Three-phase circuit: Why Z_Ye is parallel with Z_Δ?

• Engineering
• Curiosity 1
In summary, the conversation discusses the conversion of a delta network to a wye network and the addition of a neutral wire. The participants also discuss the parallel arrangement of two wye subcircuits and its effect on circuit behavior.

Homework Statement

This problem is taken from Fundamentals of Electric Circuits (chapter 12, problem 15).

Homework Equations

As shown in the first blue box, I have trouble seeing why after the conversion, Z_Ye will be parallel with Z_Y. I think it's because the voltages 'emitting' from the converted network is like another 3-phase voltage source.

The Attempt at a Solution

The diagram on the right-hand side is my sketch in Microsoft Paint, where I fail to see why are them parallel after the conversion.

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Perhaps you should introduce a 4th line, the neutral. See whether you can draw the Y network (in place of the ∆ arrangement) when you have a neutral line available.

(I suggest that you self-impose a constraint of no oblique lines; draw all lines in your Y network horizontally or vertically.)

I think I disagree with what NascentOxygen just posted. The circuit doesn't require a 4th wire to handle the neutral point.

If you draw a wye inside a delta, the neutral point of the wye will be sufficient without the addition of a neutral wire.

It's true that the circuit doesn't require a 4th wire, but the OP is having trouble seeing why the two wye subcircuits are in parallel. Adding a wire between the center nodes of the pre-existing wye (composed of Zy) and the wye resulting from the delta-wye conversion of ZΔ will make it clear why each element of one wye is in parallel with the corresponding element of the other wye. This wire will carry no current because of the symmetry, so adding it doesn't change the behavior of this very symmetrical circuit. In a circuit without this symmetry, adding this 4th wire would probably change the circuit behavior.

NascentOxygen
Thanks everyone for answering!

1. What is a three-phase circuit?

A three-phase circuit is an electrical system that uses three alternating currents, each with a phase difference of 120 degrees, to power a load. It is commonly used in industrial and commercial settings due to its efficiency and ability to handle high power loads.

2. What is the difference between Z_Ye and Z_Δ in a three-phase circuit?

Z_Ye refers to the equivalent impedance in a three-phase circuit when the loads are connected in a Y or star configuration, while Z_Δ refers to the equivalent impedance when the loads are connected in a Δ or delta configuration. In other words, Z_Ye and Z_Δ are two different representations of the same circuit.

3. Why is Z_Ye parallel with Z_Δ in a three-phase circuit?

In a three-phase circuit, the loads are connected in parallel to each other. Since Z_Ye and Z_Δ represent the equivalent impedance of the same circuit, they must also be connected in parallel to each other. This allows for the distribution of power from the source to the loads in an efficient and balanced manner.

4. What is the significance of Z_Ye being parallel with Z_Δ in a three-phase circuit?

The parallel connection of Z_Ye and Z_Δ in a three-phase circuit ensures that the loads receive equal amounts of power, leading to a balanced and stable system. This also helps to prevent any overload or voltage imbalances that could damage the electrical equipment.

5. Are there any exceptions to Z_Ye being parallel with Z_Δ in a three-phase circuit?

In some cases, a three-phase circuit may use a different type of connection, such as a corner-grounded configuration, which may not follow the standard Y or Δ connection. In these situations, Z_Ye and Z_Δ may not be parallel, but they will still represent the equivalent impedance of the circuit.