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.)

I felt I had to "start from someplace in the middle", as I didn't
have enough of a maths background to start "properly from the ground up".

I started out by spending some time reading primers on the
S-parameters
and the
Smith chart
(kudos to Microwaves101.com):
what variables are on each axis, why the grid is curved,
the distinction between the impedance vs. admittance version of the
Smith chart. I was wondering what some typical circuits looked like
in the Smith chart - and the interwebs provided surprisingly few
answers. That's when I started to squint back at Qucs:
"Oh wait... I can sketch some circuits in Qucs. And, Qucs contains
some Smith charts in the graphing tabs. Now how do I go about this,
does it actually work? I could then try fiddling with some
component values to see the effect of my changes..."

The current version of Qucs (0.0.19 RC as of this writing) turns out to be an excellent educational toy when it comes to S-parameters and the Smith chart.

Essentially, you need a "power source" for each port of your circuit. This component looks like a voltage source with a series resistor in one package. Next, you need an "S-parameter simulation". This will automatically set up the matrix of N by N ports. Once you compute the simulation, in the graphing tab you can select the two versions of the Smith chart, or a "cartesian" Bode plot. You will be offered the S-matrix members for use in your charts/plots.

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

Note: speaking of Q, this is affected by the resistors
in the schematic. The resonant cell itself consists of ideal L and C,
but the resistance from the signal sources (and also the one intrinsic
to the signal sources) plays a role in the Q.

Try playing with the values yourselfs (data download follows below).

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.

My experiments in Qucs have resulted in some trivial example circuits and simulations = a Qucs project directory containing some ready files. Have them :-) and use them as a basis for your own tweaks/experiments.

Qucs project homepage

Wikipedia on Admittance (and its relationship to Impedance)

Microwaves101 on the Smith chart, including a "nomogram trick" to calculate a matching compensation TML stub

Wikipedia's Smith chart (a beautiful vector image in SVG format)

By: Frank Rysanek [ rysanek AT fccps DOT cz ] in January 2017