The reflection coefficient is then related to the load impedance (ΖL), characteristic impedance (Ζ0) and VSWR (S) through the next two equations. The absolute value bars in the second equations are necessary to force the VSWR value to a positive number.
A perfect short is nothing more than an impedance of 0Ω, while a perfect open would be an impedance of infinite Ω. As we move from the right side of the chart to the left side, the normalized resistance values decrease from infinity to zero. Thus a perfect open is represented by the point along the horizontal axis on the right side of the chart while a short is the corresponding point on the left side of the impedance chart.
Both of these locations represent a reflection coefficient with a magnitude of 1, so the outer circle of the Smith chart corresponds to a reflection coefficient magnitude of 1. A wave encountering an open circuit is reflected back in phase, while a short is 180 degrees out of phase. So we can see that rotating around the outer ring of the chart relates to the resulting phase shift, and we are able to plot points and directly read the phase shift.
The locations on the outer ring of the chart up and down from the center location, which has an impedance of 1+j0Ω, the reference value, are a 90-degree phase shift. These locations intersect the 0Ω resistance circle at +j1 and -j1Ω, resulting in normalized impedances of 0+j1 and 0-j1Ω. These are an inductor and capacitor respectively and correspond to the known phase relationship between voltage and current in these components.
There is also a scale for the transmission coefficient, which is the complex sum of 1 plus the reflection coefficient. It is not as widely used as the reflection coefficient. Outward from these two scales are the wavelength scales, used in transmission line calculations. They show that one full rotation around the chart corresponds to one-half wavelength.
Below the chart in Figure 1 there are several radial scales. These scales function as rulers, and can be utilized to make measurements based on items already illustrated on the chart. Typically included are scales for SWR, attenuation, return loss, and transmission and reflection coefficients.
Figure 2. The ZY chart (click image to enlarge.)
Calculations may be simplified using a ZY chart (Figure 2). On the ZY chart, the impedance Smith chart is overlaid by its admittance Smith chart, which is created by rotating the impedance chart by 180 degrees. Addition of components in series moves the impedance along a constant resistance circle, while shunt or parallel addition moves the impedance along a constant conductance circle, which is the analog to the resistance circle. With a T or pi configuration, nearly any impedance can be matched to another.
Even when not performing design calculations, the Smith chart provides an excellent method to visualize issues with antenna systems. A simple sweep may illustrate a greatly elevated VSWR. This measured quantity, flat across a range of frequencies, certainly demonstrates the presence of an issue with the system; but where is it located? From a Smith chart plot a relative location may be discerned. If the plot shows large diameter circles centered on the midpoint, then the impedance mismatch is distant from the generator. If the trace results in a compact circle slid off center, then the reflection is closer to the transmitter. In an ideal world, antenna systems would closely approximate a dot in the center of the chart.
For almost 75 years the Smith chart has been a mainstay of RF engineers. Its construct really is quite remarkable, and its ease of use is one reason for its continued relevance in today''s world of calculators and computers.
Ruck is a senior engineer with D.L. Markley and Associates, Peoria, IL.