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Fast Harmonic Balance of SIS Mixers with Multiple Junctions and Superconducting Circuits

F. Rice, John S. Ward, Jonas Stasys Zmuidzinas, Goutam Chattopadhyay

Published Mar 1, 1999• 12 citations
OpenAlex

Abstract

We have extended the spectral domain harmonic balance method of Withington and Kollberg [1] to handle quasiparticle mixer circuits of arbitrary complexity containing any number of superconducting tunnel junctions. Our algorithm uses a multidimensional Newton-Raphson technique to achieve rapid convergence, even when many harmonics are included in the analysis. Another improvement is to perform all linear circuit calculations using the scattering matrix representation typical of modern circuit modeling techniques. This algorithm has been implemented as part of an extensive C++ class library for modeling the signal and noise performance of superconducting heterodyne receivers (SuperMix). Using the common, free C++ compiler "g++" (Free Software Foundation), impressive results can be achieved on modern workstations of moderate capability. Such speeds will make complex nonlinear optimization calculations a routine part of SIS receiver design.

Related papers

Simulation of the superconducting quasiparticle mixer using a five-port model

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A harmonic Newton technique has successfully been used in the simulation of superconducting quasiparticle mixers in a five-port model. The steady-state large-signal waveform impressed across the tunnel junction by the incident local oscillator (LO) power is computed in the frequency domain. The three-frequency approximation provides the initial trial solution, which, together with the superlinear convergence property of Newton's method, ensures rapid convergence. A wide range of simulations based on the response function of a real superconductor-insulator-superconductor (SIS) junction is presented. The case of an SIS harmonic mixer (n=2) is also studied. Using a five-port model, the simulation explicitly takes the effect of the first harmonic into account. The algorithm and the principal results obtained for a hypothetical mixer incorporating the measured current-voltage characteristics of a real (SIS) junction are presented. The first results indicate that harmonic quasiparticle mixing may be very attractive in the submillimeter-wave band.>

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An algorithm was developed for calculating the harmonic performance of superconducting quasiparticle millimeter-wave mixers. The scheme uses harmonic balance to determine the steady-state waveform of the large-amplitude voltage which is induced across the tunnel junction by the local oscillator source. A key feature of the algorithm is that the large-signal tunneling-current calculations are done in the frequency domain rather than in the time domain, and this approach leads to numerically efficient computer simulations. The superiority of the spectral-domain method is particularly pronounced when modeling mixers that incorporate high-quality tunnel junctions with very sharp DC nonlinearities. A simplified mixer simulation was performed to determine the range of omega CR products for which harmonic effects are likely to be important.>

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To assist the development of superconducting millimetre and submillimetre-wave receivers, the authors have devised algorithms and written in the C programming language modular software for analysing the small-signal, large-signal and noise behaviour of SIS tunnel-junction circuits. Typical applications include the modelling of mixers, subharmonic mixers and direct detectors. To analyse the complex behaviour of a nonlinear SIS tunnel-junction circuit, it is first necessary to model the large-signal steady-state dynamics of the circuit; the underlying system of equations can then be perturbed, either numerically or analytically, to gain information about the linear relationships between signal and noise variables. A general-purpose simulator must therefore be capable of solving large systems of coupled nonlinear equations. >

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We present a new software package, called QMix, for simulating the behavior of superconductor/insulator/ superconductor (SIS) mixers. The software uses a harmonic balance procedure to calculate the ac voltage across the SIS junction and multitone spectral domain analysis to calculate the quasiparticle tunneling currents. This approach has two major advantages over other simulation techniques: 1) it can include an arbitrary number of higher order harmonics, and 2) it can simulate the effect of multiple strong nonharmonic frequencies. This allows the QMix software to simulate a wide range of SIS mixer operation, including the effects of harmonics in the local-oscillator (LO) signal and gain saturation with respect to the RF signal power. In this article, we compare the simulated results from QMix to the measured performance of a 230-GHz SIS device, both to validate the software and to investigate the experimental data. To begin, we simulated the conversion gain of the mixer and we found excellent agreement with the experimental results. We were also able to recreate the broken photon step effect by adding higher and lower order harmonics to the LO signal. We then simulated a range of incident RF signal powers in order to calculate the gain saturation point. This is an important metric for SIS mixers because we require linear gain for reliable measurements and calibration. In the simulated results, we found both gain compression and gain expansion, which is consistent with other studies. Overall, these examples demonstrate that QMix is a powerful software package that will allow researchers to simulate the performance of SIS mixers, investigate experimental results, and optimize mixer operation. We have made all of the QMix software open-source and we invite others to contribute to the project.

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European Microwave Conference (EuMC)•Oct 1, 1990•0 citations
We have extended the spectral domain harmonic balance method of Withington and Kollberg [1] to handle quasiparticle mixer circuits of arbitrary complexity containing any number of superconducting tunnel junctions. Our algorithm uses a multidimensional Newton-Raphson technique to achieve rapid convergence, even when many harmonics are included in the analysis. Another improvement is to perform all linear circuit calculations using the scattering matrix representation typical of modern circuit modeling techniques. This algorithm has been implemented as part of an extensive C++ class library for modeling the signal and noise performance of superconducting heterodyne receivers (SuperMix). Using the common, free C++ compiler "g++" (Free Software Foundation), impressive results can be achieved on modern workstations of moderate capability. Such speeds will make complex nonlinear optimization calculations a routine part of SIS receiver design.
A harmonic Newton technique has successfully been used in the simulation of superconducting quasiparticle mixers in a five-port model. The steady-state large-signal waveform impressed across the tunnel junction by the incident local oscillator (LO) power is computed in the frequency domain. The three-frequency approximation provides the initial trial solution, which, together with the superlinear convergence property of Newton's method, ensures rapid convergence. A wide range of simulations based on the response function of a real superconductor-insulator-superconductor (SIS) junction is presented. The case of an SIS harmonic mixer (n=2) is also studied. Using a five-port model, the simulation explicitly takes the effect of the first harmonic into account. The algorithm and the principal results obtained for a hypothetical mixer incorporating the measured current-voltage characteristics of a real (SIS) junction are presented. The first results indicate that harmonic quasiparticle mixing may be very attractive in the submillimeter-wave band.>
An algorithm was developed for calculating the harmonic performance of superconducting quasiparticle millimeter-wave mixers. The scheme uses harmonic balance to determine the steady-state waveform of the large-amplitude voltage which is induced across the tunnel junction by the local oscillator source. A key feature of the algorithm is that the large-signal tunneling-current calculations are done in the frequency domain rather than in the time domain, and this approach leads to numerically efficient computer simulations. The superiority of the spectral-domain method is particularly pronounced when modeling mixers that incorporate high-quality tunnel junctions with very sharp DC nonlinearities. A simplified mixer simulation was performed to determine the range of omega CR products for which harmonic effects are likely to be important.>
To assist the development of superconducting millimetre and submillimetre-wave receivers, the authors have devised algorithms and written in the C programming language modular software for analysing the small-signal, large-signal and noise behaviour of SIS tunnel-junction circuits. Typical applications include the modelling of mixers, subharmonic mixers and direct detectors. To analyse the complex behaviour of a nonlinear SIS tunnel-junction circuit, it is first necessary to model the large-signal steady-state dynamics of the circuit; the underlying system of equations can then be perturbed, either numerically or analytically, to gain information about the linear relationships between signal and noise variables. A general-purpose simulator must therefore be capable of solving large systems of coupled nonlinear equations. >
We present a numerical procedure for simulating the non-linear behaviour of Superconductor-Insulator-Superconductor (SIS) tunnel-junction mixers. Mixers of this kind are used extensively in high-performance submillimetre-wave astronomical receivers. The procedure is complete in that both the signal and local-oscillator voltages can be at high levels, with significant amounts of harmonic content; also, the full quantum mechanical behaviour of the tunnel junction is taken into account. The initial simulations show that multi-tone analysis can be carried out in a numerically efficient manner, and that saturation, and the attendant generation of IF harmonics, can be modelled rigorously. This work constitutes the first full non-linear model of an SIS mixer, and the theoretical and numerical procedures described, will be of great importance when designing the next generation of heterodyne astronomical receivers.
We present a new software package, called QMix, for simulating the behavior of superconductor/insulator/ superconductor (SIS) mixers. The software uses a harmonic balance procedure to calculate the ac voltage across the SIS junction and multitone spectral domain analysis to calculate the quasiparticle tunneling currents. This approach has two major advantages over other simulation techniques: 1) it can include an arbitrary number of higher order harmonics, and 2) it can simulate the effect of multiple strong nonharmonic frequencies. This allows the QMix software to simulate a wide range of SIS mixer operation, including the effects of harmonics in the local-oscillator (LO) signal and gain saturation with respect to the RF signal power. In this article, we compare the simulated results from QMix to the measured performance of a 230-GHz SIS device, both to validate the software and to investigate the experimental data. To begin, we simulated the conversion gain of the mixer and we found excellent agreement with the experimental results. We were also able to recreate the broken photon step effect by adding higher and lower order harmonics to the LO signal. We then simulated a range of incident RF signal powers in order to calculate the gain saturation point. This is an important metric for SIS mixers because we require linear gain for reliable measurements and calibration. In the simulated results, we found both gain compression and gain expansion, which is consistent with other studies. Overall, these examples demonstrate that QMix is a powerful software package that will allow researchers to simulate the performance of SIS mixers, investigate experimental results, and optimize mixer operation. We have made all of the QMix software open-source and we invite others to contribute to the project.
The paper summarises preliminary work on the development of a general-purpose superconducting nonlinear-circuit simulator. After contrasting, from a CAD perspective, time and frequency domain descriptions of the macroscopic quantum behaviour of superconducting tunnel junctions, it is reported that the large-signal steady-state dynamics of a tunnel-junction circuit can be conmpletely decomposed by using the method of harmonic balance, and that the resulting system of coupled nonlinear algebraic equations can be solved through harmonic Newton. The principle is illustrated by briefly mentioning an analysis of inadvertent harmonic pumping in quasiparticle mixers.