Editing Smith chart (section)

==Overview==
[[Image:NetworkAnalyzer.jpg|400px|thumb|A [[Network analyzer (electrical)|network analyzer]] set up to display measured data on a Smith chart.]]

The Smith chart is a mathematical transformation of the two-dimensional Cartesian complex plane. [[Complex numbers]] with positive real parts map inside the circle. Those with negative real parts map outside the circle.
If we are dealing only with impedances with non-negative resistive components, our interest is focused on the area inside the circle. The transformation, for an impedance Smith chart, is:

<math display="block">\Gamma = \frac{Z - Z_0}{Z + Z_0} = \frac{z - 1}{z + 1},</math>

where {{math|1= ''z'' = {{sfrac|''Z''|''Z''{{sub|0}}}}}}, i.e., the complex impedance, {{mvar|Z}}, normalized by the reference impedance, {{math|''Z''{{sub|0}}}}. The impedance Smith chart is then an [[Argand plot]] of impedances thus transformed. Impedances with non-negative resistive components will appear inside a circle with unit radius; the origin will correspond to the reference impedance, {{math|''Z''{{sub|0}}}}.

The Smith chart is plotted on the [[complex number|complex]] [[reflection coefficient]] plane in [[dimension|two dimensions]] and may be scaled in normalised [[Electrical impedance|impedance]] (the most common), normalised [[admittance]] or both, using different colours to distinguish between them. These are often known as the Z, Y and YZ Smith charts respectively.<ref name="Gonzalez_1997"/>{{rp|page=97}} Normalised scaling allows the Smith chart to be used for problems involving any [[characteristic impedance|characteristic]] or system impedance which is represented by the center point of the chart. The most commonly used normalization impedance is 50&nbsp;[[ohm]]s. Once an answer is obtained through the graphical constructions described below, it is straightforward to convert between normalised impedance (or normalised admittance) and the corresponding unnormalized value by multiplying by the characteristic impedance (admittance). Reflection coefficients can be read directly from the chart as they are unitless parameters.

The Smith chart has a scale around its [[circumference]] or periphery which is graduated in [[wavelengths]] and [[degree (angle)|degree]]s. The wavelengths scale is used in [[distributed-element model|distributed component]] problems and represents the distance measured along the transmission line connected between the [[signal generator|generator]] or source and the load to the point under consideration. The degrees scale represents the angle of the voltage reflection coefficient at that point. The Smith chart may also be used for [[lumped-element]] matching and analysis problems.

Use of the Smith chart and the interpretation of the results obtained using it requires a good understanding of [[alternating current|AC circuit theory]] and transmission-line theory, both of which are prerequisites for RF engineers.

As impedances and admittances change with frequency, problems using the Smith chart can only be solved manually using one [[frequency]] at a time, the result being represented by a [[point (geometry)|point]]. This is often adequate for [[frequency range|narrow band]] applications (typically up to about 5% to 10% [[Bandwidth (signal processing)|bandwidth]]) but for wider bandwidths it is usually necessary to apply Smith chart techniques at more than one frequency across the operating frequency band. Provided the frequencies are sufficiently close, the resulting Smith chart points may be joined by straight lines to create a [[locus (mathematics)|locus]].

A locus of points on a Smith chart covering a range of frequencies can be used to visually represent:
*how [[capacitance|capacitive]] or how [[inductance|inductive]] a load is across the frequency range
*how difficult matching is likely to be at various frequencies
*how well matched a particular component is.
The accuracy of the Smith chart is reduced for problems involving a large locus of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these.