STUDY OF THE MECHANICAL RESPONSE

Transcription

1 STUDY OF THE MECHANICAL RESPONSE OF THE LTP S TEST MASSES DUE TO THE ACTION OF CONTROL HEATERS IN LISA PATHFINDER - PROJECT REPORT - PROJECTE FINAL DE CARRERA D ENGINYERIA AERONÀUTICA AUTHOR: FERRAN GIBERT GUTIÉRREZ TUTORS: ALBERTO LOBO AND EDWARD CHESTER SCHOOL TUTOR: ELENA FANTINO Terrassa, January 2011

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3 Contents List of Figures v List of Tables viii Acknowledgements x Acronym list xi Abstract xii I Introduction 1 1 Introduction Aims of the project Organization of the document Scope Statement of the problem Project scheme i

4 2 The LISA mission LISA and gravitational waves LISA mission as a space-borne observatory LISA Pathfinder mission overview Details of the mission Details of the LTP Thermal Data and Diagnostics System Thermal effects on the TM Radiation pressure effect Radiometer effect Outgassing effect DDS thermal items description Temperature sensors Heaters Experiment Master Plan ESATAN Thermal Model II Algorithm specification 27 4 EH heaters model Resistive heaters ii

5 4.2 NTC thermistors as heaters Thermistor contact model Expressions for NTC thermistors as heaters Temperature response Data provided Data fit Temperature time responses Digital filter implementation Total temperature increment Final temperature distribution Forces and torques model description Thermal effects implementation Alpha coefficient Force computation Torque computation III Simulation 62 7 EMP summary review EH exercise iii

6 7.2 Side effects on EH Torques on both TMs from EH heaters activation Forces on the other TM from EH heaters activation Consequences in the EH due to OW heater activation Consequences in the EH due to different struts heater activation Transient periods analysis Proposed modifications Summary and conclusion 79 Bibliography 83 iv

7 List of Figures 1.1 Scheme of the project GW detection scheme LISA drag-free test masses distribution [14] LISA orbit [11] LISA Pathfinder inner distribution. The LISA Technology Package, LTP, is seen in the middle, with the two Vacuum Enclosures (VE), that host the Electrode Housings with the Test Masses, and the Optical Bench between them [7] General picture of the LPF while in operations [7] LISA PathFinder orbit [7] Layout of elements present in the VE Artist impression of the LTP IFO performance, emphasyzing the light beams on the Optical Bench (OB) and the location of the Test Masses (TMs) [6] General view of the LISA Technology Package (LTP), where the LPF Interferometer (IFO) and the Inertial Sensors (IS) are located [7] LTP frame Force requirement for each TM v

8 3.1 Scheme of distribution of the different EH thermal DDS items. The different items are attached to the outer faces of the EHs, inside the VE Distribution of the DDS items in the Optical Window (OW) Distibution of the different DDS items located in the struts Electrode Housing discretisation in the ESATAN thermal model. Red nodes refer to Molybdenum nodes from the EH while green nodes represent the sapphire electrodes [3] Model for the NTC heater contact. R and C are the thermal resistance and capacitance respectively Bode plots of different order fits for the transfer function relating the response at output node due to H3 activation Example of bode plots from EH heaters Example of bode plots of from OW heaters Example of bode plots from struts heaters Temperature response at a set of thermal nodes from electrodes EH1- MXF-MY and EH1-MXF-PY (see Table 5.2). Upper image: General view of the run. Lower image: Zoom on the second peak, where it can be observed that nodes from each electrode have grouped in two groups, the one with higher temperatures is the one just behind the heater. As can be seen, the maximum difference is of about 3mK between opposite nodes from the same electrode and about 10mK considering both electrodes. The input power has been a double pulse of 40mW and period 1000s only to H1-1, what is an extreme case because H1-1 and H1-2 will always be activated simultaneously, so differences between opposite nodes from same face will never be that far Discretisation proposed for the temperature distribution in the inner faces of the EH where the electrodes are set, with the notation for each facet. This discretisation allows the representation of the temperature asymmetries required to calculate forces in X axis and torques in Y and Z vi

9 5.7 Example of a surface partition or facet and its representative node Scheme of the equivalent forces on the TM force calculation TM partition proposed with numeration for each facet Example of torque in Z axis created by a resulting force on a TM subsection Example of pair of facets Example of the torques resulting of the application of two opposite forces Input wave for the EH heaters. H3 and H4 follow equivalent activation sequences Absolute temperature increments at different EH nodes from the X faces of TM Temperature difference between opposite faces of the EH Force response at TM1 due to H1 and H2 activation at a power of 10mW Torques in Y and Z axes on TM1 by the activation of the EH heaters Torques in Y and Z axes on TM2 by the activation of the EH heaters Temperature at EH1 when heaters from EH2 are activated Temperature difference at EH1 when heaters from EH2 are activated Force on TM1 when heaters from EH2 are activated Input wave for the OW heaters. Heaters H7 and H8 have identical heat pulses Temperature response at EH1 due to the activation of H5 and H Temperature response at EH1 due to the activation of H7 and H vii

10 7.13 Temperature difference at EH1 due to the activation of H5 and H Temperature difference at EH1 due to the activation of H7 and H Force on TM1 due to the activation of OW1 heaters (H5 and H6) Force on TM1 due to the activation of OW2 heaters (H7 and H8) Torques on the Y axis of TM1 due to the activation of the different OW heaters Torques on the Z axis of TM1 due to the activation of the different OW heaters Input wave for each Suspension Strut heaters (from H9 to H14) Averaged temperatures at +X face of TM1 due to the activation of Suspension Strut heaters H9 to H Differential temperature at X faces of TM1 due to the actication of strut heaters H9 to H Forces created on the TM1 due to the activation of H9 to H Torques on the Y axis of TM1 due to the activation of H9 to H Torques on the Z axis of TM1 due to the activation of H9 to H viii

11 List of Tables 2.1 Requirements for DC values of forces and torques. They have been obtained from acceleration requirements applying a mass of 1.96kg and a moment of inertia of kg m Location of the different TS. TS from the EH are aimed to study the thermal effects in the TM environments, while all the other TS are aimed to study the thermoelastic effects that could distort the optical path of the IFO, what is out of the scope of this document Steinhart-Hart parameters for temperature sensors. Units are [1/K] Notation of the heaters. Each heater from the EH is composed by 2 physical heaters List of physical heaters that compound each logical heater and their locations Heaters classification Parameters of the heater model List of Electrode Housing (EH) Temperature Sensors (TS) output nodes of the ESATAN transfer functions provided List of Electrode Housing nodes of the ESATAN transfer functions provided. More information can be found at Appendix A ix

12 5.3 Fit error of different order fits at specific frequencies. The error at each frequency considers the absolute value of the variation between each gain and the gain from provided data normalised with the provided data gain Position of each facet in the EH discretisation with their representative nodes from the ESATAN Thermal model. These nodes have been obtained from [3] Values of the considered parameters Constants used in the algorithm Visibility table between the different TM and EH elements. A facet from TM column has visibility link with EH facet from the same row (Example: TM facet 18 is visible from EH facets 17, 18, 19 and View factor values matrix for the torque calculation process under current notation for EH1-TM1. Same view factors can be applied to EH2-TM2. They have been obtained by a generic form for not-aligned parallel panels in Appendix B Areas of each TM facet Table with the correspondances between facet directions and vector notation Vector d k values. Parameter d is 1/4 of a TM side, i.e. d = m List of TM pairs of opposite facets. Same pairs can be made with EH facets EH heater activation parameters. This exercise is applied to both EHs but in separate experiments OW and Struts heater activation parameters. These sequences are applied simultaneously to both heaters in the same OW and separately for each OW and for each strut heater x

13 Agraïments Aquest projecte final de carrera que teniu a les mans no haguera assolit el seu final sense la valuosa ajuda i consell de diferents persones que m han envoltat durant aquest últim any. Des d aquestes modestes línies m agradaria poder agrair a tots el suport rebut. Molt sincerament, vull agrair a l Alberto Lobo el seu recolzament, esforç i fermesa a l hora de tirar endavant aquest projecte en moments gens trivials. Ell, més que ningú, es mereix tota la meva gratitud, i en ell rau gran part de l èxit d aquest treball. Vull agrair també a la gent de l IEEC que han fet del dia a dia de l elaboració d aquest projecte un conjunt de moments divertits i d aportacions trascendentals. Gràcies, doncs, Nacho, Marc, Júlia, Alberto G i Markus, gràcies també als companys del grup LISA de l IEEC escampats arreu: Ivan, Aleix, Lluís, Carlos i Pris, i especialment als components del que podríem anomenar "l IEEC a l exili", Pep i Miquel, que des de la distància han fet contribucions puntuals però l ajuda dels quals ha sigut decisiva. Aquest projecte també és un punt i final a tota una etapa com a estudiant a l escola ETSEIAT de Terrassa. Des d aquí vull donar les gràcies a tots els professors, especialment a l Ed i a l Elena pels seus importants comentaris a l hora d enllestir aquest treball. D altra banda, vull donar les gràcies a tots els amics i companys de l escola de tots aquests anys per tants moments -i de tants colors- que hi hem viscut. Mirant ja cap a casa, vull donar les gràcies a tots els amics de Manresa, per ser-hi tots els divendres quan, acabades les setmanes, enfilo el camí de volta cap a casa. Finalment, també vull donar les gràcies als meus pares i a la meva germana Marta, per tota la paciència que han hagut de tenir amb mi al llarg d aquests anys d universitat. Aquest projecte ha estat possible gràcies al contracte MICINN-CSIC ESP També vull agrair l ajuda rebuda pel Departament de Projectes d Enginyeria de la UPC a l ETSEIAT, que em va permetre participar al 8è Congrés del projecte LISA. Barcelona, Gener del 2011 xi

14 Acronyms ASD CGS CQG CSIC DDS DMU EH ESA ESAC ESATAN ESOC ESTEC ETSEIAT GRS ICE IEEC IS LISA LPF LTP LTPDA NTC OSE OB OBC OW SNR SS STOC TM TN TS UPC UTN VE Astrium Deutschland/Germany Carlo Gavazzi Space Classical and Quantum Gravity Centro Superior de Investigaciones Científicas Data Management and Diagnostics Sub-System Data Management Unit Electrode Housing European Space Agency European Space Astronomy Centre ESA Thermal Analyser European Science and Operations Centre European Space and Technology Centre Escola Tècnica Superior d Enginyeries Industrial i Aeronàutica de Terrassa Gravitational Reference Sensor Institut de Ciències de l Espai Institut d Estudis Espacials de Catalunya Inertial Sensor Laser Interferometer Space Antenna LISA Pathfinder LISA Technology Package LTP Data Analysis Negative Temperature Coefficient Offline Simulations Environment Optical Bench On Board Computer Optical Window Signal-to-Noise Ratio Suspension Struts Science and Technology Operations Centre Test Mass Technical Note Temperature Sensor Universitat Politècnica de Catalunya University of Trento Vacuum Enclosure xii

15 Abstract The main aim of the LISA Pathfinder (LPF) mission is to place two Test Masses into a very stable drag free environment and make very precise interferometric measurements of displacements between them. In order to achieve such an objective, it is necessary to characterize and model the disturbances that will appear. Amongst the different physical effects that will appear onboard, temperature fluctuations in the Electrode Housing (EH) generate disturbances on the interferometer (IFO) readouts, therefore they must be known and controlled. Consequently, a simulator of the whole LPF is being developed to provide a validation tool for the mission operations tele-commanding chain, as well as for a deeper understanding of the underlying physical processes happening in the LTP (LISA Technology Package), the instrument hosting the Test Masses. In this study, the whole algorithm required to calculate the forces and torques on the Test Masses due to the activation of the different LTP control heaters is detailed. More specifically, transfer functions relating heat input signals to temperature increments on the Test Masses (TMs) in the LTP frequency band, from 1 mhz to 30 mhz, are determined. Following, the EH environment is studied and discretised to calculate the forces and torques that appear through thermal effects (radiometer effect, radiation pressure effect, etc). Finally, the algorithm is implemented and some experiments from the EMP (Experiment Master Plan) are simulated to evaluate the associated dynamical effects on the Test Masses. A complete thermal model of the entire LPF spacecraft plus payload, elaborated and maintained at European Space Technology Centre (ESTEC), was used to obtain temperature distributions in response to heat inputs at prescribed spots (heaters). As a result of this work, a poster was presented at the 8th LISA Symposium that was held in the SLAC (Stanford Linear Accelerator Center), California (USA) in June The proceedings of this Symposium allowed the author to submit a paper to be published in a volume of the Journal of Physics: Conference Series, in xiii

16 Part I Introduction 1

17 Chapter 1 Introduction Amongst all the missions that nowadays are under development, maybe LISA Pathfinder (LPF, earlier known as Smart-2), as a precursor of LISA (Laser Interferometer Space Antenna) mission, is one of the most thrilling due to the amount of new challenges it faces. LISA will be a gravitational wave telescope, a so far unexplored tool for the observation of the Universe. LISA is a collaborative ESA-NASA mission intended to create the first space-borne gravitational wave detector. However, due to the huge technological challenge that it represents, specifically in its low frequency stability requirements -lower than in any other previous mission-, a testing mission called LISA Pathfinder (LPF) has been conceived to pave the way to the later deployment of the LISA mission. The LISA Pathfinder mission is thus conceived as a technology demonstrator intended to verify that the main requirements of the future LISA mission can be met. The LPF spacecraft hosts two test masses in a near-perfect gravitational free fall, and measures their relative position by interferometry (IFO) with an extreme accuracy. Both Test Masses and the interferometer are in the core of the LPF, in the so called LISA Technology Package (LTP). Since the early stages of LPF, IEEC has been involved in different design aspects of the LTP. The Institute is responsible for the Data and Diagnostics System (DDS), 2

18 1.1. AIMS OF THE PROJECT Introduction which include thermal and magnetic diagnostics as well as a particle counter and spectrometer, plus the Data Management Unit (DMU) which is the LTP computer. The main purpose of the DDS is to analyse the impact of various disturbances in the LTP and to model them so that improvements can be made in the sensitivity of LISA [1]. Following to these high stability requirements there is the necessity of checking and controlling every action being executed in flight. Consequently, a complete simulator of the whole behaviour of the spacecraft -the so called LTP Offline Simulations Environment (OSE)- is being designed as a key piece of the mission. One of the aims of this simulator is to recreate the thermal noise effects in the interferometer (IFO), to make sure that they are low enough to meet the requirements. Additionally, some experiments have been designed to be carried on board so as to calibrate these thermal effects. Therefore, this simulator will be used as well to previously check these experiments, that include the application of tiny and controlled amounts of heat signals to see how the LTP behaves. The present study focuses on the analysis and development of the algorithm that describes the consequences in the Test Masses (TMs) due to thermal asymmetries in the LTP. Such a process requires an analysis of the different thermal effects appearing in the TMs environment (radiation pressure effect, radiometer effect...), as well with a discretisation of the system. 1.1 Aims of the project The different objectives of this study are listed below: Define a detailed algorithm capable of calculating the forces and torques that appear in the TMs of the LPF. Implement the algorithm. Apply some exercises to explore the model s capabilities and predictions. 3

19 1.2. ORGANIZATION OF THE DOCUMENT Introduction 1.2 Organization of the document This document is divided in three blocks or parts, each of them split in different chapters. It is organized as follows: Part I introduces the mission in Chapter 2 and describes the spacecraft and the different thermal items of the thermal Data and Diagnostics System (DDS) in Chapter 3. Part II explains how the system is modelled. Chapter 4 describes how the heaters are modelled, Chapter 5 describes how the temperature distributions around the Test Masses (TMs) are obtained and Chapter 6 details the process to determine the forces and torques appearing in the TMs as consequence of thermal asymmetries. Part III is centred in the simulation of different experiments using the algorithm. In it, some exercises are applied to the model in Chapter 7 to see the forces and torques generated on two Test Masses (TMs). Finally, the results are analysed and the conclusions presented. The document is as well accompanied by a series of appendices that complement some specific aspects. 1.3 Scope This document is intended to report the whole description of the procedures that model the thermal effects on the TMs of the LTP, i.e. the forces and torques that appear on the TMs due to the presence of temperature asymmetries. A whole description of the system is presented with the features of the already identified thermal effects, with the final aim of embedding the thermal model into a simulator that recreates the behaviour of the whole spacecraft. Finally, an analysis of the consequences of the thermal effects using the developed algorithms with the current simulator will be presented. 4

20 1.4. STATEMENT OF THE PROBLEM Introduction 1.4 Statement of the problem Nowadays, the OSE model is nearly finished, and only few packages such as the thermal model designed and described in this document are still pending to be added. On the other hand, some experimental tests are being performed in UTN so as to get more accurate values for the parameters that drive the thermal effects in the model. These tests involve simulating the drag-free environment in a torsional pendulum, where radiation pressure effect, radiometer effect and outgassing can be observed and measured. Finally, for the last years IEEC has also been experimenting with the thermal hardware and its features. The electronics have already been designed and tested, and different experiments have been carried out with the thermistors and heaters that will be boarded, so as to set a precise thermal and electric model of its behaviour. 1.5 Project scheme As a summary of the different blocks composing this project, Figure 1.1 presents a scheme of the different parts, as they are organised. 5

21 1.5. PROJECT SCHEME Introduction Figure 1.1: Scheme of the project. 6

22 Chapter 2 The LISA mission LISA mission, as it is called the Laser Interferometer Space Antenna mission, and its technology demonstrator LISA Pathfinder are devoted to the final aim of observing gravitational waves. Prior to the description of both missions and their requirements, it will be useful to briefly introduce the nature of gravitational waves and the working principles of their detectors. 2.1 LISA and gravitational waves Gravitational waves were predicted by Einstein on the basis of his Theory of General Relativity in This new type of radiation presented some parallelism with electromagnetic radiation but instead of being based on the charge interaction between particles is based on the mass interaction between bodies. In other words, it was created by accelerated massive bodies, was expected to propagate through the space at the speed of light and could interact with massive bodies exposed to the gravitational radiation. The radiation of gravitational waves should be therefore detectable by placing two bodies in the space and measuring their relative distance variation created by a wave passing through, in a similar way as electromagnetic antennas do with charged par- 7

23 2.1. LISA AND GRAVITATIONAL WAVES The LISA mission ticles. The resonance frequency of the wave to observe is defined by the distance between the two bodies, being half of the wavelength the optimum length. Figure 2.1 shows a scheme of a passing of gravitational wave detection. Figure 2.1: GW detection scheme. Such an experiment is far from being trivial because the displacements created by common gravitational waves extremely tiny, and the disturbances that these bodies would receive from other energy sources would largely exceed the signal. The solution to this problem consists in isolating the two bodies or test masses in a free fall or drag free environment so the noise is reduced to a level that enables the observation of gravitational waves. Different on-ground gravitational wave detectors as LIGO [17] have been developed so far, making the length measurements by extremely high precision interferometry techniques. However, they are limited to the khz measurements, as the distance between the test masses is limited to a few km and, consequently, the detectable wavelength cannot be below khz. Additionally, for lower frequencies (at about 1Hz) the seismic noise makes gravitational wave detection unfeasible. Anyway, as huge bodies tend to move slowly, the number of GW sources in the gravitational radiation spectrum for frequencies higher than approximately 1Hz is relatively limited, and on-ground gravitational wave detectors can just observe gravitational radiation from quick bodies, such as pulsars. Consequently, a more interesting bandwidth (BW) for the observation of gravitational waves lies in a lower range of the spectrum, e.g. around 1mHz. 8

24 2.2. LISA MISSION AS A SPACE-BORNE OBSERVATORY The LISA mission 2.2 LISA mission as a space-borne observatory The LISA mission was designed on the basis of the considerations in previous section. It is aimed to set the first space-borne gravitational wave detector, by placing three satellites, each one containing a pair of test masses, in drag free conditions. The distance between spacecraft, about km and forming an equilateral triangle (see Figure 2.2), is intended to observe and measure gravitational waves in the bandwidth from 0.1mHz to 0.1Hz, covering a wide range of sources of gravitational radiation. Theses spacecraft will as well be linked by laser beams in order to measure the relative displacements of different satellite s test masses with picometer resolution by precision interferometry. The stability requirement in the test masses environment of each satellite to assure the observation of gravitational waves is given in power spectral density, as [1] S 1/2 a (w) {[ 1 + in the band from 0.1mHz to 0.1Hz. ( ) ] ω/2π 4 [ 1 + 8mHz ( )] } 1/2 0.1mHz ms 2 Hz 1/2 (2.1) ω/2π The orbit where this triangular constellation will be placed is an heliocentric one, at the same distance to the sun as the Earth but trailing it 20 behind. The plane of the constellation is inclined 60 with respect to the ecliptic plane, as seen in Figure 2.3, and the whole triangle makes a 360 rotation around its barycentre per year. This favours compliance with the noise requirements as it is far enough from the Earth and its magnetic influence, while it simplifies the communication problems. Such a mission requires a particularly high technology level that in some aspects has not been developed to date. Among other, it demands extremely stable thermal and magnetic environment conditions. This requires to develop new extra-accurate instruments for sensing and actuating under these circumstances. It includes, for example, new technology related to micro-thrusters and extremely precise laser interferometers, required for very accurate relative-distance measurements between different points in space. 9

25 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission Figure 2.2: LISA drag-free test masses distribution [14]. Figure 2.3: LISA orbit [11]. 2.3 LISA Pathfinder mission overview After realizing the amount of effort that LISA mission requires, LPF was conceived as the best option for making a previous test of the functionality and viability of LISA s 10

26 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission most critic systems that could not be tested on ground. LPF aims to demonstrate the technological capability for the future LISA mission and it was born with the specific objectives that follow: Demonstrate drag-free conditions are achievable. Test the feasibility of laser interferometry in the BW of interest. Test lifetime and performance of the different instruments in space conditions. In order to achieve that, LISA Pathfinder will carry a reduced version of LISA s drag free system: LISA s initial 5 million km separation between masses is here shrunk to 35 centimeters, what discards the possibility of detecting gravitational waves but, on the other hand, will allow the test and validation of the different items in the above list. With respect to the orbit, LISA Pathfinder will be placed in a Lissajous orbit around Lagrange point 1 (L1), at 1.5 million km from the Earth towards the sun Details of the mission The LPF spacecraft has been conceived as a compact standard octagonal-shaped structure. The power is provided by its end-of-life 650W solar panels that cover a whole side of the spacecraft. Its main instrument or payload instrument, the LISA Technology Package (LTP), is in the middle of the structure, while all the other equipment, such as the On Board Computer (OBC), the Data Management Unit (DMU), the Radiation Monitor (RM), etc. are set in the outer lateral compartments of the structure, as seen in Figure 2.4. Figure 2.5 shows the external aspect of LPF in the operation phase. The dimensions of the spacecraft are about 2m wide and 96cm in depth, and its launch weight is estimated to 1900kg, although 1100kg will be of propellant. 11

27 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission Figure 2.4: LISA Pathfinder inner distribution. The LISA Technology Package, LTP, is seen in the middle, with the two Vacuum Enclosures (VE), that host the Electrode Housings with the Test Masses, and the Optical Bench between them [7]. Figure 2.5: General picture of the LPF while in operations [7]. The spacecraft is equipped with a set of FEEP (Field Emission Electric Propulsion) micro-thrusters that will be used during the scientific phases to preserve the drag-free conditions, i.e. to make the spacecraft follow the free-fall of the test masses. On the other hand, the LTP will be initially accompanied by a propulsion module with the propellant required to reach its operation orbit. This propulsion module will be detached prior to drag-free scientific operations because the remnant propellant of the fuel tanks could generate disturbances in the inertial sensors. Figure 2.6 shows the trajectory to follow before its injection in the Lissajous orbit. The chosen launcher is the new ESA VEGA launcher, conceived to place up to 2000kg of payload in geostationary orbit (GSO). Its purpose with LPF is to launch it into a GSO. From there it will use its own propulsion module to reach Lagrange 1 (L1). The total mission length is of 6 months, and the current estimation for the mission launching is scheduled for

28 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission Figure 2.6: LISA PathFinder orbit [7] Details of the LTP The LTP, the LPF Technology Package, can be considered the payload of LPF, as it contains mainly the test masses in free fall conditions and the interferometer (IFO). The two test masses are 46mm to the side cubes made of an Au-Pt alloy, weighing 1.96kg each one. Each TM floats in the middle of an Electrode Housing (EH), which is a cubic box of 54mm to the side made of Molybdenum that hosts, in its inner faces, the sapphire electrodes used to sense TM position, and to actuate over it. The gap between the TM and its EH is 4mm. Two cylindrical vessels made of Titanium, the Vacuum Enclosures (VE), host the EHs, as shown in Figure 2.7. These VEs include "getter pumps" to maintain an internal pressure of 10 5 Pa. The two VEs are separated by the Optical Bench (OB), which contains the optical elements of the interferometer. Each VE has an Optical Window (OW) where the laser beam passes through, as seen in Figure 2.8, so that the laser beam can be reflected off the TMs. The set composed by the two VEs and the Optical Bench is held by eight Suspen- 13

29 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission Figure 2.7: Layout of elements present in the VE. sion Struts, which attach it two the spacecraft structure. The whole set is the LTP, and Figure 2.9 presents a general view of it. The IFO is intended to make different measurements: Displacement measurements: The IFO measures the X axis displacement between TM1 and the spacecraft and the displacement between both TMs (see Figure 2.10). Rotation measurements: The IFO can also measure the rotations of the TMs around their Y and Z axes (see Figure 2.10). 14

30 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission Figure 2.8: Artist impression of the LTP IFO performance, emphasyzing the light beams on the Optical Bench (OB) and the location of the Test Masses (TMs) [6]. Figure 2.9: General view of the LISA Technology Package (LTP), where the LPF Interferometer (IFO) and the Inertial Sensors (IS) are located [7]. LTP noise requirements The LPF main noise requirements are relaxed one order of magnitude with respect to LISA requirements. They are given as spectral density of noise by [18] " 1/2 S a, LPF (ω) ω/2π mhz 2 # m s 2 Hz 1/2 (2.2)

31 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission Figure 2.10: LTP frame. in the band from 1mHz to 30 mhz. Considering that the mass of each TM is 1.96 kg [2], Equation 2.2 can be translated into a tidal force requirement: S 1/2 F, LPF (ω) [ 1 + ( ) ] ω/2π 2 N Hz 1/2 (2.3) 3 mhz Figure 2.11 shows this requirement. 16

32 2.3. LISA PATHFINDER MISSION OVERVIEW The LISA mission Figure 2.11: Force requirement for each TM. On the other hand, there are a set of requirements for the DC values that forces and torques in different axes have to meet [18]. Table 2.1 shows these requirements. Magnitude Value Unit Observations Maximum DC difference of force along X N Between the TMs Maximum DC difference of force along Y N Between the TMs Maximum DC difference of force along Z N Between the TMs Maximum DC torque along φ Nm On each TM Maximum DC torque along η Nm On each TM Maximum DC torque along θ Nm On each TM Table 2.1: Requirements for DC values of forces and torques. They have been obtained from acceleration requirements applying a mass of 1.96kg and a moment of inertia of kg m 2. 17

33 Chapter 3 Thermal Data and Diagnostics System The Thermal Data and Diagnostics System (Thermal DDS) is the set of devices responsible for the characterisation of the thermal effects happening in the LTP that could alter the IFO readouts. It is composed by heaters and temperature sensors strategically distributed across the LTP, and by the electronic equipment that allows their control. The main objective of the Thermal DDS is twofold: 1. To sense with very high precision (< 10 5 K/ Hz) the temperature at various places in the LTP to obtain a temperature map. 2. To determine the relationship between temperature variations and phase changes in the interferometer readout channels. Such relationship is needed to establish the quantitative relevance of temperature fluctuations noise in the total LTP budget, Equation 2.2. In this chapter, the different thermal effects are explained before describing the thermal items -heaters and temperature sensors- and their locations in the LTP. 18

34 3.1. THERMAL EFFECTS ON THE TM Thermal Data and Diagnostics System 3.1 Thermal effects on the TM Temperature differences across the TMs environment can produce differential pressures that turn into to net forces and torques on the TMs. Three different thermal effects have been identified [11] as possible mechanisms that will create noticeable dynamic effects on the TMs: Radiation pressure effect The Radiation pressure effect is based on the temperature-dependence of the radiation emitted by a surface, following the Stefan-Boltzmann Law. The force generated is expressed as: F radiation = 8 ɛ ij AσT 3 T [N] (3.1) 3 c where σ is the Stefan-Boltzmann constant, c is the speed of light, ɛ ij is the view factor between two surfaces, A is the area of the section where the force is applied, T is the absolute temperature and T is the temperature difference between two surfaces Radiometer effect The Radiometer effect appears in rarefied atmospheres where the particles have a mean free path much longer than the distance between the surfaces of the enclosing volume. The consequent force is represented here as: F radiometer = 1 4 Ap T [N] (3.2) T where p stands for the remaining gas static pressure. A, T and T have same meanings than as Equation

35 3.2. DDS THERMAL ITEMS DESCRIPTION Thermal Data and Diagnostics System Outgassing effect The outgassing effect is caused by the detachment of gas molecules from the surface, in a low pressure environments. Its behaviour is strongly related with the surface degree of coarseness, and although its consequences could be at the same level of the ones from previous effects, current models are not as detailed and accurate. More details of all these effects can be found at [11]. Unlike previous thermal effects the outgassing effect does not have a clear model and still presents uncertainties in the parameters that drive it. It will not be implemented in the model. 3.2 DDS thermal items description The DDS thermal items consist basically of a set of 24 temperature sensors and 14 heaters located in specific points in the LTP. All of them are connected to 2 Data Acquisition Units (DAUs) inside the Data Management Unit (DMU), which processes the telecommands Temperature sensors The temperature sensors of the DDS consist of a set of 24 thermistors located in different representative points of the LTP. All of them are Negative Thermal Coefficient (NTC) thermistors, and they were selected because of their higher sensitivity coefficient at the LTP temperature range compared to other temperature sensors such as Platinum sensors. They are placed in different positions, each of them with different purposes. Table 3.1 presents the different locations and applicable notation. Figures 3.1, 3.2 and 3.3 show the position of each temperature sensor. 20

36 3.2. DDS THERMAL ITEMS DESCRIPTION Thermal Data and Diagnostics System Temperature sensors TS1 to TS8 TS9 to TS12 and TS23 to TS24 TS13 to TS16 TS17 to TS22 Location Electrode Housing (X faces) Optical Window Optical Bench Suspension Struts Table 3.1: Location of the different TS. TS from the EH are aimed to study the thermal effects in the TM environments, while all the other TS are aimed to study the thermoelastic effects that could distort the optical path of the IFO, what is out of the scope of this document. The expression which best describes the behaviour of an NTC thermistor is the so called Steinhart-Hart equation, that relates the temperature of the sensor, T, with its electric resistance R, as seen in Equation T = A + B ln (R) + C [ln (R)]3 (3.3) The parameters of this expression vary depending on the TS model and are found experimentally. For the LTP temperature sensors, the chosen model was 620k Betatherm, and its parameters A, B, C where determined in [13] are the ones in Table 3.2. A B C Table 3.2: Steinhart-Hart parameters for temperature sensors. Units are [1/K] Heaters There are 14 logical heaters in the LTP. Their locations are as follow: 4 heaters are placed in the EH. 4 heaters distributed in the 2 Optical Windows, 2 heaters in each one. 6 heaters on 6 Suspension Struts, coinciding with the temperature sensors. The notation of the different logical heaters is shown in Table

37 3.2. DDS THERMAL ITEMS DESCRIPTION Thermal Data and Diagnostics System H4 2 EH2 (IS 2) TS5 TS TS6 H3 2 H4 1 TS8 TS7 H3 1 H2 1 TS3 H2 2 z LTP frame TS1 TS4 y H1 1 H1 2 EH1 (IS 1) x TS2 Figure 3.1: Scheme of distribution of the different EH thermal DDS items. The different items are attached to the outer faces of the EHs, inside the VE. Heaters H1 to H4 H5 to H8 H9 to H14 Location Electrode Housing Optical Window Suspension Struts Table 3.3: Notation of the heaters. Each heater from the EH is composed by 2 physical heaters. Figures 3.1, 3.2 and 3.3 show the position of each heater, and Table 3.4 links physical heaters with their equivalent logical heater. The reason of the distinction between logical heaters and physical heaters is functional. For design reasons, in each X face of an EH there are two physical heaters but they are always activated simultaneoulsy, so the activation of a logical heater HX -see Table 3.4- actually represents the simultaneous activation of both physical heaters HX- 1 and HX-2 present in the same face. Therefore, although the total number of logical heaters in the LTP is 14 there are actually 18 physical heaters. 22

38 3.2. DDS THERMAL ITEMS DESCRIPTION Thermal Data and Diagnostics System Figure 3.2: Distribution of the DDS items in the Optical Window (OW). Figure 3.3: Distibution of the different DDS items located in the struts. Initially, all the heaters had to be standard resistive heaters with a linear resistive behaviour to take advantage of its simplicity, but as explained in [15] this type of heaters presented magnetic incompatibilities and outgassing related problems that could compromise the low magnetisation requirements in the EH. As a consequence, and due to the low power required to be applied in the EH, a thermistor similar to the NTC thermistors used as Temperature Sensors was selected to work as a heater in the EH faces, although it was finally shown that NTC thermistors also presented remanent magnetisation and had to be subjected to a demagnetization process [16]. 23

39 3.3. EXPERIMENT MASTER PLAN Thermal Data and Diagnostics System Logical Heater Physical Heaters Location (LTP frame) H1 H1-1 EH1 +X face -Y side H1-2 EH1 +X face +Y side H2 H2-1 EH1 -X face -Y side H2-2 EH1 -X face +Y side H3 H3-1 EH2 +X face +Y side H3-2 EH2 +X face -Y side H4 H4-1 EH2 -X face +Y side H4-2 EH2 -X face -Y side Table 3.4: List of physical heaters that compound each logical heater and their locations. Therefore, two types of DDS heaters are found in the LTP: NTC thermistors in the EH outer walls. Standard resistive heaters in the rest of heater locations (Struts and OW). The NTC thermistors are different from the ones working as TS, as they have a nominal resistance of 2kΩ (at 25 C) while TS have a nominal resistance of 10kΩ, allowing more power to be applied with less voltage. However, using NTC thermistors will involve complications in the algorithm specification, as they do not present linear behaviour (see Equation 3.3). Specifically, the relatively high amount of power that they have to dissipate (tens of mw, see Chapter 7) causes them to heat enough to vary significantly their electric resistance, until a steadystate is reached. Therefore, a specific and detailed model for the NTC heaters must be considered, and is described in Chapter Experiment Master Plan Due to the difficulty of accurately finding values for thermal effects in on-ground conditions, a series of experiments have been developed to be carried out during the mission flight with the final aim of characterising different effects. This type of experiments are 24

40 3.4. ESATAN THERMAL MODEL Thermal Data and Diagnostics System part of the so called Experiment Master Plan (EMP) [19, 2], which is the sequence of experiments/measurements to be performed during the mission lifetime. Thermal experiments in the EMP aim to characterize the weight of the various thermal effects, as consequences of asymmetric temperature fluctuations appearing in the LTP. Such experiments consist of applying well-defined heat loads at specific points of LTP and observing induced temperature variations at particular points in order to see which is their impact on interferometer readouts through all the process. Their ultimate goal is to assess the influence of these effects so that the temperature noise can eventually be subtracted out from the IFO readout. A series of power input values were proposed in [12], and simulations with these inputs will be presented in Chapters ESATAN Thermal Model In order to simulate the complete thermal behavior of the whole LTP, a combinated model linking an LTP thermal model from CGS (Carlo Gavazzi Space, a company sited in Milan and responsible for the procurement of the Inertial Sensors that contains the TMs) and a S/C general thermal model from ASD (Astrium Deutschland/Germany, the LTP Architect) was developed at ESTEC. The software used in both models was ESA- TAN, which is a finite-element program based on the Lumped Parameter method that consists of dividing an object -in this case, the whole LPF- into many elements, nodes, with specific properties (size, thermal capacitance and resistivity...) and linking them with thermal resistances that make the Fourier law to be satisfied at a local level. The main goal of this software is to determine temperature distributions in response to heat inputs applied at specific points. It produces transfer functions from one arbitrary node to another, which has been particularly useful for this project. The fact of having these transfer functions separated from the model allows the simulation of specific pointto-point responses without having to run the whole thermal model. Detailed information of the ESATAN model of the LTP can be found at [3]. 25

41 3.4. ESATAN THERMAL MODEL Thermal Data and Diagnostics System The resulting model of the whole spacecraft has >20,000 thermal nodes. Figure 3.4 partially shows how the EH with its electrodes is discretised. Figure 3.4: Electrode Housing discretisation in the ESATAN thermal model. Red nodes refer to Molybdenum nodes from the EH while green nodes represent the sapphire electrodes [3]. 26

42 Part II Algorithm specification 27

43 Chapter 4 EH heaters model The purpose of this chapter is to model the behaviour of the heaters present in the LTP. As stated in Section 3.2.2, not all the heaters are resistive and some of them are NTC thermistors that do not present a linear behaviour. Therefore, their modelisation becomes a little more complicated. Table 4.1 specifies the different heater classification: Heater H1 to H4 H5 to H14 Type NTC thermistors Resistive heaters Table 4.1: Heaters classification. 4.1 Resistive heaters Resistive heaters are assumed to present an ideal resistive behaviour, i.e, their nominal resistance is constant and the delivered power can be directly determined from the voltage applied. Neglecting the thermal contact, the power is obtained with Equation

44 4.2. NTC THERMISTORS AS HEATERS EH heaters model P i (t) = [V (t)]2 R res (4.1) where V (t) is the voltage applied to the heater. On the other hand, R res is the nominal electrical resistance which is of 45Ω for all resistive heaters [8]. As the sequence of values must be discrete, the applicable expression is where V (t K ) is the voltage applied at time t k. P i (t k ) = [V (t k)] 2 R res (4.2) 4.2 NTC thermistors as heaters NTC thermistors are more complex and require an accurate treatment. The fact that their electric resistance varies with the temperature, and that the latter keeps increasing during the experiments, is the main reason, as it makes the injected power to vary through time. Consequently, the thermal contact between the thermistors and the surface where each of them is attached must be modelled in order to determine the real power injected in the EH at each moment Thermistor contact model The solution proposed in [14] is to model the heater and the surface as a RC filter. Figure 4.1 presents the scheme of the model. The equation of the model is the energy balance P i (T (t)) = T (t) T 0 Θ dt (t) + C dt (4.3) where T (t) stands for the temperature inside the thermistor and T 0 is the temperature of the surface, that is assumed to be constant here due to the high thermal inertia of the EH compared to the thermal capacitance of the thermistor. P i (T ) is the power 29

45 4.2. NTC THERMISTORS AS HEATERS EH heaters model Figure 4.1: Model for the NTC heater contact. R and C are the thermal resistance and capacitance respectively. dissipated by the heater by Joule effect, assuming that the power is only dissipated by thermal conduction to the EH, Appendix C discusses this assumption. The purpose is to solve Equation 4.3 so as to obtain an expression of the real power injected in the EH, P r (t) for a given voltage V. Defining a new variable µ as µ = T (t) T 0 (4.4) we have and Equation 4.3 is rewritten as dµ dt = dt (t) dt (4.5) P i (µ + T 0 ) = µ Θ + C dµ dt (4.6) This new equation can be solved by linearisation, assuming small variations of temperature. The Taylor series of P i (µ + T 0 ) is Replacing in Equation 4.6 P i (µ + T 0 ) P i µ=0 + dp i dt µ=0µ + ɛ(µ 2 ) (4.7) P i µ=0 + dp i dt µ=0µ µ Θ + C dµ dt (4.8) The term dp i dt µ=0 is expanded as dp i dt µ=0 = d dt ( V 2 R(T ) ) µ=0 = V 2 d dt ( ) 1 µ=0 (4.9) R(T ) 30

46 4.2. NTC THERMISTORS AS HEATERS EH heaters model To determine the term d dt 1 R(T ) µ=0, it is necessary to differentiate the Steinhart-Hart equation (see Section 3.2.1), whose final expression is: d dt ( ) 1 = R(T ) β R(T )T 2 (4.10) where β is β = 1 [ ( )] 2 (4.11) B + 3C ln 1 R(T ) To evaluate d 1 dt R(T ) at µ = 0 it is necessary to invert the Steinhart-Hart equation, or R(T ) = exp [ ( x(t ) y(t ) 2 ) 1/3 ( x(t ) + y(t ) ) ] 1/3 2 (4.12) where y(t ) = A 1 T (4.13) C [ ( ) ] B 3 1/2 x(t ) = + y2 (4.14) 3C 4 Therefore, dp i dt µ=0 = V 2 ( ) β β R(T ) T0 2 µ=0 = P i 2 T µ=0 = P i µ=0 (4.15) 0 Equation 4.8 is now written as P i µ=0 + P i µ=0 µ = µ Θ + C dµ dt (4.16) Rearranging and using a simpler notation P i P i µ=0 and P i P i µ=0, dµ dt = P i C + 1 ( P i 1 ) µ (4.17) C Θ Defining a 1 C ( P i 1 Θ), then integrating dµ P i ac = adt + K (4.18) 31

47 4.2. NTC THERMISTORS AS HEATERS EH heaters model µ P i ac ln ( µ P i ac ) = at + K (4.19) = e (at+k) = K 1 e at (4.20) µ = P i ac K 1e at (4.21) The constant K 1 is determined considering that at t = 0 the temperature is µ = 0. Therefore, K 1 = P i ac and the final expression is: µ = P i ( 1 e at ) (4.22) ac T (t) = T 0 + P i ( 1 e at ) (4.23) ac with a = 1 ( C P i 1 ) Θ and P β i = P i. Equation 4.23 gives the temperature evolution Ti 2 when a voltage V is applied to the thermistor. It is also interesting to know the response of the heater when the voltage is discontinued. It can be done rearranging Equation 4.6 as 0 = µ ΘC + dµ dt (4.24) Equation 4.24 is solved following the same preocedure, and the obtained solution is µ = K 2 e t ΘC (4.25) T (t) = T 0 + K 2 e t ΘC (4.26) The constant is determined considering that the temperature at the beginnig of the discharge is known, say T = T max at t = 0. Therefore T (t) = T 0 + (T max T 0 ) e t ΘC (4.27) Coming back to the contact model in Figure 4.1, the expression for the power injected in the surface, can be written as P r (t) = T (t) T 0) Θ (4.28) 32

48 4.2. NTC THERMISTORS AS HEATERS EH heaters model where temperature T (t) can be either the one from Equation 4.23 (ramp up case) or the one from Equation 4.27 (relaxation case). Indicative values for the thermal resistance and the thermal capacity, Θ and C in the model respectively, were found experimentally in ground [14]. presented in Table 4.2. Their values are Parameter Value Units Θ 100 K/W C 0.01 J/K Table 4.2: Parameters of the heater model. Hence ΘC = 1s and a 1s, which characterises the time constants during the ramp-up and relaxation phases Expressions for NTC thermistors as heaters The result is that there is a short transient part in the beginning of each on/off voltage state. The length of this transient is quite short compared with the expected periods of the pulses. Its value is around few seconds and depends basically on the thermal capacitance of the heater and the contact thermal resistance. After this short transient, the values of injected power tend to a stationary state. The power expression for NTC thermistors as heaters, considering the discrete time treatment, is written as P r (t k ) = 1 Θ (T (t k) T 0 ) (4.29) where T (t k ) is given by Ramp up case, when the heater is switched on: T (t k ) = T 0 + P ( i 1 e a tk) (4.30) ac 33

49 4.2. NTC THERMISTORS AS HEATERS EH heaters model Relaxation case, when the heater is switched off: T (t k ) = T 0 + (T max T 0 ) e tk ΘC (4.31) where k is the time step counter that will have to be reset at the beginning and the end of each pulse of voltage, and t is the time step (or the inverse of the algorithm sampling frequency, f s ). Finally, the resolution of each transient part will be determined by the time step value t compared to the time constant of the system, τ = 1/a in the ramp up segment and τ = ΘC in the relaxation segment (in both cases, τ 1s). In practice, t << τ should be used to get enough resolution of the transient part. 34

50 Chapter 5 Temperature response The objective of this chapter is to describe the whole procedure that leads to the determination of the temperature increments on the inner parts of the EH, as a result of a heat input due to a heater activation. It starts with the data provided from the ESATAN thermal model, continues describing the process required to get temperature responses and finally explains how temperatures obtained in specific points can be extrapolated to the rest of surfaces and which is the error added to the results. 5.1 Data provided As explained in Section 3.4, an ESATAN program produces the frequency response between nodes in a thermal model. Under these circumstances, a set of frequency response samples were provided by ESTEC, describing the response at specific points in the inner EH electrode nodes and in the different DDS temperature sensors by the application of power in each DDS heater. Tables 5.1 and 5.2 show the list of output points of the transfer functions provided. Appendix A clarifies the location of each node. As there are a total of 18 physical heaters acting as inputs and the outputs are composed by 24 temperature sensors (1 thermal node per sensor) and the different electrode nodes represented by 176 nodes (see Appendix A and Table 5.2), a total of 35

51 5.1. DATA PROVIDED Temperature response TS identifier Location Thermal node in the model TS1 EH1, +X face, -Y +Z side TS2 EH1, +X face, +Y -Z side TS3 EH1, -X face, -Y +Z side TS4 EH1, -X face, +Y -Z side TS5 EH2, -X face, -Y -Z side TS6 EH2, -X face, +Y +Z side TS7 EH2, +X face, -Y -Z side TS8 EH2, +X face, +Y +Z side Table 5.1: List of Electrode Housing (EH) Temperature Sensors (TS) output nodes of the ESA- TAN transfer functions provided. 18 ( ) = 3600 sampled transfer functions were provided. Each of these sampled transfer function consists of a list of 100 gain and phase values at 100 logarithmically-spaced values of frequencies from 10 7 Hz to 1Hz, widely exceeding the LTP required bandwidth. It is important to observe that there are output points in the inner surface of the electrodes but also output points in the different locations where all DDS temperature sensors are located Data fit The first step consists of fitting the data so as to obtain continuous transfer functions relating temperature increments in different spots by the application of heat flux in the input nodes. This allows us to operate the provided data in an easy handy way. The tool used is an LTPDA function named sdomainfit, which applies the least squares method to provided data, allowing to set the order of the resulting function. The resulting fitted function is a polynomial fraction in s domain. Each one is represented as H h n ( x h, x n, s) [K/W ] (5.1) where h stands for the input node where the heater is attached, n for the output node, and h refers to the activated heater. It is important to observe that the fitting procedure actually extrapolates the sample 36

52 5.1. DATA PROVIDED Temperature response Output identifier Location Thermal nodes in the model EH1 MXF MY EH1, -X face, -Y side to EH1 MXF P Y EH1, -X face, +Y side to EH1 P XF MY EH1, +X face, -Y side to EH1 P XF P Y EH1, +X face, +Y side to EH2 MXF MY EH2, -X face, -Y side to EH2 MXF P Y EH2, -X face, +Y side to EH2 P XF MY EH2, +X face, -Y side to EH2 P XF P Y EH2, +X face, +Y side to EH1 MY F P Z EH1, -Y face, +Z side to EH1 MY F CZ EH1, -Y face, Z centered to EH1 MY F MZ EH1, -Y face, -Z side to EH1 P Y F P Z EH1, +Y face, +Z side to EH1 P Y F CZ EH1, +Y face, Z centered to EH1 P Y F MZ EH1, +Y face, -Z side to EH2 MY F P Z EH2, -Y face, +Z side to EH2 MY F CZ EH2, -Y face, Z centered to EH2 MY F MZ EH2, -Y face, -Z side to EH2 P Y F P Z EH2, +Y face, +Z side to EH2 P Y F CZ EH2, +Y face, Z centered to EH2 P Y F MZ EH2, +Y face, -Z side to EH1 MZF MX EH1, -Z face, -Y side to EH1 MZF CP Y EH1, -Z face, +X side to EH1 MZF CMY EH1, -Z face, -X side to EH1 MZF P X EH1, -Z face, +Y side to EH1 P ZF MX EH1, +Z face, -Y side to EH1 P ZF CP Y EH1, +Z face, +X side to EH1 P ZF CMY EH1, +Z face, -X side to EH1 P ZF P X EH1, +Z face, +Y side to EH2 MZF MX EH2, -Z face, -Y side to EH2 MZF CP Y EH2, -Z face, +X side to EH2 MZF CMY EH2, -Z face, -X side to EH2 MZF P X EH2, -Z face, +Y side to EH2 P ZF MX EH2, +Z face, -Y side to EH2 P ZF CP Y EH2, +Z face, +X side to EH2 P ZF CMY EH2, +Z face, -X side to EH2 P ZF P X EH2, +Z face, +Y side to Table 5.2: List of Electrode Housing nodes of the ESATAN transfer functions provided. More information can be found at Appendix A. data beyond the minimum and maximum frequency samples of the list. Two comments are in order here: 37

53 5.1. DATA PROVIDED Temperature response Extrapolation to f=0: The nearly constant gain and practically zero phase of provided samples at low frequencies (<1e-7Hz) suggest them to be part of the asymptote that characterises a typical filter, so the extrapolation to f = 0 is straightforward, and can be easily checked. Extrapolation to f= : The low-pass nature of a system of thermal layers makes the high frequency components to be strongly damped, so their effect is practically insignificant. Consequently, the continuous transfer functions obtained will be considered applicable for the whole spectrum of frequencies. The order of the fitted functions must be also analysed to get enough accuracy without wasting computational resources. For this purpose, different-order fits were applied to a sampled transfer function and their gain in DC and in the ends of the BW were compared with the provided data. Table 5.3 shows the resulting error of each case, and Figure 5.1 shows the Bode plots of the different fits. Case Gain DC Error Gain 1mHz Error Gain 30mHz Error Provided data Order % % % Order % % % Order % % % Order % % % Order % % % Order % % % Order % % % Table 5.3: Fit error of different order fits at specific frequencies. The error at each frequency considers the absolute value of the variation between each gain and the gain from provided data normalised with the provided data gain. 38

54 5.1. DATA PROVIDED Temperature response Figure 5.1: Bode plots of different order fits for the transfer function relating the response at output node due to H3 activation. From Table 5.3, an order 8 was considered acceptable for the fit of the different transfer functions, as its fit error is less than 1%, at least in the BW of interest. Finally, Figures 5.2 to 5.4 show some Bode plots for different input/ouput nodes combinations. 39