  The electronic circulator, and its close relative the gyrator, are invaluable
tools for noise management and signal routing in the current generation of
low-temperature microwave systems for the implementation of new quantum
technologies. The current implementation of these devices using the Faraday
effect is satisfactory, but requires a bulky structure whose physical dimension
is close to the microwave wavelength employed. The Hall effect is an
alternative non-reciprocal effect that can also be used to produce desired
device functionality. We review earlier efforts to use an ohmically-contacted
four-terminal Hall bar, explaining why this approach leads to unacceptably high
device loss. We find that capacitive coupling to such a Hall conductor has much
greater promise for achieving good circulator and gyrator functionality. We
formulate a classical Ohm-Hall analysis for calculating the properties of such
a device, and show how this classical theory simplifies remarkably in the
limiting case of the Hall angle approaching 90 degrees. In this limit we find
that either a four-terminal or a three-terminal capacitive device can give
excellent circulator behavior, with device dimensions far smaller than the a.c.
wavelength. An experiment is proposed to achieve GHz-band gyration in
millimetre (and smaller) scale structures employing either semiconductor
heterostructure or graphene Hall conductors. An inductively coupled scheme for
realising a Hall gyrator is also analysed.
