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S11 and S21 measurements on uni-directional VNA to determine impedance of DUT

This web page will cover ideas why to do S11 or S21 measurements with the uni-directional (using 3-term calibration) NanoVNA-H 3.5 (software version 1.0.64, kernel 4.0.0). This can be 1port (1p) or 2port (2p) measurements . The frequency range is from 0.1 to 30Mhz.
The following sections are covered:

Test setups

Formula (with all complex variables) are derived (somewhat extended) from https://xdevs.com/doc/HP/pub/5988-0728EN.pdf: and based on verification in real experiments with the NanoVNA.

Calibration plane

The calibration plane is at the end of the middle/earth tips of the PCB SMA connector (like the calibration kit). The DUT is soldered directly to these pins aka calibration plane (so no real extension of the plane). This all to minimize variability of reference plane, parasitic capacitance,  etc. An example is below:
Setup

The DUT is put on a Styrofoam underground, which has a dielectric close to air, this to minimize the effects of e.g. the workbench's dielectric on the measurements.
The DUT is certainly effected by nearby objects (such as hands, workbench; aka adding parasitic capacitance), so one needs to stay away;-). This hand effect is most prominent on the Zdut(2psS21), and a little less (and somewhat different behavoir) on the Zdut(1prS11) or Zdut(2psS11) measurements.

Some further setup variations have been tested:
All these variations did not really have an effect on Zdut(1prS11), Zdut(2psS11) or Zdut(2psS11) measurements. Only slightly at the end of the frequency range (from 26MHz).

Compensating for deviating Zload and Zsource

As a NanoVNA (an uni-directional VNA) can only calibrate/measure S11 and S21 and not the full set including S12 and S22 (like when using a bi-directional VNA), we need to determing Zload and Zsource to compensate the S11 and S21.
Zload is measured using a Calibration-through device between port-1 and port-2. This measures Zdut(1prS11), remember that in this case the DUT is port-2 of this particular NanoVNA-H 3.5 (software version 1.0.64, kernel 4.0.0)! For this NanoVNA, the Zload of port-2 is very close to an average 50.8-0.1iΩ (300kHz to 30MHz), which is based on a Calibration-load (Z0) of 49.7Ω (using DVM, accuracy 1.5%). As the Calibration-load is a little smaller than 50+0iΩ, the resistances measured by the NanoVNA will be a little too high.

Looking at the schematics of my NanoVNA-H, there is already a 20dB attenuation (-20*log(49.9Ω [R50]/530Ω [R23+R13+R22])) build into port-2.
Zload is: 56Ω [R24] in parrallel with (2*240Ω [R22&R23] + 49.9Ω [R13]) =530Ω. This gives a Zload = 50.6Ω. This is  very close to what is measured above (50.8-0.1iΩ). The input impedance of U8 (SA612A, Table 6) has been neglected (1.5kΩ||3pF).
In some literature it is recommended to use a 10dB attenuator on port-2, to get Zload close to 50+0iΩ. In my measurements such an extra attenuator had no influence.
The Zdut(2psS21) and Zdut(2psS11) are compensated for that Zload.
 
Zsource is measured using Owen Duffy's method. But the formula is changed (as it should be the inverse of above Zdut(2psS21) formula). Best to use a Zref close to Z0 (e.g. 100Ω), to make sure the Zref accuracy is good (based on S11 measurement):

Zsource = ((Zref/2*S21/(1-S21))^2)/Zload)

Zref: reference impedance

Zsource is not the impedance of port-1, but the impedance of the oscillator (SI5351A) circuit feeding the measurement bridge. For this particular NanoVNA, Zsource is significantly different from 50Ω; on average 43.6+3.2iΩ, 
Zsource as seen in the schematics of my NanoVNA-H is: 56Ω [R14] in parrallel with ((85Ω [SI5351A, Table 2] || 497Ω [R16+R17])) ~72.6Ω. This gives a Zsource = 31.6Ω. The input impedance of U6 (SA612A, Table 6) has been neglected (1.5kΩ||3pF).
This is different from what is measured above (43.6+3.2iΩ).
Question: Do not yet fully understand this reason!

So Zdut(2psS21) and Zdut(2psY21) is compensated for this Zsource.

With decreasing/increasing Zload or Zsource, the derived Zdut decreases/increases more or less proportional.

Zload ans Zsource of thsi particular NanoVNA-H
Warning: the above graph is likely to be different for other (Nano)VNAs, but it looks one can calculate good approximations of the Zload or Zsource by looking at the schematics!

Offset delay or parasitic capacitance relating to S11

If we use the default Offset delay  = 0ps (the dark red and blue curves in this graph), we get the following impedance (|Zdut(2psS11)| and |Zdut(2psS21)|), resistance (Rdut(2psS21)) and reactance |Xdut(2psS21)|):
Resistance, reactance and
          impedance of Choke 0ps
One can see that the
|Zdut(2psS11)| (red dashed curve) deviates from |Zdut(2psS21)| (green dashed curve).
The peak in |Zdut(2psS21)| (green dashed curve) happens at around 24.3MHz, assuming an inductance at 18H this would happen with a Cpara = 0.25pF.
The peak in
|Zdut(2psS11)| (red dashed curve) happens at around 12.1MHz, assuming an inductance at 183μH this would happen with a Cpara = 0.95pF.

The Offset delay (in NanoVNA Saver) was reduced, which has as expected no effect on Zdut(2psS21), but the R and X peaks of Zdut(2psS11) were shifted to higher frequencies. If one puts the Offset delay to around -54ps (equivalent to -7mm), the R and X peaks of S11 map the R and X peaks of S21.
The below screengrab of NanoVNA Saver is with a Offset delay of 0ps (sweep) and -54ps (reference):
S11 (delayed with -ps) and S21
Top two graphs: Above dark red (R or |Z|) and dark blue (X) 2psS11-related curves at with Offset delay at 0ps and light red (R or |Z|) and light blue (X) 2psS11-related curves at with Offset delay at -54ps.
Bottom two graphs: Important to see that (light) red (R or |Z|) and (light) blue (X) 2psS21-related are not depending on the Offset delay.
In general; the 2psS21-related curves have less noise, while the 2psS11-related curves have more noise above say 5kΩ.

When we look at Offset delay = -54ps (the lighter curves) the peak of the dashed red curve (|Zdut(2psS11)|) gets similar to the peak of the dashed green curve (|Zdut(2psS21)|).
The following impedance (|Zdut(2psS11)| and |Zdut(2psS21)|), resistance (Rdut(2psS21)) and reactance |Xdut(2psS21)|) are derived/measured:
Resistance, reactance and impedance of Choke
The |Zdut(2psS21)| curve, regardless of the Offset delay, does not change; only |Zdut(2psS11)| curve is changed to map the |Zdut(2psS21)| curve.

If adding some capacitance parallel to the DUT by using a twisted wire (Telfon coated AWG26: 0.35mm core and 2.75mm distance, gives around 0.2pF/cm) with a length from 0.5 and 12cm; the peaks of both
Rdut(2psS21) and Rdut(2psS11) are at lower (resonance) frequencies in accordance with the increasing capacitance (~2.4pF/cm) due to length increase. Reducing or increasing a parallel capacitance to DUT does not move the Zdut(2psS11) and Zdut(2psS21) peaks together, so a parallel capacitance to DUT is not able to align the two.

The question remains if the difference
in Zdut(2psS11) and Zdut(2psS21) is due to:
  1. parasitic capacitance, while the test setup is for both precisely identical?
    If the are there with Zdut(2psS21), why are these not seen when performing at the same time the Zdut(2psS11) measurement.
  2. a different port-1 reference plane, while calibrating and connecting has been done on the (assumingly) correct plane?
    Changing the reference plane can map the
    Zdut(2psS11) and Zdut(2psS21) peaks together.

As the parasitic capacitance is small, it can be 'replaced' with a reference plane move. So its assumed (hopefully correct) that the second reason (different reference port-1 plane) is correct.
According to K6JCA it shows that S21 is 'pretty good' and it 'explains the resonance frequency shift when measuring impedance with S11'.
But a third calculation method migth shet more light: W1QG's Y21 method.

W1QG's Y21 method

When including W1QG's Y21 method, it should provide another method of determining Zdut.
aA s we can only measure uni-directional: we assume (only valid when Zsource = Zload): S11=S22 and S21=S12;
Zdut(2psY21)=SQRT(Zsource*Zload)*(2*S11 +S11^2 -S21^2 +1)/(2*S21)

Remark: Zo in W1QG's Y21 formulas has been replaced with SQRT(Zload * Zsource) to try to include the compensation (emperically deducted, some analytic background is here [Frickey, 1994]).

So this method (|Zdut(2psY21)| , blue curve) was included in the Excel spreadsheet that determines the earlier discussed values (with Offset delay=0ps):
 


The
W1QG's Y21 method provides |Zdut(2psY21)| (blue) which values are very close to the |Zdut(2psS21)| (green-dashed).
Remember that the S and Y matrixes are different representations of the same circuit (if one includes Zsource and Zload)
[Frickey, 1994]. So in principle the derived Zdut should be the same.

The shunt capacitance (on port-1 and port-2) calculated with
W1QG's Y21 method is constant around 1.1pF over a large frequency range (from 3.5 to 30MHz). This is in the order of the parasitic(?) capacitance (0.95pF)  that explained the lower resonance frequency of Zdut(2psS11) and Zdut(1prS11)).
 
The theoretical simulated Zeff, which uses μ' and μ" of the FT140-43 specification, can be matched on Zdut(2psY21) and Zdut(2psS21). This was done by matching the peak of the simulated curve of Zeff on the resonance peak of the Zdut(2psS21) (so watch out for a circular reasoning!).

Comparing S11, S21, Y21 and DVM Zdut measurements

 A few 1pr, 2ps NanoVNA and DVM Zdut measurements (DVM accuracy ~1.5%) were performed on three resistors (99Ω [SMD], 330Ω [metal, with 1inch leads] and 21800Ω [metal, with 1inch leadsl]) and a 1:1 Guanella Choke:
For the resistors; the 1prS11&2psS11 and 2psS21&2psY21 Zdut are quite close, except the Zdut(1pS11) of 21800Ω resistor is deviating somewhat from the others Zdut 2psS11&2psS21&2psY21.
In case of the 1:1 Guanella Choke; the Zdut 1prS11&2psS11 deviate considerable from 2psS21&2psY21 (due to influence of parasistic capacitance?).

By looking above pictures, some consideration about accuracy (trueness) and precision (beside influences of possible parasitic capacitances):

I use above 'assumed' due to missing comparison of the same DUT using 3term and 12tem calibration, see below footnote.

Conclusions

I understand K6JCA, G3TXQ and Owen Duffy like the (Zsource&Zload compensated) Zdut(2psS21); also because Zdut(2psS21) has less noise for high impedances than Zdut(1prS11). K6JCA recommends beside this the slightly more computational Zdut(2psY21). But remember that the S and Y matrixes are different representations of the same circuit (if one includes Zsource and Zload) [Frickey, 1994]. So in principle the derived Zdut should be the same from an S or Y matrix.
Others say (e.g. Roger Need, Owen Duffy and Steve Sandler) that 1prS11 will be ok until a few thousands of ohm. These statements are not contradicting.
Zdut(1prS11) has the same value and frequency behavoir as Zdut(2psS11) (also for low impedances), if the parasitic capacitance is not removed or Offset delay not included.
According to K6JCA (pers. comm, 2022):
The effect of this parasitic capacitance is less [then S11] when measuring impedance using S21 or Y21, because, the capacitance now is in shunt with a VNA port's 50 ohm termination (rather than shunting the inductor-under-test), and so its impact should only become noticeable at significantly high frequencies, when its shunting action begins to change the VNA port's impedance.

This web site recommends (due to above mentioned experience) for a Choke's Zdut that is expected to be more than 5kΩ, to use 2psS21 (or 2psY21, but it does not seem to provide really much different results than 2psS21 for a uni-directional VNA). But see below footnote!
So if the Zdut(2psS21) (or Zdut((2psY21)) are indeed closer to reality (upto at least 30MHz), then non-corrected (Offset delay or parasitic impedance) Zdut(2psS11) and Zdut(1prS11) are off (even for low value impedances).

Important footnote!

The above might still have some circular reasoning! From the above experiments, I am not able to say for 100% that Zdut derived from S21 (or Y21) is better that Zdut derived from S11! The reason why I find S21 derived Zdut better, is that I can transpose (e.g. using parasitic capacitor or moving reference plane) all S11 measurements towards S21 (the other way around is not possible).
Seeing the difference in opinions of experts with regard to S11 or S21 based Zdut, this might need to be checked with a bi-directional VNA (with 10/12 term calibration).
If someone can compare
(
using the same DUT and test-rig) 3 term calibrated 2psS11 and 2psS21 and 2psY21 with 10/12 term calibrated VNA results, let me know (I am willing to help of course).

References

Agilent: Advanced impedance measurement capability of the RF I-V method compared to the network analysis method. In: https://xdevs.com/doc/HP/pub/5988-0728EN.pdf (2001),
Frickey, Dean A.: Conversions between S, Z, Y, h, ABCD and T parameters which are valid for complex source and load impedances, IEEE Transactions on microwave theory and techniques, Vol. 32, No. 2 Feb. 1994.
Need, Roger: Component measurement accuracy of NanoVNA-H4. In: box.
Sandler, Steve: Accurately measure ceramic capacitors by extending VNA range. EDN Network, pp.   https://www.edn.com/accurately-measure-ceramic-capacitors-by-extending-vna-range/, 2017.
Walker, Brian: Make accurate impedance measurements using a VNA. In: Microwaves&RF(2019), pp. 1-5.

Acknowledgements

I would like to thank people, such as Jeffrey Andreson, Owen Duffy, Roger Need and others for their help, encouragement and/or constructive feedback. Any remaining errors in methodology or results are my responsibility of course!!! If you want to provide constructive feedback, let me know.
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Major content related changes: June 9, 2022