Find resistance of inductor in series RLC circuit

[Pifagor]

Homework Statement

We have a resistor of 350 Ohm, an inductor of 0.15H and a capacitor of 10 microfarad, connected in series to an AC source of voltage 240 V and frequency 50 Hz. The measured effective voltage over the resistor is 180V, and over the system of the both the inductor and the capacitor it's 120V. What is the resistance of the inductor?

The answer is supposed to be 140 Ohm.

Homework Equations

Z= sqrt{ R^{2}+ (X_L-X_C)^2 }

The Attempt at a Solution

My attempt: the effective current through the resistor is 180V/350 Ohm = 0,514A. The same current flows through the rest of the circuit, whose impedance then should be 120V/0,514A = 233 Ohm. But the (absolute value of the) reactance wL-1/wC of this system is greater than this, and a resistance in the inductor would only increase the impedance. Where am I going wrong?

 

[Svein]

But the (absolute value of the) reactance wL-1/wC of this system is greater than this, and a resistance in the inductor would only increase the impedance. Where am I going wrong?

You are calculating the impedance across LC without introducing R_L (contrary to your argument, R_L does not necessarily increase the impedance).

 

[Pifagor]

Thanks, but I still don't understand. In the above equation, used over the inductor and capacitor, I meant for the R to be the resistance in the inductor. How can an R not increase the impedance? Even if we do it by complex numbers, resistance is always real, right?

 

[Svein]

The correct impedance equation is $Z=\sqrt{(R+R_{L})^{2}+(\omega L-\frac{1}{\omega C})^{2}}$. You have already calculated the current through R...

 

[Pifagor]

I tried that already, with voltage 240V and current 0,514A. That gives me a positive R_L = 30 Ohm, but that answer is still not right.

Why is it wrong to calculate the impedance with my equation, without the resistor, over only the LC-circuit (which of course contains the R_L, too, so it is really RLC), and use the given effective voltage over that?

 

[Svein]

1. The impedance across the RLC circuit is $Z_{RLC}=\sqrt{ R_{L}^{2}+(\omega L-\frac{1}{\omega C})^{2}}$
2. The current through the RLC circuit you have already calculated
3. The voltage across the RLC circuit is 120V (as given)

Now you have enough to calculate the value of Z_RLC and thus R_L.