Determine the value of the inductance ?

[Fazza3_uae]

Determine the value of the inductance ...??

Homework Statement

Consider a series RLC circuit. The applied voltage has a maximum value of 210 V and oscillates at a frequency of 76 Hz. The circuit contains an inductor whose inductance can be varied, a 900 ohm resistor, and a 1 μF capacitor.Determine the value of the inductance such that the voltage across the capacitor is out of phase with the applied voltage by 54◦, with V_max leading V_C. Answer in units of H.

Homework Equations

Cos \phi = X_L / Z
Tan \phi = X_L - X_C / R
w = 2*pi*f
Z² = R² + (X_L - X_C)²

The Attempt at a Solution

First I used the 2^nd Eq. Tan \phi = X_L - X_C / R

(R* Tan \phi)² = (X_L - X_C)²

Then I used the 1^st Eq. Cos \phi = X_L / Z

Z² = R² + (X_L - X_C)²

I replaced (X_L - X_C)² with (R* Tan \phi)²

then ,

Cos² \phi = X_L² / (R² + (X_L - X_C)²)

X_L² = Cos² \phi * (R² + (X_L - X_C)²) = w²L²

Then , L = 1.8827 H which is wrong .

Am i in the right direction , if not please point me towards it . Thanks.

 

[rl.bhat]

Since the voltage across the capacitor is out of phase with the applied voltage, angle between Vc and hence VC - VL and Vmax is 54 degrees.
So tan(54) = R/(XC - XL)
Solve for XL and find L.

 

[Fazza3_uae]

wow ..thx a ton rl.bhat.

whould u tell what expression must i use if the question says the voltage across the inductor ??

and how did you get Vc-Vl why it not Vl-Vc ??

thanks again ..

 

[rl.bhat]

Since applied voltage leads with Vc, Vc must be greater than VL. So net voltage across reactance should be VC - VL.
If VL leads with the applied voltage then you have to take VL - VC.

 

[Fazza3_uae]

Thanks again for the clarification.

Hmmmm... why tan(theta) why not cos(theta) ...??

You have answered same question before and i want to understand it 2 ..

https://www.physicsforums.com/showthread.php?t=328964


check the link .

If i was answering the question i would say the answer must be in this form ...

tan(theta)= R / XL-XC

Because it says that Vl is leading so it is greater that Vc like what you 've said ...

But you answered cos(theta)=Xc/Z ,,,,,??

 

[rl.bhat]

In both the treads VC is out of phase with the applied voltage.
So you can write either tanθ = R/(XC - XL) or cosθ = (XC - XL)/Z. In the second thread it was typo.
In LCR circuit always Vmax leads VC and VL leads Vmax. So it does not change expression for tanθ. In your relevant equation the angle φ is the phase difference between the current and Vmax.