The scariest thing you learn in Electrical Engineering | The Smith Chart

Smith Charts

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Graphical tool used to represent complex impedance and admittance. Chart allows you to plot normalized impedances and find relationships between different electrical properties of a TL, such as reflection coefficients. SWRs, and input impedance as a function of a position along the line.

A Smith Chart!

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Normalization

$$ z_L = \frac{Z_L}{Z_0} $$

• Make a plot on the Smith chart corresponding to normalized impedance

Reflection Coefficient

Relates the reflected and incident waves at a load

$$ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$

Determine Distance

Use the wavelength scale around the circumference of the chart to determine how impedance changes along the transmission line

• Input impedance as a function of distance

$$ Z_{in} = Z_0 \frac{Z_L + j Z_0 \tan(\beta d)}{Z_0 + j Z_L \tan(\beta d)} $$

• Reflection coefficient as a function of distance

  $$ \Gamma_d = \Gamma_L e^{-j2\beta d} \text{, where } \beta = \frac{2 \pi}{\lambda} $$

De-normalize

Get the actual impedance

• Multiply $z_{in}$ by $Z_0$