Star Delta and Delta Star Conversion

STAR DELTA AND DELTA - STAR CONVERSION

Consider the circuit in figure 1.19. The total effective resistance between A and B is easily found since R₁ and R₂ are in series and R₃ and R₄ are also in series. The two combinations are in parallel. If a resistor R₅ is now added between CD as in figure 1.20, the resistance between A and B cannot be found out as easily as before.

However, if we can convert the three resistors R₁, R₅ and R₃ which form a closed loop, (also called a Δ) in to a star (Y) as shown in dotted lines, then RA, RC and RD can replace R₁, R₅ and R₃. This is shown in figure 1.21. It is easy to calculate the effective resistance between A and B now. (R_c + R₂) is in parallel with (R_D + R₄). This parallel combination is in series with R_A.

Delta to star conversion

Let the three resistors RAB, RBC and RCA from a A. It is required to convert this into its equivalent Y consisting of R_A, R_R and R_C. This implies that for all purposes, the resistors RAB, RBC and RCA can be the replaced at terminals, A, B and C by RA, RB and RC. So, the problem reduces to finding out the values of Y i.e. the values of R_A, R_B and R_C, for a given R_AB, R_AB and R_CA. It is shown in figure 1.22.

To do this, we start by finding out the equivalent resistance between A and B in the A and Y.

For the Δ, the effective resistance between A and B is the equivalent of R_AB, in parallel with the series combination of R_BC and R_CA.

Effective resistance between A and B in the Δ.

The effective resistance between A and B in the Y

Star to delta conversion

To obtain the relationships necessary in the conversion from star to delta, first divide equation (31) by equation (32).

EXAMPLE 77: In the Wheatstone bridge circuit of figure find the effective resistance between PQ. Find the current supplied by a 10 V battery connected to PQ.

Solution :

The circuit then reduces to the following form.

The two 1.5 Ω resistors are in series and this combination is in parallel with the

0.6 2 and 0.4 Ω resistors which are in series.

Therefore, the total effective resistance between P and Q. gnited adve

= 0.75 + 1 + (3 × 1) / ( 3 × 1)

= 2.5 Ω

If a 10 V battery is attached to PQ, current drawn

10/2.5 = 4 A

EXAMPLE 78: Transform a set of identical resistors connected (a) in A to Y and (b) in Y to Δ.

Solution:

(a) Figure shows three equal resistors (value R) connected in A. The star equivalent to be attached to each of the terminals will be

(b) Figure shows the three resistor in Y. The equivalent is

EXAMPLE 79: In figure, find the effective resistance between P and Q by Star – Delta conversion.

Converting the 102, 2002 and 40 Q resistors forming a Y at A (star-point) in to a A, we have the circuit in figure.

The resistance between P and Q is calculated from figure as

EXAMPLE 80: In the circuit given in figure, find the effective resistance between A and B.

Solution:

This problem can be solved by converting the two 20 22 and 60 22 resistors in A into a Y as shown in figure. The values of the equivalent Y are

EXAMPLE 81: In the circuit of figure six resistors are connected to form Δ and a Y. Find the effective resistance between (a) A and B (b) A and N.

Solution :

(a) Convert the star connected resistor (29, 30 and 60) to delta. It is shown in figure.

Now, the circuit becomes

(b) To find the effective resistance between A and N, convert the 29, 69 and 129 forming a Δ to Y

 EXAMPLE 82: Find the equivalent resistance between A and B of the circuit shown in figure.

EXAMPLE 83: Convert the given A network in to star connection.

EXAMPLE 84: For the circuit shown in figure, determine the equivalent resistance by using star-delta transformation.

Solution :

The above circuit consists of two star circuits, one consisting of 5 Ω, 3 Ω and 4 Ω resistors, and the other consisting of 6 Ω, 4 Ω and 8 Ω resistors. We convert the star circuits into delta circuits, so that the two delta circuits are in parallel.

In the above circuit, the three resistors, 9.4 Ω, 17.3 Ω and 10 Ω are connected in parallel.