Development and validation of the thermal diagnostics instrumentation in lisa pathfinder

Development and validation of the thermal diagnostics instrumentation in LISA Pathfinder Thesis submitted in fulfillment of the requirements for the degree of doctor Josep Sanju´ an Mu˜ noz Advisors: Juan Ramos Castro Jos´ e Alberto Lobo Guti´ errez Barcelona, May 2009 El f´ısico tiene que estudiar el lugar de la misma manera que el infinito, a saber: si es o no es, de qu´e modo es, y qu´e es. Arist´oteles, F´ısica, 208a25-28 Contents Agra¨ıments v Acronym list vii 1 Introduction 1 1.1 Gravitationalwaves .................................... 1 1.2 Ground-basedGWdetectors ............................... 3 1.2.1 Resonant mass detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Interferometric detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 LISAmissionoverview .................................. 5 1.4 LISA Pathfinder mission overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 DiagnosticsintheLTP .................................. 15 1.5.1 Thermal disturbances in the LTP . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.1.1 Noise effects inside the GRS . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1.2 Noise effects inside the OMS . . . . . . . . . . . . . . . . . . . . . . 19 1.5.2 Temperature measurement subsystem sensitivity requirement . . . . . . . . . 19 1.6 Structureofthethesis ................................... 20 2 The LTP temperature measurement subsystem 23 2.1 Stateoftheart....................................... 23 2.2 Temperaturesensors .................................... 24 2.2.1 Resistance temperature detectors (RTDs) . . . . . . . . . . . . . . . . . . . . 25 2.2.2 NTCthermistors ................................. 25 2.3 Signalprocessingchain .................................. 27 2.3.1 Principle of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.2 Analog signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2.1 Wheatstone bridge circuit . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2.2 Drive bridge circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2.3 Self-heating effect error considerations . . . . . . . . . . . . . . . . 34 2.3.2.4 Multiplexers, amplification and low-pass filter . . . . . . . . . . . . 35 2.3.2.5 Analog-to-digital conversion stage . . . . . . . . . . . . . . . . . . . 39 2.3.3 Digital signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.4 Theoretical noise equivalent temperature . . . . . . . . . . . . . . . . . . . . 41 2.3.5 Temperature coefficient and uncertainty . . . . . . . . . . . . . . . . . . . . . 42 2.3.6 Scales changing scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Thermal diagnostic items positioning and connections . . . . . . . . . . . . . . . . . 44 2.4.1 Temperature sensors location . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.2 Heaterslocation .................................. 46 2.4.3 Summary of sensors and heaters location . . . . . . . . . . . . . . . . . . . . 47 ii CONTENTS 3 Interactions with LTP subsystems 49 3.1 Thermistors magnetic polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.1 Force fluctuations in the TM due to magnetic field and magnetic field gradient 51 3.1.2 Magnetic field and magnetic field gradient in the TM due to NTCs. Measurement of the magnetic properties of the NTCs . . . . . . . . . . . . . . . 53 3.1.3 NTCs magnetic characterisation . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.4 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.5 NTCs first magnetisation curve . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1.6 Excess TM noise calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Interferences in the thermistors in the GRS . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.1 Analog signal processing chain . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.2 Signal demodulation process . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.3 Digital filtering and downsampling . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.4 Generalisation for any frequency . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.5 Experimentalresults ............................... 68 3.2.6 Permitted amplitudes of the interferences . . . . . . . . . . . . . . . . . . . . 69 3.2.6.1 Sine wave interferences . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.6.2 Electrical ground fluctuations . . . . . . . . . . . . . . . . . . . . . 70 4 Test bed design for the TMS validation 73 4.1 Test bed for the LTP requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1 Insulator design: theoretical analysis . . . . . . . . . . . . . . . . . . . . . . 74 4.1.2 Insulatorsizing .................................. 75 4.1.3 Heat leakage through thermistors’ cables . . . . . . . . . . . . . . . . . . . . 77 4.2 TestbedfortheLISAMBW ............................... 81 4.2.1 Solution: differential measurements . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.2 Test bed non-idealities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.3 Temperature control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2.3.1 Feedforward filter design . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.3.2 Feedback-feedforward control . . . . . . . . . . . . . . . . . . . . . 91 5 TMS performance tests 93 5.1 Tests in the LTP measurement bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 93 5.1.1 PrototypeTMStest ................................ 93 5.1.2 Engineering model test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.1.2.1 Excess noise due to non-ideal analog-to-digital conversion . . . . . . 103 5.1.2.2 Excess noise in the thermistors . . . . . . . . . . . . . . . . . . . . 105 5.1.3 Flightmodeltest .................................107 5.2 TestsintheLISAMBW .................................108 6 Performance improvements of the TMS 111 6.1 ADC non-linear errors correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1.1 Non-ideal quantisation noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1.1.1 ADC bit error description . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1.1.2 Dither signal effect in ideal quantisers . . . . . . . . . . . . . . . . 114 6.1.1.3 Dither signal effect on non-ideal quantisers . . . . . . . . . . . . . . 116 6.1.1.4 INL effect on general signals . . . . . . . . . . . . . . . . . . . . . . 118 6.1.2 Mitigation of ADC errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.1.2.1 Gaussian noise dither injection . . . . . . . . . . . . . . . . . . . . 120 6.1.2.2 Triangular wave dither injection . . . . . . . . . . . . . . . . . . . . 122 6.1.3 Test set up and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.1.3.1 Testsetup................................124 CONTENTS iii 6.1.3.2 Testresults ...............................125 6.2 Floornoisereduction ...................................127 6.2.1 Thermistor nominal resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2.2 Power increase in thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7 Thermal experiments in the GRS 135 7.1 Estimation of errors on parameters: the Cram´er-Rao bound . . . . . . . . . . . . . 135 7.2 Purpose of the thermal experiments in the GRS . . . . . . . . . . . . . . . . . . . . 136 7.3 Thermal coupling within the LTP dynamics . . . . . . . . . . . . . . . . . . . . . . 138 7.3.1 Available hardware description . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.3.2 Signalsdefinition .................................141 7.3.3 Parameters estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.4 Estimation of α..................................143 7.3.5 Resolving radiation pressure and radiometer effect . . . . . . . . . . . . . . . 145 7.4 Simulations .........................................150 7.4.1 x-axisLTPdynamics................................150 7.4.2 Implementation in Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.4.3 Thermal excitation and parameters estimation . . . . . . . . . . . . . . . . . 154 8 Conclusion 161 A Temperature coefficient and uncertainty analysis 165 A.1 Voltageleveladaption ...................................165 A.1.1 Voltagefollower ..................................165 A.1.2 Voltagedivider...................................167 A.2 Drivebridgecircuit.....................................168 A.2.1 Resistors ......................................168 A.2.2 Operational amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.3 Wheatstonebridge.....................................170 A.4 2-wireconfiguration ....................................171 A.5 Multiplexers.........................................172 A.6 Instrumentation amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 A.7 Low-passfilter .......................................174 A.8 Analog-to-digital converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 A.9 Temperaturesensor.....................................177 A.10 TC and uncertainty of the measurement chain . . . . . . . . . . . . . . . . . . . . . . 177 B Heaters in the GRS 179 B.1 Thermistor temperature increase due to the SHE . . . . . . . . . . . . . . . . . . . 180 B.2 Second-order model for the thermistor response . . . . . . . . . . . . . . . . . . . . 185 C Bit rate reduction 189 D Spectral density function for the fluctuating force due to magnetic disturbances 193 E Insulator construction 195 F Heater transfer function 197 F.1 Frequency-domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 F.2 Time-domainanalysis ...................................198 Chapter 1 Introduction This thesis focuses on a very specific issue: the measurement of temperature at the micro-Kelvin level in the frequency range of the milli-Hertz. Such measurements are required as a part of a space mission intended to put to test key technologies for a space borne gravitational wave detector. The technological mission is LISA Pathfinder (LPF), which is devoted to pave the way to the LISA (Laser Interferometer Space Antenna) mission. LISA is a space-based gravitational wave observatory with the primary scientific goal of detecting and observing gravitational waves (GWs) from astronomical sources in a frequency range of 0.1 mHz to 0.1 Hz. Observation of gravitational waves will be a new way to explore the Universe and it will convey new rich information about its structure and evolution. However, the detection of GWs at low frequency requires differential measurements of distances of pico-meters between two bodies separated by 5 million kilometres. In view of the very demanding requirements LISA must attain, a precursor mission, LISA Pathfinder, has been decided to be launched first. The work presented in this thesis is devoted to the measurement of the thermal perturbations in the main subsystems of LISA Pathfinder. Such measurements, as the whole mission itself, will be useful as a debugging tool for LISA. Before entering into the details of the issues related to the measurement of temperature and its purpose on-board LISA Pathfinder, an introduction about the nature of the gravitational waves and the efforts made to detect them since the 1960s is required. In this chapter we give a brief description of the gravitational wave properties and an overview of the on-ground based detectors: resonant mass detectors and interferometric detectors. Later, an overview of LISA and LISA Pathfinder missions is presented. Finally, the thermal effects in critical subsystems of LISA Pathfinder are discussed and they are used to define the requirements of the temperature measurement subsystem. 1.1 Gravitational waves In the Newtonian Gravitation theory, gravitational fields propagate instantly to remote places, no matter how distant from the source. This is of course untenable, as it would appear that gravity is not subject to the laws of causality every other interaction complies with. A consistent alternative to the Newtonian Gravitation theory is the Einstein’s General Theory of Relativity where gravitational waves (GWs) describe the propagation of gravitational fields. GWs were predicted by Einstein in 1916 as a consequence of the General Theory of Relativity. They are originated in a source of gravitational field whose structure varies with time, for instance, when a mass distribution moves in an asymmetric way. In this case the gravitational field variation propagates away into the surrounding space at some finite speed and thus, a body at a certain distance will feel such variation after some finite time. Gravitational perturbations travel outwards as ripples of the 2 1 Introduction spacetime geometry and, therefore, they distort the spacetime geometry1. The basic properties of GWs are: (i) travel through empty space at the speed of light, (ii) have two polarisations modes (’+’ and ’×’) and (iii) are transverse to the propagation direction [93]. Consequently, the distance between freely falling bodies changes in the perpendicular direction of the passing GW in two different polarisations —see Figure 1.1— and this is exactly the key aspect to measure them. GWs were already inferred by Hulse and Taylor in 1978 [69, 136]. They measured the decrease of the rotation period of a binary pulsar (PSR 1913+16) discovered in 1974, and attributed the observed loss of energy of the system to gravitational radiation [137]. time Figure 1.1: Freely falling macroscopic bodies (represented as black dots)—GW interaction. The GW direction is perpendicular to the plane of the circles. Both polarisation modes are shown (’+’ and ’×’). The main problem of measuring GWs is the exceedingly small relative change in the distance between freely floating bodies due to a passing GW where the current (or potential) detectors can be placed (on Earth or at some point in outer space close to the Earth). The amplitude of a GW is given by the strain, h, which is defined as 2δL /Land represents the relative distance change between two test bodies separated by an initial distance L. For gravitationally bounded systems, an order of magnitude estimation of the strain, h, can be calculated as2[127] h∼` RGM `c22 (1.1) where Mis the mass of the source, Ris the distance from the source, `is the linear size of the source and G= 6.67·10−11 m3kg−1s−2and c= 3·108m s−1are the gravitational constant and the speed of light, respectively. For GW detectors on Earth the values for the strain, h, are ∼10−18 in the most favourable conditions3. Like electromagnetic waves the GW spectrum covers a wide frequency range: from 10−16 Hz (quantum fluctuations in the very early Universe) to GWs radiating in the kilo-Hertz (and higher). The GW spectrum can be divided into four rather different bands [68]: (i) the ultra low frequency band (10−18 Hz to 10−13 Hz); (ii) the very low frequency band (10−9Hz to 10−7Hz); (iii) the low frequency band (10−5Hz to 1 Hz); and (iv) the high frequency band (1 Hz to 104Hz). On the one hand, the detection of GWs in the high frequency band is covered by the on-ground detectors —see § 1.2. The candidate sources to be detected by these instruments are rotating neutron stars (due to small ellipticities in their structure), massive objects like binary systems of neutron stars or black holes (specially, the last epoch of the inspiral process) and supernova explosions [68]. On the other hand, the low frequency band will be covered by the space-based GW detector: Laser Interferometer Space Antenna (LISA) —see § 1.3— which has a list of guaranteed binary systems if 1The geometry is, thus, determined by the distribution of mass and energy and it cannot be determined a priori. 2More specifically, it is proportional to the source’s quadrupole moment acceleration, ¨ Qij(t). 3If we assume the free floating bodies are at a distance of L=1 km, the change in distance is of 5 ·10−16 m. 1.2 Ground-based GW detectors 3 the design sensitivity of the detector is achieved. These binary systems are known as the verification binaries [134]. Among the binary systems, the large amount of white dwarf binaries detectable by LISA produce a background confusion noise around the 5 mHz. LISA also will be able to detect GWs coming from several other sources such as super massive black holes (SMBHs) [15] and extreme mass ratio inspirals (EMRIs) [55]. 1.2 Ground-based GW detectors At present two different types of detectors have been used for the detection of GWs. They are resonant mass detectors and interferometric detectors [109] and both work in the high frequency band. The following sections describe how they work and their main properties. 1.2.1 Resonant mass detectors Large efforts have been done since the early 1960s in the detection of GWs. J Weber [156, 157] initiated his work by means of resonant mass detectors. The development of such detectors has continued until today achieving sensitivities almost three orders of magnitude (in amplitude) higher than those exhibited by the first ones. Resonant mass detectors are solid bodies (cylinders or spheres) designed specifically to have a very high Qmechanical resonator [52, 46]. This resonance is excited by the passing GW —see Figure 1.2—, thus enabling detection. ω0 m 2 dL 2 dL GW m L Sh −1/2 ( ) [Hz ] −1/2 ω 10−18 10−19 10−20 allegro auriga explorer nautilus 10−21 frequency [Hz] 800 840 880 920 960 1000 Figure 1.2: Left: resonant mass detector scheme. The gravitational wave Fourier component at a frequency ωois amplified by the spring-masses system. Right: strain sensitivities of different resonant mass detectors [108]. Current resonant mass detectors4reach very impressive strain sensitivities: ∼10−21 Hz−1/2 (∼10−21 m Hz−1/2in distance since they are around one meter long) at the kilo-Hertz frequency range with a bandwidth of tens of Hertz —see Figure 1.2. 1.2.2 Interferometric detectors An alternative method to detect GWs rose in the early 1970s. This method was based on laser interferometry and, more specifically, in the classical Michelson interferometer. Roughly speaking, such detectors measure the phase differences between two laser beams reflected in two bodies caused by the motion of the bodies when passing a GW —see Figure 1.3 (left). The change of phase, δφ, 4Allegro in USA [64], Auriga in Italy [162], Explorer in Switzerland [10], Nautilus in Italy [9], Niobe in Australia [17], MiniGRAIL in the Netherlands [36] and Mario Schenberg in Brazil [4]. 4 1 Introduction in a Michelson interferometer of arm-length Ldue to a passing GW is [85] δφ = 2ωlaser ωGW h+sin ωGWτ 2,(ωGW ωlaser) (1.2) where ωlaser is the angular frequency of the laser light, ωGW is the angular frequency of the GW, and τis the round-trip time: τ=2L /cwith c= 3 ·108m s−1the speed of light and Lthe arm-length of the interferometer. The optimum arm-length is L=πc /ωGW and the transfer function of the antenna rolls off at frequencies above the inverse of the round-trip time τ(= 2L/c) —see Figure 1.6. Laser Recycling mirror Photodiode Fabry−Perot cavity S−1/2 ω h] −1/2 ( ) [Hz shot noise thermal noise seismic noise 10−16 10−18 10−20 10−22 10−24 10 100 1000 10000 frequency [Hz] Figure 1.3: Left: schematic of a GW detector using Fabry-Perot cavities and power recycling. Right: sensitivity curves of the LIGO detector. The solid black trace is the design sensitivity [108]. However, the high sensitivity demanded to the interferometers complicates their design. The configurations adopted are the power-recycled Michelson interferometer with Fabry-Perot cavities in the arms [3, 1, 32], and the dual recycling technique [87, 91]. The use of these techniques increase the interaction time between the light and the GW and reduces the photon shot noise. The sensitivity of ground-based GW interferometric detectors is limited by different sources of noise (seismic noise, thermal noise, photoelectron shot noise, gravity gradient noise and quantum effects). These factors prevent the detection of GWs in the low frequency range, essentially, due to gravity gradient and seismic noise. At high frequency the shot noise dominates the measurement [88] —see Figure 1.3. The ground-based interferometric detectors are summarised in Table 1.1. detector arm-length strain sensitivity GEO600 [87] 0.6 km ≃7·10−22 Hz−1 LIGO Hanford [130] 2 km ≃4·10−23 Hz−1 LIGO Hanford [130] 4 km ≃2·10−23 Hz−1 LIGO Livingston [130] 4 km ≃2·10−23 Hz−1 TAMA300 [135] 0.3 km ≃3·10−21 Hz−1 VIRGO [2, 3] 3 km ≃5·10−23 Hz−1 Table 1.1: Ground-based interferometers sensitivities. Second generation instruments for the LIGO detector are planned [138], which promise an order of magnitude increase in broadband strain sensitivity, i.e., ∼2·10−24 Hz−1/2. This means an increase in the probability of detecting GWs by a factor of 1000. So far the high frequency band of the GW spectrum is covered by the ground-based detectors. On the contrary, the low frequency band remains still uncovered since ground-based detectors are 1.3 LISA mission overview 5 not able to work at frequencies below 1 Hz due to seismic noise. Hence, the logical solution to avoid such limitations is to place an interferometric GW detector in space, where the gravity noise is strongly reduced and the length of the arm can be orders of magnitude longer than in Earth. This detector is the so called Laser Interferometry Space Antenna (LISA) which has been designed to detect GWs with high signal-to-noise ratio in the milli-Hertz range. An overview of this mission and its precursor mission, LISA Pathfinder (LPF), are given in the next sections. 1.3 LISA mission overview The primary objective of the LISA mission [14] is to detect GWs from massive black holes and galactic binaries5in the low frequency range, i.e., from 0.1 mHz to 0.1 Hz. More specifically, the main purpose of the mission is to learn about the formation, growth, density and surroundings of massive black holes (from 106Mto 108M). Observations of these sources can provide unique new information about these objects, and would test General Relativity and black hole theory to unprecedented accuracy. Low frequency GW detection is not possible to interferometric groundbased detectors due to the local gravitational noise (moving objects, seismic movements, gravity gradient noise, meteorological phenomena, etc.) and due to the short interferometer arm-length (few kilometers). Nevertheless, a space-based detector is free from such noise sources and can have very long arm-length, hence, it should be able to work in the low frequency range, where the most certain and powerful GW sources radiate (large scale mass motions imply typically long time scales). Figure 1.4 shows the expected sensitivity for LISA and LIGO and the expected sources of gravitational waves at different frequencies. Figure 1.4: Strain sensitivities and gravitational wave sources for LISA and LIGO [14]. The LISA mission consists in three identical spacecraft separated by 5 ·106km (in comparison, the longest arm-length of a ground-based detector is 4 km, the LIGO detector) forming an equilateral triangle. This triangular constellation turns around the Sun in an Earth-like orbit about 20obehind the Earth. The plane of this triangle needs to have an inclination of 60owith respect to the ecliptic to provide the most stable size of the triangle —see Figure 1.5 (left). 5Pairs of close white dwarfs, pairs of neutron stars, pairs of neutron star-black hole, etc. 6 1 Introduction Figure 1.5: Left: orbit LISA configuration. Right: LISA’s equilateral triangle configuration. The distances between the vertexes of the equilateral triangle shown in Figure 1.5 are not constant since the LISA constellation is affected by the gravitational field of the Solar System. Actually, the arm-lengths and the angles of the triangle change ∼120 000 km (peak-to-peak) and ∼1oper year, respectively. However, length and angle variations occur in time scales of months and LISA is interested in length variations in time scales of hours, hence, LISA can surely detect GWs in the submilli-Hertz frequency range, albeit the distance between spacecraft is not constant over long time scales. LISA works, basically, as a giant Michelson interferometer with an extra arm added to give independent information on the GW polarisations and for redundancy. In LISA, the arm-length of the interferometer (5 ·106km) determines the frequency range in which observations can be made [the arm-length needs to be close to λGW/2 —see Eq. (1.2)] and it has been chosen to allow observations of most of interesting sources of gravitational radiation. In LISA each of the three spacecraft contains two optical assemblies. The two assemblies on one spacecraft are each pointing towards an identical assembly on each of the other two spacecraft to form a pseudo-Michelson interferometer —see Figure 1.5 (right). To achieve the desired sensitivity the interferometric system must keep the noise (shot noise, mainly) in measuring the differences in the round-trip path length between two arms below ∼40 pm Hz−1/2(or an equivalent GW strain of ∼10−20 Hz−1/2) in the milli-Hertz frequency range. Each spacecraft contains two vacuum enclosures housing a platinumgold cube, 46 mm in size, in nominally perfect free fall (or geodesic motion), known as the test mass (TM), which serves as an optical reference or mirror for the light beams of the interferometer. A passing GW will change the length of the optical path between the test masses of one arm relative to the other arms. The spacecraft is used to isolate the test masses from the environment, specially from the solar radiation pressure and solar wind, to maintain them in perfect geodesic motion or free fall. Consequently, the spacecraft will not be able to maintain its movement along the geodesic whereas the tests masses do. The spacecraft position, thus, does not directly translate into the GW signal, however, it is totally necessary to keep the spacecraft well centred on their respective test masses to reduce spurious local noise forces and, obviously, to avoid a spacecraft-test mass crash. To do this, the relative motion of the spacecraft with respect to the TMs can be measured precisely by means of a capacitive sensor that measures the change in the electrical capacitance between the TM and a set of electrodes surrounding it, fixed to the spacecraft. This measurement is then converted into a force-command which activates a set of micro-thrusters6which exert forces on the spacecraft. This technique is known as drag-free control [77]. The required resolution of the capacitive sensors 6The needed force is micro-Newton. The used thrusters are field emission electric propulsion (FEEP) and they operate by accelerating ions in an electric field, and ejecting them to thrust. 1.3 LISA mission overview 7 is ∼10 nm Hz−1/2and the disturbing accelerations induced by the sensor back-action and by the parasitic forces on the test mass must be lower than 3 fm s−2Hz−1/2in the frequency range of 0.1 mHz to few milli-Hertz [14]. The sensitivity of LISA is determined by the response of the interferometer to a GW of strain h compared with the effects of various noise sources that fake the GW. A Michelson type interferometer measures the phase difference between the two beams after they have returned from the two arms of length L, i.e., 2Lplus noise effects (δLn). The transfer function of the antenna is7—cf. Eq. (1.2) and see Figure 1.6—, H(ωGW) = sincωGWτ 2(1.3) where ωGW is the frequency of the incoming GW and τis the round-trip time for the light beam in each arm of the interferometer. 10−4 10−3 10−2 10−1 100 10−3 10−2 10−1 100 frequency [Hz] GW antenna transfer function /fτ α1 Figure 1.6: Magnitude of the LISA transfer function assuming L= 5 ·106km and a passing GW perpendicular to the LISA plane. However, LISA interferometry is considerably different from the classical Michelson interferometer. In such interferometers a single light source is split and recombined after travelling similar pathlengths. Therefore, when they are combined and measured by the photodetector the noise is rejected since it is common to both laser beams. On the contrary, LISA uses six different light sources: one per test mass —see Figure 1.5 (right). The noise from the light sources is uncorrelated and the time for a photon to travel from one spacecraft to another is ∼15 s. The uncorrelated noise of the lasers will dominate the measurement of the distance between the test masses, hampering the detection of GWs. However, it is possible to form combinations of different phasemeter outputs from different optical benches (there is one optical bench per test mass) with suitable time delays that cancel the laser phase noise and meet the requirements to detect GWs. This technique is known as time-delay interferometry (TDI) and requires accurate knowledge of the distance between the spacecraft and generation of time-delayed copies of the phasemeter signals [139, 31]. The sensitivity limiting noise effects in LISA can be summarised in [14, 33, 126]:  Shot noise The effect is a spurious path difference δLninversely proportional to the square root of the received laser power. Due to the low level of light power received by the interferometer telescopes, the shot noise plays a major role in the total noise budget of spurious displacements. This noise defines the flat centre band of the LISA sensitivity curve, 7In the case of LISA, the two arms do not form a right angle, but one of 60o, thus, decreasing the response of the antenna. A factor sin 60o=0.866 appears in the transfer function. 8 1 Introduction  Antenna transfer function The antenna response rolls off as (fτ)−1at frequencies above the inverse of the round-trip time τ—see Eq. (1.3) and Figure 1.6. Thus, at these frequencies the sensitivity decreases proportionally to ω,  Acceleration noise At frequencies below 1 mHz the noise is dominated by the acceleration of the test mass that cannot be shielded by the drag-free control, i.e., internal forces due to phenomena related to temperature fluctuations, magnetic forces, charging of the test mass due to cosmic and solar radiation, microgravity effects, back actions related to the drag-free control, etc. Residual noise accelerations have a rather white spectral distribution, which results in position errors rolling off approximately as ω−2, since S1/2 ∆x=S1/2 ∆F mTMω2(1.4) where mTM is the mass of each test mass. Considering these noise sources and the frequency dependence of the interferometer response, the LISA typical sensitivity curve is the one shown in Figure 1.7. 10 −4 10 −3 10 −2 10 −1 10 0 10 −22 10 −20 10 −18 10 −16 10 −14 frequency [Hz] S−1/2] ] (ω) function antenna transfer hHz acceleration noise shot noise Figure 1.7: Sensitivity of LISA. Noise related to disturbances acting on the test masses such as temperature fluctuations are proportional to ω−2. The shot noise defines the noise in the centre of the band. The high frequency noise is due to the antenna transfer function and increases proportionally to ω. At this moment, the permitted noise in the low frequency range, <2 mHz, must be defined. The noise at such frequencies is determined by the acceleration noise, leading to a decrease in sensitivity towards lower frequencies proportional to ω−2. Above ∼2 mHz the noise is dominated by the shot noise of the laser, whereby the decline of the antenna transfer function above 10 mHz causes a decrease in sensitivity proportional to ω. Low frequency stray forces inside the spacecraft tend to move the test masses away from their geodesics and, therefore, they prevent to put them in perfect free fall. A residual noisy acceleration can be converted to strain noise as S1/2 ∆F(ω) 1 m −→ S1/2 ∆a(ω) 1 ω2 −→ S1/2 ∆x(ω) 2 L −→ S1/2 h(ω),(1.5) i.e., S1/2 h(ω) = 2 mTML 1 ω2S1/2 ∆F(ω).(1.6) As the effect decays with ω2, above ∼3 mHz the performance of LISA is expected to be set by the nearly shot noise limited displacement sensitivity of the laser interferometer of ∼40 pm Hz−1/2. 1.4 LISA Pathfinder mission overview 9 This is true only if stray forces have a power spectral density of S1/2 ∆a(ω) = S1/2 ∆F(ω) mTM ≤3·10−15 "1 + ω/2π 3 mHz2#m s−2Hz−1/2(1.7) down to a frequency of 0.1 mHz. This residual noise is shown in Figure 1.8 and it determines the free fall accuracy level required. 10−4 10−3 10−2 10−1 100 10−20 10−18 10−16 10−14 10−12 10−10 10−8 frequency [Hz] S∆a 1/2 (ω) [m s −2 Hz−1/2] SHOT NOISE ACCELERATION NOISE Figure 1.8: Power spectral density of the residual noise acceleration allowed for LISA to achieve the sensitivity curve shown in Figure 1.7. Shot noise stands for the noise in the relative position measurement of the interferometer (assumed to be flat and of ∼40 pm Hz−1/2). From this figure we can infer the permitted noise levels (in terms of acceleration) due to thermal fluctuations, magnetic fields, etc. and that they become important at frequencies below the milli-Hertz. Up to now the difficulties and the great challenges to detect GWs have been exposed. These difficulties arise from the fact that the interaction between GWs and the measuring apparatus is exceedingly small. 1.4 LISA Pathfinder mission overview During the various studies for LISA, the need for a technological mission demonstration was recognised: the so-called LISA Pathfinder (LPF) mission. It includes the European Space Agency (ESA) science module LISA Technology Package (LTP) [145] plus parts of its National Aeronautics and Space Administration (NASA) counterpart, the Disturbance Reduction System (DRS) [47]. The main goal of LPF is to check if the residual acceleration in Eq. (1.7) can be achieved, i.e., it must verify the ability to set a test mass in purely gravitational free fall to the level required by Eq. (1.7), although in the LPF mission the requirements have been relaxed one order of magnitude both in amplitude and in frequency, i.e., S1/2 a(ω) = S1/2 F(ω) mTM ≤3·10−14 "1 + ω/2π 3 mHz2#m s−2Hz−1/2,1 mHz ≤ω/2π≤30 mHz (1.8) The basic idea of the LTP is to squeeze one LISA arm from 5 ·106km to 30 cm and place it aboard a single spacecraft —see Figure 1.9. This scheme, thus, prevents LPF from detecting GWs due to the short arm-length, 30 cm, and converts it into a test bench for the different technologies to be used in LISA8. The whole instrument is designed to contain the essence of the construction 8Specially, the two main concepts: the GRS (and drag-free and low-frequency control) and the interferometric system. 16 1 Introduction strumental and environmental origin. One of these perturbing elements is temperature fluctuations, for which a stability requirement for the LTP experiment has been set to [86, 145, 84] S1/2 T(ω)≤10−4K Hz−1/2,1 mHz ≤ω/2π≤30 mHz (1.10) in the absolute temperature range of 10  to 30  . Because temperature stability is important, high precision temperature thermometers will be placed in different strategic points over the LTP. Such measurements will be useful to identify the fraction of the total system noise which is due to temperature fluctuations only, and this will in turn provide important debugging information to assess the performance of the LTP. In addition, by applying controlled temperature signals and measuring their effect on the system read out, the transfer function between temperature fluctuations and displacement (real or faked11) in the test masses will be estimated. In this manner we want to assess whether or not the modeling of the thermal effects in the subsystems of the LTP are accurate. In the following the temperature stability requirement expressed in Eq. (1.10) is analysed. The sources of temperature fluctuations that contribute to the total noise budget of Eq. (1.8) are identified and quantified. Afterwards we discuss how the temperature stability requirement results in the definition of the temperature measurement subsystem performance. Random temperature fluctuations in the LTP introduce noise in the system through different mechanisms. Proper characterisation of these effects will set the limits of temperature fluctuations compatible with the LTP requirements. As a rule of thumb, the total contribution of temperature fluctuation noise to the total acceleration noise —Eq. (1.8)— should not exceed 10%. Thus, it is required that S1/2 a,T (ω)≤3·10−15 "1 + ω/2π 3 mHz2#m s−2Hz−1/2(1.11) within the LTP MBW. This assumption is in fact somewhat conservative, as it has been estimated that more than twice this value is actually compliant with the overall LTP noise budget [155]. However, here Eq. (1.11) is adopted as reference to ensure that we are on the safe side. The influence of the temperature in the whole LTP experiment has been separated into two subsystems: the GRS and the OMS. 1.5.1.1 Noise effects inside the GRS Temperature differences between the walls of the electrode housing inside the GRS cause differential pressures on opposite faces of the test masses, which in turn result in net forces on them, and real motion of the test masses. Three different mechanisms have been identified: radiation pressure, radiometer effect and asymmetric outgassing [145, 84, 26]. Radiation pressure A body at any (absolute) temperature Temits thermal radiation. This exerts pressure on any surface the radiation hits —see Figure 1.15. According to electromagnetic theory, such pressure is given by pe.m.=4 3 σ cT4(1.12) where σ=5.67·10−8W m−2K−4is the Stefan-Boltzmann constant, and cis the speed of light. Consequently, if there are temperature fluctuations around the test mass, a noisy net force will appear on it —see Figure 1.15. The pressure gradient through xis easily inferred from Eq. (1.12) as: dpe.m. dx =dpe.m. dT dT dx =16σ 3cT3dT dx ,(1.13) 11For instance local temperature fluctuations in different components of the OMS cause errors in the measurement susceptible of being confused with actual test masses’ displacement. 1.5 Diagnostics in the LTP 17 hence ∆pe.m.=16σ 3cT3∆T(1.14) where ∆pand ∆Tare the differences of pressure and temperature between the sides of the test masses, respectively. Associated acceleration noise is easily obtained multiplying Eq. (1.14) by the TM surface area, `2 TM, and dividing by its mass12,mTM, i.e., ae.m.=16`2 TMσ 3mTMcT3∆T . (1.15) electrode housing T1T2 p 1p2 test mass Figure 1.15: Effect of the different pressure on opposite faces of a test mass due to differences of temperature in the walls of the EH. Radiometer effect [96] This effect happens in rarefied gas atmospheres. In low pressure atmospheres, where the gas particles have a mean free path well in excess of the dimensions of the containing vessel, equilibrium conditions do not happen when pressure is uniform, but rather when the ratios of pressure to square root of temperature equal each other. Let Aand Bbe the two sides 1 T p T p2 2 1 S A B Figure 1.16: Receptacle with different temperature and pressures. of a receptacle with temperatures T1and T2, separated by a wall with a hole S—see Figure 1.16. The equilibrium condition between both sides occurs when the number of molecules per time unit passing from Ato Band from Bto Ais the same, i.e., n1c1=n2c2(1.16) where n1and n2are the number of particles per unit volume of both receptacle sides, and c1and c2are the average speeds of the particles. They can be expressed as ni=pi kBTi ,(1.17a) ci=r8RTi πM (1.17b) 12In [26] the radiation pressure factor is modelled as (8σ/3c)T3∆T. 18 1 Introduction where kB= 1.38 ·10−23 J K−1is Boltzmann’s constant, R= 8.314 J K−1mol−1is the ideal gas constant and Mis the molar mass of the gas. Substituting Eqs. (1.17a) and (1.17b) into Eq. (1.16) the equilibrium condition is obtained: p1 √T1 =p2 √T2 .(1.18) Equation (1.18) can be expressed in the form of p(T) = p0rT T0 (1.19) and its gradient through xis easily derived from Eq. (1.19), i.e., dp(T) dx =dp(T) dT dT dx =1 2p(T)1 T dT dx .(1.20) Re-arranging we obtain ∆pr.m. p(T)=1 2 ∆T T(1.21) and considering ∆pr.m.=(mTMar.m.) /`2 TM we arrive to13 ar.m.=1 2 p(T)`2 TM mTM ∆T T.(1.22) Outgassing Outgassing is one of the causes of the presence of gas within the walls of the GRS. In the present context, outgassing problems actually derive from temporal fluctuations in rate, which once more result in pressure fluctuations, thence in noise. Partial evidence has been gathered that outgassing might be in practise a small effect in the LTP [25, 26]. Moreover, a baking process to the GRS will be done in order to reduce the potential effects of this issue. Therefore this effect will be omitted in further analysis. Total temperature fluctuation noise in the GRS Radiometer and radiation pressure acceleration fluctuations are of course totally correlated since they have the same noisy source, ∆T. If we neglect outgassing then Eqs. (1.15) and (1.22) are linearly added, and hence the spectral densities of acceleration and temperature in the GRS are related by S1/2 a,GRS(ω) = 16`2 TMσ 3mTMcT3+p`2 TM 2mTM T−1S1/2 ∆T , GRS(ω) (1.23) Nominal conditions in the GRS are as follows, `TM=46·10−3m, mTM = 1.96 kg, T=293 K, p=10−5Pa which yield S1/2 ∆T , GRS(ω) = [2.2·1010 K (m s−2)−1]·S1/2 a, TF GRS(ω) (1.24) with 60% coming from the radiation pressure term, and 40% from the radiometer effect term in Eq. (1.23). Equation (1.24) gives 70 µK Hz−1/2as the permitted fluctuations in the worst case that all the thermal acceleration budget, Eq. (1.11), is allocated to temperature fluctuations in the GRS. 13In [26] this effect is modelled as (1 /4)p(T)`2 TMm−1 TM∆T/T . 1.5 Diagnostics in the LTP 19 1.5.1.2 Noise effects inside the OMS The optical metrology system is affected by temperature fluctuations basically through three distinct effects [145, 84, 97]:  the index of refraction of optical components depends on the temperature,  temperature changes cause dilatation (and contractions) of optical elements, which in turn cause light’s optical path to change accordingly and,  the temperature effect on the analog electronics, in particular a change of capacitances in the photodiodes which converts into a phase change of the measured signal. It is not difficult to characterise how individual components are influenced by the above effects. For instance a set of on-ground tests in the optical window (OW) and the optical bench (OB) of the OMS have been done in order to estimate the relationship between temperature and interferometer performance when this thermally disturbed. In the case of the OW it has been estimated ∼6.5 nm K−1, which allows a maximum temperature fluctuations in the OW of ∼10−4K Hz−1/2 not to exceed the noise budget allocated for the interferometer [98, 58]. However, the assessment of the behaviour of the fully integrated optical metrology is a complicated task. Significant progress has been made since the early design proposals, and improved materials and designs more immune to temperature fluctuations are now available. Altogether, it appears that S1/2 T ,OMS(ω).10−4K Hz−1/2(1.25) in the LTP MBW is a requirement which should guarantee the performance of the OMS against temperature fluctuations in flight. Again, the noise level in Eq. (1.25) has been estimated for about 10% of the total LTP acceleration noise. 1.5.2 Temperature measurement subsystem sensitivity requirement Estimates so far indicate that both GRS and OMS noise must be in the order of 10−4K Hz−1/2, as shown in the requirement in Eq. (1.10). Noise in the GRS should be, in principle, uncorrelated with the noise in the OMS since they are of different nature: temperature gradient fluctuations across the test masses cause noise in the GRS, local temperature fluctuations affect the OMS read out. Then, the zero-correlation hypothesis implies that both kinds of noise add quadratically: S1/2 ∆T,T (ω) = [S∆T , GRS(ω) + ST , OMS(ω)]1/2⩽10−4K Hz−1/2(1.26) Equation (1.26) sets the maximum temperature fluctuations permitted in the LTP to respect the requirement given in Eq. (1.11). We are thus reassured that Eq. (1.10) is a sensible requirement for the temperature fluctuations which can be tolerated in the LTP, and this will be designed to ensure such temperature stability. Nevertheless, it is necessary to know if the stability is actually met during the mission. For this reason it is required to monitor the temperature across different places in the LTP by means of temperature sensors able to detect small temperature fluctuations: 10−4K Hz−1/2in the frequency range of 1 mHz to 30 mHz. The temperature measurement subsystem (TMS) should be one order of magnitude less noisy than the maximum noise level that has to be measured in the temperature range from 10  to 30  [83, 119], or S1/2 T , TMS(ω)≤10−5K Hz−1/2,1 mHz ≤ω/2π≤30 mHz, 10 oC≤T≤30 oC (1.27) which, in fact, becomes a mission top level requirement [146]. There are mainly two reasons which support this decision: (i) Eq. (1.10) defines the maximum acceptable level of temperature fluctuations in the LTP. This, of course, must be satisfied by proper satellite design. Hence, actual 20 1 Introduction fluctuations will be, in principle, less than that. Requirement (1.27) sets a 10% minimum discrimination capability for the measuring system, a standard approach which is compatible with better performance and (ii) LISA is more demanding than LPF as regards thermal stability (by one order of magnitude [145, 14, 126]). If we require (1.27) for LTP then we are in a position where analysis of thermal sources of noise of relevance for LISA can be identified and tagged for improvement. This prospect is in line with the very concept of LPF as a precursor mission. 1.6 Structure of the thesis Temperature fluctuations are an important physical phenomena to take into account for LISA and for LPF. In § 1.5.1 we have presented the different mechanisms whereby temperature fluctuations cause forces in a macroscopic body and thus, challenge the free fall of the body. For this reason in LISA and LPF temperature fluctuations must be kept within certain limits. The required temperature stability is based on calculations (and on ground experiments) assuming certain feedthrough factors14. In view of this, the role of the thermal diagnostic subsystem in the LTP on-board LPF is twofold: on the one hand it is used to excite thermally different subsystems of the LTP (mainly, the GRS and the OMS) to estimate the transfer functions between temperature and TMs acceleration and OMS performance. On the other hand, it must monitor the temperature stability in the LCA which is ensured by a suitable thermal shield that keeps the fluctuations at the level of ∼10−4K Hz−1/2 in the milli-Hertz range —see § 1.5.1. With this information the purpose of the thermal diagnostic subsystem is to provide information to identify the fraction of noise in the test masses motion caused by thermal effects, with the goal of diagnosing the LTP performance, and guiding the search for the final sensitivity leap from Eq. (1.8) to (1.7). This is the reason why the LTP diagnostics in general is such an important subsystem: it would surely be less relevant should LPF be the ultimate mission, i.e., with no further projection into LISA. The following chapters focus on aspects of the thermal diagnostics on-board LISA Pathfinder, specially, on the temperature measurement subsystem (TMS). They are organised as follows:  In chapter 2 we describe the TMS designed for the LTP. The system must reach a noise equivalent temperature of 10−5K Hz−1/2in the LTP measurement bandwidth. Aspects related to the positioning and connections of the items forming the LTP TMS are also discussed.  Chapter 3 focuses on potential problems of placing temperature sensors in the GRS. First, we analyse the compatibility of the sensors with the required magnetic cleanliness in the GRS. Temperature sensors in the LTP are thermistors which exhibit a tiny ferromagnetic behaviour. Second, we take into account the capacitive coupling between the cables of the thermistors and the high frequency signals present in the capacitive sensor since they can degrade the TMS performance.  In chapter 4 we describe the test bed designed in order to validate the LTP TMS. The validation requires to place the temperature sensors in an environment where fluctuations are kept below 10−5K Hz−1/2for f≥1 mHz. Another test bed to validate the noise performance of the sensors in the LISA band (f≥0.1 mHz) is also described.  Chapter 5 presents the results of the different test campaigns for the validation of the TMS. The results belong to the prototype validation campaign, to the engineering model (EM) campaign and to the flight model (FM) campaign. In addition, results of the noise investigations of the TMS in the LISA band are presented. 14Feedthrough factors stand for the coefficients relating thermal effects with TM motion (radiation pressure, radiometer effect, etc.) or error read out in the OMS, e.g., the index of refraction temperature dependence, etc. 1.6 Structure of the thesis 21  Chapter 6 focuses on the improvement of the LTP TMS in view of the LISA mission which is more demanding in terms of bandwidth and sensitivity. Effects related to the analog-to- digital converters that limit the sensitivity at low frequency are discussed. The reduction of the floor noise of the TMS is also analysed.  Chapter 7 deals with the issues related to the thermal experiments to be performed in the GRS in order to estimate the feedthrough factors relating temperature and forces induced in the test masses. An analysis of the suitable signals to extract the maximum information of the thermal experiments is presented altogether with simulations.  Finally, in chapter 8 we present the conclusion of this work.  A few appendices are added to expand technical details. Chapter 2 The LTP temperature measurement subsystem In this chapter we describe in detail the LTP temperature measurement subsystem (TMS). Once the noise requirement of the measurement has been defined the design of the system can be addressed. The TMS must comply with the noise requirement given in Eq. (1.27) —cf. § 1.5. We write it down again S1/2 T, req(ω)≤10−5K Hz−1/2,1 mHz ≤ω/2π≤30 mHz (2.1) in the temperature range of 10  to 30  . This means a root-mean-square (rms) noise of ∼0.3µK in a bandwidth of 1 mHz. The chapter is organised as follows: first, we review the state of the art in low noise temperature measurements. Second, the LTP TMS is described: in § 2.2 the temperature sensor itself and in § 2.3 the analog and the digital signal processing chains [119]. The last section of the chapter deals with the positioning and the connections of the thermal diagnostic devices in the LTP. 2.1 State of the art A variety of techniques are available to detect temperature variations [28]. Our interest lies in the detection of temperature fluctuations of small amplitude (<10−4K Hz−1/2) at the frequency of the milli-Hertz and at room temperature. Different techniques, configurations and temperature sensors have been designed for this purpose [28, 111, 159, 54, 161, 60]. Considerable efforts have been done in the development of bolometers for applications such as infrared astronomy and microcalometry. Basically, all these designs are based on an ac bridge and the subsequent signal demodulation at the bias frequency. This configuration allows measurements of temperature fluctuations of ∼µK Hz−1/2 at low frequencies (tens of milli-Hertz). However, most of these designs have been used at low temperature. Table 2.1 summarises the main results reported in the literature. Low noise temperature measurements are also required in microdegree temperature controllers which are important devices in many areas of research and applications (investigations in pure fluids, laser heterodyne research [45], study of weak teleseismic signals [40], microgravity experiments on fluids, etc.) [144]. Most of the designs described in the literature use ac techniques altogether with lock-in amplifiers and thermistors as the sensing element. Attempts with dc techniques are also done, however, the temperature stability achieved is worse due to problems related to offset drifts, thermally induced voltages, 1/felectronic noise, etc. The results reported of controllers at room temperature are of tens of micro-Kelvin during a few hours1[144, 40, 45]. 1They are not usually expressed in terms of power spectral density. 24 2 The LTP temperature measurement subsystem Ref. T[K] NET [K Hz−1/2]f[Hz] Technique [29] 2.17 .10−10 &0.01 SQUID [35] 2.17 .5×10−11 (not specified) SQUID [53] 2.17 .5×10−10 (not specified) SQUID [158] 3.31 .10−9&0.1 SQUID (for low temperature experiment in Earth orbit) [159] 0.3 6 ·10−80.01 NTC (Ge:Ga)+ac bridge [37] 0.3 10−60.035 NTC (Ge:Ga)+ac bridge [110] 87.4 8 ·10−610 YBCO thermometers [54] 0.1 — 0.1 thermistor+capacitive load in ac bridge [60] 90 10−40.001 YBCO+ac bridge [161] 300 6 ·10−710 LSMO thermometers Table 2.1: Summary of high-sensitivity temperature measurement system designs reported. The ones in the first rows use SQUIDs (superconducting quantum interference device) in conjunction with materials that change their magnetisation with temperature. All in all temperature fluctuations measurements at the micro-Kelvin level and at the milli-Hertz range at room temperature have been hardly done. Results for uncooled (at room temperature) bolometric detectors have been reported in [161], where noise levels of 0.6 µK Hz−1/2at 10 Hz and dissipating 10 mW in the sensor are shown. Another design is reported for an active cavity radiometer where platinum resistance thermometers and yttrium barium cuprate (YBCO) superconducting thermometers were tested at room temperature exhibiting noise levels of 10 µK Hz−1/2 at 0.1 Hz [60]. In view of this, dedicated investigations in the design of a system able to meet the requirement set in Eq. (2.1) are mandatory. Furthermore, the design of the system is subject to space missions constraints such as availability of space qualified components, power constraints or room. 2.2 Temperature sensors Two different temperature sensors are commonly used for sensing small temperature variations, although both are based on the same principle: the change of a resistive element under temperature changes. The two available types are resistance temperature detectors (RTDs) and negative temperature coefficient (NTC) thermistors. Their characteristics are summarised in Table 2.2. RTD (Pt) Thermistor Temperature span -250 to 900  -100 to 450  Sensitivity 0.00385 K−1≃0.04 K−1 Accuracy ±0.01 K ±0.1 K R−Tcurve Linear Exponential Excitation Current or voltage source Current or voltage source Typical size 6 mm×6 mm 3 mm×3 mm Table 2.2: Typical properties of platinum RTD sensors and NTC thermistors. Usually in low-noise temperature measurements thermistors are the preferred option because their high sensitivity. In the following we consider thermistors and platinum resistance temperature detectors. 2.2 Temperature sensors 25 2.2.1 Resistance temperature detectors (RTDs) RTDs are resistive elements manufactured with different metals such as platinum, nickel or copper. These metals exhibit a known change in their resistance with changes in their temperature. The change of the resistance in these materials is large enough to detect small temperature variations [101]. Commercial RTDs are, however, manufactured mainly with platinum, hence, we only describe these ones here. The relationship between resistance and temperature can be expressed, in general, as R(T) = Ro[1 + α1(T−To) + α2(T−To)2+. . . +αn(T−To)n] (2.2) where Tis the temperature, α1,α2,. . . ,αnare constant coefficients and Rois the resistance at a reference temperature, To, usually, 273 K. In the linear region (which extends from −250  to 850  ) Eq. (2.2) reduces to [28] R(T) = Ro[1 + αPt(T−To)] (2.3) where αPt is the sensitivity of the sensor defined as αPt =1 Ro dR(T) dT (2.4) which for platinum RTD is 0.00385 K−1. The value of Rovaries from 25 Ω to 10 kΩ, however, the use of large values of Rodoes not produce better results in the noise performance of the system as one could expect —see § 2.3.2.1—, but does reduce the errors due to the 2-wire measurement set up —see appendix § A.4. Two important features of the platinum sensor are [72, 28]: (i) its longterm stability (manufacturers specify values around 0.05 K yr−1) and (ii) its excellent repeatability. However, as it will be shown in § 2.3.2.1 the sensitivity of the platinum RTD is not enough to reach the demanding requirements of the measurement under certain power limitations. 2.2.2 NTC thermistors NTC thermistors are also resistive elements that change their resistance with temperature. NTC thermistors are manufactured mixing and synthesising oxides doped with metals. The usual oxides used are of manganese, nickel, cobalt, iron, copper or aluminium. The oxide proportion determines the resistance and the temperature coefficient of the sensor [123, 101]. Thermistors are characterised by (i) exhibiting a large change in their resistance under small temperature variations, (ii) a negative temperature coefficient and (iii) an exponential relationship between temperature and resistance, which implies a non-constant, but high sensitivity —see Figure 2.1. The resistance-temperature relationship of a thermistor can be expressed by means of the Steinhart-Hart equation [133], T−1=A+Bln R(T) + Cln3R(T) (2.5) where Tis the temperature in Kelvin units and A,Band Care constant coefficients that depend on the thermistor type and are given by the manufacturer. Nevertheless, Eq. (2.5) can be simplified, if the working temperature range is small (∆T≃50 K) to [123, 28, 101] R(T) = Roeβ(T−1−T−1 o)(2.6) where βis the temperature characteristic of the material and is constant for small temperature ranges and Rois the resistance of the thermistor at a given reference temperature, To, usually 298 K. The values of Rofor space qualified thermistors are in the kΩ range (2 kΩ to 20 kΩ [16]), thus, appropriate for the 2-wire measurement configuration [75, 101, 28] —see appendix § A.4. The value of βdepends on the thermistor and the typical values are around 3500 K [16]. 32 2 The LTP temperature measurement subsystem  the greater the dissipated power in the sensor, P, the lower the noise,  the greater the relative sensitivity, α, the lower the noise,  the value of the nominal sensor resistance, Ro, does not reduce or increase the noise,  in the case of the platinum sensors the increase of R2translates into a lower noise, although the noise is not further reduced for a relationship of R2/Ro>100. Evaluation of Eq. (2.15) for both sensors is shown in Figure 2.8. The results confirm that the noise levels when using a thermistor are one order of magnitude lower than those of the platinum RTD. Furthermore, the noise levels of the latter are non-compliant with the requirement. The thermistor noise levels are five times lower than the requirement. Consequently, at this point, platinum RTDs were discarded as an option capable of achieving the requirements and the NTC thermistor was the option chosen for the LTP TMS although some concerns had first to be cleared —see § 2.2.2. 10 12 14 16 18 20 22 24 26 28 30 1.94 1.96 1.98 2 2.02 2.04 2.06 2.08 temperature [oC] ST,NTC [ K Hz −1/2] 21 21.5 22 22.5 23 23.5 24 24.5 S T,PRTD [ K Hz −1/2] NTC PRTD µ µ Figure 2.8: Wheatstone bridge noise equivalent temperature when using a thermistor and a platinum RTD. The latter is already above the requirement —see Eq. (2.1)— and, therefore, it is discarded. This noise figure is valid for all the frequency range since Johnson noise, i.e., white noise, is considered. The dissipated power in the sensor is 10 µW. The noise of the bridge when using a platinum sensor can be slightly reduced by using a higher value of R2. For R2=100 kΩ the noise goes down a factor of ≃2, still not meeting the requirements. The noise of the bridge when using thermistors is five times lower than the requirement. Another important issue to take into account with respect to the Wheatstone bridge is its temperature coefficient (TC), αb. Temperature fluctuations in the resistors forming the bridge show up as an error in the measurement. The coefficient for the whole bridge is (in V K−1) —see appendix § A.3, αb(T) = VbR2Rref (Rref +R)2+RNTC(T) (RNTC +R)2αR(2.16) where R=R1=R2= 10 kΩ and αRis the temperature coefficient of the resistors. The maximum bridge resistors’ TC, αR, permitted in the MBW can be calculated by using the following expression αb(T) sb,NTC S1/2 T, FEE(ω)≤S1/2 T, req(ω) (2.17) where S1/2 T, FEE stands for the temperature fluctuations in the electronics, i.e., in the Wheatstone bridge and S1/2 T, req = 10−5K Hz−1/2. Thus, the maximum permitted resistors’ TC, αR, depends 2.3 Signal processing chain 33 on the temperature fluctuations of the electronics on board the satellite. In the design we have assumed to be less than 0.1 K Hz−1/2which appears to be comfortably met in the satellite. This leads to a maximum value of αRof 2 ppm K−1. The resistors of the bridge (R1,R2and Rref ) used in the system have a lower temperature coefficient (Vishay Metal Foil resistors of 0.6 ppm K−1) in order to be on the safe side. In summary, the Wheatstone bridge designed consists of two high stability resistors (R1= R2=10 kΩ), six high-stability reference resistors (Rref0 to Rref5) to centre the output of the bridge at different temperatures and a NTC thermistor as the sensing element. Table 2.3 summarises the references used and the associated centre for each of the scales and their temperature span considering a gain of 200 in the amplification stage —see § 2.3.2.4. reference label resistance [kΩ] centre of scale [  ]Tmin—Tmax [  ] 0 17.5 12 8.28—15.87 1 15 15 11.79—19.29 2 12.5 20 15.99—23.41 3 11 22.65 18.97—26.54 4 10 25 21.21—28.89 5 9.1 27.5 23.47—31.29 Table 2.3: References used and their corresponding centres of scale in the Wheatstone bridge circuit. All the resistors have very low temperature coefficient (0.6 ppm K−1). The selection of the references is done by means of multiplexers —see Figure 2.10. Tmin and Tmax are calculated assuming the gain of the amplification stage is 200. The temperature expected in the LCA of the LTP is from 10  to 30  . The bridge exhibits a sensitivity of ≃6.5 mV K−1in the temperature range from 10  to 30  with P≃10 µW. In terms of noise equivalent temperature the Wheatstone bridge contributes with white noise9of amplitude ≃2µK Hz−1/2(20% of the requirement). Finally, we calculate the needed number of bits of the ADC by using the dynamical range (DR) definition [102], i.e., DR = 20 log temperature span resolution ≃6Nbit (2.18) where the temperature span is Tmax −Tmin ≃7 K and the resolution is set to 10−6K. These numbers yield a needed number of bits, Nbit, of ≃22 —see § 2.3.2.5. 2.3.2.2 Drive bridge circuit As explained in § 2.3.1 and shown in § 2.3.2.4 the 1/fnoise introduced by the amplification stage prevents achieving the requirement given in Eq. (2.1). For this reason and also to reduce dc errors (thermoelectric voltages, offset voltages, and bias current), the Wheatstone bridge circuit is powered with an ac square signal which keeps quasiconstant the electrical power dissipated in the thermistor10 In this section we describe the circuit designed for the ac square wave voltage generation which modulates the temperature signal of the bridge. The demodulation of the signal is done digitally and is described in §2.3.3. The drive bridge circuit is a circuit based on two operational amplifiers (OAs) and two analog switches controlled by digital signals —see Figure 2.9. 9The noise is white since Johnson noise is assumed as the noise model for all the discrete elements of the bridge, including the sensors, i.e., we consider that in principle excess noise is not present. 10This reduces non-linear effects due to the non-linear relationship between the dissipated power in the sensor and the self-heating effect —see 2.3.2.3. Moreover the generation of a square wave is simple and, usually, exhibits higher stability than a sine wave. With this technique the bridge signal (and also the excess noise of the components of the Wheatsone bridge) is modulated at a specific frequency while the Johnson noise and the noise introduced in the amplification stage is not. Thus, when demodulating, the signal returns to the baseband while the 1/fnoise of the amplification stage is modulated and kept away from the baseband —see § 2.3.1 and § 2.3.4. 34 2 The LTP temperature measurement subsystem fb 10 kΩ Ω10 k 10 kΩ10 kΩ pol (0,1) k .Vref Vb=2 −2 kVref ref Vk if pol=1 if pol=0 Figure 2.9: Drive bridge circuit scheme for the square wave bridge voltage supply. The switches in Figure 2.9 permit reversing the output voltage of the bridge: for one polarity the output of the bridge is Vband for the other is −Vb. The polarity is reversed each 90 ms, thus, the frequency of the square wave is (180 ms)−1=5.55 Hz, which minimises the 1/fnoise of the amplification stage —see § 2.3.2.4. Another important feature of this circuit is that the bridge voltage supply is the ADC voltage reference (VADC=2.5 V) with a scale factor, k(=0.25), to adjust the power in the sensor. This technique permits a voltage reference-independent conversion and, thus, highly accurate conversions using voltage references of modest quality are performed. The voltage reference-independent conversion can be expressed, in general, as [49] D=2Nbit −1 VADC vo=2Nbit −1 VADC αVADC = (2Nbit −1)α(2.19) where Dis the digital code output and αVADC is the output of the bridge. The temperature coefficient (TC) of the resistors of the drive bridge circuit (the 10 kΩ resistors in Figure 2.9) also affects the performance of the measuring system —see appendix § A.1.2. The value must be below 30 ppm K−1(assuming ambient temperature fluctuations lower than 0.1 K Hz−1/2). The resistors used are of 0.6 ppm K−1(the same type of the ones used in the Wheatstone bridge). Moreover, the offset, the bias current, the open-loop gain, and the common-mode rejection ratio (CMRR) of the OAs introduce a gain error and their dependence with temperature can cause fluctuating errors in the measurement. These effects are, in principle, tiny (tens of µK), specially, if the temperature of the environment is stable —see appendix § A.1. 2.3.2.3 Self-heating effect error considerations The self-heating effect (SHE) introduces an error in the measurement due to the non-zero thermal contact resistance [71, 16], θ, between the thermistor chip and the temperature of the sensed body11. In terms of power spectral density it can be expressed as12 S1/2 TSHE (ω) = θS1/2 P(ω) (2.20) where S1/2 Pstands for the power fluctuations in the thermistor. 11Experimental tests have shown that θ <100 K W−1when attaching the sensors to an aluminium block surrounded by polyurethane foam. This value might be different in the satellite since it depends on the environmental conditions, however, it must be taken into account that an intrinsic thermal resistance of ∼50 K W−1is always present —see appendix § B. 12The thermal mass of the thermistor is very small and we are interested in the milli-Hertz region. For this reason we can consider the transfer function between power and temperature as a constant, i.e., the bandwidth of the transfer function is much larger than the frequency of interest. Thus, we omit the frequency dependence of θ. 2.3 Signal processing chain 35 Power fluctuations, S1/2 P, come in two different ways: (i) bridge voltage fluctuations that in the end result in voltage fluctuations in the thermistor, S1/2 V, and (ii) NTC resistance fluctuations due to actual temperature fluctuations. Manipulation of Eq. (2.20) leads to the error introduced by both mechanisms, S1/2 TSHE,1(ω) = θ2V RS1/2 V(ω) (2.21a) S1/2 TSHE,2(ω) = θV 2β RoT2 o S1/2 T(ω) (2.21b) where V(= Vb/2) is the voltage in the NTC, Ris the resistance of the NTC, S1/2 Vare the voltage fluctuations in the thermistor, and S1/2 Tare the temperature fluctuations in the thermistor. Equations (2.21a) and (2.21b) are used to calculate the maximum voltage fluctuations permitted in the NTC and the maximum temperature fluctuations. Both contributions must be lower than 10−6K Hz−1/2in order to keep this effect one order of magnitude below the requirement. The values obtained are (assuming P=10 µW, T=293 K and θ=100 K W−1) S1/2 V(ω).0.15 mV Hz−1/2, ω/2π≥1 mHz,(2.22a) S1/2 T(ω).0.025 K Hz−1/2, ω/2π≥1 mHz,(2.22b) Voltage fluctuations are about 2 µV Hz−1/2at 1 mHz, i.e., they met comfortably the requirement given in Eq. (2.22a); and, obviously, the temperature fluctuations condition is also met — Eq. (2.22b)— since the fluctuations to measure are of 0.1 mK Hz−1/2. In conclusion, self-heating effect does not affect the measurement due to the low power dissipated in the thermistor. In chapter 6.2 we show the effect of dissipating higher power in the thermistors. 2.3.2.4 Multiplexers, amplification and low-pass filter The blocks described in this section are placed at the output of the Wheatstone bridge. They are:  the multiplexers (MUX),  the instrumentation amplifier (IA) and,  the low-pass filter. A total of 24 channels (and 36 measurements —see § 2.4) are needed to monitor temperature at different places of interest of the LTP [113, 83] —see § 1.5 and § 2.4. Weight, size and power limitations prevent a single conditioning chain for each temperature sensor13. For this reason six independent boards have been used to perform all the needed measurements. Each electronic board must allow measurements of four channels with respect to six reference temperatures (defined by Rref ) and one differential measurements between two sensors —see § 2.4.3. The circuit designed to deal with this requirement is shown in Figure 2.10. The arms of the bridge are selected by two one-to-eight channel solid state MUXs (Maxim DG408 in the prototype; Intersil HI-508 in the FM design). Each half of the bridge is selected by the MUXs with digital signals and connected to the non-inverting and inverting inputs of the IA. The MUXs contribution to the whole noise system performance is negligible14. The following block is the instrumentation amplifier. The IA is used to amplify the Wheatstone bridge output, vo—see Eq. (2.8). The output voltage of the bridge when measuring at the level 13This would imply a bridge circuit with six references and a drive bridge circuit for each of the 24 sensors. Also an amplification stage, a low-pass filter and a dedicated ADC for each sensor. This, clearly, appears unfeasible. 14The MUXs can be modelled as a simple resistance, RON, which is in the order of 400 Ω as maximum, which, assuming Johnson noise results on a equivalent noise of ≃0.4µK Hz−1/2, well below the noise of the Wheatstone bridge itself. 36 2 The LTP temperature measurement subsystem IA Rref0 R ref5 R (T) 0R (T) 3 RRRR 5 069 0 6 5 7 0 3 6 9 MUX−B MUX−A sensors ref.ref. sensors sensorsref. Figure 2.10: Wheatstone bridge and multiplexer connections. This scheme permits absolute measurements (using different scales: from Rref0 to Rref5) and differential measurements. Also a measurement to check the electronics itself independent of the sensors is possible by measuring one reference against another one. of the micro-Kelvin is very weak (tens of nV), thus, amplification is mandatory before the digital conversion is done by the ADC (16-bit) —see § 2.3.2.5. The noise introduced by the 16-bit ADC to the system (referred to the input) is15 S1/2 T, ADC(T, ω) = σADC 216 VFS 1 pfs/2 1 GIAsb(T)(2.23) where σADC=1 LSB (least significant bit) is the noise specified by the manufacturer, VFS=10 V is the full-scale voltage, fsis the sampling frequency (38.4 kHz in the prototype, 50 kHz in the FM design), GIA is the gain of the IA and sbis the sensitivity of the bridge (≃6.5 mV K−1). Now we calculate the needed gain of the IA in order to keep this noise below 1 µK Hz−1/2, i.e., σADC 216 VFS 1 pfs/2 1 GIAsb(T)≤10−6→GIA ≥170 .(2.24) The IA used in the prototype is the AD624 of Analog Devices (the one used in the FM design is very similar, the AD620 from Analog Devices). This solution reduces the use of discrete resistors, eliminates in some cases the necessity of external trims, and maintains a low temperature coefficient of the amplifier, specially for the gain. The gain of the IA, GIA, is set to 200 which is easily configured in the AD624 (and in the AD620) and is slightly larger than the minimum required calculated in Eq. (2.24). Therefore, the sensitivity of the system at the output of the IA is sIA =sbGIA ≃1.35 V K−1(2.25) around 20  . The IA also introduces noise in the measurement due to its inherent voltage and current noise sources. In order to analyse the contribution of the IA to the total noise, the bridge output impedance (the MUX impedance can be neglected) has to be taken into account. Figure 2.11 shows the noise sources coming from the IA. The equivalent spectral voltage density referred to the input (RTI) is e2 IA,RTI(T, ω) = e2 n(ω) + i2 n(ω)"RRref R+Rref 2 +RRNTC(T) R+RNTC(T)2#(2.26) where e2 nand i2 ncan be modelled as [102] e2 n(ω) = e2 ni +e2 no G2 IA =K2 v1 + ωcv ω,(2.27) i2 n(ω) = K2 i1 + ωci ω.(2.28) 15Note this is not the quantisation noise. This is known as the transition noise in ADC data-sheets. The quantisation noise is considered later on. 2.3 Signal processing chain 37 * * * * in in eni Rref v o eno R R R(T) IA Figure 2.11: IA + Wheatstone bridge noise equivalent circuit. The noise parameters of the IA AD624 are Kv=4 nV Hz−1/2,ωcv/2π=3 Hz, Ki=0.3 pA Hz−1/2, and ωci/2π=100 Hz. The noise levels introduced by the IA can be converted into noise equivalent temperature dividing Eq. (2.26) by the sensitivity of the bridge, i.e., S1/2 T, IA (RTI)(T, ω) = eIA,RTI(T, ω) sb(T).(2.29) Evaluation of Eq. (2.29) leads to the results given in Figure 2.12. 10−4 10−3 10−2 10−1 100101102 1 10 100 frequency [Hz] ST,IA [ K Hz −1/2 −1/2 ] en source in source total µ( )ω Figure 2.12: Noise introduced by the amplification stage. The noise in the measurement corresponds to that at the modulating frequency (=5.55 Hz) and at all the odd harmonics —see § 2.3.1. The modulating frequency cannot be set arbitrarily high due to the settling time of the anti-alias filter. Two important aspects must be noted from Figure 2.12: (i) the dominating noise source comes from the noisy current sources of the IA [which translates into noise voltage when coupling with the Wheatstone bridge impedance —see Eq. (2.26)] and, (ii) the noise in the MBW, i.e, at the milli- Hertz region exceeds the requirement of 10−5K Hz−1/2if no lock-in amplification is used. Actually, the noise at 1 mHz is one order of magnitude higher than that. However, we have to keep in mind that the the noise of the IA is modulated by the demodulation of the Wheatstone bridge signal, thus, the 1/fnoise of the IA is shifted to the modulating frequency (and to the odd harmonics), ωb, and the IA noise present at the modulating frequency is shifted to the baseband —see §2.3.1. This means that the noise introduced in the MBW during the amplification stage must be calculated at f=ωb/2π, i.e., at 5.55 Hz(and at the odd harmonics). Figure 2.13 shows the actual noise introduced by the IA as a function of the measured temperature. The discontinuities are caused by the change of scale —see Table 2.3. 38 2 The LTP temperature measurement subsystem 10 12 14 16 18 20 22 24 26 28 30 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 temperature [oC] S T,IA [ K Hz −1/2 −1/2] ( )ωµ Figure 2.13: Noise introduced by the IA after the demodulation process. This is the noise at f=ωb/2π= 5.55 Hz. Other sources of error that might affect the performance of the system are the common-mode voltage error and the gain temperature coefficient of the IA. The common-mode voltage at the input of the IA is not cancelled by the modulation/demodulation process, hence, this error appears at the output of the IA attenuated by the IA CMRR. The errors can be up to 140 µK in a very worst case. However, we are not interested in absolute temperature values but in their relative changes, thus, the common-mode voltage can only be a problem if the temperature coefficient of the CMRR of the IA is high —see appendix § A.6. The other important parameter to take into account is the temperature coefficient of the IA gain, αGIA . The temperature coefficient of the amplification stage is αIA(T) = vo(T)αGIA .(2.30) The maximum permitted TC of the IA gain is readily calculated by vo(T)αGIA sb(T)S1/2 T, FEE(ω)≤S1/2 T, req(ω) (2.31) where the value obtained for αGIA , assuming S1/2 T, FEE lower than 0.1 K Hz−1/2, is 3.5 ppm K−1. The value for the AD624 is 3.5 ppm K−1(the FM model IA, the AD620, is slightly higher, 10 ppm K−1). The low-pass filter stage is included to limit the signal bandwidth before sampling and, thus, reduce as much as possible the effect of the aliasing due to high frequency noise and interference signals. The cut-off frequency of the filter cannot be reduced arbitrarily: there is a compromise between the settling time of the filter and the aliasing errors. The low-pass filter implemented is a second order Sallen-Key Butterworth —see Figure 2.14— with R1=2.84 kΩ, R2=10.7 kΩ, C1=100 nF and C2=33 nF. This results in a cut-off frequency of 500 Hz. C1 R1R2 C2vo vi Figure 2.14: Second order Sallen-Key Butterworth low-pass filter. 2.3 Signal processing chain 39 Each time the polarity of the bridge is reversed or another channel is selected, the filter has to settle to the new value. Thus, it is necessary to introduce a delay after a change in the input before acquiring the signal with the ADC. For this reason the data from the first 10 ms are discarded after a change in the bridge polarity or in the channel selected. This dead-time reduces the error of this effect at the level of the pico-Kelvin —see appendix § A.7. 2.3.2.5 Analog-to-digital conversion stage The analog-to-digital conversion circuit has been implemented using a complete integrated complementary metal oxide semiconductor (CMOS) sampling ADC (AD977 of Analog Devices for the prototype and ADS7809 of Texas Instruments for the FM design)16. This stage contains a 16-bit capacitive-based successive approximation register (SAR) ADC with sample-and-hold, reference voltage, clock and serial data interface. This implementation reduces the external components to a minimum. As mentioned in § 2.3.2.2, the voltage reference of the ADC is also used to feed the Wheatstone bridge to obtain a voltage reference-independent conversion —see Eq. (2.19). Two different sources of noise appear during the analog-to-digital conversion: (i) the noise (transition noise) calculated in Eq. (2.23) which for GIA = 200 is (at most) ≃0.8 µK Hz−1/2and, (ii) the quantisation noise, which for an ideal ADC is17 Sq(ω) = ∆2 12 1 fs/2(2.32) where ∆ = VFS/2Nbit is the quantisation step or least significant bit (LSB). The quantisation noise with fs=38.4 kHz, Nbit=16 and VFS=10 V is ≃0.2 µK Hz−1/2from dc to fs/2. Thus, the noise introduced by the ADC to the system is within the requirements, i.e., below 10 µK Hz−1/2. Finally, the full-scale error drift due to the temperature affects the measurement. The temperature coefficient of this stage is αADC(T) = vo(T)αGADC (2.33) where αGADC is the temperature coefficient of the ADC gain. The following expression permits to calculate its maximum permitted value, vo(T)αGADC sb(T)S1/2 T, FEE(ω)≤S1/2 T, req(ω).(2.34) The value obtained for αGADC , assuming S1/2 T, FEE lower than 0.1 K Hz−1/2, is 35 ppm K−1. The AD977 gain temperature coefficient is around 7 ppm K−1(the FM ADC, ADS7809, has the same TC). 2.3.3 Digital signal processing The digital signal processing consists, basically, in the digital demodulation of the signal vo. The process is the following: during one polarity Npoints of Mpoints are averaged (the first 10 ms are discarded to avoid errors due to the low-pass filter settling time —see § 2.3.2.4). When the polarity of the bridge is reversed the same process is done and then the two averaged samples are subtracted and divided by 2 to obtain the final value. The digital processing chain extends from the ADC output to the final temperature value. In the prototype design, it has been implemented in the firmware of the electronics [using a 16F877 PIC of Microchip] and in a computer running a Labview application software [in the FM design it is implemented in a field programmable gate array (FPGA) in the DAUs of the DMU and in the 16Space qualified constraints force to use 16-bit ADC [44]. Furthermore, ADCs with higher number of bits are Delta-Sigma which, in general, are not suitable for multiplexed systems nor for signals with steep changes (such as a square wave) due to long stabilisation times. 17Problems with the non-ideality of the quantisation transfer curve of the ADC are discussed in § 6. 40 2 The LTP temperature measurement subsystem DPU —see Figure 1.12]. The digital demodulation described above is shown in the block diagram of Figure 2.15, which is an alternative representation of the typical demodulation scheme —see Figure 2.3— that permits to easily analyse the noise in the system. averaging N2 Mz−1 + − decimator difference averaging 1/2 x[n] y[n] Figure 2.15: Digital demodulation block diagram. d[n] is the signal quantised by the ADC. The analog signal processing block is shown in Figure 2.5. The digital output from the ADC, x[n], consists of Msamples (=3456, or 90 ms), however, only Nsamples (=3072, or 80 ms) are averaged —see Table 2.4. This is equivalent to a low-pass filter and a subsequent downsampling by M. This process results on an equivalent resolution of18 [74, 22, 70] Neq bit =Nbit ADC +1 2log2N≃22 bit (2.35) which meets the required resolution in Eq. (2.18) —see § 2.3.2.1. The equivalent transfer function of the digital averaging is |HN(ω)|=sin πNω/ωs Nsin πω/ωs (2.36) where Nis the number of averaged points and ωs/2πis the ADC sampling frequency. The power spectral density of the downsampled (by M) signal is [100] SM↓(ω) = M−1 X k=0 SNω−kωs M(2.37) where SNis the spectrum of the digitised data after the averaging, i.e., SN(ω) = |HN(ω)|2Sd(ω) (2.38) with Sdthe spectrum of the digitised analog signal, i.e., Sd(ω) = ∞ X k=−∞ ST,analog(ω−kωs).(2.39) The spectrum of the discrete signal is essentially free of aliasing since the sampling frequency —see Table 2.4— is well above the cut-off frequency of the anti-alias filter (500 Hz). Therefore, Sd(ω) = ST,analog(ω) from dc to fs/2, or Sd(ω) = ST,analog(ω) = ST, b(ω) + ST, IA(ω) + ST, ADC(ω) (2.40) where ST, b,ST, IA and ST, ADC are the noise of the Wheatstone bridge, the noise of the IA and the noise of the ADC19, respectively. The actual temperature value is obtained by taking the average difference of two consecutive samples. This is equivalent to an average difference and a subsequent downsampling by 2. The transfer function of the average difference is Hdiff (ω) = e−iω/ωsM−1 2→ |Hdiff (ω)|=|sin πMω/ωs|(2.41) 18Due to the oversampling and considering white noise greater than the quantisation noise at the input of the ADC. 19The high frequency transition noise of the ADC is aliased because is not low pass filtered. 2.3 Signal processing chain 41 and the power spectral density of the signal downsampled by 2, i.e., the output temperature measurement, is S2↓(ω) = 1 X k=0 Sdiff ω−kωs 2M(2.42) where Sdiff (ω) = |Hdiff (ω)|2SM↓(ω).(2.43) The parameters involved in the digital processing are summarised in Table 2.4. Parameter Value Description N3072 Number of averaged samples M3456 Number of samples per polarity ωs/2π38.4 kHz ADC sampling frequency ωb/2π5.55 Hz Frequency of the square wave Table 2.4: Parameters and their values involved in the digital signal processing of the prototype TMS. The final read out frequency of the TMS is the frequency of the square wave, i.e., 5.55 Hz (if only one temperature is acquired). The resultant total noise after the analog and signal processing chain is given in next section. 2.3.4 Theoretical noise equivalent temperature Once all the signal processing chain (analog and digital) has been described we proceed to the evaluation of the theoretical expected noise of the system. It is calculated by using Eqs. (2.36) to (2.43). Results are shown in Figure 2.16. 10−3 10−2 10−1 10 0 1 1.5 2 2.5 3 3.5 4 4.5 5 frequency [Hz] S−1/2 T [ K Hz−1/2] 10 oC 30 oC ( )ω µ Figure 2.16: Theoretical noise for the TMS at T=10  and T=30  after the analog and digital signal processing. The noise is slightly higher when measuring low temperature since the equivalent resistance of the bridge is higher at low temperatures. 48 2 The LTP temperature measurement subsystem IS−1 IS−2OB OW−1 OW−2 TS3TS1 TS13 TS15 TS17 TS19 TS4TS2 TS6TS8 TS14 TS16 H1 H9 H11 H4 temperature sensor connected to DAU−1 heater connected to DAU−1 heater connected to DAU−2 temperature sensor connected to DAU−2 TS10 TS12 TS9 TS11 TS5TS7 H2 H3 H7 H8 H5 H6 TS23TS24 TS20 TS21 TS22 TS18H10 H12 H13 H14 Figure 2.22: Summary of thermal diagnostics location and connections. implies a bit rate of 1386.67 bps, which is considered too high. The bit rate has been reduced by a factor of four by means of a digital downsampling. This process is described in appendix C. Chapter 3 Interactions with LTP subsystems The interaction of the temperature measurement subsystem with the other subsystems on board the LTP is another issue that needs to be analysed. The thermal diagnostic subsystem might perturb other subsystems nearby and vice versa, i.e., some subsystems might degrade the measurement of temperature. We focus on the potential problems in the gravity reference sensor (GRS), since it is the most sensitive subsystem of the LTP with diagnostics devices in it. Two basic potential problems have been identified. On the one hand, the magnetic polarisation of the thermistors might be incompatible with the magnetic cleanliness requirements of the GRS and thus, they could cause an excess force noise in the test mass (TM) [120]. On the other hand, the capacitive coupling through stray capacitance between the electrodes of the capacitive sensor in the GRS (or other unforeseen interferences) and the temperature sensors might degrade the temperature measurement. Both problems are discussed in the following sections. 3.1 Thermistors magnetic polarisation Four thermistors are attached to the outer face of the electrode housing (EH) surrounding the TMs. More specifically, two acting as sensors (Ro=10 kΩ) and two acting as heaters (Ro=2 kΩ) —see § 2.4 and appendix § B— on each of the two EH face perpendicular to the LTP sensitive axis. All in all, eight devices are laid down around each of the TMs at a distance of only ∼13 millimetres —see Figure 3.1. NTCs however, as previously stated, are manufactured by mixing and synthesising oxides doped with metals such as manganese, nickel, cobalt, iron and copper [123, 160, 101]. Except manganese and copper, which show paramagnetic and diamagnetic behaviours, respectively, these materials show ferromagnetic behaviour. In spite of their tiny size, NTC magnetic properties can jeopardise the performance of the LTP, as they are placed quite near the TMs. Magnetic cleanliness in the LTP must comply with the requirements that limit the acceptable values of magnetic field and magnetic field gradient in the region occupied by the TMs. These requirements are set both on dc values and fluctuations of these quantities —see § 3.1.1. Previous missions such as the Advanced Composition Explorer (ACE)1currently measuring the magnetic field in the Lagrange point L1 (where LPF will operate) have shown that the interplanetary magnetic field will not pose a problem in this sense since its dc values and fluctuations are orders of magnitude below the LTP magnetic cleanliness requirements —see Figure 3.2 and Table 3.2. 1Data extracted from www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA MAG.html. 50 3 Interactions with LTP subsystems Figure 3.1: Top: EH perspective (thermistors are not drawn). Bottom: layout of the eight NTCs in the EH around one TM. Axes are labelled with distances in millimetres. The approximate distance of each thermistor to the TM is 13 mm. 0 2 4 6 8 10 x 10 4 −12 −10 −8 −6 −4 −2 0 2 4 6x 10 −9 Time [s] B [T] Bx By Bz 10−5 10−4 10−3 10−2 10−1 10−12 10−11 10−10 10−9 10−8 10−7 10−6 Frequency [Hz] Bx By Bz B [T Hz−1/2 ] Figure 3.2: ACE mission data. Left: magnetic field in the Lagrange point L1. Right: magnetic field power spectral density in the Lagrange point L1. The requirement of the background dc value in the LTP is 10 µT, at L1 the values are around the nano-Tesla. The fluctuations in L1 are also two orders of magnitude lower than the expected inside the LTP due to electronic systems [141] —see Table 3.2. We thus only need to consider the magnetic field generated by the sources inside the LPF spacecraft and, among these, specially those that are close to the TMs, i.e., the thermistors. All magnetic sources will be outside the LTP core assembly (LCA) and are designed not to exceed the magnetic requirements in the TM location. Inside the LCA, only thermal diagnostic items will be close enough to perturb the magnetic cleanliness, in case of being magnetically active. The noise 3.1 Thermistors magnetic polarisation 51 acceleration budget assigned to magnetic effects has been set to [146] S1/2 a, magn(ω)≤12 fm s−2Hz−1/2(3.1) in the LTP MBW. The following sections address the problem of the potential excess noise caused by the magnetic behaviour of the thermistors. First we review the mechanism whereby magnetic fields induce forces on the TM. Second, we present the measurements performed to characterise magnetically the NTCs and the relationship between their magnetic moment and the created magnetic field and magnetic field gradient in the TM location. Finally, we estimate the effect of the thermistors on the TM noise, and suggest actions to minimise the risk of possible excess noise of this nature [120]. 3.1.1 Force fluctuations in the TM due to magnetic field and magnetic field gradient If a magnetic field Bacts on a small volume d3xof magnetic material with low magnetic susceptibility χand density of magnetic moment Mthen the force on that small volume is given by [73] dF d3x=∇M+χ 2µ0 B·B=M+χ µ0 B·∇B(3.2) where µ0= 4π×10−7m kg s−2A−2is the magnetic constant. The second equality follows from the first as a consequence of Maxwell’s equation ∇×B= 0. To calculate the total force on the TM, Eq. (3.2) must be integrated to its volume, V= (46 mm)3. It is expedient to express the integral in terms of averaged quantities, defined by hfi ≡ V−1ZV f(x)d3x(3.3) where fis any function, scalar, vector or tensor. We are mostly interested in the x-component of the force, as it is the relevant one for the mission science. With this notation, Fx=VhM·∇Bxi+χV µ0hB·∇Bxi.(3.4) The fluctuations of the force are, from Eq. (3.4), given by δFx=VhδM·∇Bxi+VhM·δ(∇Bx)i+χV µ0hδB·∇Bxi+χV µ0hB·δ(∇Bx)i(3.5) where δrefers to temporal fluctuations, and provided second order terms are neglected2. The magnetic field Bin Eq. (3.5) is the sum of the background field, Bbg, and the field created by the thermistors (if any) BNTC: B=Bbg +BNTC (3.6) Fluctuations of the density of magnetic moment of the TM, δM, can be neglected since the environmental temperature is very stable [84], and the same applies to the magnetic field and magnetic field gradient fluctuations created by the NTCs. Consequently, Eq. (3.5) can be simplified to δFx=VhM·δ(∇Bx)i+χV µ0hδB·∇Bxi+χV µ0hB·δ(∇Bx)i(3.7) 2Terms of the form δB·δ(∇Bx), etc. These may be relevant in the presence of high frequency fields which, due to the quadratic coupling, can generate low frequency fluctuations. We do not consider this possibility here. 52 3 Interactions with LTP subsystems where δBand δ(∇Bx) refer only to the background magnetic field. A worst case estimate of the force fluctuations in the TM (in terms of spectral density) due to the magnetic properties of the TM and the magnetic field and magnetic field gradient in it is —see appendix § D, SδFx(ω) = V2|hMi|2S∇Bx(ω) + χV µ02 |h∇Bxi|2SB(ω) + χV µ02 |hBi|2S∇Bx(ω) (3.8) where S(ω) refer to power spectral densities. The values of SBand S∇Bxare assumed uniform throughout the TM volume. This approach is, again, based on the assumption that fluctuations come only from the background field which can be considered homogeneous in the TM volume. The same applies to its fluctuations —see Table 3.2. From Eq. (3.8) we note that the potential excess noise due to the presence of the NTCs surrounding the TM will only come from the last two terms, since the first term is only affected by the magnetic field gradient background fluctuations, S∇Bx, and the density of magnetic moment of the TM, M. Numerical evaluation of Eq. (3.8) depends on the values of the fluctuations of the environmental magnetic field and gradient, as dc magnetic fields couple to fluctuating fields to generate noise, due to non-null magnetic susceptibility, χ, of the TMs. The magnetic nominal properties of the TM are given in Table 3.1. The measurement and estimation of the averaged values in Eq. (3.8) caused by the presence of the NTCs, |hBNTCi| and |h∇BNTCi|, are detailed in § 3.1.2, while the dc requirements for Bbg and ∇Bbg,x are given in Table 3.2. |χ| |M|SM(ω) 2·10−52·10−4A m−1∼0 Table 3.1: TM magnetic properties [34, 124, 141, 146]. dc Req. PSD est. ∀x |Bbg| ≤ 10 µTS1/2 B(ω)≤100 nT Hz−1/2 |∇Bbg, x| ≤ 5√3µT m−1S1/2 ∇Bx(ω)≤50√3 nT m−1Hz−1/2 Table 3.2: Magnetic dc requirements in the TMs [146] and estimated magnetic fluctuations in the TMs location [141, 145]. Note that S1/2 B=S1/2 Bbg and S1/2 ∇Bx=S1/2 ∇Bbg, x , since the magnetic field background and its gradient are the only time dependent terms. The numbers shown in Tables 3.1 and 3.2 yield a nominal acceleration noise due to magnetic effects in the TM in absence of thermistors, i.e., for B=Bbg and ∇Bx=∇Bbg, x. The values are given in Table 3.3, which will be the noise reference when considering the force noise added by the magnetisation of the thermistors. It is thus clear that the requirement set by Eq. (3.1) is comfortably satisfied in the absence of thermistors. The last row in Table 3.3 is the total magnetic noise spectral density, and is evaluated by the expression S1/2 total mag(ω) = V"χ µ0|h∇Bxi|2 SB(ω) + |hMi|+χ µ0|hBi|2 S∇Bx(ω)#1/2 (3.9) where we have assumed that SBand S∇Bxare uncorrelated. 3.1 Thermistors magnetic polarisation 53 Term SFx[fN Hz−1/2]Sax[fm s−2Hz−1/2] V|hMi|S1/2 ∇Bx(ω) 1.68 0.86 (χV/µ0)|h∇Bxi|S1/2 B(ω) 1.67 0.85 (χV/µ0)|hBi|S1/2 ∇Bx(ω) 1.67 0.85 S1/2 total mag(ω) 3.74 1.91 Table 3.3: Nominal (in absence of NTCs) noise values in terms of force and acceleration (mTM=1.96 kg) within the MBW. 3.1.2 Magnetic field and magnetic field gradient in the TM due to NTCs. Measurement of the magnetic properties of the NTCs G10K4D and G2K7D BetaTherm NTCs are small devices, 6 mm in diameter, and 2 mm thick [16]. Eight NTCs are surrounding each of the TMs at a distance of ≃13 mm as shown in Figure 3.1. In the following, in order to estimate the magnetic field and magnetic field gradient caused by the thermistors in the TM, we shall make the assumption that the NTCs behave like magnetic dipoles of remanent magnetic moments, ma,a=1,...,8. Given their small size, corrections to this hypothesis may only be tiny. Under this assumption, the magnetic field created by these dipoles will be given by [73] BNTC(x) = µ0 4π 8 X a=1 3(ma·na)na−ma |x−xa|3(3.10) and its gradient by ∂BNTC,i ∂xj =µ0 4π 8 X a=1 3 |x−xa|4[(ma,ina,j +ma,jna,i)+(ma·na)(δij −5na,ina,j)] (3.11) where xais the position of the a-th NTC, and nais the unit vector in the direction from the a-th NTC to the field point x, or na=(x-xa)/|x-xa|.δij is the usual Kronecker symbol. In order to evaluate Eqs. (3.10) and (3.11) and, thus, be able to calculate |hBNTCi| and |h∇BNTC, xi| we need to know the magnetic moments of the NTCs, ma. In the following sections we describe the measurements done to characterise magnetically the NTCs. 3.1.3 NTCs magnetic characterisation The instrument used to characterise the magnetic properties of the thermistors is a Quantum Design MPMS XL SQUID of the Serveis Cient´ıfico-t`ecnics of the Universitat de Barcelona. The tests consisted in measuring the magnetic moment, m, of the NTCs when subjected to an external varying magnetic field, H, and thus obtain the hysteresis cycle of the device. In order to fully characterise the thermistor, the hysteresis curve was measured in two different orientations of the NTCs relative to the direction of the external field (configurations parallel and orthogonal) —see Figure 3.3. 54 3 Interactions with LTP subsystems parallel orthogonal H NTCs Figure 3.3: Orientation configurations of the NTCs inside the SQUID with respect to the applied external magnetic field, H, for the NTC magnetic moment measurement. We first characterised a BetaTherm 10 kΩ NTC (intended to be used as temperature sensor) and a BetaTherm 2 kΩ NTC (intended to be used as heater in the EH). The results are shown in Figure 3.4. All the measurements were done at 300 K since is the expected temperature at the TM location. −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1x 10 −4 µ0H [T] m [A m2] parallel orthogonal −1.5 −1 −0.5 0 0.5 1 1.5 −1 0 1 0.5 −0.5 x 10 −4 µ0H [T] m [A m2] parallel orthogonal Figure 3.4: Hysteresis curve at 300 K for the 10 kΩ BetaTherm NTC thermistor (left) and the same for the 2 kΩ thermistor (right). Results given in Figure 3.4 show that both items exhibit a ferromagnetic behaviour (coercive field, µ0|Hcoer|, is about 10 mT), and that the remanent magnetic moment, mr, of the 10kΩ NTC is slightly higher than that of the 2 kΩ one. Table 3.4 summarises these results. 2 kΩ 10 kΩ Conf. |mr| |msat| |mr| |msat| Par. 16 µA m250 µA m226 µA m290 µA m2 Orth. 7µA m250 µA m29.4 µA m2100 µA m2 Table 3.4: BetaTherm (2 kΩ and 10 kΩ) NTCs magnetic properties. Coercive field, |µ0Hcoer.|, for both sensors and configurations is 10 mT. The values of the remanent magnetic moment, mr, are indicative of the very worst case since they reflect the magnetisation of the sensor after being saturated. The saturating magnetic field is around 500 mT, a considerably large one. The assessment of the effect of NTCs on the LTP acceleration noise depends on the quality of the determination of the magnetic properties of the thermistors, specially the remanent magnetic moment parameter, mr. In view of this, measurements of a set of BetaTherm NTC thermistors (four samples) and another type of thermistor, YSI NTC (five samples of the YSI 44031 bead type 3.1 Thermistors magnetic polarisation 55 model), were performed to ratify the parameters given in Table 3.4 and to compare them with another type of thermistor (the YSI one). Thermistors tested were all of 10 kΩ. NTCs of 2 kΩ were not tested because they are less magnetic than the 10 kΩ ones —see Table 3.4. Moreover, the magnetic moment of the BetaTherm thermistor was measured only for the parallel configuration since this corresponds to the worst case —see Table 3.4. The orientation in the YSI thermistors was irrelevant due to its spherical symmetry. Hysteresis curves for both sets of thermistors are shown in Figure 3.5. −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −1 0 1 0.5 −0.5 x 10 −4 m [A m ] 2 H [T] 0 µ −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −1 0 1 0.5 −0.5 x 10 −4 ] m [A m2 H [T] 0 µ Figure 3.5: Left: hysteresis curve at 300 K for the set (four samples) of 10 kΩ NTC thermistors of BetaTherm and parallel configuration. Right: idem for the set of YSI thermistors. Figure 3.5 (left) confirms the behaviour observed in the measurements done previously with the BetaTherm NTCs, i.e., ferromagnetism. However, Figure 3.5 (right) shows that YSI thermistors get easily demagnetised after removing the external magnetic field, i.e., they have a comparatively small coercive field (≃1 mT) and a small susceptibility, too. Finally, Figure 3.5 shows that the results for each set of thermistors are consistent with one another (small standard deviation values). Results are summarised in Table 3.5. Sensor |mr| |msat.|µ0|Hcoer.| BetaTherm 24±2µA m283±2.5 µA m210 mT YSI 1.8±0.5 µA m2110±5µA m21 mT Table 3.5: Magnetic properties for the two sets of thermistors tested. BetaTherm values are for the parallel configuration (see Figure 3.3) and for the 10 kΩ of nominal resistance only. Four BetaTherm samples and five YSI samples were measured. 3.1.4 Numerical calculations In order to evaluate Eq. (3.8) we first calculate the averaged values of the magnetic field and magnetic field gradient created by the NTCs —Eqs. (3.10) and (3.11). We make the simplifying assumption that all 8 NTCs are 10 kΩ, even if only 4 are. We accordingly overestimate the magnetic effect, which gets us on the safe side. The evaluation of the averaged values has been done by means of numerical methods. More specifically, by a finite element method (FEM) approach: the volume of the TM, (46 mm)3, has been divided into Nvolume elements, ∆V(=V/N), and for each ∆VEqs. (3.10) and (3.11) have 56 3 Interactions with LTP subsystems been used, then the average value and modulus calculated, i.e., hBNTCi ≃ N−1 N X k=1 BNTC(xk) (3.12) h∇BNTC,xi ≃ N−1 N X k=1 ∇BNTC,x(xk) (3.13) where xkis the position of the k-th volume element. This has been chosen of 8 mm3, which corresponds to N=12167. Decreasing the size of the volume elements did not improve the results of the computations. Calculations considering different orientations of the eight magnetic moments of the NTCs allow us to know the worst possible magnetic moment orientation combinations for each axis. These are summarised in the four configurations shown in Figure 3.6. In terms of magnetic field, configurations B and D give the highest mean magnetic field value. However, the magnetic field gradient is zero. On the contrary, configurations A and C yield the highest mean value of the magnetic field gradient, but the mean magnetic field is zero. Intermediate configurations have been seen to produce milder effects. A m=−6 A m2 9.4x102.6x10 −5 A m2 m= x z B C D Figure 3.6: Configurations analysed for the evaluation of Eqs. (3.10) and (3.11). Configurations A and B assume the magnetic moments of all NTCs are oriented along the z-axis. However, in configuration A magnetic moments take opposite directions, while in B all the orientations are equal. Configurations C and D assume all the NTCs are oriented along the x-axis. In configuration C magnetic moment orientations are antiparallel, while in configuration D all the orientations coincide. This four configurations cover the worst cases. The magnetic moment values used are extracted from Table 3.4. The results obtained for the configurations given in Figure 3.6 are summarised in Table 3.6, and they can be seen graphically for configurations A and B in Figure 3.7. Conf. hBi;|hBi| [µT] h∇Bxi;|h∇Bxi| [µT m−1] A(0,0,0); 0 (0,0,15.5); 15.5 B(0,0,-0.25); 0.25 (0,0,0); 0 C(0,0,0); 0 (-11.3,0,0); 11.3 D(0.18,0,0); 0.18 (0,0,0); 0 Table 3.6: Mean values and modulus for the magnetic field and magnetic field gradient caused by the eight NTCs in the TM. The magnetic moment used in configurations A and B is |mr|= 26 µA m2, and |mr|= 9.4µA m2for configurations C and D —see Table 3.4 and Figure 3.6. 3.1 Thermistors magnetic polarisation 57 20 30 40 50 −20 −10 0 10 20 −0.1 0 0.1 x [mm] z=0 y [mm] Bz [µT] 20 30 40 50 −20 −10 0 10 20 −100 −50 0 x [mm] y [mm] ∂Bz/∂x [µT m−1] 20 30 40 50 −20 −10 0 10 20 −0.3 −0.2 −0.1 0 x [mm] z=0 y [mm] Bz [µT] 20 30 40 50 −20 −10 0 10 20 −100 0 100 x [mm] y [mm] ∂Bz/∂x [µT m−1] Figure 3.7: Left: configuration A (see Fig. 3.6) and |mr|=26 µA m2. Top: Representation of the z-component of the magnetic field in the equatorial plane of the TM, i.e., xy-plane for z=0 —see Figure 3.1. The parity symmetries of field and gradient, Bz(x, y) = Bz(x, −y) = −Bz(−x, y), and ∂xBz(x, y) = ∂xBz(x, −y) = ∂xBz(−x, y), respectively, are clearly visible in the plots. In view of these symmetries, the field at the top averages to zero, while the gradient at the bottom does not. Right: configuration B (see Figure 3.6) and |mr|=26 µA m2. In this case the parity symmetries of field and gradient are slightly different: Bz(x, y) = Bz(x, −y) = Bz(−x, y), and ∂xBz(x, y) = ∂xBz(x, −y) = −∂xBz(−x, y), and are also reflected in the plots. In view of these symmetries, the gradient at the bottom averages to zero, while the field at the top does not. See Table 3.6 for numerical results. 3.1.5 NTCs first magnetisation curve The magnetic moments used in the calculations of the previous section are remanent magnetic moments after saturating the NTCs. However, it must be realised that the magnetic fields the LTP will go through, from the launch pad to the operation orbit, are orders of magnitude below the saturation field of the thermistor µ0Hsat ∼500 mT. For instance, the Earth magnetic field is in the order of 50 µT, the magnetic field in the van Allen belts is of the same order of magnitude3, and the interplanetary magnetic field is in the order of 10-100 nT [21] —see Figure 3.2. These values suggest that we are heavily overestimating the effect of the NTCs on the TM when using the remanent magnetic moment after saturation, which fully magnetises the device4. To fine-tune the previous estimates, we proceeded to analyse the first magnetisation curve (FMC) of the NTCs. The FMC is obtained by means of a two step procedure [129]: (i) an alternate magnetic field with decreasing amplitude is applied to the sample in order to demagnetise it to the level of the instrument resolution, and (ii) an external magnetic field which slowly increases in small steps (∼0.1 mT in our case) is applied to the sample. With this method, we first erase the magnetic moment of the NTC to then evaluate the magnetic response of the thermistor. We thus obtain the first response to the magnetic field, which will be lower than the response after applying a strong magnetic field. The FMC has been measured for the two sets of 10 kΩ NTC thermistors studied previously, the BetaTherm and the YSI ones. However, the demagnetisation was only meaningful for the BetaTherm NTCs due to its ferromagnetic behaviour, though not so for the YSI NTCs due to their narrow hysteresis curve —cf. § 3.1.3. Results are shown in Figure 3.8 and summarised in Table 3.7. 3Data extracted from the mission Champ, www.gfz-postdam.de/pb1/op/champ/results/index RESULTS.html. 4The highest risk of exposition to stronger fields actually resides in the transport phases of the LTP on-ground, when stronger magnetic fields in transporting vehicle machinery may cause problems. 64 3 Interactions with LTP subsystems v(t) c(t) y(t) Figure 3.14: Signal demodulation. The interference, v(t), is not modulated by the square wave excitation but it is modulated by the digital signal processing. This process consists in multiplying the interference, v(t), by the square wave, c(t). The square wave signal, c(t), is not a perfect square wave since dead times are incorporated to avoid errors due to the settling time of the anti-alias Butterworth filter —see appendix § A.7. The square wave signal is shown in Figure 3.15 where ais the dead-time, bis the integration time (for each polarity) and T(= 2a+ 2b) is the period. −1 1 −T/2 +T/2 −T/2+a/2 −a/2 a/2 +T/2−a/2 T b T/2T/2 a/2 b a/2 a 0 Figure 3.15: Square wave signal, c(t), used in the demodulation process. It is convenient to convert the signals, v(t) and c(t), to the frequency domain. To do this we first calculate the Fourier series of c(t), c(t) = ∞ X n=1 bnsin n2π Tt(3.23) where bnis bn=1 T/2ZT/2 −T/2 c(t) sin n2π Ttdt =−4 nπ sin nπ 2sin nπ 2T(T−2a).(3.24) Substituting Eq. (3.24) into Eq. (3.23) yields c(t) = −4Co π ∞ X n=1 K(n) nsin n2π Tt(3.25) where, for clarity, we define K(n) as K(n) = sin nπ 2sin hnπ 2T(T−2a)i(3.26) 3.2 Interferences in the thermistors in the GRS 65 which is zero for neven. The other signal involved in the process is the interference, v(t). We assume the interference is a sine wave of frequency Ω, phase φ, and amplitude V, i.e., v(t) = Vsin(Ωt+φ).(3.27) We have the two waves of interest, c(t) and v(t), defined in the time domain. The latter is given in Eq. (3.27) and the former is given in Eq. (3.25). We write it down again with Co= 1 —see Figure 3.15—, c(t) = −4 π ∞ X n=1 K(n) nsin(nωct) (3.28) where ωc/2πis the fundamental frequency of the square wave, i.e., 1 /T. The Fourier transforms of both signals are now readily obtained. They are: ˜v(ω) = −iπV eiφδ(ω−Ω) −e−iφδ(ω+ Ω),(3.29a) ˜c(ω) = i4∞ X n=1 K(n) n{δ(ω−nωc)−δ(ω+nωc)}.(3.29b) These two signals are multiplied during the demodulation process in the time domain —see Figure 3.14—, which in the frequency domain is equivalent to the convolution, y(t) = v(t)c(t)→˜y(ω) = 1 2π˜v(ω)∗˜c(ω).(3.30) The result of the convolution, ˜y, is ˜y(ω) = 1 2πZ∞ −∞ ˜v(ω0)˜c(ω−ω0)dω0 = 2V"∞ X n=1 K(n) neiφδ(ω−Ω−nωc)−eiφδ(ω−Ω + nωc)− −e−iφδ(ω+ Ω −nωc) + e−iφδ(ω+Ω+nωc)#.(3.31) Two cases of interest as a result of the obtained output, ˜y, are presented below: (i) when the interference signal is an even multiple of ωcand (ii) when it is an odd multiple of ωc. For the first case (Ω = nΩωcwith nOmega even), the interference, ˜v, will appear at the output, ˜y, at the following frequencies ω=±(nΩ+n)ωc,(3.32a) ω=±(nΩ−n)ωc(3.32b) which implies that for ω= 0 the output is zero [K(n) = 0 for neven]. Moreover all the frequency lines are multiples of ωc. This is important because they will be killed by the posterior averaging —see § 3.2.3. For the second case, i.e., when Ω = nΩωcwith nΩodd Eq. (3.31) becomes ˜y(ω) = ˜y(0)n=nΩ + 2V  ∞ X n=1 n6=nΩ K(n) n[eiφδ(ω−(nΩ+n)ωc) + e−iφδ(ω+ (nΩ+n)ωc)] +∞ X n=1 n6=nΩ K(n) n[−eiφδ(ω+ (−nΩ+n)ωc)−e−iφδ(ω+ (nΩ−n)ωc)]  (3.33) 66 3 Interactions with LTP subsystems where ˜y(0)|n=nΩstands for the signal at ω= 0, i.e., the gain at dc, or ˜y(0)n=nΩ =−4VK(nΩ) nΩ cos φ . (3.34) All the terms in Eq. (3.33) will be killed by the digital filtering (since they are located at multiples of ωc) —see § 3.2.3— except the term ˜y(0)n=nΩ . 3.2.3 Digital filtering and downsampling The digital filter consists of an average of Nsamples, with N= (2b+2a)fs. We keep the continuous time analysis for simplicity, however, it must be noticed that ˜y(ω) is repeated at multiples of fs (the sampling frequency of the ADC, 38.4 kHz in the prototype and 50 kHz in the FM). The filter, in the continuous time domain, is H(ω) = T 2b sin ωT/2 ωT/2(3.35) where the factor T/2bappears because the sum of all the samples is divided by 2binstead of T, i.e., the points within the dead times (2a) are not averaged. The filter is zero at integer multiples of ωc. After the filtering, the outputs for the two cases mentioned in the previous section are ˜yF(ω= 0) = −T 2b4VK(nΩ) nΩ cos φ, (3.36) and ˜yF= 0 for ω6= 0. Note that ˜y(0) is zero for nΩeven since K(n) = 0. When nΩis odd the gain at dc depends on the phase, φ, of the interference, v(t). Once the signal is filtered a downsampling by Nis performed. The process of downsampling consists in ˜yDS(ω) = N−1 X `=0 ˜yF(ω−`ωs N) = N−1 X `=0 ˜yF(ω−`ωc) (3.37) where ωc/2πis the fundamental frequency of the square wave. The output of the downsampled signal is exactly the same of the one obtained after the filtering if the interference is a multiple of ωcsince the filter has zeroes at all the harmonics (a generalisation for any frequency is given in § 3.2.4), i.e., ˜yDS(ω) = ˜yF(ω). At this moment we have assumed that the interference complies with the Nyquist requirement, i.e., Ω <ωs/2and, thus, aliasing is not present. Later on this is taken into account. Figure 3.16 shows the gain at dc as a function of the phase, φ, and the frequency of the interference, nΩωc. The plot in the right represents the worst-case, i.e., only the maximum value for each nΩbin (this occurs at φ= 0 and φ=π). Different vales of dead times, a, are shown to observe their effect on the interference signal. The gain at dc of the shifted component is calculated by evaluating the ratio between ˜y(0) and ˜v(ω), or7 Hdc =˜yDS(0) 2˜v(nxωc)=2 π K(nΩ) nΩ cos φ(3.38) 7The factor of 2 in the denominator is present because at dc the negative and the positive frequencies coincide. 3.2 Interferences in the thermistors in the GRS 67 20 40 60 80 100 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 a=20 ms φπ /nx gain at dc 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n gain at dc a=0 ms a=20 ms a=50 ms Ω Figure 3.16: Left: gain at dc as a function of the frequency and the phase of the interference [Eq. (3.36)] with a dead time a=20 ms. Right: maximum gain for each frequency bin and different dead times (0 ms, 20 ms and 50 ms). If the interference is an even multiple of ωcno signal is present at dc (nor at any other frequency) after the demodulation and the digital filter. However, if the frequency is an odd multiple of ωca signal at dc will appear due to the demodulation. The gain depends on the value of a. For instance for a=50 ms the gain of a sine wave interference Ω = ωcis 0.9, for a=20 ms is 0.75 and for a=0 ms is 0.63. This occurs because a square wave with dead times is more similar to a sine wave that an ideal square wave is. 3.2.4 Generalisation for any frequency So far we have analysed the problem considering the interference signals are at multiples of the frequency of the square wave. In this section we give an expression for an interference signal at any frequency Ω. The downsampled output signal is ˜yDS(ω) = N−1 X `=0 ˜yF(ω−`ωs N) = N−1 X `=0 ˜yF(ω−`ωc) (3.39) with ˜yF(ω−`ωc) = H(ω−`ωc)˜y(ω−`ωc) (3.40) where H(ω) is the filter given in Eq. (3.35). It can be shown that an interference at Ω will appear at a frequency within 0 and ωc/2 given by ωo=|Ω−nΩωc|(3.41) where nΩ=round Ω ωc.(3.42) The signal that appears at ωois8 |˜yDS(ωo)|=   4VT 2bK(nΩ) nΩcos φ, ωo= 0 VPN−1−nΩ k=1,3,5 K(k) kH(−Ω−kωc)−PN−1+nΩ k=1,3,5 K(k) kH(−Ω + kωc), ωo6= 0 (3.43) 8The spectra of the modulated signal, ˜y, is repeated at mfs/2π(the ADC sampling frequency) for m=±1,±2, .... In order to obtain a simple expression we have assumed that it is only repeated at ±fs. If fsfcthe assumption is accurate enough since we are only discarding the high frequency components of the signal that are weighted by K(n)/n. 68 3 Interactions with LTP subsystems and the gain is calculated as in Eq. (3.38), i.e., |˜ HDS(ωo)|=     2 π T 2bK(nΩ) nΩcos φ, ωo= 0 1 πPN−1−nΩ k=1,3,5 K(k) kH(−Ω−kωc)−PN−1+nΩ k=1,3,5 K(k) kH(−Ω + kωc), ωo6= 0 (3.44) Figure 3.17 represents the gain function given in Eq. (3.44). The peaks (at ωc, 3ωc, 5ωc, etc.) are gains at dc where the negative and positive frequency coincide. The rest of the signals are originated by intermediate frequencies and the frequency bin at which they will appear is calculated using Eq. (3.41). 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 ω0 HDS 0 5 10 15 20 25 30 35 40 45 50 0 0.5 1 frequency [Hz] a=0 ms a=20 ms a=50 ms () Figure 3.17: Gain after the digital demodulation and the digital filtering. Top: a=0 ms. Centre: a=20 ms. Bottom: a=50 ms. The frequency where the interferences will appear in ˜y(ω) is calculated using Eq. (3.41). For instance, an interference at 24 Hz appears at 1 Hz with a certain attenuation depending on the the dead time value. The gain pattern shown in Figures 3.16 and 3.17 is repeated at multiples of the sampling frequency of the ADC, fs(38.4 kHz in the prototype and 50 kHz in the FM version), due to the aliasing introduced by the sampling of the ADC. For instance, a signal at fs+fcwill appear at fc and then when demodulating will appear as a signal at dc —see Figure 3.18. 3.2.5 Experimental results A test to verify the analysis was performed. The test included the whole measurement chain, i.e., the analog and digital signal processing, thus, the final output is the combination of the analog acquisition chain plus the digital demodulation. The test consisted in applying a sine wave of different frequencies and then measuring the output to estimate the gain of the whole system. The sine wave was injected as viin Figure 3.11 with Cp=3 pF which represents the worst-case scenario9. The experimental results and the theoretical values are shown in Figure 3.18. The agreement 9The 3 pF stray capacitance is the one between the small aluminium plate of the thermistor and its cables. In the case of the ground fluctuations the stray capacitance will not differ too much from this value, however, in the case of the interference coming from the electrodes of the capacitive sensor it can be much smaller. This depends on the harness configuration. The higher the distance between both cables (thermistors and electrodes) the lower the stray capacitance between them. 3.2 Interferences in the thermistors in the GRS 69 between the experimental and theoretical gain is good. It can be seen that high frequency signals appear in the MBW. For instance, an interference at fs+fc(38 400 Hz+5 Hz) and another at fc (5 Hz) have approximately the same effect in the MBW. 100101102103104105 10−8 10−7 10−6 10−5 10−4 10−3 frequency [Hz] gain at dc theoretical experimental fs+fc 2fs+fc Figure 3.18: Experimental and theoretical results for the effect of a sine wave interference at one of the arms of the Wheatstone bridge. The plot represents the gain at dc, i.e., how a signal at high frequency appears in the measurement of temperature at dc (or at low frequency). The agreement between theoretical calculations and experimental results is good (the dashed trace is the envelope of the actual gain curve). The interferences used were at frequencies of 5 Hz, 105 Hz, 205 Hz, ..., etc. Note that due to aliasing an interference at 38 400 Hz+5 Hz (fs+fc) exhibits a gain similar to an interference at 5 Hz. fsis the ADC sampling frequency and fcis the frequency of the square wave. 3.2.6 Permitted amplitudes of the interferences Once the transfer function between the effect of an interference signal and the temperature measurement has been found, we can set upper limits of the permitted amplitudes of the interference and of the ground fluctuations. Tentative numbers are given in the following sections which are split into two independent scenarios: ground fluctuations and sine wave interferences. 3.2.6.1 Sine wave interferences We first consider the case of a pure sine wave as the interference at the input of the system. The fact that the capacitive sensor uses a signal of 100 kHz with an amplitude of 1 V makes relevant the analysis. By taking into account the gain curve shown in Figure 3.18 we can calculate the maximum permitted amplitude of the interference which does not affect the temperature measurement in a very worst-case, i.e., with Cp=3 pF. The power spectral density of the temperature read out is estimated as [13] b S(ω) = 1 tmZttot 0 x(t)e−iωtdt 2 (3.45) where tmis the time of the measurement. In the case of the presence of an interference signal, x(t) is x(t) = v(t) + n(t) (3.46) 70 3 Interactions with LTP subsystems where v(t) is the interference and n(t) is the nominal noise of the measurement. Substituting Eq. (3.46) into Eq. (3.45) yields b S(ω) = b Sn(ω) + 1 tm|˜v(ω)|2+1 tm [˜n(ω)˜v∗(ω) + ˜n∗(ω)˜v(ω)] (3.47) where b Snis the power spectral density of the noise. ˜vand ˜nare the Fourier transform of the interference and the noise signals, respectively. We want to find the maximum amplitude of the interference not to degrade the measurement, i.e., b S(ω)∼ =b Sn(ω) in Eq. (3.47). The “∼ =” symbol in our case means that b S(ω)<b Sn(ω) /10, or 1 tm|˜v(ω)|2+1 tm [˜n(ω)˜v∗(ω) + ˜n∗(ω)˜v(ω)] <b Sn(ω) 10 (3.48) where if we consider that10 |˜v(ω)|2˜n(ω)˜v∗(ω) + ˜n∗(ω)˜v(ω) (3.49) we obtain 1 tm|˜v(ω)|2<b Sn(ω) 10 .(3.50) The Fourier Transform of v(t) (a sine wave of amplitude Vand angular frequency Ω) during an integration time, tm, is ˜v(ω) = V tm 2i[sinc(ω−Ω)tm/2−sinc(ω+ Ω)tm/2] (3.51) thus, the maximum permitted amplitude of the interference for a given measurement time, tm, is V(ω)<1 p5tm/2 1 |H(ω)|b S1/2 n(ω) (3.52) where |H(ω)|is the transfer function shown in Figure 3.18, i.e., the gain of the whole measurement chain for an interference signal assuming a stray capacitance of 3 pF. Equation (3.52) is plotted in Figure 3.19. The measurement time, tm, considered is 10 ks. Theoretically, a signal of 100 kHz should not affect the measurement: the ADC (in the EM and the FM system) samples at 50 kHz, thus, a 100 kHz signal appears at dc due to aliasing, however, when demodulating by the 5 Hz square wave signal the interference is shifted to the 5 Hz frequency bin. Nevertheless, in practise the frequency of the interference will fluctuate around 100 kHz, and therefore the noise introduced in the MBW will not be exactly zero as it should ideally be. The maximum amplitude for an interference at 100 kHz±5 Hz considering a stray capacitance of 3 pF and an integration time of 10 ks is ∼10 mV. However, the stray capacitance between the cables of the capacitive sensor and the thermistors’ cables will be much lower than 3 pF, thus the maximum amplitude of the 100 kHz can be much higher than 10 mV. Moreover, if the frequency of the 100 kHz signal is stable, its effect on the TMS will be negligible. 3.2.6.2 Electrical ground fluctuations For the ground fluctuations case we consider them as white noise. The noise introduced in the measurement bandwidth due to this effect is b Sn(ω) = Sn,GND(ω)1 ∞ X n=0 |H(nωc)|2 (2n+ 1)2.(3.53) 10For long time measurements and assuming v(t) is a sine wave, this is true since |˜v(ω)|2is proportional to t2 m whereas ˜v(ω) is proportional to tmand ˜n(ω) is white noise which its amplitude is constant independently of the measurement time. 3.2 Interferences in the thermistors in the GRS 71 100101102103104105 10−4 10−3 10−2 10−1 100 101 frequency [Hz] Vmax [V] fs+fc 2fs+fc Figure 3.19: Maximum amplitude permitted for a sine wave interference not to affect the temperature measurement performance. The sampling frequency of the ADC is 38.4 kHz (in the FM is 50 kHz). The measurement time is 10 ks and the stray capacitance considered is 3 pF. The concern about ac interference signals comes, mainly, from the 100 kHz signal of the electrodes. The values shown in this figure are a very-worst case: actually, Cp should be much lower than 3 pF and the signal is at 100 kHz (2fs), not 100 kHz+5 Hz (2fs+fc) which alleviates the risk of this problem. The maximum permitted fluctuations between both grounds are readily obtained by considering that Sn,GND(ω)∞ X n=0 |H(nωc)|2 (2n+ 1)2<b Sn(ω) 10 .(3.54) If we truncate nto the 100 kHz range the term of the summatory is ≃2·10−10 and, therefore, the noise levels permitted are readily calculated: Sn,GND(ω)<200 mV Hz−1/2(from zero to 100 kHz) which seems comfortable. The analysis presented in this section describes a potential problem in the TMS due to the capacitive coupling between different systems in the GRS. Such problem mainly depends on the harness configuration of the subsystems in the LTP. One may not predict whether this problem will affect the temperature measurements or not until the test of the TMS within the assembled satellite will be performed, but at least the analysis presented here can provide a clue in case extra noise or anomalous behaviours are observed during such tests. The tests performed on ground to validate the TMS showed that it is very sensitive to capacitive couplings. In view of this, during such tests all the metallic bodies were always connected to the same ground of the electronics and, in addition, the cables were shielded —see chapter § 5. Any missconfiguration led to noticeable interferences in the measurement, specially in long runs. For this reason capacitive coupling is an important issue to take into account in the performance of the TMS. Once more, such fears cannot be discarded until tests with the entire satellite will be performed. Chapter 4 Test bed design for the TMS validation This chapter describes the test bed designed to perform a meaningful test to determine the noise of the temperature measurement subsystem described in § 2. Two configurations are discussed. The first one is intended to assess whether or not the requirements for the LTP TMS are met. The second configuration focuses on noise investigations in the thermistor and the electronics in the frequency range of LISA going one order of magnitude below that of LISA Pathfinder. The results obtained during the investigations in the MBW of LISA are, obviously, also useful for the temperature measurements of the LTP. Both test bed configurations are based on the reduction of all possible environmental disturbances, mainly the temperature fluctuations at the location of the thermistors. If we ensure that the thermistors are placed in a sufficiently stable environment, we can ascribe all the fluctuations in the measurement to noise coming from the sensor or the electronics, not to temperature fluctuations. Consequently, the limits in the temperature measurements in the LTP can be established. 4.1 Test bed for the LTP requirements A specific on-ground test to confirm whether or not the theoretical noise levels of the TMS given in § 2.3 are accurate is absolutely necessary [84]. The sensors and the associated electronics must be able to make measurements capable of discerning temperature fluctuations at the level of 10−4K Hz−1/2—cf. § 1.5—, which implies a maximum equivalent temperature noise level in the sensor and electronics one order of magnitude lower, i.e., 10−5K Hz−1/2—see § 1.5 and § 2. In order to do a meaningful test a stable thermal environment must be guaranteed for the sensors not to confuse real temperature fluctuations with sensor and/or electronic noise. In other words, real temperature fluctuations in the sensors must be at least one order of magnitude smaller than the limit imposed in Eq. (2.1), or S1/2 test bed(ω)≤10−6K Hz−1/2,1 mHz ≤ω/2π≤30 mHz.(4.1) The required stability figure of Eq. (4.1) is not available in standard climatic chambers. The analysis and design of a special thermal insulator jig to guarantee the levels required by Eq. (4.1) have been necessary. The concept idea of the insulator is shown in Figure 4.1 [84, 97]: a block of aluminium inside a layer of polyurethane (or another insulating material) of suitable dimensions. The aluminium block ensures the temperature stability of the sensors attached to it, while the surrounding polyurethane shields them from external temperature fluctuations in the laboratory. The thermal insulator must be designed taking into account the necessity of placing and removing 80 4 Test bed design for the TMS validation different configurations to enhance the effect on one of the sensors and to mitigate it on another one were similar. Figure 4.6 shows the response of a thermistor when exciting the end of the cables. The cables used are described above. The increase of temperature in the thermistors is clear. The input signal was a triangular wave of 0.6 mHz and peak-to-peak amplitude 2 K. The increase of temperature in the thermistor was ≃20 µK which corresponds to a gain of ≃10−5. 5 10 15 20 25 30 −2 0 2 −1 1 temperature [K] time [ks] 5 10 15 20 25 30 −20 0 20 temperature [µK] Figure 4.6: Thermal excitation of the cables (blue trace) and the response of the thermistor placed inside the insulator (green trace). The input signal has a peak-to-peak amplitude of 2 K and frequency 0.6 mHz. The response is a similar waveform of amplitude peak-to- peak 20 µK. The attenuation is ∼10−5at 0.6 mHz. This test was done with cables of different radii, however, not significant differences were observed. In summary, the capability of screening the laboratory temperature fluctuations by the insulator considering non-idealities and heat leakage through the cables is ≃10−5at 1 mHz. Therefore, the insulator system is compliant with the requirements to perform meaningful tests —see Eq. (4.11). In order to fully overcome the potential effect of the cables a precaution was taken: the use of long and thin cables8wound a few turns around the aluminium block [40] —see Figure 4.7. 00000000 00000000 00000000 11111111 11111111 11111111 Al block cable NTCs Figure 4.7: Thermal trap concept. The ambient temperature fluctuations are attenuated in the aluminium block prior to reach the sensing head of the thermistor. 8Length of the cables was ∼2 m and the radius was of 0.1 mm instead of 0.255 mm. 4.2 Test bed for the LISA MBW 81 4.2 Test bed for the LISA MBW Investigations of the behaviour of the TMS in the LISA measurement bandwidth have been also done. The LISA frequency range goes down to 0.1 mHz, i.e., one order of magnitude lower than LTP. In principle, the electronic noise should remain flat due to the measurement method used —see § 2.3. However, unexpected noise might appear at these very low frequencies, specially, due to the sensor itself. Thermistors are of semiconductor nature —see § 2.2.2—, therefore susceptible to exhibit 1/fnoise. The test bed designed to assess whether or not the system is compliant with the requirements in the LTP band has been described in § 4.1. However, the test bed design in the submilli-Hertz region is considerably more complicated. In this section we describe the problems that appear when measuring at 0.1 mHz and the methods adopted to overcome them. We have to keep in mind that the main concerns of excess noise in the system come from the thermistor itself, thus the aim of the following investigations is the study of potential 1/fnoise in the thermistors. We remark that 10−5K Hz−1/2at 0.1 mHz is equivalent to temperature variations of ∼10−7K in time scales of hours9. First, we consider the possibility of using the same concept of test described in § 4.1, although re-scaling the size of the insulator. It was soon discovered that this option is not practical. At 0.1 mHz the ambient temperature fluctuations become much larger than those at 1 mHz and, in addition, the passive insulator attenuation appears to be poor10, unless a prohibitively large one is built. The nominal laboratory temperature fluctuations and those required inside the insulator are, at 0.1 mHz, S1/2 T, amb(0.1 mHz) ∼1 to 10 K Hz−1/2,(4.18a) S1/2 T, ins(0.1 mHz) ∼10−5K Hz−1/2.(4.18b) From Eqs. (4.18a) and (4.18b) the needed attenuation at 0.1 mHz is readily calculated, i.e., |Hins(0.1 mHz)| ≤ 10−5to 10−6.(4.19) In order to obtain such attenuation the needed passive insulator would consist in a aluminium sphere of 0.4 m diameter surrounded by a 2 m diameter polyurethane foam layer which appears inconvenient. However, the strongest reason to discard using a passive insulator is its time constant which is about 10 days. Therefore, temperature stabilisation down to 10−5K Hz−1/2at 0.1 mHz by using a mere passive insulator appears rather problematic. 4.2.1 Solution: differential measurements A straightforward solution to avoid the need of a giant insulator is the use of differential measurements instead of absolute measurements11. For differential measurements we have ST1(ω) = |Hins(ω)|2ST,amb(ω) + nT1(ω),(4.20a) ST2(ω) = |Hins(ω)|2ST,amb(ω) + nT2(ω) (4.20b) where nT1and nT2are the noise (in power spectral density) of each of the thermistors, plus the electronic noise. Thus, the differential measurement fluctuations are12 S∆T(ω) = nT1(ω) + nT2(ω).(4.21) 9Or voltage fluctuations in the Wheatstone bridge of ∼5·10−10 V in time scale of hours. 10The insulator designed for the LTP tests exhibits a gain of ∼10−3at 0.1 mHz —cf. Figure 4.2. 11“Differential measurements” stands for measuring the temperature difference between two thermistors very close to one another in a more or less thermally stable environment while “absolute measurements” stands for measuring the temperature of individual thermistors. 12Assuming the noise of the two thermistors is uncorrelated. 82 4 Test bed design for the TMS validation Ideally, when differential measurements are considered, only noise from the thermistors and the electronics is measured, the common ambient temperature fluctuations cancel out. Therefore, a less demanding passive insulator would be enough for our purposes. However, different non-idealities arise in the practical implementation. These are discussed in next section. 4.2.2 Test bed non-idealities Different non-idealities preclude a repetition of the test proposed in § 4.2.1 at 0.1 mHz. The main limitations of the test set up are:  power fluctuations in the thermistor (kV),  temperature coefficient of the electronics (αFEE),  cables connecting the thermistors to the electronics (kc),  intrinsic differences between thermistors (kNTC). All these effects couple into the measurement and disturb the 1/f thermistor noise investigations. The apportioning of these effects appears in the differential measurements as follows: S∆T(ω)≃nT1(ω) + nT2(ω) + k2 VSV,ref (ω) + α2 FEEST,FEE(ω) + k2 cST,amb(ω) + k2 NTCST,ins(ω) (4.22) where SV,ref ,ST,FEE,ST,amb and ST,ins stand for the voltage reference fluctuations of the Wheatstone bridge, the temperature fluctuations in the electronics, in the laboratory and inside the insulator, respectively. The main objective is to determine nT1+nT2, hence, all the other contributions must be minimised. More specifically, we assign ≃5µK Hz−1/2to each of the four disturbing terms13. In the following we detail these non-idealities and the solutions adopted to overcome them. Reference voltage fluctuations (kV)Fluctuations in the voltage supply of the thermistor cause temperature fluctuations in the sensors by the the self-heating effect —see § 2.3.2.3. When differential measurements are considered, the temperature fluctuations in the measurement are (assuming both sensors are exactly at the same temperature) S∆T(ω) =  2VNTC RNTC ∆θ 2 SVNTC (ω) (4.23) where ∆θis the difference between the thermal resistances of the two thermistors (the voltage feeding both thermistors is, in principle, correlated), VNTC is the voltage applied to the thermistors, SVNTC are the voltage fluctuations and RNTC is the resistance of the thermistor (the same for both thermistors since they are at the same temperature). The term within brackets is kVin Eq. (4.22). Using Eq. (4.23), an upper limit for the voltage fluctuations, SVNTC , can be calculated. The numbers involved are VNTC ≃0.3 V, RNTC ≃10 kΩ, θ≃50 K W−1(we use θinstead of ∆θto be in a worstcase, i.e., assuming the voltage fluctuations in both thermistors are not correlated) —see appendix § B— and S1/2 ∆T≤5·10−6K Hz−1/2. The required stability for the reference voltage stability, S1/2 VNTC , results in 1.5 mV Hz−1/2for f≥0.1 mHz. The measured stability at 0.1 mHz is ∼0.1 mV Hz−1/2. With these numbers the contribution of this effect at 0.1 mHz is ≃0.3µK Hz−1/2, thus negligible. 13This results in a total noise of p4(5 ·10−6)2= 10−5K Hz−1/2. 4.2 Test bed for the LISA MBW 83 Temperature coefficient of the measurement system (kFEE)The electronics temperature coefficient causes read out errors —see § 2.3.4: S∆T(ω) = α2 FEEST,FEE(ω) (4.24) where ST,FEE are the temperature fluctuations of the electronics, αFEE is the temperature coefficient of the system, and S∆Tare the fluctuations in the differential temperature measurement due to this effect. The theoretical estimation of αFEE is given in § 2.3.4. However, in order to determine it more precisely a simple test has been done. The test consisted in connecting a high-stability resistor14, instead of a thermistor to the measurement system. The high-stability resistor acts as a thermistor at a constant temperature of 25  . The FEE was thermally excited by using a heater and the temperature of the electronics and the equivalent temperature of a 10 kΩ resistor were measured. Using these two measurements the value of the electronics temperature coefficient is readily estimated. A scheme of the test set up and the obtained measurements are shown in Figure 4.8. 000 000 000 111 111 111 FEE T FEE insulator voltage heater RHS −0.2 −0.1 0 0.1 0.2 TFEE [K] 10 20 30 40 50 60 70 −2 0 2 ∆RHS [µK] time [ks] Figure 4.8: Electronics temperature coefficient estimation. The resistance of a highstability resistor is measured while applying a thermal excitation to the electronics. The change in the high-stability resistor is attributed to the TC of the electronics. The panel in the right shows the temperature of the electronics when being thermally excited and the measurement of the resistance (low-pass filtered to reject the high frequency noise of the measurement). From the measurements shown in Figure 4.8 (right) the estimation of the parameter αFEE is bαFEE = 13.5µK K−1(4.25) which is in good agreement with the theoretical prediction —cf. § 2.3.4 (in the very worst-case scenario at 25  it is 40 µK K−1). This value limits the maximum temperature fluctuations of the electronics. The budget for this effect is ≃5µK Hz−1/2, thus, the fluctuations in the electronic must be .0.35 K Hz−1/2at 0.1 mHz. The fluctuations in the electronics are approximately the same as those in the ambient, which at 0.1 mHz are ∼1 K Hz−1/2—see Eq. (4.18a). In order to overcome this problem, the temperature of the electronics was controlled by means of a feedback control (a proportional control) that maintained the temperature stable by controlling a heater attached to the electronics. Cables connecting the thermistors to the electronics (kc)Cables are a fair path for the ambient temperature fluctuations to show up in the thermistors’ readings. In the case of the 1410 kΩ Vishay resistor with of 0.6·10−6K−1placed inside the insulator 84 4 Test bed design for the TMS validation differential measurements the effect of the cables is very attenuated because the ambient temperature fluctuations are exactly the same for both. The only fluctuations that can appear are due to asymmetries in the cables (for instance, if they are of different length) and in the thermistors themselves. In the case of absolute temperature measurements the effect of the cables does not affect the performance of the insulator provided long cables and a good thermal contact is ensured. In the case of the differential measurements, this effect is even milder. A test to assess whether or not the differential measurement rejects the common-mode temperature fluctuations was performed. A thermal excitation (triangular wave of peak-to-peak amplitude 0.2 K) was applied to the cables of both sensors while measuring the differential temperature. The results are shown in Figure 4.9 where no signal correlated with the excitation is observed. The thermal excitation main frequency component was at 2 mHz. In the test in the LISA MBW we are interested at 0.1 mHz, however, the response at 2 mHz and 0.1 mHz should be similar since the transfer function describing the heat leakage through the cables does not differ too much between 1 mHz and 0.1 mHz —see Figure 4.5. Thus, the effect of the cables should not pose a problem for the test. 10−4 10−3 10−2 10−1 100 10−5 10−4 10−3 10−2 10−1 100 101 frequency [Hz] ST 1/2 (ω) [K Hz −1/2 ] cables excitation differential meas. Figure 4.9: Thermal excitation at the cables of both thermistors during differential measurements. The solid trace shows the excitation at the cable’s end, and the dashed line is the differential measurement which is free of the effect of the heat leakage. Intrinsic differences between thermistors (kNTC)The inherent mismatch between thermistors, causes an error in the differential temperature due to differences in the response of each thermistor. The apportioning of this effect is: S∆T(ω) = |H1(ω)−H2(ω)|2ST,ins(ω) (4.26) where here H1−H2is15 H1(ω)−H2(ω) = q1(ω) sinh aq1(ω)−q2(ω) sinh aq2(ω)(4.27) with athe radius of the thermistor head and qi=ρici κi iω (4.28) 15Assuming the head of the thermistor is spherical [27]. 4.2 Test bed for the LISA MBW 85 where ρi,ciand κiare the density, the specific heat and the conductivity of the thermistors, respectively. Equation (4.27) is difficult to evaluate due to not well-known characteristics of the thermistors. However, it can be seen that Eq. (4.27) is actually a high-pass filter, i.e. H1(ω)−H2(ω)≃kNTCiω. (4.29) The experiments to estimate kNTC consisted of exciting thermally the aluminium block and measuring the absolute and the differential temperature. These measurements allow us to find the relationship between the absolute temperature in the centre of the aluminium block and the differential measurement between two thermistors very close to one another16, also in the centre of the block. The thermal excitation consisted in dissipating a constant power in a heater placed in one of the faces of the block. The evolution of the absolute temperature is shown in Figure 4.10 (top left panel), and the differential measurement is shown in the top right panel. The estimation of kNTC is done by calculating the time derivative of the absolute temperature measurement and then fitting it to the measured differential data. The estimated parameter is bkNTC ≃6 s. Figure 4.10 (bottom) shows the measured differential temperature (solid trace) and the fit with kNTC = 6 s. The agreement is good, thus confirming the behaviour predicted by Eq. (4.29). 5 10 15 25.8 26 26.2 Tins [oC] 5 10 15 −2 −1 0 1 2x 10−4 ∆T [ oC] 8 10 12 14 16 −2 −1 0 1 2x 10−4 ∆T [ C] experimental fit time [ks] time [ks] 2 4 6 o time [ks] Figure 4.10: Top left: absolute temperature measurement in the aluminium block. Top right: differential temperature measurement between two thermistors very close one to each other in the aluminium block. Bottom: differential temperature measurement (solid trace) and that obtained by the time derivative of the absolute temperature and adjusting the parameter to kNTC=6 s (dashed trace). This effect implies that the temperature fluctuations of the aluminium block at 0.1 mHz must be less than 5 ·10−3K Hz−1/2not to perturb the test. This figure is used to calculate the attenuation required for the the thermal insulator, i.e., |Hins(ω)|=S1/2 T,Al(ω) S1/2 T,amb(ω)0.1 mHz ∼10−4(4.30) The insulator constructed for the LTP test exhibits a gain of 10−3at 0.1 mHz —see Figure 4.2. Instead of designing another insulator with such attenuation, a combination of the built insulator in conjunction with an active temperature control in the aluminium block was implemented. 16For this reason, the temperature difference between the two sensors is interpreted as a mismatch between them. 86 4 Test bed design for the TMS validation Finally, another source of error is the difference between the temperature characteristic of the thermistors, β. The tolerance specified by the manufacturer is ±0.3% [16]. This represents a coefficient of ∼10−3K K−1in the differential measurements. If a temperature stability of 5 · 10−3K Hz−1/2is required in the aluminium block, the contribution of this effect is negligible. 4.2.3 Temperature control design The implementation of an active system to control the temperature [48, 60, 65, 78, 144, 42, 40] of the aluminium block inside the insulator appears as the only feasible option to deal with the investigation of the 1/fnoise in the thermistors in the submilli-Hertz frequency band. In this section we describe the design and different aspects of the implementation. An active temperature control based on a feedback-feedforward (FB-FF) scheme17 [104, 12] —see Figure 4.11— has been used to keep the temperature fluctuations of the insulator less than ∼5·10−3K Hz−1/2at 0.1 mHz —see § 4.2.2. The active controller attenuates the fluctuations in the submilli-Hertz region (and at lower frequencies) while the passive insulator screens out the fluctuations in the milli-Hertz region (and at higher frequencies) —see Figure 4.16. Our main concern is the rejection of ambient temperature fluctuations in the aluminium block. The effect of the disturbance (the ambient temperature) on the system, in principle, can be highly reduced by using the feedforward compensation shown in Figure 4.11, which can be implemented provided the disturbance signal can be measured and good knowledge of the transfer functions involved in the system are available. The former condition is possible, we only have to place a thermistor outside the insulator; the latter condition is also met with reasonable accuracy. The control system works as follows: a reference temperature for the aluminium block, Tref (a few degrees over the ambient temperature), is set; the control tries to maintain this temperature by dissipating power through a heater attached to the aluminium block. The required power is calculated by a computer using the data coming from the ambient temperature (feedforward) and from the aluminium block temperature (feedback). The needed power is converted to units of Volt and sent to a programmable power supply connected to a heater placed onto one of the aluminium block faces. The block diagram of the control system is given in Figure 4.11. Gc(s) Gp(s) GF(s) Hins. (s) ref. T Tins Tamb. sensor noise feedfoward digital filter insulator heater actuator noise control (P) Figure 4.11: Feedback-feedforward temperature control system block diagram. The closed-loop response of the FB-FF system is (in the s-domain, we omit the sdependence 17With only a proportional feedback control an attenuation of four orders of magnitude is difficult to attain. The needed gain of the controller, Gc, has to be large and the system is of very high order. This combination results in an under-damped system that spoils the purpose of the test. 4.2 Test bed for the LISA MBW 87 argument in the rest of the section for notation simplicity) e Tins =GcHh 1 + GcHhe Tref +Hins −GcHhGFF 1 + GcHhe Tamb +Hh 1 + GcHh ˜nact +GcHh 1 + GcHh ˜nsens (4.31) where:  Hhis the heater transfer function, i.e., the transfer function that relates the power dissipated in the heater to the increase of temperature in the aluminium block. The units are K W−1. The estimation of this transfer function is presented in § 4.2.3.1,  Hins is the passive insulator transfer function. This transfer function is accurately known for f≤0.1 mHz —see § 4.1 and Figure 4.2,  Gcis the controller transfer function, in our case a mere constant, Kp[W K−1],  GFF is the feedforward filter. The main purpose of this filter is to fully reject ambient temperature fluctuations from the aluminium block. By inspection of Eq. (4.31) we notice that the term multiplying e Tamb can be nulled if the feedforward filter is GFF =1 Gc Hins Hh (4.32) where all the transfer functions in the right hand side of the equation are known to a certain accuracy. The filter design and implementation is described in § 4.2.3.1,  e Tref ,e Tins and e Tamb are the set point temperature for the aluminium block, the measured temperature of the aluminium block and the laboratory temperature, respectively,  ˜nact and ˜nsens are the noise introduced by the programmable power supply and the noise of the absolute temperature sensor. They are negligible for the measurement of interest (the differential measurement). 4.2.3.1 Feedforward filter design In this section the different steps needed to design and implement the feedforward filter are described. The filter is defined in Eq. (4.32) where it is clear that the transfer functions of the passive insulator, Hins, and of the heater, Hh, are needed. The former has been described in § 4.1.1 and § 4.1.2. In the following the heater transfer function is estimated and then the feedforward filter design is described. Heater transfer function (Hh)This transfer functions relates the power dissipated in the heater to the increase of temperature in the aluminium block. In order to obtain an analytical transfer function we consider the problem as shown in Figure 4.12: all the faces of the aluminium block are considered adiabatic except the one with the heater attached to it. The insulating material is modelled by a lumped thermal resistance, θ, and a lumped capacitance, C. Therefore, the power dissipated in the heater heats up the aluminium block, but, also the insulating material. Our interest is in the temperature increase at x=`/2 (where the sensors are placed) due to the power dissipated in the heater, P. The transfer function of the system is —see appendix F—, e H(x, s) = e T(x, s) e P(s)=θcosh q(`−x) Aκθq sinh q` + cosh q`(q2KθC + 1) (4.33) where, as usual, K=κ/(ρc), q2=s/K and θand Care the lumped thermal parameters of the insulating layer —see Table 4.2. Ais the contact area of the heater with the aluminium block. In 88 4 Test bed design for the TMS validation insulating wool Al block NTC heater c, κ, ρ Cθ Al block x=lx=0 heater P insulator Figure 4.12: System to heat up the temperature of the aluminium block. We are interested in the transfer function between the electrical power dissipated in the heater and the increase of temperature at x=`/2 (at the centre of the block). The figure in the left is the actual system whereas the figure in the right is the model used for the analysis. order to assess whether the model is accurate or not we have to compare it to experimental data. A simple way to obtain the response of the system is to apply a power step to the system and observe the temperature evolution at x=`/2. However, the expression given in Eq. (4.33) is in the frequency domain, therefore it must be transformed to the time domain to compare it with the experimental data. The input signal, P, is the Heaviside function. The response of the system in the time domain to the Heaviside function can be obtained by using Eq. (4.33) and the Inversion theorem. The resultant expression is [27] —see appendix § F for details—, T(t) = Pθ "1−2∞ X n=1 e−Kα2 nt αn cos αn(`−x) θαn(Aκ` + 2KC) cos αn`+ (`+Aκθ −`θKα2 nC) sin αn`#(4.34) where αnare the solutions of αntan αn`=1−α2 nθKC Aκθ .(4.35) The experimental temperature and analytical responses for P= 0.095 W are shown in Figure 4.13 (left panel). The expression given in Eq. (4.34) has been fitted to the experimental response with θand Cas free parameters. The values found are given in Table 4.3. Figure 4.13 (right panel) shows the transfer function in the frequency domain which is the one needed in the design of the feedforward filter. parameter value K ρc/κ ≃10−4s m−2 A 0.005 m2 θ7.5 K W−1 C10 kJ K−1 `0.15 m Table 4.3: Parameters used for the evaluation of Eqs. (4.33) and (4.34) —see Figure 4.13. θand Crepresent the properties of the insulating material surrounding the aluminium block. Their values have been found by fitting Eq. (4.34) to the experimental temperature response. However, a rough estimation of them leads to similar values: for the case of the thermal resistance, if we calculate θconsidering a hollow sphere, the obtained value is ∼9 K W−1and the heat capacity is ∼12 kJ K−1. 4.2 Test bed for the LISA MBW 89 0 50 100 150 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time [ks] ∆T [K] measured theoretical 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 −12 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 10 2 frequency [Hz] |H(ω)| [K W−1 ] Figure 4.13: Left: temperature response to a power step of 0.095 W. Solid trace represents the experimentally measured response, while the dashed trace is the one estimated by means of Eq. (4.34). Right: The transfer function in the frequency domain (at x=`/2). The parameters used for both plots are those shown in Table 4.3. Feedforward filter (GFF)The feedforward filter is —see Eq. (4.32), GFF =1 Kp Hins Hh (4.36) where Hhand Hins are the heater transfer function and the insulator transfer function, respectively, which have been characterised before. The term Kpis the gain of the proportional control, and is a mere constant to be set later on—see § 4.2.3.2. The feedforward filter is implemented digitally in a computer. To do so, we first approximate it to a pole-zero filter (still in the s-domain) to then convert it into a discrete filter. The term Hins/Hhis approximated by the following pole-zero expression18: Hins(s) Hh(s)=1 θ low pass filter z }| { 1 (as + 1)7(bs + 1)3 all pass filter z }| { (−cs + 1)2 (cs + 1)2 (−ds + 1) (ds + 1) (−es + 1) (es + 1) (4.37) with a=320, b=3800, c=185, d=100 and e=50. Equation (4.37) is formed by two parts, the first is a low-pass filter of 10-th order and is used to adjust the gain, and the second is an all-pass filter which adjusts the phase —see Figure (4.14). The approximation is good but they do not quite match, although the errors at 0.1 mHz are small. At higher frequencies the needed accuracy of the feedforward filter, GFF, is relaxed since the passive insulator is the one which screens the temperature fluctuations. 18The pole-zero model chosen was obtained by minimising the error of the gain and the phase at the region of 0.1 mHz with a filter of reasonable order. 96 5 TMS performance tests Figure 5.2: Anechoic chamber at UPC premises. This chamber was used due to its good temperature stability and to avoid disturbances during the test (such as people passing by, interferences, etc.). analog RS232 PIC ambient sensor anechoic chamber NTCs under test LabView ADC prototype measurement system Figure 5.3: Set up for the TMS prototype test. Three thermistors are attached to the aluminium block which, in turn, is surrounded by the insulating material. The cables connecting the thermistors to the electronics are twisted and shielded to avoid interferences. The aluminium block, the cable shields and the electronic boxes are all connected to the same ground also to avoid interferences. The digital signal processing is done by a PIC that sends out data via serial interface to a computer running a LabView application. The ground of the serial interface and the ground of the electronics are kept separated to avoid interferences related with the ground of the computer. Opto-isolators are used for this purpose. This set up is placed into an anechoic chamber at Universitat Polit`ecnica de Catalunya (UPC) premises to work in a quiet environment. The results in Figure 5.4 show that the system exhibited the same noise levels when using thermistors as when measuring HS resistors, for frequencies higher than 1 mHz. Therefore, excess noise is not observed in the thermistors in the LTP MBW [119]. These results show that the temperature measurement noise levels required for the LTP mission are met. However, at frequencies below 5.1 Tests in the LTP measurement bandwidth 97 26.366 26.368 26.37 26.372 Tabs [µK] 012345678 −5 0 5 time [hour] Tabs [µK] 10−5 10−4 10−3 10−2 10−1 100 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 frequency [Hz] ST 1/2 (ω) [K Hz −1/2 ] NTC NTC HS 10 k Ω NTC HinsS1/2 Tamb S1/2 Tamb αfeeS1/2 Tamb Figure 5.4: Left panel: the plot in the top shows the temperature evolution of one of the measurements. The bottom plot shows the same data band-pass filtered (between 0.5 mHz and 10 mHz) to remove the trend. The rms noise is ∼10−6K in the bandwidith between 0.5 mHz and 10 mHz. Right panel: power spectral density for different measurements are shown. The upper dashed trace stands for the temperature fluctuations in the anechoic chamber. The three superimposed black traces are the measurements with the thermistors inside the insulator and the lowest dash-dotted trace is the measurement with a 10 kΩ high-stability resistor. The red dashed trace stands for the ambient temperature fluctuations projection in the aluminium block. The magenta dashed trace is the noise projection of the temperature fluctuations caused by the TC of the electronics together with ambient temperature fluctuations. The power spectral density in the MBW is within the requirement, i.e., 10−5K Hz−1/2. 1 mHz the fluctuations in the thermistor are higher than the expected ones. The excess noise for f < 1 mHz could be, in principle, ascribed to different sources. One of them is the temperature coefficient of the electronics together with ambient temperature fluctuations. This is immediately discarded since the ambient temperature was stable enough not to degrade the measurement —see dashed magenta trace, or αFEES1/2 T,amb(ω)<10−5K Hz−1/2, ω/2π > 0.1 mHz.(5.3) Another possible source of noise is real temperature fluctuations of the aluminium block, however, this possibility can be also discarded: the projection of the laboratory temperature fluctuations in the aluminium block considering the insulator described in § 4.1 does not agree with the experimental results —see the dashed red trace in Figure 5.4. They only match at very low frequency (<0.1 mHz). Moreover the excess noise in the thermistors presents peaks (in the PSD) which are not observed in the ambient temperature fluctuations. After having discarded the previous possible explanations, there is still one possibility: the excess noise at frequencies below 1 mHz could have been caused by some unforeseen noise coming from the electronics. This hypothesis is, in principle, difficult to maintain since when measuring with high-stability resistors the performance of the system is the expected one. Nevertheless, there is an important difference between both measurements: the value of the high-stability resistor remains constant, i.e., the system is always measuring 10 kΩ (or 25  in equivalent temperature); on the contrary, when measuring temperature with the thermistor its resistance is drifting with time. In other words, the slope of the input signal when measuring the high-stability resistor is zero, while when measuring with thermistors it can be of micro-Kelvin per second. Drifting signals at the input of the ADC add noise to the signal due to non-idealities of the ADC. Different tests have confirmed this hypothesis. An example of this is shown in Figure 5.5 where it is clear that the power spectral density increases with the slope of the input signal and 98 5 TMS performance tests that this effect spreads over the MBW. The PSD of the solid trace in Figure 5.5 (right) has been estimated using the data shown in the left panel of Figure 5.5. The slope (in absolute value) of such data varies from 15 to 8 µK s−1. The PSD of the dashed trace has been estimated using data where the temperature slope is lower than 0.5 µK s−1. It is clear that, under the latter conditions, most of the excess noise in the LTP MBW vanishes. For this reason the maximum slope permitted in the input signal not to degrade the system performance in the frequency range of the LTP is ≃0.5 µK s−1(this value depends on each specific ADC, however, it should not differ significantly when using another ADC of the same kind). Higher slopes result in an increase of the noise at frequencies around the milli-Hertz6. Temperature slopes of ≃0.5 µK s−1are not foreseen in nominal conditions7in the locations where the sensors are attached to. Moreover, it is also important to note that differential measurements are, in principle, free of this problem since these measurements should exhibit very low slopes. Consequently, the problem of the non-ideal quantisation of the temperature signal must not pose a problem in the LTP MBW. Although in views of LISA this effect must be minimised since the measurement bandwidth goes down to 0.1 mHz and any small slope in the signal can degrade the measurement at that frequency. Further results on this issue are shown in § 5.1.2 and a dedicated chapter describing the identification and minimisation of this problem is presented in § 6.1. 25 25.2 25.4 25.6 T [oC] 0 5 10 15 20 8 10 12 14 time [hour] |dT/dt| [µK s−1] 10−5 10−4 10−3 10−2 10−1 100 10−6 10−5 10−4 10−3 10−2 10−1 100 frequency [Hz] S1/2 T(ω) [K Hz−1/2] Figure 5.5: Left panel: absolute temperature measurement and its temperature slope (in absolute value). Right panel: Power spectral density of two measurements. The solid trace corresponds to the data shown in the left panel where the slope varied from 14 µK s−1to 8µK s−1. The dashed trace is the noise curve of an absolute measurement where the slope did not exceed 0.5 µK s−1. The former exhibits noise levels one order of magnitude higher than those obtained with the measurement with lower slopes. It is thus clear that in order to stick to the TMS requirements of the LTP, the slopes of the measured temperature must be less than ≃0.5 µK s−1. 5.1.2 Engineering model test The TMS in the engineering model (EM) of the data management unit (DMU) was also tested [115] —see Figure 5.7. The system design is exactly the same of the prototype with minor changes in some parts due to space qualified component constraints. The test was performed with half DMU, i.e., 6Roughly speaking the frequency at which this effect appears is proportional to the slope of the input signal. 7The worst-case rate change of temperature at the LCA strut interface is 0.4 µK s−1. Since the thermal noise requirements within the LCA are more stringent than those at the interface, it is likely that they will be better within the LCA [154]. High slopes are, however, foreseen during the thermal experiments to be performed on-board the LTP —see chapter § 7. However, in those experiments the temperature signal is also very strong and this effect becomes negligible. 5.1 Tests in the LTP measurement bandwidth 99 only one of the data acquisition units (DAU) was tested —see Figure 1.12. Twelve EM thermistors were tested since each DAU permits connections for 12 sensors. The test set up was very similar to that used for the prototype qualification campaign —see Figure 5.3—, however, this time the anechoic chamber used was the one at the Centre Tecnol`ogic de Telecomunicacions de Catalunya (CTTC) —see Figure 5.7. A significant difference from the prototype tests was that during this test the 12 thermistors were attached to the aluminium block by means of the glue (CV-2496 Nusil Tech.) intended to be used for the final attachment of the sensors at the specific locations of the LTP. The attachment process consisted in gluing each thermistor onto the aluminium block surface. The glue needs a cure time of about one week at ambient temperature. In order to keep the sensor securely attached during the curing time, a lid of methacrylate was used to exert pressure over them —see Figure 5.6— and, thus, ensure a good thermal contact between the sensors and the aluminium block. Figure 5.6: 12 thermistors attached onto the aluminium block surface. A small lid of methacrylate is used to exert pressure in the thermistors to keep them tightly fixed during the curing time of the glue. The measurement flow chart is the one shown in Table 2.9: 12 absolute temperature measurements (four in each of the three boards in the DAU), 3 differential measurements (one in each of the three boards in the DAU) and 3 reference measurements (one in each of the three boards in the DAU). As stated in § 5.1.1 the input temperature slope must be below a certain limit to ensure the performance of the measurement due to problems related with the ADC. The limit of such slope was set, initially, to |dT/dt|.0.5µK s−1. However, the ADC of the EM (ADS7809 of Texas Instruments) TMS performed slightly worse than the model used in the prototype design (AD977 of Analog Devices). The former was more sensitive to drifting input temperatures than the latter. For this reason, the maximum temperature slope of the input signal not to degrade the EM (and FM) TMS performance in the MBW is 0.4 µK s−1instead of 0.5 µK s−1. 100 5 TMS performance tests Figure 5.7: Left: engineering model of the DMU. Right: CTTC anechoic chamber. Figures from 5.8 to 5.10 show the results for the 12 absolute temperature measurements. The results for the three independent electronic boards in the DAU are given separately. The plots in the left display the temperature and the temperature slope evolution with time. The plots in the right are the power spectral density estimated from the data where the temperature slope was lower than 0.4 µK s−1. The noise level of the sensors attached to the first independent board (G11 in Table 2.9) of the DAU are within the expectations. The floor noise is ≃7.5 µK Hz−1/2, and stays flat for f≥1 mHz. The floor noise of the EM system is slightly lower than the prototype one. The reason is the higher sampling frequency (in the ADC) of the EM system with respect to the prototype. The EM sampling frequency is 50 kHz while the one of the prototype is 38.4 kHz: the higher the sampling frequency the higher the immunity to aliasing. The noise levels of the four sensors are essentially the same, except for one which presents a tiny increase in the milli-Hertz region. µ 10−4 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 frequency [Hz] ST 1/2(ω) [K Hz−1/2] Figure 5.8: Measurements of the first group of the EM DAU, G11. The left panel shows one absolute temperature measurement (top) and its slope (bottom). The data segment used for the PSD estimation shown in the right panel (for all four sensors) is that where the slope is less than 0.4 µK s−1(dashed horizontal line in the plot on the left). 5.1 Tests in the LTP measurement bandwidth 101 Figure 5.9 shows the results for the absolute temperature measurements of the second board of the DAU, G12, where an unexpected behaviour in one of the sensors is observed. µ 10−4 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 frequency [Hz] ST 1/2(ω) [K Hz−1/2] Figure 5.9: Idem as Figure 5.8 for the four sensors connected to the second board of the DAU, G21. The results for the measurements of the third independent board in the DAU, G31, are given in Figure 5.10. Again, one of the sensors exhibits an anomalous behaviour which translates into unacceptable noise levels. The analysis of such behaviour and the possible causes of it are detailed in § 5.1.2.2. Like in the previous measurements, the noise performance of the system is obtained using data with temperature slope lower than 0.4 µK s−1. The rest of the sensors performed properly within the MBW. µ 10−4 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 ST 1/2(ω) [K Hz−1/2] frequency [Hz] Figure 5.10: Idem as Figure 5.8 for the four sensors connected to the third board of the DAU, G31. Figures 5.11 and 5.12 show the differential and the reference measurements of the three boards, respectively. The former corresponds to the measurement between two thermistors in the aluminium block. Two of the measurements (those in G21 and G31) are very noisy because one of the thermistors involved in the differential measurements was not working properly, as it can be seen in Figures 5.9 and 5.10. The differential measurement of G11 performs as expected8. This 8The differential measurements are intended to be done between locations with similar temperatures, therefore, the differential measurement should exhibit small drifts with time since the common mode temperature is rejected. During the on-ground tests the differential measurements were performed between two sensors attached to the 102 5 TMS performance tests measurement presents noise levels of ≃7µK Hz−1/2at frequencies close to 0.1 mHz. The reference measurement consists in measuring two high-stability resistors forming the Wheatstone bridge. This measurement permits to observe any anomaly in the performance of the electronics independently of the sensors. This measurement will be also done in flight in order to identify any malfunctioning of the electronics. The results of the reference measurements during the EM test showed that the electronics were working properly. The noise is flat at frequencies close to 0.1 mHz with an amplitude of ≃7µK Hz−1/2. The anechoic chamber temperature fluctuations did not perturb the measurement since the temperature was very stable during the tests9 (for this reason the reference and differential measurements are almost flat at frequencies close to 0.1 mHz). In addition, the DMU metallic box also acts as a thermal insulator and screens out ambient temperature fluctuations from the electronics. 0.0285 0.0286 0.0286 0.0286 Tdiff [K] 2 4 6 8 10 12 14 0.01 0.02 0.03 20 time [hour] |dT/dt| [µK s−1] 20 µK µΚ 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 frequency [Hz] ST 1/2 (ω) [K Hz −1/2 ] G11 G21 G31 Figure 5.11: Idem as Figure 5.8 for the three differential measurements (one per group in the DAU). Two measurements are much noisier than the nominal floor noise. This is so because the sensors in the differential measurements were the malfunctioning sensors detected in the absolute measurements. The differential measurement of the first group, G11, is performing correctly and it is free of the non-ideal quantisation excess noise since the two sensors are measuring similar temperatures, i.e., the slope of the differential signal is small, e.g., 0.02 µK s−1. The time domain data shown here is not filtered, thus the band goes from 0.1 mHz to 0.5 Hz. The rms noise in such band is ≃3µK. aluminium block, hence, at the same temperature. 9At 1 mHz fluctuations were ≃0.05 K Hz−1/2. 5.1 Tests in the LTP measurement bandwidth 103 22.772 22.772 22.772 Tref [K] 2 4 6 8 10 12 14 0.01 0.02 0.03 20 time [hour] |dT/dt| [ µK s−1] µΚ 20 µΚ 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 frequency [Hz] ST 1/2(ω) [K Hz−1/2] Figure 5.12: Idem of Figure 5.8 for the three reference measurements (one per group in the DAU). The three boards worked perfectly. The electronic noise is equivalent to a rms noise of 20 µK in a band of 0.1 mHz to 0.5 Hz. Two important conclusions emerge from the EM test. On the one hand, the effect of the analog- to-digital conversion when measuring signals drifting with time was confirmed, although it should not be a problem in the LTP measurement bandwidth, as previously stated. On the other hand, and more worrying, is the fact that two sensors out of 12 were not performing properly for unknown reasons. The following sections deal with these two issues. 5.1.2.1 Excess noise due to non-ideal analog-to-digital conversion As discussed in § 5.1.1 the noise in the absolute temperature measurements is affected by the slope of the measured temperature. In this section we present more examples confirming this effect in the EM. These examples are intended to help understand the nature of this effect. However, the problem is described and investigated at length in § 6.1. Figure 5.13 (top left) shows the temperature slope for the data used in the PSD estimation of the EM test, i.e., data with a slope lower than 0.4 µK s−1—see Figures 5.8, 5.9 and 5.10. The bottom left plot shows the temperature band-pass filtered10. The correlation between both plots is clear: when the slope is close to zero (from t=1.5 h to t=3 h) random noise with a peak-to-peak amplitude of ≃2µK is present. On the contrary, as the temperature slope grows (from t=0 h to t=1.5 h and from t=3 h to the end), the noise in the measurement becomes periodic with a peak- to-peak amplitude of ≃8µK. This effect is thus responsible for the increasing noise at frequencies around the milli-Hertz (and lower) observed in Figures 5.8, 5.9 and 5.10. For comparison, the same plots for a differential measurement are given in Figure 5.13 (right): the slope of the signal is almost zero during the whole measurement, and the extra noise does not show up but it is rather kept with a peak-to-peak noise amplitude of ≃2µK, which further confirms the correlation between temperature slope and noise. 10The filter used is a Butterworth of second order with cut-off frequencies at 0.5 mHz and 10 mHz. The filter is applied forward and backward in order obtain a zero-phase filter. 104 5 TMS performance tests 0.1 0.2 0.3 0.4 0.5 |dT/dt| [µK s−1] 0.5 1 1.5 2 2.5 3 3.5 4 −10 −5 0 5 10 Tabs [µK] time [hour] 0 0.2 0.4 |dT/dt| [µK s−1] 0 1 2 3 4 5 6 7 −10 −5 0 5 10 time [hour] Tdiff [µK] Figure 5.13: The plots in the left show the slope of an absolute temperature measurement (top) and the band-pass filtered absolute temperature measurement. The plots on the right are the same, but for a differential measurement. Higher noise is observed in the absolute measurement due to the increasing slopes at the beginning and at the end of the data segment. Figure 5.14 shows another example of the quantisation problem when sampling signals of high slope. In this case the temperature slope varies from ≃2µK s−1to ≃1µK s−1. The performance of the measurement is drastically reduced, i.e., the noise is about one order of magnitude higher than that under nominal conditions for f < 5 mHz. The time evolution of the measurement when different slopes are present is shown in Figure 5.14 (left panel). The noise is somehow modulated by the slope of the measured temperature, i.e., the frequency of the periodic noise signal decreases as the temperature slope decreases, and vice versa. Consequently, if the temperature slope is changing with time the noise due to this effect is spread over the measurement bandwidth. For this reason, the power spectral density of this measurement presents a plateau at frequencies below 5 mHz. If for instance, the temperature slope is constant, the noise in the spectrum appears as peaks and is not spread over the band. The theoretical analysis, more experimental evidences and methods to overcome this problem are given in § 6.1. 1 1.5 2 |dT/dt| [µK s−1] 2 4 6 8 10 12 14 −20 −10 0 10 20 Tabs [µK] time [hour] 10−3 10−2 10−1 10−6 10−5 10−4 10−3 10−2 10−1 frequency [Hz] ST 1/2(ω) [K Hz−1/2] Figure 5.14: Left panel: the plot in the top shows the temperature slope. The plot in the bottom shows the temperature band-pass filtered. The relationship between the slope and the noise is clear. The higher the slope the higher the frequency of the periodic components of the noise. Right: power spectral density. The noise is clearly higher than usual in the MBW. 5.1 Tests in the LTP measurement bandwidth 105 5.1.2.2 Excess noise in the thermistors During the engineering model test an unexpected behaviour in 2 of the 12 sensors was detected. The noise in these two measurements was about one order of magnitude higher than the expected one in the measurement bandwidth. The measured data in the time domain are shown in Figure 5.15. The plot in the top corresponds to a correct thermistor. The plots in the middle and in the bottom correspond to the measurements of the two malfunctioning sensors connected to the boards G21 and G31. Spurious steps appear in the measurement which translate into high noise levels when estimating the PSD —see Figures 5.9 and 5.10. −100 0 100 T [µK] −100 0 100 T [µK] 1000 2000 3000 4000 5000 6000 −100 0 100 time [s] T [µK] −100 −50 T [µK] −100 −50 0 T [µK] 3000 3200 3400 3600 3800 4000 −150 −100 −50 0 time [s] T [µK] Kµ 20 Kµ 20 Kµ 20 Kµ 20 Kµ 45 Figure 5.15: Top: absolute temperature measurement of a thermistor working properly. Middle: idem for the malfunctioning thermistor of the second board of the DAU, G21. Bottom: idem for the malfunctioning thermistor of the third board of the DAU, G31. Data is detrended for clearer visualisation. The plot in the right is a zoom. Different potential sources were investigated in order to discern the actual reason of such behaviour. The causes considered were: (i) electronics of the DAU, (ii) bad soldering, (iii) poor thermal contact and, (iv) mechanical stress. Electronics of the DAU This cause was very unlikely since all the other measurements were working properly, and the reference measurements were also correct. However, in order to fully discard this possibility, the two sensors exhibiting extra noise were connected to other channels of the DAU which were working OK with other thermistors, and vice versa. The results obtained were exactly the same, i.e., the sensors working correctly continued to work properly and the malfunctioning ones failed in the same manner as when they were connected to their original channels. Therefore, the excess noise was definitely ascribed to the thermistors themselves. Bad soldering The cables of the EM sensors were not long enough to connect them directly to the DAU connections. For this reason, they had to be soldered to other longer cables. A possible bad soldering could have caused the behaviour observed in the two sensors. In order to discard this possibility, a visual inspection of the soldering was first done; no sign of bad soldering was detected. Another test consisted in measuring two high-stability resistors using the cables and solders of the two two malfunctioning sensors. The obtained measurements did not exhibit anomalies, either. Consequently, the soldering was discarded as the source of the problems in the measurement. Poor thermal contact The third possible cause considered was a possible poor thermal contact between the sensors and the aluminium block, this also was very unlikely since the thermal resistance should be about one order of magnitude higher than the usual ones to provoke such noise. The thermal resistances for the two malfunctioning sensors were ≃60 K W−1, which are typical values 112 6 Performance improvements of the TMS where Nis the number of bits, VFS is the full-scale voltage of the ADC and ∆ is the least-significant bit (LSB) or quantisation step. The error introduced by the quantisation step can be treated as an independent random variable with uniform probability density function (pdf) and uniform spectral density when a large number of bits is present [74]: Sq(ω) = ∆2 12 1 fs/2,0≤ω/2π≤fs/2 (6.3) where fsis the sampling frequency. Nevertheless, in a non-ideal ADC the quantisation steps are not uniform due to mismatches in its internal topology, specifically, due to tolerances in the capacitors [66, 76, 8, 150]. This nonuniformity is specified in two parameters: differential non-linearity (DNL) and integral non-linearity (INL). The DNL error is defined as the deviation between the actual difference between midpoints and a least significant bit (LSB) for adjacent codes. The additional noise related to this effect can be bounded by proper dithering [24, 59, 125, 149, 151, 152, 11, 153, 19]. The inherent analog noise of our system can be considered as a dither source, enough to make this effect negligible —see § 6.1.1.3. On the other hand, the INL error is defined as the discrete integration of the DNL and can be understood as the difference between the non-ideal and the ideal (uniform quantisation steps) ADC transfer curves —see Figure 6.1. The noise introduced by this error is usually less noticeable, although much more difficult to reduce. The DNL and INL are expressed as [92, 89] DNL(i) = Ki+1 −Ki−∆i= 0,1,2, ..., 2N−1 (6.4a) INL(i) = i X j=0 DNL(j)i= 0,1,2, ..., 2N−1 (6.4b) where Kiis the i-th transition voltage of the ADC. The left panel of Figure 6.1 shows the transfer curves of an ideal 8-bit ADC and a simulated non-ideal 8-bit ADC. The difference between both results in the error signal is shown in the bottom plot. The panel in the right shows the DNL and the INL of the non-ideal 8-bit ADC. Such curves are different for each individual ADC and they appear due to non-uniform quantisation steps. 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 code output ideal real 0 0.2 0.4 0.6 0.8 1 −1 0 1 analog input real−ideal [LSB] 50 100 150 200 250 −0.4 −0.2 0 0.2 0.4 0.6 0.8 output code [LSB] DNL INL Figure 6.1: Left: ideal and non-ideal (simulated) 8-bit ADC transfer curves. The difference between both is shown in the plot in the bottom. Right: DNL and INL of the non-ideal ADC. The non-idealities of the ADC have been simulated considering a 1% tolerance in the capacitors of the ADC. We focus on the INL error since it is the one limiting the performance of the measurement when slow rate input signals, i.e., with slopes of µV s−1, are considered, which is a common situation in 6.1 ADC non-linear errors correction 113 temperature measurements. The sensitivity of the TMS is 1.35 K V−1, therefore, drifts of µK s−1 are equivalent to drifts of µV s−1at the input of the ADC. 6.1.1.1 ADC bit error description In this section we describe the effect of a faulty bit in a SAR ADC. Afterwards, we will apply these results to actual measurements to validate whether or not the origin of the extra noise is due to ADC non-linearities. To do so, a simple model for the ADC is considered. The analog-to- digital conversion is done (in SAR ADCs) by comparing the sampled signal with an analog voltage generated by a digital-to-analog converter (DAC) and a SAR3[66]. The topology of the DAC is based on a switching capacitor bank composed of 16 capacitors, scaled from 216Cto C, where 216C is the most-significant bit (MSB) and Cis the LSB. A faulty bit is that with a capacitance value different from the ideal one. The transition voltage of the ADC, i.e., the DAC output voltage is [74] VDAC =PN−1 k=0 bk2kC PN−1 k=0 2kCVFS =PN−1 k=0 bk2kC (2N−1)CVFS (6.5) where bkis the binary digit (k= 0 stands for the LSB and k=N−1 for the MSB); it is set to 0 or 1 depending on the sampled voltage4, and Nis the number of bits. An error δCk=Ck,real −2kC in the k-th capacitor results in an error in the DAC output when the corresponding bit is 1, i.e., the transition voltage is slightly different from the ideal one. The error for a faulty bit is k. =VDAC,real(k)−VDAC,ideal(k) ≃bkδCk (2N−1)CVFS =bk δCk C∆ (6.6) The erroneous bit produces a superimposed periodic pattern in the quantisation error of the ADC with 1 LSB of amplitude when the input voltage varies between zero and VFS, and exhibits a periodicity of 2k+1∆. An example of this is shown in Figure 6.2 where it can be seen how a faulty bit introduces low frequency components superimposed on the typical sawtooth error function of an ideal ADC (red trace). In the left panel of Figure 6.2 the effect of a faulty bit in the transfer curve is clearly seen. The transition voltages in the ideal ADC (red trace) and the non-ideal ADC (black trace) are different. 3Successive approximation register (SAR) uses the algorithm put forth Tartaglia in 1556 to determine an unknown weight by a minimal sequence of weighting operations [76]. The algorithm is the following: for instance, we assume x=7. Then, is x≥23? no, the bit is set to zero and we do not retain 23. Is x≥22? Yes, the bit is set to 1 and we retain 22. Is x≥(22+ 21)? Yes, the bit is set to 1 and we retain 21. Is x≥(22+ 21+ 20)? Yes, the bit is set to 1. The result is 01112. 4The values of bkare the ones obtained by means of the SAR algorithm. In the example of footnote 3, bk=(1 1 1 0). 114 6 Performance improvements of the TMS 0 0.2 0.4 0.6 0.8 1 0 5 10 15 0 0.2 0.4 0.6 0.8 1 0 5 10 15 analog input (v in/VFS) code output [LSB] ideal non−ideal (δC1=0.15) ideal non−ideal (δC2=0.15) ∆ ∆ 1 ε2 ε 0 0.2 0.4 0.6 0.8 1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 error [LSB] 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 analog input (vin/VFS) δC1=0.15 δC2=0.15 ideal δC1=0.15 δC2=0.15 ε2 ε1 ε1 ε2ε2ε2 23∆ 22∆ Figure 6.2: Three simulated 4-bit ADC: ideal, 1= 0.3∆ and 2= 0.6∆. Left: transfer curves. Right: in the top we show the quantisation error functions. The plot in the centre and in the bottom show the differences between the ideal and the non-ideal transfer curves. Note the low frequency components superimposed (black and magenta traces) to the ideal ADC transfer curve (red trace). The differences between the ideal ADC transfer curve and the real ADC ones are also shown and represent the extra noise due to the non-idealities of the ADC. Another example is shown in Figure 6.3. The Fourier transforms of the quantisation error functions of an ideal 8-bit ADC and an 8-bit ADC with a faulty bit in the fourth LSB (k=3) are shown —see § 6.1.1.2 and § 6.1.1.3. It is clear the extra components introduced by the non-idealities of the ADC. 100101 100 10−2 10−4 10−6 10−8 |Q(ξ)| [V] 100101 100 10−2 10−4 10−6 10−8 [V−1] |Q(ξ)| [V] ideal non−ideal ∆−1 2∆−1 (24∆)−1 3(24∆)−1 ξ/2π Figure 6.3: 8-bit ADC. Top: Fourier transform of the ideal quantisation noise. Bottom: idem as the plot in the top for a non-ideal ADC: 3=0.4∆ The components of the non-ideal ADC are the ones of the ideal quantiser plus the ones due to k. 6.1.1.2 Dither signal effect in ideal quantisers When a dither voltage, d, with a certain probability density function (pdf), p(d), is added to the input signal of the ADC, v, the average of the quantisation error observed at the ADC output, 6.1 ADC non-linear errors correction 115 hq(v)i, is given by [149] hq(v)i=Z∞ −∞ q(v+z)p(z)dz (6.7) where q(v) stands for the quantisation error of an ideal ADC. If we define the Fourier transform in voltage domain by5 Q(ξ) = Z∞ −∞ q(v)e−iξvdv (6.8) and apply it to Eq. (6.7), the following ensues: hQ(ξ)i=Q(ξ)P∗(ξ) (6.9) The quantisation error for an ideal ADC (a sawtooth waveform of amplitude ∆/2 —see Figure 6.2) is a periodic function of vwith period ∆. It can be expanded in Fourier series: q(v) = ∆ π ∞ X n=1 (−1)n+1 nsin 2πnv ∆(6.10) and its Fourier transform is Q(ξ) = −i∆∞ X n=−∞ (n6=0) (−1)n+1 nδξ−2πn ∆(6.11) where δ() is Dirac’s delta function. If we consider zero-mean Gaussian noise as the dither voltage, its pdf and Fourier transform are, respectively, p(d) = 1 (2π)1/2σe−d2 2σ2(6.12) P(ξ) = e−ξ2σ2/2(6.13) where σ2is the variance of the noise. Substituting Eqs. (6.11) and (6.13) into Eq. (6.9) and taking the modulus we obtain |hQ(ξ)i| = ∆ e−ξ2σ2/2∞ X n=−∞, (n6=0) 1 nδξ−2πn ∆,(6.14) From Eq. (6.14) we note that the Gaussian dither signal, d, low-pass filters the quantisation error —see Figure 6.6. For instance, for σ= ∆ the attenuation of the first term in the series is 2.7·10−9. In our case, we have a 16-bit ADC with VFS=10 V, quasi-white noise at the input of the ADC S1/2 V≃7µV Hz−1/2and a noise equivalent bandwidth (NEBW) of ∆f≃1.2fcut−off =600 Hz where fcut−off is the cut-off frequency of the anti-alias filter —see § 2.3.2.4. Thus, ∆ and σcan be readily calculated, i.e., ∆ = 1 2N−1VFS = 0.15 mV,(6.15) σ=S1/2 V(ω)·f1/2 cut−off ≃0.17 mV.(6.16) In our case σ≃∆, thus the quantisation noise from the ideal ADC is suppressed by the inherent noise of the analog processing chain. In Figure 6.4 an example of this effect is shown. We have added white Gaussian noise of σ= ∆ to the simulated 8-bit ADCs shown in Figure 6.3. The effect is clear: the peaks in the spectrum of the ideal ADC are gone whereas the low frequency components introduced by the non-ideal ADC still remain. In Figure 6.6 (top) the shape of the dither filter is shown together with the frequency components of an ideal ADC. 5ξis the Fourier-conjugate variable of v, and is accordingly measured in 2πvolts−1. 116 6 Performance improvements of the TMS 100101 10−6 10−4 10−2 10−3 10−5 |Q(ξ)| [V] 100101 10−6 10−4 10−2 10−3 10−5 [V−1] |Q(ξ)| [V] non−ideal ideal (24∆)−1 ξ/2π Figure 6.4: Idem as Figure 6.3. However, this time Gaussian noise with σ= ∆ has been added to the signal before the quantisation process. All the peaks of both quantisers vanish except those at low frequency introduced by the faulty bit of the non-ideal ADC. 6.1.1.3 Dither signal effect on non-ideal quantisers The error of a real ADC is formed by the ideal quantisation error function —see Eq. (6.10)— plus a term related to the non-idealities of the ADC, i.e., q(v) = qi(v) + qk(v) (6.17) where qi(v) and qk(v) are the ideal quantisation error and the quantisation error due to the faulty bits, respectively. As seen in the previous section, the ideal quantisation error is filtered out by the analog noise in the measurement chain, which acts as a dither signal of Gaussian pdf. Instead, the quantisation error due to the non-ideality of the ADC is not reduced by the same analog noise, which causes it to show up as extra noise. In this section we show how it can be identified in the temperature measurements and how it limits the performance of the system. The quantisation error caused by faulty bits, qk(v), is the waveform shown in Figure 6.5. q0(v) ε0ε0ε0 ε1ε1ε1ε1 ε2ε2ε2ε2ε2ε2ε2ε2 23∆ q(v) q(v) 1 2 ∆ ∆ 2 ∆ 2 21∆ v ∆ ∆ ∆ Figure 6.5: qk(v). This signal is the difference between the quantisation error of an ideal ADC and the quantisation error of a real ADC. kis given in Eq. (6.6). 6.1 ADC non-linear errors correction 117 The modulus of the Fourier transform of the non-ideal noise for a single defective bit, k, say, is [152] |Qk(ξ)|= ∆ ∞ X n=−∞ sin nπk 2k+1∆ n sin nπ 2 sin nπ 2k+1 δξ−πn 2k∆(6.18) where kis given in Eq. (6.6). The effect of Gaussian dither on qk(v) is readily calculated with Eq. (6.9): |Qk(ξ)|= ∆ e−ξ2σ2/2∞ X n=−∞ sin nπk 2k+1∆ n sin nπ 2 sin nπ 2k+1 δξ−πn 2k∆(6.19) Equations (6.18) and (6.19) are plotted in Figure 6.6 (centre and bottom). The plot in the top represents the components introduced by an ideal ADC. The plot in the centre corresponds to errors associated to k= 0, i.e., the LSB. Here, the fundamental period is 2∆. The plot in the bottom shows the same for k=3 (the 4-th bit) with a fundamental period of 16∆. For each plot, the low-pass filter generated by Gaussian dither of σ= ∆ is also plotted (dashed lines). It can be seen that the dither signal suffices to suppress the noise associated to a faulty LSB, k=0 (this corresponds to the DNL effect which, is readily mitigated). However, it cannot attenuate the low frequency lines due to errors in the higher bits. In fact, when k= 3 the dithering with σ= ∆ is clearly insufficient. 0 2 4 6 8 10 0 Qi(ξ) (2∆)−1 [V−1] 0 2 4 6 8 10 0 Q0(ξ) ∆ ∆/2 ∆ ∆/2 ∆ ∆/2 (2∆)−1 [V−1] 0 2 4 6 8 10 0 Q3(ξ) (24∆)−1 [V−1] Figure 6.6: Top: quantisation error for an ideal ADC, Qi(ξ). Centre: quantisation error for a non-ideal ADC with k= 0 as the faulty bit. The error is 0= 0.5∆. Bottom: same as the plot in the middle but for k= 3. The error is 3= 0.5∆. Note that the x-axis scale of the plot in the bottom is different from the one in the top and the centre plots; for instance, the main frequency component for Q0is 1/2∆ while for Q3is 1/24∆. This implies that errors in the MSBs appear at lower frequencies than those in the LSBs errors. The dashed red trace is the equivalent low-pass filter of the dither signal which is Gaussian noise with σ= ∆. The filter is sufficient to reduce the frequency components of the ideal ADC and of the non-ideal ADC with the faulty bit k=0. However, it is useless for the faulty bit k=3. The term at “dc” is only a constant error, and therefore does not affect the measurement. From Figure 6.6 we note how faulty bits introduce spectral components in the measurement if the signal spans a large enough fraction of the range of the ADC transfer curve. Conversely, if the input signal is a constant dc then no extra noise will be seen, since no bit, whether faulty or not, will change its state. Therefore, the INL effect becomes perceptible in our system only when the input signal runs through a sufficiently wide fraction of the ADC range. 118 6 Performance improvements of the TMS 6.1.1.4 INL effect on general signals As we have just seen, when the input signal, v, is not a dc constant (e.g., a ramp), the errors related to the INL of the ADC tend to introduce low frequency components which degrade the performance of the system. In fact, this is seen to happen in our device for voltage variations in the order of few milli-volts. Moreover, if the input signal changes rate, i.e., ¨v6= 0, the frequency components introduced by the INL errors spread across the frequency band. In order to deal with non-constant signals and to estimate INL induced noise, a simplification is expedient. First, let us assume the input signal in straight line for a certain time interval, i.e., v(t) = (a+bt if 0 ≤t≤tmax 0 if t≤0 or t > tmax (6.20) where ais the initial value of the signal v(t), and bis the slope of the input signal [V s−1]. If we note ˜qk(ω) the usual time-frequency Fourier transform of the non-ideal quantisation noise, it immediately follows from Eq. (6.20) that ˜qk(ω) = eiωa/b |b|Qkω |b|.(6.21) In this case, Eq. (6.18) becomes |˜qk(ω)|=∆ |b| ∞ X n=−∞ sin nπk 2k+1∆ n sin nπ 2 sin nπ 2k+1 δω−πn|b| 2k∆(6.22) and the main frequency component for an ADC with an error in the k-th bit is located at: ω1k=π|b| 2k∆(6.23) Before we proceed further, a technical comment is in order. In several equations above, Dirac δ-functions appear. They are the result of infinite length integration intervals in Fourier transform calculations, which are of course idealisations. In any practical situation, such intervals are limited to the experimentally available data ranges, so that the δ-functions are actually sinc-functions: they have identical centre points but spread around those centres depending on the integration interval lengths. In order to make sense of e.g. Eq. (6.22) we must ask which is the minimum required integration time to obtain a meaningful spectrum or, in other words, which is the minimum duration, tmax, of the ramp signal in Eq. (6.20). This is easily inferred from the frequency of the lowest spectral line ω1kin Eq. (6.23): for a conventional ten cycle integration time we get tmax >10 2π ω1k = 10 2k+1 ∆ |b|(6.24) These considerations apply to the analysis of the ADC response to signals which drift slowly with time, where slow drift means such signals can be conveniently approximated by a series of concatenated ramps with suitable slopes, and lengths complying with Eq. (6.24). In these circumstances, we can generalise Eq. (6.23) as follows: ω1k=π2N 2kVFS |˙v(t)|,˙ω1k=π2N 2kVFS |¨v(t)|(6.25) 6.1 ADC non-linear errors correction 119 From Eqs. (6.23) and (6.25) some conclusions can be drawn:  High-slope signals at the ADC input translate into high-frequency components at the output of the ADC.  Errors in the higher bits show up as noise at low frequency.  Consequently, high-slope input signals combined with errors in high bits may show as noise peaks in the measurement bandwidth.  The fundamental frequency associated to a faulty bit varies with the variations of ˙v(t), hence the error in a faulty bit spreads across the frequency band when ¨v6= 0. The above can be validated by looking at real data from the TMS. We took a long time series of temperature data, about 4 ·105sec, and subdivided it into shorter stretches of 16 000 sec. With this, we calculated the short-time Fourier transform (STFT) [100], also known as spectrogram, and plotted the results as shown in Figure 6.7 —see caption for details. The data are amenable to the analysis described above, and the results indicate the presence of the foreseen frequency peaks associated to faulty bits. They are clearly visible in Figure 6.7, bottom graph as the darker areas, which show how the fundamental frequency of the INL errors for different faulty bits is varying with time. A straightforward fit to the first Eq. (6.25) permits the identification of the corresponding faulty bit, thus confirming that the extra noise in the temperature measurements is indeed due to the INL errors of the ADC. 1 2 3 4 0 5 10 15 time (×105) [s] dV/dt [µV s−1] 10−4 10−2 100 10−4 10−2 10−3 10−5 frequency [Hz] S1/2 v(ω) [V Hz−1/2] non−stationary stationary time (×105) [s] frequency [mHz] k=2 k=3 k=4 k=5 k=6 k=7 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 Figure 6.7: Top left: Absolute value of the input signal slope, |˙v(t)|, of the measurement. It varies between 0 and 12.5 µV s−1. The sensitivity at the input of the ADC is ≃1.35 V K−1, hence the values of the voltage slope almost directly provide the actual temperature drifts in [K s−1]. Top right: power spectral density for: (i) the signal shown in the top left plot, i.e., a non-dc signal (solid black trace) and, (ii) a dc signal (dashed red trace), i.e., with |˙v(t)|.0.5µV s−1. The excess noise when measuring non-constant signals is very noticeable, and degrades the performance of the measurement by about one order of magnitude. Bottom: STFT (or spectrogram) of the measurement. The energy of the signal is concentrated in specific frequencies which change with time precisely following |˙v(t)|, as predicted by Eq.(6.25). The latter is represented by dashed superimposed traces, and are labelled by the order of the corresponding faulty bit (from k= 2 to k= 7). Experimental results and theoretical estimates are in excellent agreement. Once the problem has been identified, we focus on possible solutions to solve it. The following section describes two different methods tested successfully to avoid the consequences of INL errors in real ADCs. 120 6 Performance improvements of the TMS 6.1.2 Mitigation of ADC errors Two different techniques have been used to deal with the non-linearities of the ADCs. The first one is based on the injection of Gaussian noise out of the MBW, taking advantage of the oversampling involved in the measurement. The second option is based on the addition of a triangular wave of high frequency to the signal of interest before quantisation. The following sections describe both techniques as well as their practical implementation. 6.1.2.1 Gaussian noise dither injection In § 6.1.1.2 we saw that Gaussian noise can be used as a dither signal in order to mitigate the nonidealities of the ADC transfer curve. We have also seen that the amount of noise generated by the measurement system itself (σ≃∆) is not enough to suppress the periodic components of erroneous bits for k&2 —see Figure 6.7. Thus, a natural solution is to add more Gaussian noise to the ADC input. Obviously, this noise should be added out of the MBW in order not to disturb the frequency range of interest. The ADC sampling frequency is 38.4 kHz (in the prototype), thus leaving a large frequency slot to accomodate the required additional noise. Care must however be taken to also place it away from the fundamental frequency of the modulating square wave (and its harmonics) so as to avoid bringing the Gaussian noise back into the MBW in the demodulation process. The required amount of noise to be injected, characterised by σ, basically, depends on the input signal slope, ˙v(t), and on the frequencies of interest, and is limited by the digital processing performed after quantisation —see below. Figure 6.8 shows the relationship between the input signal slope and the main frequency component for each of the faulty bits. It is useful to identify the faulty bits potentially affecting the measurement as a function of the input signal slope. k=15 k=12 k=6 k=3 k=0 k=9 LISA LPF |dv/dt| [V/s] frequency [Hz] 10−7 10−6 10−5 10−7 10−6 10−5 10−4 10−3 10−2 10−1 Figure 6.8: Relationship between the input signal slope, the faulty bit and the fundamental frequency affected by the error in the k-th bit. When high input signal slopes are present, more faulty bits will affect the measurement quality in the MBW. Clearly, LSBs show up at higher frequencies than MSBs. The shaded areas span the MBWs of LISA and LPF, as indicated. Note that LPF’s MBW is a subset of LISA’s. Once the problematic bits are identified, the needed amount of Gaussian dither must be calculated. For this, we define the σneeded to filter out the noise components associated to the k-th faulty bit. If we (conventionally) adopt a damping factor of 10 for the first harmonic then σis easily derived from the condition —see Eq. (6.19), e−ξ2σ2/2≤1 10 for ξ=2π 2k+1 ∆(6.26) 6.1 ADC non-linear errors correction 121 whence σ≥√2 ln 10 2π2k+1 ∆ (6.27) Practical implementation In this section we briefly describe the hardware implementation of the Gaussian dithering scheme. The circuit scheme is given in Figure 6.9 (left panel). The first stage amplifies the noise of an operational amplifier (OP-07 of Analog Devices); the second stage is a high-pass filter of 4-th order (two Sallen-Key filters in cascade) with a cut-off frequency of 100 Hz; the third stage is an adder that sums the signal of interest (the amplified output of the Wheatstone bridge) to the dither signal. In the fourth stage, the sum of signal and dither are low-pass filtered with a 4-th order low-pass filter (again, two Sallen-Key filters in cascade) with cut-off frequency 3 kHz. The output of this chain is fed to a 16-bit SAR ADC. The noise shape of the dither signal is given in Figure 6.9 (right panel) where the characteristic frequencies of the high- and low-pass filters are clearly visible. 10 0 LPF (fc=3 kHz)HPF (fc=100 Hz) G=1500 d(t) High−pass filter v(t) Low−pass filterAmplification x2 10 x2 210 4 10 −6 10−5 10−4 frequency [Hz] SV 1/2 (ω) [V Hz −1/2] extra dithering FEE noise Figure 6.9: Gaussian noise dither signal generator. d(t) is the Gaussian noise dither and v(t) is the amplified output of the Wheatstone bridge. Right: PSD of the Gaussian noise (solid trace). The dashed trace shows the floor noise of the measurement system. As mentioned before, the amount of Gaussian dither must be kept under control to avoid excessive noise folding back into the MBW during the digital demodulation stage. The limit of usable Gaussian noise amplitude, σ, can be easily estimated assuming a flat spectrum, SV,dither, from ωm/2π(the corner frequency of the high-pass filter) to ωM/2π(the corner frequency of the low-pass filter). The demodulation by a square wave of frequency ωcentails a certain gain at dc of the odd harmonics of ωcof the dither noise which, under the just mentioned assumption, results in the following —see § 2.3 and § 3.2: SV, extra(ω→0) ≃SV, dither ·4 π2ωM 2ωc X n=bωm/2ωcc 1 (2n+ 1)2(6.28) where b c is the floor function. In order to cut down the noise leaking into the MBW, we impose a requirement that it be less than 10% the floor noise, in the absence of ADC errors, SV,FEE, i.e., SV, extra(ω→0) ≤SV, FEE(ω) 10 (6.29) In our case, ωm/2π≃100 Hz, ωM/2π≃3 kHz, and ωc/2π≃5.55 Hz, so that the term multypling SV,dither is approximately 0.015, hence SV(ω→0) ≃0.015SV,dither. The value of SV,FEE is 1.5· 128 6 Performance improvements of the TMS where  ST,bis the Johnson noise of the Wheatstone bridge,  ST,b+IAiis the noise caused by the current noise of the instrumentation amplifier (IA) together with the equivalent resistance of the Wheatstone bridge,  ST,IAvis the inherent voltage noise of the IA,  ST, ADC is the noise introduced by the analog-to-digital converter (transition noise). Equation (6.38) can be expressed as8 S1/2 T(ω) = 2T2 (RP)1/2β4kBT R +i2 n(ω)R2 2+e2 n(ω) + e2 ADC(ω)1/2 (6.39) where Rand βare the nominal resistance and the temperature characteristic of the thermistor, respectively, Tis the temperature in Kelvin, Pis the power dissipated in the sensor, i2 nand e2 nare the noise figures of the IA —cf. § 2.3.2.4, e2 ADC is the noise of the ADC —cf. § 2.3.2.4— and kBis Boltzmann’s constant. For a given IA and ADC, the noise is inversely proportional to the square root of the power, and also depends on the nominal resistance of the thermistor. The effect of these two variables, Pand Ris analysed in the following sections. 6.2.1 Thermistor nominal resistance The optimum value of Rthat minimises Eq. (6.39) is found by ∂ST ∂R = 0 (6.40) which yields Ropt =2 i2 n(ω)e2 n(ω) + e2 ADC(ω)1/2 .(6.41) The optimum value of the thermistor nominal resistance, Ropt, depends only on the IA noise properties and on the noise introduced by the ADC. The expressions of e2 n,i2 nand e2 ADC are [102] —see § 2.3.2.4 and § 2.3.2.5, e2 n(ω) = K2 v1 + ωcv ω(6.42a) i2 n(ω) = K2 i1 + ωci ω(6.42b) e2 ADC(ω) = σ 216 VFS 1 √6fs 1 GIA 2 (6.42c) For the case of the IA AD624 the values of e2 nand i2 nat fc=5.55 Hz are 2.47 ·10−17 V2Hz−1and 1.71 ·10−24 A2Hz−1, respectively. The noise introduced by the ADC9is e2 ADC = 3 ·10−17 V2Hz−1. Substituting these values into Eq. (6.41) yields Ropt ≃8 kΩ which is very close to the value used in the TMS, 10 kΩ. However, the IA in the FM system is the AD620 which is slightly different from the AD624. The AD620 noise figures are: e2 n= 2.26 ·10−16 V2Hz−1and i2 n= 2.1·10−25 A2Hz−1 and the optimum nominal resistance is ≃50 kΩ. However, differences in the noise figure when using a 50 kΩ thermistor or a 10 kΩ one are tiny. Figure 6.16 shows the floor noise when using the AD624 and the AD620 IAs for different nominal resistance thermistors. 8For simplicity, we assume R1=R2=Rref =R(T). 9Considering a gain in the amplification stage of 200 and the sampling frequency 38.4 kHz. 6.2 Floor noise reduction 129 102103104105 10−6 10−5 10−4 resistance (R) [Ω] S1/2 T(ω) [K Hz−1/2] AD624 10 µW 1 channel AD620 10 µW 1 channel Figure 6.16: Noise levels as a function of the nominal resistance of the thermistor. Two instrumentation amplifiers are considered: AD624 (the one used in the prototype design) and AD620 (the one used in the flight model system). The analysis shown in this section indicates that the noise of the system is already at its minimum for a given power and that the only chance to reduce it is by dissipating more power to increase the sensitivity of the Wheatstone bridge. This possibility is discussed in the next section. 6.2.2 Power increase in thermistors Equation (6.39) indicates that the floor noise of the temperature measurement is inversely proportional to √P. In order to assess this, the electronics was modified to dissipate 100 µW in the thermistor, instead of 10 µW. Moreover, only one channel was acquired, i.e., no multiplexing was present. Differential measurements were taken since they (i) reduce the stabilisation time in the measurement, (ii) reduce the effect of the ambient temperature fluctuations and (iii) increase the immunity to the problems of non-linearity in the ADC since the slopes of the measurement are very small. The results for a 10 µW-1 channel measurement and for a 100 µW-1 channel measurement are shown in Figure 6.17. The floor noise is reduced by a factor of p100 /10 = 3.3, thus, Eq. (6.39) is validated. The noise is flat for frequencies down to ≃1 mHz in both measurements. At lower frequencies the effect of the temperature coefficient (TC) of the electronics degrades the measurement in both configurations. The dashed grey traces represent the projection of the laboratory temperature fluctuations in the aluminium block when measuring differential temperature10. The magenta dashed trace is the effect of the TC of the electronics and it is the one limiting the measurement for f < 1 mHz. At frequencies close to fs/2the noise increases slightly due to the differentiation process involved in the measurement —see § 2.3.4. 10ST,ins(ω) = k2 NTCω2|Hins(ω)|2ST, lab(ω) —see § 4.1.2 and § 4.2.2. 130 6 Performance improvements of the TMS 10−4 10−3 10−2 10−1 100 10−8 10−6 10−4 10−2 100 10−7 10−5 10−3 10−1 101 frequency [Hz] ST 1/2 (ω) [K Hz −1/2 ] insulator insulator TC electronics lab 10 µW 100 µW Figure 6.17: Noise levels when sampling only one channel, i.e., with no multiplexing, and for P= 10 µW and P= 100 µW. The measurement sensitivity is limited at low frequency (f < 1 mHz) by the TC of the electronics (magenta trace). The results in Figure 6.17 lead to the following conclusions:  the noise in the measurement is reduced by dissipating more power in the thermistor,  measurement with no multiplexing avoids aliasing —see § 2.3.4,  no excess noise has been detected in the thermistors or electronics at 1 µK Hz−1/2and at ∼1 mHz. The noise in the measurement in terms of power and number of multiplexed channels can be expressed as S1/2 T(ω)≃4µK Hz−1/210 µW P1/2 N1/2 ch (6.43) where Pis the power dissipated in the thermistor and Nch is the number of channels multiplexed. However, noise cannot be reduced as much as desired by increasing the power. The self-heating effect (SHE) restricts the amount of power dissipated in the thermistor; if this limit is exceeded the noise introduced by the SHE is higher than the noise modelled by Eq. (6.43). The SHE has been described in § 2.3.2.3. Here we use the same analysis to estimate the maximum power permitted in the thermistor. The temperature fluctuations caused by the SHE are ST(ω) = θ2SP(ω)≃θ2V R(T)2 SV(ω) (6.44) where R(T) is the resistance of the thermistor, θis the thermal resistance between the sensor and the body, SVare the voltage fluctuations in the thermistor and Vis the voltage in the NTC, V=√PR. Substituting V=√P R into Eq. (6.44) yields S1/2 T(ω)=2θP R1/2 S1/2 V(ω).(6.45) Combining Eq. (6.43) and (6.45) we obtain an expression for the maximum power, i.e., P≤4µK Hz−1/2·(10 µW)1/2R1/2 2θS1/2 V(ω)= 1.25 ·10−8R1/2 2θS1/2 V(ω)(6.46) 6.2 Floor noise reduction 131 where we notice that the maximum power is (i) inversely proportional to the thermal resistance, θ, and to the voltage fluctuations in the thermistor, SV, and, (ii) proportional to the square root of the nominal resistance of the thermistor, R. Consequently, the better the voltage reference the higher the power that can be dissipated in the thermistor and, therefore, the lower the noise in the measurement. The thermal resistance even assuming an ideal contact between the thermistor head and the body is ∼50 K W−1due to the inherent properties of the thermistor —see appendix § B. Large values of Rpermit higher values of P. However, the range of resistance values of space qualified NTC thermistors is from 2 kΩ, to 20 kΩ. In our system we are using R=10 kΩ. The total noise in the temperature measurements considering the electronics noise and the SHE can be expressed, in general, as: S1/2 T(ω) = 2 R1/2T4 Pβ24kBTR +i2 n(ω)R2 2+e2 n(ω) + e2 ADC(ω)+θ2P SV(ω)1/2 (6.47) Equation (6.47) is plotted in Figure 6.18 for different values of Rand Pfor a given IA (AD624), ADC (AD977), θ(=100 K W−1) and SV(= 10−10 V2Hz−1). We notice that for a given Rthe power cannot be increased arbitrarily since then the SHE becomes important. However, large values of R permit higher values of Psince in this case the SHE becomes less critical. If SVand θare large, the measurement is dominated by the SHE. R [Ω] P [W] 1000 10000 20000 500002000 5000 10−6 10−5 10−4 10−3 10−2 2 4 6 8 10 12 14 16 18 x 10−6 ST 1/2 [K Hz−1/2] Figure 6.18: Noise in temperature measurements for different values of Rand P. The dark blue region is the one exhibiting the lowest noise. In order to assess this analysis, measurements with P=1 mW and P=10 mW (both with R=10 kΩ) were performed. The results are shown in Figure 6.19. In both measurements the SHE was detected. When P=1 mW (solid black trace) the noise differed from the expected one (dash-dotted black trace) at ∼20 mHz. The projection of the ambient temperature in the measurement (dashed grey traces) and the TC of the electronics (dashed magenta trace) could not explain such behaviour between 1 mHz and 20 mHz. For P=10 mW the noise due to the SHE starts to dominate at 200 mHz and at 30 mHz is already higher than the one for P=1 mW, thus confirming the fact that the power cannot be increased arbitrarily: it becomes counterproductive due to the voltage fluctuations in the thermistors which in turn cause temperature fluctuations due to the SHE. This is specially noticeable at the milli-Hertz range. 132 6 Performance improvements of the TMS 10−5 10−4 10−3 10−2 10−1 100 10−8 10−6 10−4 10−2 100 10−3 10−5 10−7 10−1 101 frequency [Hz] ST 1/2(ω) [K Hz−1/2] lab TC electronics insulator insulator 1 mW 10 mW Figure 6.19: Noise levels of the TMS with P= 1 mW and P=10 mW (in both cases only one channel is sampled). The projection of the ambient temperature fluctuations through the insulator (grey dashed trace) and the TC of the electronics (magenta dashed trace) are also shown. None of them explain the extra noise from ∼1 mHz to ∼50 mHz. The extra noise comes from the SHE which dominates the measurement when high power is dissipated in the sensor. Finally, there is a physical limitation independent of the method of measurement imposed by the fluctuation-dissipation theorem which states that for a thermodynamic system in equilibrium the fluctuations around the equilibrium point are random. Thus, they limit the temperature measurement to a certain level. The limit is given by [29, 35] ST(ω)=4θkBT2(6.48) where θis the thermal resistance, Tis the absolute temperature and kBis Boltzmann’s constant. In the case of thermistors the minimum thermal resistance is ∼50 K W−1—see appendix § B— and the temperature we are interested in is ∼300 K. This leads to a limit of 16 nK Hz−1/2. The TMS achieves noise levels of ≃1µK Hz−1/2in the milli-Hertz region when dissipating 100 µW and of ≃0.2 µK Hz−1/2at 1 Hz with P=10 mW. We are still a factor of ∼60 above the theoretical limits in the milli-Hertz band and a factor of 12 at ∼1 Hz. All in all, it seems difficult to achieve measurements with noise levels below 1 µK Hz−1/2in the milli-Hertz with the measurement method described in this thesis. The SHE spoils the measurement when increasing the power to improve the temperature sensitivity of the system. In summary, we have shown that the noise of the TMS is compatible with the measurement of temperature fluctuations at the level of µK Hz−1/2in the milli-Hertz band if the power is increased by a factor of 10 with respect to the one in the LTP (10 µW), and if no multiplexing is used. However, the power dissipated in the thermistor does not only concern the SHE. The power also must be small enough not to perturb other subsystems nearby. In the LTP it has been limited to 10 µW. Such fears mainly concern the GRS since the power dissipated in the thermistors can generate thermal fluctuations in the walls of the EH and thus, exert forces in the TMs. A simple approach to obtain an idea about the effect in the temperature stability of the GRS is presented herein. First we consider that the temperature evolution of one wall of the EH is described by (during the transient response) P=C˙ T(6.49) where Cis the thermal mass of the EH wall, Tis the temperature of the EH wall and Pis the power dissipated in the thermistor. We consider that the cut-off frequency of the transfer function 6.2 Floor noise reduction 133 between power and temperature increase (of the EH walls) is, in principle, lower than the MBW11. Therefore, we Fourier transform Eq. (6.49) to obtain the transfer function in the MBW. In power spectral density it is12: SP(ω) = ω2C2ST(ω) (6.50) Power fluctuations, SP, can be approximated to: SP(ω) = 2V R2 SV(ω)V=√P R = 4P RSV(ω) (6.51) where Pis the power dissipated in the thermistor, Ris the resistance of the thermistor and SVare the voltage fluctuations in the thermistor. Now we combine Eqs. (6.50) and (6.51) to obtain the relationship between voltage fluctuations and increase of temperature of the EH walls, i.e., ST(ω) = 4P RC2ω2SV(ω) (6.52) The value of Chas been estimated by simulations performed with the CGS thermal model. Details on the thermal model and the simulations can be found in [50, 51, 97]. The value is C∼70 J K−1, Ris 10 kΩ and S1/2 Vis ∼10−4V Hz−1/2in the LTP MBW in the very worst-case. Evaluation of Eq. (6.52) yields temperature fluctuations of ∼10−8K Hz−1/2and ∼5·10−8K Hz−1/2at 1 mHz for P=10 µW and P=100 µW, respectively. The stability in the GRS must be kept lower than 10−4K Hz−1/2—cf. § 1.5. From this analysis it is clear that the effect of the power dissipated in the thermistors should not pose a problem to the required temperature stability in the GRS and thus, higher power (at least 100 µW) can be dissipated in the thermistors in order to achieve higher sensitivity in the temperature measurements. Dedicated experiments to assess the effect of dissipating power through a thermistor attached to the walls of the EH will provide accurate information on the thermal response of the EH walls. Such tests must be performed in view of the definition of the thermal experiments in the GRS which are described in chapter § 7. Anyway the effect of the power fluctuations in the thermistor of the TMS can be considered negligible. 11Simulations performed with the LCA thermal model provided by CGS suggest this hypothesis [50, 51, 97]. 12This expression is valid in the MBW if and only if the cut-off frequency of the thermal response of the EH walls is smaller or of the samer order than 1 mHz. The cut-off frequency is approximated to fc=1 /(2πCθ)where Cis the thermal mass of the EH wall and θis the thermal resistance with the surroundings Chapter 7 Thermal experiments in the GRS A set of 14 heaters will fly in the LTP as a part of the thermal diagnostic subsystems. Their objective is to generate thermal gradients in thermally sensitive locations on-board the LTP. The purpose of these controlled disturbances is to induce a perturbing signal of high amplitude in different subsystems to then estimate the relationship between the temperature and the subsystem thermally perturbed. The temperature disturbances are monitored by 24 thermistors distributed across the LTP. The final scope of such measurements is to characterise the thermal noise contribution to the LTP noise performance curve. In the following sections we focus on the definition of the thermal experiments in the GRS. Thermal effects in the GRS are those causing a force on the test mass when a temperature gradient is present across the opposite faces of the electrode housing. The most relevant derived effects in the LTP conditions are: the radiation pressure, the radiometer effect and the outgassing from the housing walls [84, 26] —cf. § 1.5.1. We mainly discuss the possibility of disentangling two thermal effects (radiation pressure and radiometer effect) by using suitable thermal signals generated by the heaters. The analysis is a continuation of the work presented in [97]. The fist part focuses on the definition of suitable signals. The second part of the chapter describes simulations within the LTP dynamics control loop in order to assess the quality in the estimation of the parameters and to determine the motion of the TMs within the control loops when thermally perturbing them. 7.1 Estimation of errors on parameters: the Cram´er-Rao bound The approach followed to design experiments on-board the LTP is based on the evaluation of the errors in the estimation of the parameters. The figure of merit is the variance of the parameters in terms of the experiment variables, i.e., the power dissipated in the heaters and the integration time (duration of the experiment). A suitable tool for this purpose is the Fisher information matrix, whose inverse sets lower bounds on the variance of the parameters [63, 163]. In order to introduce the notation and the main concepts required for the analysis, we consider a known deterministic signal s[n, θ] with an unknown parameter θ, to produce an output, y[n], which is measured with a noisy component, w[n], i.e., y[n] = s[n, θ] + w[n], n = 0,1, . . . , N −1.(7.1) where Nis the number of samples. If we can assign a probability density function to the process y[n], i.e., p(y;θ), then the Cram´er-Rao bound (CRB) states that the variance of any unbiased 136 7 Thermal experiments in the GRS estimator b θ, satisfies var(b θ)≥ − 1 E"∂2ln p(y;θ) ∂2θθ=θtrue #(7.2) where E{–}stands for the expectation value taken with respect to p(y;θ) and the derivative is evaluated at the true value of θ,θtrue. The term ln p(y;θ) is also known as the log-likelihood ratio and, in the case of a set of parameters Θ={θ1, θ2, . . . , θM}, the denominator of Eq. (7.2) becomes a matrix, known as the Fisher information matrix I(Θ) = −E"∂2ln p(y;Θ) ∂2ΘΘ=Θtrue #(7.3) and the CRB is var( b Θ)≥diag[I(Θ)]−1(7.4) saying that the lower bound for the variance in the estimation of the parameters Θis the inverse of the Fisher matrix. 7.2 Purpose of the thermal experiments in the GRS The nominal thermal feedthrough factor for the case of infinite planes, α, is1[84] —cf. § 1.5: α(T, p) = ATM mTM 16 3 σ cT3+1 2 p T+Q(T) Ceff Θ T2(7.5) where the three terms correspond to the known effects: radiation pressure, radiometer effect and outgassing processes. The acceleration caused by this joint effect in a free floating test mass is proportional to the thermal gradient between the faces of the walls of the EH, or aTM =α∆T. (7.6) In Eqs. (7.5) and (7.6) the variables stand for (see also Figure 7.1):  ATM: surface of each of the TM faces,  mTM: mass of the TMs,  σ: the Stefan-Boltzmann constant,  c: the speed of the light,  p: the pressure inside the vacuum enclosure,  Q(T), Θ and Ceff : parameters to describe the outgassing effect,  T: absolute temperature of the EH walls facing the TM,  ∆T: differential temperature between the EH walls facing the TM. 1Other factors can be found in the literature that are slightly different [26, 67]. 7.2 Purpose of the thermal experiments in the GRS 137 electrode housing T T 1T 2 test mass ∆ Figure 7.1: Scheme of the TM and EH walls surrounding it. We assume a constant pressure, p, and ignore the outgassing effect due to the uncertainty of this effect which, in principle, will be negligible —see § 1.5.1. Thus, the feedthrough factor, α, is simplified to α(T)≃ATM mTM 16 3 σ cT3+1 2 p T.(7.7) In addition, the temperature surrounding each of the TMs, T, will be quasi-constant during the experiment. The increase of this temperature should be around ∼0.5−1 K maximum during each run —see Figure 7.11. This fact suggests considering the absolute temperature constant during the experiments, whence, Eq. (7.7) can be further simplified: α≃ATM mTM 16 3 σ cT3 o+1 2 p To(7.8) where Tois the temperature of the EH walls during the experiment (assumed constant). However, this assumption does not allow to discern the radiation pressure effect and the radiometer effect since the parameter αbecomes a mere constant. In order to resolve both effects we must know accurately the value of αand consider Tis not constant. The accuracy needed is determined by the change of αwith respect to the absolute temperature, and the variation of the latter during the experiment2, or dα dT =ATM mTM 16σ cT2−1 2 p T2(7.9a) ∆αGRS(T) = 1 αo dα dT [T(tf)−T(t= 0)] (7.9b) where [T(tf)−T(t= 0)] is the variation of the absolute temperature during the experiment and αois αat T(t= 0). If the accuracy in the estimation of the parameter is higher than the change observed in αduring the experiment due to absolute temperature changes —Eq. (7.9b)— then we will be able to disentangle the two effects. Now we evaluate Eqs. (7.8), (7.9a) and (7.9b) to have an idea about the needed accuracy. The values used are [99]:  ATM=(46 ·10−3)2m2  mTM=1.96 kg  To=293 K  p= 10−5Pa 2The change in absolute temperature permits to distinguish between the T3behaviour from the 1/Tone in Eq. (7.7).