# When should we use $V_L=V$ when we prove $P^1_{LOSS}=2P^3_{LOSS}$.?

The book said when the transmission power and wire resistance of both one phase and three phase are the same,the power loss for one phase system is equal to two times power loss for three phase ,that is ,$$\P^1_{LOSS}=2P^3_{LOSS}\$$.

$$\P^1_{LOSS}=\$$ power loss for one phase system

$$\P^3_{LOSS}=\$$ power loss for three phase system

However,when i try to prove this relation ,i found that we will regard line voltage as the voltage in the one phase system.I mean In this proof,i will replace $$\\frac{P}{Vcos\theta }\$$ with $$\x\$$, As we can see,

if we regard phase voltage,$$\V_P\$$, as the voltage when i am calculating the power loss,that is

$$\V_P=V\$$,i will have

$$\P^1_{LOSS}=2x\$$,and $$\P^3_{LOSS}=\frac{1}{3}x\$$,so it will become $$\P^1_{LOSS}=6P^3_{LOSS}\$$.

if we regard line voltage,$$\V_L\$$, as the voltage when i am calculating the power loss,that is

$$\V_L=V\$$,i will have

$$\P^1_{LOSS}=2x\$$,and $$\P^3_{LOSS}=x\$$,so it will become $$\P^1_{LOSS}=2P^3_{LOSS}\$$.

This means we should assume $$\V_L=V\$$ when we are trying to prove $$\P^1_{LOSS}=2P^3_{LOSS}\$$,but why?why should we assume $$\V_L=V\$$,not $$\V_P=V\$$?The reason i think we should use $$\V_P=V\$$ is that when use $$\P=IV=\frac{V^2}{R}=I^2R\$$ ,this $$\ V\$$ should be the voltage of a resistance,this means it should be phase voltage in the three phase system

• How can transmission power and wire resistance are the same - one is measured in watts and the other is measured in ohms - they can never be the same because their respective units are different. They are incompatible in terms of making a comparison. – Andy aka May 14 at 9:15
• OH I mean the transmission power and wire resistance of both one phase and three phase are the same,i have edited the question – shineele May 14 at 10:22