Relation between phase voltage & line voltage in case of balanced star connected load

Q: Derive the relation between phase voltage and line voltage in case of balanced star connected load.

OR

Derive an expression for total power input for n balance 3 phase load in terms of line quantities and power factors.

Star-connection

As show in figure given below, three equal impedances (load) are also connected in star across star-connected supply. If all the 3 impedances arc equal, then it is known as balanced load.

All the three voltages will have same magnitude but 1200 phase difference. These voltages are measured between neutral and phase points (i.e. R, Y, B) and hence known as phase voltages (V ph ). Three more combinations are also possible. These arc voltages between (i) pts. R and Y VRY (ii) pts. Y and B,VYB (iii) pts. B and R, VBR . These are known as line voltages ( VL )  .

Relation between  VL and Vph

Phasor diagram  for the phase voltages will be as shown below. We will calculate any one of the line voltage. VRY is the voltage between R and Y .From the figure,

Hence, VRY  is obtained by adding ( -VY ) and The resultant VRY  will be 300 ahead of VR . Hence, VL is 300 ahead of V ph . The magnitude of VRY  can be calculated as,

Hence, magnitude of line voltage is magnitude under root 3 of phase voltage.

Relation-between  IL and Iph

The three currents IR , IY and IB can be calculated as,

Hence, all the three currents have same magnitude but 1200 phase difference Now, current through neutral wire ( IN) will be the sum of all the three currents.

Expression for power : For 3-phase load

Total power , P = 3  phase power


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