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Published on January 18, 2013
A Smith Chart is a graphical aid designed for RF engineers to solve transmission line and matching problems. Impedance values (real and reactive) are plotted on a graph of Reflection Coefficient, Γ, using the following relationship:
Z is the input impedance of your system (for example, the impedance of the antenna you are trying to match). The value Z usually contains real and reactive components. As an example, the antenna impedance may be 30+j56. Z0 is the characteristic impedance of the measurement system such as a network analyzer. In most cases, Z0 is equal to 50 Ohms and does not have a reactive component. The antenna impedance, 30+j56, is plotted on a Smith chart and used to help determine the proper matching components and transform the load to match the 50 Ohm transmission line.
Vincident is the incident voltage, Vreflected is the reflected voltage. The ratio between these two values, and consequently Γ, will always be greater than 0, but less than one. This is because the incident voltage will always be greater than reflected due to at least some power transfer to the load antenna, no matter how poor the match. Γ, Vincident, and Vreflected are complex values.
The amount of reflected power is measured by the quantity |Γ|2 (please note that |Γ|is the magnitude of Γ). Typically, reflected power is represented in dB by the equation shown below.
For example, if ΓdB = -10 dB, then |Γ|2 = 0.1 and 10% of the incident power is reflected. In this case, 90% of the available power is transmitted into the antenna. It is typical to define the impedance bandwidth of an antenna as containing the range of frequencies in which
ΓdB £ -10 dB
Γdb is the parameter most often measured on a network analyzer. Another commonly used term is Return Loss, which is simply = -ΓdB.
Finally, this leads to an explanation of the Voltage Standing Wave Ratio (VSWR). VSWR is determined by the ratio of the peak amplitude of a standing wave to the minimum amplitude of a standing wave along the transmission line. VSWR can be calculated by:
Low values for VSWR indicate a good antenna match. For a description on how standing and traveling waves are formed, please see our blog on 'Antenna Matching Within an Enclosure Part I: Theory and Principle’.