In what condition,we can have $P_{\Delta}=P_{Y}$,not $P_Y=\frac{1}{3}P_{\Delta}$

The power consumption of three phase circuit, which is connected with $$\ \Delta\$$ type, is $$\3000W\$$,now if we modified it with $$\ Y\$$ type,what is the power consumption of this circuit?

The answer is 1000W,because when we have the same impedance for each three wire,we can have $$\P_Y=\frac{1}{3}P_{\Delta} \$$

But now i am confused that in what condition,we can have the same power of $$\ \Delta\$$ type and $$\ Y\$$ type ,I mean,$$\P_{\Delta}=P_{Y}\$$,obviously if the impedance are the same between the $$\ Y\$$ type connection and $$\ P_{\Delta}\$$ type connection,the $$\P_{\Delta }\neq P_{Y}\$$

So in what condition,we can have $$\P_{\Delta}=P_{Y}\$$,just like the formula below:

$$\P_{\Delta}=3\times \frac{V_L}{\sqrt{3}}\times I_Lcos\theta=\sqrt{3}V_LI_Lcos\theta\$$

$$\P_{Y}=3\times V_L\times \frac{I_L}{\sqrt{3}}cos\theta=\sqrt{3}V_LI_Lcos\theta\$$

$$\V_L\$$ means line voltage,$$\I_L\$$ means line current

$$\V_P\$$ means phase voltage,$$\I_P\$$ means phase current

In the $$\ Y\$$ type,$$\\sqrt{3}V_P=V_L\$$,$$\I_L=I_P\$$

In the $$\ \Delta\$$ type,$$\V_P=V_L\$$,$$\I_L=\sqrt{3}I_P\$$

• What does $V_L$ mean in your equations? Hit the edit link. – Transistor May 12 at 6:18
• @Transistor i have modified the question – shineele May 12 at 7:16
• What is "i" tag? – Shadow May 12 at 7:27
• "i"??which part do you mention? – shineele May 12 at 7:49
• You tagged the question with "i". What does that mean? You can use the "current" tag. – Shadow May 12 at 9:32

So in what condition,we can have $$\P_Δ=P_Y\$$
• In a delta connected load, the line current is $$\\sqrt3\$$ times greater than that for a star load using the same limb impedances.
So if you want a star load to match a delta load in terms of power (VA), each limb of the star has to have an impedance that is $$\\sqrt3\$$ times lower than each limb of the delta load. This will then make the phase current (also line current) $$\\sqrt3\$$ times bigger and therefore matches the delta load line current.