A Smith chart is a plot in the complex plane giving the real and imaginary parts of the complex voltage reflection coefficient Γ. The contour lines on the chart represent values of the normalized resistance
![](./The Smith Chart - Wolfram Demonstrations Project_files/1.gif)
(orange circles) and reactance
![](./The Smith Chart - Wolfram Demonstrations Project_files/2.gif)
(green arcs) of a circuit element. The relevant formulas are
![](./The Smith Chart - Wolfram Demonstrations Project_files/3.gif)
, where
![](./The Smith Chart - Wolfram Demonstrations Project_files/4.gif)
,
![](./The Smith Chart - Wolfram Demonstrations Project_files/5.gif)
,
![](./The Smith Chart - Wolfram Demonstrations Project_files/6.gif)
,
and
![](./The Smith Chart - Wolfram Demonstrations Project_files/7.gif)
is a characteristic impedance, often chosen as a resistance of 50Ω. Note that electrical engineers use
![](./The Smith Chart - Wolfram Demonstrations Project_files/8.gif)
for
![](./The Smith Chart - Wolfram Demonstrations Project_files/9.gif)
to avoid confusion with the current
![](./The Smith Chart - Wolfram Demonstrations Project_files/10.gif)
. The circumference of the chart gives the relative phase of the reflection coefficient, expressed as a multiple of the signal wavelength λ. This is useful in determining the length of transmission lines, so as to enable standing waves.
In this Demonstration, you can vary the resistance (
![](./The Smith Chart - Wolfram Demonstrations Project_files/11.gif)
in ohms Ω), inductance (
![](./The Smith Chart - Wolfram Demonstrations Project_files/12.gif)
in nanohenries nH=
![](./The Smith Chart - Wolfram Demonstrations Project_files/13.gif)
H), and capacitance (
![](./The Smith Chart - Wolfram Demonstrations Project_files/14.gif)
in picofarads pF=
![](./The Smith Chart - Wolfram Demonstrations Project_files/15.gif)
F) of the circuit element, as well as the frequency of the propagated signal (
![](./The Smith Chart - Wolfram Demonstrations Project_files/16.gif)
in megahertz MHz=
![](./The Smith Chart - Wolfram Demonstrations Project_files/17.gif)
Hz). The complex reduced impedance
![](./The Smith Chart - Wolfram Demonstrations Project_files/18.gif)
is then marked on the Smith chart.