Way, way back in 1986, when I was developing the method of moments software that we are selling today [1], [2], I encountered (and overcame) many problems. One problem first became apparent when I was demonstrating prototype software at HP (now Agilent, soon to be Keysight Technologies) in Santa Rosa, one of my research sponsors. The results did not look so good. When we compared the results to measurement, the differences steadily increased as we went up in frequency. My funding from HP could be in jeopardy. Not good.

At that time, HP was the center of the microwave measurement universe. They were quite proud, and justifiably so, of their automated network analyzer (ANA, the word “vector” was added later, but even then the measurements were of full S-parameter vectors) calibration algorithms. Take, for example, the HP 8542B. I used one literally for hundreds of hours measuring and designing filters at GE Space Systems in Valley Forge, PA in the mid-1970s. It consisted of four full racks, each 6 feet tall, of a quarter million dollars worth of microwave equipment. The computer that drove everything had, as I recall, an amazing 32 kB of RAM (that’s kB, kids, not MB, or GB!). I would connect a short, open, through (i.e., a transmission line connecting the input port straight through to the output port), and sliding load. The magical calibration algorithm would then do its duty, and when I measured my filter, a beautiful, very accurate result was immediately plotted. So easy!

Another project I had at GE Space Systems (now Lockheed Martin) was the development of a millimeter wave network analyzer. There were no commercial ANAs for those frequencies at that time. So, I had to dive into all of the calibration theory. It was scary at first, to look at all those equations. After a bit, however, those equations started to become my friends. I found lots of neat math there. I was impressed by the elegance of the theory and the different forms it could take.

Early microwave measurements measured standing waves on a slotted line. In fact, in my very first microwave class (at Cornell, Prof. G. Conrad Dalman), this is exactly how we were introduced to S-parameters. We would first connect a short circuit to the open end of a rectangular waveguide and measure the exact position of the standing wave nulls and the maximum amplitude of the standing wave. Then, we would connect a device under test (DUT) to the waveguide port and repeat the measurement. By noting the change in the location of the standing wave nulls and the change in the standing wave amplitude, we could calculate the S-parameter of the device we had connected. Having gone through all this work for one S-parameter at one frequency, I really appreciated the automation of the 8542B.

Back to HP in Santa Rosa. My funding, I felt, was on the line. I mumbled something about the difference between the measurement and analysis looking like a small shunt capacitance connected to each port. Maybe it was due to fringing fields around the EM port? Maybe we could develop a calibration algorithm, just like we do for ANAs, to remove that capacitance? Maybe … maybe I was grasping at straws? I went home very discouraged and highly motivated to solve the problem.

The problem was actually solved easily. Based on the ANA calibration theory I had learned, I was quickly able to work up the equations for an EM calibration theory. Practically the first idea I tried actually worked. For ANAs, we would measure, in those days, a short, open, load, and through. Later algorithms might also include one or more delays, which are just longer throughs. So, I tried working the equations for doing just a through and a delay. Pretty quickly, that morphed into the equations for a through of length *L*, and a through of length 2*L*.

To do the equations, I used something called cascading parameters, also known as ABCD-parameters. Once you put some simple equations on a computer, you can convert your measurements from S-parameters to ABCD-parameters. For example, if you have measured S-parameters of two devices, you can convert them both to ABCD-parameters. For a two-port, the ABCD-parameters are a 2×2 matrix, one at each frequency, just like S-parameters. When you multiply the two ABCD-parameter matrices (typically, order matters), you get the ABCD-parameter matrix of the cascade of the two devices. Now, you can convert the resulting ABCD-parameter matrix back to S-parameters (all the complete equations are in [3]), and you can predict what you will measure if you actually cascade the two devices you measured.

This becomes really valuable when we realize that we can calculate the ABCD-parameter matrix of many ideal circuit theory and transmission line components. For example, the ABCD matrix for a shunt capacitor, C_{cap}, is

(1)

where ω is 2πf and C_{cap} is the amount of the capacitance. Figure 1 shows the schematic of the shunt capacitor.

**Figure 1. Schematic of a 2-port shunt capacitor.**

What happens when we invert an ABCD parameter matrix? If we invert any matrix and then multiply the result by the original matrix, we end up with the identity matrix. What kind of circuit has an ABCD matrix that is an identity matrix? Look at eq. (1) and set C_{cap} to zero. That is an identity matrix. Look at Figure 1 and set C_{cap} to zero. That is a zero length through, where port 1 is connected directly to port 2. When the ABCD matrix is the identity matrix, we have a zero length through.

What kind of component can we cascade with a shunt capacitor to get a zero length through, i.e., to get zero capacitance? Easy, a capacitor with a capacitance of –C_{cap}, a negative capacitance! If the reason for the measurement versus analysis error is due to fringing capacitance on the EM analysis port, all I have to do is figure out how much that capacitance is and connect a negative capacitance of the same value in shunt across that port. Then my EM data will be de-embedded.

Somehow, I was thinking that I might get the value of that port fringing capacitance by EM analyzing an *L* length through and a 2*L* length through. The port capacitance would be the same for both through lines. Any change would be due to only the through line. The constant part would be the port capacitance. Could I do something to separate out the constant port capacitance? I drew up some circuit schematics of both the *L* length and the 2*L* length lines and tried to see what would happen if I calculated ABCD-parameters and inverted and multiplied them in various ways. The way that worked is shown in Figure 2.

**Figure 2. Pre- and post-multiplying the un-inverted ABCD-parameters of the 2L length line by the inverted ABCD matrix of the L length line, illustrated schematically here (top), leaves us with a cascade of two of the port capacitances (bottom) — what we call a double port discontinuity … with negative capacitance. From [3].**

Just by drawing the schematics, I found that I could invert the ABCD-parameters of the *L* length line (calculated by the EM analysis) and then pre- and post-multiply the un-inverted ABCD-parameters of the 2*L* length line (which were also calculated by the same EM analysis), and the result would be the ABCD-parameters of twice the port capacitance — only with a negative sign.

But we need the capacitance of one port discontinuity, not two! Very easy, see eq. (1). Just cut the value of C in half. Now, we can use this single negative port discontinuity data to de-embed the device under test (DUT). The DUT is our filter, amplifier, inductor, or whatever, for which we need de-embedded EM analysis data. This is illustrated schematically in Figure 3.

**Figure 3. Given the ABCD-parameters of the embedded device under test (DUT), i.e., the circuit for which we need accurate EM analysis data, pre- and post-multiply the DUT data by the single negative port discontinuity data (left), shown here schematically, to realize the de-embedded DUT data (right). From [3].**

This is the basic idea that we have been using for nearly 30 years now, first published in [4]. It is so simple, and it works really well, for EM data anyway. So, why don’t we use it for measurements? Because this technique requires that the port discontinuity have no series inductance, only shunt capacitance. (In general, there can be shunt conductance too, but no series resistance.) As I will describe in a later column, we can have zero inductance port discontinuities in EM analysis, but it is not practical in actual measurements.

You do not have to take my word that our EM analysis port discontinuities have no series inductance. If you implement this technique, you can look at the actual values that are calculated for the ABCD-parameter matrix for the double port discontinuity. If there is no series component, then A and D must both be exactly equal to one and B must be exactly equal to zero. If they are not, then this technique does not work. This is because, in that case, the single port discontinuity (one that also has series inductance) can no longer be uniquely determined from the double port discontinuity, because it does not have A and D equal to one and B equal to zero.

One note on terminology before we go further. “Calibration” is the process of quantitatively figuring out what the systematic errors are in the measurement equipment (or, in this case, in the EM analysis). That is what Figure 2 shows. De-embedding is the process of removing that error from the measurement (or EM analysis). De-embedding is shown in Figure 3. Thus, we calibrate the EM analysis and we de-embed the result. These terms are often confused, so do try to be careful.

My discussion above is for de-embedding a single port. Next time we will discuss what we do for calibrating and de-embedding closely spaced coupled ports, any number of them, all as close as you like. But don’t get worried that you might have to wade through huge complexities. It is really easy to do.

**References**

[1] J. C. Rautio, and R. F. Harrington, “An efficient electromagnetic time-harmonic analysis of shielded microstrip circuits,” IEEE International Microwave Symposium Digest, pp. 295–298, June 1987.

[2] J. C. Rautio, and R. F. Harrington, “An electromagnetic time-harmonic analysis of shielded microstrip circuits,” IEEE Trans. Microwave Theory Tech., Vol. 35, pp. 726–730, Aug. 1987. http://www.sonnetsoftware.com/support/downloads/publications/TransAug87_InitialTheoryHarrington.pdf

[3] G. Crupi, and D. M. M. -P. Schreurs, editors, **Microwave De-Embedding – From Theory to Applications**, Elsevier, 2014, Chapter 4, pp. 151–187. http://store.elsevier.com/product.jsp?isbn=9780124017009&pagename=search

[4] J. C. Rautio, “A de-embedding algorithm for electromagnetics,” International Journal of Microwave & Millimeter-Wave Computer-Aided Engineering, Vol.1, No. 3, pp. 282-287, July 1991. http://www.sonnetsoftware.com/support/downloads/publications/IntJMMCAD_DoubleDelay91.pdf