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:
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.
1port reflection (1pr) measurement
Zdut(1prS11)=Z0*(1+S11)/(1-S11)
Z0: The impedance of the Calibration-load (should
be 50+0iΩ)
2port series through (2ps) measurement
Zdut(2psS11)=((Z0-Zload)+(Z0+Zload)*S11)/(1-S11)
and if Zload close to Z0:
Zdut(2psS11)
~ (Z0+Zload)*S11/(1-S11)
or even:
Zdut(2psS11)
~ 2*Z0*S11/(1-S11)
The Zdut(2psS11) measurement, should be
same/close to the Zdut(1prS11) measurement. As
described by K6JCA, NanoVNA Saver and above link; with Zload
is Z0.
The Zdut(2psS11) formula does include the Zsource
of port-1 (is calibrated away). With NanoVNA Saver (with Zload
is Z0): R+jX or |Z| S11 .
Zdut(2psS21)=2*SQRT*(Zsource*Zload)*(1-S21)/S21
Zsource: The impedance of port-1
Zload: The impedance of port-2
SQRT background is here
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:
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:
clip-on ferrites on the port1, port2 or NanoVNA USB cable;
shortened the port-2 cable (from 16cm to 2cm);
located the test setup as far away from other equipment (the
NanoVNA is though always some 16cm nearby, see above picture).
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.
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)|):
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 183μH 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):
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:
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:
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.
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:
99Ω ± 1.5% (100Ω 5% SDM resistor)
330Ω ± 1.5% (330Ω 5% metal resistor, with 1inch leads)
21800Ω ± 1.5% (22kΩ 5% metal resistor,
with 1inch leads)
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?).
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).
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.