star-delta transformation

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This note is for private circulation only. Compiled by Dr. A. Kar Gupta, [email protected]

Star Delta Conversion

What is a STAR Network?

Given a T-type connection of resistors (or capacitors or impedances) as follows, we can make an equivalent STAR-type connection from that.

Fig.1

What is a DELTA Network?

Given a Pi-type connection of resistors (or capacitors or impedances in general), we can make an equivalent DELTA-type connection from that.

Fig.2

STAR- DELTA Transformation

We can solve simple series, parallel or bridge type resistor networks using Kirchhoff’s current and voltage laws and other techniques. But for 3-phase circuits like above, we can simplify our analysis by some mathematical techniques and tricks.

Now if a 3-phase, 3-wire supply or even a 3-phase load is connected in one type of configuration, it can be easily transformed or changed into an equivalent configuration of the other type: from STAR to DELTA or from DELTA to STAR. This is called Star Delta conversion or

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Delta Star conversion. In this process, we produce mathematical relationships between the various resistors of the configurations.

DELTA STAR

To convert a delta network to an equivalent star network we need to derive a transformation formula for equating the various resistors to each other between the various terminals. Consider the circuit below.

Fig.3

Resistances between terminals 1 and 2:

(1)


(2)


(3)

Now a little manipulation gives us the following.

From (3) – (2),

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(4)

Adding (1) and (4),

(5)

Similarly,

(6)

(7)

Note: When we convert a STAR network into a DELTA one, the denominator of the all the transformation formulas are the same, that is the sum of all the resistors (or impedances), in the DELTA configuration.

Also note, if the three resistors are same, , we have

.

STAR DELTA

Fig.4

𝑃 𝐴𝐵

𝐴 𝐵 𝐶

𝑄 𝐴𝐶

𝐴 𝐵 𝐶

𝑅 𝐵𝐶

𝐴 𝐵 𝐶

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Now we can easily find the impedances A, B, C in the Delta configuration as P, Q, R are given in Star network. From expressions (5), (6) and (7), if we consider (5)x(6)+(6)x(7)+(7)x(8), we get

(8)

(8) (7) leads to (9)

Similarly,

(10)

(11)

Note: When , we have .

Star-Delta conversion with Capacitors:

If the above formulae are with resistors, we have to put in place of resistance, where is

capacitance.

For example, if we have a Star network with the capacitors, , and in place of , and

respectively in Fig.4, we will have the equivalent capacitors

(

) and so on.

[Note: For a series combination:

and for a parallel combination:

]

𝐴 𝑃𝑄 𝑄𝑅 𝑅𝑃

𝑅

𝐵 𝑃𝑄 𝑄𝑅 𝑅𝑃

𝑄

𝐶 𝑃𝑄 𝑄𝑅 𝑅𝑃

𝑃

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Problems to be solved by using the STAR-DELTA transformation:

Ans.:

Ideas are given below how to go about: (follow the next two pages)

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star-delta transformation

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