Damped Resonance Frequency OF series LCR circuit

In summary, for a series LCR circuit, the solution for current I is given by I = (E/Z)sin(wt+\phi), where Z = \sqrt{R^2 + (X_{L}-X_{C})^{2}}. For resonance, the necessary condition is X_{L}=X_{C}, which gives \omega=1/\sqrt{LC}. However, there is some confusion about the damped resonance frequency, which is the frequency of the sinusoidal component in the expression for the oscillations of an underdamped, undriven system. This can be derived by solving the harmonic motion equations with a damping term, and for an LCR circuit, the specific coefficients correspond to L, R
  • #1
I_am_learning
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the solution for current I, for series LCR circuit is
I = (E/Z)sin(wt+[tex]\phi[/tex])
Where Z = [tex]\sqrt{R^2 + (X_{L}-X_{C})^{2}}[/tex]
So for Resonance (i.e. maximum Current Amplitude) of LCR Circuit the necessary condition seems to be
[tex]X_{L}[/tex]=[tex]X_{C}[/tex]
Which gives [tex]\omega[/tex]=1/[tex]\sqrt{LC}[/tex]

But some text-books and wikipaedia have given that the damped resonace frequency is
dd12e89af3c3a6d9b0352bb6a316a798.png

where
fefd9016ff9e5960ac7486df3f17bbe8.png

How is this relation Derived ?
 
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  • #2
Not sure about your solution. What driving voltage are you assuming?

Anyway the undamped resonant frequency is the frequency the undriven system would oscillate with when there is no damping ([tex]R=0[/tex]). When the system is underdamped and undriven then its oscillations consist of a sinusoidal component multiplied by an exponentially decaying envelope. The frequency of the sinusoidal terms in this expression is called the damped resonant frequency. It is found just by solving the harmonic motion equations with a damping term and then identifying the sinusoidal component.

See http://en.wikipedia.org/wiki/Damping" [Broken] for how to do this for a mass-spring-damper oscillator. You should be able to get the result for an LCR circuit just by replacing the terms used there with the specific coefficients of the damped harmonic oscillator differential equation that arise in LCR circuits (e.g. L corresponds to m, R to c, and C corresponds to 1/k).
 
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  • #3
The first frequency is the frequency of the undamped oscillator. It is also the frequency of the resonance for a driven damped oscillator.

The second one is the frequency of a "free" (not driven) damped oscillator.
 
  • #4

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1. What is a damped resonance frequency in a series LCR circuit?

A damped resonance frequency in a series LCR circuit is the frequency at which the circuit reaches its maximum current and the voltage across the circuit is at its minimum. This occurs when the inductive reactance and capacitive reactance cancel each other out, resulting in a purely resistive circuit.

2. How is the damped resonance frequency calculated in a series LCR circuit?

The damped resonance frequency in a series LCR circuit can be calculated using the formula fd = 1 / (2π√(LC)), where fd is the damped resonance frequency, L is the inductance in henries, and C is the capacitance in farads.

3. What factors affect the damped resonance frequency in a series LCR circuit?

The damped resonance frequency in a series LCR circuit is affected by the values of inductance and capacitance in the circuit. It can also be affected by the resistance in the circuit, as well as any external forces or disturbances that may be present.

4. What is the significance of the damped resonance frequency in a series LCR circuit?

The damped resonance frequency in a series LCR circuit is important because it indicates the frequency at which the circuit will have the most efficient energy transfer. It is also useful for determining the bandwidth of the circuit and can be used in applications such as tuning radio receivers.

5. How does the damped resonance frequency change in a series LCR circuit when the values of the inductance and/or capacitance are altered?

When the values of inductance and/or capacitance are altered in a series LCR circuit, the damped resonance frequency also changes. Increasing the inductance will decrease the damped resonance frequency, while increasing the capacitance will increase the damped resonance frequency. This is because the inductive reactance and capacitive reactance have an inverse relationship with frequency.

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