Resonance frequency in an LCR circuit

In summary, Daniel is seeking help in deriving the expression for resonance frequency (w) in a circuit consisting of an inductor (L), a resistor (R), and a capacitor (C). He is unsure how to approach the problem and is looking for resources to guide him. The conversation also mentions finding the omega value at which the load becomes purely resistive and resonance is achieved.
  • #1
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
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Boxcutter said:
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.
 
Last edited by a moderator:
  • #3


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,
 

FAQ: Resonance frequency in an LCR circuit

What is resonance frequency in an LCR circuit?

Resonance frequency in an LCR circuit is the frequency at which the reactance of the inductor and capacitor in the circuit cancel out, resulting in a purely resistive circuit. At this frequency, the circuit exhibits its maximum impedance and the current in the circuit is at its minimum.

How is resonance frequency calculated in an LCR circuit?

Resonance frequency in an LCR circuit can be calculated using the formula: fr = 1 / (2π√(LC)), where fr is the resonance frequency, L is the inductance of the circuit, and C is the capacitance of the circuit.

What is the significance of resonance frequency in an LCR circuit?

Resonance frequency is significant in an LCR circuit because it allows for efficient transfer of energy between the inductor and capacitor. At resonance frequency, the circuit has its maximum energy storage capacity and is able to sustain oscillations for a longer period of time.

How does the value of inductance and capacitance affect resonance frequency in an LCR circuit?

The resonance frequency in an LCR circuit is directly proportional to the square root of the product of inductance and capacitance. This means that increasing the value of either inductance or capacitance will increase the resonance frequency, while decreasing the value of either component will decrease the resonance frequency.

What are some practical applications of resonance frequency in LCR circuits?

Resonance frequency in LCR circuits has many practical applications, such as in radio and television receivers, electronic filters, and in wireless charging systems. It is also used in medical devices like MRI machines to produce and detect radio frequency signals.

Similar threads

Back
Top