How to Determine Values of L and C in a Series RLC Circuit at Resonance?

[Rijad Hadzic]

Homework Statement

A series RLC circuit with a resistance of 120 ohms has a resonance angular frequency of 4x10^5 rad/s

At resonance the voltages across the resistor and inductor are 60 V and 40 V respectively.

Determine the values of L and C

Homework Equations

I_t = Imax sin(ωt)
w_o = 1/(LC)^1/2

Z= R when we're given the resonance frequency
Imax(w_o) = εmax/ R

The Attempt at a Solution

4x10^5 rad/s = 1/(LC)^1/2

ωL - 1/ωC = 0

ωL = 1/ωC

L = 1/ωCThe problem is I'm not given the AC εmf.

Ac emf = (120 Ohm * 60 V ) + (40 V * ω_0 L ) + ( x V * (1/(ω_0)(c)) )

so I don't know: c, L, voltage of capacitor, or the voltage of the AC generator.

I'm lost as to how to proceed from here. Any help would be appreciated.

 

[BvU]

Ac emf = (120 Ohm * 60 V ) + (40 V * ω_0 L ) + ( x V * (1/(ω_0)(c)) )

Hi,
you write this as if all three are in the same phase ...

 

[Rijad Hadzic]

Hi,
you write this as if all three are in the same phase ...

In the question it says

"at resonance, the voltages across the resistor and inductor are 60 v and 40 v respectively"

does that mean

I(t) = Imax sin(wt)

and ε(t) = εmax sin (ωt)

will have ω= ω_o? (the resonance frequency given in the problem)

?

Also for ε(t) = εmax sin (ωt) I eliminated φ because arctan (xl-xc / r ) = 0

 

[Rijad Hadzic]

Also will the fact that the resistor is in phase with I(t), and inductor is +pi/2 in phase with I(t) help me out here?

didn't want to triple post so I will just edit:

do the equations

(60/120) = Imax sin(ω_0 t )

and

(40/ω_o L) = Imax sin (ω_0 t + pi/2 )

make sense/lead me to answering this problem?

Thanks

another edit:

my trouble here is finding the max emf by the ac generator. I just have no clue how I would be able to find that with the given information..

 

[BvU]

You know the impedance of the circuit at resonance, right ? What does that maen for V_s (your source) ?emf

 

[Rijad Hadzic]

You know the impedance of the circuit at resonance, right ? What does that maen for V_s (your source) ?emf

Impedance is = to resistance right? So at resonance, impedance is going to = R.

Therefore, ε(t) = I(t)R

which gives me the expression ε(t) = 120*imax*sin(ω_0 t)

I still don't understand how to find imax or εmax. We were only given 3 voltages,

60 = V(r), 40 = V(l) and then x = V(c)

I suppose the fact that V(I) + V(c) are 90 degrees ahead of the resistors voltage is suppose to help me..

Meaning when V(r) = 0, V(I) + V(c) Gives me the voltage produced by the AC source.

so V(I) + V(c) = ε_ac

But I'm not given any information about V(c)...

 

[Rijad Hadzic]

You know the impedance of the circuit at resonance, right ? What does that maen for V_s (your source) ?emf

Holy hell thank you brotha. Idk how you stuck through with me through this but you are a legend.

I know

60 = I(t) R and 40 = I(t)Xl

so 1/2 = I(t) and 40 * 2 = 80 = X l = ωl = 80/ω = l

so I just found l

now Xl = Xc and I can just find c from there.

What the hell man. I wasted about 2 hours just looking at this problem. And I didn't realize it was that simple. I thought I had to do a bunch of stuff with functions and all kinds of stupid things wow I am heated ℙℙI'm pretty mad. I want to change majors at this point but I can't stop won't stop.

Thanks a lot dude cheers

 

[Rijad Hadzic]

Okay from the same problem..At what frequency does the current in the circuit lag the voltage by 45 degrees?

I have the equation φ = arctan (Xl - Xc / R)

and I get tan(φ)* 120 = 120 = Xl - Xc

and I solve this quadratic equation, but my answer isn't right.

What am I doing wrong here?

 

[Rijad Hadzic]

Nvm found the answer to the above also.

Looks like I was doing the right thing except I was doing my calculations wrong.

Don't do physics when sleep deprived you may go insane