### Star - Delta Transformations

The three terminal network shown in figure 1.10 (a) is referred as delta (Δ) network and the network shown in figure 1.10 (b) is referred as star or wye (Y) network. For the analysis of an electrical network and yo reduce a complex network in to a simpler one, the the star to delta and delta to star transformation is highle useful. The expression for the conversion of star to delta and vice versa is given below

Diagram Coping Problem

Delta (Δ) to Star or Wye (Y) Network:

The equivalent impedance of the delta network between terminals while open circuit the other terminal is expressed in equations (29), (30) and (31).

The equivalent impedance between the terminal (1) and (2) is

$\small =\frac{Z_{12}(Z_{23}+Z_{31})}{Z_{12}+Z_{23}+Z_{31}}$                           (29)

The equivalent impedance between the terminal (2) and (3) is

$\small =\frac{Z_{23}(Z_{12}+Z_{31})}{Z_{12}+Z_{23}+Z_{31}}$                           (30)

The equivalent impedance between the terminal (3) and (1) is

$\small =\frac{Z_{31}(Z_{12}+Z_{23})}{Z_{12}+Z_{23}+Z_{31}}$                            (31)

The equivalent impedance of the wye network between terminals while open circuit the other terminal is expressed in equations (32), (33) and (34).

The equivalent impedance between the terminal (1) and (2) is

$\small =Z_1+Z_2$                           (32)

The equivalent impedance between the terminal (2) and (3) is

$\small =Z_2+Z_3$                            (33)

The equivalent impedance between the terminal (3) and (1) is

$\small =Z_1+Z_3$                           (34)

Equating the respective equations of delta and wye network results in

$\small Z_1+Z_2=\frac{Z_{12}(Z_{23}+Z_{31})}{Z_{12}+Z_{23}+Z_{31}}$                           (35)
$\small Z_2+Z_3=\frac{Z_{23}(Z_{12}+Z_{31})}{Z_{12}+Z_{23}+Z_{31}}$                           (36)
$\small Z_1+Z_3=\frac{Z_{31}(Z_{12}+Z_{23})}{Z_{12}+Z_{23}+Z_{31}}$                           (37)

Substracting equation (36) from (37), we get

$\small Z_1-Z_2=\frac{Z_{31}Z_{12}-Z_{23}Z_{12}}{Z_{12}+Z_{23}+Z_{31}}$                           (38)

Adding equation (38) and (35), we get

$\small Z_1=\frac{Z_{12}Z_{31}}{Z_{12}+Z_{23}+Z_{31}}$                           (39)

Similarly, we get

$\small Z_2=\frac{Z_{12}Z_{23}}{Z_{12}+Z_{23}+Z_{31}}$                           (40)
and

$\small Z_3=\frac{Z_{23}Z_{31}}{Z_{12}+Z_{23}+Z_{31}}$                           (41)

Equations (39), (40) and (41) are used for delta to wye network conversion.

Delta (Δ) to Star or Wye (Y) Network:

Cross multiplying the equations (39), (40) and (41) we get,

$\small Z_1(Z_{12}+Z_{23}+Z_{31})=Z_{12}Z_{31}$                           (42)
$\small Z_2(Z_{12}+Z_{23}+Z_{31})=Z_{12}Z_{23}$                           (43)
$\small Z_3(Z_{12}+Z_{23}+Z_{31})=Z_{23}Z_{31}$                           (44)

Dividing equation (43) by (44) we get

$\small Z_{12}=\frac{Z_2Z_{31}}{Z_3}$

Similarly, we get

$\small Z_{23}=\frac{Z_2Z_{31}}{Z_3}$