Thursday, August 30, 2012

What's a Smith chart?

       What is a Smith chart? It's really just a plot of complex reflection overlaid with an impedance and/or admittance grid referenced to a 1-ohm characteristic impedance. That's it! Transmission coefficient, which equals unity plus reflection coefficient, may also be plotted (see below). You can find books and articles describing how a Smith chart is a graphical representation of the transmission line equations and the mathematical reasons for the circles and arcs, but these things don't really matter when you need to get the job done. What matters is knowing the basics and how to use them, like always.
The Smith chart contains almost all possible impedances, real or imaginary, within one circle. All imaginary impedances from - infinity to + infinity are represented, but only positive real impedances appear on the "classic" Smith chart. Yes, it is possible to go outside the Smith chart "unity" circle, but only with an active device because this implies negative resistance.
       One thing you give up when plotting reflection coefficients on a Smith chart is a direct reading of a frequency axis. Typically, plots that are done over any frequency band have markers calling out specific frequencies.
Why use a Smith chart?  It's got all those funny circles and arcs, and good ol' rectangular plots are much better for displaying things like VSWR, transmission loss, and phase, right? Perhaps sometimes a rectangular plot is better, but a Smith chart is the RF engineer's best friend! It's easy to master, and it adds an air of "analog coolness" to presentations, which will impress your friends, if not your dates! A master in the art of Smith-charting can look at a thoroughly messed up VSWR of a component or network, and synthesize two or three simple networks that will impedance-match the circuit in his head!

Impedance and admittance

          A quick refresher on the basic quantities that have units of ohms or its reciprocal, Siemens (sometimes called by its former name, mhos), is helpful since many of them will be referenced below. We all think of resistance (R) as the most fundamental of these quantities, a measure of the opposition to current flow that causes a potential drop, or voltage, according to Ohms Law: V=I*R. By extension, impedance (Z) is the steady state AC term for the combined effect of both resistance and reactance (X), where Z=R+jX. (X=jwL for an inductor, and X=1/jwC for a capacitor, where w is the radian frequency or 2*pi*f.) Generally, Z is a complex quantity having a real part (resistance) and an imaginary part (reactance).
We often think in terms of impedance and its constituent quantities of resistance and reactance. These three terms represent "opposition" quantities and are a natural fit for series-connected circuits where impedances add together. However, many circuits have elements connected in parallel or "shunt" that are a natural fit for the "acceptance" quantity of admittance (Y) and its constituent quantities of conductance (G) and susceptance (B), where Y=G+jB. (B=jwC for a capacitor, and B=1/jwL for an inductor.) Admittances add together for shunt-connected circuits. Remember that Y=1/Z=1/(R+jX), so that G=1/R only if X=0, and B=-1/X only if R=0.
        When working with a series-connected circuit or inserting elements in series with an existing circuit or transmission line, the resistance and reactance components are easily manipulated on the "impedance" Smith chart. Similarly, when working with a parallel-connected circuit or inserting elements in parallel with an existing circuit or transmission line, the conductance and susceptance components are easily manipulated on the "admittance" Smith chart. The "immittance" Smith chart simply has both the impedance and admittance grids on the same chart, which is useful for cascading series-connected with parallel-connected circuits.

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