## Wednesday, October 21, 2015

### Notes on Measuring Antenna Tuner Power Loss

Important Note:  12 August 2018:

This post has been superseded by a new post on measuring Antenna Tuner loss.  Please click on the following link to go to the new post:

The original post, which contains much background information, continues, below...

*****************************************

These are my notes on measuring Antenna Tuner Loss that I've made while working on my own tuner design (here:  http://k6jca.blogspot.com/2015/09/antenna-auto-tuner-design-part-7-build.html).

But first, please note...

>>  Click on any image to enlarge! <<

And now, on to my notes...

Over the years, several different methods for measuring Tuner Power Loss have been mentioned in ARRL publications.  Let's look at two of these.

Technique 1:  Measure Loss via SWR, the AI1H Method:

The first mention of Tuner Loss Measurements (that I'm familiar with) is a technique described by Frank Witt, AI1H, published in the April, 1995 issue of QST.  This technique can be found in these following references:
"How to Evaluate Your Antenna Tuner, Part 1", Frank Witt, AI1H, QST, April, 1995  (Part 2 is in the May, 1995 issue -- the link associated with this text goes to a site that has combined both articles into one).

"Evaluation of Antenna Tuners and Baluns -- an Update", Frank Witt, AI1H, QEX, Sept/Oct, 2003

This technique uses either an SWR meter or an Antenna Analyzer to measure SWR for load resistances that are either twice or half of the value (Rload) being analyzed, and the measured SWR values are then applied to the following formula:

LEST = 5*log(((S1 + 1)(S2+1))/(9*(S1-1)(S2-1)))

Where:
• LEST =Estimated Loss, in dB
• S1 = SWR with load resistance equal to Rload/2
• S2 = SWR with load resistance equal to Rload*2
There are a number of factors which will affect the accuracy of the results, though...

Of course, accuracy of the SWR reading will be a major influence.  But also important is the actual impedance of the loads being measured, at the test frequency.  Impedances measured at one frequency might be different at another, depending upon load fixture parasitics and the characteristics of the resistances, themselves.

As the ARRL points out in "Antenna Tuner Testing Methods vs Accuracy" by Michael Tracy, KC1SX in the February, 2003 issue of QST (page 75), "the bottom line is that the original test method was reasonably accurate, but not necessarily reproducible.  To the extent practical, the Lab will continue to use the more accurate "direct" method for further testing.  The AI1H method will still be used for some testing, but only with careful cross-checking of the instrumentation used"

Note their mention of the "more accurate 'direct' method."  Let's look into this further...

Technique 2:  Measure Loss via the ARRL "Direct" Method:

The ARRL's "direct" method of Loss Measurement is first mentioned (to my knowledge) in the article
"Antenna Tuner Testing Methods vs Accuracy", by Michael Tracy, KC1SX,  and which appeared in QST, February, 2003, on page 75.

From the article, here is what their test fixture looked like.  Note the power resistors.

Unfortunately, the article's description of the test setup is sketchy at best.  But later articles do a better job describing the fixture and test technique.

For example, Phil Salas, AD5X, has a more detailed description of this testing technique in a product review of two MFJ auto-tuners published in QST  ("The MFJ-994BRT and MFJ-998RT Remote Automatic Antenna Tuners", Phil Salas, AD5X, QST, Aug, 2012).  The following two images are quotes from the article.  First, a description of the basic setup:

And here's a description of the Load Box used for the testing.  Note that the resistors need to be able so handle the RF power sourced by the transmitter:

And, from the article, here's a picture of the Resistive Load Box:

So, for example, if the desired Tuner test-load were 200 ohms (4:1 SWR), the series 150 ohm resistor and no shunt resistance would be selected.  This 150 ohm resistor, in series with the 50 ohms presented to the fixture by the 40 dB pad, adds to create the 200 ohms seen by the tuner.

And the voltage at the input of the 40 dB pad should be 1/4 of the applied voltage.  That is, there should be a 12.041 dB loss due to this fixture.  And thus any measured loss beyond the "ideal" attenuation of 12.041 dB would be due to Tuner Loss.

Similarly, if the desired Tuner test-load were 12.5 ohms (again, 4:1 SWR), the series element would be the short (0 ohms) and the shunt resistance would be be 16.67 ohms, which, when shunted across the 50 ohms presented to the fixture's output by the 40 dB pad, creates a resistance, as seen by the tuner, of 12.5 ohms.

Again there's 12.041 dB loss due to the fixture (because 3/4 of the power is shunted into the 16.67 ohm resistor).

[Note that in the actual fixture schematic above 15 ohms is used in lieu of  16.67 ohms.  This creates an 11.54 ohm impedance in lieu of 12.5 ohms, for an SWR of 4.33: 1 and an "ideal" attenuation of 12.74 dB.]

I took a similar approach to my initial power-loss testing of my own tuner (here).  But I didn't have any high-power resistors suitable for testing at RF frequencies, and so I thought, why not use my HP 8753A Vector Network Analyzer, instead?

Power could be kept low (on the order of 0 dBm), and so I could minimize unwanted fixture "parasitic" inductance and capacitance by using small, low-power components placed on a small piece of PCB with SMA connectors on either end.  Pads on the PCB would allow me to connect a resistor in series or in parallel with the SMA that connects to Port 2 of the S-Parameter Test Set.
Below is my test setup for this method of power-loss testing.  Note that:

1.  To test loads (Rload) greater than 50 ohms, Rp should not be stuffed and Rs should be equal to Rload - 50.

2.  To test loads (Rload) less than 50 ohms, Rs should be stuffed with a short (0 ohms) and Rp should be equal to 1/(1/Rload - 1/50).

I measure S21 with the fixture in place and compare it to the "ideal S21" that should exist if:
• The tuner is lossless, and
• There was no reflected power from the tuner's input (SWR = 1:1)

Here's my fixture:

I used this method to measure the loss of two tuners:  one of my own design and my Elecraft KAT500.  Here are the results:

Notes regarding these tables:
• Column A is the SWR measured at the Tuner's input (Red means appreciably high SWR).
• Column B is the Return Loss equivalent of the measured SWR in Column A.
• Column C is the effect, in dB that this Return Loss has on the S21 measurement (and is equal to -10*log10(1-10^(- Return_Loss/10)). [Note that Return Loss, in the table above, is positive, and so I negate the Return Loss term in the equation].
• Column D is the S21 Measurement.
• Column E is the actual S21 loss due to the tuner itself with the loss due to Return Loss removed (i.e. Column E = Column D + Column C).
• Column F is Column E represented as a percent of total power delivered to the tuner assuming an ideal S21 of -10 dB.  That is, Column F = 100*(1-(10^((Column_E - Ideal_S21)/10))).  Note that "Ideal_S21" should be -10.00 dB for 500 and 5 ohm loads.
• And the remaining columns are simply the tuner settings that gave me minimum SWR for the given load.
Note that there are issues with this measurement technique!

The biggest problem with this technique that it assumes that resistor divider actually has the "desired" voltage division ratio and that it's presenting the "desired" impedance to the tuner.

For example, if, for a 500 load, Rs on the test fixture were exactly 450 + j0 ohms, and if the impedance presented by Port 2 of the S-Parameter Test Set to the SMA were exactly 50 ohms, and if the Load Fixture had no parasitic elements, then the ideal S21 would be exactly -10 dB, and any derivation from this value would represent actual tuner loss (as long as Return Loss is compensated for).

But my Load Fixture, even though I took care to minimize lead lengths, does have parasitic reactance, and an S11 measurement of the impedance presented by the input of this Fixture will show a small but noticeable rotation clockwise, off of the horizontal axis, on a Smith Chart.

So we already know there is some reactance -- how much of this is due to parasitic elements (e.g. series inductance and shunt capacitance) is difficult for me to say.  How close is the division ratio to the ideal 10:1?

Again, tough for me to say.  The reactance just mentioned will undoubtedly have some effect, depending upon how it is distributed between fixture components, as well as the "not-exactly-50-ohms" impedance presented to the fixture's output by the S-Parameter Test Set (I think on my test set Port 2's Return Loss is somewhere between 30 and 40 dB).

And I'll note that the ARRL fixture has another potential issue:  the resistors in this fixture could be dissipating significant power.  If they heat up because of this power dissipation, their values will change and thus the voltage division ratio will change.  The amount of this change depends upon the resistor's characteristics, its heat-sink, and the amount of power it's dissipating, but any change will add uncertainty to the assumed "ideal" S21.

So there is some uncertainty regarding what the ideal S21 really ought to be.  And although there might be ways to calibrate this uncertainty down to acceptable levels, they aren't obvious to me!  (If obvious to you, please let me know).

Anyway, there is some amount of assumption (i.e. the "ideal" division ratio versus actual component resistances/impedances and interconnect parasitic elements ) upon which this technique depends, making it, in my opinion, less than ideal.

But what to do instead?  I happened to mention this problem to a friend of mine, Dick Benson, W1QG, an engineer of many years and vast experience.  It piqued his interest and he started emailing me (while I was gallivanting about on vacation) some interesting ways to measure tuner loss.

Here's his first effort.  Note that, like my first attempt, all of his techniques will depend upon the use of a Vector Network Analyzer...

Technique 3: W1QG, Measure Tuner Loss with "Passive-T" (or Pi) Interface:

What better way to explain this technique than to simply copy Dick's email to me:

The current DUT is simply a (well characterized) toroid L with a fixed mica C shunted by a trimmer:

The “trick” was to carefully create a very accurate 3 resistor network that provided:
1: a 500 ohm load to the DUT
2: a 50 ohm source Z to port 2 of the VNA
3: a precisely known attenuation

The setup is:

The resistors to accomplish this are:
Ra =  478.158     (477.0)
Rb =  31.942       (31.86)
Rc =  19.0679     (19.08)

The values in () are what I used.  The resistors were carefully crafted from 1% films and measured (1kHz) with GR 1658.

With these values, the attenuation works out to be 19.995 dB, where 20.000 was the goal.  This (known) attenuation was compensated for in the M code.

The 8753a VNA also has a small error when measuring a 20 dB loss. It is off by -0.028 dB, and this also must be factored in since it is large relative to the loss of the matching network.

How one determines this correction is yet another saga J

The “raw” s-parameter measurements:

The current scheme works for a 500 ohm Zload, but obviously it won’t work for a 5 ohm load.
A pi-network rather than a T should do it for the 5 ohm version.  But it is a pain to create these T networks.

Therefore I will be investigating an “Active” approach rather than the above “Passive”.
The hope being that only ONE resistor (the load)  needs to be changed.

Note that this technique depends upon an assumption that is difficult to verify, which is that the T-network's component values (measured by Dick at 1 KHz with a GenRad 1658 Digibridge), and thus the T-network's attenuation and load characteristics, are the same at the RF test frequency (e.g. 30 MHz).

With that, let's look at Dick's next approach...

Technique 4: W1QG, Measure Tuner Loss with an "Active Load Interface":

Dick's next approach again uses a Vector Network Analyzer (the HP 8753A), and utilizes an "Active Interface" (or, as I call it, an "Active Load Interface") that terminates the tuner with the desired load impedance (in this case, 500 or 5 ohms).  An op-amp senses the voltage across this load and forwards it on to Port 2 of the VNA's S-Parameter unit, thus providing isolation between the load being tested and port 2 of the S-parameter Test Set.

Here's a description, in Dick's words:

An Active Interface (“AI”) based on the AD9632 op-amp was created.  The measurement setup is:

The idea is to provide a known termination for the DUT, a known gain, and a known Zout to port 2 of the VNA.  The charm of this is that no complicated passive network is required, just a single “Rterm” resistor as a load.

The DUT power loss in dB is:  dBloss = 10*log10(Rterm/50) - measured_S21

Step 1:  Calibrate the VNA with the AI in place and Rterm= 50 ohms, no DUT.
This removes gain errors in the VNA measurement as well as gain errors in the AI.

Step 2:  Replace the 50 ohm termination on the AI input with the desired termination. In this case a 500 ohm (499 1%) was used.  Then MEASURE the actual input impedance. Here is the result:

VNA Port 1 must be calibrated otherwise this result will be useless. The AI input looks like 5.5 pF and 494.4 ohms (Cp and Rp above).  This result is expected due to the physical interface to the DUT as well as parasitic losses that are not accounted for. Since the capacitance ends up in parallel with the matching network output tuning capacitance, it is of no consequence.  The important part is the load is NOT 500 ohms, it is 494.4 (ok +/- measurement noise).   We are indeed splitting hairs here.

Step 3: Insert the DUT between Port 1 of the VNA and the AI input with the 500 ohm (nominal) termination.  Use the VNA calibration and setup in Step 1.

Here is the result:

Now, if Rterm were 500 Ohms, the DUT loss would be 10.0 - 9.9089 =  0.0911 dB.
This is a bit on the high side, but not atrocious.  But per Step 2, Rterm is closer to 494.4 ohms, therefore the DUT loss is:

10*log10(494.4/50)-9.9089 =  0.042 dB.

Ok, it is not in perfect agreement with the 0.057 dB measured with the Passive T,but in reality, who gives a damn about .015 dB !!!

A more realistic test is when the L gets “de-Q’d” to make for a more realistic finite loss situation.

Here is the AI measurement with the same 3.07 ohm resistor in series with the L that was used in the Passive T measurement:

Note that the DUT input SWR is not longer perfect at 5.9 MHz since the DUT input now looks like 50+3.07 ohms,and what we see above in the S11 window is 53.1 ohms!

From the above, the DUT loss is therefore:  10*log10(494.4/50)-9.6351 =  0.316 dB.
The Passive T method gave 0.319 dB so the measurements are within 3/1000th of a dB.
It just does not get any better!

A single supply can be used as long as it is dedicated to this device.  In other words, no other stuff is being powered.

And the (more or less) end result:

The resistor on the top side is the load Z.  As previously described, the fixture needs to be measured to get a better estimate of the actual DUT load for maximum accuracy.

The gain is quite flat (circa +/- .01 dB) over the HF frequency range:

The reverse isolation (S12) is pretty remarkable for what that is worth J

The DUT power loss in dB is:  dBloss = 10*log10(Rterm/50) - measured_S21

It is a matter of doing vna calibrations which includes the AI per this recipe:

Step 1:  Calibrate the VNA with the AI in place and Rterm= 50 ohms, no DUT.
This removes gain errors in the VNA measurement as well as gain errors in the AI.

Step 2:  Replace the 50 ohm termination on the AI input with the desired termination. In this case a 500 ohm (499 1%) was used.  Then MEASURE the actual input impedance.

Here is the result:
VNA Port 1 must be calibrated otherwise this result will be useless. The AI input looks like 5.5 pF and 494.4 ohms (Cp and Rp above).  This result is expected due to the physical interface to the DUT as well as parasitic losses that are not accounted for. Since the capacitance ends up in parallel with the matching network output tuning capacitance, it is of no consequence.  The important part is the load is NOT 500 ohms, it is 494.4 (ok +/- measurement noise).   We are indeed splitting hairs here.

Step 3: Insert the DUT between Port 1 of the VNA and the AI input with the 500 ohm (nominal) termination.

Use the VNA calibration and setup in Step 1.  Here is the result:
Now, if Rterm were 500 Ohms, the DUT loss would be 10.0 - 9.9089 =  0.0911 dB.
This is a bit on the high side, but not atrocious.  But per Step 2, Rterm is closer to 494.4 ohms, therefore the DUT loss is:

10*log10(494.4/50)-9.9089 =  0.042 dB.

Here is the measurement setup with DUT in place:
The Rq is set to zero for this run and the L C tuned to provide a 1.01:1 SWR:
The point is the loss is dBloss = 10*log10(Rterm/50) - measured_S21 = 10 – 9.9998 = ZERO.

Now, with a inductor with a Q of 380, Rq= 0.392 ohms.
The loss is now 10-9.9655= 0.0345 dB.

I implemented the same circuit that Dick designed.  Here's its schematic:

And here's my implementation of the "ALI" (Active Load Interface):

ALI attached to Tuner:

The supply current into the unit should be kept to around 10-15 mA (i.e. 10 - 11 volts) to prevent excessive power dissipation within the ALI fixture.  If the unit feels like it's getting hot, it's drawing too much current from the power supply.  Remember, heat will change the value of Rterm and thus your measured S21.

Here are my measurement results for my own tuner design and for my KAT500

Notes on these measurements:
• Column A is just the Return Loss of the ALI with its Rterm set to 50 ohms (made while calibrating the S21 response).  This measurement simply tells me that the fixture was properly terminated when I did the VNA calibration and therefore S21 of 0 dB represents an Rterm of 50 ohms.
• Column B identifies the value of Rterm (ideal) used for the measurements -- either 500 or 5 ohms.
• Columns C-F are a log of a tuner's settings for minimum SWR for the selected frequency range.
• Column G is just a reminder to myself to set the VNA IF bandwidth to 30 Hz (to reduce measurement noise).
• Column H is the frequency at which I measured minimum SWR (via S11) for the tuner settings in columns C-F.
• Column I is the Return Loss associated with the minimum SWR (and logged in lieu of SWR).
• Column J is the S21 of 'Tuner plus ALI' measured with the VNA.
• Columns K-N represent the actual measured impedance of the ALI's input for the selected Rterm.  This impedance is represented either in series form (Rs + Ls) or parallel form (Rp + Cp)  (These values come from one of Dick's Matlab programs).  Either this measured Rs or Rp will be used to calculate the "ideal S21 (i.e. Ideal S21 = 10*log10(Rs_or_Rp / 50))
• Column O identifies if Rs or Rp was used to calculate the "ideal" S21.  (Note that Rp is used for all -- this will be explained in just a bit).
• Column P is the calculated "ideal" S21 ( = 10*log10(Rp/50)).  (Note that Rp is used for all -- this will be explained in just a bit).
• Column Q is the amount of loss due to Return Loss.  This is not loss dissipated by the tuner and thus should not be included in the Tuner's loss measurement, but it is a part of the S21 number, so it must be subtracted out.  This value = -10*log10(1-10^((Return Loss (dB))/10)), where Return Loss  is expressed as a negative number.
• Column R is the calculated Tuner Loss, which is equal to:  Column_P - Column_J - Column_Q.
• And Column S is the loss in Column R represented as a percent of Total Power applied to the Tuner, and is equal to 100*(1-10^((-Column_R)/10)).
Here's a summary of the equations used:

Ideally, the ALI and  its Rterm have no parasitic elements nor delay and its measured impedance is strictly resistive without any reactance.  If this were the case, you'd simply use the measured value of Rterm to calculate the ideal S21.

But suppose there is reactance in the impedance that the ALI presents as a load to the tuner?

Here's an example:

At 29.67 MHz my ALI's impedance with an Rterm of 5 ohms looks like 5.144 + j1.614 ohms.

The series and parallel equivalent-circuits of this impedance would be:
• Series:  Rs = 5.144 ohms, Ls = 0.008659 uH.
• Parallel:  Rp = 5.65 ohms, Cp = -297.9 pF (that is, because of the negative sign it really should be modeled as an inductance, but this is the value that Dick's routine give's me).
Note that Rs and Rp are about 10% different.  This is a significant difference, and it will result in a difference in loss of roughly 0.5 dB (that is, roughly 10 percentage points), depending upon which resistance, Rs or Rp, is used to calculate the "ideal S21."

So which one to use?  It turns out that Rp will give very close results (although I will admit that it isn't obvious to me why Rp should be used).  But I "brute forced" my way to this conclusion, as follows:

First, I assumed that the op-amp in the ALI is measuring the voltage across the entire S11 impedance, which in this case is 5.144 + j1.614 ohms, and not some portion thereof.  (This assumption is very dependent upon ALI implementation and is not easily verified (if it's verifiable at all), and thus the reason why this is an assumption).

What would be the voltage across this impedance (and measured by the op-amp), if this load were perfectly matched to 50 ohms with a lossless network and driven with 1 watt of power?  And how would this voltage compare to the voltage across a 50 ohm resistor driven with 1 watt?  After all, this difference is the heart of the "ALI" measurement technique.

To calculate the voltage across 5.144 + j1.614 given 1 watt of drive, I used an Excel spreadsheet I had created (before I purchased Matlab) to do simple network analysis.

Here's the spreadsheet with the values of the lossless network (i.e. lossless because component Q is defined to be 10^10).

Note that Return Loss is 128 dB (the last line).  The lossless network has an excellent match to 50 ohms!

This spreadsheet also returns voltage and current for every element, given my input power of 1 watt:

I've highlighted the voltage across Zload.  Note that it is has a magnitude of 2.377 volts.

Now, if there were no network and the load was 50 ohms, driven with 1 watt, the voltage across the 50 ohm load would be 7.071 volts.

The difference between these two voltages is:

20*log10(2.377/7.071) = -9.4689 dB

OK, so now let's look at the "ideal S21" formula for this ALI technique of measuring Tuner Loss (refer to Dick's post, above):

Ideal_S21 = 10*log10(Rterm/50)

If  either Rs or Rp (or |Z| = |5.144 + j1.614|) were plugged into this equation for Rterm, would the result be the same as the above result of -9.469 dB?

Doing the math...

Ideal_S21(|5.144 + j1.614|) = -9.673 dB.  Nope, not close.

Ideal_S21(Rs = 5.144) = -9.877 dB.  Even further off!

Ideal_S21(Rp = 5.65) = -9.4692 dBVery close!!

(There's a difference of about 0.0003 dB between the Ideal_S21 calculation using Rp and the "target" of -9.4689, so, although very close, they are not identical.  But close enough for me.)

Anyway, my conclusion is that even this technique is not ideal.  Again, assumptions are made that are difficult to verify, and so some amount of "hand-waving" is taking place.  And the results very much depends upon how one actually builds the ALI fixture.  Ideally, Rterm should always look resistive, with no additional reactive component (or reactance minimized to be insignificant).  And its measured S11 impedance should be the same impedance across which the op-amp is measuring voltage.

(Also, note that when changing values of Rterm, if soldering them to the fixture they should first be allowed to cool before measurements are made, as heat (from soldering) will temporarily change the value of a resistor).

But Dick hadn't stopped in his search...

Technique 5:  W1QG, Measure Gp (Operating Power Gain) with a VNA:

Dick then came up with simplest (and potentially, in my opinion, the most accurate) approach to measuring tuner loss -- use the Vector Network Analyzer itself to measure Gp (Operating Power Gain) without any loads attached.  Just connect the VNA to the Tuner's In and Out ports.

First, though...what is Gp, "Operating Power Gain"?

It is simply the power going to the load divided by the power going into a 2-port network (such as the tuner):

Gp = PL / Pin

Note that the power into the network is exactly that, the power that is actually going into the network (or tuner).  That is: Pin =  PFORWARD - PREFLECTED.

Gp can be expressed in terms of S-Parameter measurements and therefore can be measured with a Vector Network Analyzer.  Its accuracy will depend upon the VNA's accuracy and how well it was calibrated.

Here's a terse mathematical explanation of Gp from Matlab (type "edit powergain" in Matlab's Command Window if you have their RF toolbox):

%   G = POWERGAIN(S_OBJ, ZL, 'Gp') calculates the operating power
%   gain of a 2-port network by
%
%     Gp = Pl/Pin = (|S21|^2) * (1 - |GAMMAL|^2) /
%         ((1 - |GAMMAIN|^2) * (|1 - S22 * GAMMAL|^2))
%
%   where Pin is the input power and:
%     GAMMAIN = S11 + (S12 * S21 * GAMMAL)/(1 - S22 * GAMMAL)

%   The reflection coefficients are defined as:
%
%     GAMMAS  = (ZS - Z0)/(ZS + Z0)
%     GAMMAL  = (ZL - Z0)/(ZL + Z0)
%
%   The function arguments are:
%   S_PARAMS is a complex 2x2xK array of K 2-port S-parameters.
%   Z0 is the reference impedance of the S-parameters. The default is 50 ohms.
%   S_OBJ is a 2-port sparameters object
%   ZS is the source impedance. The default is 50 ohms.
%   ZL is the load impedance. The default is 50 ohms.

(Derivation of Gp can also be found here: http://rfic.eecs.berkeley.edu/~niknejad/ee242/pdf/eecs242_lect7_powergain.pdf ).  And other explanations/derivations can be found via Google.

Note:  Gp is significantly different from Gt, "Transducer Gain".  Gt, Transducer Gain, also includes loss due to power being reflected back from the input of the two-port network.  So, if a network's SWR is not 1:1, Gt would not accurately represent the power lost within the two-port network itself, and so it should not be used as a measure of Tuner loss (unless the SWR at the Tuner input is 1:1).

Let's follow Dick's line of thought...

Yesterday it struck me that one should be able to directly measure the s-parameters of the DUT / Coupler and then mathematically predict loss due to the DUT without any other attachments.  After all, the DUT s-parameters should describe behavior when the DUT is connected to arbitrary source and load impedances.

It is a three step procedure:
1) Connect the desired test load (eg 500 ohms) to the DUT and tune the DUT for best input match.
2) Then, remove the test load and carefully measure the s-parameters of the DUT in this “tuned” configuration.
3) M-code is used to predict the loss due to the DUT.

I started (naturally) with a model and connected a 500 ohm test load:
The inductor Q was set by the addition of a series resistor of 3.16 ohms which corresponds to my physical DUT.  The L in my physical DUT is fixed, but the C can be varied over a small range.

Note that only s11 has meaning. The C in the model was tuned to attain a rather perfect match,

Next, the 500 ohms is removed, port 2 of the VNA is connected:
And the 2 port s-parameters are measured:
Next this M-Code uses the DUT s-parameters to predict the loss:
% Compute the insertion loss of a DUT for arbitrary source and load
% impedances.
% Dick Benson, October 2015

clear
clc
close all
path='';

% Choose synthetic or measured data by editing:
FileName='analyze_LC_Match_NO_Interface_with_Loss.s2p';
% FileName='LC_Match_Q42.s2p';  % Measured Data

[rf_obj,Notes,State]=spar_read(path,FileName);
spar=rf_obj.S_Parameters;  % this should completly describe the DUT
Fvec=rf_obj.Freq;

Zo=50;     % the system impedance
Zs=50;     % the source
Zl=500;    % note the LOAD is set to 500 ohms.

% Check the input relection coefficient when the 500 ohm load is apppled.
figure;
gamma = gammain(spar,Zo,Zl);
smithchart(gamma);

% Compute the DUT VOLTAGE transfer function and compensate for Zl
figure;
tf=s2tf(spar,Zo,Zs,Zl,1);
plot(Fvec*1e-6,20*log10(abs(tf))-10*log10(Zl/Zo),'marker','+');
h=title(FileName);
set(h,'Interpreter','none');

% Use the "analyze" method to compute "Transducer Gain" and compare it to
% the above method.
% They sdhould be identical and they are, which is comforting.
hold on
analyze(rf_obj,Fvec,Zl,Zs,Zo);
plot(rf_obj,'Gt');
xlabel('MHz');
ylabel('dB');
legend('s2tf','analyze() Gt')
h=title(FileName);
set(h,'Interpreter','none');

The first result is the DUT input reflection coefficient with the 500 ohm load:

Note the nice match. Sorry no useful cursor for this crummy stock smith chart!

That result is in excellent agreement with the other (passive and active) techniques.

The rubber meets the road with real-world measurements.

A full 2 port calibration was done on the 8753A.

A 500 ohm (a 499 ohm film resistor, that happens to be 500 ohms!) is attached to the physical DUT.

The 2 resistors in || make up the 3.16 ohm series R.  The resistor on the 3rd SMA is the 500 ohms test load.
The capacitor is tuned for match:
The 500 ohm load is removed and port 2 of the the 8753a is attached:
The 2 port parameters are measured:
And this measured data is then analyzed with the above M code:
Nice !!

And this is even nicer:
So we have 0.308 dB loss measured and 0.318 dB loss from the Simulink circuit model.

WOW ….  0.01 dB difference ….  it just does not get any better than that.
I think my work is done here, it was quite a journey J

Note that Dick is using Gt for his measurements.  This parameter is OK to use as long as the DUT's SWR is very low, but note that Gt doesn't just represent DUT loss, it also includes power loss due to power reflected from the DUT's input.  If SWR is low, then this reflected power is negligible and Gt will represent the DUT's power loss.  In the case of non-negligible SWR, a better measure of DUT power loss (as Dick later determined) is Gp, as this value is a measure of only the DUT's power loss and does not include power reflected from its input.

As Dick mentioned above, with a VNA the test procedure is very simple.  Here are the steps I take:

Before I start the actual test, on my 8753A I select the frequency range I want to test over (e.g. 21 - 21.45 MHz), set the IF bandwidth to 30 Hz (to minimize measurement noise) and do a full two-port calibration.

Then I attach the desired test-load to the output (antenna) port of the Tuner and tune the Tuner for minimum SWR (VNA S11), as measured across the frequency range I've selected on the VNA.

Here's my setup with load attached to my Tuner's Antenna port.  The Tuner's "XMTR" port is driven by Port 1 of the VNA's S-Parameter unit.

Next, I replace the load attached to the Tuner's output with a coax cable (the same used for calibration!) connected to Port 2 of the VNA's S-Parameter unit.  I try to keep Port 2's Reference Plane as close as I can to the point where the load was attached to the Tuner (see photo above) -- in actuality, the delta between these two is about 0.36 inches.  I could compensate for this delta by adjusting the location of Port 2's reference plane, but the measurement results seem to be close enough, in my opinion, without any compensation.

Then I run Dick's Matlab routine, which automatically captures the S-parameters through my GPIB interface and plots Gp and S11 between the VNA's start and stop frequencies.

Here's an example of the output from Dick's Matlab routine.  (I've modified the routine slightly to show Return Loss (RL) on the Smith Chart "readout", for example):

At resonance (3.77 MHz), the Tuner's loss is 0.065196 dB.  (That is, Gp (gain) is -0.065196 dB).

Percent Loss can be calculated from Gp as follows:

Power Loss (percent) = 100 * ( 1 - 10^(Gp/10) )

Which, in this case,  works out to be 1.5%.  (And note, too, that to apply the formula above correctly, Gp should be negative).

Here are my Tuner-Loss results using the "Gp" technique for both my tuner and my Elecraft KAT500:

Of course, Dick's Matlab routine makes these measurement a snap.  But you can still run this test without Dick's routine.  However, I would strongly recommend using Matlab.  Their home version is, in my opinion, a great deal compared to what a Matlab "seat" normally costs.  (If you are interested in using Matlab, please note that for this application I'm using Matlab plus the "RF Toolbox" and the "Instrument Control Toolbox").

But you don't need Matlab.  If not running Matlab, instead collect S11, S12, S21, and S22 from the network analyzer and use these values to solve the Gp equation.

Please note that this technique does not rely upon any assumptions!  (Apart from an assumption on the accuracy of calibration.)  So, unlike the other four techniques where some amount of "hand-waving" takes place, this technique has the potential to be the most accurate of the five techniques described above.  Of course, its accuracy depends greatly upon the accuracy of the VNA calibration, so care must be taken during this step and accurate Reference Standards should be used.

Additional Notes:

1.  Measurement uncertainty of my 8753A:
• When run with a bandwidth of 30 Hz (default is 3000 Hz), the "noise" on the measurements was about 0.01dB peak-to-peak, IF Bandwidth = 30 Hz.  (Per "eyeball" measurement).
2.  Use of Adapters:

For testing low-impedance loads, try to minimize the number of adapters between a Tuner's output (typically a PL-259) and the test-fixture/load, as each will add some small amount of resistance and thus additional loss.

For example, for Gp testing, I have, between the Tuner's antenna port and the coax going to Port 2 of the S-Parameter Test set, the following adapters:
• PL-259 to BNC (female)
• BNC (male) to SMA (female)
• SMA (male) to SMA (male)
• SMA (female) to SMA (female)
The latter two adapters were used to ensure that that the reference plane for Port 2 was near where the load had been when I'd tuned for minimum SWR.
3.  Here's a summary of the Power Loss found via three different measurement techniques for both my tuner and my Elecraft KAT500:

Note that I recorded Return Loss in lieu of SWR for Technique 4 (ALI).  You can easily convert Return Loss to SWR, if you'd like to, using on-line calculators such as the one found here.

Also, the Technique 2 measurements were the first Power Loss measurements that I made and I did not note the frequency where SWR was lowest.  I corrected this omission in my later  measurements.

As you can see, the results for the three different techniques vary significantly.  I believe that technique 5 (Gp measurement) is the most accurate (but I cannot prove it), because it relies on no assumptions, and that the results from the other techniques differ from those of technique 5 because the assumptions I made (but could not verify) were incorrect.

Never the less, if we assume that an acceptable error is, say,  +/- 5 percentage points in the "percent loss" column (which isn't bad!), then all three are acceptable.

4.  A note on the sign of Return Loss:

In this blog post you will sometimes see Return Loss expressed as a positive number and sometimes as a negative number.  Given that Return Loss is a loss, my belief is that it should be expressed as a positive number when expressing actual attenuation.

However, the 8753A displays S11 (when expressed as magnitude, in dB) as a negative number, and I lazily usually just copy this value down as Return Loss, sign included.  Anyway -- if the inconsistent use of signs bothers anyone, my apologies, and I will add that it bothers me, too, but I have no desire to go back and fix all of the sign inconsistencies.  Just remember that, in this blog post, Return Loss always represents a loss, never a gain.  And where Return Loss is used in a formula, I have tried to identify if the equation requires it be expressed as a positive or as a negative number (please let me know if I've missed any equation!).

OK, that's it for this blog post!

Antenna Tuner Blog Posts:

A quick tutorial on Smith Chart basics:  http://k6jca.blogspot.com/2015/03/a-brief-tutorial-on-smith-charts.html

The L-network:  http://k6jca.blogspot.com/2015/03/notes-on-antenna-tuners-l-network-and.html

A correction to the usual L-network design constraints:  http://k6jca.blogspot.com/2015/04/revisiting-l-network-equations-and.html

Calculating L-Network values when the components are lossy:  http://k6jca.blogspot.com/2018/09/l-networks-new-equations-for-better.html

A look at highpass T-Networks:  http://k6jca.blogspot.com/2015/04/notes-on-antenna-tuners-t-network-part-1.html

More on the W8ZR EZ-Tuner:  http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-more-on-w8zr-ez.html  (Note that this tuner is also discussed in the highpass T-Network post).

The Elecraft KAT-500:  http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-elecraft-kat500.html

The Nye Viking MB-V-A tuner and the Rohde Coupler: http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-nye-viking-mb-v.html

The Drake MN-4 Tuner: http://k6jca.blogspot.com/2018/08/notes-on-antenna-tuners-drake-mn-4.html
Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.