What is a Smith Chart ? by Robin Hodgson 9H1ZZ

Prior to the arrival of the electronic calculator, generations of engineers, through practical usage of their “slide rules”, gained an inherent understanding of logarithms. RF engineers have also a tool which assists in the understanding of transmission lines.  This is called the Smith Chart, after the mathematician, P. Smith, who developed and published it originally in 1939.

Unfortunately, engineering schools always fail to teach students how to make practical use of  Smith Charts (or even slide rules).  Now, with PCs readily available, there is less motivation to self-learn how to use these tools, which do provide an insight and simple means of handling the calculations involved in transmission lines and components at radio frequencies.

As Hams, we have all learnt that an electronic component can be represented as an impedance which is a combination of (pure) resistance and (inductive or capacitive) reactance. Any such complex impedance can be visualized as a point on the Argand diagram.

Argand Diagram for representing Complex Impedance

Further, we have learnt that a RF transmission line has a characteristic impedance (Z0) dependent on its physical cross-sectional construction. Now, if this line is terminated by a load impedance of the same value, Z0,  then all the power is transferred into the load.

Any other load impedance will cause some power to be reflected back (the proportion is defined as the reflection coefficient) resulting in standing waves along the lines. There are complex mathematical relationships between the reflection coefficient, the standing wave ratio (SWR) and the normalised load impedance ZL/Z0. 

Schematic of the transmission line model

It is well-known that a specific reflection coefficient can be represented graphically as being on a circle of radius equal to magnitude of the reflection coefficient. This reflection coefficient is defined as lying between the values 1 (total reflection) to 0 (no reflection). The angular position, on a circle of constant reflection coefficient, represents the measured phase angle of the reflection, thereby defining the polar coordinates of the reflection. At the measuring point, the phase is going to depend also on the electrical length of the line, so in shifting up and down the line, the measured value moves around the circle representing the reflection coefficient, rotating once for every half-wavelength.

Chart showing Reflection Coefficient

The Smith Chart is the chart above, used to represent the reflection coefficient, on which is also traced the impedance values of resistance and reactance. These traces are no longer the familiar x – y rectangular coordinates of the Argand diagram, but follow different circular paths on the reflection coefficient chart.

In practice, once the reflection coefficient (and its phase, θ) is known, it can be plotted on the Smith Chart and impedance immediately becomes available in the familiar terms of resistance and reactance.

To simplify the printed chart, it is usual to omit the concentric circles representing the reflection coefficients, since the user will place these himself (with a ruler and compass) as he performs the measurement. This usually involves measuring the SWR, identifying a phase reference point (probably an open circuit, replacing the device under test) and the electrical length of the transmission line.

Once the concept has been grasped, the Smith chart becomes an essential tool in computing the complex impedance of combinations of components, both discrete and distributed. Problems of matching loads, tuning antennas, optimizing bandwidths, and even complex number arithmetic can easily be handled.. For those who insist on using their PC, there is a very useful and educational program for manipulating the Smith Chart, by Marian van Westen (PA0MVW), called PASAN. You can download it (free) from http://members.home.nl/mvanwesten/progs/pasanse.zip

As with all techniques, you will only learn and realize its full potential by using it.  Try manipulating the Smith Chart in a few examples, I’m sure you will find the results obtained make it worth the effort.

The Complete Smith Chart (Click to enlarge)

Additional  charts for printing can be downloaded (free) from http://www.engj.ulst.ac.uk/sidk/graph/smith.pdf

 Robin, 9H1ZZ