Editing Smith chart (section)

==Using the Smith chart to analyze lumped-element circuits==
The analysis of [[lumped-element]] components assumes that the wavelength at the frequency of operation is much greater than the dimensions of the components themselves. The Smith chart may be used to analyze such circuits in which case the movements around the chart are generated by the (normalized) impedances and admittances of the components at the frequency of operation. In this case the wavelength scaling on the Smith chart circumference is not used. The following circuit will be analyzed using a Smith chart at an operating frequency of 100&nbsp;MHz. At this frequency the free space wavelength is 3 m. The component dimensions themselves will be in the order of millimetres so the assumption of lumped components will be valid. Despite there being no transmission line as such, a system impedance must still be defined to enable normalization and de-normalization calculations and <math>Z_0 = 50 \ \Omega\,</math> is a good choice here as <math>R_1 = 50 \ \Omega\,</math>. If there were very different values of resistance present a value closer to these might be a better choice.

[[Image:SmithCctEx1.png|thumbnail|A lumped-element circuit which may be analyzed using a Smith chart.|alt=|325x325px]]
[[Image:SmithEx5.png|thumbnail|Smith chart with graphical construction for analysis of a lumped circuit.|alt=|325x325px]]
The analysis starts with a Z Smith chart looking into R<sub>1</sub> only with no other components present. As <math>R_1 = 50 \ \Omega\,</math> is the same as the system impedance, this is represented by a point at the centre of the Smith chart. The first transformation is OP<sub>1</sub> along the line of constant normalized resistance in this case the addition of a normalized reactance of -''j''0.80, corresponding to a series capacitor of 40 pF. Points with suffix P are in the ''Z'' plane and points with suffix Q are in the ''Y'' plane. Therefore, transformations ''P''<sub>1</sub> to ''Q''<sub>1</sub> and ''P''<sub>3</sub> to ''Q''<sub>3</sub> are from the Z Smith chart to the Y Smith chart and transformation ''Q''<sub>2</sub> to ''P''<sub>2</sub> is from the Y Smith chart to the Z Smith chart. The following table shows the steps taken to work through the remaining components and transformations, returning eventually back to the centre of the Smith chart and a perfect 50 ohm match. 
{| border="1" cellpadding="2"
|+Smith chart steps for analysing a lumped-element circuit
!width="100"|Transformation
!width="100"|Plane
!width="100"|''x'' or ''b'' Normalized value
!width="100"|Capacitance/Inductance
!width="150"|Formula to Solve
!width="100"|Result
|-
|<math> O \rightarrow P_1\,</math>
|<math>Z\,</math>
|<math>-j0.80\,</math>
|Capacitance (Series)
|<math>-j0.80 = \frac{-j}{\omega C_1 Z_0}\,</math>
|<math>C_1 = 40 \ \mathrm{pF}\,</math>
|-
|<math> Q_1 \rightarrow Q_2\,</math>
|<math>Y\,</math>
|<math> -j1.49\,</math>
|Inductance (Shunt)
|<math>-j1.49 = \frac{-j}{\omega L_1 Y_0}\,</math>
|<math>L_1 = 53 \ \mathrm{nH}\,</math>
|-
|<math> P_2 \rightarrow P_3\,</math>
|Z
|<math>-j0.23\,</math>
|Capacitance (Series)
|<math>-j0.23 = \frac{-j}{\omega C_2 Z_0}\,</math>
|<math>C_2 = 138 \ \mathrm{pF}\,</math>
|-
|<math> Q_3 \rightarrow O\,</math>
|Y
|<math>+j1.14\,</math>
|Capacitance (Shunt)
|<math>+j1.14 = \frac{j \omega C_3}{Y_0}\,</math>
|<math>C_3 = 36 \ \mathrm{pF}\,</math>
|}