Star to Delta Conversion: Transformation, Formula, Diagram

In an electrical network, the connection of three branches can be done in different forms however most commonly used methods are star connection otherwise delta connection. A star connection can be defined as the three branches of a network can be commonly connected to a mutual point in Y -model. Similarly, a delta connection can be defined as; the three branches of a network are connected in a closed loop in the delta model. But, these connections can be changed from one model to another model. These two conversions are mainly used to make simpler of complex networks. This article discusses an overview of the star to delta conversion as well as a delta to star connection.

Star to Delta Conversion and Delta to Star Conversion

The typical three-phase networks use two main methods by names which specify the way in which resistances are allied. In a star connection of the network, the circuit can be connected in symbol ‘∆’ model, similarly in a delta connection of the network; the circuit can be connected in symbol ‘∆’. We know that we can change the T-resistor circuit into the Y-type circuit for generating equivalent Y- model network. Similarly, we can change the п-resistor circuit for generating equivalent ∆- model network. So now it is very clear what is a star network circuit and delta network circuit, and how they transform into Y- model network as well as ∆- model network by using T-resistor and п-resistor circuits.

Star to Delta Conversion

In star to delta conversion, the T-resistor circuit can be transformed to Y-type circuit to generate an equivalent Y- model circuit. The star to delta conversion can be defined as the value of the resistor on any one side of the Delta network, and the addition of all the two resistor product combinations in the stat network circuit separate with the star resistor which is placed straightly opposite to the delta resistor being found. The star-delta transformation derivation is discussed below.

Star to Delta Conversion
Star to Delta Conversion

For resistor A = XY + YZ + ZX/Z

For resistor B = XY + YZ + ZX/Y

For resistor C = XY + YZ + ZX/X

By separating out every equation with the denominator value we finish with 3-separate conversion formulas that can be utilized to change any Delta resistive circuit into an equivalent star circuit that is shown below.

For resistor A = XY + YZ + ZX/Z = XY/Z + YZ/Z + ZX/Z = (XY/Z) +Y+X

For resistor B = XY + YZ + ZX/Y = XY/Y + YZ/Y + ZX/Y= (ZX/Y) + X+Z

For resistor C = XY + YZ + ZX/X = XY/X + YZ/X + ZX/X= (YZ/X) +Z+Y

So, the final equations for the star to delta conversion are

A= (XY/Z) +Y+X, B= (ZX/Y) + X+Z, C= (YZ/X) +Z+Y

In this type of conversion, if the entire resistors values in the star connection are equal then the resistors in the delta network will be thrice of the star network resistors.

Resistors in Delta Network = 3 * Resistors in Star Network

For Example

The star-delta transformation problems are the best examples to understand the concept. The resistors in star network are denoted with X, Y, Z, and the values of these resistors are X= 80 ohms, Y= 120 ohms, and Z = 40 ohms, then the A and B and C values are followed.

A= (XY/Z) +Y+X

X= 80 ohms, Y= 120 ohms, and Z = 40 ohms

Substitute these values in the above formula

A = (80 X 120/40) + 120 + 80 = 240 + 120 + 80 = 440 ohms

B= (ZX/Y) + X+Z

Substitute these values in the above formula

B = (40X80/120) + 80 + 40 = 27 + 120 = 147 ohms

C= (YZ/X) +Z+Y

Substitute these values in the above formula

C = (120 x 40/80) + 40 + 120 = 60 + 160 = 220 ohms

Delta to Star Conversion

In delta to star conversion, the ∆-resistor circuit can be transformed to Y-type circuit to generate an equivalent Y- model circuit. For this, we require to derive a conversion formula for comparing the different resistors to every other among the different terminals. The delta star transformation derivation is discussed below.

Delta to Star Conversion
Delta to Star Conversion

Evaluate the resistances among the two terminals like 1 & 2.

X + Y = A in parallel with B +C

X + Y = A (B+C)/ A+B+C (Equation-1)

Evaluate the resistances among the two terminals like 2 & 3.

Y + Z = C in parallel with A + B

Y + Z = C (A + B)/ A + B + C (Equation-2)

Evaluate the resistances among the two terminals like 1 & 3.

X + Z = B in parallel with A + C

X + Z = B (A + C)/ A + B + C (Equation-3)

Subtract from equation-3 to equation-2.

EQ3- EQ2 = (X + Z) – (Y + Z)

= (B (A + C)/ A + B + C) – (C (A + B)/ A + B + C)

= (BA + BC/ A+B+C) – (CA + CB/ A+B+C)

(X-Y) = BA-CA/ A + B+C

Then, rewrite the equation will give

(X + Y) = AB + AC/ A +B +C

Add (X-Y) and (X+Y) then we can get

= (BA-CA/ A + B+C ) + (AB + AC/ A +B +C)

2X = 2AB/A +B+C => X = AB/ A+B+C

Similarly, the Y and Z values will be like this

Y = AC/ A+B+C

Z = BC/ A+B+C

So, the final equations for delta to star conversion are

X = AB/ A+B+C, Y = AC/ A+B+C, Z = BC/ A+B+C

In this type of conversion, if the three resistor values in the delta are equal then the resistors in the star network will be one by third of the delta network resistors.

Resistors in star network = 1/3(Resistors in delta network)

For Example

The resistors in delta network are denoted with X, Y, Z, and the values of these resistors are A= 30 ohms, B= 40 ohms, and C = 20 ohms, then the A and B and C values are followed.

X = AB/ A+B+C = 30 X 40/ 30 +40 +20 = 120/90 = 1.33 ohms

Y = AC/ A+B+C = 30 X 20/30 +40 +20 = 60/90 = 0.66 ohms

Z = BC/ A+B+C = 40 X 20/30 +40 +20 = 80/90 = 0.88 ohms

Thus, this is all about the star to delta conversion as well as the delta to star conversion. From the above information, finally, we can conclude that these two conversion methods can allow us to change one kind of circuit network into other kinds of circuit network. Here is a question for you, what are the  star delta transformation applications?

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