Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 2C: Introduction to the Smith Chart

The Smith chart is a graphical aid or nomogram [1] designed as an aid in radio frequency engineering to assist in solving problems with transmission lines and matching circuits. The reflection coefficient is the relationship based on which the Smith chart is constructed:

(2C.1)$$\mathrm{\Gamma }=\frac{Z_{\text{L}}-Z_0}{Z_{\text{L}}+Z_0}.$$

The x-axis is the real part of the reflection coefficient (Γ_r) and the y-axis is the imaginary part of the reflection coefficient (Γ_i), where

(2C.2)$$\mathrm{\Gamma }={\mathrm{\Gamma }}_{\text{r}}+\text{j}{\mathrm{\Gamma }}_{\text{i}}.$$

The impedance plotted on the chart will be normalized with respect to the characteristic impedance Z₀:

(2C.3)$$z_{\text{L}}=r+\text{j}x=\frac{Z_{\text{L}}}{Z_0}=\frac{R_{\text{L}}+\text{j}{X}_{\text{L}}}{Z_0}.$$

The reflection coefficient can be represented as

(2C.4)$$\mathrm{\Gamma }=\frac{z_{\text{L}}-1}{z_{\text{L}}+1}$$

and the load impedance as

(2C.5)$$z_{\text{L}}=\frac{1+\mathrm{\Gamma }}{1-\mathrm{\Gamma }}.$$

In polar ...