Quick question about impedance paradox

[Tusike]

Homework Statement

Hi!
We have a parallel circuit on 230V, 60Hz voltage. One of the parallel's has a 0.0768H coil, the other has a 50 Ohm resistance on it. The question is: What is the effective I in the main current?

Homework Equations

Z = Lw

The Attempt at a Solution

OK, so I have two ways of solving this, and they lead to different results, I don't know which isn't correct and why.

#1). The impedance of the resistance is the same, 50 Ohms, while the impedance of the coil is L*w = 0.0768H * 120*pi = 28.868 Ohms. And so for the overall impedance: 1/Z = 1/Z(R) + 1/Z(L) = 1/50 + 1/28.868 = 0.05464 and so Z = 18.3 Ohms. For the effective I in the main current; I(eff) = U/Z = 230V/18.3Ohms = 12.567A.

#2) Since the coil's current is 90 degrees behind the resistance's, it is true that the main current i^2 = i(ZL)^2 + i(ZR)^2, where i(ZL) is the current in the coil and i(ZR) is the current in the resistance at a given point in time. Divide this equation by sqr(2)*sqr(2) to get the same equation for the effective currents: I^2 = I(ZL)^2 + I(ZR)^2.
We have calculated before that Z(L) = 28.868, so I(ZL) = 230/28.868=7.9673A. Z(R) = R = 50, so I(ZR) = 230 / 50 = 4.6A. And so the effective current in the main current is sqr(7.9672^2 + 4.6^2) = 9.2A.

Can anybody please tell me what I did wrong and why is it wrong? Thanks!

-Tusike

 

[ehild]

Homework Equations

Z = Lw

#1). The impedance of the resistance is the same, 50 Ohms, while the impedance of the coil is L*w = 0.0768H * 120*pi = 28.868 Ohms. And so for the overall impedance: 1/Z = 1/Z(R) + 1/Z(L) = 1/50 + 1/28.868 = 0.05464 and so Z = 18.3 Ohms.

You can not simply add up impedances or reciprocal impedances. If you learned about complex impedances, you should handle them as complex numbers. Z_L=iwL where i is the imaginary unit. So the reciprocal resultant impedance is 1/Z=1/(iwL)+1/R = 1/R-i/wL. The magnitude is obtained as |1/Z|=sqrt((1/wL)²+(1/R)²), |Z|= 25 ohm

 

[Tusike]

Oh OK I see now. I didn't learn yet about calculating with imaginary numbers, so for now I'll just use my other method, but I'll definitely look into this after I'm finished.

Thanks!