Star Delta Transformation

There are two types of system available in electric circuit, single phase and three phase system. In single phase circuit, there will be only one phase, i.e the current will flow through only one wire and there will be one return path called neutral line to complete the circuit. So in single phase minimum amount of power can be transported. In 1882, new invention has been done on polyphase system, that more than one phase can be used for generating, transmitting and for load system. Three phase circuit is the polyphase system where three phases are send together from the generator to the load. In three phase circuit, connections can be given in two types:

• Star or Wye(Y) connection
• Delta(Δ) connection

Circuit configurations are often encountered in which the resistors do not appear to be in series or parallel. Under these conditions, it may be necessary to convert the circuit from one form to another to solve for any unknown quantities if mesh or nodal analysis is not applied. Two circuit configurations that often account for these difficulties are the Star (Y) and Delta (Δ) configurations depicted in [Fig. 1]. The purpose of this section is to develop the equations for converting from Star to Δ, or vice versa. This type of conversion normally leads to a network that can be solved using techniques such as those described in previous chapters. It is our purpose (referring to [Fig. 2]) to find some expression for $R_1$, $R_2$, and $R_3$ in terms of $R_A$, $R_B$, and $R_C$, and vice versa, that will ensure that the resistance between any two terminals of the Y configuration will be the same with the Δ configuration inserted in place of the Y configuration (and vice versa). If the two circuits are to be equivalent, the total resistance between any two terminals must be the same. Consider terminals a-c in the $Δ-Y$ configurations in [Fig. 3]. Let us first assume that we want to convert the Δ ($R_A$, $R_B$, $R_C$) to the Y ($R_1$, $R_2$, $R_3$). This requires that we have a relationship for $R_1$, $R_2$, and $R_3$ in terms of $R_A$, $R_B$, and $R_C$. If the resistance is to be the same between terminals a-c for both the Δ and the Y, the following must be true: so that Using the same approach for a-b and b-c, we obtain the following relationships: and Subtracting Eq.(1) from Eq.(2), we have so that Subtracting Eq. (4) from Eq. (3) yields so that Following the same procedure for $R_1$ and $R_2$, we have and

Wye to Delta Conversion

To obtain the relationships necessary to convert from a Y to a Δ, first divide Eq. (5) by Eq. (6): or Then divide Eq. (5) by Eq. (7): or Substituting for $R_A$ and $R_B$ in Eq. (7) yields Placing these over a common denominator, we obtain and We follow the same procedure for RB and RA: and Let us consider what would occur if all the values of a Δ or Y were the same. If $R_A = R_B = R_C$, Eq. (8) would become (using RA only) the following: In general, therefore, or which indicates that for a Star(Wye) of three equal resistors, the value of each resistor of the Δ is equal to three times the value of any resistor of the Star (Y).