This article canvasses a possible improvement of the s21 series-through measurement of impedance to compensate for errors in VNA port impedances that are not corrected in simpler calibration / correction schemes.

The diagram above is from (Agilent 2009) and illustrates the configuration of a series-through impedance measurement.

Agilent gives the expression for s21 as \(S_{21}=\frac1{1+\frac{Z_x}{2 Z_0}}=\frac{100}{Z_x+100}\) (taking Zo=50).

We can rearrange that making Zx the subject \(Z_x=\frac{100}{S21}-100\) and expanding the values 100 to their components \(Z_x=\frac{Z_s+Z_l}{S21}-\left(Z_s+Z_l\right)\) where Zs and Zl are the equivalent Thevenin impedances of the source and load.

That begs the question whether we might compensate for small errors in Zs and Zl by measuring their values and substituting into the calculation.

Measuring Zl is simple enough, after SOLT calibration of the VNA, an s11 reflection measurement of Port 2 gives a value from which Zl can be calculated: \(Z_l=50 \frac{1+S11}{1-S11}\).

Zs cannot be measured directly, but can be found indirectly by measuring s21 of a known series impedance… so the first step is to measure Z of this calibration part, let's call it Zc. Depending on the fixture, it might be possible to use the same part used in the SOLT calibration.

We can measure Zc using a s11 reflection measurement and calculate Zc as \(Z_c=50 \frac{1+S11}{1-S11}\).

Now we can measure s21 with Zc in series from Port 1 to Port 2. Zs is given by \(Z_s=\frac{Z_l-s21\left(Z_c+Z_l \right)}{s21-1}\).

Having determined Zs and Zl, we can now measure s21 of an unknown Zu in series from Port 1 to Port 2 and calculate \(Z_u=\frac{Z_s+Z_l}{S_{21}}-\left(Z_s+Z_l\right)\).

A small ferrite cored test inductor was measured with a ‘bare' nanoVNA SOLT calibrated, firstly using s11 reflection.

Above is the test fixture.

Above is the R,X,|Z| plot from the s11 reflection measurement.

### 7.09MHz

Above is a calculation of Z at 7.90MHz from the saved .s1p file, it reconciles with the cursor data on the plot.

Above is a calculation of Z using the values from a series-through sweep saved .s2p file using Agilent's formula.

Above is a calculation of Z correcting for Zs and Zl as discussed in this article.

In this case there is not a lot of difference in the values obtained. Measurement noise is an issue, all measurements were single captures with no averaging (the averaging function in the firmware does not have the expected outcome so it is not used).

### 29.7MHz

Above is calculation of a simple s11 reflection measurement.

Above is calculation of a simple s21 series-through measurement. The R value is significantly lower.

Above is calculation of enhanced s21 series-through measurement. There is a small difference in both R and X compared to the simple s21 method.

Note there is greater departure of calculated Zs and Zl from the ideal 50+j0Ω

## A closer look at Zl

Above is a screenshot from Simsmith basically to plot a very expanded view of the Smith chart of the s11 reflection measurement Port 2. There is measurement noise, but the underlying response is a curve is reasonably clear and while it doesn't exactly follow a curve of constant G, it is a reasonable first approximation and suggests the cause appears to be a small shunt capacitance… around 2pF. A short series section of low Zo transmission line has quite similar effects.

So even though Port 2 has very good InsertionVSWR at 7MHz, it is somewhat poorer at 30MHz.

## Analysis

The study charted departure from ideal of Port 2 impedance. Though not charted, departure of Port 1 impedance was calculated at 29.7MHz, and departure in both Port 1 and Port 2 impedances impacts the simple s21 series-through measurement.

Online experts opine that s11 reflection measurement is not capable of good accuracy above perhaps 500Ω, and that Agilent's s21 series-through configuration yields much better results… but that is not borne out in the 7MHz example which is typical of a high impedance ferrite cored common mode choke.

The Zs Zl corrected measurement is a lot more complicated to perform, and gave very similar results to Agilent's s21 series-through method at lower frequencies.

The fixture is all important in obtaining good results. There may be cases where it is easier to build a good s21 series-through fixture than a good s11 reflection fixture, particularly for larger parts, and that might drive a preference for s21 series-through measurement.

Further, the enhanced algorithm may provide improved accuracy, especially for high Zu in fixtures where departure from ideal is unavoidable.

The technique described here might give improved accuracy for VNAs where 12 term correction is not available.

More at Improving ‘s21 series-through' measurement of high impedances – more detail.

## References

- Agilent. Feb 2009. Impedance Measurement 5989-9887EN.
- Agilent. Jul 2001. Advanced impedance measurement capability of the RF I-V method compared to the network analysis method 5988-0728EN.
- Derivation of the expression for the unknown impedance in a s21 series through measurement
- Calculate impedance using S11 reflection, S21 series & S21 shunt
- Calculate Z from series s21 – enhanced