Star Delta Transformation (AC)

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. In three phase circuit, connections can be given in two types:
  • Star or Wye(Y) connection
  • Delta(Δ) connection
The $\Delta -Star (Y)$ or $Star (Y) - \Delta$, (or $\pi -T$, $T- \pi$ as defined in Chapter 8) conversions for ac circuits will not be derived here since the development corresponds exactly with that for dc circuits. Taking the $\Delta -Y$ configuration shown in [Fig. 1],
Delta - Y configuration
Fig. 1: $Star - \Delta$ configuration
we find the general equations for the impedances of the Y in terms of those for the $\Delta$:
$$Z_1 = {Z_BZ_C \over Z_A + Z_B + Z_C}$$
$$Z_2 = {Z_CZ_A \over Z_A + Z_B + Z_C}$$
$$Z_3 = {Z_AZ_B \over Z_A + Z_B + Z_C}$$
For the impedances of the $\Delta$ in terms of those for the Y, the equations are
$$Z_A = {Z_1Z_2 + Z_1Z_3 +Z_2Z_3 \over Z_1}$$
$$Z_B = {Z_1Z_2 + Z_1Z_3 +Z_2Z_3 \over Z_2}$$
$$Z_C = {Z_1Z_2 + Z_1Z_3 +Z_2Z_3 \over Z_3}$$
Note that each impedance of the Y is equal to the product of the impedances in the two closest branches of the $\Delta$, divided by the sum of the impedances in the $\Delta$.
Further, the value of each impedance of the $\Delta$ is equal to the sum of the possible product combinations of the impedances of the Y, divided by the impedances of the Y farthest from the impedance to be determined.
Drawn in different forms ([Fig. 2]), they are also referred to as the T and $\pi$ configurations.
Delta - Y configurationDelta - Y configuration
Fig. 2: The T and $\pi$ configurations.
In the study of dc networks, we found that if all of the resistors of the $\Delta$ or Y were the same, the conversion from one to the other could be accomplished using the equation
$$ R_{\Delta} = 3 R_Y \,\, or \,\, R_Y = {R_{\Delta} \over 3}$$
For ac networks,
$$ Z_{\Delta} = 3 Z_Y \,\, or \,\, Z_Y = {Z_{\Delta} \over 3}$$
Be careful when using this simplified form. It is not sufficient for all the impedances of the $\Delta$ or Y to be of the same magnitude: The angle associated with each must also be the same.
Example 1: Find the total impedance $Z_T$ of the network of [Fig. 3].
Fig. 3: Converting the upper $\Delta$ of a bridge configuration to a Y
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