## Smith Chart

RF/Photonics Lab

Jared Alves

November 2019

*Smith Chart*

The Smith Chart, named after laboratories engineer Phillip Smith, is a graphical tool for solving RF transmission line problems. There are many specific uses for a Smith Chart, but it is most commonly used to visually represent impedance matching problems. Although paper Smith Charts are outdated, RF equipment such as Network Analyzers display information using the chart as well.

The Smith Chart is a unit circle (radius of one) plotted on the complex plane of the voltage reflection coefficient (ᴦ – gamma). As with any complex plane, the vertical axis is the imaginary and the horizontal axis the real. The Smith Chart can be used as an admittance or impedance chart or both. For a load impedance to be plotted on the chart, it must be normalized (divided by) the characteristic impedance of the system (Z_{o}) which is the center of the chart. With this information in mind, it is apparent that a matched load condition would result in traveling to the center of the chart (where Z_{L}=Z_{o}). Along the circumference of the chart, there are two scales: wavelength and degrees. The degrees scale can be used to find the angle of the complex reflection coefficient. Since the plot is the polar representation of the reflection coefficient, if a line is drawn from the load impedance point to the center of the chart this would be considered the magnitude of the reflection coefficient. By extending the line to the circumference of the circle, the angle (in degrees) can be found. The wavelength scale shows distance across a transmission line in meters. A clockwise rotation represents moving towards the generator whereas a counter-clockwise rotation represents moving towards the load side.

It is important to note that a Smith Chart can only be used at one specific frequency and one moment in time. This is because waves are functions of both space and time as shown by the equations:

V_{F} is the forward propagating voltage wave and V_{R} is the reverse propagating voltage wave. If a transmission line system is not impedance matched, a reflected wave will exist on the line which will cause partial or fully standing waves to occur on the line (the reflected wave will add to the incident wave). For the matched condition the reflected wave is zero. Because the Smith Chart can only be used at a specific instant in time and at one frequency the first exponential term in each equation drops out. Because the reflection coefficient is the ratio of the reflected wave to the forward propagating wave, the reflection coefficient becomes:

Where C is the ratio of the amplitudes of both waves. For a passive load, the reflection coefficient must be equal to one or less because the reflected wave cannot be greater in amplitude than the incident wave.

Many transmission lines can be approximated as lossless and therefore have zero attenuation. This leads to:

The propagation constant is a complex number that describes how a wave changes as it propagates down a transmission line. The real part is attenuation constant (Nepers/meter) and the imaginary part is the phase constant or wave number (radians/meter).

For the lossless condition the attenuation is zero, as stated previously.

On the Smith Chart, the wavelength λ = 720. This is because the reflected wave must travel the roundtrip distance moved (it must propagate forward and then back again). Using the piece of information, a half wavelength distance is one complete revolution on the chart. This leads to the conclusion that a transmission line that is a half wavelength long does not transform impedance.

The following image shows common points on the Smith Chart.

The left-hand side of the chart (lying on the real axis) represents a short circuit load. This makes intuitive sense because the reflection coefficient must be real and negative for a short circuit. This is because short circuits have a voltage drop of zero across them which would require a same-amplitude wave with a 180-degree phase shift to cancel the forward propagating wave. The right-hand part of the real axis represents the open circuit load, where the reflection coefficient is purely real but has no phase shift. For an open circuit, the current wave would have to be phase shifted by 180-degrees, but since the reflection coefficient is a voltage reflection coefficient it is not necessary for it to be phase shifted. As shown in the image, the upper half plane is inductive (positive reactance) and the lower half is capacitive (negative reactance).

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