ELL 100 - Introduction to Electrical Engineering
LECTURE 4 : CIRCUIT ANALYSIS
DELTA – STAR TRANSFORMATIONS
Outline
Introduction
Star Connection
Delta Connection
Delta to Star Transformation
Star to Delta Transformation
Equivalent Resistance of Circuit
Exercise/Numerical Analysis
2
INTRODUCTION
Star connections is generally used in long distance transmission lines as
insulation requirement is less in star connection.
Transmission Network
3
INTRODUCTION
Delta connections are generally used in distribution networks for short
distances.
Distribution Network
4
INTRODUCTION
Alternators and generators are usually star connected.
600 MW Turbo-Generator at power plant
5
INTRODUCTION
Transformer windings are connected in Star/Delta Connections.
A Three Phase Transformer with Name Plate
6
INTRODUCTION
Generating transformer near to power plant generator are connected in
star connection to provide grounding protection.
Generating Transformer
7
INTRODUCTION
AC motors winding are connected in star/delta connection depending on
requirement and application.
Three Phase Induction Motor Winding
8
INTRODUCTION
Star and Delta connections are used in starting of three phase induction
motors using STAR-DELTA Starter.
Star Delta Starter for Three phase Induction Motor
9
INTRODUCTION
Delta-Star Starters are installed in cement industries for high inertial load
applications.
Cement Industry
10
INTRODUCTION
Power capacitors in 3 phase capacitor bank connections are either delta
connected or star (wye) connected.
Delta connected capacitor bank
11
INTRODUCTION
The application of such connection is also used in high voltage direct
current (HVDC) systems.
Generating Transformer
12
INTRODUCTION
The application of such connection is also used in Wheatstone bridge
resistance measurement device.
Wheatstone Bridge
13
INTRODUCTION
STAR/Delta transformations and equivalent circuit calculations help in
simplification and understanding of complex electrical circuits.
Complex Electric Circuits
14
INTRODUCTION
Star/Delta connection is an arrangement of passive elements R, L and C
such that the formed shape resembles a star or a delta symbol.
These connection are neither series and nor parallel.
Such connections are simplified using star-to-delta or delta-to-star
conversion.
Such connections are found in complex DC circuits, full bridge
rectifiers.
Such connections has larger application in three phase AC system.
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STAR CONNECTION
A star network is rearranged form of Tee (T) network.
16
STAR/WYE (Y) CONNECTION
Three ends of resistors are connected in wye (Y) or star fashion. A
common node point of star connection is known as neutral.
17
STAR CONNECTION
Three ways in which star connection may appear in a circuit.
18
DELTA CONNECTION
When three resistors are connected in a fashion to form a closed mesh Δ,
connection formed is known as Delta Connection.
19
DELTA CONNECTION
Three ways in which delta connection may appear in a circuit.
20
DELTA TO STAR TRANSFORMATION
Three resistors RAB , RBC and RCA connected in delta form and its equivalent
star connection is shown below.
Delta and its equivalent Star
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DELTA TO STAR TRANSFORMATION
• Two arrangements shown are electrically equivalent.
• Resistance between A and B for star = Resistance between A and
B for delta.
• Therefore,
RA RB RAB ||( RBC RCA ) (1)
RAB ( RBC RCA ) (2)
RA RB
RAB RBC RCA
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DELTA TO STAR TRANSFORMATION
• Similarly for resistance between two terminals B-C and C-A,
RBC ( RCA RAB )
RB RC (3)
RAB RBC RCA
RCA ( RAB RBC )
RC RA (4)
RAB RBC RCA
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DELTA TO STAR TRANSFORMATION
• The objective is to find RA , RB and RC in terms of RAB , RBC and RCA .
• Subtracting (3) from (2) and adding to (4) we obtain,
RAB RCA
RA (5)
RAB RBC RCA
RBC RAB
RB (6)
RAB RBC RCA
RCA RBC
RC (7)
RAB RBC RCA
24
DELTA TO STAR TRANSFORMATION
• Easy way to remember delta to star transformation is,
Product of two adjacent arms of
Any arm of star connection =
Sum of arms of
25
STAR TO DELTA TRANSFORMATION
Three resistors RA , RB and RC connected in star formation and its
equivalent delta connection is shown below
Star and its Equivalent Delta
26
STAR TO DELTA TRANSFORMATION
• Dividing (5) by (6) we obtain,
RA RCA
(8)
RB RBC
RA RBC
RCA (9)
RB
• Dividing (5) by (7) we obtain,
RA RAB
(10)
RC RBC
RA RBC
RAB (11)
RC
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STAR TO DELTA TRANSFORMATION
• Substituting (9) and (11) into (5),
RB RC
RBC RB RC (12)
RA
• Similarly,
RA RB (13)
RAB RA RB
RC
RC RA
RCA RC RA (14)
RB
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STAR TO DELTA TRANSFORMATION
• Easy way to remember star to delta transformation is,
Resistance between two terminals of Δ =
Sum of star resistances connected to those terminals +
product of same two resistances divided by the third
29
STAR\DELTA TRANSFORMATION
• If a star network has all resistances equal to R, its equivalent
delta has all resistances equal to ?
• If a delta network has all resistances equal to R, its equivalent
star has all resistances equal to ?
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STAR\DELTA TRANSFORMATION
• If a star network has all resistances equal to R, its equivalent
delta has all resistances equal to 3R.
• If a delta network has all resistances equal to R, its equivalent
star has all resistances equal to R/3.
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SUMMARY
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EQUIVALENT RESISTANCE
• The equivalent resistance of a circuit or network between its any
two points (or terminals) is that single resistance which can replace
the entire circuit between these points (or terminals).
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DEFINITIONS
• STAR/DELTA CIRCUITS: These circuits generally possess
star/delta configurations and needs to be simplified using necessary
transformations and are converted into series parallel circuits.
Neither Series Nor Parallel Circuit
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EQUIVALENT RESISTANCE OF STAR/DELTA CIRCUIT
• The circuit is a combination of neither series nor parallel circuits.
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EQUIVALENT RESISTANCE OF STAR/DELTA CIRCUIT
• RULE: Such circuit form star/delta. Use star delta transformation
and convert to equivalent series-parallel circuit.
• Changing Delta formed by points A,B,C into equivalent Star,
R R
RY R/3
RRR
• The circuit formed is now combination of series-parallel circuit.
• The series-parallel circuit is further simplified to series circuit.
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EQUIVALENT RESISTANCE OF STAR DELTA CIRCUIT
The circuit now becomes a simple series-parallel circuit and can be
solved easily. A
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EXERCISE/NUMERICAL ANALYSIS
Q. Convert the Y network to an equivalent Δ network.
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EXERCISE/NUMERICAL ANALYSIS
Soln:
7.5 5
Ra 7.5 5 25 ohms
3
7.5 3
Rb 7.5 3 15 ohms
5
5 3
Rc 5 3 10 ohms
7.5
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EXERCISE/NUMERICAL ANALYSIS
Q. Convert the Δ network to an equivalent Y network.
40
EXERCISE/NUMERICAL ANALYSIS
Soln:
Rb Rc 10 25
R1 5 ohms
Ra Rb Rc 15 10 25
Rc Ra 25 15
R2 7.5 ohms
Ra Rb Rc 50
Ra Rb 15 10
R3 3ohms
Ra Rb Rc 50
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EXERCISE/NUMERICAL ANALYSIS
Q. Using delta/star transformation, find equivalent resistance across AC.
42
EXERCISE/NUMERICAL ANALYSIS
Soln: Delta can be replaced by equivalent star-connected resistances,
RAB RDA 10 20
R1 2.86 ohms
RAB RDA RBD 10 40 20
RAB RBD 10 40
R2 5.72 ohms
RAB RDA RBD 10 40 20
RDA RBD 10 40
R3 11.4 ohms
RAB RDA RBD 10 40 20
43
EXERCISE/NUMERICAL ANALYSIS
Figure now becomes,
(30 5.72) (15 11.4)
RAC 2.86 18.04 ohms
30 5.72 15 11.4
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EXERCISE/NUMERICAL ANALYSIS
Q. Calculate equivalent resistance across terminals A and B.
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EXERCISE/NUMERICAL ANALYSIS
Soln: Converting inner STAR (3 ohms, 3 ohms and 1 ohms ) into Delta.
3 3
R1 3 3 15 ohms
1
3 1
R2 3 1 5 ohms
3
1 3
R3 1 3 5 ohms
3
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EXERCISE/NUMERICAL ANALYSIS
Circuit now becomes,
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EXERCISE/NUMERICAL ANALYSIS
Delta-connected resistances 1 Ω, 5 Ω and 8 are converted in star,
1 8 4
R
'
ohms
1 5 8 7
1
5 1 5
R2
'
ohms
1 5 8 14
8 5 20
R3 '
ohms
1 5 8 7
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EXERCISE/NUMERICAL ANALYSIS
Circuit now becomes,
4 5 || 20 20
RAB 2.5 7.6 10 ohms
7 14 7 9
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EXERCISE/NUMERICAL ANALYSIS
Q. Calculate equivalent resistance across terminals A and B.
50
EXERCISE/NUMERICAL ANALYSIS
Soln: Replacing inner STAR into DELTA.
51
EXERCISE/NUMERICAL ANALYSIS
15.8 ohm is in parallel with 5 ohm and 26.3 ohm is in parallel with
4 ohm, circuit becomes
52
EXERCISE/NUMERICAL ANALYSIS
Converting upper delta into star,
53
EXERCISE/NUMERICAL ANALYSIS
Now equivalent resistance can be calculated as,
Req (3·8 + 2·98) || (1·99 + 3·5) + 1·2
4.23 ohms
54
EXERCISE/NUMERICAL ANALYSIS
Q. Obtain the equivalent resistance Rab for the circuit and use it to
find current i.
55
EXERCISE/NUMERICAL ANALYSIS
Soln: In this circuit, there are two Y networks and three Δ networks.
Transforming just one of these will simplify the circuit.
56
EXERCISE/NUMERICAL ANALYSIS
We convert the Y-network comprising the 5-Ω, 10-Ω, and 20-Ω resistors
into delta.
5 10
R1 5 10 17.5 ohms
20
(comes in parallel with 12.5 Ω)
5 20
R2 5 20 35 ohms
10
(comes in parallel with 15 Ω)
10 20
R3 10 20 70 ohms
5
(comes in parallel with 30 Ω)
57
EXERCISE/NUMERICAL ANALYSIS
Combining the three pairs of resistors in parallel, we obtain.
(12.5 || 17.5 Ω) Rab (7.292 10.5)||21
17.792 21
=9.632 ohms
17.792 21
(30 || 70 Ω)
(15 || 35 Ω) vs 120
i 12.458 A
Rab 9.632
58
EXERCISE/NUMERICAL ANALYSIS
Q. Determine the load current in branch EF in the circuit shown.
59
EXERCISE/NUMERICAL ANALYSIS
Sol. ACGA forms delta, Converting it to equivalent star.
200 500 500 200 200 200
RAN 111.11ohms RGN 111.11ohms RCN 44.44ohms
900 900 900
60
EXERCISE/NUMERICAL ANALYSIS
Circuit can be redrawn as
RNEF 111.11 600 711.11ohms
RND 600 44.44 644.44ohms
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EXERCISE/NUMERICAL ANALYSIS
Branches NCD and NEF are in parallel, 711.11 || 644.44=338 ohms.
V 100
Total current I in the circuit = I 0.222 A
Req 111.11 338
62
EXERCISE/NUMERICAL ANALYSIS
To obtain current in branch EF, we apply current division formula.
RNCD
I NEF I
RNCD RNEF
644.44
0.222
711.11 644.44
0.1055 A
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EXERCISE
Q. A square and its diagonals are made of a uniform covered wire. The
resistance of each side is 1 Ω and that of each diagonal is 1·414 Ω.
Determine the resistance between two opposite corners of the square.
64
EXERCISE
Q. Determine the resistance between the terminals A and B of the network.
65
EXERCISE
Q. Find the current in 10 Ω resistor in the network shown by star-delta
transformation.
66
EXERCISE
Q. Using star/delta transformation, determine the value of R for the
network shown such that 4Ω resistor consumes the maximum power.
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REFERENCES
[1] Charles. K. Alexander and Matthew Sadiku “Fundamental of Electric Circuits”,
McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121, ch. 2 and 3.
[2] Edward Hughes, John Hiley, Keith Brown and Ian McKenzie Smith Hughes
“Electrical & Electronic Technology”, Pearson Education Limited, Edinburgh Gate,
England, ch. 2 and 3.
[3] V. K. Mehta and Rohit Mehta “Basic Electrical Engineering”, S. Chand & Company
Pvt. Ltd., Ram Nagar, New Delhi, ch. 2.
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