Star Delta Transformation

In this article, we will learn about the start connection and delta connection of resistors and how to convert a delta-connected resistive network into a star-connected network and vice-versa.

Sometimes, during the analysis of electrical networks, we find many resistive networks that are neither connected in parallel nor in series like in the case of a Wheatstone bridge. These networks are the star network or Y-network or T-network and delta network or ∆-network or ∏-network. Both star-network and delta-network are three-terminal networks.

Generally, the star and delta networks are used in three-phase systems, electrical filters, and matching networks.

In this article, our main interest is in how to identify the star and delta network when they occur as part of a network and how to apply the star-delta transformation to simplify the network analysis.

Basically, the star-delta transformation is used to determine the equivalent resistance of a complicated electrical network, where it is not possible to determine it using the series and parallel simplification methods.

What is a Star Connection?

A star connection or star network is a type of connection where one end of each of three resistors is connected to a common point and another end of each resistor is taken out as the terminal of the network. A typical star network of resistors is shown in the figure.

What is a Delta Connection?

A type of resistive network in which three resistors connected in series form a triangular arrangement and the supply is input at the three junction points is called a delta network or delta connection. The following figure shows a typical structure of the delta network.

Delta to Star Conversion

Sometimes it is more convenient to work with a star network instead of a delta network. In such a case, we can replace a delta network with a star network by finding the equivalent resistances.

In order to determine the equivalent resistances in the star network, we compare the two networks and make sure that the resistance between each pair of terminals in the delta network is the same as the resistance between the same pair of terminals in the star network.

Explanation:

Consider a star network and a delta network as shown in the figure-3. We will convert the delta network into its equivalent star network.

In the star network, the resistance between each pair of terminals is given by,

`\R_{ab}=R_1+R_2\ \ \ \ \ …(1)`

`\R_{bc}=R_2+R_3\ \ \ \ \ …(2)`

`\R_{ca}=R_1+R_3\ \ \ \ \ …(3)`

Similarly, in the delta network, the resistance between each pair of terminals is given by,

`\R_{ab}=(R_A (R_B+R_C ))/(R_A+R_B+R_C )=(R_A R_B+R_A R_C)/(R_A+R_B+R_C )\ \ \ \ …(4)`

`\R_{bc}=(R_B (R_A+R_C ))/(R_A+R_B+R_C )=(R_A R_B+R_B R_C)/(R_A+R_B+R_C )\ \ \ \ …(5)`

`\R_{ca}=(R_C (R_A+R_B ))/(R_A+R_B+R_C )=(R_A R_C+R_B R_C)/(R_A+R_B+R_C )\ \ \ \ …(6)`

Now, in order to be the two networks equivalent, the resistances in the star and delta across appropriate terminals should be equal. Therefore, we have,

`\R_1+R_2=(R_A R_B+R_A R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(7)`

`\R_2+R_3=(R_A R_B+R_B R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(8)`

`\R_1+R_3=(R_A R_C+R_B R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(9)`

Next, in order to find the equivalent resistance between terminals ab of the star network in terms of resistances of the delta network, we subtract the eq. (8) from (7) and adding the difference to the (9), we get,

`\(R_1+R_2 )-(R_2+R_3 )+(R_1+R_3 )=`

`\((R_A R_B+R_A R_C)/(R_A+R_B+R_C ))-((R_A R_B+R_B R_C)/(R_A+R_B+R_C ))+((R_A R_C+R_B R_C)/(R_A+R_B+R_C ))`

`\⟹R_1=(R_A R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(10)`

Similarly,

`\R_2=(R_A R_B)/(R_A+R_B+R_C )\ \ \ \ \ …(11)`

And

`\R_3=(R_B R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(12)`

Hence, the resistances R₁, R_2, and R₃ are the equivalent resistances in the star network of the resistances connected in the delta network and denoted by R_A, R_B, and R_C.

How to Remember?

In order to remember the formula of delta-to-star conversion, consider delta and star networks as shown in the above figure. Now, the resistances of an equivalent star network can be remembered as

`\R_k=("Product of delta connected resistances connected to terminal "k)/("Sum of all delta connected resistances")`

Therefore, for terminal k = a:

`\R_1=(R_A R_C)/(R_A+R_B+R_C )`

For terminal k = b:

`\R_2=(R_A R_B)/(R_A+R_B+R_C )`

For terminal k = c:

`\R_3=(R_B R_C)/(R_A+R_B+R_C )`

Hence, in this way, the delta-to-star transformation of a resistive network can be achieved.

Star to Delta Conversion

We can transform a star network into an equivalent delta network, if the resistances R₁, R_2, and R₃ are known in the given star network.

If the resistances of the star network (from the above section) are: 

`\R_1=(R_A R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(1)`

`\R_2=(R_A R_B)/(R_A+R_B+R_C )\ \ \ \ \ …(2)`

`\R_3=(R_B R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(3)`

Now, we multiply each two equations, and adding the three products, i.e. (1) × (2) + (2) × (3) + (3) × (1), we get,

`\R_1 R_2+R_2 R_3+R_3 R_1=(R_A R_C×R_A R_B)/(R_A+R_B+R_C )^2` 

`\+(R_A R_B×R_B R_C)/(R_A+R_B+R_C )^2 +(R_B R_C×R_A R_C)/(R_A+R_B+R_C )^2`

`\⟹R_1 R_2+R_2 R_3+R_3 R_1=(R_A^2 R_B R_C)/(R_A+R_B+R_C )^2` 

`\+(R_A R_B^2 R_C)/(R_A+R_B+R_C )^2 +(R_A R_B R_C^2)/(R_A+R_B+R_C )^2`

`\⟹R_1 R_2+R_2 R_3+R_3 R_1=(R_A R_B R_C (R_A+R_B+R_C ))/(R_A+R_B+R_C )^2`

`\⟹R_1 R_2+R_2 R_3+R_3 R_1=(R_A R_B R_C)/(R_A+R_B+R_C )\ \ \ \ \ …(4)`

Now, in order to get the value of any resistance of the delta network equivalent to the resistance of the star network. If we divide eq. (4) by (1), we get,

`\(R_1 R_2+R_2 R_3+R_3 R_1)/R_1 =(((R_A R_B R_C)/(R_A+R_B+R_C )))/(((R_A R_C)/(R_A+R_B+R_C )) )`

`\∴R_B=(R_1 R_2+R_2 R_3+R_3 R_1)/R_1\ \ \ \ \ …(5)`

Similarly, by dividing equation (4) by (2), we get,

`\R_C=(R_1 R_2+R_2 R_3+R_3 R_1)/R_2\ \ \ \ \ …(6)`

And dividing equation (4) by (3), we get,

`\R_A=(R_1 R_2+R_2 R_3+R_3 R_1)/R_3\ \ \ \ \ …(7)`

In this way, we can find the resistances R_A, R_B, and R_C, i.e. the equivalent resistances of the delta from the resistances of the given star network.

How to Remember?

In order to remember the formula of star-to-delta transformation, consider the network as shown in Figure-4. Then, the equivalent delta resistance can be remembered as

`\R_Δ=("Sum of all possible products of resistances of star network")/("Resistance of star network opposite to considered delta resistance")`

Therefore, we have,

`\R_A=(R_1 R_2+R_2 R_3+R_3 R_1)/R_3`

`\R_B=(R_1 R_2+R_2 R_3+R_3 R_1)/R_1`

`\R_C=(R_1 R_2+R_2 R_3+R_3 R_1)/R_2`

Therefore, in this article, we have studied about star-network, delta network, and the conversion of a star network into an equivalent delta network.