Smith Charts

Overview

Smith charts are used to display RF-relevant information about a particular circuit, device, or antenna trace. Smith charts show information such as the real and imaginary parts of both impedance and admittance and the reflection coefficicent (S11).

Impedance Smith charts are good for matching the impedance with series components, while admittance Smith charts are good for matching the impedance with parallel (shunt) components.

Smith charts can be generated by most network analysers. They normally perform a frequency sweep and plot a line on the graph rather than a single point.

In many case, it is desired so that the impedance is matched to \( (50 + j0)\Omega \) (the impedance is purely real). This is normally the point at the centre of the chart.

Terminology

Term Definition
\( Z_L \) The load impedance.

Solving Impedance Matching Problems

Most circuits (e.g. GPS antennas and WiFi antennas on PCB’s) are matched with capacitors between 1-10pF and inductors between 1-10nH. Make sure the inductor is O.K. at the operating frequency. This means you will probably need to get “high-frequency” or “RF” inductors, assuming the frequency of operating will be between 100MHz-6GHz. Power inductors, ferrite beads, and other types of inductors with ferrite cores are not suitable! You can also make your own air-core inductor suitabe for matching relatively easily with wire and a non-ferrous core (there are many calculators on the internet that will tell you how many turns you will need).

Series Inductors

Moves the operating point clockwise along circles of constant resistance.

The equation to calculate inductance from a normalised Smith chart is:

$$z_{IND} = \frac{i2\pi fL}{Z_0}$$

where:
\( Z_{IND} \) is the inductors impedance, \( \Omega \)
\( f \) is the frequency, \(Hz \)
\( L \) is the inductors inductance, \( H \)
\( Z_0 \) is the normalised impedance, \( \Omega \)

Series Capacitors

Moves the operating point anti-clockwise along circles of constant resistance.

Parallel Inductors

Moves the operating point anti-clockwise along circles of constant conductance.

Parallel Capacitors

Moves the operating point clockwise along circles of constant conductance.

Transmission Lines

The addition/modification/removal of a transmission line moves the operating points along circles of constant VSWR.

What If My Point Is Outside The Chart?

The means that the real part of the impedance is negative. This is only possible with an active (powered) circuit (all passive circuits must have 0 or positive real resistance). Saying that, I have had times when the Smith Charts goes outside of the unit circle when measuring something like a passive antenna? Many online calculators will give you an error or no result if you enter in a negative real impedance.

Online Calculators

The idea behind the online calculators is that you enter the actual impedance (as measure by a network analyser), the desired impedance, and the operating frequency, and the calculator will work out the values and locations (e.g. series or parallel) of capacitance and inductance to match the circuit so it is at the desired impedance.

Analog Devices has a simple impedance matching calculator here.

This one is interactive and shows you where your actual and desired impedances are on the Smith chart.