Optimal Impedance Taper in the Presence of Nonlinear Voltage Reflections

A signal introduced into a load-bearing transmission line will partially reflect back along the input line if an impedance mismatch exists at the injection interface. Placement of a tapered impedance transformer at the interface can reduce these reflections. [1] In earlier work we showed how to determine the optimal shape of a continuous lossless taper of wide frequency bandpass. [2] Our theory assumed impedance mismatch sufficiently small that large voltage reflections within the taper could be considered negligible. Here we extend our earlier theory, allowing for arbitrarily sized impedance mismatches and large voltage reflections, reporting results for a lossless taper. Our extended theory has important application to the design of tapered transformers for electronic devices with unavoidably large impedance mismatches, and can be leveraged to construct on-chip Purcell filters used in the fast readout of superconducting transmon and Xmon qubits.
[1] R. W. Klopfenstein, Proc. of the IRE, 31 (1956); D. M. Pozar, Microwave Engineering, 4th ed. (J. Wiley & Sons, New York, 2012) pp. 261-267.
[2] R. P. Erickson, J. of Appl. Physics. 123, 074501 (2018).