Resonance frequency in an LCR circuit

[Boxcutter]

Hello everyone!

I've been trying to derive the expression

w = sqrt( (1/(LC)) - (R_l^2 / L^2) )

where w is the resonace frequency, L is the inductance of the inductor, R_l is the resistance in the inductor, R is the resistance of the resistor and C is the capacitance of the capacitator.

for this circuit:

http://web.telia.com/~u18412273/lcr.JPG

I'm not sure how to attack the problem and I can't find any good texts about it.
I know how to do it in series circuits. I've been trying to do it in a correspondning way for this one but I can't quite do it.

Any help is appreciated
/Daniel

 

[Curious3141]

Hello everyone!

I've been trying to derive the expression

w = sqrt( (1/(LC)) - (R_l^2 / L^2) )

where w is the resonace frequency, L is the inductance of the inductor, R_l is the resistance in the inductor, R is the resistance of the resistor and C is the capacitance of the capacitator.

for this circuit:

http://web.telia.com/~u18412273/lcr.JPG

I'm not sure how to attack the problem and I can't find any good texts about it.
I know how to do it in series circuits. I've been trying to do it in a correspondning way for this one but I can't quite do it.

Any help is appreciated
/Daniel

Set up an expression for the complex impedance of the circuit (you can forget about the R since it doesn't affect the analysis). Find the omega for which the impedance of the capacitor in parallel with (series inductance + resistance) becomes a pure real number. At this point the reactive component disappears, the load is purely resistive and resonance is achieved.

 

[kyle8921]

Hello Daniel,

The resonance frequency in an LCR circuit is a fundamental concept in electrical engineering. It is defined as the frequency at which the circuit has maximum impedance, or minimum current. This means that the energy stored in the circuit is oscillating back and forth between the inductor and the capacitor at this frequency, resulting in a resonance effect.

To derive the expression for resonance frequency in this circuit, we can use the principles of Kirchhoff's laws and Ohm's law. Starting with Kirchhoff's voltage law, we can write:

V_L + V_R + V_C = 0

where V_L is the voltage across the inductor, V_R is the voltage across the resistor, and V_C is the voltage across the capacitor.

Using Ohm's law, we can write:

V_L = L di/dt

V_R = R_L i

V_C = 1/C ∫ i(t) dt

Substituting these into the first equation and rearranging, we get:

L d^2i/dt^2 + R_L di/dt + 1/C i = 0

This is a second-order differential equation, which can be solved using standard techniques. The solution will have the form of a sinusoidal function with a frequency of w, the resonance frequency. Plugging this into the equation, we get:

-Lw^2 sin(wt) + R_L w cos(wt) + 1/C sin(wt) = 0

Simplifying, we get:

w^2 = 1/LC - R_L^2/L^2

Taking the square root of both sides, we get the expression you were looking for:

w = sqrt( (1/(LC)) - (R_l^2 / L^2) )

I hope this helps you in understanding the derivation of resonance frequency in an LCR circuit. If you need further assistance, I suggest consulting your textbook or a trusted online resource for more detailed explanations and examples. Keep up the good work in your studies!

Best regards,