Smith Chart

The Smith Chart is named after its inventor Phillip Smith, who developed it at Bell Telephone’s Radio Research Lab during the 1930s.

It is believed that Mizuhashi Tosaku developed this tool independently around similar times. Smith Chart is a tool that enables the computation of complicated equations related to transmission lines and circuits for matching.

Nowadays those calculations could be resolved with computer software but over the years, the Smith Chart method has retained its appeal and is preferred by many.

This type of chart essentially provides a mapping of Reflection Coefficient (in Polar Form) to Load Impedance (in Rectangular Form), based on the equation:

where z=Z/Z0 , Z is the complex impedance (Z = R +jX) and Z0 is the reference impedance.

For a basic impedance, Z could be referred to as the complex impedance of the load while Z0 is the impedance of the transmitter, or whatever is delivering power to the antenna.

The Smith Chart is a graphical way of showcasing the impedance of an antenna based on the frequency, either as a single value or as a selection of values.

Figure 1. Examples normalized Smith Chart. Each point on the chart simultaneously represents both a value of z and the corresponding value of Γ. Relationship between those values defined by above equation (1) or it’s alternative z=(1 + Γ)/(1 − Γ). Impedances with non-negative resistive components will appear inside a circle with a unit radius.

Basic Impedance

The appearance of a regular Smith Chart can be intimidating due to its crisscrossing lines. Once the meaning of each line is comprehended, the text becomes simpler to appreciate.

Impedance contains two major elements. That is, the two sets of circles/arcs, which define the shape and data represented by the Smith Chart. These circles are known as:

1. The Constant R Circles
2. The Constant X Circles

R and X is referred to the real and imaginary parts of normalized impedance z = R +iX (eq. 2)

1. The Constant Resistance (R) Circles

The first set of lines referred to as Constant Resistance circles, set of different diameter circles all touching the right-hand side on the horizontal axis (z = ∞).

When the Resistance (R) of Impedance is kept the same, the Constant R Circles are the result of the variable X. As result, all the points on a particular Constant R circle represent the same resistance value (fixed Resistance).

The value of the resistance represented by each Constant R Circle is marked on the horizontal line (where the circle intersects with a horizontal line). That is also the diameter of the circle.

Figure 2. A standardized grid that only displays the real part of the impedance. The red and green line is generated when the Resistance (R=1 and 0.21 respectively) of the Impedance is held constant, while the value of X varies from -∞ to +∞.

For example, consider a normalized impedance equation 2, if R was equal to one and X was equal to any real number such that, z = 1 +i0, z = 1 +i3, and z = 1 -i1.5, a plot of the impedance on the smith chart will look like the figure 2.

This should give a clear idea of how the giant circles in the smith chart are generated. The inner and outer Constant R Circles define the limits of the Chart.

The inmost circle is referred to as the infinite resistance, which is reduced to one point (figure 1, z=∞). While the outmost circle is referred to as the zero resistance (figure 2, biggest circle). There should be no negative resistance.

2. The Constant Reactance (X) Circles

The Constant Reactance (X) Circles are circles, which are also all touching the right-hand side on the horizontal axis (z = ∞).

X Circles are more of arcs when circles are restricted to positive R only. Curves are generated when the impedance has a fixed reactance but a varying value of resistance.

Figure 3. Normalized Smith Chart only with ‘imaginary’ grid. The line is generated when the Reactance (X=1) of the Impedance is held constant, while the value of R varies from -∞ to +∞. On the left, the complete circles of Reactance (X) are shown. On the right, only the arc is shown then the chart is clipped to positive R.

The lines in the upper half represent positive reactances while those in the lower half represent negative reactances. An important curve is given by Im[z]=0, a straight line across the Smith Chart.

That is the set of all impedances given by z = R, where the imaginary part is zero and the real part (the resistance) is greater than or equal to zero.

For example, let us consider a curve defined by equation 2. If X = 1 and is kept constant, while R represents a real number, which is varied from 0 to infinity, a plot similar to the one in Figure 3 is obtained.

Different versions of the Smith Chart

Smith chart is plotted on the complex reflection coefficient plane in two dimensions and is scaled in impedance (the most common), admittance, or both.

The Impedance chart is terrific when working with load in series because one just simply needs to add the impedance up. However, the math becomes really tricky when working with parallel components (parallel inductors, capacitors, or shunt transmission lines).

To allow the same simplicity, the admittance chart was developed. The basic electricity course teaches that admittance is the inverse of impedance.

Therefore, an admittance chart makes sense for the complex parallel situation as all one will need to do is examine the admittance of the antenna rather than the impedance.

Below is an equation for the relationship between admittance and impedance:

Where y is the admittance of the load, z is the impedance, C is the real part of the admittance known as Conductance, and S is the imaginary part known as Susceptance.

The Admittance Smith Chart has an inverse orientation to the Impedance Smith Chart. There are the constant Conductance Circles (instead of Resistance circles) and the Constant Susceptance Circles (instead of Reactance circles) in the Admittance Smith Chart.

Admittance value could be calculated graphically by using the Basic (Impedance) Smith Chart. To invert a complex impedance z, plot the impedance z on the Smith Chart and then find the point on the chart that is equidistant from the origin but on the exact opposite side of the chart (red line in figure 4). 

Figure 4. Given the impedance z = 3 +i2, the admittance determined by using the graphic method is y ≈ 0.23 – i 0.15.

LightningChart Smith Charts

LightningChart Smith charts are specialized diagrams specifically designed to be used in radio frequency applications, such as creating and assessing the stability of oscillators.

The LightningChart Smith chart makes it possible to see several characteristics at once, including impedance, admittance, reflection coefficients, and scattering parameters.

• Distinctions between absolute and normalized measurements.

• Smith charts are able to adjust the distance and size of the grid markings depending on the zoom level. Zooming in will generate precise ticks and grids to help users better comprehend the values without making the display too busy.

• Automatic zoom-to-fit for the whole area or data only.

• Zooming and panning via ready-made mouse operations.

• Tick values are always visible, even when zoomed in close.

• Data and grid clipping to graph area. You can show data also outside of the graph.

Conclusion

In conclusion, the advantages of using LightningChart Smith charts for scientific data visualization are numerous. LightningChart Smith charts offer an efficient and accurate method for representing complex impedance data.

With their user-friendly interface and customizable features, LightningChart Smith charts enable users to easily interpret data and make informed decisions.

Additionally, LightningChart Smith charts are highly interactive. Interactivity allows users to explore data in real time, zoom in on specific points, and quickly identify trends and patterns.

Overall, the use of LightningChart Smith charts can greatly enhance the accuracy, speed, and effectiveness of impedance analysis. This makes LightningChart Smith charts a valuable tool for professionals in a wide range of industries.

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