The mathematics of the smith chart

The Smith chart is a graphical tool used in electrical engineering, particularly in RF (radio frequency) and microwave engineering, to represent complex impedance(~Z~), admittance(~Y~), and reflection coefficients(~\Gamma~). It provides a way to visualize and solve problems involving transmission lines and matching circuits without requiring extensive calculations.

Key Aspects of the Smith Chart:

  1. Complex Impedance and Admittance: The Smith chart allows you to plot complex impedances and admittances, where each point on the chart represents a unique complex number.
  2. Reflection Coefficient: It can also represent the reflection coefficient, which is a measure of how much of a signal is reflected by an impedance discontinuity in a transmission line.
  3. Normalization: Impedances are typically normalized to a characteristic impedance (usually 50 ohms) before being plotted on the Smith chart. ~z=Z/Z_0 y=Y/Y_0~
  4. Transmission Line Problems: It is used to solve transmission line problems, including impedance matching, finding the input impedance of a transmission line, and designing matching networks.
  5. Circles and Arcs: The Smith chart consists of circles and arcs that represent constant resistance, reactance, and SWR (standing wave ratio).

Applications:

  • Impedance Matching: Design of matching networks to ensure maximum power transfer between a source and a load.
  • Transmission Line Analysis: Analyzing how impedance changes along a transmission line.
  • Network Design: Used in the design and tuning of RF and microwave circuits.

By using the Smith chart, engineers can quickly visualize and solve complex impedance-related problems, making it an essential tool in RF and microwave engineering. The mathematics is a little difficult for the smith chart. Here we summarize some of them.

Assumption:

The characteristic impedance ~Z_0~ is always real.

The normalized impedance ~z=Z/Z_0=r+jx~.

The normalized admittance ~y=Y/Y_0=g+jb~.

The reflected coefficient ~\Gamma=\Gamma_r+j\Gamma_i~

The reflected coefficient in terms of z: ~\Gamma=(z-1)/(z+1)~

The inverse transform: ~z=(1+\Gamma)/(1-\Gamma)=(1-\Gamma_r^2-\Gamma_i^2+2j\Gamma_i)/((1-\Gamma_r)^2+\Gamma_i^2)~

The reciprocal of a complex number:

~1/(a+bi)=(a-bi)/((a+bi)(a-bi))=(a-bi)/(a^2+b^2)~

CircleCentre at ~\Gamma~ planeradius
Equal r(~r/(1+r),0~)~1/(1+r)~
Equal x(~1,1/x~)~|1/x|~
Equal g(~-g/(1+g),0~)~1/(1+g)~
Equal b(~-1,-1/b~)~|1/b|~
Transmission line(~(z-1)/(z+1),0~)
Equal circles

The angle: Math.atan2(~\Gamma_i-y_0, \Gamma_r-x_0~) 180/pi

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