# Which matrix index should be used when Y (admittance) is shown to compose a circuit segment mapped to an ABCD matrix? Y21? Other?

I would like to cascade a serial capacitor with a shunt inductor to form an L-match network by multipling two ABCD parameters and convert the result to S-parameters that represent the composite network. As a starting point, the capacitor's S-parameters were converted to an ABCD matrix using transforms from here. Now I need to create an ABCD matrix for the shunt inductor.

To avoid the X/Y problem, ultimately this is my question, but please also address the questions at the bottom to help my understanding of the situation: How do you transform an ABCD matrix to form a shunt so I can multiply it (chain) with another ABCD matrix? (related questions below)

Here is what I've discovered so far:

This video shows how to create an ABCD matrix for various circuit segments. For my purpose, this frame shows the ABCD matrix for the shunt component (the inductor, in my case):

The Y parameters for the 120 nH inductor were converted to Y from series-modeled S-parameters at 100MHz as provided by Coilcraft.

### 3x Serial 120nH Coilcraft Inductors

Update: In the original post I had posted s-parameters for a composite of 3 serial inductors. Using these numbers, @ThePhoton expertly composed his answer but noted that the numbers in that original post did not match the .s2p files from the Vendor's website.

These are the original numbers @ThePhoton used for his calculations:

These are the inductor's Y parameters at 100MHz in mag-angle format:

Y11: [0.00440205824472129, -86.149908981413]
Y21: [0.00440205824186984, 93.8500909570987]
Y12: [0.00440205824186984, 93.8500909570987]
Y22: [0.00440205824472129, -86.149908981413]


These are the inductor's original S parameters at 100MHz in mag-angle format:

S11: [0.893392359379373, 23.1032289305104]
S21: [0.393276519863971, -63.0466801187767]
S12: [0.393276519863971, -63.0466801187767]
S22: [0.893392359379373, 23.1032289305104]


### 1x 0402DC-121 coilcraft 120nH Inductor

These are the inductor's Y parameters at 100MHz in mag-angle format for a single component. (Note that the values above in the 3x-serial version were used by @ThePhoton to calculate his numbers):

Y11: [0.013083525177408, -86.1734582308486]
Y21: [0.0130835251738178, 93.8265417327069]
Y12: [0.0130835251738178, 93.8265417327069]
Y22: [0.013083525177408, -86.1734582308486]


These are the 1x inductor's original S parameters at 100MHz in mag-angle format:

S11: [0.588600478, 50.2086462]
S21: [0.770096917, -35.9648121]
S12: [0.770096917, -35.9648121]
S22: [0.588600478, 50.2086462]


Questions:

• Given a two-port Y-parameter matrix representing the inductor, which Y matrix index would I use for the "C" value of the ABCD matrix pictured above?

• Intuition says to use $$\C=Y_{21}\$$, but I think that means the remaining indexes (11, 12, 22) are lost information so, for example, the $$\Y_{11}\$$ input admittance behavior would not present in the composite circuit after the two ABCD matrixes that represent the serial capacitor and shunt inductor are multiplied. Is this a a correct interpretation?

• If so, Is there a more accurate way to transform the 2-port 2x2 Y matrix into a shunt ABCD matrix where no information is lost?

• The Y-matrix you were given for the inductor, is it for shunt or series configuration? Can you share the actual Y-matrix? May 29 at 4:38
• @ThePhoton, the component is series-modeled. The question has been updated with the Y-matrix and the original S-matrix that it was derived from. May 29 at 4:49
• The numbers you shared don't match what's in the s-parameter files on the vendor website. For the 120 nH part (0402DC-R12X), at 100 MHz they give 0.589<50.2 for S11, not 0.893<23.1. May 29 at 6:25
• @ThePhoton, you are correct: the s-params I posted were for 3x serial 120nH inductors. Oops! I am writing a command line tool to do s2p transforms and didn't realize the serial test bits were still in place. I've corrected them on the question. May 30 at 21:35

The shortest route to a solution is probably to convert the S-parameters you have for the series-connected inductor to ABCD parameters. Then, since the ABCD parameters have a very simple structure for a simple series-connected or shunt-connected device, you can convert to the ABCD-parameters for the shunt-connected device.

To convert from S-parameters to ABCD parameters you can use the formula I found here. For the case where the reference impedance is 50 ohms on both sides of the 2-port,

$$A = \frac{(1+S_{11})(1-S_{22})+S_{21}S_{12}}{2S_{21}}$$ $$B = 50\frac{(1+S_{11})(1+S_{22})-S_{21}S_{12}}{2S_{21}}$$ $$C = \frac{1}{50}\frac{(1-S_{11})(1-S_{22})-S_{21}S_{12}}{2S_{21}}$$ $$D = \frac{(1-S_{11})(1+S_{22})+S_{21}S_{12}}{2S_{21}}$$

Using these formulas, for your device I calculated (after rounding appropriately)

$$\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 1 & 15.253 + j226.65 \\ 0 & 1 \end{bmatrix}.$$

This is consistent with the expectation that for a single series element we should have

$$\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 1 & Z \\ 0 & 1 \end{bmatrix}.$$

(The reason I did all this math myself rather than just explain it in rough terms was to make sure that the numbers you had would actually give an ABCD matrix in this form—it wouldn't have surprised me if, for example, the vendor measurements captured some parasitic shunt elements in their test fixture and produced an ABCD matrix with a non-zero C value)

Now you can convert this to the ABCD matrix for the same device in shunt configuration

$$\begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ Y & 1 \end{bmatrix}.$$

using $$\Y=\frac{1}{Z}\$$

$$\begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0.0002956-j0.0004392 & 1 \end{bmatrix}.$$

Note: This result ($$\Y=0.0002956-j0.0004392\$$) is pretty far from the value for an ideal 120 nH inductor, which would have $$\Y=-0.0133j\$$. I'm not sure exactly what causes the discrepancy, given the charts in the component datasheet indicate this inductor should be well below resonance at 100 MHz.

Solution: You don't seem to be using the right data for the 120 nH part. If I look at the vendor provided S2P file for the 120 nH part, for 100.93 MHz it gives $$S = \begin{bmatrix} 0.3767+j0.4523 & 0.6233-j0.4523 \\ 0.6233-j0.4523 & 0.3767+j0.4523 \end{bmatrix}$$ and if I calculate from these values I get $$\Z\approx 76.3\ \Omega\$$, which is pretty close to the $$\75.4\ \Omega\$$ we'd expect for an ideal 120 nH inductor at 100 MHz.

• Where you said "This is consistent with the expectation that for a single shunt element we should have [[1 Z], [0 1]]" did you mean for a single series element ? May 30 at 21:41
• Oh, I think I get it! Since the s-params came from a series model, you converted to ABCD, interpreted it as a series-ABCD matrix (because it is), and then re-arranged those ABCD values (Z, in this case) to create the ABCD matrix for a shunt using Y=1/Z. Question: is it the case that $A^\prime=A$, $B^\prime=C$, and $D^\prime=D$? For example if A, C, and D are not 1, 0, 1 then where should those values be moved in the $ABCD^\prime$ shunt matrix? Or is this only valid when A,C,D=1,0,1? I'm trying to keep as much original data as possible to minimize conversion loss. May 30 at 21:49
• " if A, C, and D are not 1, 0, 1 then where should those values be moved in the ABCD′" If A, C, and D are not 1, 0, 1, then you don't have a network composed of a single series element. It doesn't make sense to ask what matrix you'd get if you moved the series element to shunt configuration, because there is no single series element to be moved. May 31 at 15:32
• "I'm trying to keep as much original data as possible to minimize conversion loss. " The deviations in this matrix from the ideal values (1, 0, 1) are most likely either small measurement errors, numerical rounding errors in converting from s-parameters to ABCD, or caused by parasitic elements in the test fixture used by the vendor when measuring the s-parameters. In any of those cases, they don't tell you anything about the actual device being measured, and it doesn't make sense to try to preserve them when calculating the matrix for the same element in shunt configuration. May 31 at 15:35
• @KJ7LNW, Yes, I think you're right, in the series configuration $Y_{11}$ will just be equal to the admittance of the device. Jun 1 at 22:11