
2
J/m
3
Volume of airgap D Al D 500 ð 2.0 mm
3
D 500 ð 2.0 ð 10
9
m
3
.
Hence the energy stored in the airgap,
W
f
D 9.95 ð 10
2
J/m
3
ð 500 ð 2.0 ð 10
9
m
3
= 9.95 × 10
−4
J 0.995 mJ 995 mJ
Problem 24. Determine the strength of a uniform electric field if
it is to have the same energy as that established by a magnetic field
of flux density 0.8 T. Assume that the relative permeability of the
magnetic field and the relative permittivity of the electric field are
both unity.
From equation (40.26), energy stored in magnetic field,
ω
f
D
B
2
2
0
D
0.8
2
24 ð 10
7
D 2.546 ð 10
5
J/m
3
From equation (40.17), energy stored in electric field,
ω
f
D
1
2
ε
0
ε
r
E
2
Field theory 753
Hence, if the current stored in the magnetic and electric fields is to be the
same, then
1
2
ε
0
ε
r
E
2
D 2.546 ð 10
5
, i.e.,
1
2
8.85 ð 10
12
1E
2
D 2.546 ð 10
5
from which electric field strength,
E D
22.546 ð 10
5
8.85 ð 10
12
D
5.75 ð 10
16
D 2.40
× 10
8
V=m or 240 MV=m
Further problems on energy stored in an electromagnetic field may be
found in Section 40.9 following, problems 33 to 37, page 756.
40.9 Further problems
on field theory
Field plotting by curvilinear squares
1 (a) Explain the meaning of the terms (i) streamline (ii) equipotential,
with reference to an electric field.
(b) A field plot between two metal plates is shown in Figure 40.23.
If the relative permittivity of the dielectric is 2.4, determine the
capacitance of a 50 cm length of the system. [23.4 pF]
2 A field plot for a concentric cable is shown in Figure 40.24. The
relative permittivity of the dielectric is 5. Determine the capacitance
of a 10 m length of the cable. [1.66 nF]
3 The plates of a capacitor are 10 mm long and 6 mm wide and are
separated by a dielectric 3 mm thick and of relative permittivity 2.5.
Determine the capacitance of the capacitor (a) when neglecting any
fringing at the edges, (b) by producing a field plot taking fringing
into consideration.
[(a) 0.44 pF (b) 0.60 pF 0.70 pF,
depending on the accuracy of the plot]
Capacitance between concentric cylinders
4 A coaxial cable has an inner conductor of radius 0.4 mm and an
outer conductor of internal radius 4 mm. Determine the capacitance
per metre length of the cable if the dielectric has a relative permit-
tivity of 2. [48.30 pF]
Figure 40.23
5 A concentric cable has a core diameter of 40 mm and an inner sheath
diameter of 100 mm. The relative permittivity of the dielectric is 2.5
and the core potential is 50 kV. Determine (a) the capacitance per
kilometre length of the cable and (b) the dielectric stress at radii of
30 mm and 40 mm.
[(a) 0.1517
µF (b) 1.819 MV/m, 1.364 MV/m]
754 Electrical Circuit Theory and Technology
Figure 40.24
6 A coaxial cable has a capacitance of 100 pF per metre length. The
relative permittivity of the dielectric is 3.2 and the core diameter is
1.0 mm. Determine the required inside diameter of the sheath.
[5.93 mm]
7 A single-core concentric cable is to be manufactured for a 100 kV,
50 Hz transmission system. The dielectric used is paper which has
a maximum safe dielectric stress of 10 MV/m and a relative permit-
tivity of 3.2. Calculate (a) the core and inner sheath radii for the most
economical cable, (b) the capacitance per metre length and (c) the
charging current per kilometre run.
[(a) 10 mm; 27.2 mm (b) 177.9 pF (c) 5.59 A]
8 A concentric cable has a core diameter of 30 mm and an inside
sheath diameter of 75 mm. The relative permittivity is 2.6, the loss
angle is 2.5 ð10
3
rad and the working voltage is 100 kV at 50 Hz
frequency. Determine for a 1 km length of cable (a) the capacitance,
(b) the charging current, and (c) the power loss.
[(a) 0.1578
µF (b) 4.957 A (c) 1239 W]
9 A concentric cable operates at 200 kV and 50 Hz. The maximum
electric field strength within the cable is not to exceed 5 MV/m.
Determine (a) the radius of the core and the inner radius of the
sheath for ideal operation, and (b) the stress on the dielectric at the
surface of the core and at the inner surface of the sheath.
[(a) 40 mm, 108.7 mm (b) 5 MV/m, 1.84 MV/m]
10 A concentric cable has a core radius of 20 mm and a sheath inner
radius of 40 mm. The permittivity of the dielectric is 2.5. Using two
equipotential surfaces within the dielectric, determine the capacitance
of the cable per metre length by the method of curvilinear squares.
Draw the field plot for the cable. [200.6 pF]
Capacitance of an isolated twin line
11 Two parallel wires, each of diameter 5.0 mm, are uniformly spaced
in air at a distance of 40 mm between centres. Determine the capac-
itance of a 500 m run of the line. [5.014 nF]
12 A single-phase circuit is comprised of two parallel conductors each
of radius 5.0 mm and spaced 1.5 m apart in air. The p.d. between
the conductors is 20 kV at 50 Hz. Determine (a) the capacitance
per metre length of the conductors, and (b) the charging current per
kilometre run. [(a) 4.875 pF (b) 30.63 mA]
13 The capacitance of a 300 m length of an isolated twin line is 1522 pF.
The line comprises two air conductors which are spaced 1200 mm
between centres. Determine the diameter of each conductor.
[10 mm]
14 An isolated twin line is comprised of two air-insulated conductors,
each of radius 8.0 mm, which are spaced 1.60 m apart. The voltage
between the lines is 7 kV at a frequency of 50 Hz. Determine for a
Field theory 755
1 km length (a) the line capacitance, (b) the value of charge carried
by each wire, and (c) the charging current.
[(a) 5.248 nF (b) 36.74
µC (c) 11.54 mA]
15 The charging current for a 1 km run of isolated twin line is not to
exceed 30 mA. The p.d. between the lines is 20 kV at 50 Hz. If
the line is air insulated and the conductors are spaced 1 m apart,
determine (a) the maximum value required for the capacitance per
metre length, and (b) the maximum diameter of each conductor.
[(a) 4.775 pF (b) 5.92 mm]
Energy stored in an electric field
16 Determine the energy stored in a 5000 pF capacitor when charged to
800 V and the average power developed if this energy is dissipated
in 20
µs. [1.6 mJ; 80 W]
17 A 0.25
µF capacitor is required to store 2 J of energy. Determine the
p.d. to which the capacitor must be charged. [4 kV]
18 A capacitor is charged with 6 mC. If the energy stored is 1.5 J deter-
mine (a) the voltage across the plates, and (b) the capacitance of the
capacitor. [(a) 500 V (b) 12
µF]
19 After a capacitor is connected across a 250 V d.c. supply the charge
is 5
µC. Determine (a) the capacitance, and (b) the energy stored.
[(a) 20 nF (b) 0.625 mJ]
20 A capacitor consisting of two metal plates each of area 100 cm
2
and spaced 0.1 mm apart in air is connected across a 200 V supply.
Determine (a) the electric flux density, (b) the potential gradient and
(c) the energy stored in the capacitor.
[(a) 17.7
µC/m
2
(b) 2 MV/m (c) 17.7 µJ]
21 A mica capacitor is to be constructed to have a capacitance of 0.05
µF
and to have a steady working potential of 2 kV maximum. Allowing
a safe value of field stress of 20 MV/m, determine (a) the required
thickness of the mica dielectric, (b) the area of plate required if the
relative permittivity of the mica is 5, (c) the maximum energy stored
by the capacitor, and (d) the average power developed if this energy
is dissipated in 25
µs.
[(a) 0.1 mm (b) 0.113 m
2
(c) 0.1 J (d) 4 kW]
22 A 500 pF capacitor is charged to a p.d. of 100 V. The dielectric has
a cross-sectional area of 200 cm
2
and a relative permittivity of 2.4.
Determine the energy stored per cubic metre in the dielectric.
[0.147 J/m
3
]
23 Two parallel plates each having dimensions 30 mm by 50 mm are
spaced 8 mm apart in air. If a voltage of 40 kV is applied across the
plates determine the energy stored in the electric field. [1.328 mJ]
756 Electrical Circuit Theory and Technology
Inductance of a concentric cable
24 A coaxial cable has an inner core of radius 0.8 mm and an outer
sheath of internal radius 4.8 mm. Determine the inductance of 25 m
of the cable. Assume that the relative permeability of the material
used is 1. [10.2
µH]
25 A concentric cable has a core 12 mm diameter and a sheath 40 mm
diameter, the sheath having negligible thickness. Determine the
inductance and the capacitance of the cable per metre assuming
nonmagnetic materials and a dielectric of relative permittivity 3.2.
[0.291
µH/m, 147.8 pF/m]
26 A concentric cable has an inner sheath radius of 4.0 cm. The induc-
tance of the cable is 0.5
µH/m. Ignoring inductance due to internal
linkages, determine the radius of the core. Assume that the relative
permeability of the material is unity. [3.28 mm]
27 The inductance of a concentric cable of core radius 8 mm and inner
sheath radius of 35 mm is measured as 2.0 mH. Determine (a) the
length of the cable, and (b) the capacitance of the cable. Assume
that nonmagnetic materials are used and the relative permittivity of
the dielectric is 2.5. [(a) 5.794 km (b) 0.546
µF]
Inductance of an isolated twin line
28 A single-phase power line comprises two conductors each with a
radius of 15 mm and spaced 1.8 m apart in air. Determine the induc-
tance per metre length, ignoring internal linkages and assuming the
relative permeability,
r
D 1. [1.915 µH/m]
29 Determine (a) the loop inductance, and (b) the capacitance of a
500 m length of single-phase twin line having conductors of diameter
8 mm and spaced 60 mm apart in air.
[(a) 0.592 mH (b) 5.133 nF]
30 An isolated twin power line has conductors 7.5 mm radius. Deter-
mine the distance between centres if the total loop inductance of
1 km of the line is 1.95 mH. [765 mm]
31 An isolated twin line has conductors of diameter d ð 10
3
metres
and spaced D millimetres apart in air. Derive an expression for the
total loop inductance L of the line per metre length.
L D
0
1
4
C ln
2D
d
32 A single-phase power line comprises two conductors spaced 2 m
apart in air. The loop inductance of 2 km of the line is measured as
3.65 mH. Determine the diameter of the conductors. [53.6 mm]
Energy stored in an electromagnetic field
33 Determine the value of the energy stored when a current of 120 mA
flows in a coil of 500 mH. What value of current is required to
double the energy stored? [3.6 mJ, 169.7 mA]
Field theory 757
34 A moving-coil instrument has two airgaps each 2.5 mm long and
having a cross-sectional area of 8.0 cm
2
. Determine the total energy
stored in the magnetic field of the airgap if the flux density is 100 mT.
[15.92 mJ]
35 Determine the flux density of a uniform magnetic field if it is to
have the same energy as that established by a uniform electric field of
strength 45 MV/m. Assume the relative permeability of the magnetic
field and the relative permittivity of the electric field are both unity.
[0.15 T]
36 A long single core concentric cable has inner and outer conductors
of diameters D
1
and D
2
respectively. The conductors each carry a
current of I amperes but in opposite directions. If the relative perme-
ability of the material is unity and the inductance due to internal
linkages is negligible, show that the magnetic energy stored in a
4 m length of the cable is given by
0
I
2
ln
D
2
D
1
joules
37 1 mJ of energy is stored in a uniform magnetic field having dimen-
sions 20 mm by 10 mm by 1.0 mm. Determine for the field (a) the
flux density, and (b) the magnetic field strength.
[(a) 0.112 T (b) 89200 A/m]