IEEE Solid-States Circuits Magazine - Fall 2021 - 116

loaded quality factor of the transmission
line, the quality factor of the individual
varactors, and the permittivity
modulation index [37]. Although not
discussed in [37], phase noise in the
modulation signal can also impact
the noise performance. Circulators
based on permittivity modulation
can be made to appear externally LTI
and thus avoid the problem of noise
folding, as was shown in [40].
Inspired by how ferrite circulators
operate, a nonreciprocal response
can also be achieved using angularmomentum
biasing of a resonant
ring [38]. An effective electronic spin
is enabled by the spatiotemporal permittivity
modulation with a traveling
wave along the ring (Figure 8). In addition,
resonators can be used to miniaturize
the size of the ring significantly
while boosting the weak permittivity
modulation effect because of the limited
modulation ratio, resulting in a
stronger nonreciprocal response at
the subwavelength scale. The resonators
(bandpass or bandstop) can be
connected together in various forms,
such as delta or wye topologies [39],
[40], [57]. The resonators must exhibit
a high loaded Q-factor to achieve a
significant impedance change, and
this limits the operation bandwidth.
Additionally,
the inductors and the
varactors must have an even higher
unloaded Q-factor to achieve low loss,
which is fundamentally challenging,
particularly on integrated semiconductor
substrates.
Approaches Based on
Permeability Modulation
Similar
to permittivity modulation
based on varactors, parametric
effects based on inductance moduIt
is shown in [58] that the bandwidth
of any pseudo-LTI nonmagnetic
circulator is smaller than twice
the modulating bandwidth. This
implies that, for larger bandwidths,
a higher modulation frequency is
required. However, by increasing ,fm
the dynamic power consumption is
increased. Additionally, the varactors
and the permittivity modulation in
general also exhibit a tradeoff between
the modulation index and loss, particularly
as the operating frequency
is increased. Similar to parametric
approaches, the noise performance
in angular-momentum-based devices
depends on the resonators'
loaded
Q-factor and the unloaded Q-factor of
the varactors. Phase noise in the modulation
signal can also impact the noise
figure by creating random variation in
the harmonic S-parameters. A proper
design of the modulation circuitry
can lower such undesired effects, and
a noise figure close to the insertion
loss can be achieved [39]. The angular
momentum biasing can also be implemented
at mm-wave frequencies, an
example of which is demonstrated in
[59] using three parametric mixers in
a loop. However, the poor Q-factor of
varactors at mm-waves results in high
loss levels.
lation have also been explored in
the past [60]. However, inductance
modulation is very challenging in
IC platforms. Permeability modulation
has also been investigated, specifically
using Josephson junctions
(JJs) and superconducting quantum
interference devices (SQUIDs)
as nonlinear inductors in cryogenic
superconducting platforms [25], [26],
[44], [45]. Nonreciprocal components
play a crucial role in the readout circuitry
of qubits in superconducting
quantum computing systems. Ferrite
circulators cannot be used in such
applications because their bulky
nature limits the number of qubits
that can be operated simultaneously
[61]. Nonmagnetic cryogenic
nonreciprocal circulators and isolators
based on parametric frequency
conversion using JJs and SQUIDs as
the modulated element are becoming
popular in these scenarios [25], [26].
Conductance Modulation
More recently, conductivity modulation
was introduced as an alternative
approach to achieve nonreciprocity
in time-varying systems. In contrast
to the low permittivity modulation
index, a transistor switch can provide
an orders-of-magnitude-higher
conductivity modulation index.
Temporally modulated systems
are commonly known as linear periodically
time-varying (LPTV) circuits.
One of the earliest reports on LPTV circuits
dates back to 1949 [62], in which
P1
P1
ε(t)
ε(t-Tm/3)
Cc
Cc
P3
ε(t-2Tm/3)
(a)
P2
P3
(b)
P2
(c)
FIGURE 8: (a) Spatiotemporal permittivity modulation in a ring resonator. (b) The implementation of a ring resonator by three separate LC
tanks and (c) PCB photo of the implementation in [57].
116
FALL 2021
IEEE SOLID-STATE CIRCUITS MAGAZINE
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IEEE Solid-States Circuits Magazine - Fall 2021

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