Quality Factor and Frequency Bandwidth in a matching network

it is known that the Quality Factor appears in the formulas for transformation of an impedance series bipole to a parallel one. A simple example is this one:

Now I have three questions:

1) I have always been told that the quality factor is the reciprocal of the fractional bandwidth (calculated by using as reference an attenuation of 3dB with respect to the peak of the impedance/admittance of the circuit considered). Is this relation true for any circuit with (at least) one reactive element and a resistance? Or is it necessary to prove it for each circuit?

Now, let's consider a L network, for instance this one (ignore the values of Cp and Ls):

It may be used, as in this case, if we have a certain resistance RL and we want that an external device sees a value R. Therefore:

2) We insert a parallel capacitance and then a series inductance. The second one is used to balance the reactive part introduced by the capacitance. Precisely, its admittance has to cancel that of the capacitance. In this way the total impedance seen by the external device would be totally resistive and equal to:

Rp = RL (1 + Q^2).

So we may impose that Rp = R (known); we know RL. Therefore Q^2 is determined.

Have I understood it correctly?

3) The quality factor is function of frequency. Therefore, it seems to me that a certain transformation from series model to parallel one (or vice versa) is true only for a certain frequency. But we have said that Q is related to the fractional bandwidth, so it seems to me that the transformation is true for all frequencies inside that bandwith. Which is the truth?

• Q is basically impedance ratio of a single component but for 2nd order circuits it is equivalent to fc/BW(-3dB) Caps also have a Q , more important > 30MHz. The Q of each component must be >> Q of a 2nd order filter – Tony Stewart Sunnyskyguy EE75 Oct 11 '19 at 18:50

1) I have always been told that the quality factor is the reciprocal of the fractional bandwidth (calculated by using as reference an attenuation of 3dB with respect to the peak of the impedance/admittance of the circuit considered).

Good for series RLC to parallel RLC, because we calculate Q @ resonance.

2) We insert a parallel capacitance and then a series inductance. The second one is used to balance the reactive part introduced by the capacitance.

These matching networks are often designed for low Q. Q > 1 is always required when an impedance-matching solution is needed. However, the frequency response is not symmetrical above and below the frequency where impedance is matched....frequency response takes on a high-pass or low-pass form. Q is ill-defined here, especially where Q is low.
But the matching point, where all reactances cancel is very well-defined.

3) The quality factor is function of frequency. Therefore, it seems to me that a certain transformation from series model to parallel one (or vice versa) is true only for a certain frequency.

At resonance the transformation using Q applies: RLC (parallel) <> RLC(series).
For the "L" network matching RLC, resonance and Q are ill-defined.