Unbalanced Three-phase Four-wire, Y-Connected Load

For the three-phase, four-wire, Y-connected load of Fig. 1, conditions are such that none of the load impedances are equal-hence we have an unbalanced polyphase load.
Unbalanced Y-connected load.
Fig. 1: Unbalanced Y-connected load.
Since the neutral is a common point between the load and source, no matter what the impedance of each phase of the load and source, the voltage across each phase is the phase voltage of the generator:
$$\mathbf{V}_{\phi}=\mathbf{E}_{\phi}$$The phase currents can therefore be determined by Ohm's law:$$\mathbf{I}_{\phi_{1}}=\frac{\mathbf{V}_{\phi_{1}}}{\mathbf{Z}_{1}}=\frac{\mathbf{E}_{\phi_{1}}}{\mathbf{Z}_{1}} \text { and so on }$$
The current in the neutral for any unbalanced system can then be found by applying Kirchhoff's current law at the common point $ n $:
$$\mathbf{I}_{N}=\mathbf{I}_{\phi_{1}}+\mathbf{I}_{\phi_{2}}+\mathbf{I}_{\phi_{3}}=\mathbf{I}_{L_{1}}+\mathbf{I}_{L_{2}}+\mathbf{I}_{L_{3}}$$
Because of the variety of equipment found in an industrial environment, both three-phase power and single-phase power are usually provided with the single-phase obtained off the three-phase system.
In addition, since the load on each phase is continually changing, a fourwire system (with a neutral) is normally employed to ensure steady voltage levels and to provide a path for the current resulting from an unbalanced load.
industrial supply.
Fig. 2: 3p/1p, 208-V/120-V industrial supply.
The system of Fig. 2 has a three-phase transformer dropping the line voltage from 13,800 V to 208 V. All the lower-power-demand loads such as lighting, wall outlets, security, etc., use the single-phase, 120-V line to neutral voltage. Higher power loads, such as air conditioners, electric ovens or dryers, etc., use the single-phase, 208 V available from line to line. For larger motors and special high demand equipment, the full three-phase power can be taken directly off the system, as shown in Fig. 2.
In the design and construction of a commercial establishment, the National Electric Code requires that every effort be made to ensure that the expected loads, whether they be single or multiphase, result in a total load that is as balanced as possible between the phases, thus ensuring the highest level of transmission efficiency.