De-Embedding 2 Whole Degrees of Freedom

In the last post we discussed de-embedding a simple transmission line, namely a phase shifter and attenuator. Surprisingly, most structure that we might need to de-embed will be far from that, to say the least. To maintain a methodical approach, however, the model will be complicated one order at a time, so in this post I’ll add one more degree of freedom. In many cases, the impedance of the De-Embedded transmission line will be different than expected. This is the case discussed in this post. Below is an example for the same configuration in the last post, but with 40 Ohm transmission lines.

1. Mathematical Definition of The Model

The scattering matrix for this kind of transmission line is

S_l = \frac{1}{\Delta} \begin{bmatrix}  -\left({\frac{Z_c}{Z_0} - \frac{Z_0}{Z_c}}\right) \sinh \gamma l & 2 \ 2 & - \left({\frac{Z_c}{Z_0} - \frac{Z_0}{Z_c}}\right) \sinh \gamma l \end{bmatrix}

where

\Delta = 2\cosh\gamma l -  \left({\frac{Z_c}{Z_0} + \frac{Z_0}{Z_c}} \right )\sinh \gamma l.

Here, Zc is the unknown characteristic impedance of the transmission line, and Zo is the normalization impedance (port impedance). For now, however, we will deal with the ABCD form of the scattering matrix. Let us recall from the previous post that we are using the butterfly\mirror setup,

so we use the matrix equivalent to a double length transmission line,

\mathbf{M}_{2l} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} \cosh \gamma 2l & Z_c \sinh \gamma 2l \\ \frac{1}{Z_c} \sinh \gamma 2l & \cosh \gamma 2l \end{bmatrix}

It is important to remember that the measured S-parameters need to be converted to the ABCD form. The reader may recall that in the previous de-embedding post, the unknown to de-embed was simply

\mathrm{e}^{-2\gamma l}.

In this case it is a bit more tricky, but keep faith! Half of it is already available with

A = \cos 2\gamma l

and to obtain the hyperbolic sine, the quick student will write

\sinh 2 \gamma l = \sqrt{\cosh^2 2 \gamma l - 1},

immediately ignoring the perils this encapsulates. The square root of most numbers, especially complex numbers, has two solutions. So the question is how to choose which one? The answer should arise from the physics of the model, rather than from mathematics. In this specific case, we are de-embedding a passive transmission line. So we assume it will be lossy or lossless, but certainly not amplifying. So we define two possible solutions:

\sin 2 \gamma l = \begin{cases} \sqrt{1 - \cos ^ 2 2 \gamma l} \\ -\sqrt{1 - \cos ^ 2 2 \gamma l} \end{cases}

All that remains is to choose between the solutions by demanding

\left|{\mathrm{e}^{-2 \gamma l}}\right| = \left|{\cosh 2 \gamma l - \sin 2 \gamma l}\right| \le 0

So now we have the cosine and the sine, while the exponent in this case was only used to choose the correct complex solution. The characteristic impedance can now be extracted by

Z_c = \frac{B}{j \sin 2 \gamma l}.

If you desire to build the entire scattering matrix of the transmission line, you now have everything you need, as all of the arguments in the first equation of the post are available.

2. Let’s Check If It WorksUsing the mirror setup described above, we employ the algorithm above, implemented in my github. First, let’s observe the selection of the correct solution, by looking at the amplitude of the exponent

So we see indeed that the solutions are interchangeable, Namely only one of the solutions apply the condition at each time. Calculating the characteristic impedance yields a perfect straight line:

This line can never be obtained with real measurements, but since we are still living in the ideal, no need to worry. De-embedding is finally obtained using the general formula

S_L = \frac{ S_M - S_{l_{11}} } { S_{l_{22}}\left({S_M - S_{l_{11}}}\right) + S_{l_{12}} S_{l_{21}} },

where S_L is the DUT load reflection and S_M is the measured reflection.

3. What’s Next

In the next post, I won’t immediately move to the third-order model, rather explain how this model is useful in port-extension, a complementary operation for calibration in VNAs.

Leave a Reply

Your email address will not be published. Required fields are marked *