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There is the proof about \$3S_Y=S_\Delta\$ in lots of books enter image description here

I have other way to prove the \$3S_Y=S_\Delta\$,however,there is a problem in my proof,but i don't know where am i wrong

As we can see the proof as above,we can know that proof is using the \$S=\frac{V^2}{Z}\$ to prove \$3S_Y=S_\Delta\$

However,as we know \$S=\frac{V^2}{Z}=I^2 Z\$,that is ,we can use \$S=I^2 Z\$ to prove \$3S_Y=S_\Delta\$ ,too!

Here is my proof by using \$S=I^2 Z\$

enter image description here

Now i have \$S_Y=3S_\Delta\$ ,but the answer should be \$3S_Y=S_\Delta\$,where am i wrong by using this method(\$S=I^2 Z\$)?

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    \$\begingroup\$ Your dropping the square on your VL’s. \$\endgroup\$ – relayman357 May 13 at 13:11
  • \$\begingroup\$ @relayman357 what do you mean?? \$\endgroup\$ – shineele May 13 at 13:47
  • \$\begingroup\$ When you square (VL/root3) you should get a VL^2 term. \$\endgroup\$ – relayman357 May 13 at 13:53
  • \$\begingroup\$ @relayman357 OH i know it,that is a small problem,but my problem is not that,by the way,i have improved the first figure \$\endgroup\$ – shineele May 13 at 14:13
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The phase current in the star configuration (with Z/3) converted from delta (with Z) is wrong-

enter image description here

The phase current in the star configuration (with Z/3) converted from delta (with Z)will be

$$I'_{p} = \frac{V_{L}\sqrt{3}}{Z}$$

$$I'_{p} = 3I_{p}$$

which gives you $$S^{\Delta} = 9I^{2}_{p}Z$$

and $$S^{Y} = 3I^{2}_{p}Z$$

hence

$$3S^{Y} = S^{\Delta}$$

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there is a problem in my proof,but i don't know where am i wrong

This seems an obvious mistake: -

enter image description here

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  • \$\begingroup\$ oh yes,but my problem is not that ,my proof is using P=I^2R,But get the wrong solution,the mistake you said is just a small and obvious mistake \$\endgroup\$ – shineele May 13 at 14:10
  • \$\begingroup\$ You asked if there is a problem in my proof and I have shown 1 and cannot rule out more. Why have you corrected your error in the question? It now makes my answer look faulty and I don't want people to come along and downvote because of that. If you are making a correction in your question that then counteracts an already given answer \$\color{red}{\text{YOU SHOULD MAKE THAT CLEAR}}\$ when you amend your question. \$\endgroup\$ – Andy aka May 13 at 14:12
  • \$\begingroup\$ my proof is in the second figure,the first figure is the proof in my book,i just make that mistake with careless,divyam have answered what i really want to ask \$\endgroup\$ – shineele May 13 at 14:15
  • \$\begingroup\$ well i have already said that which is the proof from the book,which part is my proof in the begining,i think you didn't understand that first,However,indeed i make a mistake in my first figure,thanks for your reminding \$\endgroup\$ – shineele May 13 at 14:19
  • \$\begingroup\$ @Andyaka shouting? \$\endgroup\$ – skvery May 14 at 6:23

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