AC power distribution systems have three categories: single-phase, two-phase, and three-phase systems. The single-phase circuit, common in residential settings, typically employs a two-wire system connecting a single AC source to various loads. These circuits support standard household appliances operating at 120 volts (V) and 240 V, such as lamps, televisions, and microwaves. The first generators, Niagara Falls hydro plant installed in 1895, were two-phase and designed by Nikola Tesla. The two-phase supply system is less common, and the three-wire system incorporates two voltage sources out of phase with each other by 90 degrees.

The three-phase, four-wire system is most prevalent in industrial applications. This system comprises three AC sources that are identical in amplitude and frequency but have a phase difference of 120 degrees between each other. The loads in such a system can be connected in either a Y (star) configuration or a delta (Δ) configuration.

The advantage of three-phase circuits lies in their efficiency and stability. They deliver power with lower pulsation than single-phase systems, resulting in smoother power transmission and minimizing vibrations—a crucial factor for industrial machinery like induction motors. Another benefit is the reduced amount of conductor material required to transmit a given amount of power, which renders three-phase systems more cost-effective than their single-phase counterparts.

The relationship between the line and phase voltages and currents in Y-connected three-phase systems is given by:

And for Δ-connected systems:

These equations show the relationship between line-to-line voltages and currents with those across each phase, which is essential when calculating the total power in three-phase systems. The total power for both Y and Δ connections can be found using:

where ϕ is the power factor angle, which affects power transmission efficiency.