# Impedance Matching Between SQUID and Warm Amplifier for TES Readout in TDM for the ATHENA X-IFU Instrument

Manuel Gonzalez, Yann Parot, Damien Prele, Christian Kirsch, Edoardo Cucchetti, Philippe Peille, David Murat, Laurent Ravera, Jan van der Kuur

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Manuel Gonzalez, Yann Parot, Damien Prele, Christian Kirsch, Edoardo Cucchetti, et al. Impedance Matching Between SQUID and Warm Amplifier for TES Readout in TDM for the ATHENA X-IFU Instrument. Journal of Low Temperature Physics, 2024, 215 (3-4), pp.201-208. 10.1007/s10909-024-03085-2. hal-04608165

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# Impedance matching between SQUID and warm amplifier for TES readout in TDM

Manuel Gonzalez

manuel.gonzalez@apc.in2p3.fr

Astroparticle and Cosmology Laboratory

Yann Parot

Institut de Recherche en Astrophysique et Planétologie, CNRS

Damien Prele

Astroparticle and Cosmology Laboratory

Christian Kirsch

FAU Erlangen-Nürnberg

Edoardo Cucchetti

Centre National d'Etudes Spatiales

Philippe Peille

Centre National d'Etudes Spatiales

David Murat

Institut de Recherche en Astrophysique et Planétologie, CNRS

Laurent Ravera

Institut de Recherche en Astrophysique et Planétologie, CNRS

Jan Kuur

SRON Netherlands Institute for Space Research

## Research Article

**Keywords:** TES readout, low noise amplifier, impedance matching, settling time

**Posted Date:** November 6th, 2023

**DOI:** https://doi.org/10.21203/rs.3.rs-3547516/v1

**License:** cc BY This work is licensed under a Creative Commons Attribution 4.0 International License.

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**Additional Declarations:** No competing interests reported.
**Version of Record:** A version of this preprint was published at Journal of Low Temperature Physics on April 7th, 2024. See the published version at https://doi.org/10.1007/s10909-024-03085-2.
Impedance matching between SQUID and warm
amplifier for TES readout in TDM

Manuel Gonzalez¹*, Yann Parot²†, Damien Prele¹†,
Christian Kirsch³†, Edoardo Cucchetti⁴†, Philippe Peille⁴†,
David Murat²†, Laurent Ravera²†, Jan Van der Kuur⁵†

¹* Astroparticule et Cosmologie, Université Paris Cité, CNRS, France.

²Institut de Recherche en Astrophysique et Planétologie, CNRS, France.

³Dr. Karl Remeis-Observatory & ECAP, FAU Erlangen-Nürnberg,
Germany.

⁴Centre National d'Etudes Spatiales, France.

⁵SRON Netherlands Institute for Space Research, The Netherlands.

*Corresponding author(s). E-mail(s): manuel.gonzalez@apc.in2p3.fr;

Contributing authors: yann.parot@irap.omp.eu; prele@apc.in2p3.fr;

david.murat@irap.omp.eu; laurent.ravera@irap.omp.eu;

j.van der kuur@sron.nl; christian.ck.kirsch@fau.de;

edoardo.Cucchetti@cnes.fr; philippe.Peille@cnes.fr;

†These authors contributed equally to this work.

Abstract

Current cryogenic instruments require an increasingly high number of super-
conducting detectors. Large multiplexing factors are thus needed, increasing the
bandwidth of the readout signals. In the specific case of transition edge sensors
(TES), a cold amplification stage using superconducting quantum interference
devices (SQUIDs) is usually coupled to a room temperature low noise amplifier
(LNA). A resistive harness up to a few meters long connects these two stages
carrying signals with bandwidth of up to a few tens of MHz. In this context, it
is reasonable to consider the possibility of impedance matching at the input of
the LNA. In this paper we present the impact of such impedance matching for
an ATHENA X-IFU like instrument [1].

Keywords: TES readout, low noise amplifier, impedance matching, settling time
# 1 Introduction

TES are widely used in astrophysics and cosmology instruments. Although they can be multiplexed in different ways, we will concentrate our attention in time-division multiplexing (TDM)[2]. ATHENA's X-IFU instrument will use TDM with a two SQUID stages configuration with a multiplexing factor of 48. A simplified diagram of 1 column of X-IFU's readout chain is presented in Figure 1. The response of the room temperature LNA and the last SQUID stage (AMP SQ) are also shown.

<figure><img src="image_3.png" /><figcaption>**Fig. 1** Left: Simplified diagram of X-IFU readout chain. Top right: frequency response of the room temperature LNA. Bottom right: *V*-*φ* and *I*-*V* curves of the series SQUID array at 2 K.</figcaption></figure>

A 2.5 m harness connects the focal plane array (FPA) at cryogenic temperature with the warm front-end electronics (WFEE). Each TDM channel, or column, contains 48 rows of TES coupled to a first SQUID stage (MUX SQUID). The rows are sequentially read by switching ON and OFF the polarization of the MUX SQUIDs.
The row switching time is 160 ns. The settling time, after switching, has to be lower than 120 ns to allow at least 40 ns for the readout of the signal. Due to the length of the harness and the bandwidth of the signals (≥ 10 MHz), the differential twisted pair connecting the AMP SQ and the LNA has to be considered as a transmission line. Impedance mismatch at both ends of the transmission line induces reflections, leading to an increase of the settling time. It is worth noting that in order to decrease the thermal load on the FPA, resistive wires are used, so a lossy transmission line has to be considered.

The twisted pair used to read out the AMP SQ is also used to provide its bias current coming from the WFEE as shown in Figure 1. As a consequence, lowering the input impedance of the LNA will have an impact on the biasing condition of the SQUID.

## 2 Lossy transmission line and impedance matching

In order to show the effect of impedance mismatch with a lossy transmission line, measurements of signal propagation were carried with different lossless and lossy transmission lines and varying the source and load impedance (Figure 2).

<figure><img src="image_4.png" /><figcaption>Fig. 2 Equivalent circuit of the setup used to perform the measurements of signal propagation with different combinations of source, load and transmission line impedance.</figcaption></figure>

As shown in Figure 3, the impedance of the lossy resistive transmission line is complex and depends on frequency:

$$Z_C = \sqrt{\frac{R + i\omega L}{i\omega C}}, \qquad (1)$$

where R, L and C are, respectively, the series resistance, inductance and capacitance per unit length of the transmission line. Here we are neglecting the conductance G due to leakage between the conductors of the transmission line, which can also cause losses in the more general case. At high enough frequency, where $\omega L \gg R$ the characteristic impedance becomes real $Z_C \approx \sqrt{L/C}$. Only at these high frequencies, the impedance can be matched with a resistive source or load impedance.

These results show that the lossy line washes out the clear reflections that are observed in the lossless case. However, a long settling time can be observed with a high impedance load. The value of the settling time strongly depends on the combination of source and load impedance.
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Top left (right): Measurements of several combinations of source and load impedance using a 2 m long lossless 50 Ω (75 Ω) coaxial cable. Bottom left: complex impedance of the lossy twisted pair as a function of frequency, measured with a VNA. R, C and L can be determined by fitting the model of Equation 1. Bottom right: Measurements of several combinations of source and load impedance using a 2 m long lossy twisted pair.</figcaption></figure>

**3 Results and discussion**

In order to study the impact of the impedance mismatch on a more realistic case, the circuit of Figure 4 has been simulated. The SQUID's response is interpolated from $V_{SQ}(T, i_{SQ}, \phi)$ measurements taken at different temperature, current and applied flux as the ones presented in Figure 1.

The LNA could be modeled as an ideal amplifier of gain $-G$ with a second order RLC filter at the input to reproduce its frequency response. The input impedance of the LNA can be lowered by implementing negative feedback, through a feedback resistor $R_{FB}$ [3]. For this work, a simplified version of the circuit has been analyzed (Figure 4). The effect of the feedback is introduced as a simple noiseless resistor at the input of the LNA which is implemented as an ideal amplifier of gain $-G$. One limitation of this simplified model is that it doesn't describe the finite bandwidth of the LNA. However, the LNA bandwidth is higher than 25 MHz (see Figure 1).

The simulation consists of applying a flux excursion to the SQUID during 160 ns and then coming back to the baseline (Figure 5), simulating the readout of an active
<figure><img src="image_6.png" /><figcaption>Fig. 4 Model circuit of the AMP SQ, harness and LNA readout block.</figcaption></figure>

pixel in a TDM column. The following ODE system was numerically solved:

$$v'_{1} = (I_{bias} - v_{1}/R_{in} - i_{1})/C_{e}$$

$$i'_{1} = (v_{1} - v_{2} - i_{1}R_{e})/L_{e}$$

$$v'_{2} = (i_{1} - i_{2})/C_{e}$$

$$i'_{2} = (v_{2} - v_{3} - i_{2}R_{e})/L_{e}$$

$$v'_{N} = (i_{N-1} - i_{N})/C_{e}$$

$$i'_{N} = (v_{N} - V_{SQ}(T, i_{N}, \phi(t)))/L_{e}$$

The parameters used for the simulation are: G = 170, and for a 2.5 m long harness: N = 16, C = C<sub>e</sub> × N = 240 pF, L = L<sub>e</sub> × N = 1.28 µH, R = R<sub>e</sub> × N = 40 Ω. The ODE system was solved using SciPy's implementation of the Runge-Kutta-Fehlberg method (RK45)[4, 5].

The simulations were performed with different values of LNA input impedance. A high impedance value of 5 kΩ, corresponding to the intrinsic input impedance of the X-IFU WFEE was considered as the case where no impedance matching is performed (i.e. no feedback). A low impedance case of 73 Ω was analyzed as the case where the input impedance of the LNA is matched to the high frequency impedance of the transmission line. Lastly, an intermediate case of 250 Ω was also simulated. The initial conditions correspond to a constant solution with the baseline flux, and the value of I<sub>bias</sub> was varied for each simulation in order to keep the SQUID at the same working point before the flux excursion occurs.

The simulations results suggest that keeping a high input impedance at the input of the LNA can lead to a settling time that is not compatible with the required row switching time of 160 ns of X-IFU. The optimal case, purely in terms of settling
<figure><img src="image_7.png" /><figcaption>**Fig. 5** Left: Flux excursion at the input of the SQUID simulating the readout of an active TES.
Right: Simulated voltage at the input of the LNA for different input impedance values.</figcaption></figure>

time, is obtained when we match the high frequency impedance of the transmission line. However, this low LNA input impedance would mean that a large fraction of the biasing current would flow through the input of the LNA instead of the SQUID, potentially affecting the performances of the LNA and inducing an offset. Finally, although active impedance matching preserves the signal-to-noise ratio[3], the dynamic range at the output of the WFEE will also depend on the LNA input impedance. This would need to be considered in the whole design of the readout chain. Interestingly, the simulations show, that an intermediate input impedance value could be a good compromise, mitigating the effects previously discussed while significantly reducing the settling time.

## 4 Conclusions

The simulations' results presented in this work suggest that decreasing the input impedance of the LNA in an X-IFU like instrument could be required in order to reduce reflections and meet the settling time requirements. Based on these results, the demonstration model of X-IFU WFEE will implement several readout channels with different feedback resistors values in order to test the performances for different values of LNA input impedance. In parallel, the simulation software XIFUSIM[6] is being modified to include these effects in the end-to-end X-IFU simulations in a similar approach to the one used in this work.

## References

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[3] Gonzalez, M., Prêle, D., Chen, S.: Fully differential broadband LNA with active impedance matching for SQUID readout. Journal of Low Temperature Physics **209**(3-4), 606–613 (2022) https://doi.org/10.1007/s10909-022-02836-3

[4] Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: Open source scientific tools for Python (2001–). http://www.scipy.org/

[5] Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics **6**(1), 19–26 (1980) https://doi.org/10.1016/0771-050X(80)90013-3

[6] Kirsch, C., Lorenz, M., Peille, P., Dauser, T., Ceballos, M., Cobo, B., Merino-Alonso, P., Cucchetti, E., Smith, S.á., Gottardi, L., et al.: The Athena X-IFU instrument simulator xifusim. Journal of Low Temperature Physics **209**(5-6), 988–997 (2022) https://doi.org/10.1007/s10909-022-02700-4