There is a fashion of seeing s21 measurements as the answer to all things, and amongst the revelations is an explanation of measuring inductor Q using s21 shunt through configuration.

Let’s explore the use of s21 shunt through to directly find the half power bandwidth of a series tuned circuit and calculate the Q from that and the resonant frequency (as demonstrated by online posters).

To eliminate most of the uncertainties of measurement, let’s simulate it in Simsmith.

The simulation has a series tuned circuit resonated at 3400kHz, and the source and plot are set to calculate |s21| in dB. Though the model specifies Q independent of frequency, the D block adjusts Q for a constant equivalent series resistance (ESR) which simplifies discussion of resonance and Q.

The markers are set to the frequencies where |s21| is 3dB higher that at the resonant frequency, and the bandwidth is 3601-3210=391kHz, which gives \(Q=\frac{f_0}{BW}=\frac{3400}{391}=8.7\).

But wait a minute, the simulation specifies Q=10!

Why is the measurement technique not giving the expected result?

## Answer

It goes to understanding some basic concepts, and applying them to the measurement problem.

### Series resonant circuit

The response of a simple series resonant RLC circuit is well established, when driven by a constant voltage source the current is maximum where Xl=Xc (known as resonance) and falls away above and below that frequency. In fact the normalised shape of that response was known as the Universal Resonance Curve and used widely before more modern computational tools made it redundant.

Above is a chart of the Universal Resonance Curve from (Terman 1955). The chart refers to “cycles”, the unit for frequency before Hertz was adopted, and yes, these fundamental concepts are very old.

### Q factor

(Terman 1955) gives a general definition of Q that is widely accepted:

circuit Q is 2π(energy stored in the circuit)/(energy dissipated in circuit in one cycle) .

Q is easily assessed for a single reactor:

- inductor: energy stored is \(L=\frac{I_{pk}^2}{2}\) and energy lost per cycle is \(\frac{I^2R}{2 f}\) so \(Q=\frac{2 \pi f L}{R}=\frac{X_l}{R_s}\);
- capacitor: energy stored is \(\frac{C V_{pk}^2}{2}\) and energy lost per cycle is \(\frac{V^2}{2 f R_p}\) so \(Q=2 \pi f C R_p=\frac{R_p}{X_c}\).

A RLC series resonant circuit is nearly as easy as at the instance when the inductor current is maximum, capacitor voltage is zero and hence the energy stored is as per the simple inductor case and \(Q=\frac{X_l}{R_s}\).

Likewise for a parallel RLC resonant circuit, at the instant of maximum capacitor voltage, inductor current is zero and hence the energy stored is as per the simple capacitor case and \(Q=\frac{R_p}{X_c}\).

Note that this model is not valid where the inductor component (which is more completely a resonator itself) approaches its own self resonant frequency (SRF).

### Q and bandwidth

Q and bandwidth are related, high Q circuits have a narrow response, narrow bandwidth. Q is a factor used in normalising the Universal Resonance Curve (URC) above, and appears in several of the equations on the chart.

Lets focus in particular at the point on the URC corresponding to where |X|=R. When |X|=R, |Z| is 2^0.5 times that at resonance, and the current response is 2^-0.5 (or 0.707) times that at resonance. Since power=I^2*R, power at that point is half the maximum response. These are known at the half power points.

If you look at the URC, you will seen that the current response is 0.7 of maximum when a=0.5. Taking the case for a=0.5 (the half power points) and noting that Bandwidth (BW) is measured between the upper and lower points, BW=2*CyclesOffResonance, we can substitute into a=Q*(CyclesOffResonance/ResonantFrequency) to obtain BW=ResonantFrequency/Q. This is a well known relationship.

For a circuit where R is approximately constant with frequency, we can find the approximate BW by finding the frequencies at which |X|=R.

### Applying these concepts to the VNA simulation

Lets repeat two important statements from the above:

- for a circuit where R is approximately constant with frequency, we can find the approximate BW by finding the frequencies at which |X|=R; and
- when |X|=R, |Z| is 2^0.5 times that at resonance, and the current response is 2^-0.5 (or 0.707) times that at resonance. Since power=I^2*R, power at that point is half the maximum response. These are known at the half power points.

This does not mention s21, though s21 is a means to a solution, but it is not by itself the solution.

We can use (12 term corrected) s21 shunt through measurement to find R, X, |Z| of the series circuit LC1. Remembering that s21 is a complex quantity, \(z_x=\frac{25 s_{21}}{1-s_{21}}\).

Above is the Simsmith model extended to calculate zx, the impedance of the series circuit LC1, and to plot some relevant quantities. Remember that the simulation Q is 10, and ESR is independent of frequency.

Note that your VNA may not be able to calculate zx in the s21 shunt thru configuration.

The chart is quite busy, it contains:

- the originally plotted |s21| as s21m in blue;
- R as zx.R in red;
- |X| as zx.I.M in yellow; and
- |Z| as zx.M in magenta.

We can calculate Q using the method used at the start of this article, and two methods based on the discussion of concepts.

Method | Bandwidth (kHz) | Calculated Q |

|s21| 3dB points | 391 | 8.7 |

\(|Z|=\sqrt{2}Z_0\) | 340 | 10.0 |

\(|X|=R\) | 340 | 10.0 |

The |s21| 3dB points clearly fails to reconcile, it is not based on sound concepts. It might give results that appear correct for some scenarios, but it is not soundly based.

The last two methods both reconcile with the simulated value of Q.

Measurements using a spectrum analyser and tracking generator to measure the equivalent of |s21| shunt through will also fail for the same reasons.

### Real world components

The simulation forced ESR to be independent of frequency as is done is most textbook discussion of the concepts discussed above.

Real world inductors at RF are not quite that simple, a first approximation is that \(R \propto \sqrt{f}\) by virtue of skin effect. Close wound solenoids and layered windings are complicated by Proximity Effect which increases R. Self resonance of the inductor (which is really of itself a resonator) causes a change in R at frequencies above about a tenth of self resonant frequency.

Let’s change the model to make \(R \propto \sqrt{f}\) to demonstrate a more practical scenario.

Note the curves are not so symmetric, a result of R varying with frequency.

We can calculate Q using the method used at the start of this article, and two methods based on the discussion of concepts.

Method | Bandwidth (kHz) | Calculated Q |

|s21| 3dB points | 391 | 8.7 |

\(|Z|=\sqrt{2}Z_0\) | 340 | 10.0 |

\(|X|=R\) | 342 | 9.9 |

Again, the |s21| 3dB points clearly fails to reconcile, it is not based on sound concepts. It might give results that appear correct for some scenarios, but it is not soundly based.

Even though the resonance response is a little skewed, methods two and three still produce a result that reconciles well with expected Q @ 3.4MHz. The skew warns the measurer to not measure only one half power point, but to measure both to offset some of the error due to the skewed response.

Measurement of real world series resonant circuits will be complicated by the above effects, parasitic inductances and capacitances of fixtures, and measurement error.

Application to inductors where the equivalent series inductance is not approximately independent of frequency is questionable, the response will not follow the URC (including the half power bandwidth inference), the broader inference of Q is not valid, it is possibly specious (ie superficially plausible, but wrong).

Measurement of Q is not quite a no-brainer, but application of sound concepts and an intelligent approach in experiment design should yield valid results.

## Real world measurement

Accurate measurement of s21 with a VNA system depends on the accuracy of the input impedance or Port 2, errors in which can be corrected by 12 term correction.

The measurements need to demonstrate the the inductor is sufficiently far below its SRF to be adequately described as a frequency independent inductance in series with some resistance.

## Measuring Q of a parallel resonant circuit using s21 series through

Just as in the case above that you cannot determine half power bandwidth from |s21| 3dB points, the same applies to a parallel resonant circuit using s21 series through measurement.

Applying the information in this article to the parallel resonant circuit using s21 series through is left to the reader as an exercise.

## References

- Terman, Frederick. 1955. Electronic and Radio Engineering – 4th ed. New York: McGraw-Hill.