Prior to performing VNA measurements on a two-port device, a VNA's calibration steps will often include using a THRU adapter to connect together the cables attached to the VNA's two ports. 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 (it has length (i.e. delay) and/or loss or mismatch), then the VNA, through its calibration, will compensate for the non-idealness of the THRU adapter -- that is, it will correct the deficiencies introduced by the THRU by including these deficiencies as part of the 12-term error correction process. (See my previous blog post on 12-term error correction: here).

But what happens if the THRU is

*removed*following the calibration process, prior to actual DUT measurements being made?

Because the THRU adapter is no longer present, there needs to be some way to

*remove*its effect on the 12-error terms derived during calibration when it

*was*present. After all, its deficiencies are no longer present in the test setup, and thus they no longer need to be included in the 12-term error model. If its effects are

*not*removed, the two-port measurements of the DUT will be incorrect.

In this blog post I will explore compensating the 12-term error model to remove the effect of a no-longer-present THRU adapter (i.e. THRU standard) used during VNA calibration.

Let's get started...

__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.

Let's look more deeply into this model. To start, I'm going to reference the model for a

**one-port**standard in my previous blog post (here). Note that, because this model is a one-port and not a two-port device, the second port (right-hand side) is terminated.

Next, I'll remove the model's termination to transform the one-port model into a two-port transmission line...

...leaving the Signal Flow Graph for just the transmission line:

Note the following Keysight definitions:

And here is how Keysight relates their "offset" definitions to the above model's variables:

As you can see, the model of a THRU (above) is a bit complex.

**And frankly, I don't know how to "de-embed" something this complex from a VNA's measurements!**(But I suspect the technique is similar to what is discussed in this Keysight App Note: [Keysight, De-embedding and Embedding])

But, if I make the assumption that Zc = Zr (that is, the THRU's characteristic impedance matches the system impedance), then Γ

_{1}= Γ

_{2}= 0 and the model collapses down to something much more manageable, a "matched" transmission line:

Thus,

*the S-Parameter matrix of a non-ideal THRU is:*

**assuming that the THRU's "Offset Zo" equals the system's characteristic impedance,**Note that S

_{11}and S

_{22}are zero (a "matched" THRU), and the S

_{12}and S

_{21}terms consist solely of an exponential equal to e

^{(-γ*length).}

Again, the γ in the exponent equals α + j*β and represents both loss and delay. Note that this exponential term can also be expressed as:

e

^{(-γ*length)}= e

^{(-(α+jβ)*length)}=e

^{(-α*length)}*e

^{(-jβ*length)}

Next -- given a THRU with an actual length, how do we compensate the two-port error-correction equations and remove the effect of a THRU that might have delay and attenuation?

__Compensating Two-Port Measurements for a Non-Ideal THRU:__Interestingly, although there are quite a few sources on the web that describe the 12-term error model, almost none of these sources describe how to compensate for (i.e. de-embed) a non-ideal THRU.

I finally found some equations for THRU de-embedding in the User Manual for

*DeEmbed*[Schreuder, DeEmbed User Manual, 15Feb2017], a software tool from Schreuder Electronics.

Per this User Manual, Schreuder de-embeds the THRU's effect from the following four error-terms calculated in the 12-term calibration process: e

_{22}, e'

_{11}, e

_{10}e

_{32}, and e'

_{23}e'

_{01}(see here for more on the 12-term calibration process).

(Note: I've changed Schreuder's nomenclature for these four error terms to be the names used by Rytting).

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:

- e
_{22}= ELF (Load Match of Port 2 as seen at Port 1) - e
_{10}e_{32}= ETF (Transmission Tracking, Port 1 to Port 2) - e'
_{11}= ELR (Load Match of Port 1 as seen at Port 2) - e'
_{23}e'_{01}= ETR (Transmission Tracking, Port 2 to Port 1)

Schreuder compensates the equations for e

_{22}, e'

_{11}, e

_{10}e

_{32}, and e'

_{23}e'

_{01}for a non-ideal THRU by

**the appropriate S-parameter measurements in these equations measurement by the THRU's S**

*normalizing*_{21}or S

_{12}value.

"Normalizing" is just dividing an S-parameter measurement by the THRU's S

_{21}or S

_{12}value, depending upon whether the model being corrected is the Forward model or the Reverse model.

So, if the THRU had loss, this loss is cancelled from the S-parameter measurement by this division (i.e. we are multiplying the measurement by the inverse of the loss, thus cancelling it). If the THRU had delay, this delay is also cancelled-out from said S-parameter measurement by this division.

Per Schreuder, the steps below should remove the effect of a non-ideal THRU used in calibration.

__Non-ideal THRU Error-Compensation Equations:____Step 1__: This step should remove the non-ideal THRU's influence from e

_{22}(Port 2's match as seen from Port 1) and e'

_{11}(Port 1's match as seen from Port 2).

First, I'll rename the THRU's S

_{21}and S

_{12}to be the following: S

_{21(THRU)}and S

_{12(THRU)}.

Next, using Schreuder's normalization method, the original error equations for e

_{22}and e'

_{11}, become (using Rytting's nomenclature):

(13) e

_{22}= ((S

_{11M}/S

_{21(THRU)}) - e

_{00})/((S

_{11M}/S

_{21(THRU)})*e

_{11}- Δ

_{e(Port1)})

(14) e'

_{11}= ((S

_{22M}/S

_{12(THRU)}) - e'

_{33})/((S

_{22M}/S

_{12(THRU)})*e'

_{22}- Δ

_{e(Port2)})

__But wait, is Schreuder's normalization correct?__I believe there is a flaw in Schreuder's method...

Note that e

_{22}and e'

_{11}error terms are errors of

*Reflection*measurements (i.e. S

_{11}or S

_{22}). That is, the signal for the measurement must pass through the THRU standard not once, but

*twice*(first from the signal-generating port, through the THRU, to the second port, and then reflected back from the second port, back through the THRU, to the first port).

So, for these two error terms, the THRU should introduce

*twice*the attenuation and

*twice*the delay, compared to a single-pass through the THRU.

Equations (13) and (14), above, only remove the effect of a single pass through the THRU from the S11 and S22 measurements.

*They do not remove the effects of the second pass.*Therefore, I believe Schreuder's equations 15 and 16

**to include an additional normalization by S**

__should be modified___{21(THRU)}or S

_{12(THRU)}:

(15) e

_{22}= ((S

_{11M}/(S

_{21(THRU)}*S

_{12(THRU)})) - e

_{00})/((S

_{11M}/(S

_{21(THRU)}*S

_{12(THRU)}))*e

_{11}- Δ

_{e(port1)})

(16) e'

_{11}= ((S

_{22M}/(S

_{12(THRU)}*S

_{21(THRU)})) - e'

_{33})/((S

_{22M}/(S

_{12(THRU)}*S

_{21(THRU)}))*e'

_{22}- Δ

_{e(port2)})

And, given that S

_{21(THRU)}= S

_{12(THRU)}= e

^{(-γ*length)}, these two equations then become:

(17) e

_{22}= ((S

_{11M}/e

^{(-2γ*length)}) - e

_{00})/((S

_{11M}/e

^{(-2γ*length)})*e

_{11}- Δ

_{e(Port1)})

(18) e'

_{11}= ((S

_{22M}/e

^{(-2γ*length)}) - e'

_{33})/((S

_{22M}/e

^{(-2γ*length)})*e'

_{22}- Δ

_{e(Port2)})

With equations (17) and (18), above, we have now compensated e

_{22}and e'

_{11}for a non-ideal THRU. Our next step (step 2) will be to compensate e

_{10}e

_{32}, and e'

_{23}e'

_{01}for a non-ideal THRU...

__Step 2__: This compensation is performed with equations 17 & 18 from Schreuder's User Manual

We remove the effect of the THRU's delay and loss effect on e

_{10}e

_{32}by

**S**

*normalizing*_{21M}by the THRU's S

_{21(THRU)}, which gives us the following equation (again, expressed using Rytting's nomenclature):

(19) e

_{10}e

_{32}= ((S

_{21M}/S

_{21(THRU)}) - e

_{30})*(1 - e

_{11}e

_{22})

In a similar fashion, the reverse path's e'

_{23}e'

_{01}becomes:

(20) e'

_{23}e'

_{01 }= ((S

_{12M}/S

_{12(THRU)}) - e

_{03})*(1 - e'

_{22}e'

_{11})

And given S

_{12(THRU)}= e

^{(-γ*length)}:

(21) e

_{10}e

_{32}= ((S

_{21M}/e

^{(-γ*length)}) - e

_{30})*(1 - e

_{11}e

_{22})

(22) e'

_{23}e'

_{01}= ((S

_{12M}/e

^{(-γ*length)}) - e

_{02})*(1 - e'

_{22}e'

_{11})

__A few notes...__

1. Both e

_{10}e

_{32}, and e'

_{23}e'

_{01}use the newly compensated values of e

_{22}and e'

_{11}, calculated above in equations (17) and (18)).

2. Schreuder doesn't use e

^{(-γ*length)}to define the THRU's S

_{21}and S

_{12}characteristics. Instead, he defines a THRU with the following equation:

Γ

_{THRU}= e

^{(-j2πlength*f/c)}*10

^{(-Loss(dB)/20)}*10

^{(-Loss(dB/Hz)*f/20)}, (where 'c' is the speed of light in meters/sec).

Note that this equation is essentially an equivalent way of stating that the THRU's Transmission characteristic equals e

^{(-γ*length)}.

(A quick aside -- I don't know why Schreuder labels this term as "Gamma". Gamma refers to a Reflection Coefficient, which this equation is not -- it represents the THRU's Transmission, not its Reflection, characteristic. Also -- note that the Schreuder's original equation omitted the '-j' in e's exponent).

- (Schreuder Equation 21): There should be a '-j' in e's exponent.
- (Schreuder Equation 23): There should be a '-j' in e's exponent.
- (Schreuder Equation 25): There should be a '-j' in e's exponent.

**My notes above are based upon Schreuder's method of de-embedding a non-ideal THRU standard. It is possible that actual de-embedding methods (e.g. the method used by Keysight) are different and/or more comprehensive (e.g. they could include the effect of a THRU having non-zero S11 and S22 values).**

__References:__**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**: Good source of equations, but note error. 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-FixtureMeasurements 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.

__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.