Single stub conjugate matching non unity conductance circle

[optic37]

I'm working on an interstage matching network for an amplifier design, aiming for a conjugate match between stage 1 output (source) and stage 2 input (load) using a single shunt stub and the Smith Chart.

My understanding of the standard single-stub procedure for matching a load (ZL) to the system impedance Z0 is typically: (when working on a smith chart with both Z and Y traces)

Convert ZL to normalized impedance

Rotate clockwise on its constant |Γ| circle.

Stop rotation when intersecting the G=1 circle.

Calculate the length of the transmission line from the distance to the intersection.

Now graphically look at the admitance to find the susceptance.

Add a shunt stub to cancel the resulting susceptance.

However, when applying this to interstage conjugate matching, my target admittance isn't necessarily 1+j0, but rather the conjugate of the source's output admittance (Y_source). Using the standard procedure above (stopping at G=1) doesn't seem right because it doesn't incorporate any information about Y_source. I only plot the load admittance (YL) and rotate to G=1.

My question is: For interstage conjugate matching, should I still rotate ZL until I intersect the G=1 circle? Or is the correct procedure to rotate ZL until the conductance G matches the real part of the target conjugate admittance (G = Re{Y_source})*, which might not be 1?

Using G=1 feels incomplete because the source information isn't used, but I'm having trouble finding clear graphical examples demonstrating rotation to a non-unity G circle specifically for this conjugate matching case.

Any clarification on the correct procedure would be greatly appreciated.

Thanks.