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#  The Simons Observatory: Complex Impedance Measurements for a Full Focal-
Plane Module

Jack Lashner  Department of Physics and Astronomy, University of Southern
California, Los Angeles, CA 90007, USA  Joseph Seibert  Department of Physics,
University of California, San Diego, La Jolla, CA 92093, USA  Max Silva-Feaver
Department of Physics, University of California, San Diego, La Jolla, CA
92093, USA  Tanay Bhandarkar  Department of Physics and Astronomy, University
of Pennsylvania, 209 S 33rd St. Philadelphia, PA 19104, USA  Kevin T. Crowley
Department of Physics, University of California, Berkeley, CA 94720, USA
Shannon M. Duff  National Institute of Standards and Technology, Boulder,
Colorado 80305, USA  Daniel Dutcher  Joseph Henry Laboratories of Physics,
Jadwin Hall, Princeton University, Princeton, NJ 08544, USA  Kathleen
Harrington  Department of Astronomy and Astrophysics, University of Chicago,
Chicago, IL, USA  Shawn W. Henderson  SLAC National Accelerator Laboratory,
Menlo Park, CA 94025, USA  Amber D. Miller  Department of Physics and
Astronomy, University of Southern California, Los Angeles, CA 90007, USA
Michael Niemack  Department of Physics, Cornell University, Ithaca, NY 14853,
USA  Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
Suzanne Staggs  Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton
University, Princeton, NJ 08544, USA  Yuhan Wang  Joseph Henry Laboratories of
Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA  Kaiwen
Zheng  Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton
University, Princeton, NJ 08544, USA

###### Abstract

The Simons Observatory (SO) is a ground based Cosmic Microwave Background
experiment that will be deployed to the Atacama Desert in Chile. SO will field
over 60,000 transition edge sensor (TES) bolometers that will observe in six
spectral bands between 27 GHz and 280 GHz with the goal of revealing new
information about the origin and evolution of the universe. SO detectors are
grouped based on their observing frequency and packaged into Universal Focal
Plane Modules, each containing up to 1720 detectors which are read out using
microwave SQUID multiplexing and the SLAC Microresonator Radio Frequency
Electronics (SMuRF). By measuring the complex impedance of a TES we are able
to access many thermoelectric properties of the detector that are difficult to
determine using other calibration methods, however it has been difficult
historically to measure complex impedance for many detectors at once due to
high sample rate requirements. Here we present a method which uses SMuRF to
measure the complex impedance of hundreds of detectors simultaneously on hour-
long timescales. We compare the measured effective thermal time constants to
those estimated independently with bias steps. This new method opens up the
possibility for using this characterization tool both in labs and at the site
to better understand the full population of SO detectors.

##  1 Introduction

The Simons Observatory (SO) is a new Cosmic Microwave Background (CMB)
experiment being built on Cerro Toco in Chile, which will make precision
measurements of the temperature and polarization anisotropies of the CMB in
order to measure and constrain fundamental properties of the Universe [1]. SO
will field over 60,000 Transition Edge Sensor (TES) detectors, split between a
6 m crossed Dragone large aperture telescope (LAT)[2, 3] containing 30,000+
detectors, and an array of three 42 cm small aperture refracting telescopes
(SATs) containing a total of 30,000 detectors [4, 2, 5]. SO will use dichroic
pixels observing in six spectral bands, at frequencies 27/39 GHz (LF), 93/145
GHz (MF) and 225/280 GHz (UHF) in order to measure the CMB and remove
foreground contamination.

Advancements in readout architecture are essential to accommodate the large
detector counts of SO and future CMB experiments. High multiplexing factors
are necessary to decrease the thermal loading on the focal-plane, and decrease
the complexity of the readout wiring. SO will use microwave superconducting
quantum interference device (SQUID) multiplexing ($\mu$mux) [6], where TESs
are inductively coupled to superconducting microresonators between 4-6 GHz via
RF SQUIDs. The SQUID coupling transduces current through the TES into a change
in effective inductance of the microresonator, causing a measurable shift in
its resonance frequency. At room temperature we use SLAC Microwave RF (SMuRF)
electronics [7] to generate the tones used to interrogate resonators and track
their resonance frequencies. In addition, SMuRF provides DC TES bias voltages,
and the flux-ramp signal which is used to linearize the RF SQUID response.

The optical coupling, detector arrays, and cold electronics are all packaged
together into 150 mm hexagonal assemblies called universal focal-plane modules
(UFM) [8]. The focal planes of each telescope will be composed of tiled UFMs,
with each SAT focal-plane containing 7 modules, and each of seven LAT optics
tubes containing 3 modules. UFMs can support up to 1820 multiplexing channels
and up to 1728 optical detectors, and will be read out using two Radio
Frequency (RF) transmission lines. The UFM also provides routing for the 12 DC
bias lines used to bias to the TESs, and two flux ramp lines used to linearize
the SQUID response [9].

Before deployment, each UFM is tested and characterized in an SO testbed [10].
By measuring detector noise, current-voltage curves (IVs), and measuring the
change in current through the TES in response to a small step in DC bias
voltage (bias steps), we are able to determine detector properties such as the
normal TES resistance ($R_{N}$), the bias power required to drive detectors to
90% of $R_{N}$ ($P_{\mathrm{SAT}}$), the thermal conductance $G$, and the
effective thermal time constant $\tau_{\mathrm{eff}}$. Some detector
parameters such as the logarithmic sensitivities to temperature and current
($\alpha$ and $\beta$ respectively), the loop-gain $\mathcal{L}_{I}$, and the
TES heat capacity $C$ are difficult to measure independently from each other
due to degeneracy. The complex impedance (CI) of the TES
$\hat{Z}_{\mathrm{TES}}$ provides a method by which we can access these
parameters by measuring the response of the TES to sinusoidal bias voltage
stimuli. CI measurements have previously been performed on SO prototype
bolometers [11], however these measurements have been limited to only a few
bolometers due to the high sample rate requirements. The high bandwidth of
SMuRF allows us to take data at sample rates of up to 25 kHz for over 150
detectors at a time, and the Digital to Analog Converters (DACs) that power
the bias lines can be used to send sine waves along the TES bias lines at
frequencies approaching 5 kHz. SMuRF allows us to measure the complex
impedance of large batches of TESs simultaneously, and perform preliminary and
in-situ array-scale characterization in a matter of hours.

In this paper, we describe a procedure of measuring the complex impedance of
SO detectors at array scales using the SMuRF readout electronics. Section 2
presents the TES model used to interpret the CI measurements. Section 3
describes the methods used to characterize detectors with complex impedance
measurements, and bias step measurements. Section 4 presents the complex
impedance results for a prototype UFM, and compares the effective thermal time
constants with those calculated based on bias steps.

##  2 Background

Figure 1:  Diagram of the bolometer model used in this study. The TES is
modeled as an island thermally isolated from the bath with heat capacity C
coupled to a bath of temperature $T_{\mathrm{bath}}$ with thermal conductance
$G$. Each bias line contains hundreds of TES circuits in series, each with the
TES wired in parallel with a shunt resistor.

To model the complex impedance of the TES, we follow the formalism used by
Irwin and Hilton [12] and model the TES bolometer as a single thermal block
with temperature $T$ and heat capacity $C$. This is a very simplified model,
but measuring the complex impedance of the TES can help us understand
departures from these assumptions. The TES is coupled to a thermal bath of
temperature $T_{b}$ with thermal conductance $G=\left.\partial
P_{\mathrm{bath}}/\partial T\right|_{T_{b}}$. We bias the detectors using a
constant-voltage bias $V_{\mathrm{bias}}$ that is applied to each bias line,
which consists of a large bias line resistor $R_{\mathrm{BL}}\approx
16~{}\mathrm{k\Omega}$ in series with around 150 TES circuits. Each TES
circuit is modeled as the TES resistance in series with a parasitic inductance
$L$, and with a shunt resistor $R_{\mathrm{sh}}\approx 0.4$ m$\Omega$
connected in parallel. A depiction of this model can be seen in Figure 1.

We substitute the full bias schematic shown in Figure 1 with a Thevenin
equivalent circuit containing just a single TES, with equivalent voltage
$\hat{V}_{\mathrm{th}}$ and equivalent impedance $\hat{Z}_{\mathrm{eq}}$ where
both are complex phasors that depend on the excitation frequency $\omega$. For
the bias circuit described above, the equivalent impedance at low frequencies
is given by $Z_{\mathrm{eq}}(\omega)\approx R_{\mathrm{sh}}+i\omega L$. The
dynamics of the TES are governed by the set of coupled differential equations:

$C\frac{dT}{dt}=-P_{\mathrm{bath}}+P_{J}+P_{\mathrm{opt}}$ (1)
$L\frac{dI}{dt}=V_{\mathrm{th}}-IR_{\mathrm{sh}}-IR(T,I)$ (2)

where $P_{\mathrm{bath}}$ is the power flowing from the TES to the thermal
bath, $P_{J}$ the power from joule heating, $P_{\mathrm{opt}}$ the power
deposited onto the TES through optical coupling, and $R(T,I)$ the TES
resistance.

Under the constant-voltage bias, in-transition detectors are able to achieve
negative electrothermal feedback which keeps the TES in transition. The
detector is characterized by its logarithmic temperature sensitivity,
logarithmic current sensitivity, and the loop-gain:

$\alpha\equiv\left.\frac{\partial\log R}{\partial\log
T}\right|_{I_{0}}\qquad\beta\equiv\left.\frac{\partial\log R}{\partial\log
I}\right|_{T_{0}}\qquad\mathcal{L}_{I}\equiv\frac{P_{J_{0}}\alpha}{GT_{0}}$
(3)

where $I_{0}$, $T_{0}$, and $P_{J_{0}}$ are the equilibrium current, bolometer
temperature and bias power.

For small perturbations $\delta I$ and $\delta T$, equations 1 and 2 can be
linearized and solved[13]. These solutions contain two time constants:
$\tau_{\mathrm{el}}$ characterizing the initial electrical response, and
$\tau_{\mathrm{eff}}$ characterizing the effective thermal decay to
equilibrium. Both time constants can be seen in a TES’s response to a sudden
voltage bias step, as is shown in Figure 5. SO bolometers have a very small
inline inductance $L$, which causes $\tau_{\mathrm{eff}}$ to be much longer
than $\tau_{\mathrm{el}}$ and makes $\tau_{\mathrm{eff}}$ the determining
factor for how fast the TES can respond to an optical signal.

Solving the coupled differential equation yields

$\tau_{\mathrm{eff}}=\tau_{0}\left(1+\frac{(1-R_{\mathrm{sh}}/R)\mathcal{L}_{I}}{1+\beta+R_{\mathrm{sh}}/R}\right)^{-1}$
(4)

where $\tau_{0}=C/G=\tau_{I}(1-\mathcal{L}_{I})$ is the natural thermal time
constant in the absence of electrothermal feedback and $\tau_{I}$ is the
thermal time constant in the case of a constant-current bias source.

The complex impedance of the TES can be extracted from the general solutions,
and is given by

$\hat{Z}_{\mathrm{TES}}(\omega)=R(1+\beta)+\frac{R\mathcal{L}_{I}}{1-\mathcal{L}_{I}}\frac{2+\beta}{1+i\omega\tau_{I}}.$
(5)

where $\omega$ is the excitation frequency. $\hat{Z}_{\mathrm{TES}}$ can be
accessed by measuring the transfer function across the TES as is described in
the next section.

##  3 Methods

The measurements in this section were carried out on a prototype SO UFM in a
testbed designed for optical characterization of SO detectors [14]. Two thirds
of the detectors in the UFM are covered by a gold-plated silicon mask allowing
us to test both optical and dark detectors simultaneously. IVs taken
immediately prior are used to determine parameters such as $R_{N}$. The DC
voltages for each TES bias line were chosen based on IVs to maximize the
number of detectors with resistances in the range 30% - 60% $R_{N}$. Note that
the measurements shown in this section demonstrate a useful characterization
method, but do not reflect expected operating parameters. The measurements
were taken at a higher bath temperature (150 mK) than that expected during
site observation (100 mK), and this particular set of TESs have optical
efficiency that is too low to be deployed for observations.

###  3.1 Complex Impedance Measurement

Figure 2:  Magnitude (left) and phase (right) of $\hat{I}_{\mathrm{TES}}(f)$
for a single detector, where $f$ is the excitation frequency. The magnitude of
$\hat{I}_{\mathrm{TES}}$ is the amplitude of the sine wave through the
detector, and the phase is measured relative to the commanded bias waveform.
We show the TES current measured while the detector is superconducting,
overbiased, and in transition. The small uptick at high frequencies, is likely
due to limitations in the SMuRF waveform generation, but this is absorbed into
$\hat{V}_{\mathrm{th}}$ in 6 and is not seen in the measured
$\hat{Z}_{\mathrm{TES}}$.

Figure 3:  $\hat{Z}_{\mathrm{eq}}(f)$ (left) and $\hat{V}_{\mathrm{th}}(f)$
(right) for a single detector, determined by the TES response to sinusoidal
stimuli in the overbiased and superconducting states. In each plot the real
part is shown in blue, and the imaginary part is shown in orange. Using the
simple electrical model of the bias line described in Section 2, at low
excitation frequencies we expect the equivalent impedance to approach
$\hat{Z}_{\mathrm{eq}}(\omega)\approx R_{\mathrm{sh}}+i\omega L$. In the left
plot we are able to observe $R_{\mathrm{sh}}\approx 0.4$ m$\Omega$ from the
low-frequency limit of $\mathrm{Re}[\hat{Z}_{\mathrm{eq}}]$, and estimate the
inductance $L$ as the slope of the low-frequency limit of
$\mathrm{Im}[\hat{Z}_{\mathrm{eq}}]$. $\hat{Z}_{\mathrm{eq}}(f)$ is plotted
against a linear axis so we are able to more clearly show the linear
dependence of $\mathrm{Im}[\hat{Z}_{\mathrm{eq}}]$ on frequency.

The transfer function of the TES in response to an excitation frequency
$\omega$ is measured by sending a small sine wave along the bias line on top
of the DC bias voltage biasing the TESs into transition. For each data sample,
the commanded output of the DACs driving the TES bias line is recorded along
with the TES response. We use these commanded TES bias values as a reference
for a digital lock-in procedure[15] that allows us to extract the amplitude
and phase of the TES response relative to the commanded bias with high levels
of signal-to-noise. We measure the transfer function while the detectors are
superconducting, overbiased, and in transition which can be seen in Figure 2.
The superconducting and overbiased measurements allow us to estimate the
Thevenin equivalent voltage and impedance of the system for every TES bias
circuit at each frequency [16] and thus correct for stray impedances in the
system. $\hat{V}_{\mathrm{th}}$ and $\hat{Z}_{\mathrm{eq}}$ are given by

$\hat{V}_{\mathrm{th}}(\omega)=\frac{R_{N}}{\hat{I}_{\mathrm{OB}}^{-1}(\omega)-\hat{I}_{\mathrm{SC}}^{-1}(\omega)}$
(6)
$\hat{Z}_{\mathrm{eq}}(\omega)=\hat{V}_{\mathrm{th}}(\omega)/\hat{I}_{\mathrm{SC}}(\omega).$
(7)

and the measured values for a single detector can be seen in Figure 3.

The complex impedance of the TES is then given by

$\hat{Z}_{\mathrm{TES}}(\omega)=\hat{V}_{\mathrm{th}}(\omega)/\hat{I}(\omega)-\hat{Z}_{\mathrm{eq}}(\omega)$
(8)

where $\hat{I}$ is the complex phasor representing the TES current in
transition. $\hat{Z}_{\mathrm{TES}}$ for a single detector can be seen in
Figure 4. The measured $\hat{Z}_{\mathrm{TES}}$ for each detector is then fit
using the model in equation 5 for the parameters $R$, $\mathcal{L}_{I}$,
$\beta$ and $\tau_{I}$, which are used to compute derived parameters such as
$\tau_{\mathrm{eff}}$ and $\tau_{0}$.

Bipolar sine waves between 1 Hz and 5 kHz are applied to each bias line on top
of the DC TES bias voltage, with an amplitude of around 0.1% of the DC bias
level. The frequencies of the sine waves are limited by how quickly the two
DACs that generate the TES bias voltages can be programmed. With the default
SMuRF 1 MHz clock-rate, the bias voltage can be updated at around 16 kHz. This
update rate can be increased by either increasing the clock-rate, or by only
sending a sine wave on a single DAC while keeping the other fixed. As we
approach excitation frequencies of 5 kHz the DACs can only be updated a few
times per cycle, however using digital lock-in procedure we are still able to
extract amplitudes and phases from the TES response. There is a small amount
variation in the amplitude of the output sine wave at higher frequencies as
can be seen in Figure 2, however, as shown in Figure 3 this is absorbed into
$\hat{V}_{\mathrm{th}}$ and does not effect the $\hat{Z}_{\mathrm{TES}}$
measurement. $\hat{Z}_{\mathrm{TES}}$ approaches negative $R$ at low
frequencies and $R(1+\beta)$ at high frequencies, so high fidelity
measurements out to high frequencies are particularly important for
constraining $\beta$. We sample detectors at 25 kHz, well above the Nyquist
rate of the 5 kHz excitation.

Performing this measurement on a full UFM takes around 30 minutes, with the
measurement time being limited by the fact that we need to measure the TES
response in 12 batches, corresponding to the 12 bias lines on our UFM. The
measurement is bottlenecked by the low-frequency data points, which require
more time due to their longer periods. The measurement time can be shortened
by reducing the number of samples at low excitation frequencies, and by
reducing the number of periods we measure for those particular samples. It is
sufficient to take superconducting and overbiased measurements once per array,
and those measurements can be used for all in-transition measurements. This
can be run in parallel across multiple UFMs simultaneously, resulting in an
estimated measurement time for 30,000 detectors of approximately 2 hours.

Due to performance issues in the cryostat used to take these measurements, it
was not possible to heat the array to perform tests at multiple bath
temperatures. This causes certain parameters such as $\mathcal{L}_{I}$ and
$\tau_{0}$ to be poorly constrained, and could cause degeneracy between
parameters such as $\beta$ and $\tau_{I}$. One could improve these
measurements by taking data at multiple bath temperatures and bias levels,
forcing $\tau_{0}$ to remain constant across datasets.

Figure 4:  $\hat{Z}_{\mathrm{TES}}$ of a single bolometer on the transition.
On the left we see the semicircle that $\hat{Z}_{\mathrm{TES}}$ traces in the
complex plane, where the color of the data points corresponds to the
excitation frequency. On the right we see the real and imaginary components of
$\hat{Z}_{\mathrm{TES}}$ both plotted against frequency. In both plots, the
dashed lines show the $\hat{Z}_{\mathrm{TES}}$ fitted with equation 5.

###  3.2 Bias Step Measurement

We compare the $\tau_{\mathrm{eff}}$ measured using complex impedance to those
independently measured from bias steps taken immediately before. Bias step
measurements consist of biasing the detectors into their transition, and then
sending a small-amplitude square wave on top of the DC bias voltage [17, 11].
We perform many steps in quick succession, and align each step such that it
crosses the midpoint at $t=0$. We then use the aligned steps to compute the
mean response for each TES. We fit the thermal decay of the mean step response
with a single-pole exponential of the form $f(t)=A\exp(-t/\tau)+B$. We only
fit the response after $t=0$ to avoid the initial electrical response of the
TES. In Figure 5 we show each of the steps after alignment, the mean step, and
the exponential fit for a single TES.

To command the bias steps, we use SMuRF’s arbitrary waveform generator to play
a square wave with a period of 50 ms, and amplitude of around 1% of the DC
bias voltage. We take data on one bias line at a time to avoid crosstalk
between bias lines, and sample detectors at 4 kHz. We apply around 40
individual steps to improve the statistical significance of our measurements.
This measurement takes around 30 seconds for a full UFM, making it a very
useful tool for characterizing detectors throughout an observation.

Figure 5:  TES response to bias steps taken for a single detector biased in
the transition. This plot shows forty steps taken in quick succession, aligned
such that the step midpoint occurs at $t=0$. Only the downward steps are shown
here. The small increase in current before the step shows the initial
electrical response of the TES, while the remainder of the step shows the
slower thermal relaxation to equilibrium. Here the light grey lines show the
individual steps, the blue points show the average of all steps after they
have been aligned, and the orange line shows the fit to the average step,
starting at $t=0$ to avoid the initial electrical response.

##  4 Results

We performed the measurements described in Section 3 on an early prototype
UFM, and obtained data for 663 TESs. This population includes both 90 and 150
GHz detectors, and includes both detectors that are blocked off at the UFM,
and those that have an optical pathway out to room temperature. This is a
relatively low yield for a full UFM, partially due to the elevated bath
temperatures causing most of the optical 90 GHz detectors to be saturated, but
it is enough to observe population statistics and trends. Figure 6 shows a
comparison between $\tau_{\mathrm{eff}}^{\mathrm{BS}}$ and
$\tau_{\mathrm{eff}}^{CI}$ for each channel that passed data quality cuts. In
general we see good agreement between the two datasets, though bias step
calculations slightly overestimate $\tau_{\mathrm{eff}}$ compared to the
complex impedance estimation, with this bias being worse for faster detectors.
It is possible that this bias stems from inadequacies of using a single-pole
exponential to fit the bias data, as it has been seen that the fitted
$\tau_{\mathrm{eff}}$ is somewhat sensitive to our choice of $t=0$ [18].

Figure 7 shows all CI fit parameters for each channel, and how they compare to
one another. From these fits, it is clear that the loop-gain
$\mathcal{L}_{I}$, and thus the derived parameter $\tau_{0}$, are poorly
constrained by this dataset. For optimally biased detectors, we expect large
values of $\mathcal{L}_{I}$, at least greater than 10. In this regime the
loop-gain has a negligible effect on the complex impedance in equation 5
compared to $\beta$ and $\tau_{I}$, and so fits tend to gravitate towards the
$\mathcal{L}_{I}\rightarrow\infty$ limit.

A better estimation of $\mathcal{L}_{I}$ could be achieved by repeating this
measurement at various bath temperatures, fixing $\tau_{0}$ across the
datasets. Alternatively, one could fix the thermal coupling coefficient $G$ to
values measured by taking IVs at various temperatures [10].

Figure 6:  Comparison of $\tau_{\mathrm{eff}}$ from the complex impedance
measurements with $\tau_{\mathrm{eff}}$ estimated from bias steps. Each point
represents a pair of measurements for a single detector. This plot shows dark
and optical detectors in the 90 GHz and 150 GHz bands. 150 GHz detectors have
a higher saturation power than the 90 GHz detectors, and we expect them to
have faster time constants. Due to the elevated bath temperature, the majority
of the optical 90 GHz detectors and many of the optical 150 GHz detectors are
saturated and cannot be biased into the transition. The dashed black line is a
guide line with a slope of one to show where the measurements would be equal.
Figure 7:  Detector parameters fit to $\hat{Z}_{\mathrm{TES}}$ measurements,
with each point representing the fit for a separate detector. Loop-gain fits
are unphysically large, as they are not well constrained by measurements taken
at a single bath temperature.

##  5 Conclusion

In this work we present complex impedance results for 663 TESs on a prototype
UFM. We compare the effective thermal time constant $\tau_{\mathrm{eff}}$,
determined from the complex impedance measurement, with those estimated using
bias-steps. We observe that, though there is general agreement between the two
methods, bias-steps slightly overestimate $\tau_{\mathrm{eff}}$ relative to
complex impedance measurements. We also observe that certain parameters such
as $\mathcal{L}_{I}$ are not well constrained by these complex impedance
measurements. This is unsurprising because we were operating at high bath
temperatures due to performance issues of our cryostat.

This work demonstrates a new implementation for characterizing TESs enabled by
advances in readout technology that allows us to take in-situ complex
impedance measurements of thousands of detectors using SMuRF electronics. This
allows us to study population statistics for detector parameters that have
been historically difficult to measure, which can inform future detector
design. Because this method does not require specialized hardware, it can be
run at the site to monitor parameters so they can be accessed by the SO
analysis pipeline for improved mapping. In the future we plan on optimizing
this measurement so it takes less time, and retaking this data for SO
deployment UFMs at operational temperatures to better constrain these
parameters, and to explore how SO bolometers deviate from the simple thermal
model presented here.

##  6 Acknowledgements

This work was supported in part by a grant from the Simons Foundation (Award
#457687, B.K.).

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