High Frequencies and Microwaves 2
The complex variable is the voltage reflection coefficient in an arbitrary point. The bilinear transformations (10), (11) represent a conformal mapping between the and complex planes. Under this transformation, the family of straight lines from the plane is transformed into a family of circles in the plane ; in a similar way, the family of straight lines is transformed into another family of circles, orthogonal to the first circles family. The basic transformation can be used to localize these circles in the plane, by expressing the complex variables and through their real and imaginary parts, , .
Containing all passive impedances - maps in the plane into the inside of the unit radius circle, with the centre in origin. Essentially, the Smith chart is just this part of the plane, with circles and drawn on it.
In microwaves, the reflection coefficient and/or the normalized impedance can be measured with the slotted line.
For a given waveguide and a given frequency of the source, the distribution depends only on the normalized load impedance. This normalized load determines the (relative) values of the maxima and minima (the shape of the distribution), and their positions on the line, with respect to the load (see fig. 4).
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