Balanced Star to Delta & Delta to Star Conversion

As we know any balanced Star connected system can be completely replaced by an equivalent Delta connected system. This means that a three phase Star connected balanced system having line voltage V_L and line current I_L can be replaced by an equivalent Delta connected system having phase voltage V_L and phase current I_L / 1.732.  But for replacing a balanced Star connected system with Delta connected system we need to know how to find the equivalent Delta connected system or Star connected system. In this post we will discuss this aspect.

Let the line voltage for Star connection be V_L and line current be I_L.

So, impedance Z_Y= Phase Voltage / Phase Current

But for Star connection, Phase Voltage = Line Voltage / 1.732 = V_L / 1.732

 and Phase Current = Line Current  = I_L

Therefore,

Impedance Z_Y = V_L/ (1.732I_L)  …………………(1)

Now we will calculate impedance for Delta connection assuming same voltage and current as in Star connection.

Impedance Z _Δ = Phase Voltage / Phase Current

But for Delta connection, Phase Voltage = Line Voltage = V_L

 and Phase Current = Line Current / 1.732  = I_L / 1.732

Therefore,

Impedance Z Δ = Phase Voltage / Phase Current

                         = V_L / (I_L/ 1.732)

                         = 1.732V_L / I_L 

But from equation (1),

VL = 1.732Z_YI_L

So, Impedance Z _Δ = 1.732x(1.732Z_YI_L) / I_L

                               = 3Z_Y

Thus we see that for balanced load,

Z Δ = 3ZY