The problem when i proved $3S_Y=S_\Delta$ by using $S=I^2 Z$ in three phase circuit

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

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\$$

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\$$)?

• Your dropping the square on your VL’s. May 13, 2020 at 13:11
• @relayman357 what do you mean?? May 13, 2020 at 13:47
• When you square (VL/root3) you should get a VL^2 term. May 13, 2020 at 13:53
• @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 May 13, 2020 at 14:13

The phase current in the star configuration (with Z/3) converted from delta (with Z) is wrong-

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

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

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

This seems an obvious mistake: -

• 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 May 13, 2020 at 14:10
• 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. May 13, 2020 at 14:12
• 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 May 13, 2020 at 14:15
• 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 May 13, 2020 at 14:19
• @Andyaka shouting? May 14, 2020 at 6:23