# Geometric Characterization of Many Body Localization

W. N. Faugno, Tomoki Ozawa  Advanced Institute for Materials Research (WPI-
AIMR), Tohoku University, Sendai 980-8577, Japan

###### Abstract

Many body localization (MBL) represents a unique physical phenomenon,
providing a testing ground for exploring thermalization, or more precisely its
failure. Here we characterize the MBL phase geometrically by the many-body
quantum metric (MBQM), defined in the parameter space of twist boundary. We
find that MBQM scales linearly as a function of the inverse system length in
the MBL phase, and grows faster in the ergodic phase. We validate our theory
using the disordered hardcore Bose-Hubbard model, and characterize the ergodic
to MBL phase transition via the localization length scale defined from the
MBQM. MBQM provides an intuitive and experimentally accessible method to
characterize MBL phases.

Advancements in highly controllable experimental platforms, such as optical
lattices[1, 2, 3, 4], trapped ions[5], and Rydberg atom arrays[6, 7], have
allowed for realizations of well isolated quantum systems in the lab where
quantum dynamics and non equilibrium physics can be observed [8, 9]. Of
particular interest are nonergodic systems, which do not thermalize within a
finite time. These systems can host stable phases of matter and help to
understand thermalization mechanisms [10]. Thus far the only known robust
nonergodic phase is many body localization (MBL), which arises from the
interplay of disorder and interactions [11, 12]. In contrast, other nonergodic
phases, such as non-interacting phases, quantum scars or integrable models,
typically thermalize with the addition of some perturbation [13].

Anderson localization was the first identified nonergodic system wherein
disorder in a non-interacting system localizes all eigenstates [14]. Following
studies saw the inclusion of interactions to determine the stability of the
ground state in this localized phase, finding that indeed localization
persisted in several regimes [15, 16, 17]. More recent studies have found that
excited states, even those corresponding to infinite temperature, can remain
localized, resulting in a phenomenon termed MBL [18, 19, 20].

MBL hosts many interesting properties beyond those observed in its non-
interacting counterpart and demonstrates a unique dynamical phase transition
between the localized and delocalized regions. The phase boundary can be
determined through the entanglement entropy scaling as a function of system
size, as in the delocalized region entanglement entropy grows with volume
while in the localized phase the entanglement follows an area law [21].
Additionally the entanglement entropy growth of initial product states has
been shown to grow logarithmically, in contrast to the quickly saturating
entanglement entropy in non-interacting Anderson insulators [22]. The unique
behavior of the entanglement entropy is related to the localization that
occurs in Fock space in addition to the real space localization. This has been
shown to be related to the emergence of a set of quasilocal integrals of
motion (LIOM), which has allowed for the construction of phenomenological
models of MBL[23]. Strong evidence of MBL has been found in the disordered
Bose-Hubbard [24, 25] and Fermi-Hubbard models [26] in 1D as well as quasi-
disordered lattice systems with non commensurate periodic external potentials
[27].

An important parameter in understanding insulating states is the
characteristic localization length. To present, there have been several
proposed approaches to extracting the localization length in MBL systems. For
example, a localization length can be defined in terms of the inverse
participation ratio obtained from the single particle density matrix [28].
Other methods rely on phenomenological models constructed from the extensive
set of LIOM, which introduces a hierarchy of length scales[29, 30]. The LIOM
have been constructed explicitly for the XXZ model [31], the dirsordered
hardcore Fermi-Hubbard model [32, 33], and the Heisenberg model[34]. Although
there are several different localization lengths that appear in MBL, their
physical meaning or experimental signature is often unclear.

A natural localization length has been defined in the modern theory of
insulators, built upon the theory of polarization [35, 36, 37, 38, 39, 40]. In
the modern theory of the insulator, the conductivity of a state is related to
the variance in the position as defined in periodic systems to be the so-
called many-body quantum metric (MBQM). If in the thermodynamic limit the MBQM
diverges, then the system is a conductor while if it saturates the state is an
insulator. This has been applied in several paradigmatic contexts, including
quantum Hall insulators[41], Mott insulators[42], and Chern insulators[43].
MBQM is also experimentally observable through the response of the system upon
a periodic modulation [44].

In this work, we discuss that a natural characteristic length scale for
localization in real space can be defined through the MBQM in the MBL phase.
We find that the MBQM in MBL phase depends linearly on $1/N$, where $N$ is the
number of particles, whereas it depends more rapidly on $1/N$ in the ergodic
phase. By extrapolating MBQM to the thermodynamic limit, we can extract a
characteristic localization length. We calculate the MBQM as a function of
interaction and disorder strength to construct a phase diagram of the
disordered hardcore Bose-Hubbard model.

The many-body quantum metric (MBQM) -  The MBQM is a geometrical property of
many-body quantum states defined in the parameter space of twisted boundary
conditions. We focus on one-dimensional systems in this paper. Having a
twisted boundary condition with phase $\theta$ is equivalent to having a flux
$\theta$ inserted through the system with the wavefunction obeying the
periodic boundary condition without twist [45]. Let us denote the Hamiltonian
of the system with a flux $\theta$ inserted by $H(\theta)$. The MBQM, which we
denote by $g(\theta)$, for a state $|\phi_{0}\rangle$, is then defined as [46,
47]

$g(\theta)=\sum_{n\neq
0}\frac{\langle\phi_{0}|\partial_{\theta}H(\theta)|\phi_{n}\rangle\langle\phi_{n}|\partial_{\theta}H(\theta)|\phi_{0}\rangle}{(E_{0}-E_{n})^{2}},$
(1)

where the sum is over all the eigenstates $|\phi_{n}\rangle$, whose
corresponding eigenenergies are $E_{n}$, which are different from the state
under consideration, $|\phi_{0}\rangle$. (We note that both $|\phi_{n}\rangle$
and $E_{n}$ depend on $\theta$ in general.) The MBQM has been used to
characterize conductor-insulator transitions as in the thermodynamic limit, it
goes to a fixed value in an insulator while it diverges in a conductor [37,
48]. This difference in the bahavior of the MBQM for conductors and insulators
motivates us to characterize the MBL using MBQM. When the system is large
enough, we can expect that $g(\theta)$ does not depend on $\theta$ following
arguments similar to those in Refs. [45, 37, 49]; we will thus consider MBQM
at $\theta=0$ in the following discussion without loss of generality.

The behavior of the $g$ in the localized and delocalized regime displays
markedly different scaling properties. When the wavefunction is fully
localized within a finite region in a space, one can show that
$g=\mathrm{Var}(X_{CM})/L^{2}$ for a system with length $L$ and
$X_{CM}\equiv\sum_{n}x_{n}$ with $x_{n}$ the position operator for the $n$-th
particle [37]. Here we use the notation that for variables $X_{1}$ and
$X_{2}$, the variance is defined by $\text{Var}(X_{1}):=\langle
X_{1}^{2}\rangle-\langle X_{1}\rangle^{2}$ and the covariance, which we use
later, is defined as $\text{Cov}(X_{1},X_{2}):=\langle
X_{1}X_{2}\rangle-\langle X_{1}\rangle\langle X_{2}\rangle$ where the brackets
correspond to the expectation value.

When the wavefunction is not fully localized in a finite region, defining the
position operator $X_{CM}$ under the periodic boundary condition is a tricky
issue, but $L^{2}g$ can still be used to represent $\mathrm{Var}(X_{CM})$, or
the second cumulant of $X_{CM}$, as discussed in detail in [37]. We are then
tempted to write, by the indistinguishability of particles,

$\displaystyle L^{2}g=\text{Var}(X_{CM})\sim\sum_{i,j}\left(\langle
x_{i}x_{j}\rangle-\langle x_{i}\rangle\langle x_{j}\rangle\right)$
$\displaystyle\sim
N^{2}\left(\left[\text{Var}(x_{1})-\text{Cov}(x_{1},x_{2})\right]/N+\text{Cov}(x_{1},x_{2})\right).$
(2)

However, we note that this equation is not rigorously established, as we have
not defined the position operator $x_{i}$ for the $i$-th particle in the
periodic boundary condition. Rather, we can use the equation Eq. (2) as a
guiding principle to understand the scaling behavior of $g$ in large $N$ limit
when one is in MBL phase where variance and covariance of particle positions
are expected to take finite values. What Eq. (2) suggests, which we indeed
later numerically confirm, is that $L^{2}g/N^{2}$ scales linearly in $1/N$
with a constant offset term, analogous to $\text{Cov}(x_{1},x_{2})$. On the
other hand, if one is in ergodic region, we cannot expect the variance of
particle positions to remain finite, and thus we cannot expect $L^{2}g/N^{2}$
to scale linearly in $1/N$. We shall see later that the $1/N$ dependence of
$L^{2}g/N^{2}$ in the ergodic phase is much more rapid than that in the MBL
phase.

The Disordered hardcore Bose-Hubbard model -  As a concrete example, we use
the disordered hardcore Bose-Hubbard model at half-filling to explore the
relation between the MBQM and ergodic-MBL phase transition. The Hamiltonian is
given by

$H=-t\sum_{j}a^{\dagger}_{j+1}a_{j}+a^{\dagger}_{j}a_{j+1}+\sum_{j}\mu_{j}n_{j}+U\sum_{j}n_{j}n_{j+1}$
(3)

where $a^{\dagger}_{j},a_{j}$ are Bosonic creation and annihilation operators
on site $j$ and $n_{j}=a^{\dagger}_{j}a_{j}$ is the density operator. The
parameters $t$ and $U$ control the hopping and nearest neighbor interaction
strengths, respectively, and $\mu_{j}$ is the disordered onsite potential
taken from a random uniform distribution on the interval $[-W,W]$. This model
has been shown to host an MBL phase in limit of large disorder [50]. We will
fix $t=1$ and study the phase diagram as a function of $U$ and $W$. For
evaluation of the MBQM as defined in Eq. 1, the required derivative is

$\left.\partial_{\theta}H(\theta)\right|_{\theta=0}=-\frac{it}{L}\sum_{j}a^{\dagger}_{j+1}a_{j}-a^{\dagger}_{j}a_{j+1}$
(4)

obtained by taking periodic boundary conditions and threading a flux $\theta$.

The MBQM for One Particle -  Before considering the full problem at half-
filling, let us demonstrate the relation between the MBQM and the localization
for a single particle. When there is only one particle, the disordered Bose-
Hubbard model is just the Anderson model, where all the eigenstates are
localized for any disorder strength $W$. As stated previously, the MBQM is
related to the variance of the particle as $L^{2}g=\text{Var}(x)$, when the
wavefunction is localized. To understand the behavior of $L^{2}g$ when the
wavefunction is not localized, we calculate $L^{2}g$ for various lengths and
disorder strengths for the Anderson model. For a given disorder realization,
we calculate single-particle eigenstates and calculate $L^{2}g$ for a randomly
selected state. We repeat the procedure for $10^{6}$ realizations and take the
median value of $L^{2}g$. (We can also take the average here, instead of the
median; for the MBL phase at half filling, however, as we discuss below, we
find that median converges much faster than the average.) In Fig 1, we plot
the median of $L^{2}g$, for system sizes $L=30,40,50$ and $60$ versus the
disorder strength $W$. We find that above approximately a disorder strength of
$W\sim 1$, values of $L^{2}g$ for different lengths match, and agree with the
directly calculated values of $\text{Var}(x)$ assuming the open boundary 111In
order to calculate $\text{Var}(x)/L^{2}$, we first calculate an eigenstate of
the Anderson model under the periodic boundary condition, and then calculate
the center-of-mass of the wavefunction, which can be done using the framework
of Ref. [35]. We then shift the coordinate so that the center of mass is in
the middle of the one-dimensional chain, and we calculate
$\text{Var}(x)=\langle x^{2}\rangle-\langle x\rangle^{2}$ taking $-L/2\leq
x\leq L/2$. Namely, while we calculate the eigenstate under the periodic
boundary condition, we calculate $\text{Var}(x)$ assuming the open boundary
condition.. On the other hand, below the disorder $W\sim 1$, we observe that
$L^{2}g$ varies with the system length, and they also deviate from
$\text{Var}(x)$ calculated under the open boundary condition. The deviation
between $L^{2}g$ and $\text{Var}(x)$ happen at smaller values of $W$ for
systems with longer $L$. This length dependence of $L^{2}g$ and discrepancy
between $L^{2}g$ and $\text{Var}(x)$ are attributed to the eigenstates
touching the boundary at smaller values of $W$; when the wavefunction is
spread over the entire finite-sized chain, $\text{Var}(x)$ calculated under
the open boundary condition and $L^{2}g$ do not need to coincide. We note
that, although $L^{2}g$ loses its direct meaning as $\text{Var}(x)$ when the
wavefunction is extended, it is a perfectly well-defined quantity both in the
localized and extended region. An important observation here is that $L^{2}g$
do not depend on the system size in the localized region, whereas it depends
on the system size in the delocalized region. We confirm similar behavior of
the system-size dependence of MBQM for two-particle systems, as discussed in
detail in the Supplemental Material. We will see below that similar behavior
holds for disordered Bose-Hubbard model at half filling; $L^{2}g/N^{2}$ has
weak dependence on $L$ in the MBL phase whereas it depends on $L$ more
strongly in the ergodic phase.

Figure 1: MBQM ($L^{2}g$) as a function of the disorder strength $W$ for
single particle systems with different system lengths $L$. The plotted values
are the median of $10^{6}$ disorder realizations. Dashed lines are the medians
of $\text{Var}(x)$ calculated assuming the open boundary condition. These
quantities agree when the disorder strength is larger than approximately
$W\sim 1$, which corresponds to the region with well localized wavefunction,
and agreement improves as the system size increases. Figure 2: Median MBQM
($L^{2}g/N^{2}$) as a function of interaction strength $U$ for different
disorder strengths and different system sizes. Disorder strengths from left to
right are $W=4.5$, 6.0, and 7.5. The shadings correspond to regions where the
quantity $\ell=\lim_{N\to\infty}L\sqrt{g}/N$ is greater than the proposed
cutoff as labelled on the plot for $W=4.5$.

Bose-Hubbard model at Half-filling - Now we consider the MBQM of the half-
filled disordered Bose-Hubbard model. In our calculation we shift the energies
to $E-(E_{max}+E_{min})/2$ and consider the state with energy closest in
magnitude to 0, which corresponds to infinite temperature in the thermodynamic
limit. Here $E_{max}$ and $E_{min}$ are the largest and lowest eigenenergies
in the spectrum of a given disorder realization, respectively. We perform
computation over $10^{5}$ disorder realizations for $L=8$, $10^{4}$
realizations for $L=10,12$, and $10^{3}$ realizations for $L=14$. We consider
the median of the disorder as the average does not always converge within
these number of realizations due to the appearance of configurations with
anomalously high MBQM, resulting from accidental near degeneracies. We note
that, at half filling, the many-body wavefunction is always spread over the
entire system; we show in Supplemental Material that the density distribution
is indeed nonzero on all sites for typical states, even in the MBL phase. We
will show that MBQM, extrapolated to the thermodynamic limit, nonetheless
provides a meaningful characteristic localization length.

In Fig. 2, we present the median MBQM, scaled as $L^{2}g/N^{2}$, as a function
of interaction strength for different system sizes. We note that at half
filling, $N/L=1/2$, which means that $L^{2}g/N^{2}=4g$. We choose to
explicitly write as $L^{2}g/N^{2}$ in this section, rather than $4g$, to show
its $L$ and $N$ dependence. Each panel corresponds to a different disorder
strength. We see that, for $W=4.5$ and 6, $L^{2}g/N^{2}$ has weak dependence
on $L$ at small and large values of $U$ while it grows much more rapidly with
$L$ at the medium values of $U$. For $W=7.5$, $1/N$ dependence of
$L^{2}g/N^{2}$ always seems to be weak. Previous works have found that this
model shows MBL both at small and large values of $U$ when $W$ is not too
large, and enters into ergodic phase at the intermediate values of $U$. When
$W$ is large, the ergodic phase does not appear [52, 53]. We can thus identify
the region where $L^{2}g/N^{2}$ depends weakly on $L$ as the MBL phase, and
the region with strong $L$ dependence as the ergodic phase; we will later show
more quantitative comparison of the phase diagram with the existing theory of
MBL. We note that the strong system size dependence of $L^{2}g/N^{2}$ in the
ergodic phase is reminiscent of the behavior of entanglement entropy
previously observed [21]. The shadings in Fig. 2 correspond to different
proposed phase boundaries that we explain below.

The re-entrant MBL phase transition as $U$ is increased further beyond the
ergodic phase has been found in earlier works [26, 52, 53]. To our knowledge,
no property to distinguish the MBL phases at small and large $U$ in the
hardcore Bose-Hubbard model has been found, but there have been proposals of
MBL phases with different clustering properties when the hardcore constraint
is relaxed [54]. Here we find that the scaling behavior of $L^{2}g/N^{2}$ can
distinguish the two phases; for low $U$, $L^{2}g/N^{2}$ grows with system size
whereas for large $U$, $L^{2}g/N^{2}$ decreases with the system size. Our
finding suggests that the mechanisms behind the MBL phase in small and large
$U$ regimes have different origins.

We now quantify the scaling behavior of the MBQM by fitting $1/N$ dependence
of $L^{2}g/N^{2}$, which from Eq. 2 we expect to scale linearly in the MBL
phase. In Fig 3, we present two such example fittings in the ergodic (left)
and MBL (right) phases to illustrate their differences. In the MBL phase,
linear fitting well approximates the numerically obtained $L^{2}g/N^{2}$ for
four different values of $N$, and quadratic fitting does not improve the
fitting very much. On the other hand, in the ergodic phase, linear fitting and
quadratic fitting yield a big difference. Furthermore, from the slope of the
fit we observe that $1/N$ dependence in the MBL regime is much weaker than
that in the ergodic regime, confirming the finding from Fig. 2. In Fig. 3, we
chose cases where the difference in fitting behaviors between the ergodic and
MBL phases is visibly apparent. More generally, it is not always easy to find
the difference in the ergodic and MBL phases by eye. We have thus developed a
more quantitative way to analyze the difference between the linear and
quadratic fittings through their least squared residuals. We find that the
fitting residuals between linear and quadratic fits is qualitatively smaller
in the MBL regime than in the ergodic regime; more detailed analysis on linear
and quadratic fittings is provided in the Supplemental Material. The accuracy
of our analysis is limited by the few number of system sizes achievable with
exact diagonalization; better quantitative comparison between linear and
quadratic fits is possible if simulation, or experimental realization, with
longer system sizes is available.

Figure 3: Linear and quadratic fittings of $L^{2}g/N^{2}$ as a function of
$1/N$ in the extended (left) and MBL (right) phases. Blue solid (red dashed)
lines correspond to linear (quadratic) fitting.

From Eq. (2), we expect $L^{2}g/N^{2}$ to scale linearly in $1/N$ in the MBL
phase, which has been confirmed by the numerical results presented in Figs. 2
and 3 above. Taking the linear thermodynamic limit, we can define the quantity

$\displaystyle\ell=\lim_{N\to\infty}L\sqrt{g}/N.$ (5)

which in the MBL phase can be considered a localization, analogous to the
positional covariance, $\text{Cov}(x_{1},x_{2})$. We have calculated and
plotted $\text{log}_{10}\ell$in the $U-W$ plane in Fig. 4. Fixing the
interaction strength $U$, and increasing the disorder $W$, we see that $\ell$
decreases, showing stronger localization as $W$ increases. For fixed disorder
$W$, the behavior is more subtle; as $U$ is increased, $\ell$ increases first,
showing that the system delocalizes, but after some maximum value, $\ell$
starts to decrease showing re-entrant localization transition.

To illustrate that we can determine a phase boundary from $\ell$ similar to
those previously proposed, we plot several contours corresponding to
$\ell=0.25$, 0.5, 0.75, and 1.0 alongside a dotted phase boundary according to
the method given in Ref. [26]. Given the crossover nature of the transition
between the two phases, we cannot draw a definitive phase boundary between the
phases. Nonetheless, identifying the length scale $\ell$ to be a
characteristic localization length, it is reasonable to assume $\ell$ in the
MBL phase to be somewhat around or less than the lattice spacing at half
filling. Indeed we find that the contours corresponding to 0.75 and 0.5 match
quite well with the phase boundary previously found from the eigenvalue
spacing statistics in Ref. [26], supporting our argument that the phase
transition can be captured by the MBQM. (A complementary phase diagram based
on fitting residuals is presented in the Supplemental Material.) In Fig. 5, we
present $\ell$ as a function of interaction strength for three disorder
strengths to further elucidate the trends observed in our phase diagram.
Shadings again correspond to the different boundaries based on the contours in
the phase diagram.

Figure 4: The MBL phase diagram: log of the characteristic length,
$\log_{10}\ell$, plotted as a function of interaction $U$ and the disorder
strength $W$. Solid green curves are labelled with the contour of equal values
of $\ell$. The dotted blue contour corresponds to the predicted phase diagram
in Ref. [26], demonstrating that $\ell\approx 0.5$ or 0.75 sets the boundary
between the ergodic and MBL phases. Figure 5: Characteristic localization
length $\ell$ as a function of the interaction strength $U$ for three disorder
strengths. Shaded region correspond to the ergodic phases determined by the
cutoff $\ell=0.5$ or 0.75.

Conclusion - To summarize, we have shown that the system size dependence and
the thermodynamic limit of MBQM provide an excellent means to distinguish the
MBL phase from ergodic phase for the disordered Bose-Hubbard model. In
particular, $\ell\equiv\lim_{N\to\infty}L\sqrt{g}/N$ provides a natural length
scale to characterize the localized phase. The scaling behavior is reminiscent
of the entanglement entropy scaling and the resulting phase boundaries agree
with those previously proposed from other properties of the MBL.

The MBQM offers insight into further understanding the MBL phase.
Experimentally it has been proposed that the MBQM can be extracted from the
participation ratio in a lattice shaking experiment [44]. The MBQM should
allow for a straightforward and general method for detecting the presence of
MBL in a variety of systems, including fermionic and spin systems as well as
MBL systems in the absence of disorder. Moving beyond exact diagonalozation
methods to calculate the MBQM for larger systems will allow for improved
accuracy in determining the phase boundary as we expect our localization
length to scale completely differently in the two phases. In higher
dimensions, the MBQM has a tensorial structure reflecting the localization of
many-body quantum systems in different directions; such a higher dimensional
MBQM may be able to determine the fate of MBL in higher dimensions.

##  I Acknowledgements

We thank Ryusuke Hamazaki and Henning Schomerus for discussions on many-body
localization. This work has been supported by JSPS KAKENHI Grant Number
JP20H01845, JST PRESTO Grant No. JPMJPR2353, JST CREST Grant Number
JPMJCR19T1.

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##  II Supplementary Material

###  II.1 Two Particle Results

As in the single particle case, MBQM ($L^{2}g$) and the variance of center-of-
mass position of two particles agree when the wavefunction is well localized
in real space. In Fig 6, we plot the median of the MBQM and the variance in
the center of mass vs the interaction strength $U$ for two different disorder
strengths. We see that the MBQM and variance agree well so long as the system
is large enough and the disorder is strong enough, suggesting that we can
attribute this discrepancy to finite size effects in complete agreement with
the single particle results.

We now extract the single particle variance, $\text{Var}(x_{1})$ and
covariance $\text{Cov}(x_{1},x_{2})$ from the variance of the center of mass,
$X_{CM}=x_{1}+x_{2}$ and of the relative coordinate $X_{R}=x_{1}-x_{2}$ and
how these quantities compare with the MBQM. The variance and covariance are
proportional to the sum and difference of $\text{Var}(X_{CM})$ and
$\text{Var}(X_{R})$, respectively. In Fig 7, we present the average variance,
$\text{Var}(x_{1})=(\text{Var}(X_{CM})+\text{Var}(X_{R}))/4$, and covariance,
$\text{Cov}(x_{1},x_{2})=(\text{Var}(X_{CM})-\text{Var}(X_{R}))/4$, we have
directly calculated as well as the MBQM per particle, $L^{2}g/2$. As presented
in the main text for the case of a single particle, the direct calculation of
the variance and covariance is carried out by shifting the coordinates to
reduce boundary effects and calculating as though the system has open boundary
conditions. We observe that the covariance is roughly independent of disorder
and interaction strength, always remaining a small, negative number. The MBQM
per particle and single particle variance are roughly in agreement, suggesting
that the MBQM can capture the single particle variance even for two particle
systems. Unfortunately, this method cannot be directly extended to half-
filling as the definition of the relative coordinate above becomes ill-defined
for more than two particles. The agreement between the single particle
variance and MBQM per particle does not seem to extend to the half-filled
system. The analogue of covariance that we extract, i.e. the characteristic
length $\ell$ is not in general small.

Figure 6: Two particle center of mass variance (solid line) and MBQM (points)
versus interaction strength. Left we plot the results for a disorder strength
of $W=3$ where for smaller systems the two quantities do not agree well, but
become closer as the system size increases. Right depicts the same quantities
at disorder of $W=6$. For this larger disorder all system sizes show agreement
between the variance and the MBQM. Figure 7: Two particle variance (solid
line), covariance (dashed line) and MBQM per particle (points). Colors
correspond to system lengths as shown on the figure. Note that the covariance
is nearly zero compared with the variance and that the MBQM and the variance
agree very well.

##  III Density Profiles

We provide density profiles of the zero energy eigenstate for a realization of
the half-filled disordered hardcore Bose-Hubbard model in Fig. 8. For these
density profiles we have fixed the interaction strength to $U=5$. The profiles
provided correspond to disorder strengths $W=2,4,6,$ and $8$. Note that there
is a qualitative difference between the density profiles when the disorder
increases and the wave function becomes less uniform as the system enters the
MBL phase. Further note that while this qualitative change occurs, the density
is nonzero everywhere in both phases.

Figure 8: Density profiles of the zero energy eigenstate of the disordered
hardcore Bose-Hubbard model. The interaction strength is fixed at $U=5$ and
the disorder is varied from 2 to 8 as labelled.

###  III.1 Additional Detail for Phase Diagram Construction

In this section we present additional results that further support our phase
diagram presented in the main text. Firstly, we present our results for the
MBQM and the localization length for all disorder strengths considered for
completeness in Fig. 9 and in Fig. 10. From these plots it is clear that there
is a change in behavior of the MBQM scaling and in the degree of localization
as one crosses the phase boundary proposed above.

Figure 9: Median MBQM ($L^{2}g/N^{2}$) vs interaction strength for different
disorder strengths for different system sizes. The shadings correspond to the
ergodic phases, which we define as the region where the quantity
$\ell=\lim_{N\to\infty}L\sqrt{g}/N$ is greater than some proposed cutoff. Here
the red hatched region corresponds to $\ell>0.5$ and the shaded yellow region
corresponds to $\ell>0.75$. We identify the unshaded region as MBL phase for
both cutoff choices. Figure 10: Localization length versus interaction
strength for all disorder strengths considered here. The localization length
is extracted from the thermodynamic limit as defined in the main text, i.e.
$\ell=\lim_{N\to\infty}L\sqrt{g}/N$.

In Fig. 11, we provide further examples of the linear and quadratic fittings
in $1/N$ to highlight the subtlety in distinguishing the correct fitting with
the few available system sizes. For some parameters, it is clear that a linear
fitting is much more accurate than the quadratic fitting. Unfortunately, the
preferred fit can be ambiguous for many parameters. This ambiguity can be
attributed to the few number of system sizes available and potentially finite
size effects. While there is clearly a difference in scaling in different
regimes, we find it is best to use a linear fitting across all parameters and
compare the corresponding thermodynamic limit to draw our phase boundary.

Figure 11: Linear and quadratic fittings of the MBQM for a selection of
parameters. We label the predicted phase on each boundary including Anderson
localized (AL) for the case of non-interacting systems alongside the MBL and
ergodic phases. We set the boundary between MBL and ergodic phases at
$\ell=0.5$.

To quantitatively justify our thermodynamic limit, we calculate the least
square residuals for the linear fitting, $S_{lin}$, to quantify the goodness
of fit at each point in the phase diagram. Here the residuals are defined as

$S_{lin}=\frac{\sum_{N}(y_{pred,N}-y_{N})^{2}}{\ell^{2}}$ (6)

where $y_{N}$ is the calculated value of $L^{2}g/N^{2}$ for $N$ particles and
$y_{pred,N}$ is the predicted value for $N$ particles from the linear fit. We
have normalized the residuals by the thermodynamic limit $\ell^{2}$ to make
the comparison between residuals at different parameters meaningful. We plot
$\log_{10}S_{lin}$ in Fig. 12. We see that the least square residuals follow
precisely the same trend as in Fig. 4 in the main text. We have included
contours corresponding to $0.1$, $0.05$, $0.01$, and $0.005$. Note how these
contours agree surprisingly well with the contours drawn above based on the
localization length.

Figure 12: Many Body Localization phase diagram as a function of interaction
$U$ and disorder $W$. Color is obtained as $\log_{10}S_{lin}/\ell$. Green
curves are labelled with the contour of $0.1$, $0.05$, $0.01$, and $0.005$.
The dashed blue contour corresponds to the predicted phase diagram in Ref.
[26], demonstrating our method agrees with previously reported investigations.


# Design Analysis and Experimental Validation of Relaxation Oscillator-Based
Circuit for R-C Sensors

Mohamad Idris Wani1, Sadan Saquib Khan1, Benish Jan1, Meraj Ahmad2, Maryam
Shojaei Baghini3, Laxmeesha Somappa3, and Shahid Malik1 1Centre for Sensors,
Instrumentation, and Cyber Physical System Engineering, Indian Institute of
Technology, New Delhi, India  2James Watt School of Engineering, University of
Glasgow, United Kingdom  3Department of Electrical Engineering, Indian
Institute of Technology Bombay, Mumbai, India<EMAIL_ADDRESS>
###### Abstract.

Relaxation oscillator-based circuits are widely used for interfacing various
resistive and capacitive sensors. The electrical equivalent of most resistive
and capacitive sensors is represented using a parallel combination of resistor
and capacitor. The relaxation oscillator-based circuits are not suitable for
parallel R-C sensors. In this paper, we propose a modified circuit for
parallel R-C sensors. The proposed relaxation oscillator-based circuit is
based on a dual-slope and charge transfer technique to measure the resistance
and capacitance of parallel R-C sensors separately. In addition, the paper
provides a detailed analysis and design considerations for the oscillator
design by taking into account the various sources of non-idealities. A method
to reduce the error by using single-cycle averaging is also introduced. To
verify the analyzed design criteria, the circuit is tested with multiple
operational amplifiers with different non-idealities. Experimental results
verify the performance of the proposed circuit. The circuit is tested for a
range from 10 pF to 42 pF and 100 k$\Omega$ to 1 M$\Omega$ for parallel R-C
sensors with an error of less than 1.5%. The circuit is tested with a
fabricated water-level sensor. The result confirms the efficacy of the
proposed circuit.

Introduction
Resistive and capacitive sensors are very popular and widely used in the
scientific, biomedical, and industrial applications for measurement of force,
displacement, pressure, humidity, temperature, liquid level, and bio-medical
implants [1, 2, 3, 4, 5]. The signal conditioning circuits for these sensors
are generally based on bridge-based circuits, relaxation oscillator-based
resistance/capacitance to frequency converters, and direct digital converters
[6, 7, 8, 9].
Bridge circuits are the fundamental building block for resistance measurement.
Wheatstone bridge-based circuits are preferred for measuring a small variation
in sensor resistance. However, the single-element resistive sensors’ output
voltage is non-linear with respect to the change in the sensor resistance
[10]. This is especially critical when the sensor resistance change is large
compared to the baseline resistance. This affects the sensitivity for wide
dynamic range measurement. On the other hand, the relaxation oscillator-based
circuits provide a wide-dynamic range measurement of sensor resistance by
linearly converting the resistance into frequency.

Most of the signal conditioning circuits for resistive and capacitive sensors
are based on the assumption that the sensor is ideal and can be represented by
either a resistor in the case of the resistive sensors or a capacitor in the
case of the capacitive sensors [6, 9, 11, 12, 13]. However, the assumption is
not valid for many sensing applications. For instance, the resistive sensors
are affected by a parasitic capacitance [2]. Similarly, most of the capacitive
sensors are affected by a leakage resistance [14, 15]. Therefore, the
electrical equivalent circuit for most of the resistive and capacitive sensors
consists of a resistor and capacitor in parallel [2]. In such sensors, one
component carries information about the measurand, while the other component
is present because of the non-ideal nature of the sensor which is
unpredictable. Therefore, its effect should be compensated at the circuit
level, or it degrades the performance of the sensor system. Moreover, in some
sensors, both sensing components are a function of the measurand and need
simultaneous measurement [16, 17].

The signal conditioning circuits for the parallel R-C sensors are mostly based
on the phase-sensitive detection (PSD) technique. The PSD-based techniques
require an in-phase and a quadrature reference signal to separately measure
the resistance and capacitance of the parallel R-C sensors. Auto-nulling and
PSD-based signal conditioning circuits for the parallel R-C sensors are
reported in [18, 19, 20]. These circuits periodically provide the measurement
of the R-C parameters of the sensor. However, it requires analog multipliers
and an auto-nulling loop, which increases the complexity and power
consumption. Another PSD-based signal conditioning circuit for parallel R-C
sensors is reported in [21]. The circuit provides simultaneous measurement of
sensor capacitance and resistance. However, the output of the signal
conditioning circuit is sensitive to many circuit parameters and component
non-idealities, which affects the robustness of the sensor system. Moreover,
the output voltage is sensitive to noise, affecting the measurement’s
resolution.

Thanks to the inherent properties such as the quasi-digital output and high
noise immunity, the relaxation oscillator-based circuits are preferred for
lossless resistive, and capacitive sensors [7, 8, 22, 23] as shown in Fig. 1.
The oscillator-based circuit reported in [23, 24, 25] are designed for the
measurement of larger variation in sensor resistance with parasitic
capacitance. However, the design considerations for the desired accuracy are
not presented in [23, 24]. Another oscillator-based signal conditioning
circuit for parallel R-C sensors is reported in [14]; however, the measurement
range for the capacitance measurement is very small ($<$1 pF). This limits the
range of applications of the circuit reported in [14].

Figure 1. Schematic diagram of the relaxation oscillator circuit for
resistive/capacitive sensor. In the case of an ideal capacitive sensor,
$C_{x}$ is the sensing variable, and $R_{x}$ is the integrator resistance. In
the case of an ideal resistive sensor, $R_{x}$ is the sensing element, and
$C_{x}$ is the integrator capacitance as a reference element.

The initial results of the relaxation-oscillator circuit for capacitance
measurement are reported in [25]. In this paper, we present the extended
analysis, design considerations, experimental validation of the performance
parameters, and sensor testing with the relaxation oscillator-based signal
conditioning circuits. The features of the paper are highlighted as follows.

  * •

The proposed oscillator combines dual-slope and charge-transfer techniques to
separately measure the resistance and capacitance of the parallel R-C sensors.

  * •

Detailed analysis, including various sources of non-idealities, is included in
the paper. The effect of various component non-idealities on the measurement
of sensor parameters is analyzed.

  * •

A single-cycle averaging method is utilized, which significantly reduces the
effects of component non-idealities on the measurement of sensor parameters.

  * •

The design criteria for the selection of components for designing the proposed
signal conditioning circuit are also included in the paper.

  * •

The proposed circuit is tested for different sensing conditions and the
measurement is compared with the derived analytical solution.

Conventional Relaxation Oscillator for Resistive–Capacitive Sensors
The schematic diagram of the relaxation oscillator-based signal conditioning
circuit (with triangular and square wave output) for resistive and capacitive
sensors is shown in Fig. 1. The circuit consists of an integrator and a
Schmitt trigger. The operational amplifier $OA_{1}$ along with the resistance
$R_{x}$ and capacitor $C_{x}$ form the integrator of the oscillator. In the
case of a capacitive sensor, $C_{x}$ is replaced by the sensor element, and
$R_{x}$ is implemented using a known resistor. Similarly, in the case of
resistive sensors, $R_{x}$ is the sensor element (considered to be lossless in
Fig.1), and $C_{x}$ is the reference element. The signal conditioning circuits
based on the conventional relaxation oscillator for resistive and capacitive
sensors are reported in [7, 8, 22].

The integrator output in Fig. 1 is compared with the threshold voltage of the
Schmitt trigger (implemented using $R_{1}$, $R_{2}$, and Op-Amp $OA_{2}$). The
expression for the voltage $V_{R}(t)$ can then be derived as follows.

Figure 2.  Schematic diagram of the proposed modified relaxation oscillator
for parallel R-C sensors

(1) $V_{R}(t)=-\frac{1}{R_{x}C_{x}}\;\int_{0}^{t}V_{p}\;dt$

where $V_{p}$ is the peak voltage at the output of the Schmitt trigger. The
threshold voltage of the Schmitt trigger is given as $\pm
V_{p}\times(R_{1}/R_{2})$. The capacitor $C_{x}$ is initially charged with
voltage ${|V_{p}|}$ at $t=0$. At $t=T/2$, the voltage $V_{R}(t)$ can be
written as follows.

(2)
$V_{R}\left(t=\frac{T}{2}\right)=-\frac{T\;V_{p}}{2\;R_{x}\;C_{x}}=-2\;V_{p}\;\frac{R_{1}}{R_{2}}$

The expression of the oscillation time-period $T$ of the square-wave output
signal $V_{X}(t)$ can be derived as follows.

(3) $T=4\;R_{x}\;C_{x}\;\frac{R_{1}}{R_{2}}$

The expression (3) indicates that the period $T$ of the square-wave output of
the relaxation oscillator circuit is proportional to the sensor variables
(either $C_{x}$ or $R_{x}$ at once). However, the conventional relaxation
oscillator circuit in Fig. 1 is not suitable for the leaky capacitive and
parallel R-C sensors.

The accuracy of the relaxation oscillator based circuits for sensing
applications are highly dependent on the symmetrical output in both positive
and negative polarity. This can either be achieved using a network of Zener
diodes or by using a rail-to-rail operational amplifier for the Schmitt
trigger. Further, a current limiting resistor can also be used to ensure the
safety [26]. In this paper, we are assuming the output voltage of the Schmitt
trigger is symmetrical.

Modified Relaxation Oscillator for Impedance R-C Sensors
The modified relaxation oscillator circuit with the parallel $R_{x}$ and
$C_{x}$ (electrical equivalent of a resistive sensor) at the inverting
terminal of the integrator is shown in Fig. 2. The capacitor $C_{x}$ is active
only during the switching of square-wave voltage $V_{X}(t)$ from low to high
and vice-versa. Charge transfer occurs between the capacitors $C_{x}$ and
$C_{i}$, during each of these transitions. For the rest of the cycle, the
capacitor $C_{i}$ charges/discharges through $R_{x}$ with a linear slope.
$C_{p}$ is the parasitic capacitance of op-amp.

Oscillator Operating Principle The expression for the voltage $V_{R}(t)$ in
Fig. 2 can be derived as follows.

(4)
$V_{R}(t)=\left(\frac{R_{1}}{R_{2}}-\frac{2\;C_{x}}{C_{i}}\right)V_{p}-\int_{0}^{t}\frac{V_{p}}{R_{x}\;C_{i}}dt$

The factor $2C_{x}/C_{i}$ is due to the charge transfer between $C_{x}$ and
$C_{i}$. The sudden charge transfer during the transition affects the initial
voltage at the output $V_{R}(t)$. At $t=T/2$, $V_{R}(t)$ can be simplified as
follows.

(5)
$V_{R}(t)=\left(\frac{R_{1}}{R_{2}}-\frac{2\;C_{x}}{C_{i}}\right)V_{p}-\frac{T\;V_{p}}{2\;R_{x}\;C_{i}}=-V_{p}\frac{R_{1}}{R_{2}}$

Therefore, the oscillation period, $T$ at the output node $V_{x}$ can be
expressed as follows.

(6) $T=4\;R_{x}\;C_{i}\;\left(\frac{R_{1}}{R_{2}}-\frac{C_{x}}{C_{i}}\right)$

The oscillation period $T$ at the output node $V_{x}$ of the circuit in Fig. 2
is proportional to $R_{x}$ and $C_{x}$. However, the expression of time-period
$T$ is not enough to separately estimate both the capacitance $C_{x}$ and
resistance $R_{x}$ of the sensor. Hence, additional information is necessary
to measure both parameters separately.

Figure 3.  Timing diagram of the proposed oscillator shown in Fig. 2

To measure both the parameters $R_{x}$ and $C_{x}$, a zero-crossing-detector
(ZCD) (implemented using $OA_{3}$) and an XOR gate is added to separate the
sensor components ($R_{x}$ & $C_{x}$) into an equivalent pulse width. The ZCD
separates the ramp signal into two parts: The first part is proportional to
the charge transfer, and the second part is proportional to sensor resistance
as represented by $T_{p1}$ and $T_{p2}$ in Fig. 3. The expression for $T_{p1}$
and $T_{p2}$ can be derived as follows.

(7)
$T_{p1}=R_{x}\;C_{i}\left(\frac{R_{1}}{R_{2}}-\frac{2\;C_{x}}{C_{i}}\right)$
(8) $T_{p2}=R_{x}\;C_{i}\;\frac{R_{1}}{R_{2}}$

The expression of unknown impedance $(R_{x},C_{x})$ can be estimated as from
$T_{p1}$ and $T_{p2}$ as follows.

(9) $C_{x}=\frac{R_{1}}{R_{2}}\frac{C_{i}}{2}(1-\frac{T_{p1}}{T_{p2}})$

(10) $R_{x}=\frac{R_{1}}{C_{i}R_{2}}T_{p2}$

Oscillator Performance Analysis with Component Non-idealities
The developed relaxation oscillator circuit’s output periods are affected by
many circuit parameters and must be analyzed to design the circuit for a
desired target resolution. The following non-idealities are considered.

  * •

Op-Amp finite GBW, $\mathrm{A_{0}\omega_{0}}$

  * •

Op-Amp input bias current, $\mathrm{i_{b}}$

  * •

Op-Amp slew rate, SR

  * •

Schmitt trigger and Op-Amp offset voltage, $\mathrm{V_{os}}$

  * •

ZCD comparator offset, $\mathrm{V_{oz}}$

  * •

ZCD comparator response delay, $\mathrm{\tau_{Z,LH},\tau_{Z,HL}}$

  * •

Schmitt trigger response delay, $\mathrm{\tau_{S,LH},\tau_{S,HL}}$

The propagation time of the XOR gate will also affect the periods. The
parasitic capacitance of the XOR gate also contributes to the propagation
time. Usually, the propagation time of the XOR gate is in ns, which is smaller
than the zero crossing detector and Schmitt trigger used in the circuit.
Therefore, the effect of the XOR gate can be considered negligible for the
circuit. Further, the effect of the propagation time of the XOR gate can be
considered in the equation by adding the rise and fall time with the
comparator output.

The steps for analyzing the effect of component nonidealities on the
measurement of sensor parameters are as follows.

  1. (1)

First, we will analyze the effect of the op-amp nonidealities on the slope and
the offset of the integrator. The change in the integration slope and the
effective integrator output offset is the primary concern for the oscillators’
performance.

  2. (2)

Next, we consider the effect of the oscillation period due to the component
non-idealities in the ZCD and the Schmitt trigger with the non-ideal
integrator output applied.

  3. (3)

Finally, we will combine all the non-idealities to derive the oscillation
period at the output of the XOR gate.

The detailed analysis of the component non-idealities is as follows.
Effect of the op-amp non-idealities on the slope and the offset of the
integrator

First, considering finite GBW product, the integrator output voltage when
driven by a unit step voltage $\mathrm{V_{p}}$ can be derived as follows.

(11) $\displaystyle V_{R}(s)$
$\displaystyle=\frac{-V_{p}A_{0}\omega_{0}}{s^{2}R_{x}(C_{x}+C_{p}+C_{i})}\Bigg{(}\frac{1+sC_{x}R_{x}}{s+\Big{(}\dfrac{A_{o}\omega_{0}R_{x}C_{i}+1}{R_{x}(C_{x}+C_{p}+C_{i})}\Big{)}}\Bigg{)}$

(12)
$\begin{split}V_{R}(t)=&V_{p}(\alpha-2X)-\frac{V_{p}}{R_{x}C_{i}}\Bigg{\\{}\overbrace{\frac{t}{\Big{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Big{)}}}^{\text{{Integration
slope reduction due to finite GBW}}}-\\\
&\underbrace{\frac{1}{\Big{(}1+\frac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Big{)}^{2}}\frac{1}{A_{0}\omega_{0}}\Big{[}1+\big{(}\frac{C_{x}+C_{p}}{C_{i}}\big{)}+\frac{C_{x}}{C_{i}}(1+A_{0}\omega_{0}R_{x}C_{i})\Big{]}e^{-\Big{(}\tfrac{A_{o}\omega_{0}R_{x}C_{i}+1}{R_{x}(C_{x}+C_{p}+C_{i})}\Big{)}t}}_{\text{{Offset
voltage due to finite GBW}}}\Bigg{\\}}\end{split}$

The time domain voltage $\mathrm{V_{R}(t)}$ is obtained by transforming Eq. 11
into the time-domain and taking the initial value of $\mathrm{V_{R}(t)}$ as
$\mathrm{V_{P}(\alpha-2X)}$ where, $\mathrm{\alpha=R_{2}/R_{1}}$ and
$X=C_{x}/C_{i}$. The time-domain voltage expression at node $V_{R}$ is derived
in Eq. 12. The integrator output voltage contains additional terms due to the
Op-Amp non-idealities as seen from Eq. 12.

  * •

The first term indicates that the finite GBW product reduces the slope of the
linear ramp voltage at the integrator output by a factor
$\mathrm{1+1/(A_{o}\omega_{0}R_{x}C_{i})}$.

  * •

The second term offers insight into the offset voltage introduced at the
integrator output due to the finite GBW product and additionally due to the
impedance/leaky-capacitive sensor. The effect of the exponential transient
term is small enough to be neglected. We denote the final offset term
introduced due to finite GBW by $\gamma$ and is given by

(13)
$\gamma=\Bigg{(}1+\frac{C_{x}+C_{p}}{C_{i}}+\frac{C_{x}}{C_{i}}(1+A_{0}\omega_{0}R_{x}C_{i})\Bigg{)}\frac{A_{0}\omega_{0}R_{x}C_{i}}{(1+A_{0}\omega_{0}R_{x}C_{i})^{2}}$

Further, considering the Op-Amp in the integrator has an offset voltage of
$V_{os}$ and a bias current of $i_{b}$, the total offset voltage presented by
the integrator is then $\mathrm{V_{os}^{\prime}=V_{os}+i_{b}R_{x}}$. A similar
analysis of the integrator with a finite GBW Op-Amp in the presence of this
offset term shows that the effect is evaluated simply by shifting the input by
the equivalent offset voltage. Further, the slope of the integrator output
waveform (triangular) is limited by the slew rate (SR) of the Op-Amp used in
the integrator. Hence, the slope (SL) of the integrator output is given by Eq.
14, where the positive sign corresponds to $SL_{(-)}$, and the minus sign
corresponds to $SL_{(+)}$.

(14) $\displaystyle
SL=min\Bigg{\\{}\frac{V_{p}\pm(V_{os}^{\prime}+i_{b}R_{x})}{R_{x}C_{i}\Big{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Big{)}},SR\Bigg{\\}}$

Referring to Fig. 3, we now obtain the individual expressions for $T_{P1}$ and
$T_{P2}$ to obtain the $R_{x}$ and $C_{x}$ value, which is the measurement
parameter. Referring to Fig. 3, $T_{P1}$ can be expressed as,

(15)
$\displaystyle\begin{split}T_{P1}=max\Bigg{\\{}\Bigg{[}\frac{(\alpha-\gamma-2X-V_{oz}/V_{p})}{1+(V_{os}^{\prime}+i_{b}R_{x})/V_{p}}\Bigg{]}R_{x}C_{i}\\\
\cdot\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)},\frac{(\alpha-2X)V_{p}}{SR}\Bigg{\\}}\end{split}$

Similarly, $T_{P2}$ can be obtained as,

(16)
$\displaystyle\begin{split}T_{P2}=max\Bigg{\\{}\Bigg{[}\frac{(\alpha+\gamma+V_{oz}/V_{p})}{1+(V_{os}^{\prime}+i_{b}R_{x})/V_{p}}\Bigg{]}R_{x}C_{i}\\\
\cdot\Big{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\big{)},\frac{(\alpha-2X)V_{p}}{SR}\Bigg{\\}}\end{split}$

(17) $\displaystyle
T_{P1}=max\Bigg{\\{}R_{x}C_{i}(\alpha-2X)\Bigg{[}\frac{1-\dfrac{\gamma+V_{oz}/V_{p}}{\alpha-2X}}{(1+V_{os}^{\prime}/V_{p})}\Bigg{]}\Big{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Big{)},\frac{(\alpha-2X)V_{p}}{SR}\Bigg{\\}}+\frac{\tau_{S,LH}+\tau_{Z,LH}}{2}$
(18) $\displaystyle
T_{P2}=max\Bigg{\\{}R_{x}C_{i}\alpha\Bigg{[}\frac{1+\dfrac{\gamma+V_{oz}/V_{p}}{\alpha}}{(1+V_{os}^{\prime}/V_{p})}\Bigg{]}\Big{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Big{)},\frac{(\alpha-2X)V_{p}}{SR}\Bigg{\\}}+\frac{\tau_{S,HL}+\tau_{Z,LH}}{2}$

Timing non-idealities in the ZCD and the Schmitt trigger Considering the rise
and fall times of the Schmitt trigger and the ZCD as: $\tau_{S,HL}$ and
$\tau_{S,LH}$ be the high to low and low to high response delays of the
Schmitt trigger and $\tau_{Z,HL}$ and $\tau_{Z,LH}$ be high to low and low to
high response delays of the ZCD, the expression of $T_{P1}$ and $T_{P2}$ in
Eq. 15 and Eq. 16, are refined as Eq. 17 and Eq. 18, respectively.

Combined effect of the non-idealities on the measurement From the combined
non-idealities represented in Eq. 18, the expression for the sensor resistance
$R_{x}$ can be inferred as follows.

(19) $\displaystyle R_{x}=\frac{T_{P2}}{\alpha
C_{i}}\Bigg{[}\frac{1}{\beta_{1}}-\frac{\alpha}{A_{0}\omega_{0}}\Bigg{]}$

where

(20)
$\beta_{1}=\frac{1+(\gamma+V_{oz}/V_{p})/\alpha}{1+V_{os}^{\prime}/V_{p}}$

The expression for the $R_{x}$ shows that the sensor parameter is
significantly affected by the non-idealities of the circuit components.
Similarly, for the capacitive sensors and parallel R-C sensors, the
measurement capacitor $C_{x}$ can be derived from Eq. 17 and Eq. 18 as
follows.

(21) $\begin{split}C_{x}=\frac{\alpha
C_{i}}{2+\Bigg{(}\dfrac{A_{0}\omega_{0}R_{x}C_{i}}{1+A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)}}\Bigg{[}&1-\Big{(}\frac{T_{P1}}{T_{P2}}\Big{)}\Bigg{(}\frac{1+V_{os}^{\prime}/V_{p}}{\dfrac{1}{\beta_{1}}-\dfrac{\alpha}{A_{0}\omega_{0}}}\Bigg{)}\\\
&\Big{(}\dfrac{1}{1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}}\Big{)}-\frac{V_{oz}}{\alpha
V_{p}}\Bigg{]}\end{split}$

Eq. 19 and Eq. 21 provide the measured sensor capacitance and sensor
resistance values $R_{x}$ and $C_{x}$ with the component non-idealities. The
ideal values of measured sensor resistance and capacitance, are:
$\mathit{R_{x}}=\dfrac{T_{P2}}{\alpha C_{i}}$ and,
$\mathrm{C_{x}}=\dfrac{\alpha C_{i}}{2}(1-\dfrac{T_{P1}}{T_{P2}})$. Comparing
the $\mathrm{R_{x}}$ and $\mathrm{C_{x}}$ values obtained from Eq. 19 and Eq.
21 with the ideal values, it is evident that there is a large error introduced
due to the component non-idealities.

The analytical solution for $C_{x}$ and $R_{x}$ derived in Eq. 21 and Eq. 19
is evaluated by including the non-idealities of commercial operational
amplifiers. The results show that even for a precision op-amp such as OPA177FP
from Analog Devices, the worst-case error in the measurement due to component
non-idealities is as high as 30% for $R_{x}$ measurement and 60% for $C_{x}$
measurement.

Accuracy Enhancement with Single Cycle Averaging

(22) $\displaystyle
T_{P3}=max\Bigg{\\{}R_{x}C_{i}(\alpha-2X)\Bigg{[}\frac{1+\dfrac{\gamma+V_{oz}/V_{p}}{\alpha-2X}}{(1-V_{os}^{\prime}/V_{p})}\Bigg{]}\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)},\frac{(\alpha-2X)V_{p}}{SR}\Bigg{\\}}+\frac{\tau_{S,HL}+\tau_{Z,HL}}{2}$
(23) $\displaystyle
T_{P4}=max\Bigg{\\{}R_{x}C_{i}\alpha\Bigg{[}\frac{1-\dfrac{\gamma+V_{oz}/V_{p}}{\alpha}}{(1-V_{os}^{\prime}/V_{p})}\Bigg{]}\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)},\frac{(\alpha-2X)V_{p}}{SR}\Bigg{\\}}+\frac{\tau_{Z,HL}+\tau_{S,LH}}{2}$

The timing waveform of the proposed circuit with an offset voltage $V_{os}$
due to component non-idealities is shown in Fig. 4. The ideal timing waveform
$T_{p1}$, $T_{p2}$, $T_{p3}$, and $T_{p4}$ are shown in Fig. 4 as ideal
$V_{z}(t)$. However, as derived in Eq. 12, the zero-crossing point of
$V_{R}(t)$ is shifted by an offset voltage $V_{os}$ due to component non-
idealities such as fine GBW and offset voltage of operational amplifiers. This
affects the practical $V_{z}(t)$ as shown in Fig. 4. Consider $T_{of}$ as the
corresponding offset time between the ideal and the practical output waveform
$V_{z}(t)$ due to finite offset voltage $V_{os}$. From Fig. 4, if $T_{p1}$
switches from $-V_{p}$ to $+V_{p}$ by $T_{of}$ time earlier than ideal, then
$T_{p3}$ will switch from $+V_{p}$ to $-V_{p}$ after $T_{of}$ time compared to
the ideal transition.

Therefore, the practical expression for time periods can be written as
$T^{\prime}_{p1}=T_{p1}-T_{of}$ and $T^{\prime}_{p3}=T_{p3}+T_{of}$. The
averaging of $T^{\prime}_{p1}$ and $T^{\prime}_{p3}$ will eliminate the effect
of the mismatch in the time period between $T_{p1}$ and $T_{p3}$. Same is true
for $T_{p2}$ and $T_{p4}$. Therefore, intuitively, it is possible to reduce
the effect of component non-idealities by single-cycle averaging. It can be
seen from the waveform that the pattern repeats after $T^{\prime}_{p4}$.
Therefore, averaging more than one cycle does not further enhance the accuracy
of the sensor parameter measurement due to component non-idealities.

Figure 4.  The ideal and practical $V_{z}(t)$ of the proposed relaxation
oscillator based circuit shown in Fig. 2. The offset due to the component non-
idealities is represented as $V_{os}$. This offset affects the period
$T_{p1}$, $T_{p2}$, $T_{p3}$, and $T_{p4}$.The practical periods due the
offset is represented by $T^{\prime}_{p1}$, $T^{\prime}_{p2}$,
$T^{\prime}_{p3}$, and $T^{\prime}_{p4}$. Averaging the alternate cycles of
practical $V_{z}(t)$ significantly reduces the error.

In order to analyze the effect of averaging, we have derived the expression
for the period $T_{p3}$ and $T_{p4}$ in Eq. 22 and Eq. 23, respectively. The
measurement of $T_{p3}$ and $T_{p4}$ is useful to understand the effect of
non-idealities in the subsequent cycle of one complete cycle of $V_{x}(t)$.

The time periods $\mathrm{T_{p1}}$ and $\mathrm{T_{p3}}$ are averaged to
obtain $\mathrm{T_{1}}$ and, the time periods $\mathrm{T_{p2}}$ and
$\mathrm{T_{p4}}$ are averaged to obtain $\mathrm{T_{2}}$. The expression of
the averaged period $T_{1}$ and $T_{2}$ are derived in Eq. 24 and Eq. 25,(on
page 11) respectively. The high-to-low and low-to-high response delays are
assumed to be the same for the ZCD ($\mathrm{\tau_{Z}}$) and Schmitt trigger
($\mathrm{\tau_{S}}$) for deriving Eq. 24 and Eq. 25.

(24)
$\begin{split}T_{1}=\frac{T_{P1}+T_{P3}}{2}=max\Bigg{\\{}R_{x}C_{i}(\alpha-2X)\Bigg{[}\frac{1+(\dfrac{V_{os}^{\prime}}{V_{p}})\big{(}\dfrac{\gamma+V_{oz}/V_{p}}{\alpha-2X}\big{)}}{1-(V_{os}^{\prime}/V_{p})^{2}}\Bigg{]}\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)},\frac{2V_{p}(\alpha-X)}{SR}\Bigg{\\}}+\frac{\tau_{S}+\tau_{Z}}{2}\end{split}$
(25) $\begin{split}T_{2}=\frac{T_{P2}+T_{P4}}{2}=max\Bigg{\\{}\alpha
R_{x}C_{i}\Bigg{[}\frac{1-(\dfrac{V_{os}^{\prime}}{\alpha
V_{p}})\big{(}\gamma+V_{oz}/V_{p}\big{)}}{1-(V_{os}^{\prime}/V_{p})^{2}}\Bigg{]}\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)},\frac{2V_{p}(\alpha-X)}{SR}\Bigg{\\}}+\frac{\tau_{S}+\tau_{Z}}{2}\end{split}$

To observe and compare the enhancement in the accuracy due to averaging,
consider the case when all the offsets are zero, the SR is not limited, and
the only non-ideality is the limit on the GBW. For this case, the time
expressions without and with averaging are derived in the subsequent
subsections.

Without Averaging: $T_{p1}$ and $T_{p2}$ The Eq. 26 and Eq. 27 (on page 12)
shows that the period $T_{p1}$ and $T_{p2}$ for the measurement of the sensor
parameters are greatly affected by the gain-bandwidth products of the
operational amplifiers. For instance, for OPA177, for an Rx of 330 k$\Omega$
and Cx of 33 pF, the error in the period $T_{p1}$ and $T_{p2}$ due to the GBW
is 52.76$\%$ and 26.74$\%$, respectively.

(26) $\displaystyle T_{P1}$
$\displaystyle=R_{x}C_{i}(\alpha-2X)\Bigg{(}1-\dfrac{\gamma}{\alpha-2X}\Bigg{)}\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)}$
(27) $\displaystyle T_{P2}$ $\displaystyle=\alpha
R_{x}C_{i}\Bigg{(}1+\dfrac{\gamma}{\alpha}\Bigg{)}\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)}$

With Single Cycle Averaging: $T_{1}$ and $T_{2}$ Considering only the GBW of
the operational amplifiers, $T_{1}$ and $T_{2}$ from Eq. 24 and Eq. 25 can be
simplified as follows.

(28) $\displaystyle T_{1}$
$\displaystyle=R_{x}C_{i}(\alpha-2X)\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)}$
(29) $\displaystyle T_{2}$ $\displaystyle=\alpha
R_{x}C_{i}\Bigg{(}1+\dfrac{1}{A_{0}\omega_{0}R_{x}C_{i}}\Bigg{)}$

The offset, $\mathrm{\gamma}$ introduced due to the finite GBW of the Op-Amp
severely affects the accuracy of time measurement as observed in Eq. 28 and
Eq. 29 along with the slope reduction factor of
$\mathrm{1+1/(A_{o}\omega_{0}R_{x}C_{i})}$. With averaging, the time accuracy
is not dependant on the offset factor $\mathrm{\gamma}$ and depends only on
the slope reduction factor as observed in Eq. 28 and Eq. 29. Therefore, the
single-cycle averaging significantly reduces the error due to the GBW of the
Op-Amps. For example, in the case of OPA177, with an Rx of 330 k$\Omega$ and
Cx of 33 pF, the error in the period $T_{1}$ and $T_{2}$ due to the GBW is
1.5$\%$ and 0.72$\%$, respectively, which is significantly smaller as compared
to the measurement without averaging.

Design Criteria
The first design criterion is the selection of $\mathrm{\alpha}$, which
decides the range of measurement. A higher value of $\mathrm{C_{x}}$ results
in a higher factor $X$ (i.e. $C_{x}/C_{i}$), which might result in a cross-
over of the voltage beyond the zero value while charge transfer. To avoid
this, we get the following condition on $\mathrm{\alpha}$

(30) $\alpha>2\frac{C_{x,max}}{C_{i}}$

The error in capacitance measurement $\mathrm{\epsilon}$ results in a worst-
case error compared to the error in the resistance measurement. The error in
capacitance measurement is the deviation of time $\mathrm{T_{1}}$ from the
ideal value.

(31) $\displaystyle\epsilon=\frac{\Delta C_{x}}{C_{x}}=\frac{\Delta
T_{1}}{T_{1}}$

$\Delta C_{x}$ represents the incremental value of sensor capacitance and
$\Delta$ $T_{1}$ represents the corresponding relative change measurement
time.
Using Equations 24, 13 and, 31 the following conditions on the Op-Amp GBW, SR,
offset can be arrived at,

(32)
$\footnotesize{\epsilon<\frac{100}{1-\Bigg{(}\dfrac{V_{os}^{\prime}}{V_{p}}\Bigg{)}^{2}}\Bigg{[}\frac{1}{GBW(R_{x,min}C_{i})}+\\\
\frac{V_{os}^{\prime}}{V_{p}}\Bigg{(}\frac{1}{\dfrac{\alpha
C_{i}}{C_{x,max}}-2}+\frac{V_{os}^{\prime}}{V_{p}}\Bigg{)}\Bigg{]}}$

(33) $\displaystyle SR$ $\displaystyle>\frac{2V_{p}}{R_{x,min}C_{i}}$

For a given error tolerance in the measurement, an Op-Amp with a GBW and
offset requirement and SR that satisfies Eq. 32 and Eq. 33 must be chosen. A
comparator must also be selected with an offset value $\mathrm{V_{oz}<<V_{p}}$
and response time ($\mathrm{\tau}$) condition derived as follows.

(34) $\displaystyle\tau<\frac{\epsilon\alpha
R_{x,max}C_{i}}{100\Big{(}1-(V_{os}^{\prime}/V_{p})^{2}\Big{)}}$

Considering the propagation time of the XOR gate as $\tau_{p}$, the XOR gate
must be chosen based on the following equation.

(35) $\displaystyle\tau_{p}<\frac{\epsilon\alpha
R_{x,max}C_{i}}{100\Big{(}1-(V_{os}^{\prime}/V_{p})^{2}\Big{)}}+\tau$

Experimental Setup and Results
Several prototypes of the proposed relaxation oscillator-based signal
conditioning circuit shown in Fig. 2 are built to illustrate the circuit’s
capabilities for measuring sensor capacitance and resistance. Different
operational amplifiers are employed to implement the integrator, Schmitt
trigger, and ZCD of the proposed oscillator and are tabulated in Table 1,
further using these opamps, the effect of component non-idealities on the
measurement of sensor parameters has been studied. All the passive components
are first measured using a commercial table-top LCR meter (Agilent E4980A).

Table 1. Operational amplifiers with performance parameters | Op-Amp
---
IC
|  GBW/(2$\pi$)
---
(MHz)
| SR
---
($V/\mu s$)
| $C_{p}$
---
(pF)
| $V{of}$
---
(mV)
| $i_{b}$
---
(nA)
| $\tau$
---
(ns)
AD741 | 1 | 0.5 | – | 5 | 500 | 300
LT1360 | 60 | 800 | 4 | 0.3 | 250 | –
TL071 | 5.25 | 29 | 2 | 4 | 0.1 | 310
OPA177 | 0.6 | 0.3 | – | 0.6 | 6 | –
LT1049 | 0.8 | 0.8 | – | 0.01 | 0.05 | –

The periods of the readout signal are measured using the timer/counter module
of the microcontroller. In this experiment, we have used an Arduino Uno
microcontroller with a 16-bit timer. The data from the microcontroller is
acquired in MATLAB, and the single-cycle averaging is performed.

Multiple experiments were conducted to evaluate the performance of the
proposed signal conditioning circuit. Initial experiments were conducted by
emulating the sensor using discrete resistors and capacitors. The final
experiment is conducted on a fringing field-based capacitive water level
sensor. The sensor is emulated by placing known values of resistor $R_{x}$ and
capacitor $C_{x}$ in parallel. The time period $T_{p1}$, $T_{p2}$, $T_{p3}$,
and $T_{p4}$ are measured to calculate the value of the sensor resistor.
Resistance Measurement

Figure 5. The waveform at different nodes of the oscillator developed using
op-amp LT1360 and LTC1049. The voltage $V_{R}(t)$ is affected by the non-
idealities of the op-amps, which affects the pulse width of $V_{Y}(t)$
(ideally should be 50% duty cycle). Due to this, the output $V_{z}(t)$ has
different positive and negative pulse widths resulting in an error in the
measurement of sensor parameters. (a) The error in the measurement is
significant for LT1360, (b) the error is reduced with LTC1049 compared to
LT1360. The error is negligible after a single-cycle averaging operation.
Figure 6. Measured and Analytical error in time ($T_{2}\&T_{P2}$) for
different sensor capacitance $C_{x}$ and resistance $R_{x}$ for two the LT1360
and LTC1049. The analytical $T_{p2}$ error is calculated from Eq. 18, which
shows the error in the measurement due to component non-idealities. The
analytical $T_{2}$ error shows the effect of single cycle averaging and is
calculated from Eq. 25. The analytical and experimental error reduces
significantly with single-cycle averaging. Figure 7. Analytical and measured
worst-case error in sensor resistance $R_{x}$ with different Op-Amps. The
analytical error without averaging is calculated from Eq. 18. The analytical
error with averaging is shown the effect of single cycle averaging and
calculated from Eq. 25. Figure 8. Measured and Analytical error for different
sensor capacitance $C_{x}$ and resistance $R_{x}$ for two different Op-
Amps.Analytical and measured worst-case error in sensor capacitance $C_{x}$
with different Op-Amps. The analytical $T_{p1}$ and $T_{1}$ errors are
calculated from Eq. 15 and Eq. 24, respectively. The analytical $T_{p1}$ shows
the error due to component non-idealities. The analytical $T_{2}$ error shows
the effect of single cycle averaging. The results are experimentally verified.

The resistance ($R_{x}$) measurement is performed by measuring the period
$T_{p2}$ and $T_{2}$ (average of $T_{p2}$ and $T_{p4}$). In addition to the
measurement of the sensor resistance, the experiment also demonstrates the
effect of component non-idealities on the measurement. The experiment is
conducted as follows.

  1. (1)

A capacitor $C_{x}$ is placed in parallel with the sensor resistance $R_{x}$.
The value of $R_{x}$ is varied from 100 k$\Omega$ to 1000 k$\Omega$. The
period $T_{p2}$ and $T_{p4}$ is measured, and the percentage relative error is
calculated with and without averaging.

  2. (2)

The same process is repeated for multiple values of the capacitance $C_{x}$.

  3. (3)

The above process is repeated for different operational amplifiers, and the
effect of component non-idealities are evaluated on the measurement.

The percentage relative error for different values of the sensor resistance at
different $C_{x}$ values for the operational amplifiers LT1360 and LTC1049 are
shown in Fig. 6. The result shows that the single cycle averaging
significantly reduces the error in the measurement of the sensor resistance.
The worst-case percentage relative error with and without single cycle
averaging for Rx measurement, for different operational amplifiers is shown in
Fig. 7. The chart in Fig. 7 shows the choice of the operational amplifier and
the effect of non-idealities. The % relative error is high for the operational
amplifier OPA177, which has a low gain-bandwidth. Moreover, the % relative
error is also high for LT1360, which has a good gain-bandwidth but has high
input offset voltage. On the other hand, the error is small for LTC1049, which
has a moderate gain-bandwidth and small input offset voltage. Therefore, the
oscillator should be designed considering all the non-idealities of the
operational amplifiers. The operational amplifiers’ selection should be done
based on the Eq. (33) and Eq. (35) from the design criteria for the given
error.

Table 2. Performance Parameters of the Developed Circuit Parameters | Expression | Measurement
---|---|---
$C_{x}$ | $R_{x}$
SD($\sigma$) | $\sqrt{\sum_{n=1}^{M}\frac{(S(n)-\overline{S})^{2}}{M-1}}$ | 10.2fF | 8.1 $\Omega$
SNR | $10\log\frac{\sum_{n=1}^{M}(S(n))^{2}}{\sum_{n=1}^{M}(S(n)-\overline{S})^{2}}$ | 74.83 db | 78.96 db
Dynamic Range | - | 10pF-42pF | 100k$\Omega$-1M$\Omega$
| Worst
---
Relative Error
- | 1.6 $\%$ | 0.62$\%$
$S(n)$ is the $n^{th}$ measured value $\bar{S}$ is the average value of the
measurements

Capacitance Measurement
In case of capacitance measurement, the value of sensor capacitor $C_{x}$ is
varied and the periods $T_{p1}$, $T_{p2}$, $T_{p3}$, and $T_{p4}$ are measured
experimentally. The experiment is conducted for different values of $R_{x}$.
The experimental procedure for $C_{x}$ measurement of impedance R-C or leaky
capacitive sensors is as follows.

  1. (1)

First, $R_{x}$ is fixed at one value and $C_{x}$ is varied from 9.6 pF to 37.6
pF.

  2. (2)

Next, the same process is repeated for different $R_{x}$ values. The
experiment is conducted using different operational amplifiers.

  3. (3)

The experimental results for different values of sensor capacitor $C_{x}$ for
operational amplifiers LT1360 and LTC 1049 are shown in Fig. 8.

The error is calculated from the measured time periods with and without single
cycle averaging. The worst-case errors for different operational amplifiers
are shown in Fig. 9. The single-cycle averaging dramatically reduces the
percentage relative error for the measurement of sensor capacitance. The
experimental results showing the pattern of error for different value of
$R_{x}$ by varying $C_{x}$ from 10 pF to 42 pF is shown in Fig. 8. The
experimental results shown in Fig. 8 are obtained for Op-Amps LT1360 and
LTC1049. Results show that the percentage relative error is high for the
LT1360, which is also justified analytically. On the contrary, the error is
small for LTC1049. The experimental result follows the analytical model of the
circuit. The single-cycle averaging reduces the error to a great extent.

Figure 9. Analytical and measured worst-case error in sensor capacitance
$C_{x}$ with different Op-Amps. The analytical $T_{p1}$ and $T_{1}$ errors are
calculated from Eq. 15 and 24, respectively. The analytical $T_{p1}$ shows the
error due to component non-idealities. The analytical $T_{2}$ error shows the
effect of single cycle averaging. The results are experimentally verified.
Figure 10.  Measured $\Delta C_{x}/C_{x}$ and $\Delta R_{x}/R_{x}$ with time,
for $C_{x}$= 33 pF and $R_{x}$= 330 k$\Omega$

Other Performance Parameters
Other performance parameters such as the standard deviation and SNR of the
proposed relaxation oscillator circuit for R-C sensors is measured
experimentally. The time period $T_{p1}$, $T_{p2}$, $T_{p3}$, and $T_{p4}$ are
experimentally recorded for a $R_{x}$ and $C_{x}$ values of 33 pF and 330
k$\Omega$. The measured periods are averaged for $C_{x}$ and $R_{x}$
measurement and the $\Delta C/C_{x}$ and $\Delta R/R_{x}$ is plotted in Fig.
10. The standard deviation and the SNR are calculated from the obtained data
set using the expression mentioned in Table 2. The obtained standard deviation
and SNR for $C_{x}$ and $R_{x}$ are reported in Table 2.

The circuit is tested for a capacitance as low as ten pF. At the lower
capacitance value, the input capacitance of the operational amplifier is
comparable with the sensor capacitance. Therefore, the relative error will be
high at the lower capacitance value. However, the measurement can be performed
by calibrating the effect of the input capacitance of the operational
amplifier.

Figure 11. (a) Experimental setup for water-level leaky capacitive sensor, (b)
experimental response of variation in the capacitance measured using the
proposed circuit with multiple water levels. The sensor shows a 99.9% linear
response with respect to the water-level.

Testing with Water-Level Sensor The developed converter is utilized for the
measurement of the sensor capacitor for the leaky capacitive water-level
sensor. The water-level sensor is fabricated on a flexible substrate using the
screen-printing technique. The sensor geometry is based on the inter-digitized
electrodes. The principle of sensing is based on a change in the fringing
field around the electrodes due to the water level, which in turn changes the
capacitance of the sensor. The details of the sensor fabrication are reported
in [27, 28].

The experimental setup for the system is shown in Fig. 11(a). The sensor is
placed inside the beaker with known water levels. A controlled amount of water
is poured into the beaker. The capacitance of the sensor is measured with the
commercial LCR meter (Agilent E4980A) as well as with the proposed circuit.
The percentage relative error in the measurement of the sensor capacitance for
different water levels is shown in Fig. 11(b).

The experimental test was conducted with the oscillator circuit built using
the LTC1049 operational amplifier. In addition, the $R^{2}$ value is estimated
from the fitted experimental data and found to be 99.9%. This shows that the
sensor shows a linear response to the water level. Further, the error reported
in Table 2 is with the standard ceramic capacitors, while the error from the
plot shown in Figure 11 is with the sensor. The mismatch in the error value is
primarily because of the parasitic capacitance of the sensor-circuit
interconnection and the measurement error.

Table 3. Performance Comparison of the Proposed Circuit with Other Reported
Work

Output | Measurement Range |  $\%$ Relative Error | SNR(db) | Technique |  Power Consumption
---|---|---|---|---|---
$C_{x}$ pF |  $R_{x}$ k$\Omega$ | $C_{x}$ | $R_{x}$ | $C_{x}$ | $R_{x}$
| This Work
---
(LTC1049)
| Pulse
---
Width
10-42 | 100-1000 | 1.6 | 0.62 | 74.83 | 78.96 | Oscillator | 8.4 mW
[29] | Bit-Stream | 100-200 | 700-900 |  $\pm$0.2 |  $\pm$0.2 | - | - | Dual Slope | 175 mW
[30] | Bit-Stream | 50-1000 | 50-1000 | 0.73 | 0.82 | 64.35 | 70.87 | Swithced-Capacitor | NA
[15] | Bit-Stream | 150-206 | >5000 | 1 | - | - | - | Charge/Discharge | NA
[20] | Analog Voltage | 100-2000 | 33-3000 | 6 | 8 | - | - | PSD | 168 mW
[18] | Analog Voltage | 10-760 | 56-6500 |  $\pm$0.11 |  $\pm$0.07 | 49 | 50 | Auto-Nulling | 142 mW
[19] | Analog Voltage | 200-100k | 33-1200 | 10 | 100 | - | - | Auto-Nulling | NA

Conclusion
A modified relaxation oscillator based signal conditioning circuit for
parallel R-C sensors is presented in this paper. The proposed circuit is based
on a dual-slope and charge-transfer technique embedded into a modified
relaxation oscillator. Detailed analysis and design criteria of the proposed
circuit design are provided, considering various sources of errors. A single-
cycle averaging method is analytically derived, which significantly reduces
the effect of component non-idealities. The efficacy of the proposed circuit
is tested on the fabricated prototype. The measurement result shows the
effectiveness of the proposed circuit for parallel R-C sensors. The
performance parameters of the developed prototype are calculated. The proposed
circuit provides an SNR of around 75 dB and 79 dB for capacitive and resistive
sensors, respectively. Overall, the proposed circuit is well suited for
various sensors such as leaky capacitive sensors, resistive sensors with a
parasitic capacitor, and impedance R-C sensors.

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