How to Determine the Q Factor of a Resonator from Return Loss Measurements

It is not uncommon to want to measure the Q factor of a resonator. It may be necessary to determine its suitability for use in coupled resonant filters, or to evaluate the performance of RFID tags. Typically, this measurement is made with very light input and output coupling to reduce the loading effects of the 50-Ω source and load impedances.

1. For 2-port Q measurements of resonators, establish very light input and output coupling to reduce the loading effect of the 50Ω source and load impedances.

2. 
   

Coupling to and from the resonator can be achieved with two electrically short-circuited antennas or loops coupling the electric or magnetic fields to the resonator (Figure 1). One instrument that can make this measurement is Copper Mountain Technologies’ TR1300/1, a 1.3 GHz vector network analyzer (VNA) (Figure 2).

2. TR1300, 1 VNA can be used for resonator Q measurements.

After measuring the S21 S-parameters in this way, the data is analyzed to extract the resonant frequency and Q-factor of the resonator. Take the peak of the response as the resonant frequency and place the two markers 3 dB below the peak. The peak frequency divided by the 3 dB width of the peak equals the Q factor.

For example, scanning the circuit shown in Figure 3 results in the measurements shown in Figure 4. This plot gives us the experimental Q-factor of 13.62/(13.99 − 13.28) = 19.2.

3. Shown is a 2-port example circuit for VNA measurements.

4. This figure illustrates a 3 dB Q-factor measurement for the circuit shown in Figure 3.

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

By reducing coupling, better measurements can be obtained, bringing the S 21 peak down to around -40 dB, thus reducing loading effects. However, the S 11 reading will be very small. We will show that the Q factor is possible from S 11 measurements, but the quantities must be large enough to be used.

then what should we do? Obviously, finding a point on the S 11 curve that is 3 dB above the minimum is not a problem. The trace shown above has a minimum of -1.6 dB, so this is clearly not possible. It turns out that in lossless circuits. There is a relationship between S 11 and S 21:

From the previous plot, we can calculate the value of S 21:

if:

Then:

S 21 isn't really a value itself, but we can still use it. Calculating the value of S 21 (downward by 3 dB) means multiplying by 1/√2:

Now let's go back to S 11:

or -0.748 dB.

If we find the value of S 11 on each side of the minimum from the earlier measurement, the result is as shown in Figure 5.

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

This result is very close to the calculated value of 19.2.

Therefore, with relatively simple calculations, it is possible to determine the Q-factor of a resonator from just a return loss measurement.