Determining Resonator Q Factor from Return-Loss Measurement Alone

It’s not uncommon to want to measure the Q factor of a resonator. One may need to determine its suitability for use in a coupled resonator filter or evaluate the performance of an RFID tag. Generally, this measurement is made with very light input and output coupling to reduce the loading effect of the 50-Ω source and load impedances.

Coupling to and from the resonator might be done with two electrically short antennas or loops to couple to the electric or magnetic fields of the resonator (Fig. 1). One instrument that could make such a measurement is Copper Mountain Technologies’ TR1300/1, a 1.3-GHz vector network analyzer (VNA) (Fig. 2).

After measuring an S21 S-parameter in this fashion, the data is analyzed to extract the resonant frequency and Q factor of the resonator. The peak of the response is taken to be the resonant frequency, and then two markers are placed 3 dB down from the peak value. The peak frequency divided by the 3-dB width of the peak is then equal to the Q factor.

For example, a sweep of the circuit shown in Figure 3 results in the measurement shown in Figure 4. That graph gives us an experimental Q factor of 13.62/(13.99 − 13.28) = 19.2.

The approximate Q factor from the schematic, neglecting the effects of the 12-pF coupling capacitors and the 50-Ω source and load, is equal to the admittance of the 113-pF capacitor at 13.62 MHz divided by the conductance of the resistor, or 9.673e-03/5e-04 = 19.3. This shows that there’s reasonable agreement with the experimentally determined value.

With reduced coupling, it’s possible to obtain a slightly better measurement, allowing the S₂₁ peak to fall to −40 dB or so to reduce the loading effect. However, the S₁₁ reading would become very small. We will show that the Q factor may be derived from the S₁₁ measurement, but the numbers need to be large enough to work with.

So how can this be done? Clearly, it isn’t a matter of looking for the points on the S₁₁ curve that are 3 dB higher than the minimum. The trace shown above has a minimum of −1.6 dB, so that’s clearly out of the question. It turns out that in a lossless circuit, there’s a relationship between S₁₁ and S₂₁:

From the earlier graph we can calculate a value for S₂₁:

If:

Then:

This S₂₁ isn’t a real value per se, but we can still use it. Calculating the value of S₂₁, which is 3 dB down, means multiplying by 1/√2:

Now we take that back to S₁₁:

Or, −0.748 dB.

If we find this value of S₁₁ on each side of the minimum from the earlier measurement, we get the result shown in Figure 5.

From the three frequencies shown, we can calculate the Q factor:

That result is quite close to the calculated value of 19.2.

So, with a relatively simple calculation, one can determine the Q factor of a resonator from the return-loss measurement alone.

Brian Walker is Senior RF Design Engineer at Copper Mountain Technologies.