# The Smith Chart

The Smith chart is a graphical tool for analysing and designing transmission line circuits. It represents the reflection coefficient's complex plane.

The image below shows the the complex plane

• Point A is the reflection coefficient
• Point B is the reflection coefficient The Smith chart shows circles of constant normalised resistance , and constant normalised reactance , within the unit circle plane. Given the normalised value of a load impedance , we can find the value of the corresponding reflection coefficient, and vice-versa.

## Example

In the example below, point is plotted on the and lines, representing a normalised impedance of . • The length of the line between the and the centre corresponds to the magnitude of the reflection coefficient
• The angle between the x axis and the point is

## Phase Shifting

Based on the input impedance in terms of the reflection coefficient, we obtain

is the phase shifted reflection coefficient. at on a transmission line is equal to the reflection coefficient at the load (), shifted by :

This phase shift can be achieved on the Smith chart by maintaining constant magnitude, and decreasing the phase by the phase, corresponding to a clockwise rotation of an angle radians.

A complete rotation of radians corresponds to a change in length of . The outermost scale on the chart "wavelengths toward the generator" denotes movement on the transmission line toward the source, in units of wavelength.

### Example

Point is a normalised load of at . If the load terminates a transmission line of length , what it's input impedance?

• Move clockwise by around a constant circle
• Read the smith chart at point to get For some problems, it is more convenient to work with admittances than with impedances 