Finding values of RLC circuit given resonance frequency

[gfd43tg]

Homework Statement

A series RLC circuit is driven by an ac source with a phasor voltage Vs = 10∠30◦ V. If the circuit resonates at 103 rad/s and the average power absorbed by the resistor at resonance is 2.5 W, determine the values of R, L, and C, given that Q = 5.

Homework Equations

The Attempt at a Solution

Hello,

As I am solving this, I am substituting into the equation for the power of the resistor using some of the parameters given in the problem statement. However, when I finally solve for L, then I end up getting an imaginary number part, 0.2∠60. Do I just use the real part as my answer, or am I doing this incorrectly?

 

[gfd43tg]

Here is my attempt

 

[gneill]

The phasor angle attributed to the source voltage should be irrelevant; You can always define the source phasor to be the reference phasor for your calculations (unless other current or voltage phasors elsewhere in the circuit are also given so that the true reference phasor is something or somewhere else that you can't "get at" directly). So just take your source to be 10V @ 10³ rad/sec.

At resonance the net reactance seen by the source is what? What does that tell you about the phase angle between the voltage and current at resonance?

 

[NascentOxygen]

when I finally solve for L, then I end up getting an imaginary number part, 0.2∠60.

Average power is (magnitude of current)² * R
where you use the RMS value of the current
So current in this formula has no angle, and the inductance you calculate will not have an angle.

 

[gfd43tg]

I'm not entirely sure if this is what you guys are trying to communicate to me, so here is my interpretation of what you guys are telling me in my next attempt

 

[NascentOxygen]

Your method looks better, though I haven't checked your arithmetic.

You should give a few words' explanation why you reduced Z_L + Z_C + R to just R.

 

[rude man]

So w_0 is 10^3 rad/s, not 103 rad/s?

Anyway, start with determining R, which is already wrong.
Then L.
Then C.

Don't use current at all.