Advanced Smith Charting
Smith Charts are a beautiful way of visualizing what's going on with a circuit. The importance of Smith Charts in the design process has (arguably) waned- few professionals still design matching networks using Smith Charts outside of school. The RF design tools (with optimizers) are generally good enough to solve most problems. Unfortunately, this means many designers have forgotten or were never taught some important features of the Smith Chart.
Neat Smith Chart Fact
For circuits using only passive components (and often even those with active components), curves on the Smith Chart spin clockwise with increasing frequency.
Constant Q Lines
Smith Charts all about lines of constant... something. Smith Charts themselves are lines of constant resistance and reactance. Most people are probably familiar with lines of constant SWR, any point on the Smith Chart the same distance from the center has the same mismatch. One of the lesser known features of the Smith Chart is the lines of constant Q.
The Q of a circuit can be defined in multiple ways, often as the ratio of the frequency to the bandwidth. High Q translates into a narrow band circuit- in this case a narrow band match.
Q can also be defined as the ratio of the reactance to the resistance.
This simple relationship means that lines of constant Q can be drawn on a Smith Chart. The relationship is simple: the arc starts at R=X=0; ends at R=X=infinity; and is centered at the Cartesian coordinates of X=0, Y= +/-(1/Q).
Consider the Smith Chart below with four Q contours, Q equal to 1, 2, 5, and 10. The (admittedly crudely) drawn matching path is meant to illustrate several points:
To achieve a particular Q, the match must choose elements for the matching path to stay within curves of constant Q.
The green and red curve start and stop at the same points.
For the red line below, the path touches on (but stays below) the Q=1 line.
For the green line below, the path touches (but stays below) the Q=4 line.
Therefore, the match for the red line is four times broader in bandwidth than the green line.
Higher Q means staying closer to the real line, using less reactive components.
The green curve uses two matching element.
The red line uses eight matching elements.
More elements are needed to keep under the lower Q curve, providing that greater bandwidth.
To achieve lower Q (which corresponds to a broader band match), the curves get exponentially closer to the real line.
Technically, the relationship is 1/Q.
It gets progressively harder to get broader band designs.
The theoretic best bandwidth of the matching network is a function of the starting point on the Smith Chart
Point A is close to the real line on the Smith Chart, and has a potentially high bandwidth solution.
Point B has a stronger reactance component, and has a bound to how broadband a match can be made.
Starting SWR is only weakly coupled to the bandwidth of the match (some practical considerations exist)
Point B has a lower SWR, but a higher Q.
Point A has a broader band solution, if one is willing to use enough matching components
Real world lumped element components have losses and tolerances.
When trying to achieve a high Q solution, the losses in real components changes the trajectory of the matching path (lowering the Q of the match); this is referred to as the Q of the part.
Loss in matching elements reduces the practicality of many broadband solutions. Each added element reduces the SNR that much more.
Cascading the tolerances of parts rapidly expands the bounds over which the manufactured solution will fall.
Tapers transmission line matching networks work well by hugging a very low-Q curve.
Since tapers are printed, the tolerance is not cascaded and so are fairly manufacturable.
Tapered transmission lines still have loss issues, particularly at the high frequency end.
Click image to enlarge
As always, certain Caveats exist. For this particular scenario, I have assumed Foster components (no negative capacitors or inductors, just passive components).
Copyright 2010, Gregory Kiesel