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

Question

The power consumption of three phase circuit, which is connected with \$ \Delta\$ type, is \$3000\text{ W}\$, 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_L\cos\theta=\sqrt{3}V_LI_L\cos\theta\$

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

Answer 1

So in what condition,we can have \$P_Δ=P_Y\$

• In a star connected load, the phase current is the line current.

• 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.