Star-delta Transformation 6y1a45

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

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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) Resistances between terminals 2 and 3:

(2) Resistances between terminals 1 and 3:

(3) Now a little manipulation gives us the following. From (3) – (2),

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

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(4) Adding (1) and (4),

𝑃

𝐴𝐵 𝐴 𝐵 𝐶

𝑄

𝐴𝐶 𝐴 𝐵 𝐶

(6)

𝑅

𝐵𝐶 𝐴 𝐵 𝐶

(7)

(5)

Similarly,

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, STAR

, we have

.

DELTA

Fig.4

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

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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

𝐴

Similarly,

𝐵

𝐶

Note: When

𝑃𝑄

𝑄𝑅 𝑅

𝑅𝑃

𝑃𝑄

𝑄𝑅 𝑄

𝑅𝑃

𝑄𝑅 𝑃

𝑅𝑃

𝑃𝑄

, we have

(9)

(10)

(11)

.

Star-Delta conversion with Capacitors: If the above formulae are with resistors, we have to put capacitance. For example, if we have a Star network with the capacitors, respectively in Fig.4, we will have the equivalent capacitors

(

in place of resistance, where

,

and

in place of ,

is

and

) and so on.

[Note: For a series combination:

and for a parallel combination:

]

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

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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)

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

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

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

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