VNA: Notes on THRU-standard de-embedding

Note:  This blog post is a continuation of the topic of VNA Error Correction, which I first discuss (with a dive into "Twelve-term Error Correction") here.  I recommend first reviewing that post before delving into this one.

Prior to performing VNA measurements on a two-port device, a VNA's calibration steps will often include using a THRU adapter (such as a barrel-connector) to connect together the cables attached to the VNA's two ports to complete the end-to-end calibration process.  Following calibration, the THRU adapter is removed and the DUT (Device-Under-Test) is connected in its place, ready for its S-parameter measurements.

If the THRU adapter is not ideal (e.g. it has length (i.e. delay) and/or loss or mismatch), these impairments will affect four of the twelve error-terms calculated during calibration. 

Because the THRU adapter is removed after calibration (before DUT measurements are made), there needs to be a way to remove the effect its impairments have had on these four error terms derived during calibration when the THRU was present.  After all, with the THRU removed, its deficiencies are no longer present in the test setup, and thus they should not be included in the 12-term error model.  If these effects are not removed, the two-port measurements of the DUT will be incorrect.

HP's VNAs (e.g 8753 series) typically assume that the THRU used during calibration is ideal -- it has zero length and thus no delay.  However, these VNAs also allow the user to modify the THRU's definition to include non-ideal characteristics of the THRU used for calibration.  And their VNA software mathematically removes the effects on calibration that result from these non-ideal characteristics.

I wondered how this "de-embedding" of the THRU from the cal results was done.  A Google search turned up little -- the one document I found had errors, so its math was suspect.  So, I decided to tackle the math myself, using the 12-term error model equations in my previous blog post (here).

And thus this post!  Before getting into the math, though, let's first build a foundation...


Compensating for a non-Ideal THRU standard:

The HP 8753D User's Guide contains the following note regarding THRU connections:

In other words, if a THRU adapter is used, the HP VNA needs to know its characteristics (deficiencies) so that it can remove their effect when the THRU is removed and the DUT inserted in its place.

First, though, let's start with a definition of an "ideal" THRU:

An Ideal "Zero-Length" THRU:

A "zero-length" THRU exists when "sexless" connectors are connected together (such as APC-7 connectors), or, perhaps more commonly, when one cable has a male connector and the other a female connector, as shown, below:

An ideal THRU, having no delay or other deficiencies, does not require the VNA to make any additional adjustments to the VNA's 12-term error model when the DUT is inserted into the test system in place of the "ideal" THRU.


A Non-Ideal THRU:

For systems in which the two measurement ports have the same-sexed connector (.e.g both are male SMA connectors), a THRU adapter must be used.  But this adapter, having an actual non-zero length, introduces a delay (and possibly attenuation and/or mismatch) into the measurement:

Basic non-ideal THRU Model:

HP's model for a non-ideal THRU is shown below ([Keysight, Modifying Calibration Kit]):

This model is exactly the same transmission-line model that was used in defining the transmission-line section of the SOL Reflection Standards in my previous blog post, here.

Keysight defines these three Offset parameters to be:

Offset Delay:

For a Thru standard, this is the one-way propagation time (in seconds) from the measurement plane at one end of the Thru to the measurement plane at its other end.

Offset Loss:

Energy loss due to the Transmission Line's skin effect, specified in Ohms/Second at 1 GHz.  Keysight notes that, for many applications, setting this value to zero will not result in significant errors.

Offset Zo:

Per Keysight,"The offset impedance between the standard to-be-defined and the actual measurement plane.  Normally, this is set to the system's characteristic impedance."  I.e.  the THRU is assumed to be matched to the characteristic-impedance of the test setup.


And here are the formulas for Keysight's "offset" definitions:

A non-ideal THRU can be represented by a 2x2 S-parameter matrix whose elements are functions of these three "offset" terms defined by Keysight.  If the THRU were ideal, these four parameters would be:  S11 = S22 = 0, and S21 = S12 = 1.  A non-ideal THRU will have different S-parameter values.

Two-Port Measurement Error Terms:

Four of the twelve error-terms calculated during the VNA's calibration phase are errors calculated when Port 1 of the VNA is connected to Port 2 of the VNA.  These four errors are:  e₂₂, e'₁₁, e₁₀e₃₂, and e'₂₃e'₀₁.

These four error terms are the errors related to Port-to-port Transmission and Reflection, and they correspond to the following error-terms in the HP 8753D Manual's model:

• e₂₂ = ELF  (Load Match of Port 2 as seen at Port 1)
• e₁₀e₃₂ = ETF  (Transmission Tracking, Port 1 to Port 2)
• e'₁₁ = ELR   (Load Match of Port 1 as seen at Port 2)
• e'₂₃e'₀₁ = ETR (Transmission Tracking, Port 2 to Port 1)

In my previous blog post (which discusses how to find the twelve error terms, see here), it is assumed that these last four errors are determined when the two ports of the VNA are connected together with an ideal THRU (i.e. the THRU's S11 and S22 equal 0 and its S21 and S12 equal 1).

But suppose the two VNA ports are connected together with a non-ideal THRU that has loss, delay, and/or impedance mismatch.  For example, a barrel connector?  If we use the equations derived in the previous blog post, the results for the four error terms will be incorrect.  

We must derive new equations.

To derive these equations, I am going to assume that the THRU's s-parameters are known.  They have already been determined (e.g. either by measurement and calculation or by the THRU's manufacturer).  

Let's call the THRU's s-parameters:  S11T, S21T, S22T, S12T (where the 'T' stands for "THRU").

Given this known THRU and the other eight error terms (calculated during the calibration procedure before we connect the two VNA ports together with the THRU, see here), how do we find e₂₂, e'₁₁, e₁₀e₃₂, and e'₂₃e'₀₁ during the last phase of calibration, given a non-ideal, but fully characterized, THRU?

Equations for Two-port Errors e₂₂ and e₁₀e₃₂:

To derive the equations for the two error terms relating to Port 1 as driver and Port 2 as receiver (e₂₂ and e₁₀e₃₂), let's take a look at Doug Rytting's "Forward Model" that I presented in my previous post, here.

We can replace the unknown DUT in the drawing above with our known (i.e. characterized) THRU, as shown, below:

Note that S11T, S21T, S22T, and S12T are the THRU's known (defined) s-parameters.

If you've read the previous blog post, recall that the previous calibration steps result in all error terms  for this model being calculated except for e₂₂ and e₁₀e₃₂.

And we also already know the THRU's s-parameters.  Therefore, the equations (in the above drawing) for S11M and S21M only have one unknown each.  The unknown for the S11M equation is e₂₂, and the unknown for the S21M equation is e₁₀e₃₂.

So, using algebra, we can rearrange the S11M equation to give us an equation for e₂₂, and we can rearrange the S21M equation to give use an equation for e₁₀e₃₂. (The latter is easily derived. The former requires a bit more pencil pushing).  These two new equations are:

(If you solve these equations assuming the THRU is ideal (S11T = S22T = 0, S21T = S12T = 1), these equations reduce to equations (9) and (10) in my previous blog post, here.)

Equations for Two-port Errors e'₁₁ and e'₂₃e'₀₁:

Similarly, we can derive the equations for the two error terms relating to Port 2 as driver and Port 1 as receiver (e'₁₁ and e'₂₃e'₀₁).

This time we will use Doug Rytting's "Reverse Model" that I presented in my previous post, here.

Again, let us replace the unknown DUT in the drawing, above, with our characterized THRU:

And we can rearrange the equations for S22M and S12M to give us new equations for e'₁₁ and e'₂₃e'₀₁:

(If you solve these equations assuming the THRU is ideal (S11T = S22T = 0, S21T = S12T = 1), these equations reduce to equations (11) and (12) in my previous blog post, here.)

And that's it!  We now have four equations with which we can calculate the last four of the twelve error terms, given calibration with a non-ideal, but characterized, THRU.

Conclusion:

In my previous blog post (here) I discuss equations for calculating the four VNA error terms relating to VNA transmission errors: e₂₂, e'₁₁, e₁₀e₃₂, and e'₂₃e'₀₁.  But these equations assumed that the THRU connecting the two VNA ports was ideal -- that is, it had neither loss nor delay, and its characteristic-impedance was matched to the system.

But quite often we use non-ideal THRUs to connect the two VNA ports together during VNA calibration.  An example of a non-ideal THRU is a barrel-connector.  

And if the THRU used for calibration is not ideal and the VNA software assumed that the THRU was ideal, then, if we used the equations in the previous post to calculate e₂₂, e'₁₁, e₁₀e₃₂, and e'₂₃e'₀₁, their calculated values would be wrong.

In this blog post I derive four new equations that allow the correct calculation of e₂₂, e'₁₁, e₁₀e₃₂, and e'₂₃e'₀₁ even if the THRU used during calibration is not ideal.  The only requirement is that the THRU's impairments (loss, delay, and or impedance mismatch) are defined and expressed as S11, S12, S21, and S22. 

A Note on Determining THRU Delay:

Arguably the most important THRU parameter to accurately define is its delay.  To quote Joel Dunsmore (Handbook of Microwave Component Measurements, First Edition, section 3.3.3.3), "Failure to account for the delay of the thru (and to a lesser extent, the loss of the thru) is one of the most common causes of error in RF measurements." 

But suppose our THRU has not been characterized.  How do we determine its delay?

One technique requires the use of both male and female SHORT standards whose own delay has been accurately characterized.  The quotes in blue, below, are from the Keysight "community" website (here) from Dr_Joel:

If you have a f-f adapter, you can modify the cal kit to add the delay of the adapter to the calkit and then it will work fine for calibration.  A typical SMA "bullet" is about 90 psec of delay.

You can measure it directly by doing a 1 port cal, then adding a short from the calkit (this will be the F short), then doing a data->mem and data/mem to normalize out the short delay and get a flat line, change the format to phase.  Then add the SMA bullet (f-f adapter) and put the other sex short (male) on the output and see the phase slope.  You can use port extensions to dial in extension until the slope is flat, then this is the 1-way delay of the adapter.  (This assumes the male SHORT has the same delay as the female SHORT.  If they are different, you will need to compensate for this difference,  k6jca.)

do this from maybe 1 Mhz to 3 Ghz for the freq span (even if your DUT is narrower, as it will give some good phase slope for a short adapter).

It turns out we make the m and f shorts so that they have the same phase and delay in the 3.5 mm case.

Thanks:

A huge thanks to Christian Zietz who took the time to correct my initial approach and derived equations for finding these four error terms given a THRU whose S11T and S22T were 0 (i.e. the THRU although not ideal, is matched).  (Refer to this nanaVNA thread:  https://groups.io/g/nanovna-users/topic/solt_error_theory/77330684?p=,,,20,0,0,0::recentpostdate%2Fsticky,,,20,2,0,77330684)

Christian's thread got me thinking, and, while reviewing Doug Rytting's equations, I realized that they contained the answer for finding these four error terms even if the THRU were mismatched to the system.  All I had to do was replace the original DUT with the known THRU and solve for the error terms.

References:

8753D User Guide – Chapter 6:  Note note regarding Thru length.  Also equations and drawings.

Rumiantsev, VNA Calibration IEEE Mag June 2008 page 86:  Models use similar nomenclature to HP. 

Network_Analyzer_Error_Models_and_Calibration_Methods, Rytting: 1.  Uses e00 etc. to name the error terms.  2.  Does not give equations for solving for e00, e11 and e10e01 (but he does for e22 and e10e32).  3.  Mentions the 4 steps for determining the 12 error terms.  No de-embedding of Thru.

Time-domain thru-reflect-line (TRL) calibration errorassessment:  Nice drawings of FWD and REVERSE models (page 7).

Effect of Loss on VNA Calibration Standards, Monsalve:  Good recapitulation of HP’s lossy-standard equations (i.e. equation for gamma, etc – see Keysight’s “Specifying Calibration Standards and Kits”).

Schreuder – DeEmbed Manual:  A source of equations, but note errors, thus suspect.  Also, I don’t think he properly de-embeds the thru standard.  Includes Thru standard de-embedding (some equations might have errors).

HP Journal 1970-02:  Note that the equations (author Hand) use the e00 terminology.  Note:  does not de-embed the thru (assumed to be ideal).

Improved RF Hardware and Calibration Methods, Rytting: Recapitulation of error terms using ‘e’ nomenclature (e.g. e00).  Does not give equations for solving for e00, etc.  But does introduce “bilinear transform’ for expressing the equation for S11m.  No de-embedding of thru.

Modifying Calibration Kit Definition:  Compact summary of the Reflection Model of Standards.  Note equation for Offset Loss.  (Need to compare to equation in other HP app note).

Error Correction in Vector Network Analyzers, DG8SAQ:  Uses Moebius transform in lieu of the bilinear transform mentioned above (Rytting, Improved RF Hardware and Calibration Methods).

Applying Error Correction to Network AnalyzerMeasurements, Agilent (app note 1287-3):  Summary of 2-port error correction but with error term nomenclature Ed, Es, Ert, etc.

Keysight, Specifying Calibration Standards and Kits, 5989-4840EN:  Good model of 1 port standard.  Also note equations for alpha and beta based upon offset_delay, offset_loss, etc.

MiniCircuits -- Calibration Standards and the SOLTMethod, AN49-017:  Models of 1 port standards (but not thru).  Note equations use tanh.  And also Zc and gamma*length equations.

Agilent, Specifying Calibration Standards for the 8510, 5956-4352: Two port model in terms of Esf, Edf, Erf, etc.  Note method of determing Ceff for open (page 10).

Keysight, Modifying Calibration Kit Definition:  (Good) Models, and definition of terms, for Reflection Standards and Transmission Standard (i.e. Thru).  And note that Thru’s normally considered to be “zero length”

Keysight, In-Fixture Measurements Using Vector Network Analyzers, 5968-5329:  (Good) Describes how to characterize Short, Open, and Load standards. (Note:  Characterizing an Open’s fringing capacitance only necessary above about 300 MHz).  Also interesting Cal fixture w/zero-length Thru.

Agilent, De-embedding and Embedding S-parameter Networks Using a Vector Network Analyzer (app note 1364-1).

Handbook of Microwave Component Measurements: with Advanced VNA Techniques, 1st Edition, Joel P. Dunsmore, Wiley, 2012. 

Standard Caveat:

I might have made a mistake in my designs, schematics, equations, models, etc.  If anything looks confusing or wrong to you, please feel free to leave a comment or send me an email.

Also, I will note:

This design and any associated information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without an implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.