Resonance of a Parallel LCR network

[nathangrand]

Why does the resonance peak of a parallel LCR network (capacitor in parallel with in-series inductor and resisitor) get smaller in amplitude with increasing resistance? I have impedance/DC resistance plotted against frequency.


I'm really not sure about how to go about showing this, preferably mathematically. At resonance is the reactance 0, implying the impedance is just the resistance?

Also, what is the correct expression for the resonant frequency of this circuit? I get w$$^{2}$$ = w0$$^{2}$$ - y$$^{2}$$ when y=R/2L and w0= 1/ $$\sqrt{LC}$$ but I've seen different expressions, such as under other configurations here http://en.wikipedia.org/wiki/RLC_circuit

Any help would be massively appreciated!

 

[ehild]

Write up the impedance of the LRC circuit. Resonance is usually defined as the frequency where the magnitude of impedance is maximum(parallel circuit) or minimum (series circuit). But sometimes it is defined as the frequency where the impedance is real, the imaginary part being zero. Are you familiar with complex numbers?

ehild

 

[nathangrand]

Yeah I'm familiar with complex numbers. I can show the various how to get some of the resonance frequency equations - its the first part of my question I'm more stuck with

 

[ehild]

If you define resonance when the imaginary part is zero, w^2=1/(LC) holds for the resonant angular frequency. What is the resonant impedance then?

ehild

 

[rwisz]

The resonance peak of a parallel LCR network decreases in amplitude with increasing resistance due to the effect of damping. Damping is the dissipation of energy in a system, in this case, the dissipation of energy in the form of heat due to the resistance in the circuit. As the resistance increases, more energy is dissipated in the circuit, resulting in a decrease in the amplitude of the resonance peak.

Mathematically, at resonance, the reactance of the inductor and the capacitor cancel each other out, resulting in a purely resistive impedance. This can be expressed as Z = R, where Z is the impedance, R is the resistance, and at resonance, the reactance (X) is equal to 0. This means that the impedance is solely determined by the resistance, and as the resistance increases, the impedance also increases, leading to a decrease in the amplitude of the resonance peak.

The correct expression for the resonant frequency of a parallel LCR circuit is f = 1/2π√(LC), where f is the resonant frequency, L is the inductance, and C is the capacitance. This can also be expressed in terms of angular frequency (ω) as ω = 1/√(LC). The expression you have mentioned (w^2 = w0^2 - y^2) is for a series RLC circuit, where y represents the damping factor. The resonant frequency for a series RLC circuit is slightly different from a parallel RLC circuit and is given by f = 1/2π√(LC - R^2/4L^2).

I hope this helps to clarify your doubts. Please let me know if you need further clarification.