Smith chart is graphical technique used for calculating reflection coefficient and standing wave ratio for normalized load impedance.
Characteristics of constant R-circles and X-circles
Various impedances on Smith Chart
Above Equation (1) represents a family of constant resistance circles whose center and radius are given by
center = (rL/1+rL,0)
radius = 1/1 + rL
Equations (2) represent a family of constant reactance circles whose center and radius are given by
center (1, 1xL)
NOTE: As we move from point to point along the transmission line plotting the complex reflection coefficient which is obtained by rotating around the constant VSWR circle, we are also plotting the corresponding impedance.
Once a normalized impedance is located on the Smith chart for a particular point on t he transmission line, the corresponding normalized admittance is found by rotating 180° from the impedance point on the constant VSWR circle.
The maxima and minima location for voltages and currents along the transmission line can be located using the Smith chart given that these values correspond to specific impedance characteristics.
Impedance and corresponding Admittance on Smith Chart
Voltage maximum, Current minimum : Impedance maximum
Voltage minimum, Current maximum : Impedance minimum
Maximas and Minimas on Smith Chart
Applications of Smith Chart
Smith chart can be used for the following purposes:
Impedance Matching is the process of removing mismatch loss. i.e. the reflection coefficient should be minimized, to reduce the power reflected from the load (the antenna), and maximize the power delivered to the antenna. This is one of the fundamental tasks in getting an antenna to radiate, and hence is one of the more important topics in antenna theory. It is accomplished by placing intermediate devices:
The Z0 line has a termination = Zin of the λ/4 line of ZL load = z'o2/ZL
With the use of the quarter wave transformer the line is completely matched but the load is not completely matched but the miss match is reduced as compare without the λ/4 transformer.
A stub is a short circuit line of lt length placed at ls distance from the load such that the function impedance is Z0 and the line is matched, from that point towards the source.
Design of a stub
Identity a position on the line from the load where Z(x) = Z0 ± jx then x value is is position of stub
At this stub position and equal and opposite reactance is placed in shunt that cancels existing reactance so that the junction impedance.
This junction impedance is Z0 as the cancelling reactance is designed form a short–circuit line of ‘lt’ length
For the miss–matched region to be small ‘ls’ should be as close to the load as possible
Matching with a single stub tuner