Variational theory of the tapered impedance transformer

Superconducting amplifiers are key components of modern quantum information circuits. To minimize information loss and reduce oscillations a tapered impedance transformer of new design is needed at the input/output for compliance with other 50 Ω components. We show that an optimal tapered transformer of length ℓ, joining amplifier to input line, can be constructed using a variational principle applied to the linearized Riccati equation describing the voltage reflection coefficient of the taper. For an incident signal of frequency ωo the variational solution results in an infinite set of equivalent optimal transformers, each with the same form for the reflection coefficient, each able to eliminate input-line reflections. For the special case of optimal lossless transformers, the group velocity vg is shown to be constant, with characteristic impedance dependent on frequency ωc=πvg/ℓ. While these solutions inhibit input-line reflections only for frequency ωo, a subset of optimal lossless transformers with ωo significantly detuned from ωc does exhibit a wide bandpass. Specifically, by choosing ωo→0 (ωo→∞), we obtain a subset of optimal low-pass (high-pass) lossless tapers with bandwidth (0,∼ωc) ((∼ωc,∞)). From the subset of solutions we derive both the wide-band low-pass and high-pass transformers, and we discuss the extent to which they can be realized given fabrication constraints. In particular, we demonstrate the superior reflection response of our high-pass transformer when compared to other taper designs. Our results have application to amplifier, transceiver, and other components sensitive to impedance mismatch.