relation between line voltage and phase voltage in balanced star connected load

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

OR

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

Star-connection

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

All three voltages will have the 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 voltage (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 voltage ( VL ).

The relation between  VL and Vph

Phasor diagram for the phase voltage will be as shown below for the star connection. 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 the same magnitude but 1200 phase difference Now, the current through the neutral wire ( IN) will be the sum of all the three currents in the star connection.

Expression for power : For 3-phase load

Total power , P = 3  phase power


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