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 $\Omega$ components. We show that an optimal tapered transformer of length $\ell$, 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 $\omega_o$ 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 $v_g$ is shown to be constant, with characteristic impedance dependent on frequency $\omega_c=\pi v_g / \ell$. While these solutions inhibit input-line reflections only for frequency $\omega_o$, a subset of optimal lossless transformers with $\omega_o$ significantly detuned from $\omega_c$ does exhibit a wide bandpass. Specifically, by choosing $\omega_o\rightarrow 0$ ($\omega_o\rightarrow\infty$), we obtain a subset of optimal low-pass (high-pass) lossless tapers with bandwidth $(0,\sim\omega_c)$ ($(\sim\omega_c,\infty)$). 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.