# 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:

1. 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?

1. The quality factor is a 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 Commented Oct 11, 2019 at 18:50

There are several confusions at play here, I think.

There are at least two common applications of "Q".

When we discuss the Q of a component, a couple of assumptions apply:

• The component has a reasonable component-ness in the frequency range of interest (i.e. an inductor looks inductive ($$\Z \propto \omega^1\$$), resistor looks resistive ($$\Z \propto \omega^0\$$), or capacitor looks capacitive ($$\Z \propto \omega^{-1}\$$))
• We are concerned with the impedance of the component at its terminals

If we violate these assumptions, we get seemingly unexpected results. Suppose we take a high-Q inductor, and resonate it with a high-Q capacitor at that frequency [that the inductor Q was measured at]. The combination has extraordinarily high inductance just below resonance, Q = 0 at resonance, and extraordinarily low capacitance just above resonance. The result at resonance is trivial: inductive and capacitive reactances cancel, leaving purely resistance, and thus Q = 0 (at the terminals). This does not contradict the Q factor of the individual components: we're just not measuring individual components anymore.

The other common meaning of Q is elements embedded within a system.

Typically we analyze RF networks by placing them between ports of known impedance (typically the 50Ω that most test equipment and cabling is designed for). The system impedance can be anything of course, and where a network doesn't need to connect to such an outside system, we might design it for some arbitrary impedance (example: the input or output impedance of the amplifiers or other blocks the networks connect between, say within a radio circuit). The port impedance(s) also need not be equal; indeed the network can be used to match between different impedances. (A transformer being a trivial example!) In any case, the fact that we apply impedances to the network, means the reactances or elements within the network, bear some relation to that impedance.

That's probably overly abstract or general, so how about a somewhat worked example. While I won't hope to explain analytical filter theory here, the general idea for the most common style (a ladder network between two ports terminated at $$\Z_0\$$) is that successive each element (L and C) is designed at some ratio to $$\Z_0\$$, and the pattern of those ratios is generated by the desired filter characteristic.

That is, each element (L or C) has its reactance some ratio above or below $$\Z_0\$$. We might call this an embedded or system Q factor, for the element relative to the system.

The simplest example is the single-pole filter. We have either two ports in series with a reactance (series-first topology), or all in parallel (shunt-first):

simulate this circuit – Schematic created using CircuitLab

These being the lowpass (shunt C or series L) case.

In the left case, we have $$\R_{th} = R_1 \parallel R_2\$$ or 25Ω, and $$\F_0 = \frac{1}{2 \pi (25Ω) (1µF)}\$$ or 6.3kHz cutoff. There aren't any other special frequencies here, so we can simply note that system Q varies with frequency, with the cutoff given at the frequency where Q = 1.

On the right, we have $$\R_{eq} = R_1 + R_2\$$ or 100Ω, and $$\F_0 = \frac{(100Ω)}{2 \pi (1µH)}\$$ or 15.9MHz cutoff. Likewise, this cutoff coincides with the system Q = 1 point.

Okay, the 1st-order system is trivially simple and uninteresting. Let's try a 2nd order system and see what happens.

simulate this circuit

We have $$\Z_0 = \sqrt{\frac{L}{C}}\$$ = 50Ω*, $$\F_0 = \frac{1}{2 \pi \sqrt{L C}}\$$ = 450.1kHz, and at $$\F_0\$$, we have $$\X_L\$$ = 70.7Ω and $$\X_C\$$ = 35.36Ω. Huh, those are kind of suspicious, aren't they -- very close to $$\\sqrt{2}\$$ above and below $$\Z_0\$$ respectively, huh? Indeed, this is a Butterworth 2nd order filter, shunt-first, and that requires two elements with a system Q of $$\\sqrt{2}\$$. (The reactances are complementary because their positions are: shunt lesser than $$\Z_0\$$, series greater.)

*Which I have also not explained yet; $$\Z_0\$$ is unfortunately rather well used, referring variously to the impedance of free space, the characteristic impedance of a transmission line or surrounding system, or the characteristic impedance of a resonant circuit, filter or network. I'm using the latter case here (while also illustrating it has been chosen equal to the system impedance!), but used the system sense earlier. The sense is usually evident from context; if this were discussing fields in space, the impedance of free space would be relevant.

Note that these components ideally should have very high (component) Q, and the filter design has not accounted for any losses (component Q) associated with them. In general, loss in filter causes the transition band (frequencies near $$\F_0\$$) to droop, absorbing some power rather than purely transmitting or reflecting it. (This type of filter is reflective, i.e. any power that isn't transmitted to the load, is reflected back towards the source.) The system Q can be increased somewhat to "peak" the response, compensating for component losses, but this is only possible up to a limit equal to the component Q itself -- and that at infinite insertion loss (i.e. all the incident power is absorbed by component losses). A practical filter for signal purposes requires component Q about N times the system Q (for an order-N filter), and for power transmission purposes, many times higher still (depends how expensive your power is!).

So for filter design purposes, both system and component Q are relevant, and to achieve some desired system Q, individual component Q must be many times larger.

In conclusion, your confusion is understandable -- these meanings are context-sensitive, and lacking that context, it seems like a bunch of gibberish...

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.