  A recent experiment [Cohen et al., Science 382, 542 (2023)] observed a
quantized $e^2/2h$ conductance tunneling into a fractional quantum hall edge in
the strong coupling limit, which is consistent with the prediction of a quantum
Hall transformer. We model this quantum point contact (QPC) as a quantum wire
with a spatially-varying Luttinger parameter and a back-scattering impurity and
use a numerical solution to find the wavepacket scattering and microwave (i.e.
finite frequency) has conductance at the interface. We then compare the
microwave conductance to an analytic solution of a Luttinger liquid with an
abruptly varying Luttinger parameter and a back-scattering impurity, which is
obtained by mapping the problem to a boundary sine-Gordon (BSG) model. Finally,
we show that the quantum Hall transformer can survive the inclusion of the
domain wall which is expected to occur in a wide QPC, which is experimentally
relevant, provided the momentum change in the QPC is generated by
electrostatics. We find that back-scattering generated by the domain wall is of
an order of magnitude consistent with the experiment.
