If there is a RLC-tuned circuit designed such that an inductor and a resistor is connected to parallel each other and in series to a capacitor.
How can we find the resonance frequency and bandwidth of this circuit ?

 

If there is a RLC-tuned circuit designed such that an inductor and a resistor is connected to parallel each other and in series to a capacitor.
How can we find the resonance frequency and bandwidth of this circuit ?

Regardless of any resistance, the resonant frequency of any circuit is determined by 1/2pi(sqrt(LC)). The bandwidth of a circuit is determined by the general formula : BW=f/Q. All other things being equal, the Q is the ratio of the reactance over resistance. However, your circuit is neither a pure parallel nor pure series circuit. The proper model of this is an L network, with a resistive load. There are is a special case for determining the bandwidth of loaded L networks, covered in quite a lot of detail in the ARRL Handbook. I don't have the formula on the tip of my tongue right now, for which I am highly ashamed....I should know it. But I'll look it up. Stand by!

eric

 

If there is a RLC-tuned circuit designed such that an inductor and a resistor is connected to parallel each other and in series to a capacitor.
How can we find the resonance frequency and bandwidth of this circuit ?

Okay, this is cheating....but here's a calculator for ya. You want to use "Network 2"

http://www.smeter.net/feeding/l-network-terminating-impedance.php


Eric

 

Thanks Eric for your effort to answer my question. I really appreciate for it.

But, I m still not sure whether we have a resonance at w=1/sqrt(LC) for any combination of RLC.

For instance, I have found out that if there is a circuit like Network-3 (in the link) have a resonance at w^2 = 1/LC - (R/L)^2.

And that is not equal to the frequency which we have a series or parallel RLC ...

 

Thanks Eric for your effort to answer my question. I really appreciate for it.

But, I m still not sure whether we have a resonance at w=1/sqrt(LC) for any combination of RLC.

For instance, I have found out that if there is a circuit like Network-3 (in the link) have a resonance at w^2 = 1/LC - (R/L)^2.

And that is not equal to the frequency which we have a series or parallel RLC ...

Hi, Conclusion:

I located the elusive fomula: Q= sqrt( Rhigh/Rlow-1), where R high is the highest value of resistance and R low is the lower value.


When the Q is very low, there can actually be THREE different definitions of resonance (again, illustrated thoroughly in the venerable ARRL Handbook). You can have: 1: zero phase shift
2: Maximum voltage/minimum current
3: XL=XC

For very low Q tank (parallel) circuits, these three conditions do NOT coincide. There is such a thing as LOADED and UNLOADED Q.

Eric

 

But, I m still not sure whether we have a resonance at w=1/sqrt(LC) for any combination of RLC.

Can you provide a schematic for how you are using this circuit. I just quickly wrote out an equation for this combination assuming the output voltage is taken across the capacitor.

The transfer function is (1/RC)*(s+R/L)/(s^2+s/(RC)+1/(LC))

this is not quite a pure bandpass unless R/L is much less than 1/sqrt(LC).

Under that condition the 3 dB bandwidth is about 1/(2 pi RC) and the resonant frequency is 1/(2pi*sqrt(LC))

 

Can you provide a schematic for how you are using this circuit. I just quickly wrote out an equation for this combination assuming the output voltage is taken across the capacitor.

The transfer function is (1/RC)*(s+R/L)/(s^2+s/(RC)+1/(LC))

this is not quite a pure bandpass unless R/L is much less than 1/sqrt(LC).

Under that condition the 3 dB bandwidth is about 1/(2 pi RC) and the resonant frequency is 1/(2pi*sqrt(LC))

Indeed....you confirmed precisely my caveat about low Q circuits! It's gratifying when you get the same answer approaching from two different directions.

 

Indeed....you confirmed precisely my caveat about low Q circuits! It's gratifying when you get the same answer approaching from two different directions.

You'll also see how much straightforward this problem is if the resistance is simply in series with the other components. In this case, the loaded and unloaded Q are one and the same.

eric