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:
- 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.
- 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.
- 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~
- 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.
- 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)~
| Circle | Centre at ~\Gamma~ plane | radius |
| 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~) |
The angle: Math.atan2(~\Gamma_i-y_0, \Gamma_r-x_0~) 180/pi