# The Smith Chart

The Smith Chart

The Smith chart was developed by the electrical engineer Phillip H. Smith around 1939. It is a useful graphical aid for matching impedances of circuit elements, to optimize their performance.

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 (orange circles) and reactance (green arcs) of a circuit element. The relevant formulas are

, where , , ,

and is a characteristic impedance, often chosen as a resistance of 50Ω. Note that electrical engineers use for to avoid confusion with the current . 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.

r

x

Γ=

z-1

z+1

z=r+

j x =

Z

Z

0

Z=R+jX

X=2πfL-

1

2πfC

and

Z

0

j

-1

i

In this Demonstration, you can vary the resistance ( in ohms Ω), inductance ( in nanohenries nH=H), and capacitance ( in picofarads pF= F) of the circuit element, as well as the frequency of the propagated signal ( in megahertz MHz=Hz). The complex reduced impedance is then marked on the Smith chart.

R

L

10

-9

C

10

-12

f

10

6

z