# Why are stability circles circles on a Smith chart?

I am new to basic concepts in amplifier stability, so the question might seem obvious, sorry if it is the case.

The amplifier stability translates into the equations for the reflection coefficients in load and source plain. The stability circle in the source plain will be $$\begin{equation} |\Gamma_S - C_L| = r_L \end{equation}$$

where $$\\Gamma_S\$$ is a source reflection coefficient, $$\C_L\$$ is a circle center, $$\r_l\$$ is a radius of the circle. I know that if we plot the equation on the Smith chart, it will be a circle.

Why is it a circle? Is there a mathematical explanation for it? I can understand that in a Cartesian coordinates that would be a case, but it is not clear for the Smith chart.

• Remember how is obtained Smith chart? It is a transformation that "transforms" cartesian coordinates (x,y) onto Smith coordinates through reflection coefficient (abs(rho), phase (rho)). Remember also what is "stability" (isn't where the real part of the "impedance" became "negative")? May 24, 2022 at 12:48
• Something as this could help ? highfrequencyelectronics.com/… and this mixsignal.files.wordpress.com/2013/07/481lecture34.pdf May 24, 2022 at 12:50
• There is a rule that a circle on the polar plot of reflection coefficient will also be a circle on the Smith chart (and vice versa). But I don't think I've ever seen the actual proof of it. May 24, 2022 at 14:00
• @The Photon ... I don't remember how this transformation is called. But If I remember well ... lines, in the x,y plane, are transformed into circles into the Smith Chart. So, if a Line is a limit to a circle ... a circle transforms into a circle. When the lines are perpendicular, the resulting circles are "orthogonal". A mathematician would help. The demonstration is done in the link just before ... May 24, 2022 at 14:23
• @Antonio51, it should be conformal transform May 24, 2022 at 16:08