Playing with S-parameters and the Smith chart in Qucs

Intro

I'm not an EE graduate, but practical electronics has always been my hobby (albeit typically on the back burner) and I also happen to wade into antenna and RF technology on my job every now and then. On a recent encounter with WiFi antennas and directional couplers, as I was simulating some stuff in Qucs, I came across the S-parameters and the Smith chart (once again). I got intrigued, because this stuff seemd related to my "research into directional couplers". (Well... not all that useful, as I later concluded, but anyway.)

Charts

The following is an LC circuit, coupled by two different series resistors to two different power sources. Note the resulting shape on the Smith chart, the differences and similarities between the impedance and admittance version of the Smith chart, and the differences vs. similarities among the various S-parameters.
I've taken screenshots of two, slightly different versions of the schematic: the difference is in the L and C values, both versions resonate at 50 MHz, but each one with a different Q. Note how this affects the shape of the Bode plot and the "spacing of nodes" in the Smith chart :-)

Schematic with a higher impedance resonator (lower Q)

Charts with a higher impedance resonator (lower Q)
Impedance chart on the left, admittance chart on the right

Schematic with a lower impedance resonator (higher Q)

Charts with a lower impedance resonator (higher Q)
Impedance chart on the left, admittance chart on the right

Lessons learned

I have an itching to summarize, in lay man terms, some of the stuff I've gleaned from all the tutorials I've read. Stuff that has some practical use, without you knowing all the background maths by heart. As a consequence, some of my statements below are likely incorrect :-) or need to be further qualified by conditions.
Use with a grain of salt.

• The S-parameters generally speak about transfer of power. Not just AC voltage or current, but power. Consequently, some of the "ports" to the network under scrutiny can be e.g. waveguides, instead of just coax or balanced electrical pins/pairs. (I have not yet tried gyrators or mixing physical domains, such as electro-accoustics / vibration analysis or some such.)
• At a first glance, the matrix of S-parameters is a shallow/superficial way of saying "we want to describe the coupling of power between any two ports in the set of ports, featured in our circuit topology under scrutiny (AKA 'the network')". And, apart from coupling between different ports, we're also (or especially?) interested in coupling of an individual port unto itself = the reflection coefficient. The reflection coefficients S11, S22, S33 etc. are the "diagonal" members of the 2D square-shaped matrix. Let's start from this superficial level - each S-parameter is actually a function of frequency, but we don't absolutely need to understand the arcane math inside, for a start we can work with a Bode plot of the parameter (magnitude vs. frequency) across some frequency range that's of interest to us.
• At a closer look, i.e. once you start graphing the S-parameters, it seems that each "S-parameter" (a cell in the matrix, such as S21) is actually a set of variables or functions, superficially collectively referred to by its position (coordinate pair) in the matrix.
• An S-parameter Bode plot in Qucs displays the "power transfer" magnitude (i.e. not phase) vs. frequency. The "power transfer" is strictly in a range between 0 and 1, i.e. the network is passive and cannot amplify - unless the network is an amplifier maybe? :-) It makes perfect sense to plot several different S-matrix members in the same Bode plot, i.e. transfer and reflection related parameters all in one picture.
• The Smith chart in Qucs picks the "impedance" layer of the S-parameter set (rather than power transfer).
  Impedance as a function of frequency is a complex number, consisting of a real resistance and an imaginary reactance. A complementary function to Impedance is Admittance: Admittance is a reciprocal value of Impedance - and therefore also a complex number, consisting of a real conductance and an imaginary susceptance.
• The Impedance flavour of the Smith chart has resistance horizontally and reactance vertically.
  The Admittance flavour of the Smith chart has conductance horizontally and susceptance verticaly.
  In both flavours of the Smith chart, the upper half-plane is inductive and the lower half-plane is capacitive.
• The Impedance flavour of the Smith chart is "calibrated" for a particular nominal impedance = the center point is typically 50 Ω. Correspondingly, the Admittance flavour of the Smith chart is typically calibrated for 0.02 ʊ ("mho" - synonymous to Siemens) as the center point.
• Note how the visual mapping of the S-parameters in the Impedance and Admittance charts is identical :-) (is it?)
  This is likely a visual representation of the complementarity of impedance vs. admittance. Only the grid in the two flavours of the chart is mirrored, and also the units happen to be reciprocal :-)
• Some sources (Microwaves101 if memory serves) claim that in fact only the "reflection" S-parameters (diagonal members of the S-parameter matrix) are ever plotted in the impedance flavour of the Smith chart, "for practical reasons". And that the Admittance flavour of the chart has significant advantages for network analysis in the way of "coupling from port to port". I would re-phrase those vague statements in my own vague way:
  1) It feels perfectly allright to speak of Impedance within a particular port of the "network" (circuit): impedance between a live pin and the corresponding local gnd (coax shield), or a differential impedance in a signal pair. But, it feels awkward to think of an impedance between two different ports.
  2) On the contrary, it feels appropriate to speak of "conductivity" = Admittance from one port to another. In the case of a local reflection, Admittance doesn't feel like the right "descriptive category".
  Technically in the Smith chart the curves look all the same, whether you plot them in the Impedance or Admittance units (see above).
  Either it's all psychology, or there are additional practical tricks that only make sense in one or the other flavour of the Smith chart.
• Although I've never seen this written anywhere, it seems to me that the curved grid of the Smith chart is shaped exactly such that an LC circuit always results in a neat ring :-)
• Actually, a "quarter-wave shorted stub" (generally any wavelength-tuned transmission-line section) also results in a circular plot.
• Have you ever noticed, e.g. when working with resonators, that tuned transmission line stubs do not work exactly the same as "lumped element" LC circuits?
  While an LC cell only resonates at a single frequency, a TML stub resonates at an infinite number of higher harmonics, apart from the base frequency.
  This nuance also shows in the Smith chart. It has to do with phase.
  An LC cell results in a single "circle almost closed" in the chart. Actually completely closed, for a frequency range from -∞ to +∞ : but it's still a single "turn".
  With TML stubs, if you sweep across a broad-enough range of frequencies, you will see the Smith point circling repeatedly around some center point in the chart.
• This point should probably be someplace at the top, but it doesn't quite fit in the "sequence of thought"...
  As mentioned above, the S-parameters are functions of frequency. Consequently, frequency is present in both the Bode plot (explicitly, along the horizontal axis) and in the Smith chart (implicitly).
  You can use the Smith chart to draw an impedance at a particular frequency - the graphical representation will be a single point. Whenever you see a curve in the Smith chart, the curve is a set of points, corresponding to a frequency sweep. The frequency is not directly present in the 2D chart - you need to imagine frequency as a 3rd axis, pointing at you from the paper.
  Unfortunately there's no way to infer the frequency back from the Smith chart alone. But it probably wouldn't help you much to try and display the Smith chart combined with a Frequency axis in a "stacked Smith" kind of 3D chart... the condensed "top view" 2D representation is possibly more useful after all.
  In this particular aspect, the Smith chart is similar to the Nyquist's "gain+phase" chart used in stability analysis. (In this "implicit frequency" aspect only. The Nyquist chart is neither curved nor logarithmic and has real and imaginary gain on its two axes.)
• For some practical use of the Smith chart "nomogram" = compensating impedance mismatches with lumped LC elements or stubs, read the primary sources - see references below. Microwaves101 has a good intro on that. No point in reproducing this here.

Example Qucs data

References