Hello everybody,

Well I'm having some trouble solving one exercise concerning the Smith Chart.
The goal is to mark the given area in the Smith Chart. The area comes in the complex plane.

Z = [(100 - j*50) + r*exp(j*θ)] Ω
where 0<=θ<=∏ and 0<=r<=50.

The reference impedance is 50 Ω.

I know that this area, in the complex plane corresponds to a half circle with center in (100 - j*50) and maximum radius of 50 (without considering the normalization). My problem is how to represent this in the Smith Chart? If it would be a rectangle I would now how to represent it but not a half circle.

I already tried to obtain the reflection module/angle in order to find boundary conditions but without success.

If anyone has idea of doing this and wants to discuss this with me feel free to do it.

Thanks in advance.

Regards.

 

I don't know what level you are doing this, if you are meant to just plot lots of points and sketch the boundary then just do it. If you are meant to do it using mathematics, then read on.

The Smith Chart is a Mobius transformation from the Z plane to the \(\Gamma\) plane. Look it up and read about what circles and lines transform into.

One hint - the straight line you are mapping is already plotted on the Smith chart for you.

You might like to play with Terry Tao's applet to visualise things:
http://www.math.ucla.edu/~tao/java/Mobius.html

it will map lines and circles from the Z plane to the \(\Gamma\) plane.

 

Solved. Thanks, Tesla23.