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Calibration is a standard procedure that RF engineers perform to characterize their new designs. This procedure allows them to de-embed the effect of the measuring instrument, such as vector network analyzer (VNA), and the cables used to connect to the device under test (DUT). To do the calibration, a calibration kit is needed. There are hundreds of types of ports in the world of RF, and for every type, a specific calibration kit is needed.

Typically, calibration can be performed only when the two ports of the DUT are of the same type. This article targets cases where the RF designer wants to measure a passive device with two ports of different types, e.g., one is an SMA connector while the other port is a CPWG transmission line, or a passive device with two ports of the same type, but different frequency like the mixers.

The traditional approach is to convert one of the ports to the type of the other port, by adding additional element(s), like transformers, and consequently the measured scattering matrix is the combination of the DUT and the additional element(s). Here, we explore the option of using back-to-back measurements instead, to characterize a non-uniform DUT.

Starting first by connecting two samples of the DUT back to back, i.e., port No. 2 of the first sample to port No. 2 of the second sample, calibrate out the instrument and its cables using the calibration kit of the type of port No. 1, then measure the combination. From the measured data, we would like to extract the scattering matrices of both samples.

Instead of using another sample of the same DUT, any circuit that has the same type of ports that the DUT has can be used.

If we label the scattering matrix of the first DUT sample as “Sa,” and for the second one as “Sb” (Figure 1), then the total scattering matrix from cascading these two DUTs back-to-back is equal to:

A₁₁ = (Sa₁₁+Sa₂₁Sb₂₂W Sb₂₁) (1)

A₂₁ = Sb₂₁W Sb₂₁

A₂₁ = Sb₂₁WSa₂₂Sb₂₁+Sb₁₁

Where

W= (1.0 - Sa₂₂ Sb₂₂)^-1

Notice here that the assumption that Sa₂₁ = Sa₁₂ and Sb₂₁ = Sb₁₂ (amplitude and phase) is used to simplify the formulae. Equation 1 is a set of three complex formulae with six complex unknowns (or six real formulae with 12 real unknowns). That makes the problem ill conditioned and there is no single solution to it.

So the next step is to try to use three samples of the DUT and cascade them back to back. Assuming that the scattering matrix of the first, second and third samples are “Sa,” “Sb” and “Sc” respectively, the complex formulae for cascading these samples back-to-back can be written as, (Figure 2):

Where (Sx, Sy, ST-matrix) stands for (Sa, Sb, A-matrix), (Sa, Sc, B-matrix), and (Sb, Sc, C-matrix) respectively.

That produces nine complex formulae to solve for nine complex unknowns (Sa, Sb and Sc). So the problem is solved. Unfortunately, that is not true. By using the variable elimination technique, a formula of the following form can be reached:

a Sb₂₂ * Sb₂₂ + b Sb₂₂ + c = 0 (3)

where Sb₂₂ is the return loss of port No. 2 of the second sample. If any three samples with known scattering matrices were cascaded with each other, and the obtained A, B and C matrices were substituted in equation 3, the results will be a = b = c = 0. That means there is a redundancy in equation 2.

It was found that whatever number of samples are used to do back-to-back measurements, i.e., four, five or more, the redundancy is always there. Such redundancy prevents calculating all the unknowns, i.e., the elements of Sa, Sb and Sc matrices, without knowing one of these elements in advance.

Using the three devices back-to-back measurements, if Sb₂₂ was known, the following formulae can be used to derive the rest of the parameters:

So, the question now is how to determine the right value for Sb₂₂? And to do that, additional formulae or criterion is needed. The return losses at the two ports of a passive device are close but not equal. Hence, the approach is to look for a value for Sb₂₂ that will minimize the three following equations respectively:

ABS (|Sa₁₁|-| Sa₂₂|) tolerance (5a)

ABS (|Sb₁₁|-| Sb₂₂|) tolerance (5b)

ABS (|Sc₁₁|-| Sc₂₂|) tolerance (5c)

One way to perform that is to split the Smith Chart into a 200 × 200 grid, assign Sb₂₂ one complex value at a time, evaluate all the scattering matrices for every complex point, then choose the one that gives the minimum of the maximum of the three differences in Equation 5 — minimizing the maximum is more reliable than minimizing the summation of the differences. If a more accurate value is needed, create a 20 × 20 grid around the optimum point, with a one-tenth scale and repeat the calculations. The process can be repeated for as many decimals as needed.

This approach was tested for many random cases, where three scattering matrices were cascaded using the Advanced Design System program from Agilent (sample No. 1 with sample No. 2, sample No. 2 with sample No. 3, and sample No. 1 with sample No. 3), then the obtained matrices were fed to the program to extract the original scattering matrices. In all cases, the results were unique and close to the given ones (up to the seventh digit).

Once the scattering matrix of all the three samples are known, any one of them can be used to measure other devices or samples. This is equivalent to saying that the used scattering matrix is part of the calibration kit. Assuming there are tens of samples of a device to be measured, then the following can be done:

• measure three samples using a back-to-back setup;
• extract the scattering matrices of each of the three devices using the formulae given in the previous section;
• cascade device No. 1 with any of the other devices (fourth, fifth, …), and measure; and
• de-embed the scattering matrix of device No. 1 to obtain the scattering matrix of the samples.

Assuming that Sb is the scattering matrix of device No. 1 derived using the back-to-back technique, Sa is the scattering matrix of one of the other samples, and A is the scattering matrix of the total, one can de-embed Sb to obtain Sa using the following formulae, (Figure 3):

In short, we have demonstrated that there is a way to measure a device with two different ports and/or frequency of operation, by using three devices' back-to-back measurements. The technique presented here can be extended to the case where an additional device, with known scattering matrix, is needed in the middle to connect the samples back to back.

Hatem Akel is a senior RF and antenna designer of BreconRidge, Ottawa, Ontario, Canada.

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