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

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Discussion Overview

The discussion revolves around the relationship between the impedances Z_Ye and Z_Δ in a three-phase circuit, specifically addressing why Z_Ye is considered to be in parallel with Z_Y after a delta-to-wye conversion. The context is a homework problem from an electrical circuits textbook.

Discussion Character

  • Homework-related, Technical explanation, Debate/contested

Main Points Raised

  • The original poster (OP) expresses confusion about why Z_Ye is parallel with Z_Y after the conversion, suggesting that the voltages from the converted network resemble another three-phase voltage source.
  • One participant suggests introducing a neutral line to clarify the circuit's configuration and proposes drawing the Y network without oblique lines.
  • Another participant disagrees with the need for a fourth wire, asserting that the neutral point of the wye is sufficient without additional wiring.
  • A further reply acknowledges that while a fourth wire is not necessary, adding a wire between the center nodes of the two wye configurations could clarify the parallel relationship, noting that this wire would carry no current due to circuit symmetry.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of a neutral wire in the circuit and the clarity of the parallel relationship between the two wye subcircuits. The discussion remains unresolved regarding the best approach to illustrate this relationship.

Contextual Notes

Participants mention the symmetry of the circuit and the implications of adding a neutral wire, but the discussion does not resolve the assumptions or limitations regarding the circuit's behavior without this wire.

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Homework Statement


This problem is taken from Fundamentals of Electric Circuits (chapter 12, problem 15).
upload_2018-8-17_10-33-46.png


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.
 
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Thanks everyone for answering!
 

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