A balanced delta-wye connection system is a system with delta-connected sources with wye-connected loads.

A

balanced delta-wye systemconsists of a balanced ∆-connected source feeding a balanced Y-connected load.

*Before you start, it wise for you to read what is three-phase circuits first.*

## Balanced Delta-Wye Connection

Consider the ∆-Y circuit in Figure.(1).

Figure 1. A balanced ∆-Y connection |

Again, assuming the abc sequence, the phase voltages of a delta-connected source are

(1) |

These are also the line voltages as well as the phase voltages.

We can obtain the line currents in many ways. One way is to apply KVL to loop a AN B b a in Figure.(1), writing

(2) |

But **I**_{b} lags **I**_{a} by 120◦since we assumed the *abc* sequence; that is, **I**_{b} = **I**_{a} ∠− 120◦. Hence,

(3) |

Substituting Equation.(3) into (2) gives

(4) |

From this, we obtain the other line currents **I**_{b} and **I**_{c} using the positive phase sequence, i.e., **I**_{b} = **I**_{a} ∠− 120◦, **I**_{c} = **I**_{a} ∠+ 120◦. The phase currents are equal to the line currents.

Figure 2. Transforming a ∆-connected source to an equivalent Y-connected source. |

Another way to obtain the line currents is to replace the delta connected source with its equivalent wye-connected source, as shown in Figure.(2).

In balanced wye-wye connection, we found that the line-to-line voltages of a wye-connected source lead their corresponding phase voltages by 30◦.

Therefore, we obtain each phase voltage of the equivalent wye-connected source by dividing the corresponding line voltage of the delta-connected source by √3 and shifting its phase by −30◦.

Thus, the equivalent wye connected source has the phase voltages

(5) |

If the delta-connected source has source impedance **Z**_{S} per phase, the equivalent wye-connected source will have a source impedance of **Z**_{S}/3 per phase.

Figure 3. The single-phase equivalent circuit |

Once the source is transformed into the wye, the circuit becomes a wye-wye system.

Therefore, we can use the equivalent single-phase circuit is shown in Figure.(3), from which the line current for phase *a* is

(6) |

which is the same as Equation.(4)

Alternatively, we may transform the wye-connected load to an equivalent delta-connected load.

This results in a delta-delta system, which can be analyzed as in balanced delta-delta connection. Note that

(7) |

As stated earlier, the delta-connected load is more desirable than the wye-connected load.

It is easier to alter the loads in any one phase of the delta-connected loads, as the individual loads are connected directly across the lines.

However, the delta-connected source is hardly used in practice, because any slight imbalance in the phase voltages will result in unwanted circulating currents.

## Balanced Delta-Wye Connection Example

A balanced Y-connected load with a phase resistance of 40 Ω and a reactance of 25 Ω is supplied by a balanced, positive sequence ∆-connected source with a line voltage of 210 V.

Calculate the phase currents. Use **V**_{ab} as reference.

*Solution:*

The load impedance is

and the source voltage is

When the ∆-connected source is transformed into a Y-connected source,

The line currents are

which are the same as the phase currents.

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*Reference: Fundamentals of electric circuits by Charles K. Alexander and Matthew N. O. Sadiku*

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