Smith Chart

## Graphical Plot of Real and Imaginary Measurement Results

Measurement results are often displayed using a Smith Chart. A Smith Chart is a plot of real and imaginary part reflection overlaid with an impedance and/or admittance grid referenced to a 1 Ohm characteristic impedance. The Transmission coefficient, which equals unity plus reflection coefficient, may also be plotted. The Smith chart contains almost all possible impedances, real or imaginary, within one circle. All imaginary impedances from – infinity to + infinity are represented, but only positive real impedances appear on the “classic” Smith chart.

A Smith Chart is a direct reading of a frequency axis. Plots over a frequency band will have have labels noting the specific frequencies.

The impedance (Z) is the steady state Alternating Current (AC) term for the combined effect of both resistance (R) and reactance (X), where Z=R+jX. (X=ωL for an inductor, and X=1/ωC for a capacitor, where ω is the radian frequency or 2πf.) Generally, Z is a complex quantity having a real part (Resistance) and an imaginary part (Reactance).

Many circuits have elements connected in parallel that are a natural fit for the “Acceptance” quantity of Admittance (Y) and its constituent quantities of Conductance (G) and susceptance (B), where Y=G+jB. (B=ωC for a capacitor, and B=1/ωL for an inductor.) Y=1/Z=1/(R+jX), so that G=1/R only if X=0, and B=-1/X only if R=0.

The most common orientation of the Smith chart places the resistance axis horizontally with the Short Circuit (SC) location at the far left. There’s a good reason for this: the voltage of the reflected wave at a Short Circuit must cancel the voltage of the incident wave so that zero potential exists across the short circuit. In other words, the voltage reflection coefficient must be -1 or a magnitude of 1 at an angle of 180 degrees. Since angles are measured from the positive real axis and the real axis is horizontal, the short circuit location and horizontal orientation make sense.

For an open circuit (OC), the reflected voltage is equal to and in phase with the incident voltage (reflection coefficient of +1) so that the Open Circuit location is on the right. In general, the reflection coefficient has a magnitude other than unity and is complex. Anywhere above the real axis is inductive (L) and anywhere below is capacitive (C).

There are an infinite number of possible solutions because, at one frequency, a stub of any characteristic impedance can provide the necessary normalized susceptance simply by adjusting its length. The differences show up when looking over a frequency band. For example, the stub’s length may be increased by an integer multiple of half-wavelengths at a particular frequency and its input susceptance at this frequency will not change. But over a frequency band, the susceptance will vary considerably more than if the extra length had not been added.