Bird J. Electrical Circuit Theory and Technology
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598 Electrical Circuit Theory and Technology
(b) When a 600 j800 impedance is connected across AB, the
current I flowing is given by
I D
5.724
6
12.09
°
400 C j800 C 600 j800
D 5.724
6
12.09
°
mA
Hence the power P dissipated in the 600 j800 impedance is
given by
P D I
2
R D 5.724 ð 10
3
2
600 D 19.7mW
Further problems on Th´evenin and Norton equivalent networks may be
found in Section 33.5 following, problems 16 to 21, page 600
33.5 Further problems
on Th
´
evenin’s and
Norton’s theorem
Th
´
evenin’s theorem
1UseTh
´
evenin’s theorem to determine the current flowing in the 10
resistor of the d.c. network shown in Figure 33.85. [0.85 A]
2 Determine, using Th
´
evenin’s theorem, the values of currents I
1
,I
2
and I
3
of the network shown in Figure 33.86.
[I
1
D 2.8A,I
2
D 4.8A,I
3
D 7.6A]
3 Determine the Th
´
evenin equivalent circuit with respect to terminals
AB of the network shown in Figure 33.87. Hence determine
the magnitude of the current flowing in a 4 j7 impedance
connected across terminals AB and the power delivered to this
impedance. [E D 15.37
6
38.66
°
,
z D 3.20 C j4.00;1.97 A; 15.5W]
4 For the network shown in Figure 33.88 use Th
´
evenin’s theorem to
determine the current flowing in the 3 resistance. [1.17 A]
5 Derive for the network shown in Figure 33.89 the Th
´
evenin
equivalent circuit at terminals AB, and hence determine the current
flowing in a 20 resistance connected between A and B.
[E D 2.5V,r D 5 ;0.10 A]
6 Determine for the network shown in Figure 33.90 the Th
´
evenin
equivalent circuit with respect to terminals AB, and hence determine
the current flowing in the 5 C j6 impedance connected between
A and B. [E D 14.32
6
6.38
°
, z D 3.99 C j0.55;1.29 A]
Figure 33.85
Figure 33.86
7 For the network shown in Figure 33.91, derive the Th
´
evenin
equivalent circuit with respect to terminals AB, and hence determine
the magnitude of the current flowing in a 2 C j13 impedance
connected between A and B. [1.157 A]
8 Use Th
´
evenin’s theorem to determine the power dissipated in the
4 resistance of the network shown in Figure 33.92. [0.24 W]
Figure 33.87
Th´evenin’s and Norton’s theorems 599
Figure 33.88 Figure 33.89 Figure 33.90
Figure 33.91 Figure 33.92
9 For the bridge network shown in Figure 33.93 use Th
´
evenin’s
theorem to determine the current flowing in the 4 C j3 impedance
and its direction. Assume that the 20
6
0
°
V source has negligible
internal impedance. [0.12 A from Q to P]
Figure 33.93
10 Repeat problems 1 to 10, page 542 of Chapter 30, 2 and 3 and 11
to 15, page 559 of Chapter 31 and 2 to 8, page 573 of Chapter 32,
using Th
´
evenin’s theorem and compare the method of solution with
that used for Kirchhoff’s laws, mesh-current and nodal analysis and
the superposition theorem.
600 Electrical Circuit Theory and Technology
Figure 33.94 Figure 33.95
Norton’s theorem
11 Repeat problems 1 to 4 and 6 to 8 above using Norton’s theorem
instead of Th
´
evenin’s theorem.
12 Determine the current flowing in the 10 resistance of the network
shown in Figure 33.94 by using Norton’s theorem. [3.13 A]
13 For the network shown in Figure 33.95, use Norton’s theorem to
determine the current flowing in the 10 resistance. [1.08 A]
14 Determine for the network shown in Figure 33.96 the Norton equiv-
alent network at terminals AB. Hence determine the current flowing
in a 2 Cj4 impedance connected between A and B.
[I
SC
D 2.185
6
43.96
°
A, z D 2.40 C j1.47;0.88A]
Figure 33.96
15 Repeat problems 1 to 10, page 542 of Chapter 30 and 2 and 3 and
12 to 15, page 559 of Chapter 31, using Norton’s theorem.
Th
´
evenin and Norton equivalent networks
16 Convert the circuits shown in Figure 33.97 to Norton equivalent
networks. [(a) I
SC
D 2.5A,z D 2 (b) I
SC
D 2
6
30
°
, z D 5 ]
Figure 33.97
17 Convert the networks shown in Figure 33.98 to Th
´
evenin equivalent
circuits. [(a) E D 20 V, z D 4 ;(b)E D 12
6
50
°
V, z D 3 ]
18 (a) Convert the network to the left of terminals AB in Figure 33.99
to an equivalent Th
´
evenin circuit by initially converting to a
Norton equivalent network.
(b) Determine the current flowing in the 2.8 j3 impedance
connected between A and B in Figure 33.99.
[(a) E D 18 V, z D 1.2 (b) 3.6 A]
19 Determine, by successive conversions between Th
´
evenin and Norton
equivalent networks, a Th
´
evenin equivalent circuit for terminals
AB of Figure 33.100. Hence determine the current flowing in a
2 C j4 impedance connected between A and B.
E D 9
1
3
V,z D 1 I1.87
6
53.13
°
A
Figure 33.98
Th´evenin’s and Norton’s theorems 601
Figure 33.99 Figure 33.100
20 Derive an equivalent Th
´
evenin circuit for terminals AB of the
network shown in Figure 33.101. Hence determine the p.d. across
AB when a 3 C j4k impedance is connected between these
terminals.
[E D 4.82
6
41.63
°
V, z D 0.8 C j0.4 k;4.15 V]
Figure 33.101
21 For the network shown in Figure 33.102, derive (a) the Th
´
evenin
equivalent circuit, and (b) the Norton equivalent network. (c) A 6
resistance is connected between A and B. Determine the current
flowing in the 6 resistance by using both the Th
´
evenin and Norton
equivalent circuits.
[(a) E D 6.71
6
26.57
°
V,z D 4.50 C j3.75
(b) I
SC
D 1.15
6
66.38
°
,z D 4.50 C j3.75
(c) 0.60 A
Figure 33.102
Assignment 10
This assignment covers the material contained in chapters 30
to 33.
The marks for each question are shown in brackets at the end of
each question.
For the network shown in Figure A10.1, determine the current flowing in
each branch using:
(a) Kirchhoff’s laws (10)
(b) Mesh-current analysis (12)
(c) Nodal analysis (12)
(d) the Superposition theorem (22)
(e) Th
´
evenin’s theorem (14)
(f) Norton’s theorem (10)
Demonstrate that each method gives the same value for each of the branch
currents.
10∠0° V
(3−j4)Ω
5 Ω
(3+j4)Ω
20∠0° V
Figure A10.1
34 Delta-star and
star-delta
transformations
At the end of this chapter you should be able to:
ž recognize delta (or ) and star (or T) connections
ž apply the delta-star and star-delta transformations in
appropriate a.c. and d.c. networks
34.1 Introduction
By using Kirchhoff’s laws, mesh-current analysis, nodal analysis or the
superposition theorem, currents and voltages in many network can be
determined as shown in Chapters 30 to 32. Thevenin’s and Norton’s theo-
rems, introduced in Chapter 33, provide an alternative method of solving
networks and often with considerably reduced numerical calculations.
Also, these latter theorems are especially useful when only the current
in a particular branch of a complicated network is required. Delta-star
and star-delta transformations may be applied in certain types of circuit
to simplify them before application of circuit theorems.
34.2 Delta and star
connections
The network shown in Figure 34.1(a) consisting of three impedances Z
A
,
Z
B
and Z
C
is said to be p-connected. This network can be redrawn as
shown in Figure 34.1(b), where the arrangement is referred to as delta-
connected or mesh-connected.
The network shown in Figure 34.2(a), consisting of three impedances,
Z
1
, Z
2
and Z
3
,issaidtobeT-connected. This network can be redrawn
as shown in Figure 34.2(b), where the arrangement is referred to as star-
connected.
34.3 Delta-star
transformation
It is possible to replace the delta connection shown in Figure 34.3(a) by
an equivalent star connection as shown in Figure 34.3(b) such that the
impedance measured between any pair of terminals (1–2, 2–3 or 3–1)
is the same in star as in delta. The equivalent star network will consume
the same power and operate at the same power factor as the original delta
network. A delta-star transformation may alternatively be termed ‘ to T
transformation’.
604 Electrical Circuit Theory and Technology
Figure 34.1 (a) -connected
network, (b) Delta-connected
network
Considering terminals 1 and 2 of Figure 34.3(a), the equivalent
impedance is given by the impedance Z
B
in parallel with the series
combination of Z
A
and Z
C
,
i.e.,
Z
B
Z
A
C Z
C
Z
B
C Z
A
C Z
C
In Figure 34.3(b), the equivalent impedance between terminals 1 and 2 is
Z
1
and Z
2
in series, i.e., Z
1
C Z
2
Thus,
Delta Star
Z
12
D
Z
B
Z
A
C Z
C
Z
B
C Z
A
C Z
C
D Z
1
C Z
2
34.1
By similar reasoning, Z
23
D
Z
C
Z
A
C Z
B
Z
C
C Z
A
C Z
B
D Z
2
C Z
3
34.2
and Z
31
D
Z
A
Z
B
C Z
C
Z
A
C Z
B
C Z
C
D Z
3
C Z
1
34.3
Hence we have three simultaneous equations to be solved for Z
1
, Z
2
and Z
3
.
Equation (34.1) equation (34.2) gives:
Z
A
Z
B
Z
A
Z
C
Z
A
C Z
B
C Z
C
D Z
1
Z
3
34.4
Equation (34.3) C equation (34.4) gives:
2Z
A
Z
B
Z
A
C Z
B
C Z
C
D 2Z
1
from which Z
1
D
Z
A
Z
B
Z
A
C Z
B
C Z
C
Similarly, equation (34.2) equation (34.3) gives:
Z
B
Z
C
Z
A
Z
B
Z
A
C Z
B
C Z
C
D Z
2
Z
1
34.5
Equation (34.1) C equation (34.5) gives:
2Z
B
Z
C
Z
A
C Z
B
C Z
C
D 2Z
2
from which Z
2
D
Z
B
Z
C
Z
A
C Z
B
C Z
C
Finally, equation (34.3) equation (34.1) gives:
Z
A
Z
C
Z
B
Z
C
Z
A
C Z
B
C Z
C
D Z
3
Z
2
34.6
Figure 34.2 (a) T-connected
network, (b) Star-connected
network
Delta-star and star-delta transformations 605
Figure 34.3
Equation (34.2) C equation (34.6) gives:
2Z
A
Z
C
Z
A
C Z
B
C Z
C
D 2Z
3
from which Z
3
D
Z
A
Z
C
Z
A
C Z
B
C Z
C
Summarizing, the star section shown in Figure 34.3(b) is equivalent to
the delta section shown in Figure 34.3(a) when
Z
1
=
Z
A
Z
B
Z
A
Y Z
B
Y Z
C
34.7
Z
2
=
Z
B
Z
C
Z
A
Y Z
B
Y Z
C
34.8
and
Z
3
=
Z
A
Z
C
Z
A
Y Z
B
Y Z
C
34.9
It is noted that impedance Z
1
is given by the product of the two
impedances in delta joined to terminal 1 (i.e., Z
A
and Z
B
), divided by
the sum of the three impedances; impedance Z
2
is given by the product
of the two impedances in delta joined to terminal 2 (i.e., Z
B
and Z
C
),
divided by the sum of the three impedances; and impedance Z
3
is given
by the product of the two impedances in delta joined to terminal 3 (i.e.,
Z
A
and Z
C
), divided by the sum of the three impedances.
Thus, for example, the star equivalent of the resistive delta network
shown in Figure 34.4 is given by
Z
1
D
23
2 C 3 C 5
D 0.6 Z,
Z
2
D
35
2 C 3 C 5
D 1.5 Z
and Z
3
D
25
2 C 3 C 5
D 1.0 Z
Problem 1. Replace the delta-connected network shown in
Figure 34.5 by an equivalent star connection.
Let the equivalent star network be as shown in Figure 34.6. Then, from
equation (34.7),
Z
1
D
Z
A
Z
B
Z
A
C Z
B
C Z
C
D
2010 C j10
20 C 10 C j10 j20
Figure 34.4
606 Electrical Circuit Theory and Technology
Figure 34.5
D
2010 C j10
30 j10
D
201.414
6
45
°
31.62
6
18.43
°
D 8.944
66
63.43
°
Z or .4 Y j8/Z
From equation (34.8),
Z
2
D
Z
B
Z
C
Z
A
C Z
B
C Z
C
D
10 C j10j20
31.62
6
18.43
°
D
1.414
6
45
°
20
6
90
°
31.62
6
18.43
°
D 8.944
66
−26.57
°
Z or .8 − j4/Z
Figure 34.6
From equation (34.9),
Z
3
D
Z
A
Z
C
Z
A
C Z
B
C Z
C
D
20j20
31.62
6
18.43
°
D
400
6
90
°
31.62
6
18.43
°
D 12.650
66
−71.57
°
Z or .4 − j12/Z
Problem 2. For the network shown in Figure 34.7, determine
(a) the equivalent circuit impedance across terminals AB,
(b) supply current I and (c) the power dissipated in the 10
resistor.
Figure 34.7 Figure 34.8
(a) The network of Figure 34.7 is redrawn, as in Figure 34.8, showing
more clearly the part of the network 1, 2, 3 forming a delta connec-
tion. This may he transformed into a star connection as shown
in Figure 34.9.
From equation (34.7),
Z
1
D
Z
A
Z
B
Z
A
C Z
B
C Z
C
D
j10j15
j10 C j15 C j25
D
j10j15
j50
D j3 Z
Delta-star and star-delta transformations 607
Figure 34.9
From equation (34.8),
Z
2
D
Z
B
Z
C
Z
A
C Z
B
C Z
C
D
j15j25
j50
D j7.5 Z
From equation (34.9),
Z
3
D
Z
A
Z
C
Z
A
C Z
B
C Z
C
D
j10j25
j50
D j5 Z
The equivalent network is shown in Figure 34.10 and is further
simplified in Figure 34.11.
10 C j5 in parallel with j5 gives an equivalent impedance of
10 C j5j5
10 C j5 j5
D 2.5 j5
Hence the total circuit equivalent impedance across terminals AB is
given by Z
AB
D 2.5 j5 C j7.5 D .2.5 Y j2.5/Z or 3.54
66
45
°
Z
(b) Supply current I D
V
Z
AB
D
40
6
0
°
3.54
6
45
°
D 11.3
66
−45
°
A
(c) Power P dissipated in the 10 resistance of Figure 34.7 is given
by I
1
2
(10), where I
1
(see Figure 34.11) is given by:
I
1
D
j5
10 C j5 j5
11.3
6
45
°
D 5.65
6
135
°
A
Figure 34.10
Hence power P D 5.65
2
10 D 319 W
Figure 34.11
Problem 3. Determine, for the bridge network shown in
Figure 34.12, (a) the value of the single equivalent resistance that
replaces the network between terminals A and B, (b) the current
supplied by the 52 V source, and (c) the current flowing in the 8
resistance.
(a) In Figure 34.12, no resistances are directly in parallel or directly
in series with each other. However, ACD and BCD are both delta
connections and either may be converted into an equivalent star
connection. The delta network BCD is redrawn in Figure 34.13(a)
and is transformed into an equivalent star connection as shown in
Figure 34.13(b), where
Z
1
D
816
8 C 16 C 40
D 2 (from equation (34.7))
Z
2
D
1640
8 C 16 C 40
D 10 (from equation (34.8))
Z
3
D
840
8 C 16 C 40
D 5 (from equation (34.9))
Figure 34.12