The voltages we get from the three-phase power system are produced by a synchronous generator. In a balanced condition, the three voltages have equal amplitudes. The first type is balanced wye-wye connection.

Each voltage has the 120◦ phase angle differences. It may be delta or wye, both have a difference of 120◦.

*Make sure to read what is three-phase circuits first.*

There are four types of balanced three phase voltage:

- Balanced wye-wye connection (i.e., Y-connected source with a Y-connected load).
- Balanced wye-delta connection.
- Balanced delta-delta connection.
- Balanced delta-wye connection.

## Balanced Three-Phase Y Connection

We begin with the Y-Y system because any balanced three-phase system can be reduced to an equivalent Y-Y system.

Therefore, the analysis of this system should be regarded as the key to solving all balanced three-phase systems.

A

balanced Y-Y systemis a three-phase system with a balanced Y-connected source and a balanced Y-connected load.

Consider the balanced four-wire Y-Y system of Figure.(1), where a Y-connected load is connected to a Y-connected source.

We assume a balanced load so that load impedances are equal.

Although the impedance **Z**_{Y} is the total load impedance per phase, it may also be regarded as the sum of the source impedance **Z**_{S}, line impedance **Z**_{l}, and load impedance **Z**_{L} for each phase, since these impedances are in series.

As illustrated in Figure.(1), **Z**_{S} denotes the internal impedance of the phase winding of the generator; **Z**_{l} is the impedance of the line joining a phase of the source with a phase of the load; **Z**_{L} is the impedance of each phase of the load, and **Z**_{n} is the impedance of the neutral line.

Thus, in general

(1) |

**Z**_{S} and **Z**_{l} are often very small compared with **Z**_{L}, so one can assume that **Z**_{Y} = **Z**_{L} if no source or line impedance is given. In any event, by lumping the impedances together, the Y-Y system in Figure.(1) can be simplified to that shown in Figure.(2).

Figure 1. A balanced Y-Y system, showing the source, line, and load impedances. |

Figure 2. Balanced Y-Y connection |

Assuming the positive sequence, the *phase voltages* (or line-to-neutral voltages) are

(2) |

The *line-to-line* voltages or simply *line* voltages **V**_{ab}, **V**_{bc}, and **V**_{ca} are related to the phase voltages. For example,

(3a) |

Similarly, we can obtain

(3b) |

(3c) |

Thus, the magnitude of the line voltages V_{L} is √3 times the magnitude of the phase voltages V_{p}, or

(4) |

where

(5) |

and

(6) |

Also, the line voltages lead their corresponding phase voltages by 30◦. Figure.(3a) illustrates this. Figure.(3a) also shows how to determine **V**_{ab} from the phase voltages, while Figure.(3b) shows the same for the three-line voltages.

Notice that **V**_{ab} leads **V**_{bc} by 120◦, and **V**_{bc} leads **V**_{ca} by 120◦so that the line voltages sum up to zero as do the phase voltages.

Figure 3. Phasor diagrams illustrating the relationship between line voltages and phase voltages |

Applying KVL to each phase in Figure.(2), we obtain the line currents as

(7) |

We can readily infer that the line currents add up to zero,

(8) |

so that

(9a) |

or

(9b) |

that is, the voltage across the neutral wire is zero. The neutral line can thus be removed without affecting the system.

In fact, in long-distance power transmission, conductors in multiples of three are used with the earth itself acting as the neutral conductor.

For easier understanding, you better read about balanced three-phase voltages first.

Power systems designed in this way are well grounded at all critical points to ensure safety.

While the *line* current is the current in each line, the *phase* current is the current in each phase of the source or load. In the Y-Y system, the line current is the same as the phase current.

We will use single subscripts for line currents because it is natural and conventional to assume that line currents flow from the source to the load.

Figure 4. A single-phase equivalent circuit |

An alternative way of analyzing a balanced Y-Y system is to do so on a “per phase” basis.

We look at one phase, say phase a, and analyze the single-phase equivalent circuit in Figure.(4). The single-phase analysis yields the line current **I**_{a} as

(10) |

From **I**_{a}, we use the phase sequence to obtain other line currents. Thus, as long as the system is balanced, we need only analyze one phase.

We may do this even if the neutral line is absent, as in the three-wire system.

## Balanced Wye-Wye Connection Examples

For better understanding let us review the example below:

1. Calculate the line currents in the three-wire Y-Y system of Figure.(5).

Figure 5 |

*Solution:*

The three-phase circuit in Figure.(5) is balanced; we may replace it with its single-phase equivalent circuit such as in Figure.(4).

We obtain Ia from the single-phase analysis as

where **Z**_{Y} = (5 − j2) + (10 + j8) = 15 + j6 = 16.155 21.8◦. Hence,

Since the source voltages in Figure.(5) are in positive sequence and the line currents are also in positive sequence,

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