which lambda is used while using quarterwave transformer on microstrrip line

To design a simple microstrip patch antenna.
Leff , deltaL , Eeffective is given by Balanis

For 5.2 GHz frequency, er=4.4 h=1.6mm
λ0=300/5.2 mm=57.69mm
λnew=λ0/sqrt(Eeff)=29.28mm



So for drawing quarterwave line which lambda is to used?λ0 OR λnew?
In HFSS for airbox at quarterwave distance from model is taken, which quarterwave distance should be used?
Effective Length of patch is given by λnew/2
Why in Width we do not use effective width and physical width as in length. Is it only becuase of radiating/non radiating edge?

The quarter wavelength line refers to a quarter wavelength in the transmission line (your λ_new). Similarly, the quarter wavelength in free space refers to λ_0.

In microwave engineering, many properties scale proportionally with electrical length, hence that is often used instead of physical length.

Hi,
The airbox would be in λ0.
Width (and also height) are very small relative to the wavelength.
Furthermore, these equations assume propagation of the fundamental modes of patch and transmission line/microstrip.
These fundamental modes do not vary spatially in W and h direction of the coordinate system.