Satellite Construction

Attitude Control and Determination Systemfor DTUsat – a CubeSat contribution

byJan Hales,

–c973712–

Martin Pedersen,–c991237–

and Klaus Krogsgaard–c991365–

Midterm ProjectDepartment of Automation, Ørsted•DTU

Technical University of Denmark

Supervisor:Professor Mogens Blanke

Co-Supervisor:Process Specialist René FléronMikroelektronik Centret (MIC)

Last edited 22nd March 2002

Abstract

Different methods of attitude control and determination are considered andevaluated with respect to physical constraints, feasibility, and performance.Sun sensors and magnetic field sensors are found to be the best suited sensorsfor the DTUsat mission. The design of these have been conducted, andin the case of the magnetometer a prototype has been implemented andradiation tested. Magnetotorquers have been selected as actuators for atwo-axis stabilised attitude control system. The magnetotorquers and therequired driver circuit have been designed.

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Acknowledgements

Professor Mogens Blanke (Department of Automation, Ørsted•DTU)our main project supervisor.

Process Specialist René Fléron (Mikroelektronik Centret (MIC)) whowas the supervisor of the sun sensor part.

Associate Professor Peter Brauer (Measurement & Instrumentation,Ørsted•DTU) for guidance and magnetometer prototype testing.

M.Sc. Student Torsten Lorentzen for good cooperation in the ACDSgroup.

Staff at Mikroelektronik Centret (MIC) for expert opinions and de-sign reviews.

The entire DTUsat group and especially the system engineering groupfor its project coordination and management.

Mikroelektronik Centret (MIC) for sponsoring the sun sensors.

Rigshospitalet & Risø for performing radiation tests.

Other sponsors who make it possible to construct and launch DTUsat.See http://www.dtusat.dtu.dk/ for details.

Thank you!

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Contents

1 Introduction 1

2 Requirements & Constraints 32.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Preliminary Payload Requirements . . . . . . . . . . . . . . . 32.3 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 General Problem Analysis 53.1 Attitude Determination . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 Earth Sensor . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Sun Sensor . . . . . . . . . . . . . . . . . . . . . . . . 73.1.3 Magnetometer . . . . . . . . . . . . . . . . . . . . . . 73.1.4 GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.5 Star Imager . . . . . . . . . . . . . . . . . . . . . . . . 83.1.6 Rate gyro sensors . . . . . . . . . . . . . . . . . . . . . 93.1.7 Sensor Evaluation . . . . . . . . . . . . . . . . . . . . 9

3.2 Attitude Stabilisation and Control . . . . . . . . . . . . . . . 103.2.1 Gravity Gradient Stabilisation . . . . . . . . . . . . . 103.2.2 Permanent Magnet Stabilisation . . . . . . . . . . . . 103.2.3 Magnetotorquer Control . . . . . . . . . . . . . . . . . 113.2.4 Fly Wheel Control . . . . . . . . . . . . . . . . . . . . 113.2.5 Actuator Evaluation . . . . . . . . . . . . . . . . . . . 11

3.3 Selection of ACDS Concept . . . . . . . . . . . . . . . . . . . 123.3.1 Sensor Selection . . . . . . . . . . . . . . . . . . . . . 123.3.2 Actuator Selection . . . . . . . . . . . . . . . . . . . . 12

4 Problem Statement 154.1 Criteria of Success . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Environment Modeling 175.1 Orbit Model (SGP) . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1.1 SGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Magnetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2.1 Implementation . . . . . . . . . . . . . . . . . . . . . . 19

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5.3 Sun Positioning Model . . . . . . . . . . . . . . . . . . . . . . 19

5.4 Gravitation Gradient Model . . . . . . . . . . . . . . . . . . . 20

6 Attitude Determination: Sun Sensor 23

6.1 Sensor Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.1.1 Principle of pn junction Solar Cells . . . . . . . . . . . 23

6.1.2 Analysis of Sun Sensor Principles . . . . . . . . . . . . 26

6.2 Sensor Principle Analysis . . . . . . . . . . . . . . . . . . . . 34

6.3 Detailed Sensor Design . . . . . . . . . . . . . . . . . . . . . . 36

6.3.1 Electrical Characteristics . . . . . . . . . . . . . . . . 36

6.3.2 Implementation of the Solar Cell Areas . . . . . . . . 38

6.3.3 Micro-structure Design . . . . . . . . . . . . . . . . . . 39

6.3.4 Realisation - Mask Design . . . . . . . . . . . . . . . . 45

6.3.5 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 48

6.4 Sensor Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.4.1 Block Diagram . . . . . . . . . . . . . . . . . . . . . . 55

6.4.2 Schematic . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.4.3 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Mechanical Integration . . . . . . . . . . . . . . . . . . . . . . 59

6.6 Sensor Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7 Attitude Determination: Magnetometer 63

7.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Magnetic Sensor . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.2.1 Sensor Types . . . . . . . . . . . . . . . . . . . . . . . 64

7.2.2 Selected Sensor . . . . . . . . . . . . . . . . . . . . . . 65

7.3 Measurement Errors and Compensation . . . . . . . . . . . . 66

7.3.1 At Constant Temperature . . . . . . . . . . . . . . . . 66

7.3.2 Temperature Dependencies . . . . . . . . . . . . . . . 67

7.3.3 ±α Compensation . . . . . . . . . . . . . . . . . . . . 67

7.4 Prototype Circuit . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.4.1 Principle of Operation . . . . . . . . . . . . . . . . . . 68

7.4.2 Sensor Bridge and Amplifier . . . . . . . . . . . . . . . 69

7.4.3 MUX and PI Controller . . . . . . . . . . . . . . . . . 70

7.4.4 I-to-V Converter . . . . . . . . . . . . . . . . . . . . . 71

7.4.5 Reference Voltage . . . . . . . . . . . . . . . . . . . . 72

7.4.6 H-bridge . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.5 Magnetometer Performance . . . . . . . . . . . . . . . . . . . 72

7.5.1 Sensitivity and Linearity . . . . . . . . . . . . . . . . . 73

7.5.2 Bandwidth and Noise . . . . . . . . . . . . . . . . . . 75

7.5.3 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.5.4 Power Supply and Consumption . . . . . . . . . . . . 77

7.5.5 Radiation Hardness . . . . . . . . . . . . . . . . . . . 77

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7.6 Sensor Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 787.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8 Actuator: Magnetotorquers 818.1 Dimensioning of magnetotorquers . . . . . . . . . . . . . . . . 81

8.1.1 Driver circuit . . . . . . . . . . . . . . . . . . . . . . . 84

9 Control System 879.1 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 879.2 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899.3 Control System Model . . . . . . . . . . . . . . . . . . . . . . 91

10 Conclusion 93

Bibliography 94

A Process Sequence 99

B Wafer Masks 107B.1 Sensor without contact fingers . . . . . . . . . . . . . . . . . . 107B.2 Sensor with contact fingers . . . . . . . . . . . . . . . . . . . 113

C Ion Implantation Simulation 119

D Magnetometer Prototype: Schematic and BOM 123

E Gravity Gradient 125

F Ordered SOI Wafers 129

G Data Sheet: Magnetometer Sensor (HMC1021) 131

H Data Sheet: OP AMP (AD8552) 147

I Data Sheet: A/D Converter (LTC1285) 169

J SGP MatLab Implementation 195

K MatLab Files 205

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Chapter 1

Introduction

Today there is an increasing demand for satellite based systems – both fromthe industry and the scientific society. Examples are:

• Communication satellites that enables people in development countriesand remote areas to get access to e.g. the internet.

• Small satellites for use as probes ejected from larger spacecrafts onscientific deep-space missions.

The possibility of future employment in the space industry combinedwith personal interests in space oriented topics lead to the foundation ofa student satellite project at the Technical University of Denmark (DTU).Approximately 40 students currently participate in the phase of design andconstruction of a pico-satellite named DTUsat. These students come fromdifferent institutes and are divided into several groups. The ACDS group’spart in the DTUsat project is to evaluate and design the attitude controland determination system.

The main reason for employing a control system depends on parameterssuch as radio link, needed available power and payload(s):

• Signal to noise ratio of the radio link can be improved by orientingantennas with some directivity toward the ground station.

• Better utilisation of solar cells by increasing the effective illuminatedsolar cell area – hereby increasing the available power.

• Generally, payloads require a stabilised satellite platform or at leastprecise attitude determination for use as measurement references.

A stabilised platform can be achieved by using passive attitude stabilisa-tion or active control. Fine-pointing cannot be accomplished with passivelystabilised satellites, but in return they are simple and subject to less pos-sible errors since active interactions is not needed. The opposite applies

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for satellites using active control since they rely on attitude determinationsensors and actuators which – if needed – allows the designers to implementfine-pointing and manoeuvrability.

Launch costs – which is the driving factor of space projects – can bereduced considerably by lowering the satellites’ size and mass since moresatellites then can be launched simultaneously. The outlines for DTUsat isbased on the CubeSat concept – a collaboration between Japanese and USuniversities – introduced by Professor Bob Twiggs, Stanford University. Thisconcept provides opportunities for especially students to take part in designand construction of small satellites obeying a common standard regardingdimensions, mass, and electrical connections to ensure both electrical andmechanical stability. During launch the CubeSats are situated in a p-pod,which is ejected from the carrier rocket when the correct orbit is reached.When attitude stabilisation is obtained the satellites are ejected from thep-pod.

Since most currently available sub-systems are intended for use on e.g.large communication satellites they cannot be accommodated for a CubeSat.For the ACDS this leads to a demand for miniaturisation of the attitudestabilisation system.

This report includes the requirements capture for the ACDS of DTUsat,and a general problem analysis of the ACDS which concludes that activecontrol is needed for the DTUsat mission. Since the sensors and actuatorsneeded for this system has to be custom designed, this has been the mainfocus of this midterm project. The selected sensors for DTUsat are sun sen-sors and a magnetometer, and magnetotorquers as actuators. Specificationsfor the sensors and actuators are based on implementations of environmentmodels.

Analyses of different solutions for both the sensors are carried out, andthe best suited are selected. For the sun sensor an integrated two-axis angleslit sensor is selected, and magnetoresistive sensors for the magnetometer;thorough designs of these sensors are given. Masks and a process sequencefor the sun sensor is developed for the fabrication of the sensor chip, andthe first prototype of the magnetometer is completed and tested. Design ofthe magnetotorquers is carried out.

The attitude control part is mainly an overview of the general principleand system to be implemented. The control system is designed by the ACDSteam’s fourth member Torsten Lorentzen, and can be found in [18].

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Chapter 2

Requirements & Constraints

The requirements and constraints for the ACDS1 on DTUsat are stated inthe following.

2.1 Constraints

Since DTUsat is based on the CubeSat concept maximum permitted massis 1kg, and dimensions are required to be 10cm×10cm×10cm [23]. A mass,volume, and power budget has been made for the satellite, and the followinghas been assigned to the ACDS:

• ACDS PCB: 23g, 6cm × 6cm and a total height of 12mm.

• Actuators (not on ACDS PCB): 36g, volume should be minimised2.

• Sensors (not on ACDS PCB): 12g, volume again to be minimised.

• Attitude sensors: 10mW average.

• Actuator: 150mW during de-tumbling, 50mW in normal operation.

2.2 Preliminary Payload Requirements

DTUsat has two payloads: a tether and a camera. The preliminary require-ments from these to the ACDS are:

Tether

Before deployment:

• ±10◦ attitude pointing in the orbit plane.1ACDS: Attitude Control and Determination System.2Final dimensions are to be discussed with the structural design group.

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• As low rotation as possible outside the orbit plane when the tether isejected. A go signal is desired when the satellite is stable enough fortether ejection.

During deployment:

• Attitude control should continue with updates of centre of mass andmoments of inertia as the tether deploys further. This should minimisetumbling due to the deployment.

• Attitude change of 15◦ − 30◦ in orbit plane.

After deployment:

• Damping of rotation about the tether axis.

• If possible active control of the rotation about the tether is wanted.

Camera

The requirements from the camera group are frequently changing. The latestrequirements are:

• With respect to a fixed point on the ground the rate of the satelliteshould be less than 0.005 rads .

• Attitude estimate to enable the camera software to take pictures ofinterest.

2.3 Interfaces

Requirements for interfacing with other sub-systems are:

• Solar Panels Interface – Access to charge currents in order to allowthe solar panels to be utilised as coarse sun sensors.

• Battery Interface – Access to battery voltage level, which is to be usedfor the actuators.

• Housekeeping data from ACDS – Attitude estimates. Geomagneticfield estimate. Control status/values for debugging purposes. Distur-bance moment acting on the satellite; this can be calculated from thecorrections performed to counter act the moment3.

3This interface is not required, but wanted by the tether group if time permits it.

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Chapter 3

General Problem Analysis

The ACDS has as its name suggests the responsibility of making sure thatthe satellite has the correct attitude in space. This task can be divided into:

• Initial Acquisition – de-tumbling of the satellite which brings the at-titude of the satellite into a predefined set of conditions. The initialacquisition can also be used as a safe mode1 function.

• Operation – more precise control of the satellite’s attitude. The re-quirements for this function are payload and system dependent.

AttitudeSensor(s)

Payload

Actuator

Data Handling Unit

AttitudeDetermination

ControllerSelect

Operation

InitialAcquisition

Tele Command

Figure 3.1: ACDS overview.

Controller selection is either done autonomously or by user interaction as il-lustrated by the tele command block in figure 3.1. The operational functionsof the ACDS may also be divided into sub-functions:

1The safe mode of a satellite is a mode where only critical subsystems are operatingwith the purpose of allowing communication with the ground station.

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• Attitude Determination Sensor(s).

• Attitude Control Actuator(s).

• Data Handling Unit2 for processing sensor data and control of actua-tors.

Different solutions for attitude sensors and actuators are analysed in thischapter, and the best suitable for the DTUsat mission are selected.

3.1 Attitude Determination

Two kinds of attitude sensors exist: Reference sensors and inertial sensors.A reference sensor determines the attitude relative to one or more objects,e.g. the Earth. Inertial sensors determine the rate by which the attitudechanges and the attitude is then found by integration; the initial value for theintegration has to be calibrated, which is typically done by using a referencesensor.

A number of selected sensors are described and evaluated in the follow-ing.

3.1.1 Earth Sensor

It is intended to keep a constant orientation relative to the Earth. Sensorswith the Earth as reference are hence advantageous. An Earth sensor doesnot need upload of additional information – such as e.g. orbit parametersand date/time – after launch to determine attitude relative to the Earth.

A simple ground detector can be used to detect a unique Earth signature– typically the thermal emission in the infrared range – within the sensor’sFOV3. Another procedure is to detect the increased thermal noise from theradio receiver, however this requires a relative high directivity of the usedantenna and omnidirectional antennas for LEO4 satellites are preferred.

Higher accuracy can be obtained by using contour detection of the Earth– commonly referred to as horizon-crossing sensors. Three orthogonal sen-sors each scans a ring of the space and the found crossings of the horizon –often by detecting changes is the infrared range – can be used to calculateboth the altitude and attitude of the satellite. Fewer sensors are neededif the altitude is well known from modeling of the oblateness of the Earth,or if the relative attitude is kept within the FOV. The main obstacle of thesensor principle is the need for rotating mirrors or lenses, since moving partsin space always are considered problematic.

2A task on the main computer.3FOV: Field Of View.4LEO: Low Earth Orbit.

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3.1.2 Sun Sensor

Sun sensors are used for measuring the angle of incidence of sun rays. Thisis usually done using the relation between generated current and angle ofincidence, or by utilising shadow effects. Sensor designs differ by being ableto detect a sun direction cone or a sun direction vector.

The satellite’s solar cells could be used as a coarse sensor for determininga direction cone. However, since the obtainable accuracy cannot be used forreliable attitude determination, it is desirable to have dedicated sun sensors.These sensors come in many variants, but often with a mass of 150g or moreand a considerable price tag [2]; an example of such a sensor is given in [11].Smaller low mass solutions are available in the form of simple photo diodesetc. requiring additional interface circuits.

Analog sun sensors are available with an accuracy of 0.5◦ [7], and digitalsensors with an accuracy better than 0.017◦ [22].

Sun vectors can be used to calculate attitude if the time/date and orbitparameters are known. Till these parameters have been updated it is onlypossible to use the sensors as rate sensors for de-tumbling.

The reflection from the Earth – known as the albedo effect – introducesnoise on measurements, and care must be taken to minimise the resultingerrors.

3.1.3 Magnetometer

The magnetic field of the Earth can for LEO satellites be used for attitudedetermination – especially relevant after the recent measurements from theØrsted satellite mission. The magnetometer measures the magnetic fieldvector in the satellite frame.

If orbit parameters and time/date are known, the direction of the mea-sured field vector can be compared to the modeled direction of the Earth’sfield at the calculated position in the orbit. The spherical harmonic modelof the magnetic field of the Earth is commonly used, either directly or as alookup table depending on available processing time and program memory.

The magnetometer can be used as rate sensor without any orbit param-eters or time/date – e.g. in the first days after launch – for low rotationspeeds by using numerical differentiation to perform de-tumbling. If theorbit passes in the vicinity of the magnetic poles and the measured fieldvectors are logged, the orbit period can be calculated on basis of data froma few orbits. The passing of the South American magnetic anomaly willalso be detectable and can be used as a reference point, giving the ACDSan approximated attitude reference for obtaining a certain pointing relativeto the Earth.

The accuracy of a magnetometer depends on the used sensor, but a reso-lution of 0.5nT have been achieved. If the magnetometer is used for attitude

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determination, the resolution of both the magnetometer and the field modeldetermine the attitude accuracy. With a magnetometer resolution of 0.5nTaccuracy is determined by the resolution of the spherical harmonic modelof approximately 50nT; the maximum attitude accuracy is approximately0.16◦.

3.1.4 GPS

GPS5 is very popular for terrestrial use when determining longitude, lati-tude, and altitude, and are available at very reasonable prices. The GPSsystem consists of 24 satellites with an orbit altitude of 17500km, each car-rying a rubidium frequency standard and transmits time synchronisationcodes at 1.57542GHz.

LEO satellites are orbiting below the GPS satellites and can use thereceived signal for both position and attitude determination. The dopplershift due to the high velocity (∼ 7km/s) of LEO satellites can be up to±60kHz and poses strict requirements to the receiver with regard to lockingspeed and decoding, thus making it impossible to use GPS receivers designedfor terrestrial use. Attitude determination utilises phase shifting between anumber of antennas situated 1 − 5 metres apart; an accuracy of 0.05◦ canbe achieved.

De-tumbling and pointing operation can be initialised immediately af-ter release, if tracking of GPS satellites is unnecessary or can be done au-tonomously. The GPS signal contains accurate date/time information, mak-ing calibration of an on board clock very easy.

3.1.5 Star Imager

Many satellite missions today use a star imager for attitude determinationdue to its high accuracy (< 0.001◦). The star imager consists of a sensitivecamera and a star map with more than 1000 visible stars. High accuracy canbe obtained because of the relatively long distances to the reference objects.Comparing the star image and the map requires extensive data processing.

Attitudes calculated from the position of the stars can in most cases beregarded as inertial, however orbit position still has to be known if a fixedpointing with respect to the Earth is to be achieved. As with some of theother sensors this requires orbit and date/time information uploaded afterlaunch. A star imager will at high angular velocities e.g. when tumbling,operate with reduced accuracy due to blur and might not be usable forde-tumbling.

The risk of direct sunlight reaching the camera chip and lens is a problemthat has to be considered, since both camera saturation and permanentdamage can be the result.

5GPS: Global Positioning System.

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3.1.6 Rate gyro sensors

Rate gyro sensors measure the rate of the satellite attitude. The currentattitude is obtained by integration and by using an initial attitude providedby a reference sensor. Rate sensors are mainly used during fast attitudechanges where numerical differentiation of reference measurements no longerare reliable/adequate.

There is no need for neither accurate date/time or orbit informationwith a rate sensor, as long as the initial value can be obtained. Rate sensorsare well suited for de-tumbling where measurements can be used directly ascontrol signals.

Different principles are employed in rate gyros ranging from coarse piezo-electric vibrating gyros available as small single-chip solutions, to fine ringlaser gyros with considerable size to obtain detectable frequency shifts.

Gyros with moving parts as the piezo-electric are subject to long termdrift, and some sensors need extensive compensation of e.g. temperature.

3.1.7 Sensor Evaluation

Many of the sensors mentioned are either too heavy, or too large to beused for the DTUsat mission. Another problem is the launch and spaceenvironment – e.g. vibrations and radiation – which makes some sensorsunfit for the mission. A trade off between the properties of the selectedsensors are made to enable selection of the sensors for the DTUsat mission;see table 3.1.

The Earth sensor evaluated is a small horizon scanning sensor with therequired accuracy; such are available on the market for immediate integra-tion via appropriate interfaces.

Commercial sun sensors small enough for the mission are simple photodiodes. The evaluated sun sensor is a slit sensor or similar comparablesensors with designs feasible at DTU.

Most magnetometers are too large for DTUsat, however smaller plug-in modules do exist; these are mainly aimed at terrestrial navigation andcannot withstand the vibrations during a launch. The evaluated sensor isa design utilising similar small chip sensors in a more mechanically stableconfiguration.

GPS for attitude determination is still in the test phase, and small re-ceivers and antenna phase arrays are not available. A miniaturised systemhave to be specially developed for the mission. The following evaluation iscarried out with respect to such a system.

High accuracy star imagers are the preferred fine pointing sensors, butthey are still relatively large. A star imager with medium accuracy willcomply with the requirements and possibly reduce the size. A star imagerfeasible at DTU of this type is the basis of the evaluation.

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Integrated piezo-electric rate gyros or other comparable gyros are re-garded as feasible rate sensors and used for the sensor evaluation.

Weight Earth Sens. Sun Sens. Mag.Meter GPS Star Imager Rate Gyro

Mass 2 - ++ 0 - -- 0Size 1 - ++ 0 -- -- 0Power 2 - ++ + - -- -Accuracy 1 - + + ++ ++ --Feasibility 3 0 + ++ -- -- ++Const. Risk 2 ++ - + -- - 0Operation Risk 3 -- + ++ - -- ++De-tumbling 2 + 0 + - -- ++Add. Sensor 2 ++ + - ++ ++ --

Result -2 17 17 -15 -22 8

Table 3.1: Trade off between the different attitude sensors.

3.2 Attitude Stabilisation and Control

Achieving attitude stabilisation or control can be done in various ways; anumber of the most commonly used methods are described and evaluatedin the following.

3.2.1 Gravity Gradient Stabilisation

The gradient of the gravitational force can be utilised for attitude control,since torques as a result of different moments of inertia will tend to align theaxis of least inertia with the field direction. This method exhibit two stableconditions given by the mentioned alignment, and an unstable equilibriawith the axis of least inertia orthogonal to the field direction.

A counterweight is typically fitted to the satellite via a boom or string toincrease the difference in moments of inertia, thereby increasing the align-ment forces and minimising misalignment due to disturbances from e.g.drag.

Damping of the system can be both passive and active. Passive dampingcan be achieved with e.g. hysteresis rods or a conducting sphere generatingEddy currents. Active damping is typically done with magnetotorquersinteracting with the magnetic field of the Earth.

The mass distribution of DTUsat will be fairly uniform implying theneed for ejecting some mass after launch.

3.2.2 Permanent Magnet Stabilisation

Mounting one or more permanent magnets on the satellite will try aligningthe satellite with the magnetic field of the Earth; a very simple and reliable

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control method. The problem with this type of control is the introductionof tumbles when passing the magnetic poles.

Risk of spin up around the alignment axis is a serious problem, sincethis increases the gyro effect and results in misalignment with the field.Active and passive damping is typically carried out with interaction withthe magnetic field of the Earth using hysteresis rods or magnetotorquers.The problem of both is their capability of removing spin which decreases asthe magnetic alignment is improved, thus making it impossible to removerotation completely.

3.2.3 Magnetotorquer Control

Current carrying coils generate magnetic dipole moment; this interacts withthe magnetic field of the Earth and results in a torque on the satellite. Thistorque can be adjusted by controlling the (mean) current in the coil – thiscan be used for active damping.

Control similar to the permanent magnet control with active dampingcan be achieved with only one magnetotorquer, while full three-axis controlrequires a minimum of three magnetotorquers. If three-axis control is usedonly two axes can be controlled at a given point: Pitch and yaw can becontrolled at Equator, while pitch and roll can be controlled while passingin the proximity of the Earth’s magnetic poles.

An accuracy of ±10◦ is obtainable with magnetotorquers [7].

3.2.4 Fly Wheel Control

Newton’s third law is utilised when using fly wheels for attitude control:When applying a control torque on the wheel an equally large torque in theopposite direction will be exerted on the satellite.

Unwanted angular momentum of the satellite can be transferred to thefly wheel by changing the spin of the wheel. Angular momentum dump toavoid saturation of the fly wheels (maximum speed) are typically done withmagnetotorquers interacting with the magnetic field of the Earth.

Three-axis control can be achieved with three orthogonal fly wheels. Thiscontrol type is very fast and are used when fine pointing is required.

3.2.5 Actuator Evaluation

Most satellites use customised actuator designs, but e.g. fly wheels can bepurchased in complete three-axis units. These are however often too largeto fit within DTUsat, thus making it necessary to design this actuator typeas well. Trade off between the properties of the selected stabilisation andcontrol types are made to enable selection for the DTUsat mission; see table3.2. The gravity gradient stabilisation method is not capable of complying

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to the requirements from the tether payload, and is therefore not includedin the detailed evaluation.

Weight Pas. Damped Mag. Act. Damped Mag. M.torquers Fly Wheels

Mass 2 ++ + 0 -Size 1 0 + - --Power 2 ++ + + -Precision 2 0 0 + ++Feasibility 3 0 ++ + --Construction Risk 2 -- - 0 +Operation Risk 3 ++ + 0 -Result 10 12 6 -9

Table 3.2: Trade off between the different stabilisation and control types.

3.3 Selection of ACDS Concept

3.3.1 Sensor Selection

The trade off between the different sensors in table 3.1 shows that the sunsensors and the magnetometer are the most suitable sensors for the DTUsatmission. The expected accuracies of these are less than 1◦, which fulfil therequirements from the payloads.

A design decision had to be made shortly after project start in the sum-mer of 2001, if final integration is to start in the upcoming summer. Thetrade off table at that time was in favour of a magnetometer with additionalsun cone sensors; later in the project, when aspects of the sun sensor hadbeen further investigated, the final trade off in table 3.1 was made.

The final design includes both sun sensors and magnetometer, thusachieving the possibility of full redundance. The control system design car-ried out by Torsten Lorentzen [18] utilises the magnetometer for rate sensingwith respect to the local geomagnetic field, and field strength measurementin the spacecraft frame.

3.3.2 Actuator Selection

The actuator trade off in table 3.2 is in favour of permanent magnet stabil-isation with either passive or active damping. However, these designs arenot capable of fulfilling the requirements from the tether payload during itsdeployment. The use of a permanent magnet also introduces problems withthe dynamic range of a onboard magnetometer.

Initial requirements from the camera group were coarse enough to allowdamped permanent magnet stabilisation.

This being said the magnetotorquers are the best remaining design, al-lowing both simple two-axis and more advanced three-axis control. The

12

simple two-axis controller operates very similar to an active damped per-manent magnet, but allows rotation of the permanent offset field to meetthe tether requirements; the controller designed by Torsten Lorentzen [18]operates using this principle. Tumbling at the poles can also be avoidedwith the magnetotorquers.

The latest requirements from the camera group are however more strictthan the initial requirements; see chapter 2. The new requirements cannot befulfilled with magnetotorquers: If the satellite is stabilised with an attitudefixed in inertial coordinates, the satellite will rotate with respect to a fixedpoint on the surface of the Earth.

Earth

½ v

l = 7.56km

r =

600k

m

Figure 3.2: Surface point pointing error due to orbit motion.

The orbit of the satellite is expected to be circular with an altitude of∼ 600km, which corresponds to a tangential velocity of v =

√

GM/r =7.56km/s. If the linear approximation depicted in figure 3.2 is used, therotation required in a 1s period to point the satellite towards a fixed surfacepoint is:

v = 2 arcsin

(

12 l

r

)

= 2 arcsin

(

12 · 7.56km

600km

)

= 12.6 · 10−3rad

This illustrates that it is infeasible to fulfil the camera requirement of0.005 rads , since this requires a constant pitch rate of 0.0126

rads ± 0.005 rads .

In order to meet the latest camera requirements momentum wheels areneeded. The requirements are however still very hard to fulfil with momen-tum wheels, and in the current stage of the design a dramatic change of theACDS is not realisable. If momentum wheels were to be implemented thenmore mass would be needed from the mass budget, which is impossible atthe current state.

Since the new camera requirements cannot be fulfilled they are disre-garded in the remainder of the report.

13

14

Chapter 4

Problem Statement

The initial scope of the midterm project covers all aspects of a satelliteACDS. Subsequently one or more specific focus areas must be selected toallow in-depth treatment of these.

Some initial analyses were carried out to enable a well-founded selectionof focus areas:

• Capture of ACDS requirements.

• Overall design of ACDS.

• Selection of suitable sensors and actuator.

Based on these a number of possible focus areas were outlined, each with anextent making it interesting for further treatment:

• Design of dedicated ACDS hardware.

• Environment and system modeling for simulation.

• Design of attitude controller.

• Implementation of ACDS drivers and controller in relevant program-ming language.

The first two issues were selected with main focus on sensor hardware de-signs. Actuator design and environment modeling were selected as secondaryfocus areas in order to obtain a complete ACDS suited for flight; design ofthe attitude controller was executed by Torsten Lorentzen [18], and the im-plementation of the ACDS was left open for later treatment.

The selected sensors both have to be designed and produced since useablesun sensors and magnetometers are not available on the commercial market.Same applies for the selected actuator, since magnetotorquers and drivercircuits generally are custom designed.

15

Environment modeling was performed to achieve an estimated distur-bance torque, and to develop principles of the attitude determination algo-rithms needed for the specific sensors.

Design goals for the sensors are minimum accuracies of 1◦. For themagnetometer this corresponds to an accuracy of sin(1◦)Bmin ≈ 300nT.Design goal for the actuator is a lower limit of generated control torque of0.5µNm [7].

4.1 Criteria of Success

The DTUsat carries two payloads, but since this satellite is the first of itskind developed at DTU and it applies many non-space-proven technologies,then the implied technology mission is equally important for the DTUsatproject.

The following ordered list of success levels applies to the ACDS:

1. Receiving unambiguous sensor information from both sun sensors, so-lar panels, and magnetometer. Achieving this level implies fully oper-ational attitude determination hardware.

2. Brief test of magnetorquers by detecting field changes with the mag-netometer to insure full operation of the actuator hardware.

3. Receiving full attitude information after having updated the TLE dataand clock for the on board attitude software. Receival of such infor-mation implies a fully working attitude determination software.

4. Initial acquisition for de-tumbling of the satellite, which is the firsttest of the complete ACDS – both software and hardware.

5. Full attitude control in preparation for deployment of payloads as thefinal test of the ACDS.

6. If a fine pointing control algorithm has been developed at this pointof time, it could be tested prior to deployment of payloads, and ifsuccessful be put into operation. This could allow better stability ofthe satellite, and thus better results of the payloads.

Levels 3 and 4 can be interchanged if the TLE information is not availabletill later in the mission.

If levels 1 and 3 are achieved, then the sensors will be candidates forfuture DTUsats and other small low budget satellites.

16

Chapter 5

Environment Modeling

This chapter describes the different models used in the design of the controlsystem for the satellite. Torsten Lorentzen has implemented the controlsystem [18] and have therefore also contributed to the content of this chapter.

5.1 Orbit Model (SGP1)

It is possible to predict the position and velocity of low Earth orbiting objectswith a good accuracy by solving a two body problem if the most importantperturbations is accounted for; e.g. drag. In order to predict a satellite’sposition a reference point and the elapsed time is required. NORAD2 keepstrack of all objects above a certain size currently in orbit, and informationin form of TLE3 sets are available for the public.

Elements in the NORAD TLEs are mean values of the classical orbit pa-rameters generated by removing periodic variations; it is therefore necessaryto use a prediction method that reconstructs the variations in a way com-patible with the way they are removed. Five methods that can be used arepresented in [14] with included Fortran source code. A MatLab version ofthe first and simplest of these methods, the SGP method, was implementedin this project.

5.1.1 SGP

The MatLab implementation of the SGP algorithm seen in appendix J takesa modified TLE as input argument. The raw TLE is read from a textfile by the MatLab function loadTLE.m seen in appendix J. This functiontransforms the units in the TLE and returns a modified TLE that can beread by sgp.m.

1SGP: Simplified General Perturbation.2NORAD: North American Aerospace Defense Command.3TLE: Two Line Element.

17

The SGP algorithm solves Keppler’s equations by numeric iteration andcorrects the result by reconstructing the periodic variations. The resultingpositions and velocities from the SGP is given in Earth radii and Earth radiiper minute, which is transformed to kilometres and kilometres per secondin the runSGP MatLab script seen in appendix J.

The major difference between the MatLab and the FORTRAN imple-mentation in [14] is, that the MatLab version takes a time vector and cal-culates the orbit parameters for the entire vector. The FORTRAN versiontakes a single time and calculates the orbit parameters for that time only.The MatLab implementation was verified using the TLE example in [14],and results match the results from the FORTRAN implementation with sixdigits accuracy.

5.2 Magnetic Model

A precise model of the Earth’s magnetic field is required in order to performprecise attitude determination with a magnetometer. The magnetometermeasures a magnetic field strength vector, and this is compared with a ref-erence map of the Earth’s magnetic field to determine the attitude. Whenthe satellite’s location is known from orbit prediction, it is possible by com-paring the measured magnetic field vector with the model to retrieve thesatellites attitude.

The model used on DTUsat is a IGRF20004 model, which is the in-ternational standard model recommended for scientific use by the IAGA5

from the year 2000 and forward. The model consists of a series of truncatedspherical harmonic series describing the main field and its variations. It isbased on empirical data from several different sources including land basedmeasurements, ships, planes and satellites [15]. The current version of themodel with epoch in year 2000 has been adapted to the new precise mea-surements from the Ørsted satellite.The IGRF model is defined as:

V = a

N∑

n=1

n∑

m=0

(a

r)n+1(gmn cos(mφ) + h

mn sin(mφ))P

mn cos(θ) (5.1)

where V is the geomagnetic scalar potential, a is the mean radius of theEarth, r is the distance from the centre of the Earth, φ is the longitude, θis the colatitude defined as 90◦ minus the latitude, gmn and h

mn are spherical

harmonic coefficients (Gauss coefficients) of degree n and order m. N isthe maximal spherical harmonic order of the series expansion. P mn cos(θ)is the Schmidt quasi-normalised associated Legendre functions of degree n

4International Geomagnetic Reference Field, year 2000.5International Association of Geomagnetism and Aeronomy.

18

and order m calculated from (5.3).The associated Legendre functions are normalised with the constants:

Kn =

12n ; m = 0

12n

√

2(n−m)!(n+m)! ; m > 0

(5.2)

which gives the normalised associated Legendre functions:

Pmn =

( 12nn!)(d

dx)n(x2 − 1)n ; m = 0

√

2(n−m)!(n+m)! (1 − x2)m(

12nn!)(

ddx

)n(x2 − 1)n ; m > 0(5.3)

The model coefficients that are determined from empirical data are listedin appendix K. The model makes it possible to calculate the magnetic fieldstrength vector for any given set of geocentric spherical coordinates r, φ andθ.

5.2.1 Implementation

The IRGF-model is implemented in MatLab by using a S-function whichcalls a compiled version of the C implemented model. The C program wasmade by the Ørsted satellite team and has generously been at our disposal.

Figure 5.1 shows the geomagnetic field strength generated by the model.The field strength is calculated for an orbit altitude of 600km – the expectedaltitude for DTUsat.

5.3 Sun Positioning Model

It is necessary to implement a sun positioning model in order to determinethe satellite’s attitude from the measured sun vector in the satellite frame.The model to be implemented on DTUsat was developed by Paul Schlyterin 1979 [21]. Torsten Lorentzen from the ACDS group has implemented themodel in MatLab.

The model determines the position of the sun in the inertial referenceframe from known orbit parameters of the Earth’s movement around the sun.The time relative to epoch is the only input taken by the model, implyingthat it is necessary to achieve radio contact prior to the utilisation of thesun sensors, since everything in the satellite will be disabled during launch– including the real time clock. The obtainable accuracy depends on theprecision of the clock and is estimated to be below one arc minute, which isless than the sun sensors’ expected maximum accuracy; see chapter 6.

The vector from the TLE and the sun model gives the vector from thesatellite to the sun, which combined with the vector obtained from the sunsensor give the attitude.

19

Figure 5.1: Geomagnetic field strength

5.4 Gravitation Gradient Model

Another moment to be modeled is the gravitational gradient which is atorque acting on the satellite from all larger and closely placed masses inthe solar system. However it is shown in the following calculations that thetorques originating from the moon and the sun are negligible.

|F | =GMr2

dm ⇒

|F |Earth = G5, 972 · 1024kg

(6, 371 · 106m + 600 · 103m)2 dm ≈ 1, 2 · 1011Gdm

kg

m2

|F |Moon ≈ G7, 35 · 1022kg

(3, 844 · 108m)2 dm ≈ 5, 5 · 105Gdm

kg

m2

|F |Sun ≈ G1, 989 · 1030kg

(1, 496 · 1011m)2 dm ≈ 8, 9 · 107Gdm

kg

m2(5.4)

It is justified from the above calculations that only the gravitational momentfrom the Earth is taken into consideration since the contribution from themoon and the sun are in the magnitude of 104 to 106 less than that of theEarth.

An asymmetric satellite subjected to a gravitational field will experiencea torque tending to align the axis of least inertia with the field direction.

The derivation of the expression for the Earth’s gravitational momentson an asymmetric spacecraft can be found in appendix E, where it also is

20

shown that the maximal moment is given by:

|G|max =√

Gx2 + Gy

2 + Gz2

<3µ

2R03

√

(Iz − Iy)2 + (Iz − Ix)2 + (Ix − Iy)2

For simulation purposes the moments of inertia for DTUsat have beenestimated. This is done with a 230g chassis, 4 point masses located on thecentre of four panel sides (total 385g), and 385g uniformly distributed in asphere within the satellite; see [12] for more details. This simple model yields[Ix Iy Iz] = [1.552 1.552 2.164] · 10−3kgm2, which then gives a maximumgravity gradient of:

|G|max <3 · 3.98603 · 1014 m3

s2

2 · (600km + 6371.0km)3√

6122 + 6122 + 02[·10−6kgm2]

= 1.528 · 10−9Nm

Consequently the worst-case situation can be simulated by employing|G|max as a harmonic disturbance along the different axis separately. Theeffect applied on DTUsat by the gravity gradient6 is of insignificant magni-tude and is consequently thought of as a minor disturbance in the controlsystem.

6Due to its small dimensions and mass.

21

22

Chapter 6

Attitude Determination:Sun Sensor

In 3 General Problem Analysis it was decided that one of the attitude deter-mination sensors should be a sun sensor developed especially for the project,since the commercially available sensors are very expensive and often quiteheavy.

This chapter consists of theory, analysis of different design ideas, anddesign of the selected sun sensor with its associated circuits.

6.1 Sensor Types

How sun sensors work, and how they may be realised is the topic of thissection. Different design ideas are analysed and qualified sensor types arefound.

6.1.1 Principle of pn junction Solar Cells

A solar cell is a pn junction device that converts sun power (from photons)into electrical power. There are several photon-semiconductor mechanismsthat allows the conversion, but the most significant one is the interactionwith valence electrons [20]. The energy of a photon is given by

E = hν

where h is Planck’s constant, and ν the frequency. If the energy of a photonis larger than the bandgap energy Eg the photon might interact with avalence electron and elevate it into the conduction band. The probability ofthis happening is very high since the valence band contains many electronsand the conduction band many vacant places. The part of the energy aboveEg is converted into heat. If the photon energy is lower than the bandgapenergy the photon is transmitted through the semiconductor.

23

Si Si Si Si

Si Si P Si

Si Si Si Si

Si Si Si Si

Si Si B Si

Si Si Si Sie-

-

+

a) b)

Figure 6.1: Doped Si lattices. a) n-type material doped with electrons fromthe often used donor phosphor. b) p-type material with the often usedacceptor boron; the figure shows a case where an electron gains a smallamount of energy and moves to an acceptor.

In a pn junction there is one electron per donor ND that are not in acovalent bound (Si ∈ group IV, donor ∈ group V), and for each acceptor NAone extra electron can be bound in a covalent bound (acceptor ∈ group III);cf. figure 6.1. When a Si crystal is doped (figure 6.2 a)) donor electronswill diffuse from the n-material to the acceptors in the p-material causingan electric field as seen in figure 6.2 b). This diffusion continues till thediffusion force on the electrons equals the opposite directed force from theelectric field – thus creating a permanent electric field in the region knownas the depletion region.

N

e

D+

-

N

h

A-

+

N

e

D+

-

N

h

A-

+

a) b)

ND+NA

-

E-fieldnp np

Figure 6.2: a) pn junction. b) pn junction and depletion region.

Having clarified these physical properties the principle of a pn junctionsolar cell can easily be described (cf. figure 6.3):

• a) Sun light hits the pn device, which causes elevation of electrons tothe conduction band.

• b) The electric field in the depletion region causes a reverse bias cur-rent Iph.

• c) Iph results in a voltage drop V across the load R. This forward biaspotential decreases the E-field in the pn device, but the field does notreach zero or change direction.

• d) The potential causes a forward bias current Ifor. Since the pnjunction solar cell is a diode the current Ifor can be found with theideal diode equation: Ifor = Is[exp(

eVkT

) − 1], where e is the charge ofan electron, k Boltzmann’s constant and T the temperature.

• e) I = Iph − Ifor is then the net pn junction current.

24

a

np

b

E-field

Iph

Sun light

R

+ V –c

Ifor Id e

Figure 6.3: A pn junction solar cell.

The efficiency of a solar cell is defined by

η =PoutPin

where Pin is the sun power and Pout is the output power of the solar cell.

According to [13] it should be possible to obtain η = 0.16 in MIC’s clean-room for a simple pn junction solar cell. Since the solar cells in this projectare to be used as sensors this is sufficient. In practice better efficiencies areobtainable by using light trapping surface structures and multijunctions,which utilises photons with longer wave lengths. Figure 6.4 shows a part ofa multijunction silicon solar cell. The figure also illustrates how precise asmall scale sunsensor can be constructed.

Figure 6.4: Multijunction solar cell. Reproduced from [17].

Solar Cell’s Angle Dependency

The generated current in a solar cell is given by:

I =PA

Aeff

Vη

where PA

is the sun power per square metre, Aeff the effective illuminatedarea which the solar cell experiences, V the diode forward voltage and η

25

is the efficiency of the solar cell. If the solar cell has side lengths a and b,the generated current for angle variations around a and b can be found viafigure 6.5:

I =PA

ba cos(αa) cos(αb)

Vη = I0 cos(αa) cos(αb) (6.1)

Equation (6.1) could give the perception that the only variable for thegenerated current is the angle of incidence. However this is not true since Vis a function of the sensor temperature. Minor fluctuations of the sun powercan of course also occur.

In the coming sections the simpler one dimensional equation I = I0 cos(α)might be used instead of (6.1) in order to clarify principles.

a

α

a cos(α )

a

b

αbb

b

Figure 6.5: Effective illuminated area for angle variation around b.

6.1.2 Analysis of Sun Sensor Principles

Sun sensors can basically be divided into two categories, namely:

Angle sensors which measure the angle of the sun rays hitting the activesensor area. This sensor method requires that at least one of thesatellite’s sensors must be exposed to sun light within the sensors’FOV1 at all times; cf. the calculations below.

Intensity sensors which only measure the intensity of the sun power;hence utilising that the generated current depends on the angles ofincidence by I = I0 cos(αa) cos(αb). This approach requires three sen-sors having mutually different angles to be exposed to sunlight at alltimes; cf. Simple Intensity Configurations below. This method hasstringent requirements to both the placement and the FOV of thesensors.

It is desired to be capable of calculating a sun vector whenever thesatellite is exposed to sun light. To do this dead angles are not allowed,which requires the sensors to have a FOV angle enabling the sensors tocover a closed sphere.

1FOV: Field of View.

26

α w c

αFOV

αFOV

Figure 6.6: FOV angles for two complete sensors and worst case sun ray fora cubic configuration with one sensor on each side of the satellite

Figure 6.6 shows the FOV-angle for two complete two-axis sensors. Thefigure also shows the worst case angle of which a sun ray may hit a cubicsatellite equipped with one two-axis sensor on each side – this angle is:

αwc = arctan

(√a2 + a2

a

)

= 54.74◦ ≈ 55◦

This angle corresponds to the required FOV-angle if one two-axis anglesensor is placed on each of the satellite’s six sides; using sun sensors withαFOV ≥ 55◦ will allow complete coverage.

Doing similar calculations for a configuration with intensity sensors is notequally straightforward since there are more requirements to the placementsof the sensors; cf. the above statements.

In the following different ideas for small low weight sun sensors will beexamined.

Simple Intensity Configurations

The most simple option available to calculate a sun vector is to look atthe charge currents from the satellite’s individual solar cells. The currentgenerated in a solar cell is a function of the sun power exposed to it, andthereby a function of the angle; cf. section 6.1.1. Because of the angledependency the current value of one solar cell tells us, that the sun rays areparallel with the boundary of a distorted cone – with circular solar cells thesun rays would be parallel with a regular cone. If the sensors on figure 6.6are seen as circular solar cells the cones on the figure can be thought of asthe mentioned cones.

Clearly, it is not enough to know that the sun vector lies on a coneboundary – in order to obtain a more accurate sun vector it is necessary tomeasure and calculate cones from other illuminated sides. This enhances the

27

resolution since the sun vector must be located in the intersection betweenthe different cones, and from this idea one can easily realise that in orderto obtain reasonable resolution it is needed to have a small intersection be-tween at least three cones. Thus high resolution requires careful placement(location and mutual angles) of the sun sensors, which is often not possiblewith solar cells since their position is determined from a maximum powerpoint of view; this clearly disqualifies the use of solar cells as reliable at-titude determination sensors. However the simple determination idea canbe implemented with simple hardware if e.g. small photo diodes are usedinstead of solar cells.

a) b) c)

Top view A-A cross section Top view

A

A

Figure 6.7: a) & b) depicts a tetrahedron configuration with photo diodes.c) shows a simple cubic intensity sensor with better resolution for smallangles than a) & b); the illustration shows shadow effect from a light sourcenorth-east of the sensor .

A way of utilising the simple idea with photo diodes could be to placethree sensors as a tetrahedron as depicted on figure 6.7 a) & b). Fixation ofthe sensors is a mechanical challenge, but once this problem has been solvedit should be possible to achieve complete coverage by having one tetrahedronon each of the satellite’s sides. Ultimately, however, it must be concludedthat this design idea cannot compete with the more advanced angle sensorideas listed in the coming sections.

Simple Cubic Intensity Sensors

The simple intensity sensors discussed above have low resolution for smallangles since the generated current I = I0 cos(α) varies with

dIdα = I0 sin(α).

This problem can be solved by introducing another angle-alternating factorfor the current. There exist only one more factor that can be made angledependent: The illuminated area. This area is easily made angle dependentby introducing shadow on the active sensor area as the angle increases. An

28

example of such a device is given in figure 6.7 c).

In practice these sensors will have a smaller FOV, since the illuminatedarea is decreasing for increasing angles. Owing to the fact that the shadowintensity sensors also provides us with cones to find the sun vector they arenot advantageous either.

Slit Configurations

To avoid analysing intersections between cones it is necessary to use anglesensors for better resolution. An often used angle sensor is the two-axis slitsensor which is composed of two one-axis slit sensors; see figure 6.8. The twosensors should be perpendicular to each other, because the output currentfrom the sensors then gives the angles to the sun in the xz- and yz-plane(sensor coordinate system).

In order to make sure each sensor only measures the xz-/yz-shadow-contribution it is required that the illuminated area remains constant forvarying β with α ≤ αFOV . The illuminated area is made shadow indepen-dent of β by ensuring that g on figure 6.8 is large enough. Of course β stillhas an effect on the effective illuminated area, and hereby an effect on Ibecause of (6.1) – however this contribution can easily be removed with atechnique discussed in section 6.2.

x

α

a

c

z

b

d

k

g

g

axz

β

f

Figure 6.8: One-axis slit sensor. Orientation of sensor coordinate system isalso shown.

As in the case with the cubic sensor in the previous section, the slitsensor also has a limited FOV due to especially b in figure 6.8, since bcauses shadows. Of course a large c also contributes to undesirable large f ,which means that c must be chosen relatively small if large FOV is wanted.Another way of avoiding too large f values is by placing another medium– e.g. glass – between the slit and the active area, since this medium will

29

deflect the sun rays:

n sin(α) = nmed sin(αmed) ⇔ αmed = arcsin(

n

nmedsin(α)

)

(6.2)

n = 1 (vacuum), n < nmed

Figure 6.9 shows three different ways of shaping the active sensor area.Note that the sensor in c) utilises gray codes – or else it would be possibleto experience multiple bit flips (∼hazards) during a measurement. Each ofthe shapes in figure 6.9 has advantages and drawbacks:

• a) Advantages: Easy to implement in a simple mechanical solution.Easy to implement in a chip. Drawbacks: Has low resolution for smallangles; cf. Simple Cubic Intensity Sensors above.

• b) Advantages: Good resolution over a large angle spectrum, sincethe illuminated area on each sensor varies with the angle of incidence.Easy to implement in a chip. Drawbacks: Difficult to implement in amechanical solution.

• c) Advantages: Better accuracy can be obtained [22]. Digital outputcan easily be provided by comparing currents from each bit area withthe ATA2 current3. Easy to minimise errors due to albedo from Earthwith a sufficiently low bit threshold. Drawbacks: Impossible to imple-ment in a mechanical solution. Large sensor area is required for highaccuracy.

Option a) is eliminated because its only advantage compared to b) is,that it is easy to realise in a mechanical solution – and since facilities areavailable at MIC to make a simple sensor chip there is no reason to choosea large, heavy, and power consuming solution. The chip implementationfurthermore has the advantage, that there is no problem in placing the twoone-axis sensors perpendicular to each other in the chip.

The digital sensor c) has one major disadvantage, and that is that its di-mensions ascends by O(2n−1) with the desired bit resolution n. If FOV=70◦

and a 9bit resolution are wanted, then an accuracy on 2·70◦

29= 0.27◦ can

be obtained. This would require 29−1 = 256 mask holes for the LSB4.A good choice for spacing between the mask holes, and mask hole size is10µm (section 6.3.4 and [9]), which leads to a minimum side length on256 · 2 · 10µm = 5120µm! Since two one-axis sensor are needed on one chip

2ATA: Automatic Threshold Adjust.3If it is possible to define a low fixed threshold the digital output can be created

by conducting the generated currents through a resistor array; thus having a 0W powerconsumption. A more reliable solution is however to utilise an ATA current.

4LSB: Least Significant Bit.

30

d

e

k

g

g

a

e

k

g

g

h

d

a

a) b)

d'

h

a

c)

h k

g

g

ATA

Sign bit

Figure 6.9: a) & b) Shapes of active sensor area for a analog sensor. c)Digital sensor with mask for eight active sensor areas.

that should be smaller than 1cm2 one might have to settle for 8bit resolution,which gives an accuracy on 0.55◦ and a minimum side length on 2560µm.The required height of c on figure 6.8 for the 8bit solution would be 1581µmwith pyrex between sensor and slit, and 466µm with no medium.

Determining whether solution b) or c) is to prefer for small dimensionsis difficult. It might be feasible to make a sensor of type b) with same sizeand better accuracy than the 8bit version of c).

Complete Two-Axis Sensors

Instead of having two one-axis angle sensors as suggested in previous sectionsit is naturally possible to have just one two-axis sensor. Examples of suchdevices are given in figure 6.10.

a) b) c)

Ω

Ω

Ω

Ω

A B

CD

Figure 6.10: Two-axis sensors.

The sensors in figure 6.10 has an open spot through which the underlyingactive area get exposed to sun rays. The utilisation of the ideas are thoughtof as:

31

• a) with four separate conducting paths from each corner of the activearea to bonding pads. The idea originates from [13]. By evaluatingthe shown resistances it is possible to find out where the light spot hitsthe active area; better performance could be achieved by also using thecross resistances (A-C & B-D). Resistance calculations are extremelydifficult, which means that pre-calibrated characteristics has to be used[9].

• b) which has a CCD5 chip under the shadow mask. With this digitalsensor high accuracy can be obtained [22].

• c) where the sun vector can be computed by comparing generatedcurrent in the four cells.

Option b) can be eliminated immediately, because of limited mass androom for the sun sensor on DTUsat.

a) and c) are good candidates for a simple small sensor, but it is morecomplex to find a sun vector from these sensors than from the triangular one-axis sensor suggested in previous section. It also likely that these sensors willhave lower resolution than the triangular sensor, since they depend on smalldeviations of resistances and on the cosine dependence for the incoming lightintensity respectively.

Non Optical Solutions

It is of course also possible to construct non optical sun sensors, e.g. aMEMS6 solution. Actually a MEMS sun sensor for small satellites has beensuggested by [5]. The principle of this sensor relies on utilisation of thepiezoelectric effect. Figure 6.11 a) shows a piezoelectric beam which willdeflect upwards, when its temperature rises as a consequence of sun lighthitting the beam; it deflects upwards because the thermal expansion ratefor polysilicon is higher than for lead-titanate-zirconate (PZT). The deflec-tion creates a positive voltage across the PZT layer which is measured withgauging electronics connected via the platinum electrodes. The silicondiox-ide serves as a chemical barrier to prevent diffusion of molecules to and fromthe polysilicon layer, and the titanium is used as an adhesion layer. In prac-tice [5] suggests shadow walls around the sensor for shadow mask, but byusing a thin slit mask higher FOV can be obtained; cf. Slit Configurations.

Another idea for a digital MEMS sensor was suggested by [9]. Thisidea is sketched on figure 6.11 b), and the idea is that the light gray areashould be a conductor with higher thermal expansion rate than the beamitself; in practice however it may be necessary to use a structure similarto that of figure 6.11 a). When sun light hits the beam the pick-up is

5CCD: Charge Coupled Device.6MEMS: Micro Electro Mechanical Systems.

32

PZTPt

SiO2Ti PolySi

a) b)

Figure 6.11: Examples of MEMS sensors. a) piezoelectric beam, b) pick-up.

deflected downwards to a conducting path, hence establishing an electricalconnection and one of the sensor’s bit is set. This idea is based on IBMZürich’s Millipede project [28] where data is stored by deforming a polymersurface with a similar pick-up. Figure 6.12 shows the Millipede pick-uparray.

Common for the MEMS solutions is that the development time for asensor to DTUsat – with the same performance as the optical sensors –would be too long.

Figure 6.12: IBM Millipede pick-up array, a future data storage device.Reproduced from [28].

Conclusion

The conclusion of this section has to be, that the best realisable sensors arethe triangular slit sensor and the digital slit sensor; both chip implemented.For the final implementation it has been decided to implement both sensors,so that it is possible to compare their performance. In the analysis it wasfound, that it due to requirements on limited dimensions would not be pos-sible to construct a digital slit sensor with better ideal accuracy than 0.55◦,which indicates that it may be possible to obtain better accuracy with theanalog triangular sensor.

In addition to the slit sensors it was decided also to include two two-axissensors; type a) and c) from Complete Two-Axis Sensors. These sensors willbe included mainly for test purposes, since they are expected to have lower

33

resolution than the slit sensors.Since it is a time consuming task to do mask design etc. it was decided

only to include design of the triangular slit sensor in this project, and thenpostpone the design of the other sensors to another project.

6.2 Sensor Principle Analysis

In 6.1.2 it was decided that the triangular sun sensor should be one of thesensors that will be included on the final wafer. Figure 6.13 shows two waysof implementing the sun sensor. The angle v = 57.4◦ is due to the crystalstructure of silicon, and this angle cannot be changed to anything else thanv = 90◦; 57.4◦ is the plane angle, and v = 90◦ the planeangle.

α

f

a)

a

α

b

c

a

Si

Si

PyrexPyrex

Alυ=57,4°b

c

i

αOα i

f Si

b)

l

Figure 6.13: Implementation ideas.

Comparing the two methods yield:

• a) Advantages: No light is reflected before the active sensor area.Drawbacks: Loss of effective slit length a for α > v (reduction of FOV).Problems with partial transparency of thin thicknesses of silicon on thetop-layer tips. Not rigid.

• b) Advantages: Smaller f is required for large α due to the refractionindex of pyrex; cf. (6.2). Small loss of effective slit length a for in-creasing α since b is in the vicinity of 200nm whereas ba) ∼ 500µm.Very rigid. Drawbacks: Problems with impurities in the pyrex layer.Some light is reflected from the two vacuum-pyrex transitions. Re-quires vents to avoid explosion in vacuum.

For final implementation b) was chosen because of its advantages, andbecause vents (see section 6.3) and light reflections are not regarded asproblems. Light reflection is not a problem since high efficiency is not crucial– this, and since silicon is relatively easy to work with, is also why siliconwas chosen as solar cell material instead of the two-element semiconductorGaAs, which has better optical properties.

34

Elimination of Unwanted Parameters

From (6.1) it is known that the generated current is temperature dependent.This is unwanted since the sensors in worst case are exposed to alternatingtemperatures in the interval −70◦C− 70◦C [7]. The dependence of both αaand αb is also unwanted for the triangular slit sensor, since it is desired tohave a current that is dependent only of the illuminated area, which is afunction of the angle α; cf. figure 6.13.

If the illuminated area of one triangle is denoted At(α) then the currentis given by (cf. (6.1)):

It =PA

At(α) cos(αa) cos(αb)

V (T )η (6.3)

Equation (6.3) shows that in order to eliminate the unwanted parameters

it is needed to divide (6.3) by cos(αa) cos(αb)V (T ) . This division variable can be

obtained by introducing another active area as shown in figure 6.14. Thereference area has a constant illuminated area Aref = am.

e

k

g

g

h

d

a

d'

mdref

Figure 6.14: Triangular slit sensor with reference area.

If the difference between the generated currents in the two triangularcells is used to find α then it can be made independent by dividing with thegenerated current in the reference cell:

∆I

Iref=

It2 − It1Iref

=

PA

cos(αa) cos(αb)

V (T ) η (At2(α) − At1(α))PA

Aref cos(αa) cos(αb)

V (T ) η=

At2(α) − At1(α)Aref

(6.4)

As (6.4) shows, this method furthermore has the effect that the efficiencyfactor η is cancelled out. This means that the method also takes into ac-count that the solar cell areas degrade because of cosmic radiation. The

35

elimination of PA

also has a positive effect since this cancels disturbancescaused by sun power fluctuations, and albedo from the Earth7. These as-sumptions and the assumptions done in (6.4) requires that the three solarcell areas have the same physical properties and temperatures. In practiceit should be possible to fulfil these requirements quite good since the cellsare implemented on the same chip.

The method do however also have a negative consequence which appearswhen the sensor is in shadow. Both the ∆I and the Iref currents are verysmall, and this might yield ∆I

Iref-values that could be interpreted as real

measurements! In order to avoid this the division must be performed inthe software driver or by making it possible to read out both Iref and

∆IIref

.

Apparently it may seems like the software solution is the most preferable,but actually this solution has a disadvantage if there is no stable referencevoltage available since it then – due to the demand for limited hardware – isdifficult to read out both values at the same time. See section 6.4 for furtherinformation.

By using ∆I dark currents from the two triangular areas are eliminated,since the two areas will have approximately the same dark currents. When∆IIref

is used for α-lookup the dark current due to the divisor will however

introduce disturbances since it is a function of the temperature. Neverthelessthis problem is seen as a minor problem compared to the problems themethod eliminates, and because the dark current is insignificant: Idark,ret �Iref .

Conclusion

The slit sensor structure from figure 6.13 b), combined with the referencearea depicted in figure 6.14, is selected for the triangular sun sensor.

6.3 Detailed Sensor Design

In section 6.2 a system of two one-axis slit sensors with triangular sensorareas and a reference area was concluded to be a good candidate as a sunsensor for DTUsat. The dimensions of the sensor, the expected performanceand the practical realisation are the topics of this section.

6.3.1 Electrical Characteristics

Expressions for the characteristics for an ideal triangular slit sensor arederived in this section. These characteristics are then used to determine thedimensions of the sensor. To calculate the current in each triangular area

7Only the intensity contribution is eliminated.

36

an expression for the change in At(α), from equation (6.3), has to be found.Figure 6.15) shows the arrangement of the different variables.

α

b

c

a

i

αOα i

f=2e

a)

l

e

k

a

m

p

qp

u

t

ut

11

2 2

1

1 q

2

2

b)

c = c-l1

I

II

Figure 6.15: a) Slit from side. b) Slit from top; ui, pi, qi and ti values forαi = αo = 0

◦.

For the derivation the following geometric dependence proves to be useful:

k

2e=

tipi

=uiqi

⇒ ti = pik

2e, ui = qi

k

2e

The shadow effect that arises from the thickness of the slit is:

s = b tan αi

For increasing area on I the boundaries of the light beam are:

q1 = (e +1

2a) + c1 tan αo + l tan αi

p1 = (e −1

2a) + c1 tan αo + l tan αi + s

u1 = q1k

2e=

(

(e +1

2a) + c1 tan αo + l tan αi

)

k

2e

t1 = p1k

2e=

(

(e − 12a) + c1 tan αo + l tan αi + s

)

k

2e

This leads to the following expression for the illuminated area:

At1(α) =1

2(q1 − p1)(u1 + t1)

=1

2(a − s)(2e + 2c1 tan αo + 2l tan αi + s)

k

2e(6.5)

which by substitution in equation (6.3) gives:

It1 =PA

12(a − s)(2e + 2c1 tan αo + 2l tan αi + s) k2e cos(αi)

V (T )η(1 − R)

(6.6)

37

where cos(αb) is substituted with cos(αi) and cos(αb) constant at 0◦. The

extra constant 1 − R is due to the reflection R of the two pyrex-vacuumtransitions. Calculations for the situation where the illuminated area in IIis decreasing are similar:

q2 = (e +1

2a) − c1 tan αo − l tan αi − s

p2 = (e −1

2a) − c1 tan αo − l tan αi

u2 = q2k

2e=

(

(e +1

2a) − c1 tan αo − l tan αi − s

)

k

2e

t2 = p1k

2e=

(

(e − 12a) − c1 tan αo − l tan αi

)

k

2e

At2(α) =1

2(q2 − p2)(u2 + l2)

=1

2(a − s)(2e − 2c1 tan αo − 2l tan αi − s)

k

2e(6.7)

It2 =PA

12(a − s)(2e − 2c1 tan αo − 2l tan αi − s) k2e cos(αi)

V (T )η(1 − R)

(6.8)

The reference areas also has some dependence of α since it is given by:

Aref (α) = (a − s)m (6.9)

If equation (6.4) is used for α-lookup, then by substitution the characteristicfor the ideal sun sensor can be obtained:

∆I

Iref=

At2(α) − At1(α)Aref

=12(a − s)(−4c1 tan αo − 4l tan αi − 2s) k2e

(a − s)m

=(−2c1 tan αo − 2l tan αi − s)k

2em(6.10)

The characteristic and the current expressions for the situation when theilluminated area on I is decreasing, and II increasing can be found by mir-roring the obtained expressions around y = −x.

6.3.2 Implementation of the Solar Cell Areas

Since it is required to have the three different active solar cell areas isolatedfrom each other it was decided to use SOI8 wafers. Here a 100% isolation

8SOI: Silicon On Insulator.

38

between the cells can be guaranteed by etching away the top Si-layer be-tween the cells. A SOI wafer is depicted in figure 6.16 and specifications forthe ordered wafers can be seen in appendix F.

The much cheaper pure silicon wafers were disqualified, because they areopen for latch-up between the cells.

Bulk-Si

SiO2SihSi

hOX

hB-Si

Figure 6.16: A SOI wafer.

Since the SOI wafers are expensive some semi-SOI wafers have been pro-duced for test purposes. A semi-SOI wafer is a pure Si-wafer with thermallygrown oxide and sputtered polySi.

6.3.3 Micro-structure Design

Figure 6.17 shows the arrangement of the dimensions that will be determinedin this section. The reason for fastening the pyrex to the Bulk−Si and notthe Si layer is, that when the wafers are anodically bonded to each otherthe voltage drop should be situated across the bonding interface and not theoxide layer, which would be the case if direct bonding is applied betweenthe pyrex-Si layer; see e.g. section 6.3.5.

b

c

a

αO l

a)

α i

f

α ib

c

b)

k+d'

αF1

g g

αF1αF2

PyrexSiTi / AlSensor area

Figure 6.17: Detailed illustration of the slit sensor.

It was decided to use the configuration illustrated in figure 6.18 becauseit minimises the chip area.

The following worst case values will be used in the design: diode voltageV (T ) = 0.7V, solar cell efficiency η = 0.1 (cf. section 6.1.1), and averagesun intensity P

A= 1371 W

m2.

The reflectance R is defined by R = PrPin

, and in [10] it is deduced from

39

m

f

k

g

g

h

d

a

d'

g

g

href

f

a

Figure 6.18: Top view of the chip area.

the Fresnel equations that:

RTE(α) = rTE(α)2 =

(

cos α −√

n2 − sin2 αcos α +

√

n2 − sin2 α

)2

(6.11)

RTM (α) = rTM (α)2 =

(

n2 cos α −√

n2 − sin2 αn2 cos α +

√

n2 − sin2 α

)2

(6.12)

where n = n1n2

, and TE/TM the transverse electric/magnetic polarised lightwaves. In TE/TM the E/B vector is perpendicular to the plane of incidence,and the B/E vector is in the plane of incidence. Light comes in equalportions of TE and TM modes, which means that the total reflectance inthe sensor is9:

R(α)lm =RTE(α)lm + RTM (α)lm

2R(α) = 1 − (1 − R(αi)12) (1 − R(αo)21) (6.13)

Where subscript 12 and 21 means vacuum-pyrex and pyrex-cavity tran-sition respectively. Figure 6.19 shows the above described reflections withn1 = 1 (vacuum) and the refraction index for pyrex n2 = 1.474 [4].

9It is assumed that the reflected light in the pyrex-cavity transition will not be reflectedback in the vacuum-pyrex transition.

40

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Angle of incidence [degrees]

R(%

)

Internalreflection

Externalreflection

TE

TM

Avg.

TE

TM

Avg.

Rtot(α)

Figure 6.19: Reflectance in the vacuum-pyrex transition (external) and thepyrex-cavity transition (internal). The total reflection from (6.13) is alsoshown.

It is very likely that the AD8552 op amp will be used as I-V converterfor the sun sensor signals; see section 6.4 & appendix H. AD8552 has a max-imum input bias current of 1.5nA and a maximum offset current of 200pA,which means that a worst case leakage current is 1.6nA. To ensure high res-olution it is required to have

∆Iref∆α

∣

∣

∆α=1◦≥ 10 · 1.6nA/◦. Calculations show

that the ∆It easily fulfil this while Iref has problems fulfilling the require-ment since the current in the reference area only is a function of cos(α).

Iref varies slowest around α = 0◦ since

dIrefdα = I0 sin(α), which means that

equation (6.14) has to be fulfilled; the shadow contribution s from (6.9) hasbeen omitted.

∆Iref =PA

am(cos(0◦) − (cos(1◦))V (T )

η(1 − R(α)) ≥ 10 · 1.6nA (6.14)

Determining the size of m is a trade off between chip size and sensitivity.Wide a is needed to minimise m, but contrarily narrow a (⇒ long k) isneeded for high sensitivity due to At(α). With fixed a the height k of thetriangular areas can be selected from a desired current at α = 0◦. Equation(6.6) gives:

It,0◦ =PA

12ak

V (T )η(1 − R(α)) ⇔ k = 2It,0◦V (T )

aPA

η(1 − R(α))(6.15)

The choice of sensor-pyrex distance l is a compromise between position ofdeflection tangents on ∆It, and linearity of

∆ItIref

; see figure 6.20 and equation

(6.10). Ambiguous ∆It curves within FOV is a problem of smaller signifi-cance, since ∆It

Irefis to be used for α-lookup. Actually the only requirement

41

to the ∆It curves is that∆It∆α

∣

∣

∆α=1◦> 10 · 1.6nA/◦. Oppositely, linearity

of ∆ItIref

is of great importance, since this enhances effective resolution of the

A/D conversion. The value was selected as low as practically possible. Sincethe pyrex cavity will be etched it might be problematic to obtain very smallstructures (< 2µm), because of etching speed, which implies that it is neededto do characteristic calculations with a l higher than the one that will beimplemented in practice.

−80 −60 −40 −20 0 20 40 60 80 100−6

−4

−2

0

2

4

6x 10

−5

Angle of incidence [degrees]

Cur

rent

[A]

l=250µm

l=100µm

l=5µm

(a) ∆It graphs. Deflection tan-gents occur when O(cos(α)) =O(A(α)); equation (6.6) & (6.8).

−80 −60 −40 −20 0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

Angle of incidence [degrees]

Cur

rent

[A]

l=250µm

l=100µm

l=5µm

(b) ∆ItIref

graphs. Non-linearity is

introduced with tan(αi) sinceαi ≥ αo; equation (6.10).

Figure 6.20: Consequences of the sensor-pyrex height l.

From figures 6.16, 6.17, and 6.18 it can be seen that the minimum sizesfor f , h and href are:

c1 = c − hSi − hOX − lf = a + 2c1 tan αF2 + 2l tan αF1 (6.16)

h = k + 2c1 tan αF2 + 2l tan αF1 + d′ (6.17)

href = m + 2c1 tan αF2 + 2l tan αF1 (6.18)

where αF1 is the desired FOV. Desired FOV was selected to be 70◦, which

clearly is above the required 55◦. αF2 is determined by:

n1 sin(αF1) = n2 sin(αF2) ⇔ αF2 = arcsin(

n1n2

sin(αF1)

)

(6.19)

The parameters d and d′ are determined in section 6.3.4 and b in section6.3.5. Common for these parameters is that they are process defined. Itwas decided to use pyrex wafers of thickness c = 500µm, because this is acommonly used thickness.

The dimensions of the sensor in table 6.1 are achieved by iterativelyrepeating the above suggested calculations and arguments. Of course, the

42

dimensioning has to start with a constraint on the total size of the sensor,and a working point current for α = 0◦. The values in table 6.1 are for atwo-axis sensor on less than 5500µm×4500µm with a working point currentIt,0◦ = 100µA.

Description Value

Slit width a 550 µmSlit length h 3480 µmTriangle height k 2300 µmTriangle width f 1390 µmSensor-pyrex dist. l 5 µmRef. slit length href 2140 µmRef. area height m 1200 µm

Pyrex thickness c 500 µmAl thickness b 2000 ÅTriangle dist. d 120 µm- d′ 232 µm

Table 6.1: Dimensions of the sun sensor.

Verification

Figure 6.21 shows the ideal characteristics for a sensor with the parametersgiven in table 6.1. Calculations give ∆Iref |0◦−1◦ = 19nA, which meansthat equation 6.14 is fulfilled; g) indicates that

∆Iref∆α

∣

∣

∆α=1◦is abundantly

fulfilled, since∣

∣

dIrefdα

∣

∣ is an increasing function. Deflection tangents on c) arenot problematic since the gradient depicted on f) increases fast around itszero point; within 0.2◦ it has increased ±18nA/◦.

It is needed to verify that l has a height which ensures that the pyrexwill not touch the active sensor area under the anodic bonding process (seesection 6.3.5). The maximum deflection of a plate with its four sides fixedare given in [27]:

ymax =αFb4

Et3(6.20)

where F is a uniform force on top of the entire pyrex cavity, b = 4500µm thelength of the short sides in the rectangle10, E = 64GPa Young’s Modulus forpyrex [1], and t = c1 is the pyrex thickness minus the height of the cavity.From tables in [27] α can be determined to α = 0.0188 since a

b= 5500µm4500µm ≈

10The length of the long sides is denoted a.

43

0 10 20 30 40 50 60 705

6

7

8

9

10

11

12

13x 10

−5

Angle of incidence [degrees]

Cur

rent

[A]

(a) It1; increasing A(α).

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2x 10

−4

Angle of incidence [degrees]

Cur

rent

[A]

(b) It2; decreasing A(α).

−80 −60 −40 −20 0 20 40 60 80−8

−6

−4

−2

0

2

4

6

8x 10

−5

Angle of incidence [degrees]

Cur

rent

[A]

(c) ∆t for entire FOV.

0 10 20 30 40 50 60 703

4

5

6

7

8

9

10

11

12x 10

−5

Angle of incidence [degrees]

curr

ent [

A]

(d) Iref .

−80 −60 −40 −20 0 20 40 60 80−1.5

−1

−0.5

0

0.5

1

1.5

Angle of incidence [degrees]

Cur

rent

[A]

(e) ∆tIref

.

0 10 20 30 40 50 60 70−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−6

Angle of incidence [degrees]

A/d

egre

e

(f) d∆tdα

.

0 10 20 30 40 50 60 70−2.5

−2

−1.5

−1

−0.5

0x 10

−6

Angle of incidence [degrees]

A/d

egre

e

(g)dIrefdα

.

Figure 6.21: Ideal characteristics for the designed sensor.

44

1.22. An expression for the force F is deduced in [24]:

F = − �gAV2

(

12xd +

�g�0

xr

)2 (6.21)

where A is the area of the pyrex cavity, V bonding voltage, �g = 5�0 per-mittivity for pyrex [4], xd = 1µm thickness of the depletion layer introducedwith the bonding [24], and xr which is the pyrex-Si gap. When pyrex isbonded in the MIC Process Laboratory a bonding voltage of 1kV (or slightlyhigher) is typically used. If V = 1.5kV is used a higher degree of certaintycan be obtained.

The given values yield F = 3.7896N, and ymax = 3.9pm. This showsthat the selected l value can be used. That the vents has not been takeninto account ca