K

NASA Contractor Report 178292

Development of Response Models for the Earth

Radiation Budget Experiment (ERBE) Sensors:

Part I - Dynamic Models and Computer

Simulations for the ERBE Nonscanner,

Scanner and Solar Monitor Sensors

(NASA-CR-178292) DEVELOPRENT OF RESPONSE

MODELS FOR THE EARTH RADIATION BUDGET

EXPERIMENT 'ERBE) SENSORS. PART I: D¥_K_IC

MODELS AND COMPUTER SI_IULATIONS FOR TLE E_S Unclas

NONSCANNER, [Inforj_ation. and Co.tirol Syste=s) G3/35 0111320

Nesim Halyo, Sang H. Choi, Dan A. Chrisman, Jr.and Richard W. Samms

N88-13557

Information & Control Systems, IncorporatedHampton, Virginia 23666

Contract NAS1-16130

March 20, 1987

N/ ANallonal Aeronaulics andSpace Administration

Langley Research CenterHampton, Virginia 23665-5225

https://ntrs.nasa.gov/search.jsp?R=19880004175 2020-04-19T22:42:07+00:00Z

FOREWORD

This report entitled "Development of Response Models for the Earth Radiation

Budget Experiment (ERBE) Sensors" consists of the following four parts.

This is Part I, NASA CR-178292, entitled "Dynamic Models and Computer

Simulations for the ERBE Nonscanner, Scanner and Solar Monitor Sensors".

Part II, NASA CR-178293, is entitled "Analysis of the ERBE Integrating

Sphere Ground Calibration".

Part III, NASA CR-178294, is entitled "ERBE Scanner Measurement Accuracy

Analysis Due to Reduced Housekeeping Data".

Part IV, NASA CR-178295, is entitled "Preliminary Nonscanner Models and

Count Conversion Algorithms".

TABLEOFCONTENTS

page

FOREWORD .............................. i

LIST OF FIGURES ........................... iv

LIST OF SYMBOLS ........................... vl

LIST OF ACRONYMS .......................... x111

i. INTRODUCTION .......................... 1

2. SENSOR MODELING ......................... 5

2.1 GENERAL DESCRIPTION OF THE ERBE SENSOR MODELS ....... 6

2.1.1 MODELS ....................... 6

2.1.2 THERMAL CONDUCTIVE AND RADIATIVE INTERACTIONS .... 14

2.1.3 COUPLING ENCLOSURES ................. 21

2.1.4 GOVERNING EQUATION ................. 27

2.1.5 ELECTRONIC CONTROL SYSTEM MODEL FOR THE

NONSCANNER ..................... 28

2.1.6 SIMULATION OF THE NONSCANNER MODEL ......... 32

2.1.6.A PARAMETER AND COEFFICIENT DETERMINATION . . 34

2.1.6.B INSTRUMENT SIMULATION ........... 35

2.1.7 COUNT CONVERSION FOR NONSCANNER ........... 36

2.1.7.A NONSCANNER TOTAL CHANNEL .......... 36

2.1.7.B NONSCANNER FILTERED CHANNEL FOR

SHORTWAVE ................. 38

2.1.8 SOLAR MONITOR MODEL ANALYSIS ............ 40

2.1.9 SOLAR MONITOR COUNT CONVERSION ALGORITHM ...... 50

2.1.10 SCANNER CONTROL ELECTRONICS ............. 51

2.1.11 SCANNER STEADY-STATE ANALYSIS ............ 54

11

TABLEOFCONTENTS(CONCLUDED)

page

2.1.12 SCANNERCOUNTCONVERSION.............. 58

2.1.13 ACCURACYANDDIAGNOSTICCHECKS............ 61

3. INSTRUMENTPERFORMANCEANDCOUNTCONVERSIONTEST ......... 63

3.1 DESCRIPTIONOFROUTINES................... 68

3.1.1 ROUTINEI ....................... 68

3.1.2 ROUTINEII ...................... 69

3.1.3 ROUTINEIII ...................... 69

3.1.4 ROUTINEIV ..................... 70

4. DISCUSSIONOFPARAMETERESTIMATION................ 71

REFERENCES.............................. 76

APPENDIXA: PRELIMINARYRESULTSONERBENONSCANNERINSTRUMENTMODELING,

COUNTCONVERSIONANDSENSITIVITYANALYSIS ........ 78

APPENDIXB: ERBESCANNERSIMULATIONS................. 166

APPENDIXC: ERBENONSCANNERTHERMALMODEL............. 184

APPENDIXD: ERBESCANNERTHERMALMODEL................ 202

APPENDIXE: ERBESOLARMONITORTHERMALMODEL............. 250

iii

FIGURE I.

FIGURE 2.

FIGURE 3.

FIGURE 4.

FIGURE 5.

FIGURE 6.

FIGURE 7.

FIGURE 8.

FIGURE 9.

FIGURE i0.

FIGURE ll.

FIGURE 12.

FIGURE 13.

FIGURE 14.

FIGURE 15.

FIGURE 16.

FIGURE 17.

FIGURE 18.

LIST OF FIGURES

NONSCANNER MEDIUM FIELD-OF-VIEW TOTAL (MFOVT) ......

NONSCANNER MEDIUM FIELD-OF-VIEW SHORTWAVE (MFOVSW) . . .

NONSCANNER WIDE FIELD-OF-VIEW TOTAL (WFOVT) .......

page

7

8

9

NONSCANNER WIDE FIELD-OF-VIEW SHORTWAVE (WFOVSW) .... I0

SCANNER NARROW FIELD-OF-VIEW TOTAL (NFOVT) ........ ii

SOLAR MONITOR ...................... 12

A SIMPLE THERMAL CONTACT RESISTANCE MODEL ........ 15

RADIATION FLUX INTERCHANGE ............... 18

INTERACTIVE ENCLOSURE MODEL ............... 23

ACTIVE CAVITY RADIOMETER (ACR) CONTROL SYSTEM MODEL . . . 29

NONSCANNER HEAT SINK HEATER CONTROL SYSTEM MODEL .... 32

FLOW DIAGRAM INDICATING FILE AND PROGRAM USAGE IN

SIMULATION ....................... 33

THE SHUTTER MODEL .................... 41

SHUTTER TEMPERATURE AT x = 1.5 cm WITH 20% ABSORPTION

OF SOLAR FLUX Qs = 0.142 w/cm 2 ............. 43

SHUTTER TEMPERATURE AT i00 SECONDS WITH 20% ABSORPTION

OF SOLAR FLUX Qs = 0.142 w/cm 2 .............. 44

A SIMPLIFIED CONFIGURATION OF SOLAR MONITOR BARREL . . . 45

BARREL TEMPERATURE WITH RESPECT TO TIME AT x = 3 cm

FROM THE SECONDARY APERTURE ...... _ ........ 47

BARREL TEMPERATURE ALONG THE AXIAL DIRECTION AT THE

TIME OF 200 SECONDS ................... 48

iv

LIST OF FIGURES (CONCLUDED)

page

FIGURE 19. SCANNER ELECTRONIC CONTROL MODEL ............ 53

FIGURE 20. INSTRUMENT PERFORMANCE AND COUNT CONVERSION TEST .... 65

FIGURE 21. ERBE INSTRUMENT VALIDATION PLAN ............. 66

FIGURE 22. EXPLANATIONS ...................... 74

v

LIST OF SYMBOLS

Symbol

Ac

Ar

A i

Ar

Ac

Definition

Area of conductive transfer in an element

Area of radiative transfer in an element

Contacting area between elements k and i

Radius of a contacting point, e.g. Fig. 7

thInside surface area for i surface

Constant defined in Eq. 14

Area of radiative interaction

Area of nodes covered by a heater

Thermal wave velocity, e.g. Eq. 58

b

B

BQ

bE

Bk

Bias term, e.g. Eq. 25

Vector of coefficients for irradiance from the source

Coefficient which describes heating contributions

from the control systems

Electronic bias

Constants related with the irradiance at a FOV

limiter

Ck

C

AC B

Specific heat capacity of element k

Constant defined in Eq. 15

Number of counts

Baseband signal

D Enclosure coupling factor through the interactiveboundaries

vi

LIST OF SYMBOLS(CONTINUED)

STmbol

ek

ETM

n

mn

e

AEos

El(t)

ei(t)

EBi(T B)

EBx(T B)

Fk- i

Fk_ i

Definition

Energy in element k

Rate of incoming radiant energy to surface n inenclosure m

Rate of incoming radiant energy to non-opaque

boundary surface n in enclosure m from a neighboringsurface

Noise term, Eq. 25

Total incident radiation esitmate

Baseband signal

Spectral irradiance at the instrument due to radia-

tion from the footprint at time t

Combined error of the measurement and count con-

version accuracy

Output of count conversion when viewing the black-

body at temperature TB

Spectral irradiance from the blackbody at TB

Configuration factor between elements k and i

Exchange factor between elements k and i

F

FE(S)

System boundary condi_tions imposed by surrounding_

Control electronics transfer function.

hk- i

Hz

Thermal conductance of the contact between elements

k and i

Hertz

vii

LIST OF SYMBOLS (CONTI_._UED)

Symbol

I

i

Definition

Identity Matrix

Calibrat ion sequence

Number of measurements in the scanner mode.

kk

K.l

kfo

Thermal conductivity of element k

Electronic gain for component i

Post amplification gain

i

Lk- i

L

I

Longwave

Shortest distance between the center of mass of

element k and the contacting point of element k

to its neighboring element i

Length of a solar monitor shutter

Exposure length to solar radiation

M_.l

M°

l

ms

m

Opaque radios[ty - the rate of outgoing radiant

energy from surface i from diffuse and specular

reflectance

Total radiosity - opaque radiosity plus energy flux

transmitted through surface i

Milliseconds

Meter

n

n i

Number of contacts per unit area

Noise from source i

viii

LIST OFSYMBOLS(CONTINUED)

S)nnbol

,l!

qc

Definition

Dynamic heat transfer due to behavior of element k

as a heat source or sink

Heat flux by conduction through the contacting area

between two elements

R i

Ro

Thermal resistance through the contacting area be-

tween elements k and i

Inside radius of solar monitor barrel

Outside radius of solar monitor barrel

Smn

Heat flux arriving at the system boundary at surface

n in enclosure m

Shortwave

T

T k

Tk

T(i)

Total radiation channel

Temperature (vector) of element k

Governing equation for a radiometer with a dynamic

response to an incoming radiance at element k

Temperature change for thermal node i or voltage

change for electronic node i

T

T b

At

tk

t

Ambient temperature

Current shutter temperature in solar monitor

Total scan period

Time at end of space-look

Sampling instant

ix

LIST OFSYMBOLS(CONTINUED)

Symbo I

T B

Definition

Temperature of a blackbody

u(x) Unit step function

V k

v(t), v(t)

_(tk)

V(tki)

VB(t)

Vd(t k)

VBi(T B)

Volume of element k

Instrument output (in volts) at time t

Space clamp value at time tk, i.e. at the end ofspace look

Instrument output (in volts) when viewing space

at time tk

Detector bridge bias voltage at time t or most re-

cent value (v)

Drift balance DAC voltage measurement at time t

or most recent value (v)

.thSteady-state output of the i

blackbody temperature T B

channel when viewing

W Watts

x

x

x

A_ i

Distance from end of barrel in a solar monitor

Vector consisting of electrical signal processing

variables, e.g. Eq. 73

Vector consisting of the offset temperature and

radiative constant, e.g. Eq. 109

Lower frequency signal portion of Ax for component

i, e.g. Eq. 79

Noise and irradiance portion of Ax for component

i, e.g. Eq. 79

x

LIST OF SYMBOI,.q (CONTINUED)

Greek

S_ mbol

Yk

1

_k

n k

_k

Ok

P

o

2

° k

"rk

T.1

T

i(k)

Definition

Density of element k

Surface roughness of surface 1

Thickness of solar monitor shutter

Estimate of unaccounted drift during k th scan

period (v)

Emissivity of element k

Exchange factors between specular surfaces kand i

Attenuation factor for element k

Wavelength k

Reflectivity of surface k

Density of solar monitor shutter

Stefan-Boltzmann constant

Noise variance estimate during space look

Transmissivity of surface k

Time constant for electronic element i

Average time lag (in seconds)

Transmittance of ith channel

Radiative energy supplied to surface k

Superscript

d

S

m

Diffuse

Specular

Number of an enclosure in a system

xi

LIST OFSYMBOLS(CONCLUDED)

Superscript

M

W

P

Definition

Medium field-of-view

Wide field-of-vlew

Photo-diode

Subscript

s

1

s

k-i

n

H

A

F

B

A

E

v

B

o

s

H

Shortwave band

Longwave band

Surface, Eq. 6

between k and i elements

Number of an enclosure's surface in a system

Heater, Eq. 35

Precision aperture

Field-of-vlew llmlter

Bridge

Amplifier

Electronic

Volatage

Blackbody

Initial

Solar, Eq. 58

Heat sink, Eq. 99

xii

LIST OFACRONYMS

Acronym

ACR

A1 Heat Sink

A/D

B.C.

CCA

Cu Heat Sink

DAC

ERBE

EXP

FOV

FOVL

G.E.

HK

HKD

IBB

I/0

I.C.

LW

MFOV

MFVOT

MFOVSW

MAM

MRBB

Definition

Active Cavity Radiometer

Aluminum IIeat Sink

Analog-to-Digital (conversion)

Boundary Conditions

Count Conversion Algorithm

Copper Heat Sink

Digital-to-Analog Conversion

Earth Radiation Budget Experiment

Exponential

Field-of-view

Field-of-view limter

Governing Equations

Housekeeping

Housekeeping data

Internal blackbody

Input/Output (data processing)

Initial Conditions

Longwave

Medium Field-of-View

Medium Field-of-View Total

Medium Field-of-View Shortwave

Mirror Attenuator Mosaic

Master Reference Blackbody

xiii

LIST OF ACRONYMS (CONCLUDED)

Acronym

ms

NS

NFOV

RHS

RMS

SW

SWICS

SiPD

T

WFOV

WFOVT

WFOVSW

WRT

Def inlt ion

Millisecond

Nonscanner

Narrow Field-of-View

Right hand side

Root mean square

Shortwave

Shortwave Internal Calibration Source

Silicon Photo-Diode

Total radiation channel

Wide Field-of-View

Wide Field-of-View Total

Wide Field-of-Vlew Shortwave

With respect to

xiv

i. INTRODUCTION

The radlometrlc measurement system for the Earth Radiation Budget Experi-

ment (ERBE) is designed to measure the earth's radiation fluxes which result

from the earth and its atmosphere's emmltted energy as well as from reflected

energy, i.e., the earth's albedo. A general overview of the concept behind the

three-satellite ERBE radiometric system and its instruments has been described

in many internal NASA documents, such as the Science Team Minutes and Contractor

Status Reports, as well as in journal puhlications, (e.g., Ref. i).

The radiometer package on each ERBE satellite consists of eight instruments

distinguished by their mode of operation, field-of-view and spectral response.

Four of the eight instruments are nonscanner (NS) type radiometers, two with

a Medium Field-of-View (MFOV) and two a Wide Field-of-View (WFOV). Two of the

nonscanner radiometers, one MFOV and one WFOV, have a suprasil filter which

eliminates the instrument's response to longwave radiative effects so that they

only remain sensitive to shortwave radiative effects. Shortwave (SW) radiative

effects are defined here as having wavelengths between 0.2 and 5 _m and long-

wave radiative effects as having wavelengths between 5 and 50 _m. Since radio-

meters without filters measure total radiation, they are designated "Total" (T).

The four nonscanner radiometers are therefore the Medium Field-of-View Total

(MFOVT), Medium Field-of-View Shortwave (MFOVSW), Wide Field-of-View Total

(WFOVT) and Wide Field-of-View Shortwave (WFOVSW). Three of the remaining in-

struments or "channels" are Narrow Field-of-View (NFOV) scanning radiometers,

two of which have filters. One scanner radiometer has a filter which eliminates

longwave radiative effects and another has a filter which eliminates shortwave

radiative effects. Hence the three scanner radiometers are the Narrow Field-of-

View Shortwave (NFOVSW), the Narrow Field-of-View Longwave (NFOVLW) and the

Narrow Field-of-View Total (NFOVT). The eighth instrument in the ERBE radio-

metric package is the solar monitor radiometer.

After the ERBE radiometers perform their earth, space and/or sun view and

calibration measurements, the measurement results undergo an analog-to-digital

(A/D) conversion and the data now in counts is transmitted to the earth through

the spacecraft telemetry system. Counts for the nonscanners are measured

every 0.8 seconds. Heat sink temperatures are read every 8 seconds and other

housekeeping (HK) temperatures every 16 seconds. Therefore, in 16 seconds

the data record for one nonscanner consists of 20 radiometric count readings,

2 heat sink temperature readings and I reading for each of the other HK

temperatures.

The scanners perform a measurement cycle once cv=Ly _ seconds. During a

normal, earth-viewing scan each scanner performs 8 space-look measurements, 62

earth-view measurements and 4 calibration measurements. The Longwave and Total

NFOV scanners utilize the Internal Blackbody (IBB) for calibration measurements,

whereas the Shortwave NFOV scanner utilizes the Shortwave Internal Calibration

Source (SWICS). During a Mirror Attenuator Mosaic (NAM) scan cycle, the NFOVT

and NFOVSW scanners measure solar flux performing 8 space-look measurements,

46 MAM measurements and 4 calibration measurements. Housekeeping (HK) data is

sampled once per scan.

The solar monitor uses the same type of Active Cavity Radiometer (ACR) that

is used by the nonscanners. The three major differences between these ACR's

are as follows:

a) the solar monitor has a shutter to cut off the solar flux every

32 seconds,

b) the configuration of the solar monitor housing is totally different

and

c) the solar monitor has no heat sink heater.

Solar Monitor count measurementsare madeevery 0.033 seconds and HKmeasurements

are madeevery 4 seconds.

The radiometric instruments described above will have inherently different

behavior and responses to arbitrary input signals since no instrument can be

exactly duplicated or subjected to exactly the samecalibration or operating

environment. Evenwhenviewing the problems of sensor performance from th_s

macroscopic point of view, there are many things that have to be dealt with and

accounted for. For example, environmental contamination and aging effects in-

fluence the material degradation of the sensors. Sensor behavior may also be

questioned when the thermal load and leakage exceed the design level. One ex-

ample of abnormal thermal load is in the sun-blip case, where the sensor is

subjected to a sunrise and sunset during its orbit.

In attempting to account for such problems found in the sensor's radiometric

performance and capability, it is therefore necessary to develop a radiometer

sensor model. A sensor model should describe both a transient and steady-state

sensor response for a given radiative input. It must provide sufficiently simple

analytical expressions for parameters of particular interest and describe the

model's behavior, e.g., the model's sensitivity to changes in the modeling assump-

tions. The model must also be able to incorporate changes resulting from in-

strument design requirements during its construction as well as instrument aging,

i.e., sensor material degradation and unexpected thermal load and leakage during

its long-term orbital operation. The products obtained from the sensor model and

simulation must be in a form that is useful to the users. The sensors' response

must therefore be validated by updating model parameters and converted into data

with useful engineering units.

This report entitled "Development of Response Models for tile Earth Radiation

Budget Experiment (ERBE) Sensors" consists of tlle following four parts.

Part I, NASA CR-178292, is entitled "Dynamic Models and Computer Simulations

for the ERBE Nonscanner, Scanner and Solar Monitor Sensors".

Part II, NASA CR-178293, is entitled "Analysis of the ERBE Integrating

Sphere Ground Calibration".

Part III, NASA CR-178294, is entitled "ERBE Scanner Measurement Accuracy

Analysis Due to Reduced Housekeeping Data".

Part IV, NASA CR-178295, is entitled "Preliminary Nonscanner Models and

Count Conversion Algorithms".

4

2. SENSOR MODELING

The following ERBE radiometer sensor models were developed by Information &

Control Systems, Incorporated (ICS) to consider thermal and radiative exchange

effects. These models consider the surface specularity and spectral dependence

of a filter as well as the thermal conductive and radiative interactions among

the nodes in an enclosure of a radiometer. An enclosure is a domain in which

an instrument's geometry is divided into appropriate elements. These enclosures

in an instrument's model are set up using real or fictitious boundaries to ef-

fectively alleviate the computational load for defining configuration factors.

The instrument's model also includes the electronic thermal control sytems which

are coupled with the energy equations. A general form of the energy equations

can be used for all the sensors, despite differences in the mode of operation,

field-of-view and spectral response, as long as the radiative enclosures and

thermal nodes in the sytem's sensor configuration comply with the first law of

thermodynamics.

A sensor model can be tested and evaluated by comparing its behavior with

the actual results obtained from the sensor's ground and inflight calibrations.

The sensor model includes additional uncertainty since it incorporates outputs

from the calibration source models. Once the sensor and calibration source models

prod,lce sufficiently accurate results, they can be used to define the parameters

for the count conversion procedure.

The characteristics of a sensor'_ _imulation model are such that it has

a dynamic response to the time dependence of an arbitrary input signal. The

model also has a spectral response which is due to the spectral dependence of

a sensor's filter and its exposed surfaces. Thermal effects for the model can

be determined from the conductive and radiative properties of the sensor. _le

model's measurementerror can therefore be determined after its time response,

spectral response and thermal effects have been obtained. Themodel can simu-

late the sensor's response to the temporal incident radiation received through

the field-of-view limiter in both the transient and steady-states.

The purpose of model_ngand simulating is therefore I) to understand sensor

performance, 2) to improve measurementaccuracy by updating the model parameters

and 3) to provide appropriate constants necessary for the count conversion

algorithms.

2.1 General Description of the ERBE Sensor Models

2.1.1 Models

A sensor's geometry can be appropriately divided by considering its ther-

mal and radiative interactions as a consequence of its design. Figures 1

through 6 provide diagrams of how sensor elements have been geometrically

partitioned for the nonscanner, scanner and solar monitor. Each element has

a time resonse to the energy flowing to and from its neighboring elements.

One might assume that maximizing the number of sensor nodes would maximize

the accuracy of the model's prediction of energy from a target source. However,

increasing the number of nodes would introduce additional sources of error

since the large set of equations required would incorporate additional para-

meter approximations. Such equation approximations would need to be made to

calculate constants such as thermal resistance, exchange factors and surface

spectral and specular characteristics. There is an accuracy tradeoff between

increasing the number of nodes and approximating the sensor's size, material

properties, geometry, enclosure surface characteristics, temporal incident

NONSCANNER MEDIUM FIELD-OF-VIEW TOTAL (MFOVT)

5

H

4 \ E¥,

2

4

• 2 P

!

H

1 ACTIVE CAVITY

2 Cu HEAT SINK

3 REFERENCE CAVITY

4 Al HEAT SINK

5 BASE PLATE

6 FOV LIMITER

7 Al HEAT SINK

8 APERTURE

FIGURE I. H HEATER

P TEMPERATURE PROBE

NONSCANNER MEDIUM FIELD-OF-VIEW SHORTWAVE (MFOVSW)

S4

H 2

I

|,

|

• 2 P

i!

H

FIGURE 2.

1 ACTIVE CAVITY

2 Cu HEAT SINK

3 REFERENCE CAVITY

4 AI HEAT SINK

5 BASE PLATE

6 FOV LIMITER

7 AI HEAT SINK

8 APERTURE

9 SW FILTER DOME

H HEATER

P TEMPERATURE PROBE

NONSCANNEIt ",VIDE FIELD-OF-VIEW TOTAL (WFOVT)

/7

PI

'lS

_._ 8 8!

i I

M

4

m

l

2

1

AA P

A P

H

i

i

• 2 P I.I

l

7

I ACTIVE CAVITY

2 Cu HEAT SINK

3 REFERENCE CAVITY

4 Al HEAT SINK

5 BASE PLATE

| 6 FOV LIMITER

7 AI HEAT SINK

8 APERTURE

FIGURE 3.

H HEATER

P TEMPERATURE PROBE

NONSCANNER WIDE FIELD-OF-VIEW SHORTWAVE (WFOVSW)

/'!.J

H

•, .y

4 EV.

2

7 "

r

m

4

• 2 P

[ !

FIGURE 4.

H

5.r-- [

1 ACTIVE CAVITY

2 Cu HEAT SINK

3 REFERENCE CAVITY

4 AI HEAT SINK

5 BASE PLATE

6 FOV LIMITER

7 Al HEAT SINK

8 APERTURE

9 SW FILTER DOME

H HEATER

P TEMPERATURE PROBE

lO

o

_o

ZZ

!

I

11

SOLAR MONITOR

'+-h

III

-.-4-I!IIIIIII

11IIIIIIII

IJ

11

12 13

II j ' |

,oP

-j

+,III

IItIIIIII

1,

6-_, __-_-,,f /-6 ,.I

1

I..,,....

2

|9 19

14

f

|. ACTIVE CAVITY2. REFERENCrCAVITY

3. BUTTON.DUPLEXCONECAVITIES

4. HEAT SINK._.AFERTURE.PRIMARY6, SHIELD. APERTURE1. BODY IREAR BARRELI8. BODY IFRON! blARRELt9, APERTURE.SE.CO_DARY

OR FOV LI&ll IER t|0. h_OTORHOUSIhG||. CHOPPERgLADE|2, CHOPPERPENDLU/_,I13. MOTOR. CHOPPER|4, FOVLIMITERiS. FRONTTOP HOUSING COVER|6. REARTOPHOUSING COVER17. FRONTHOUSING|8. REARHOUSING19. HOLDER. HEAT SINK

?0 FICTIIIOUS SURFACE

p. TI'_4PERATURE PROBES

FICUR_ 6.

12

energy flux intensity and electronic control system. If the temperature

gradients within a node and between nodes are small enough, a model approx-

imation with only a few nodes would not Jeopardize the accuracy required to

perform the parameter estimates and error analysis. Factors which would de-

crease these temperature gradients within a sensor include uniform thermal

properties, simple and symmetrical geometry, uniformity in directional and

spectral properties of an enclosure surface and small temporal change in in-

cident energy flux.

A nonscanner sensor has 9 nodes if it has a suprasil filter dome (Figures

2 and 4) and 8 nodes if it does not (Figures i and 3). The sensor's geometry

is divided into nodes based on material homogeneity, overall sensor size, thermal

conductivity of materials, thermal resistance between materials and surface

radiative properties. The 9 nodes of the _OVSW model (Figure 2) are there-

fore, as numbered: i) the active cavity with a heating and sensor coil 2) the

copper heat sink with a heater, 3) the reference cavity with a sensor coil, 4)

the aluminum heat sink, _lich is partly covered by a heater, 5) the base plate

or substrate, 6) the field-of-view limiter, 7) the aluminum heat sink which

houses the reference cabity, which is partly covered by a heater, 8) the pre-

cision aperture and 9) the filter dome for shortwave channels.

Similarly, the scanner model (Figure 5) is divided into 12 nodes for the

total channel, 13 nodes for the longwave channel and 14 nodes for the short-

wave channel. The solar monitor is divided into 19 nodes (Figure 6).

In addition to their nodes, these sensor models incorporate fictitious sur-

faces to define and simplify enclosure geometries- Since the specular prop-

erty of each surface in an enclosure is taken into consideration, the simple

geometry facilitates the exchange factor calculation, alleviating the compu-

tational work load for determining the exchange factors. A net radiation

13

analysis is performed for each enclosure which has surfaces with specular and

spectral characteristics. This analysis appropriately deals with any partial

specular and diffuse characteristics of an enclosure surface (Ref. 2). The

model can also easily include transient response and thermal conduction. This

net radiation method utilizes the wavebandapproximation to an enclosure's

surface which has a specific spectral dependence.

2.1.2 Thermal Conductive and Radiative Interactions

A volumetrically defined node, or control volume, complies with the first

law of thermodynamics as in tile following equation

de

d--_= Ac qc'"+ Ar _ (I)

The first and second terms of the right hand side of Equation i represent the

conductive transfer through contacting surfaces and radiative transfer from

surrounding elements of an enclosure, respectively. The rate of energy change

in a single node, therefore, depends on the difference between the ingoing and

outgoing energy fluxes through a control surface or "boundary". This rate of

energy gain or loss is composed of the rate of energy stored in a node as well

as the rate of energy source or sink as indicated in the following equation

de = VyC dTd--_ _- _ (2)

where V is the volume of a node, y the density of a node, C the specific heat,

T the temperature and Q the rate of energy, or heat, at a source or sink.

Combining Equations 1 and 2 yields

dT •,,VyC _-_ = Ac qc + Ar _ + _ (3)

14

Equation 3 represents an energy balance in a single node. In the geometrically

partitioned model, the conduction term can be written in the following form

A _ T. - T.c _'' = j i

c/i-j Ri-j

(4)

where Ai_ j is the contacting area of i element to j element, qc the heat flux

by conduction through the contacting area between i and j elements. Ri_ j is the

thermal resistance between i and j elements. The thermal resistance between nodes

is defined by the following realationship

L 1 Li-j .I-!

Ri-j = k + +i Ai-j hi_ j Ai_ j k.3 AJ -i

(s)

where Li_ j is the distance from the thermal center of a node to the center of the

contacting surface, k the thermal conductivity and h the thermal conductance of

the contact between i and j elements. The thermal center is normally coincident

with the geometrical center if a node material has homogeneous and isotropic ther-

mal properties. The contact resistance (1/hA) is mainly dependent on the surface

roughness and the contacting pressure. The well-machined surface has a regular

and uniform roughness. In such a case, the contact resistance can be determined

by the following empirical relations (Ref. 3) as functions of the surface rough-

ness and the thermal conductivities of two contacting materials.

Figure 7.

//

J\

\\\

---------m

A Simple Thermal Contact Resistance Model

15

Let surfaces contact each other with surface roughness 61 and _2 and thermal

conductivities kI and k2, respectively. Assume the contacting area per periodic

2 i

contact is _a . Then the number of contacts per unit area n = _ and the

Atotal number of contacts for two surfaces with total contacting area A is _ .

_a-

Conductance can therefore be calculate,d:

h=T +n (6)

where

ks

6 = 61 + 62 (= 261 or 262 )

n = 2.13_ _ , where q is a parameter for roughness

a+c, where c is the half-width of the surface cavity

1n = _ = number of contacts per unit area

kl,k 2 ffithermal conductivity of nodes i and 2

61,82 = root mean square surface roughness of nodes 1 and 2

A = total node interface area

a ffiradius of a contacting point

For the radiative interchanges in an enclosure including both specular

and diffuse surfaces, a radiative property such as a reflectance, 0, can be

divided into a diffuse and a specular component as well as shortwave and long-

wave bands.

That is, for shortwave bands:

16

d sPs -- Ps + Ps

and for longwave bands:

d s

p£ -- p£ + P£

where the superscripts, d and s, signify the diffuse and specular, and the sub-

scripts the short and longwave bands. For a semi-transparent or an absorbing

medium, the radiative properties have the following relationship among the

reflectance 0, emittance or absorptance _:, and transmittance, T.

_s + Ps + Ts = i for shortwave

EZ + 0£ + T£ = i for longwave

The energy fluxes incident upon and leaving a typical seml-transparent

surface of the enclosure are described in Figure 8. Consider the ith inside

surface area, Ai, of the enclosure which neighbors with other enclosures. The

quantities E i and M_ are the rates of incoming and outgoing radiant energy

while considering diffuse and specular reflectance of the surface per unit in-

side area, respectively. The quantity M_ does not include the rate of trans-

mitted incident radiant energy from the other neighboring enclosures. The

quantity _i is the rate of incoming radiant energy to the non-opaque boundary

surface from the neighboring enclosure. The quanltity #i is the energy flux

supplied by some external means to the surface to make up for the net radiative

loss and thereby maintain the specified surface temperature. A heat balance

at the surface provides the relation

_i = El - Mi (7)

A second equation results from the fact that the energy flux leaving the

17

_th

SURFACE

z_.EiIIIII

Ei

FIGURE 8. Radiation Flux Interchange

18

surface having diffuse and specular characteristics is composed of directly

emitted plus reflectcd energy. Th_s gives

4 d sM_ = Ei O T i + Oi Ei + Pi Ei (8)

Equation 8 is called an opaque radiosity. The total radioslty, considering

the rate of transmitted radiant energy which is originally coming from the

neighboring enclosure, is composed of the opaque radiosity M_ plus the energy

flux transmitted through the surface with transmissivlty, T i. This gives

M i = M_I + Ti _i (9)

The diffuse total radiosity considering only a diffuse factor in Eq. 9

and excluding the specular reflection term is described as the following

d 4 dM i = E i 0 T i + O i E i + Ti _i (I0)

The contribution to Ei from the other diffusely reflecting surfaces in

an enclosure can be represented in accordance with the relationship defined by

the configuration factors in an enclosure with diffuse surfaces, except that

now the configuration factors are replaced by the exchange factors, or

= E - M_ (11)Ei j Fij S

where F.. is the exchange factor defined between i element and j element through13

j's images on the element i. Equations 7 through ii can be represensted by the

short and longwaves, respectively.

To obtain the heat balance (Eq. 7) as functions of surface temperatures Jn

an enclosure and the irradiance, or incident radiation, from the target source at

the field-of-view limiter of the radiometer models, first Eq. 9 is substituted

into Eq. 7, yielding

19

d sin which Pi + Oi = Pi where Oiis the total reflectance of the surface and

_=OT 4.

Thus

¢i = (i - pl) EI - gl _I - _i _i (12)

Then, combining Eqs. i0 and Ii and rearranging, yields the following equation

El = _ (_iJ - Fij 0_ )-I FIj(¢J _j + Tj _j). (13)

For brevity, we introduce new variables from Eq. 13, that is

AIj = (_lJ - Fij P_ )-I FiJ Ej-

Thus, Eq. 12 with the above constants can be written

(14)

(15)

EI = E Gj + Rj (16)J A±j Clj

Equation 16 has an important role to couple an enclosure with the neighboring

enclosures through a semi-transparent or transparent boundary surface.

The above relation (Eq. 16) can be used for the cases when an enclosure

is partly or totally surrounded by the neighboring enclosures and also _len

part or all of'the surfaces of the enclosure are non-opaque, i.e., seml-trans-

parent or transparent. To include all the configurations of the enclosures and

to couple them together through interactive boundary surfaces, the above rela-

tion (Eq. 16) can be generalized in the matrix form, that is,

E (17)

20

where A and C denote Eqs. 14 and 15, respectively. The bars above the notation

A and C are to distinguish Eq. 14 and 15 from the notation for the area and

the conduction constant of governing equation to be discussed in the Section 2.1.4.

2.1.3 Coupling Enclosures

In the above Eq. 17, E can be directly related to E by defining the geo-

metrical coupling factor. The relationship between E and E is apparently de-

pendent on the geometrical formation of enclosures through the interactive

(non-opaque) surfaces as in the following example. Consider the system shown

in Fig. 9 which consists of nine enclosures and interacts at the boundaries with

the incident radiation from the environment. The incoming radiation E can be

E =

written

"]17

21]

(18)

•,m l

L -

where the superscript m denotes the number of enclosures in the prescribed

system• Each enclosure of the system has a certain number of interacting sur-

faces. In such a case, the incident radiation upon the surfaces of the ith

enclosure can be described by

Ei J" ill=!EI

1,

E 2

E trl

and 1 _ i K m (19)

21

where the subscript n denotes the number of a surface in an enclosure of the

i 2

system. As shown in Fig. 9, E4 and E 2 define the incident radiation on an in-

teractive boundary surface between the first and second enclosures. The sub-

scripts 4 and 2 here, respectively denote the surface numbers as seen or counted

from each enclosure. These surfaces designated by 4 and 2 in the enclosure 1 and

2, respectively, are supposed to be the interactive boundary surface co-owned by

enclosures 1 and 2. On the contrary, incident radiation E coming from the op-

posite, or neighboring, enclosure upon the boundary surface is defined by ~2E2 and

~1E4, respectively. Accordingly, observing enclosures i and 2 with respect to the

incident radiation on the interactive boundary surface, we find the following

coincidences:

1 2 2 1E4 -- _2 and E2 = _4

Equation 20 tells us that the incident radiation

(20)

1E4, upon the interactive

boundary surface of the first enclosure is virtually regarded as the opposite of

2incident radiation _2' on the back surface of the same boundary, whether the

second enclosure is considered to be formulated or vice versa. Thls coincidence

on each interactive boundary surface is uniquely determined by the formation

of the enclosure in a given system.

To generalize the above relationship, Eq. 20, for the enclosure which

has interactive boundaries and also system boundaries, let us consider the

first enclosure shown in Figure 9. The first enclosure has not only two inter-

active boundaries at 3rd and 4th surfaces, but also two system boundaries at

ist and 2nd surfaces (Fig. 9). Extending the above relationship (Eq. 20) to

include the system boundaries and describing it in a matrix form, we have

22

$2

7

$2

l2 ~i(= E_ = E l) s_

_tE_

4t

2

E_i _Jt

E_ ~_Es

7

E_#

E_ = _S

E s = El

5 ~_ q'

E_,

EiS = _S

E8 = _7 #

_ g

E._

, _

'tE_

_3

E_ = E_

@

E_ _E9 = _.8 4.

2 c,_ @ E_

3

S_A

6

S_

9

7

S, 3 3

FIGURE 9. Interactive Enclosure Model

23

E 1 =

0 0 0 0

0 0 0 0

0 0 0 0

0 1 0 0

E2 +

"0 0 0 O"

0 0 0 0

1 0 0 0

0 0 0 0

E4 +

1 0 0 O"

0 i 0 0

0 0 0 "0

0 0 0 0

s i21)

Likewise, the other enclosures have the same expression as depicted for the

first enclosure. Thus, introducing a new varlable.D for the enclosure coupling

factor, the relationship (Eq. 21) can be written

= DE + FS (22)

where D is a matrix representing the enclosure coupling factors through the

interactive boundaries, F is a matrix representing the system boundary conditions

imposed by surroundings, and S is a column vector of the heat fluxes arriving

at the system boundaries. Now substituting Eq. 22 into Eq. 17 and collecting

the same variables, we have, with I being an Identity matrix

E = (I - CD) -I _Q + (I - CD) -I _FS (23)

Equation 23 illustrates that tile incident radiation E upon a surface of the

enclosure accounts for all the effects of radiation from the surfaces with

different temperatures not only in its own enclosure, but also in neighboring

enclosures, and the heat flux at the system boundaries.

The energy balance at a node or a surface in the enclosure due to the

incident radiation is then described by inserting Eqs. 22 and 23 into Eq. 12.

Then, by eliminating E and _ from Eq. 12, the radiative energy balance is

obtained as only functions of the temperatures of enclosure surfaces and the

irradiance from the target source or any other kinds of heat flux which inter-

acts with the system boundaries. That is,

24

(24)

When the radiative energy balance described by Eq. 24 is applied to the

enclosures obtained by properly dividing the geometry of the system, the

number of the radiative energy balance equations, _, obtained from Eq. 24 is

more than the number of nodal points because consideration is given twice to

the interactive boundary between two enclosures. Accordingly, in the multiple-

enclosure system the radiative energy balance is calculated for each surface

of enclosures. Then, for the overlapped boundary surface, the radiative energy

balances from both enclosures are added together. The radiative energy balance

(Eq. 24)is then plugged into the energy equation (Eq. 3) of the system.

The enclosure coupling technique developed here has many potential

applications and advantages over the conventional methods currently used.

Among the advantages, this technique can be used for any kind of geometry

by only setting up enclosures via dividing the geometry and defining the D

and F matrices in Eq. 22. The enclosure space with irregular geometry can be

properly divided into many smaller regular sub-spaces to adopt the enclosure

coupling technique. For an enclosure with complicated geometry, the division

of the space geometry into well-deflned sub-spaces facilitates the computation

of the configuration factor or the exchange factor, especially for an enclosure

with specular surfaces which is totally dependent on the geometry.

This technique can also be extended to treat radiation interactions in-

cluding the scattering and absorption with optically thin or thick participating

media in an enclosure. Even though there are temperature and density gradients

in participating media, by dividing the enclosure space into sub-spaces which

are small enough to assume the sub-spaces to be isothermal and isotropic, the

25

enclosure theory can be directly applied for the system without loss of gener-

ality.

In short, tile advantages and potential applications of th_ enclosure COul)li,,g

technique are listed below,

ADVANTAGES:

i. It minimizes the effort required to define the exchange factors

among the surfaces in an enclosure with complex geometry, and so

may be able to decrease a degree of possible approximation in the

computation of exchange factors.

2. It may take the wavelength characteristics of materials or participating

media into account in a simple manner.

3. It can be easily coupled with the thermal model of the system so that

the material and thermal properties are considered in the coupled

dynamic system without approximations and assumptions such as those

in the boundary definitions.

4. It enables the system to deal with any imposed initial conditions

and prescribed boundary conditions.

5. It may enhance the accuracy in the computation of a geometrically

partitioned conductive and radiative model.

6. As a consequence, this technique can avoid the length of time for

computation and a cumbersome manipulation which is often revealed in

the other methods such as ray tracing or Monte Carlo methods.

APPLICATIONS:

i. It can be used for any enclosure geometry.

2. It can be used for either the enclosure with a vacuum _pace or the

enclosure with participating media (solid, liquid or gas).

26

3. It can be applied to any optical system's anslyses and design regard-

less of the directional, spectral, and/or temperature dependencesof

its enclosure spaces.

4. Applications of the enclosure coupling technique to optical systems

mayvary from a typical radiometer to any wavegu_desystem.

2.1.4 Governing Equation

The governing equation which describes the performance of a radiometer that

has a dynamic response to an incoming radiance can be obtained by substituting

Eqs. 4 and 24 into Eq. 3 and collecting terms. Then we can separate the in-

coming radiation flux from tlle radiation balance term. The equation can be

written in a general form to describe tile energy balance in a node as fo]low_

= AT + DT 4 + BE + BQQ + b + e

where

i

MR

OAr rI.(I- P-D=--_-- L

D)(I - _m)-1 _ - E]

Ar [ D)(I CD)-I C T F]B ----M--(Z,, p - - -

BQ = _M

b = a bias term

e = a noise term

R = the thermal resistance between nodes (see Eq. 5)

M = VyC

(25)

27

Ar

AC

= the area for the radiative interaction

-- the node area covered by the heat sink heater

Equation 25 represents a general form that consists of a set of equations

for a multinode system. Considering only the thermal model for the nonscanner

total radiometer which is divided into 8 nodes, we have a set of 8 equations

which represent the energy balance in each node separately. The inclusion of the

electronic thermal control system equations will increase the total system by

the number of the transfer functions. The constants of Eq. 25 for the nonscanner,

the scanner, and the solar monitor are described in Appendices C, D, and E.

2.1.5 Electronic Control System Model for the Nonscanner

The block diagramshown in Fig. i0 illustrates the nonscanner ACR control

system model. The control electronics transfer function is of the form (cf.

ERBE Instruments, Monthly Status Report - Vol. i, Nov. 1980, unpublished)

KE(TIS+ i)(T2S+ i)

FE(S) = S(T3S + I)(T4S + i) (26)

To obtain the differential equations describing the electronics transfer

function, we define the state variables TI0, TII , TI2, as follows

• [.. ]TI0 = VB Ka _ Ra(TI) + Rr(T3) - BE + KE nE (27)

R4

Note BE = R3 + R4 is an input constant, where R3 and R4 are the bridge

resistances.

R i(Tj) = A i + B.TIj + CiT_' i = a, r; j = 1,3 (28)

TII" =-_31 (_Tl I + Tl0 + TITI0) (29)

28

l.,¢

_f

I

c_u

o_

T-t-

I

i.,4o

F-4#

,-40o"

uo

o

g

0

_._

o

<_

k__J

go

-r.I

1.1.r,I

IlJ

u

29

Substituting Eq. 27 into Eq. 29 gives

TII = T_ (-TII + TI0) +_3 VB Ka KE Ra(TI) + Rr(T3) +_3 KE nE (30)

• 1 (_TI 2 + + T 2 TII ) (31)TI2 - _3 TII

For full expression, substitute TII in Eq. 30 intoEq. 31.

Note that TI2 is now the output of the electronics, so that the voltage

output is

v = TI2 + nv (327

To convert to counts, C, use the following relation:

C = KcV = Ke TI2 + Kc nv, K = 819.1 counts/volt.C

The heat flow provided to the active cavity to maintain its temperature

at a constant level is given by

Q1 K v 2 K 2 + 2K n + K n 2 (337= a = a TI2 a TI2 v a v

This expression for QI brings to light the major effects that the noise

terms introduce into the operation of the radiometer• Note that due to the

use of square feedback, the linear noise terms acquire a gain of 2K a TI2 ,

while a quadratic noise term (having non-zero mean) is introduced. Coils

which are wrapped around the active and reference cones generate heat and are

expressed by the following terms•

2 Ra (TI)

QJI = VB [Ra(Tl) + Rr(T3)]2

• 2 Rr(T3)

QJ3 = VB IRa(T1) + Rr(T3)]z

30

To obtain the closed-loop system, the expression Eq. 33 is simply

substituted into the equation governing the active cavity temperature.

_ a 2 + a TII vT1 = AI2(T2 TI) + E 4 k 2K n

k Dik Tk + BI E +_i TII M1

2K n 2 R __(TI)

+ a---mY+ [Ra(TI )aM 1 VB + Rr(T3)] z(34)

The block diagram shown in Fig. II illustrates the non-scanner ACR heat

sink heater control system model.

The control electronics transfer function can be approximated in the

form

(THS + i)

FilE(S) = KHE s (35)

The, differential equations describing the heat sink heater controller can be

expressed as

T13 = KH13 KHE (TH - Z2) + KHE nH (36)

VH = T13 + TH KHE KH13 (T H - T2) + T H KHE n H (37)

In Eq. 25 the heat source term, Q, is substituted by Eq. 38 for the aluminum

and copper heat sinks.

2.1.6 Simulation of the Nonscanner Model

The simulation of the ERBE nonscanner sensors is divided into two major

parts; the first determines the coefficients of the system, the second solves

the 3ystems over time and plots the results. Figure 12 is a flow diagram

31

32

I

N_

H

.CX

o 8

o

e-4o

o

.r_c_

b40J

O

OZ

,-"4

c.0

TZ

33

indicating file and program usage. The equation describing the change of temp-

erature in a given node at time i is

DT4(i) Bnn (39)T(i) = AT(i) + + BE + BQQ +

Where A is the conductance coefficient matrix, D is the radiative coefficient

matrix, B is the irradiance source vector and BQ is a vector of coefficients

which describe heating contributions from the control system. T(i) represents

either the temperature change for a thermal node or a voltage change for an

electronic node. The general method of solution is a fourth order Runge-Kutta

algorithm, all of which is discussed in great detail in the following sections.

2.1.6.A Parameter and Coefficient Determination

The coefficient matrices and vectors are determined in the first program

in the sequence ERBEPRM. The data inputted to this is determined from material

specifications and engineering drawings supplied by the manufacturer. The code

is partitioned into 5 major sections, not including the program input/output,

The first of these is the subroutine "COMPUTA". This routine takes the heat

capacities and node to node thermal resistances and returns the conductance

coefficient matrix. The program then enters the routine ADDELC which is designed

to modify the conductance matrix in such a way as to allow the ACR heater con-

trois to use the same equations in the simulation as the node temperatures.

The output from this routine is an enlarged conductance matrix and several elec-

tronic vectors. Once these two routines have been computed COMPUTH calculates

the radiative exchange coefficient between surfaces for each cavity. Due to

the linking of cavities a and b, as defined in Appendix C, dome filter or

fictitious surfaces, the radiative exchange coefficients for these cavities

34

are stored for later use. The results of COMPUTH are then added together to

give the total radiative exchange coefficient matrix _n COMPUTD, COMPUTB and

CMPUTBP. The results are then issued to two files, one formatted for in-

spection and one binary for use by the simulation program.

2.1.6.B Instrument Simulation

The program consists of four logical sections, an Input, the Integration,

the intermediate Output and the Plotting section. The actual solution to the

system of equations that describes each instrument is accomplished by the pro-

gram ERBESIM. The program steps through a determined time interval using an

evenly spaced discrete time step and solves each system using a Fourth Order

Runge-Kutta scheme to evaluate the system at the next step. Each system is

sampled at another rate controlled by the user until 20 samplings are taken

or the end of simulated time is reached. The results are then stored on mag-

netic medium for use by the plotting routines. The final product is the sen-

sor's resposne described by plots and tables of the modeled response of the

instrument, i.e., time-dependent temperatures of the nodes, instrument response

(in counts), electronic voltages (in volts) and input source irradiance (in

W/m2).

The first section of the program reads the coefficient matrices and vectors,

intitial conditions from binary files, and the simulation control variables

from a namelist file. The Namellst file contains variables for controlling

simulation time step size, sampling rates and the combination of instruments

and radiation profiles, etc. which are to be simulated. The simulation is

35

capable of allowing the user to choose either a sawtooth, Fourier series, step

function, constant or ramp radiance profile.

The Runge-Kutta method used was developed by Fehlberg and solves the system

5

Yi,m+l = Y" + h Ajl, m j _0 Kj i'

where

i = i, 2, 3, ... number of nodes

i

Kji = f(t m + C.3 h, Ym + h E_Z_O bjE KEj)

where f is the function describing T(i) from Eq. 39 as functions of time and

temperature, a.,j bjE, and C.3 are constants determined by Fehlberg for solving

first order non-stiff systems of ordinary differential equations.

2.1.7 Count Conversion for Nonscanner

2.1.7.A Nonscanner Total Channel

In the active cavity radiometers, the electronics control the amount of

heat added to the active cavity to maintain the temperature of the cavity

about half a degree above the reference cavity. The control of the heat input

to the active cavity is inversely proportional to the incident radiation en-

tering through the field of view and precision aperture from a given source.

In other words, the control system provides more heat input for the weak incident

radiation and less heat input for the strong incident radiation. Thus, the

voltage driving the active cavity heater provides the sensor output proportional

to the incident radiation. The voltage reading, after A/D conversion, provides

the sensor output in counts. The sensor output in counts is then converted into

engineering units. The algorithms for the count conversion are of the gain/offset

type, and are derived based on the sampled instrument data, e.g., sensor response

(in counts), HK temperatures, etc. and the instruments radiative and conductive

interactions.

36

The following is the count conversion equation for NS total channels.

0 ¸ C,

(40)

The constants in Eq. 40 are defined as follows

B Ka a

A --v B 1

4 3

k_AE,F Dlk TkO

DI8 3

-

DI6 3

A F = - 4 -_i T6o

AI2 bE n Dlk 4" -

2V B

4R B M I B 1

where

V = instrument output (in volts)

(41)

(42)

(43)

(44)

(45)

TH = measured heat sink temperature (OK) at time t

TA = measured precision aperture temperature (OK) at time t

TF = measured FOV limlter temperature (OK) at time t

The remaining quantities, i.e., Ba, Ka, BI, T, Dik, AI2 and bE are con-

stants which are specified for each instrument. These constants for the count

donversion equations are determined for the simulation study of the instrument

model using ground and inflight calibration data. These parameters or constauts

37

may be altered at various points in time, usually after a f] ight calibration,

in order to reflect any changes in the instrument characteristics.

2.1.7.B Nonscanner Filtered Channel for Shortwave

For the filtered channels, the count conversion has a set of two equations

for short and longwaves as follows:

all Es + a12 AT9 = 81 (46)

a21 _s + a22 AT9 = 82 (47)

The coefficients of the terms in the above equations are given by the

following equations,

all ffiBls - BI£(48)

3

a12 = 4DI, 9 T9 (49)O

a21 = B9£ - B9s (50)

a22 = A9, 4 + A9, 8 - 4D9, 9 T_

O 2 ns 23 VB 8r

ns 4 VB ) - 4(k_'6 Dik Tko 16_ M IBI = (AI2 bE - k_ I Dil Tk ° 4R B M 1 - )

(51)

• 33 (TA TA ) _ 4D16 T6 (TF. - TF )* (TH - TH ) - _DI8 T8

0 0 0 0 0

2- B K v -a a s Bie (52)

n! 3ns 3 + A98(TA T9 ) + Ad8(TH T9 ) + 4(k_4 8,9 D9k Tk )_i ffi k--E1D9k Tk - -O O O O

3 3

, (TH - TH ) + 4D98 T 8 (TA - TA ) - 4D16 T6 (TF - T F ) + BI£ E0 0 0 O 0

(53)

38

The filtered channel count converstion is obtained from the solution of

two simultaneous equations (Eqs. 46 and 47). Thus the solution to the equations

is given by

= is all (_22 - a12 a21 (a22 Bl - a12 82) (54)

T9 = I (- a21 81 + all B2) . (55)all e22 - _12 _21

The longwave incident radiation estimate is then defined by subtracting

the shortwave incident radiation from the total incident radiation estimate.

That is,

The constants in Eqs. 46 - 56 are

v = shortwave channel output (in volts)

^ m2E ffishortwave incident radiation estimate (W/ )s

E£ = longwave incident radiation estimate (W/m 2)

= total incident radiation estimate (W/n$)

The temperatures of the heat sink, aperture and FOV limiter, TH, TA, and TF,

are measured as before in total NS channel.

39

,

2.1.8 Solar MonitorModel Analysis

The solar monitor is different from tile nonscanner in size, shape and

design feature as shown in Figure 6. It has a barrel structure ~ 9.5 cm long

between the primary and secondary apertures. The tail of the core body of the

instrument is attached to the housing and includes the primary and secondary

apertures, barrel, heat sink and active and reference cavities. This instru-

ment has a shutter located in front of the secondary aperture. The shutter

operates on a 64 second "on and off" cycle in which it is open for 32 seconds

and then closed for 32 seconds.

Since this instrument deals with a strong intensity of the incident

radiant energy and does not have the heat sink temperature controller, the

heat sink has a larger heat capacity for damping out the effect of large in-

put temperature gradients.

llowever, Jt is desirable to check whether there is any transient effect

due to the periodical on and off mode of the shutter, the strong intensity of

solar radiation, and the excessively long barrel geometry. In other words,

it is desirable to check whether the steady-state approximation is appropriate

to develop the count conversion algorithm for the solar monitor. Thus, we

introduce two simplified geometries here to check how fast the temperature

reaches a stable condition, as every 32 seconds the solar radiation is

viewed.

First, let us assume that the shutter has a rectangular shape of thin

aluminum plate and the shutter is partly exposed to the incident solar radiation

for a 32 second interval of the full 64 second cycle and has a radiation inter-

action with the isothermal neighboring components at any time. Figure 13 shows

4O

the shutter model.

£

!

I

II

I

!

!i

FLGURE i3. The Shutter Model

It is assumed that the ]eft end of tile shutter is insulated and the right

end is attached to a temperature reservoir which has a large heat capacity.

Then the energy balance equation, the initial and boundary conditions govern-

ing the above model can be written as

Dr k _2T _" (57)G.E . PC _ " = pC Bx---_+ 6 -

I.C . T('r = o) = 'rO

B.C • Tx(X = 0, 3) = 0

Tx(X = L, 3) = Tb

The above equation with Neumann and Dirichlet boundary conditions can be

easily solved by introducing a new variable and separating the variables.

Thus the solution to the equation is described by

(-1) n -aAn2T- e

T(x, z) = T b + 2(To-T b) n=O (%nL)

._L2 {+-2k_- [I - (L)2] -4 nZ0

COS(Xnx)

(-1) n -aXn2T }(XnL)_ e COS(Xnx) (58)

41

where T is an initial temperature set by 273.16 OK, L the total length of theO

shutter (6.5 cm), 6 the thickness of the shutter (0.2 cm), k the thermal con-

ductivity (1.557 w/cm°K), _ the exposure length to solar radiation (2.5 cm),

0 the density of the shutter (2.713 g/cm3), c the specific heat of the

shutter (0.887 J/g°K), a the thermal wave velocity (0.647 cm2/sec)

n 2L n = 0, i, 2, ...

"l!

qs is the solar flux (0.142 w/cm 2) x 0.2

= F c O [T__ - T4 ]qo_

'I!

q" = qs'"u(%-x) u(32-z) + qs u(_-x) u(z-64) u(96-T) ... + 2 4"

F the configuration factor between the shutter and the neighboring

components (0.85)

the emissivity of the shutter surface (0.8)

o = the Stephan-Boltzmann constant

To= = the ambient temperature (use T = Tb)

Tb = the reservoir temperature set by 273°K,

T = current shutter temperature, but use T(T-I)

u(x) = the unit step function.

The graphical description of the solution is shown in Figures 14 and 15.

Figure 14 shows that the shutter temperature not only responds to the on and off

mode of the solar flux, but the peak temperatures as well as the minimum temp-

eratures of each cycle (64 sec) also stay at the same ranges. That is, the time

averaged temperature of the shutter does not change sifnificantly. And Figure

14 also shows that the temperatures before reaching minimum and maximum points

become stable. Figure 15 shows the temperatures along the length of the shutter

after i00 seconds passed.

Secondly, the long barrel which is placed in the housing and is between

the pr£mary and secondary apertures is considered by the followlng assumptions:

i) no direct exposure to the solar fl_x

42

t_S

_rm

_a

S

I I I I I I I I I I

TIME( 8EC.t FOR X = 1.5 CH

FIGURE 14. SHUTTER TEMPERATURE AT X ,, 1.5 CM WITH 20% ABSORPTIONt*

OF SOLAR FLUX Qs = 0.142 W/CM 2.

43

I

GOLLI

w

ta.J

b--

o

I I I I I I•6 1.0 ! .5 2.0 2.8 3.0

XICYL_ FORTIME = 100 SEC

FIGURE 15. SHUTTER TEMPERATURE AT I00 SECONDS WITH 20% ABSORPTION

OF SOLAR FLUX Qs = 0.142 W/CM 2.

44

2) radially uniform temperature since the wall thickness is thin

3) no circumferential temperature gradient

4) radiation exchanges only at inside and outside walls

The configuration of the model is shown in Figure 16.

ell

-i...... ..... .....

0 x L --,

FIGURE 16. A SIMPLIFIED CONFIGURATION OF SOLAR MONITOR BARREL

The equations and boundary conditions governing the thermal conditions

imposed upon the above geometry can be written as

aT a2T

G.E. pc-_ = k_-2x + Ro_R i

I.C. T(T=0) = Tb

(59)

B.C. Tx(X=O) = q_'" (set to be constant)

T(x=L) = Tb .

Then the solution to the equation Is obtained as

T ,= T b + 2k(Ro_Rt ) 1 - (L) 2 - _!" L (1 - _)

2k(Ro-R i) el:: 1__.z.)._ _"(L),n) _ ..... L (L_n) CoB (_,nx)l / (60)

45

where

{_11 " |I _ " II= qi qo

43 = FI_IeI_[(T4(x , t-5) - T4(x, t-l)) l

qi'"= ['"-qs "'q_[£2FI-2

q" = Fl_3e3_[T4av(in x) -T4(x=0, t-l)][u(64-T) u(T-32) + u(128-T) u(T-96) + ...]

Tb = 273.16 OK

L -- I0 cm

k -- 1.557 w/cm OK

a = 0.647 cm2/sec

Ro-Ri = 0.5 cm

An = (2n+l_ 7r2L

_s = 0.146 [u(32-T)+ u(T-64)u(96-T)+ ...]

FI_ 1 = 0.75

eI = 0.8

FI_ 2 = 0.4

e2 = 0.08

FI_ 3 = 0.4

e 3 = 0.5

The above values are used in both the shutter and barrel cases. These estimates

were selected to best represent the solor monitor based on the material and

thermal properties of the shutter and barrel. The results of the calculations

described in Figures 17 and 18 demonstrate that the temperatures do not change

very much with respect to time or in the axial direction of the barrel since

they exhibit only about a half degree of temperature change in both cases.

46

FIGURE 17. BARREL TEMPERATURE WRT TIME AT X=3 CM FROM THE SECONDARYAPERTURE.

47

m

Le75.207_

27q.O_r_ _

0

I I I I I I I I I I I I I.77 l._a e.3o s.o7 s.M q.eo s._ s.t_ s.st 7.s_ s.q4 s.et s.y7

XT(CHt FORTIME = 200 SEC

FIGURE 18. BARREL TEMPERATURE ALONG THE AXIAL DIRECTION AT THE TIME OF 200

SECONDS.

48

Therefore, these results show that the steady-state approximation for the

count conversion algorithm does not contain any crucial errors. Therefore,

the open and closed modesof the shutter were considered in the algorithm

development. Namely, the output responds to the direct radiation which falls

into the active cavity according to the shutter's on and off modes. Even

though the instrument has a small time constant and hence it is stabilized

quickly, within the 32 seconds of a half cycle, the output data which is used

for the conversion algorithm has to be picked up at the near-end of a half

cycle just before the shutter closes the aperture.

49

2.1.9 Solor Monitor Count Conversion Al_orlthm

The approximations made here including the steady-state approximation are

the following:

i. The nodal points are selected to be adjacent to where the temperature

probes are so that the temperatures measured by the probes may be

regarded as the temperature of the nodes.

2. T 1 = T 2 + b E where b E is the electronical bias.

3. The temperature of the reference cavity is the same as that of the

heat sink, that is, T 2 = T 3. Accordingly, Tlo = T3o + b E where the

subscript o signifies the initial temperature.

4. T 1 - Tlo can be replaced by T 3 - T3o.

With the above approximations, the equation for count conversion was derived as

v 3 3 - T3o 5 (t) - TSo

+ A 7 [T7(t) - T7o ] + B (61)

where T3, T5, and T 7 are temperatures in OK which are measured in the instrument.

These temperatures are measured and telemetered every 16 seconds. If these

temperatures are measured twice during the open period of the shutter, then the

second measured temperatures can be used for the count conversion algorithm.

Otherwise, as the temperatures are measured arbitrarily without regard to the

shutter on-and-off mode, the temperature measured at or near the time before the

shutter closure must be used. As well as the temperatures measured, the count

has to be picked up at every 40th data point or near the time before the shutter

closure. V(t) in the equation is the instrument output in volts and is tele-

metered at every 0.8 second intervals. The constants in the equations are

obtained from the thermal control electronics and conductive and radiative

heat transfer, all of which interact with the active cavity and the nodes

50

which have temperature probes. In the equation,

Av'- BI

[ 434 DI_IT31o + DI_2T_o + DI_3T3o + DI-4T o ÷ DI-6T6oA3 " - B-_

+ DI_ISTI8 o + DI_x9T319o

---[ + D1-6T6 3 ]4 DI_5T3o oA 5 - BI

= _--- DI_I4TI4 oA7 BI 1-7T o + DI-8T8° DI-9T9° + DI-IIT_I°+ 3

3 + DI_I7T37o]+ DI-15TI56

B _ bE Ek Dlk T4 _ E= BI - BI ko - BI - B_"

(62)

(63)

(64)

(65)

(66)

the quantities such as Ba, Bj, Ka, Kj, V, BI, T, Dik, Tk , AI3 and b E are theo

constants to be specified for the instrument.

2.1.10 Scanner Control Electronics

The control electronics for the scanner have a different feature from

that of the nonscanner. The difference between them is attributed to the

type of sensor in the scanner. The scanner uses thermister blolometers for

both active and reference flakes. When the temperature of the thermister

varies due to the energy exchange, the current through the thermister varies

with the change of the temperature-dependent thermister resistance. In the

scanner instrument, the active flake that is attached to the heat sink in-

directly responds to the incident radiation from the source through its op-

tical arrangement. The reference flake is also attached to the heat sink

and is placed where the space is totally sealed by the surrounding materials

such as the heat sink and housing. Thus it is assumed that the reference flake's

temperarture is equal to that of the heat sink and its housing. Accordingly,

the observed difference in resistances between the active and reference flakes,

is proportional to the incoming radiation.

51

Figure 19 is the block diagram for the scanner control electronics and

their transfer functions.

The thermister resistance as a function of temperature is given by

RI(T ) ffiRi(To) EXP 8i(_i - _) , i = I, 3 (67)O

Using the polynominal expansion of the exponential in Eq. 67, the difference

in resistances between the active and reference flakes is approximated as

Rlo _l R 8 81 833o 3 RIo - R3o

RI(T I) -R3(T 3) ffi_ TI + T_ ° T3 + To(68)

The constant in the bridge is defined by

2VB k_l + k)

K B =-_-- (f + 2k) 2O

The transfer function of the bridge is transformed into a differential

equation as shown in the following

(69)

TI _i + Xl = KB(RI - R3)(70)

where T 1 is a time constant which, for the case of the scanner is less than

I0 ms.

For the preamplifier, we end up with the equation

T2 x2 + x2 = Xl + T3 Xl - Kc CB + (Eos + n) + T3(Eos + *_) (71)

Thus, the following equations describe the energy balance equation (Eq. 72) and

the differential forms of the transfer functions, (Eq. 73) which include general-

ized forms of the amplifier and filter and a model of each scanner channel

= AT + DT 4 + BQ (_8 + BE (72)

52

,,-41

0

i,--I

"1"

÷÷

0

v

i i i i ! I I I i I I I I ! i I I I I

°1°•t" ÷

Co mmml_m

V

I

4-

0

r._

o

,.-4o

l,J

ou

u.r4

0

I.-4 ,sJr._ U

U

!,,.-4

v

m imm _lm m _ mmmo _Qo_m e i_m°|m_m !

53

_: = A x + BT + B (E + n) + B CB + b (73)X OS OS C

_ere T is a 14 component vector consisting of node temperatures, x is a 6

vector consisting of electrical signal processing variables, Q8 iscomponent

the heater heat flow rate, Eos, CB, and b are electrical and flake related vari-

ables, n is the total noise as viewed at the preamplifier input, and E is the

incoming radiant incidence vector. _le measured quantities which are telemetered

to ground are the raw counts, x6, and the heat sink temperature TT, measured by

a thermister.

Ix61y = + v (74)

T 7

where v is the measurement noise including quantization error due to A/D

conversion.

2.1.11 Scanner Steady-State Analysis

Two types of transient behavior will be no¢iceable in the instruments.

One type occurs when the heater Q8 is activated, i.e., when the instrument

is turned on. The transient behavior in this case is rather slow, as the

heat sink time constant is large and the conductivity of the filters (for

SW and LW instruments) is low. Thus, the time period necessary for the

various node temperatures to reach their relatively steady operating temper-

atures, particularly the filters, will be long. The precise period can be

determined by simulation or measurement.

The second and more important transient behavior is due to quick

variations in the input (or observed) radiation incidence, E, after the opera-

ting condition has been reached. The time constants, in this case, are

determined largely by the electronic signal processing parameters and the

54

flake characteristics. After the operating condition has been reached,

the temperature variations of most nodes should be negligible. The active

flake temperature then varies with the incoming radiation. The flake

resistance and other electrical variables also vary with the incoming radiation

and can produce transient behavior which can be observed if the target level

is changed suddenly, as a step function. On the other hand, if the incoming

radiation varies slowly relative to the system time constants, then the system

will operate in quasi-steady state condition. A preliminary analysis indicates

that, for the parameters selected by the designer, the applicable time constants

are in the vicinity of i0 ms. Thus, if the input radiation does not contain

frequencies above i0 Hz, the transient will be negligible, while frequencies

above 17 Hz, will_be attenauted. Whereas the earth scan may have low content

in the high frequencies (_lich are of less interest to ERBE), the transition

from space to Earth may produce a necessarily short period of transient behavior.f

It may be of interest to analyze the magnitude and duration of such transients,

which probably last less than 50 ms.

Since the quasl-steady state condition is of interest to this analysis,

let us first consider one scan period. The scanner looks at space for a period

"sufficient" for the transient effect to die down and obtains an average of the

last few values (to reduce noise and errors) as the "space clamp". Then the

scanner sees the Earth and atmosphere, then looks at two black bodies at ambient,

but known, temperatures as well as the SWICS in the off condition. The k_t

correspond to the kth space clamp, so that T(kAt) and x(kAt) are the temperature

and electronic vectors at the space clamp measurement. Now let

Ax(t) = x(t) - x(kAt), AT(t) = T(t) - T(kAt), kAt < t <_ (k+l)At. (75)

From Eq. 72 and Eq. 73

55

3_AT+ BQA(_8 + BEAT = AAT + 4D Tkt.

A_ = AxAX + BTAT + Ad + BosAn,

(76)

(77)

where it has been assumed that the thermal and radiative properties are in-

cluded in A, D, BQ and B and the time constants for the electronics in Ax, Bos

and B have remained unchanged over the scan period of 4 seconds• Also we canc

define Ad, the drift component of the voltage as,

= + Ab + BC AC BAd AB T T(kAt) + Bos AEos (78)

Considering the amplifier and Bessel filter to be in a quasl-steady

state condition, it can be seen that

Ax6(t) = Kf A--'_2(t)+ Kf _6(t) (79)O o

Q

where Ax2(t) i's the part of Ax2(t) with lower frequency content (below I0 Hz)

consisting of an averaged part of Ax2(t), which may be considered the signal

and _x6(t) is the part of Ax6(t ) consisting largely of noise and irradiancepart,

variations over short distances at the top of the atmosphere with zero mean.

As the filter attenuates high frequencies, Ax6(t ) will have a low amplitude.

The preamp equations are expressed as

T2 A_2 + Ax2 ffiAXl + T3 A_I + AEos + An + T3(AEos + A6), - Kc ACB (80)

Now assume that

T2 _x 2 - T 3 AEos = 0 (81)

then

_x 2 = A-_1 + T3 A--_1 + AEos - Kc ACB + A-_ (82)

56

where E 1

that AEos and AC B are baseband signals.

is the low frequency portion of AXl, and where it has been assumed

Note that

Ax2 = 7x2 + fx2 (83)

where _x 2 contains the higher frequency components of Ax 2 and causes the term

_x 6. The bridge equations are now expressed as

T I ALl AXl TI ATI TI AT3 + Abl + Tl TI+ ffi BTI 1 + BTI 3 ABTII

+ T3 TI ATI +i AT3TIABTI3 + ABTI I + ABTI 3

(84)

It is assumed that the only variable on the RHS containing higher

frequency components is T I, whereas the remaining variables are slow moving

drift components.

_x I = +I(BTII AT I + BTI 3 AT 3 - Ax I)(85)

Combining Eqs. 85, 82, and 79,

_-_2 ffi _-@i AT3) + (+3+I(BTII + BTI 3

+Eos

- +i) +2

- K AC B + Anc(86)

Ax6 Kf ° Tl _i + AT3)= (BTI I BTI3

+ Kf [(T3 - +i ) A--Xl- +2 _2 + Agoso

- K ACB +_'n+ _x6]C

Now consider the thermal equation

N 3 AT k + BI EAT I = AI2(AT2 - AT I) + 4 k_ I Dik T k

N 3 AT k + _IB1E ffi A12(AT 1 - AT2) - 4 k_ 1 Dlk T k

(87)

(88)

(89)

57

1 i

AT1 = Kf T 1 "_x6 AT3 T 1 _Io BTII BTII BTII

(90)

.... K AC B + A_ + A_ 661 (T3 TI) _i T2 _2 + AEos c (91)

Substituting Eq. 88, 90, and 91 into Eq. 89, we have

BIE = (AI2 - 4 DI1 T_)TI Ax 6 - AI2 AT 2 BTI3 (AI2 - 4 DII T_)AT 3

Kf ° BTII BTII

N 3 AT k + A+ I - (AI2 - 4 DI1 T_)

- k_ 2 4 Dik Tk TI BTII 61 (92)

2.1.12 Scanner Count Conversion

The scanner count conversion algorithm below Is based on steady state

approximation. The structure (i.e., the form of the equations) of the algorithm

is the same for each of the three scanner channels. The parameters have

different values for each channel.

The average of scan points during space clamp is given below

V(tk ) 1 n tki)= n i=Zl v( ' nffi 8 (93)

The mean variance of the counts compared with the space clamp during a scan

cycle is obtained by the equation

2 1 n [V(tk) V(tk)]2Ok ffi--ni=_l (94)

The scanner count conversion equation is then defined by the following

equations

t - tk

At

tk. I _ t • tk (95)

58

The coefficients in Eq. 95 are the following

3

AI2 - 4 DII T1 VBo

Av = K Kf zI BTI1 VB(t)o

14 Fl_ 3+ 3+ 3+ 3 3 3

AH = - k _-11 T1 DI2 T2 DI3 T3 DI4 T4 + DIS T5 + DI6 T6

3 + 3 ]- Al2(1+ _) I+ DIST8 DII3TI3J

BTII

A = c

3

AI2 - 4 DII T 1 VBo

K T I BTI I VB(t ) ffic Kfo Av

0

T 1

- _-- (BTII T1o

+BTI3 T3)[VB(tk)- VB(tk_l)]:

+ Kd[Vd(tk)- Vd(tk_l) ]

(96)

(97)

(98)

(99)

tk = tk_ I + At (100)

In the last term of Eq. 95, (t-tk)/At can be replaced by the number of measure-

ments in the scan mode as follows:

t - tk= 0.0135 (5-8) (I01)

At

where J is the number of measurements in the sca_ mode. The constants in the

above equation are the following:

_(tk) = space clamp value at time tk

V(tki ) ffiinstrument output (in volts) when viewing space at tk

2Ok = noise variance estimate during space look

59

v(t) = instrument output sample at time t

TH(t ) = heat sink temperature measurementat t or most recent value (OK)

At = total scan period (4 sec.)

t k = time at end of space look (see.)

t = sampling instant (sec.)

VB(t ) = detector bridge bias voltage measurement at time t or most recent

value (v)

Vd(t k) = drift balance DAC voltage measurement at time t or most recent

value (v)

6k = estimate of unaccounted drift during kth scan period (V)

T = average time lag (sec.)

kf = post amplification gaino

Note that to compute 6k, it is necessary to have the following space clamp

value; i.e., _(tk). If the term with _k in Eq. 95 is neglected, then the count

conversion may be done without using V(tk). The variance o_ is computed for

diagnostic purposes and should be output as described below. The converted

value E.(t - T), i = s,£, T (for shortwave, longwave, and total, respectively)i

can be interpreted as follows

EiCt) = / _i(X) ExCt) d_ + ei(t) (lO2)

th

where Ti(%)__ is the transmittance of the i channel, E%(t) is the spectral

irradlance at the instrument due to radiation from the footprint at time t]

60

and e.(t) is the error in the measurementcombinedwith the count conversioni

process, and can be estimated through analysis of calibration data and model

results.

2.1.13 Accuracy and Diagnostic Checks

During each scan, the instruments view two black bodies at ambient, but

known temperatures. These data can be used to determine if the basic operation

of the instruments have changed as well as provide an approximate accuracy check

of the combined error el(t) in Eq. i01.

i. Store a table of elements

(VBs(TB) , VB_(TB) , VBT(TB) , EBs(TB) , EB_(TB) , EBT(TB)) where T B = TBj,

i _ j K N. The values VBi(T B) represent the steady state output of the ith

channel when viewing the black body at temperature TB, and may be obtained in

calibration tests. The values EBi(T B) represent the output of count conversion

when viewing the black body at temperature TB, given by

EBi(TB) = fTi(X ) EBX(TB) dX (lo3)

where EBx(T B) is the spectral irradiance from the black body at temperature T B.

2. Compute EBi using the measured black body temperature TB(t) as

TB(t)- TBI [EBi(TB2) - EBi(TBI) ] (104)EBi(TB(t), -- EBi + TB2 TBI

where TBI £ TB(t ) _ TB2

3. Compute E.(t) using the count conversion algorithm with the correspondingI

value v.(t).I

4. Plot vi(t) and VBi(TB(t)) on same axes.

61

6. Compute and plot 0_, and

evi,k+l = avi k + (i - a)(vl(t ) - VBi(TB(t)) )

ei,k+l = a eik + (i - a)(Ei(t ) - EBI(TB(t)) )

2 02 [ 1] 2Ox.z,k+l = a vi k + (i - a) (vi(t) - VBi(TB(t)) - _i,k +

m,k+l = a Oik + (i - a) (t) - EBi(TB(t)) - el,k+ 1

(1o5)

(I06)

(107)

(I08)

62

3. INSTRUMENT PERFORMANCE AND COUNT CONVERSION TEST

The instrument models developed for the non-scanner, the scanner, and

the solar monitor are based upon the theoretical approach with a certain

degree of approximation. For instance, the energy equations for the instru-

ments were formulated with the assumptions that the node (or control volume)

has uniform temperature and homogeneous thermal and radiative properties.

However, this lumped geometry of the node, in reality, cannot hold the

uniform temperature through the node geometry unless the temperatures of the

neighbol nodes are tile same, or the node boundaries are perfectly insulated,

or the thermal conductivity of the node material is infinity. Otherwise,

if there is any heat flux allowed into or from the node, a temperature grad-

ient from the center of node to tile boundary will exist.

The radiative properties that are used in the ERBE radiometer instrument

models may be the values picked out of the handbook or measured and averaged

values over a certain band of wavelength. No matter where these values were

obtained, they would introduce some errors when they are used for the compu-

tation of the radiation exchange between the surfaces. The spectral distri-

bution of the emissive energy flux from a source of radiation such as MRBB

may lie on the limited range of wavelength in which the averaged radiative

properties used may be far above or below the real values within the spectral

energy distribution. Therefore, the averaged values may not represent

reality in instrument performance. Also, during long term operation in space,

these properties will be subject to changes due to various reasons such as

thermal and environmental contamination, and aging effect.

The electronic systems are used in the radiometer to measure the temper-

ature difference between the active and reference cavities or flakes, and to

63

digitize the signal, then to transmit it through a telemetry system. The

parts such as resistors of this electronic system are sensitive to a wide

range of temperature changes. For instance, if the resistance of the bridge

slightly changes from the reference point, then the error in Irradiance

measurementdue to this temperature change maybe in error by a few watts

per square meter. Although the instrument models are well-conceptualized and

designed, the real behavior of the radiometers are, as expected, different

from what is was designed to be.

Other factors to be considered, which influence the sensor performance

and measurementaccuracy, are the inflight changes from the states that the

sensor normally operates in and the calibration source variability in its

operation modeor the changes in source characteristics. An environmental

change was expected in the long-term operation of the instrument due to the

orbital changeswhich caused cyclic variations in the heat load of the in-

strument. Another change, was due to the contamination or degradation of

then senosr surfaces due to radiative interactions with other surfaces.

The internal calibration sources such as MRBBand SWICSare also sub-

ject to change. The monitoring of the change in intensity of these sources

by possible variation in power supply and shift of the spectral distribu-

tion in the source output is possible by using the SiPD, although it is also

subject to change, and intercomparing calibration sources.

To account for the possible errors due to the reasons discussed above

a procedure was developed to detect those errors. _,e procedure uses the

algorithm shown in Figs. 20 and 21 for that purpose. This algorithm is

capable of evaluating the sensor model, adjusting the simulation model para-

meters for minimum errors, and establishing confidence intervals for the

64

I|

ll!

|

l

I"

L_I,-I

ORIGINAl] PKGE IS

OF POOR QUALITY.

65

4_

.|

00000 _i

IIIII

III

__ I __ I----_

I

- _N

66

converted data reflecting sensor characteristics and calibration sources.

The sensor simulation model responses, using known calibration sources

as inputs to the model, are checked against the instrument calibration out-

put. Any differences in the slope, offset and curvature between the simula-

tion and calibration results were interpreted as uncertainties of the model

simulating the instrument. These differences, however, generally reflect

the problems discussed earlier. In the algorithm, these differences are

narrowed by checking and adjusting the sensor model, source, and environmental

parameters to minimize the errors. However, the estimated parameters must

stay within the tolerance that can be allowed by the approximations made in

the modeling or by the nature of reasons described earlier.

The algorithm consists of three parts (see Fig. 20):

The first part is a routine to adjust the sensor model parameters by com-

paring sensor simulation outputs such as counts and HK temperatures to the

infllght and ground calibrations obtained for the known sources.

The second part is a procedure to check the errors generated by simulating

a sensor by comparing them to the calibration data for the known sources. It

checks the error generated from the sensor simulation target source, or envrion-

ment at different time intervals for the same known irradlance source as in

the infllght calibration.

The third part consists of routines to search for any changes in the target,

environment and sensor performance due to the degradation of the instrument

surface, vibration, contaminations, and thermal and aging effects on the radia-

tion interactive surfaces.

To do the above job, the algorithm has been designed to have the following

capabilities:

67

i. to adjust the sensor model parameters in comparison with the

calibration data from known sources to minimize the error be-

tween the sensor model and the calibration data,

2. to periodically check any changes in sensor, target and environment

parameters between inflight calibrations by comparing the calibration

data for the same sources at different time intervals (e.g., i or 2

week perlods),

3. to re-define the sensor, target and environment parameters, if

they have been changed,

4. to test and check any differences between calibration data and

responses from the simulated sensor model for the same known

sources,

5. to repeat the routines i through 4 until the errors produced by

step 4 are minimized,

6. to generate the count conversion parameters.

3.1 Description of Routines

3.1.i Routine I

The calibration sources identified during either ground or infllght cali-

brations are numbered by j = i, N at time ti and ti+ 1 where i = I, n. The sensor

simulation model generates temperatures and sensor responses corresponding to the

calibration sources at a time ti. These temperatures and sensor responses, which

are obtained from the model simulation, are compared with measured data to com-

pute errors of the simulation model. Once these errors fall within the pre-

determined error bounds (after repetitive refinement of simulation parameters),

then these errors will be used as criteria to determine the measurement errors

68

against knowntargets on the ground. If not, the parameters for both the tar-

get source and environment effects will be checked; then in turn tile sensor

parameters will be checked and adjusted, _n case when the computedparameters

are not within the acceptable range based on the materials properties, e.g. its

optical properties. Sucha routine will inlcude a decision block to check targets

and environmental effects to avoid unnecessary adjustments of sensor parameters

whenthe errors maybe caused by and comefrom tile target and environment para-

meters.

3.1.2 Routine II

If the errors are within bounds, the sensor responses Vj, j = i, N at dif-

ferent times ti and ti+ 1 will be sorted and evaluated in Routine I to check the

errors from the sensor simulation model for the targets at different times t i

and ti+ 1 .

3.1.3 Routine III

After finishing Routine II, using all the target sources at different times,

ti and ti+l, Routine III checks the sensor respones of the same target at a dif-

ferent time and computes the errors. If the error of the same target at a dif-

ferent time falls within the error bounds, the Routine will check the next tar-

get. If not, the history of target source variation will be studied, and at

the same time the sensor and environmental conditions will be rechecked to deter-

mine the source of errors. This evaluation in Routine I is performed whenever the

error signals the flag. The process continues until the error source is recog-

nlzed and its correction of error is made by adjusting the related parameters in

the simulation model. When this Routine finishes satisfactorily evaluating all

of the target sources, the program will print all the decisions, errors, sensor

69

and calibration source IlK data, sensor parameters and counts and will supply the

final results to the count conversion algorithm.

3.1.4 Routine IV

If the error does not fall within the prescribed error bounds, the target

and environment parameters are checked in order to find whether changes in these

parameters created pseudo-errors in the sensor responses. If there has been a

change in either the target or environmental parameters, they are re-defined

and re-input to Routine I. In spite of repeated checks, if the error cannot be

minimized, then the thorough review of instruments will be required to determine

whether the error bounds are to be reassessed. In this routine, when the target

and environment are checked and relatively free from the erroneous performance

then we would postulate that the instrument's measurement capability might be

deteriorated due partially to the thermal and environmental contamination or by

other means such as aging of the parts and so on.

70

4. DISCUSSION OF PARAMETER ESTIMATION

The simulation code was developed to study the sensor instrument responses

and behavior during the ground and inflight calibration procedures as described

earlier in Section 2.1.6. This simulation code uses the irradiance at the FOV

limiter of the instrument as the input data. All the constants representing

the instrument geometry, material, thermal and radiative properties and the

electronic thermal control systems are parameterized to be used as coefficients

in the system of equations. Then the simulation code generates the counts and

the nodal temperatures of the model corresponding to tile _nput irradiance.

When the simulation code uses the input irradiance which is the same as the

calibration source, the results of counts and housekeeping temperaures from the

simulation are compared to the calibation results. If the differences from these

comparisons for various levels of irradlance are wide, then the simulation model

parameters are adjusted to minimize those differences. This procedure for esti-

mating the model parameters is repeated until the differences can no longer be

reduced. These estimations also continue for various calibration sources until

systematic response patterns for each calibration source are established. All

procedures mentioned above are performed for each individual radiometer sensor.

Even though two instruments of the same type are used in the calibration pro-

cedure, for example, the results may differ due to the inherent differences in

the manyfactors invoZved, e.g. slight differences in the sensor assembly process

of uniform surface roughness, coatings, glue or contact areas.

The following is an example that shows how the parameters for the nonscanner

total channel are selected. Assume that the instrument output derived from

calibration input irradiamces are linear; then the factors which determine the

slope and offset of the cur_e for simulated results would be judged to be the

71

offset temperature between the active and reference cavities, and tile radiative

constant for tlle term of irradlance in Eq. 34, i.e. BI. To determine these two

factors using the measurementdata, the linear form of Eq. 34 can be written

=x -- 8 (I09)

where x is a column vector consisting of the offset temperature and radiative

constantj and a and 8 are defined later• The solution to Eq. 108 is

x = (a' _)-i a' 8

where _' is the transpose of =•

The constant a is a matrix defined by

alj = AI2 + 4 DII(T2) I, E 1

3 E2AI2 + 4 DII(T2) 2,

AI2 + 4 DII(T3)i , Ei

(lZO)

where i is the calibration sequence•

The constant 8 is a column vector

Bi = S1

B2

sll

72

8 as an explicit form is

_i = - Ba Ka V2i - _i - BJ1 QJI

4

_i = kZ Dlk Tki '

(in)

only when k = i, replace Tli by T2i . (112)

The nominal values are used for unmeasured Tki and QJI except T3 = T2 + ST.

The temperature of the reference cavity is a little higher than the heat sink

temperature due to the Joule heating by electric current flowing probe wire

of reference cavity. The temperature difference, _T, due to the Joule heating

cannot be directly measured but can be observed from the simulation results.

The plots of the calibration and simulation results show that they are

not actually linear but could be considered to be linear for estimation (see

for instance, the steady-state response comparison of simulation of calibration

data for the NS WFOVSW sensor on page 100 in Appendix A). Therefore, the above

approach can still adjust the overall slope and offset of the curve by regarding

them as a linear. The linearity and adjustment can be made by adjusting other

parameters. But since almost all parameters are interrelated, the selection

of a parameter that mostly influences the linearity of the curve in a plot of

simulation results must be cautiously taken by testing the linearity with

various parameters•

Since the simulation code was developed, the calibration data for the

nonscanner radiometers using MRBB as a calibration source was the only source

available. Hence, the nonscanner instrument models have been simulated using

MRBB data which was used in the calibration. Figure 22 shows the calibration

data for the MRBB temperatures in the ist column and the count outputs for

the four nonscanner channels in the 6th through 9th columns. The 2nd columm

shows the blackbody radiation corresponding to the temperatures in the ist

73

0_Jv

r.J

t,J

i--i

I

N

_ •

0 I_ C) O_ C) _1 0 '._" ,-I I_ ,'-I I O_ l"_l ,-I

0 (_ C) ,-I 0 ¢'_ 0 "<t 0 u_ (_ ! ,,.0 0 O00 0 0 0 0 0 0 0 0 0 0 0 0

I

O0 U'_ 0 e5 O0 O_ O_

o_ d g d d ,_ ,_

u_

la

r.J

O_f.,.l

6o ._

t_0

_8ZI-I

0

,-I

0

_4

I-4

74

column. The 3rd and 4th columns show the percentages for the shortwave and

longwave portions of the blackbody radiation described in the 5th column,

respectively. These shortwave and longwave portions were calculated by the

integration of Planck's equation for blackbody radiation with respect to the

wavelength from 0 to 5 _m for shortwave and from 5 Mm to i000 _m for longwave.

These portions were used for the MFOV and WFOV shortwave instrument simulations.

The shortwave range selection of 0 to 5 _m was simply made by the shortwave

filter (suprasil) characteristics. In the 6th through 9th columns, the numbers

in the bottom part of the blaock signify the output (in counts) corresponding

to MRBB radiation from the simulation. The 10th and llth columns show the

differences in count between calibration and simulation tests.

75

REFERENCES

i. Barkstrom, B. R., and J. B. Hall, Jr., Earth Radiation Budget Experiment

(ERBE): An Overview Journal of Energy, Vol. 6, No. 2, March - April, 1982,

pp. 141-146.

2. Sparrow, E. M., and R. D. Cess, Radiation Heat Transfer, Hemisphere

Publishing Corporation, 1978.

3. Rohsenow, W. M., Developments in Heat Transfer, The MIT Press, 1964.

76

77

APPENDIX A

PRELIMINARY RESULTS ON ERBE NON-SCANNER INSTRUMENT

MODELING, COUNT CONVERSION AND SENSITIVITY ANALYSIS

, _EC_I, ING PAGE BLANK _O_ __

78,_-. i.,_,_ _L;,NK NOT _$I_11"_1_

GOALS

• DEVELOPA DYrlAMICSENSORMODELFOR EACHINSTRUMENT

• DEVELOPAN ADAPTIVECOUNTCONVERSIONALGORITHMTHAT

CANACCOMMODATECHANGINGCO!IDITIORS

• DETERMIrETHEACCURACYOF THE ERBE INSTRUMENTS

o DETEP_INETHEACCURACYOF THE ERBEMEASUREMENTS

80 .....tNG PAGE BLANK NoT _"II4fED

APPROACHTO ASSESSMENTOF INSTRUMENTACCURACY

• DEVELOPA MATHEMATICALMODEL AND COMPUTERSIMULATIONFOR EACH INSTRUMENT

• COMPAREGROUND CALIBRATIONDATA TO CORRESPONDINGSIMULATIONDATA

• MINIMIZEDIFFERENCEBY ADJUSTINGMODEL PARAMETERS

• PERFORMA SENSITIVITYANALYSIS

• ESTIMATECONFIDENCEINTERVALSFROM GROUNDCALIBRATIONDATA

• REPEATSAME PROCEDUREFOR FLIGHTCALIBRATIONDATA (ERRORANALYSIS)

• ADJUST ESTIMATEOF CONFIDENCEINTERVALSAT EACH FLIGHTCALIBRATION

81

APPROACHTO ADAPTIVECOUNTCONVERSION

• DEVELOPA MATHEMATICALMODEL AND COMPUTERSIMULATIONFOR EACH INSTRUMENT

• COMPAREGROUND CALIBRATIONDATA TO CORRESPONDINGSIMULATIONDATA

• MINIMIZEDIFFERENCEBY FINE TUNINGMODEL PARAMETERS

• DEVELOPA COUNT CONVERSIONALGORITHM(CCA)BASEDON THE MODEL

• SIMULATETHE IrlSTRUMENTAND CCA

• ESTIMATEERROR DUE TO CCA

• REPEATPROCEDUREFOR EACH FLIGHTCALIBRATION

• FINE TUNE MODEL AND CCA PARAMETERS

• OBTAINNEW ERROR ESTIMATES

82

NON-SCANNER MFOV TOTAl.

p

H 2

u

,iV---

"2 P H

1

1 ACTIVE CAVITY

2 Cu HEAT SINK

3 REFERENCE CAVITY

4 AI HEAT SINK

5 BASE PLATE

6 FOV LIMITER

7 AI HEAT SINK

8 APERTURE

H HEATER

P TEMPERATURE PROBE

83

NO N- SCANNER WFOV SW

m

/7e/,I

-, "--I

r

5

m

4

H 2

I

1"I!

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• 2 P H

7

t

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6 FOV LIMITER

7 AI HEAT SINK

8 APERTURE

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H HEATER

P TEMPERATURE PROBE

84

NON-SCANNERDYNAMICRESPONSE

_OV TOTAL SIMULATION

(D 40"6F

_ 40-'H--.,(

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L_ 40.21

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WFOV TOTAL SIMULATION

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g00N

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WI=0V FILTERED SIMULATION

48,2O(.)Q 46.83t_

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W'FOV FILTERED SII_JI_TION

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WFOV FILTERED SIMULATION

30N

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"n_E(SEC)

RADIA_OH ERRQRZERO Lf_

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91

NON-SCANNERDYPIAMICRESPONSE

HFOV TOTAL SIHULATION

40.6

o 40.5 -

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40.1 ------

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cu HEAT SINKTOP AL HEATBOT AL HEAT SII,IK

B.,I_£ PLAll[FOY UMITER

I I100 150

TIME (SEO)

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92

NON-SCANNERDYNAMICRESPONSE

HFOV TOTAL SIMULATION

0.60Q

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MFOV TOTAL SIMULATION

_°° B-

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ac: 300 _--_., "'_-

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co 8200

0o"-"7900

775O

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7450 -

3.33N

II/-3.$31"--I I II

"-5. I I I I0 50 100 150 200

TIME (SEC)

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94

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MFOV FILTERED SIMULATION

48,20 ¸_

(D" _.83--bJ

F-

,'v 4.4.10ILl£L

LIJ42.73I.-.

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oO .60

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flOr|-SCANI'IERDYNAMIC RESPONSE

MFOV FILTERED SIMULATION

2o

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IRRAOIANCECONVERTED COUNTS

A

6500 --Z-_ 645O --00_'_6400 --

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I I I I100 150 200 250

TIME (SEC)

I3OO

97

NOrI-SCANNERSTEADYSTATE RESPONSE

COMPARISONOF SIMULATIONTO CALIBRATIOHDATA

WFOV TOTAL

0 CALIBRA_ONO MODEL

7000--

68(X_

6600--

OvI--

6200

5200; ---o loo 200 _oo 4oo soo eoo 700 =.,u

MRBB IRRADIANCE W/M 2

98

NON-SCANNERSTEADYSTATE RESPONSE

COMPARISONOF SIMULATIONTO CALIBPATIONDATA

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16

12

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I I I I3OO 4OO 5O0 600

MRBB IRRADIANCE W/M 2

I I700 BOO

99

NON-SCANNERSTEADYSTATE RESPONSE

COMPARISONOF SIMULATIONOF CALIBRATIONDATA

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0 CALIBRATION[] MOOEL

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7375

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03 72.75

,<.,7250

7225

720O0

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MRBB IRRADIANCE W/M =

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NON-SCANNERSTEADYSTATE RESPONSE

COMPARISONOF SIMULATIONTO CALIBRATIO_IDATA

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MRBB IRRADIANCE W/M 2

I I700 BOO

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COMPARISONOF SIMULATIONTO CALIBRATIONDATA

MFOV TOTAL

0 CALIBRA_ON0 MODEL

102

82oo

81oo

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G"7gO0

_:_7700

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7,400

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720Oo

I I I I I, I I I1 O0 200 300 4.00 500 600 700 BOO

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NON-SCANNERSTEADYSTATE RESPONSE

COMPARISONOF SIMULATIONTO CALIBRATIONDATA

MFOV TOTAL

fa3l"-Z

0(0V

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NON-SCANNERSTEADYSTATE RESPONSE

COMPARISONOF SIMULATIONTO CALIBRATIOIIDATA

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NIRBB IRRADIANCE W/M z

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104

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NON-SCANNERDYNAHICRESPONSE

WFOV TOTAL SIMULATION

TIME DELAY (LAG) ESTIMATION

RRARANCE.... _ COtWm

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106

NON-SCANNERDYNAMICRESPONSE

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TIME DELAY (LAG) ESTIMATION

40.6

0o 40.5

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_3n

40.2

40.1

40.0

m

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ACTIVE CAVITYREFERENCE CAVI"FYAPERTURE

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MFOV TOTAL SIMULATION

TIME DELAY (LAG) ESTIMATION

450

_,.375

0

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150

75

IRRADIANCECONVERTED COUNTS

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.oolTn A

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TIME (SEC)

NON-SCANNERDYNAMICRESPONSE

MFOV TOTAL SIMULATION

TIME DELAY (LAG) ESTIMATION

40.4

n

_l_j40.2

p

........._ I

1--

40.1 ........

40.t __

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TIHE DELAY (LAG) ESTINATION

0.600

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TIME DELAY (LAG) ESTIMATION

IRR/_ANCECONVERTED COUNTS

_S.Ol"-

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112

NON-SCANNERDYNAMICRESPONSE

WFOV FILTERED SIMULATION

TIME DELAY (LAC) ESTIMATION

48.20

0o 46.83

W0C

2 45.47

_bj 44.10O.

41.37

40.00

m

m

ACTIVE CAVITY

REFERENCE CAVITY

APERTURE

SW FILTER

40.05

Oo 40.00

LOC__9.95I--

Q.

39.80

39.75

m

CU HEAT _NKTOP AL HEAT _NK

BOT AL HEAT _NK

BASE PLAI_FOV UUITER

26.10

(.)o 25.88

ILln,,:::)25.67I---<C

IZ25,45

L_J25,231--

25.02

24.800

B

m

I • I I I I I1O0 200 300 400 500 600

TIME (SEC)

NON-SCANNERDYNAMICRESPONSE

WFOV FILTERED SIMULATION

TIME DELAY (LAG) ESTIMATION

b-•< .43rvtO

.35 --

Lo_- .27 --b-tuo,I.18 --LL.

0! i I

11111

k£T CAV - REF CAVACT CAV-CU HEAT SINK

200 30O

TIME ($EC)

I i I4OO 500 800

114

NON-SCANtIERDYNAMICRESPONSE

MFOV FILTERED SIMULATION

TIME DELAY (LAG) ESTIMATION

15.0

12.5

10.0m

7.5

5.0

2.5

0

RRN)IkNC(CONVERTED COUNTS

_ -_ I ° ___ -.. /_..-

u) 65O0F-Z

(_ 6450ov 6400F-

6350

63O0

Cn 8250ZLU(n 8200

m

m

w

m

m

RADIATIONERRORZERO LJNE

5"00 f!'V" _

3.33

,,,oL___\, / \-1.6"zi-- _.-.-.----._-

I

_"-5.00! I ! J t ,! I0 100 200 300 400 500 600

T_ME(sEc)

IIS

NON-SCANNERDYNAMICRESPONSE

MFOV FILTERED SIMULATION

TIME DELAY (LAG) ESTIMATION

48.2O

(.,)• 48.83t.d

::)45.47

_j44.10

9.

42.73

41.37

40.00

40.05

• 40.00ILln,

_3g.g5.<

I_J39.909.

_ 39.85

3g.80

39.75

26.10

t)o 25.88n,n,v

25.67

_1_j25.459.

n_,125.2.3

m

am

m

.,.. _ .,, ..

m

m

m

u

m

I--

25.02 --

24.8O0

AClNI CA_IYREFERENCE CATTYAPERIU_SW FILIER

w

I , , I I i

CU HEAT SINKTOP AL HEAT SiNKBOT AL HEAT SINK

BASE PLAIIFOV UMI'II[R

"" t..

]100

I I200 3O0

•nM_"(sEc)

I I I .,400 500 600

116

NON-SCANNERDYNAMICRESPONSE

MFOV FILTERED SIMULATION

TIME DELAY (LAG) ESTIMATION

Q

LU .52 --r_Dk-

.43

LIJ_.._--

_ .27 --

F-

i.

0.100

ACT CAV - I_ CA','ACT CAV-CU HEAT SINK

I I I I I JtO0 2OO 3OO

TIME (SEC)

400 500 600

117

SENSITIVITYANALYSISUSINGDYNAMICMODEL

NFOV TOTAL SIMULATION

ACTIVE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .92)

n- 318.67 _-_...,-- ---

'=E

_S200 F

0761_ -

(/) 7450 --

(I) 73001 _

15FT

IL,M_A'IION ERRORZERO UNE

0 50 I00

I150

"riME(sEc)

I2OO

I25O

!

118

SENSITIVITYANALYSISUSlrIGDYNAMICMODEL

DflPOV TOTAL SIMULATION

ACTIVE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .92)

40.6

Oa 40.5Ldn,D 40.4I--.<

_uJ40.311.

uj_40.2

40.0

40.05

O" 40.00tUn-

39.%5<

39.90O-

39.B0

39.7;

26.15Oo 26.10b.IC_

26.O5.<

b._26.00a.

_t_j25.g5

25.8,5O

A

mm_mNmmm. .....

m

m

--mmmm_mm--.

!5O

!IO0

1150

liME (SEC)

I

AC11VECAVITYREFERENCE CAVITfAPERTURE

OU HEAT S_IKTOP AL HEAT SINKBOT N. HEAT SINK

BASE PLATEFOV UMITER

1250

I

119

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

ACTIVE CAVITY EMISSIVITY CIt_NCED BY .02 (NOMINAL .92)

.60 --

V_

A-WO..35 --

bU.27 --

F-W(_ .IS --b.b.O .I0

0

I I I I50 100 150 200

"riME(SEC)

ACT CAV - _ OAVACT CAV-OU HEAT SINK

I J

120

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

REFERENCE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .95)

eSO.OO

_ 791.67

Z 633.330

__475.00,

<:rv, 316.67

158.331

i"T

mADIANCE_TEO COUNTS

f-%

_o 8200

0"-" 7900I--

n 7750I--

O 76OOn,oul 7450zLUu_ 73oo

m

m

D

I_IATION ERRORZERO IJNE

I300

12!

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SI_LATION

REFERENCE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .95)

40.6--

0

" 40.5 ___.b_

40.4 --<n- 40.3--bUn

i_.i40.2--F-

40.1 ----

40.0--

40.05 --

0o 40.00tUiv

39.95 --<

_139.90 --n

_ 39.85 --

39.80 --

39.75 _

26.15 --

0" 26.10-wiv"_ 26.05 --

_ 26.00 _n

_ 25.95 --

25.90 --

25.850

CU H(AT SINKTOP N. FEAT SINKBOT N. HEAT SINK

BASE PLA11[FOV UI411T.R

I I I I50 100 150 200

TIME (SEC)

I25O 30O

122

SENSITIVITYANALYSISUSINGDYNAMIC,_IODEL

MFOV TOTAL SIMULATION

REFERENCE CAVITY EMISSIVITY CHANGED BY .02 (NOMINAL .95)

(.) .600

.52

_.43

m

i

v

b.I0...35 _-,_

.27

I--

.18

1_ .100

I

CAV - RE}"cAVACT CAV.-_ HEAT

I I I I I100 150 20O 25O

TIME (SEC)

123

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

DOME FILTER LONC WAVE TRANSMITTANCE. (LEAK) CHANGED BY .005 (NOHINAL .01)

950.00

_791.67

Z 633.33O

<C 475.00

<:IZ 316.67

158.33

!

m

IRRADIANCECONVERTED COUNTS

f-%

ul 8200I--Z

805000

7900I-

a. 775OI--

O76OO

n-OUl 7450ZI.iJ(/) 730O

m

m

RADIATION ERRORZERO LINE

15-

i 10:

5

-I0 --I

'_-,5 _1 I I 1 I I0 50 I O0 150 200 250

TIME ($EC)

I3O0

124

SENSITIVITYANALYSISUSINGDYrIAMICMODEL

MFOV TOTAL SIMULATION

DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)

40.6

0

o 40.5 =-.--.----Wo,

40.4 --

<

o- 40.3 --LOtL

_L_40.2 --

40.1 m_.

40.0 _

40.05 --

0

o 40.00

tvZ3 39.95I--

_bJ39.90 --

_t_j3g.85 --F-

3g.80 --

39.75 --

26.15 --

0o 26.10-

hJo"D 26.05 --I--.<

_1 25.g5 --I.,-

25.g0 --

25.850

ACTIVE CAVITY

REFERENCE CAVITY

APERTURE

CU HEAT SINK

TOP AL HEAT SINK

BOT AL HEAT SINK

BASE PLATEFOV UMITIER

I50

I100

I150

TIME (SEC)

I200 250

J3OO

•. 125

SENSITIVITYANALYSISUSINGDYNAMICMODEL

HFOV TOTAL SIHULATION

DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)

O .600

.52

._.43n-W_.35

WI--- .27I..-WU') o18b-

O .100

I50

ACT CAV- REF CAVACT CAV-CU HEAT SIHK

I100 150

TiME (SEC)

I i I200 250 300

12_

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV FILTERED SIMULATION

DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)

3O

_25

Z 200I-'<15F,.<oc10

IRRADIANCECONVERTED COUNTS

(/) 6500--i--Z

6450-0

6400 --

S6350 --"-

6300 --

0u') 6250-z

6200-

15Fn- 5

oo_c_LO 0

z0 -5I-

-lo -

n.-15

o

RADIATION[RRORZERO LINE

I5O

I llO0 15o

TIME (SEC)

I200

L250

J3GO

.Ci_3'

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV FILTERED SIMULATION

DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)

48.20

0o 46.83

b.I

45.47I--_CC_ 44.10L_JCL

41.37

m

m

m

40.00 ,,,-

40.05

Oo 40.00 _,-ILl

=) .19.95 --I--

OC 3g.90WO."s

39.85 w

39.80

3g.7_ m

26.10 m

0o 25.88WC_:D 25.67

L_j 25.45

[1.

_L_j25.23

ACIWE: CAVITY

REFERENCE CAVITY

APERTURE

SW FILTER

I-"

25.02

24.800

CU HEAT SINKTOP AL HEAT SINK

BOT AL HEAT SINK

BASE PLATE

FOV UMITIER

I5O

1 I !100 150 200

TIME (SEC)

I25O

JY_.O

128

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV FILTERED SIMULATION

DOME FILTER LONG WAVE TRANSMITTANCE (LEAK) CHANGED BY .005 (NOMINAL .01)

O .60o

W .52Q::

<[; .43

m

m

L_

11..35--Iw

.27 --p-bJ(,/) .18 --I,.i,.h0.10

0 50

ACT CAV - RIEF CAVACT CAV-CU HEAT SINK

100I

150

TIME (SEC)

t t I200 250 3uO

129

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)

950.00N

_E791.67

3=

Z 633.330F--_.< 475.00r-_<rr- 316.67

158.33

I

I%I!

3-

IRRADIM,_ECONVERTm COLt,ITS

B200 --Z

8050 --c.)"_ 7900 I

_ 7750

7600,4--

_7450 -Z

15_

n." 50

n--O_-

Z0 -5--F

-,O-L/I_ -15

0

R_DIA_tC,N£RROFZERO L_

I I_o !_.

t I i150 2bO 25G

TIME (SEC)

J3:>0

130

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

DOHE FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOHINAL .946)

(.)40.6_. _

F

P.,AVII'YREFERF.N_ cAvn'¥/F'ER'I_

£.)40.00

hin-O 39.95 --

b.I0.

39.80--

28.15

rJo 26.10hin-

26.05.<n-"26._1W

_ 25.95p-

25.00

0

CU HEAT SINKTOP ,14.HEAT 9NKBOT hi. HEAT _INK

m

50 lClO

I L 1150 2b_ 250

TIME (SEC)

131

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)

O .60Q

l.d .5?.n--

_.43n-Ld(1- .35 --

L_

I- .27 --

t--tU

.18 --I.Lb_o .I0

o

.P_ mw

ACT CAV - REF C_kVACT CAV-CU HEAT SINK

I IIoo

I150

TIME (SEC)

I I i21)0 250 300

SENSITIVITYANALYSISUSINGDYNAMICMODEL

NFOV FILTERED SIMULATION

DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)

Oo 48.83W

iv 44.10ILlI1.

LU42.73I-"

41.37

40.00

m_ m_m

m

m

o 40.O5 F

_ _._op-

_ 3g.85 _-

:.:L

20.10Uo 25.88bJn_:D 25.67I-<_

_ 25.45D.

LJ 25.23 --1--

25.02 --

24.800 5O 100

I I15o 200

TIME (SEC)

.*t"llVl: _wrlYIt_F[ItEN_ cAvn'Y.IJ_RTIRE_V FILTER

CU N[AT SINKTOP AL Nr.AT SIl_BOT #4. H[AT SNK

I250

i3OO

133

SENSITIVITYANALYSISUSIIIGDYNAMICMODEL

MFOV FILTERED SIMULATION

DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)

_ON

Z20OD

n, 10

5

0

D

A

t-I-

I!

.-I-!II

"rI!

IRRAI_ANCr"CON'VERTEDCOUNTS

(.J"-" 6400

_i_

Z

o'_ 6200'__

o 5o

RADIATION ERRO_ZERO LIN£

IOOI

150

TINIE(SEC)

I I I200 250 300

134

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV FILTERED SIMULATION

DOME FILTER SHORT WAVE TRANSMITTANCE CHANGED BY .005 (NOMINAL .946)

0.600

I.,.I0-.35 --_E

.27 -

I--

.18b_

0 .lC0

AC'TCAV - R[F CAV^C7 CA,V-CU HEAT SINK

I II00

I I I J150 2DO 25O _00

TIME (SEC)

i

SENSITIVITYANALYSISUSIflGDYNAMICMODEL

MFOV FILTERED SINULATION

DOME FILTER LONG WAVE EMISSIVITY CHANGED BY .005 (NOMINAL .69)

20

< 1.5

!!

TI

.-L.III

-I-II

IRRADIANCE

CONVERTED COUNTS

65OO--I--7

6450 --O(D

_-_ 6400 --p-

11. 6350

O6300--

n-OCO 6250 --ZLd(Jl 6200 __

-10_-_ -,s I0 50

1150

lIME (SEC)

I200

RADIA'rIDh ERROR

Z][RO _NE

i250

J

136

SENSITIVITYANALYSISUSINGDYNN'IICMODEL

M_'OV FILTERED SIMULATION

DOME FILTER LONG WAVE EMISSIVITY CHANGED BY .005 (NOMINAL .69)

m

4O.O5

0" 40.00LUcf

31)J5<

30,1_

30.75

D

n

m

m

m

CU HEAT SINKT0P N. HEAT SINKBOT AL HEAT

26.100• 25_sLU

m

<ew 25.45 --tuft."stu 25.2.3--k-

25.02 --

24.8O0

I5O

I I I iloo 150 200

TIUE (SEC)

I3OO

137

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV FILTERED SIMULATION

DOME FILTER LONG WAVE EMISSIVITY CHANGED BY .005 (NOMINAL .69)

0.60o

m

m

_KWO._18 -

_.?.7 -

_ .100

ACT CAV - _ CAVACT CAV-CU HEAT _RNI(

I50

I100

I I I2OO 2.5O _00

138

SENSITIVITYANALYSISUSINGDYNAMICMODEL

HFOV TOTAL SIMULATION

SENSOK BRIDGE VOLTAGE CHANGED BY ,01 v (NOMINAL 10 v)

950.00

"_. "/91.67

633,33

_ 475.00

(2C316.67

158.33

m

r

m

ImADUUW_CONVUtlIO COUNTS

Ul 82OOI--

_8050C.)"'79OO

m

300

139

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

SENSOR BRIDGE VOLTAGE CHANGED BY .01 v (NOMINAL 10 v)

40.6 m

vO

* 40.5 _...._t.u

40.4

_j 40.3

40.2 m

40.1 ----

40.0 --

40o0_ m

0• 40.00bJ

39.95 --

_L_j3g.90

_ 39.85

28.15

0o 26.10i.d

_ 26.05

e_ 26.00 "_W

I_ 25.95 _F-

25.90 _

25.85 i0

ACTIVE CAVITYREFERENCE CAVITYk1_rR11.11_

CU HEAT SBIIKTOP AL HEAT'SB'IKBOT AL HEAT SINK

EIASE PLA11[FOV JMITER

I 1 I ISO 100 150 2O0

•nuv(sEc)

I J30()

140

SENSITIVITYANALYSISUSINGDYNAMICMODEL

TOTAL $II_J_TZ_I

SENSOR BRIDGE VOLTAGE CHANGED BY .01 v (NOMINAL I0 v)

O.OO

"_ .52n-

_.43

Cc.lq --

_.27--

k-

.18 --

_.I00

I

CAV - R_ CAVCAV-.CU HEAT SINK

ItOO

I I I I200 250 ,_

141

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMIfl_TION

ACTIVE CAVITY HEATER RESISTANCE CHANGED BY .I fl (NOMINAL 1492.3 fl)

g_00t_

_ 833.33

_ 475.00

tv 316.67

158.33

m

m

B

"1"

m

m

• (:_IVE:Rll_ C0UN_

8200--

(3"'_ 7900 m

::)0-7780I-

O 7600 -iv,Ocn 7450 -zLUV) 7300 _

15

o-I0=,slL/

_AI_N ERRORZIRO UN_

I I5O 100

I

TIME (SEC)

I I J2OO 25O 3OO

142

SENSITIVITYANALYSISUSINGDYNAMICMODEL

14FOV TOTAL SIMULATTON

ACTIVE CAVITY HEATER RESISTANCE CHANCED BY . 1 L'I (NOMINAL 1&92.3 _)

.,C

,'1

40.1

4O.O

m

R

_lliJlli

RE]_"IE_E]_XcA_rY

wiv

39.05

o.

Immm_,_m,m.

n

QU Frr.AT g, iKTGP AL HEAT SINKBOT N. HEAT _gNK

r,_ 26"15 r

I50

I100

I I I200 25O _mO

143

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

ACTIVE CAVITY HEATER RESISTANCE CHANGED BY .I fl (NC_INAL 1492.3 fl)

b4

WP'.27 --

_.lS --

.10o

I

ACT CAV- _ O.AVACT CAV--CU HEAT _NK

I100

I1.50

"hUE(s_C)

I I I2._ 30O

144

SENSITIVITYANALYSISUSINGDYNARICMODEL

MFOV TOTAL SIMUlaTION

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANCED BY .01 i_ (NOMINAL 2319.1 f_ e 40"C)

IImAOL4_[¢ON_TI_ COUNTS

_ B200 --

_OOm

7eO0 -

_7450 -

_730C _

. 15

-I0

n- -15

RABIAIION ERRORZERO LINE

0 50I

tOOI I I

150 2DO 2.50

_.E (SEC)

!

i43

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .01 _ (NOMINAL 2319.1 _ @ 40eC)

O 40.6_....___

'oE.......

ADTI_ CAVII_REFERENCE C_kVIT*_APERTURE

4O.O5

U6 40.00w

m

mmmm.m.m..

NEAT SINKTOP ,'41.HE,AT SNIp,BOT N. HEAT SII,_

26.1,5 --

U• 26.10w

w

I-

_._0--

0I

5O tOO 150

TIME ($EC)

i2OO

l2_

i3OO

146

SENSITIVITyANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANCED BY .01 _ (NOHINAL 2319.1 l_ @ 40"C)

OAV - REF OAr'CAV-¢IJ HEAT _,_K

.27

_"'1-

.,ol I I I I , I I50 tO0 150 _ _ 3_

"riME(SEC)0

i47

SENSITIVITYANALYSISUSINGDYNAMICMODEL

HFOV TOTAL S_TION

REFERENCE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .01 _ (N{_4IR_ 2327.1 _ _ 40"C)

I_O.00

_,79! .67

Z 433..3.1O

r_ 316.6"/

1_S.33

m

i

_ _ _ _ ..... immmw_wm,amm,,m,em_

i

IRR._N_C(CONM[RTm COUNTS

7900 I

7eO0 -

_ 74,50 --ZL_(n 73o(

15-

0:: ___00£n"t_ 0 "_

Rb/)IA'TIONERRORZERO L_r

.L I I150 200 2.5O

_E (SEC)

148

SENSITIVITYANALYSISUSINGDYN_IC MODEL

TOTAL SI_[JI_TZOI¢

_Cg CAVITY S_SO& _ RESISTANCE CHA_ED BY .01 fl (_ONI_,L 232_.1 fl _ 400C)

40.6_._.

::r-................

RErlD_NCE CATTY

,U'EI_TURE

40_5

°4OO0

W

30.8O

3g.75

20,15

rj• 2_.1oL:.In..

,<

e" 28.1 _W

,-,25.95

0

m

m

ra-

m

m

m

CU HEAT SINK

TOP AL NEAT SINK

BOT N. I.fAT gNK

PL,ATEF'OV UldnT.R

ISO

ItOO

_ME (SEC)

!2OO

1I ,

149

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

REFERENCE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .0l _ (NOMINAL 2327.1 _ @ 40"C)

0.60O

LU ,52£E

_.43,wWO..15

LU(nb.

_S mbmma

m

m

.27--

.16 --

.10 I0 _o

CAV-CU l,,(A'r._.N_,_

I I I I I1oo 150 290 25o

TIME (SEC)

1.50.

SENSITIVITYANALYSISUSINGDYNAMICMODEL

IO_V TOTAL SIHULATION

SENSOR BRIDCE RESISTANCE CHANGED BY .01 it (NOHINAL 2210 it)

9riOJ30N

79! .67

_ 316.a7

IRRADIANC[CONMgTII_ COUNTS

(Jv 7000

N 15

Ld 0

-10-15

-J

I ,I I50 tOO

R.N_^'ll_N ERRORZ_RO LIH£

l I I200 25O 3OO

151

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SII_LATION

SENSOR BRIDGE RESISTANCE CHANGED BY .01 i_ (NOMINAL 2210 i_)

Jm

0'.01.--...............

(.Io 40.00h3

_39_

_30_0

(I.

38_0

30.75

2_.15Ue 28.10bJ

,<r,- 26.OObJ0.

m

m

m

m

m

m

minE,

ACTIVE CAVITYREFE)_D4C( CAVITYAPERIURE

CU H£kT SINKTOP AL HF.AT 9NKBOT #4. Hr_T 5NK

I I I I I5O 100 150 200 250

_.E (sEc)0

I

152

SENSITIVITYANALYSISUSINGDYriAMICMODEL

MFOV TOTAL SIMULATION

SENSOR BRIDGE RESISTANCE CHANCED BY .01 _ (NOMINAL 2210 _)

,,vLU

I-I._(r1.16b.b.0,10

0

I 1 I I_ME(SEC)

CAV - _ CAVAG"TOA¥-CU I,,ll_T SINK

I I

153

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)

. 95O.OO

Tgl.e7

316.47

1M.._

ItV"- ..................T

m

(n s2oo

m

::3

::3

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0 ----

15

_ 5

1_-10

r,, -15

m

m *,

t0 50

ZERO UI_

IIO0

I I I150 2OO ZSO

_ME(sEC)300

154

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

ACTIVE CAVITY SENSOR WIRE RESZSTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)

40.6--

Ld _ ....

40.4 --

n'40 _ --LU0.=i

40.1 m ........

4_Om

• 40.00L_

.......

26.15 --

O• 26.10-LUn,

26.05 --,<

0.

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0

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ak_E PI.ATEFOV IJMITER

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TIME (SEC)

I25O

I300

155

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SI_..ATION

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)

ACT OAV- REF C,AVACT CAV-,.CUFEAT

.,o_ I I0 BO 100

I150

TIME (SEC)

I2OO

I . I

156

SENSITIVITYANALYSISUSINGDYNAMICMODEL

9,50.00

'_791.8"/'

_I:_475.00

n. 316.67

158.33

iI

7-II

.4II

J

HFOV TOTAL SIHUTATION

REDUCED CONDUCTANCE BETWEEN NODES 1 AND 2

IRRN)Ik_ICE(:OI_[RTED C'_JNT_

!

I t_

'_IVI!

; !3

Z8O5O --

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f,Z

5

°!

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ry -15 10

l

I

m

.m

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_ _ _i___.......

j t/ , ,200 300 400

TIME (SEC)

I_ATI_ ERRORZI:RO

5OO 6O0

157

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

REDUCED CONDUCTANCE BETWEEN NODES I AND 2

40.6

Q 4_LU

n

40.1

4O.O

40.05

O* 4_00LU

_._

(I..

LU 39.85F-

39.80

_.75

F'-V

AC'I_ C.AVITYmc[ cAvrr_APL_'n_

m

m

n

m

D

CU I._ATTOP AI. IEATBOT AL HF.AT

m

m

m ,v

BASE PLA11[FOV LIIml[R

26.15 --

Oa 26.10-LU,w

26.05 --

r,, 26.00Ld(3.

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2-_gO--

"OJ

1ooI I I

200 3O0 4O0

•n_E(sEc)

I J5OO 600

158

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

REDUCED CONDUCTANCE BETWEEN NODES I AND 2

O .60

ivLOU..35

b.O

.IS

.100

ACT CAV - REF CAVACT CAV-CU IEAT SNK

m

I

I I I I100 200 30O 400

TIME (SEC)

I I5OO SO0

159

SENSITIVITYANALYSISUSINGDYN_IC MODEL

MFOV TOTAL SD4ULATION

REDUCED CONDUCTANCE BETWEEN NODES 1 AND 2

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 f_ (NOMINAL 2319.1 f_)

I_0or7"3

N,it'

" i-rl ;/158.33 _- i uoU_

f

M;llll3[CIIIVER'IED

" 8200 i --

ao,sol

7gOOi --I-,'-

0. 775O --

O7600 -

(z:O

7,k_i -Zu,I

RADIATION ERROR

I

"hUE(SEC)

I I I6_

160

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

REDUCED CONDUCTANCE BETWEEN NODES I AND 2

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)

40.6

Oe 40.5bJ

4_4

n:E

4Q.I

4O.O

4O.O5

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_-g'_

(I.

3.9.85

_.80

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CAVITYREFERENCE CAVITYAPERTURE

m

CU HEAT "INKI"QP AL HEAT _MKBOT AL HEAT _NK

m

m

w

BA._E PLA11[FOV LIMITER

26.15

0• 26.10-Ld

_ 26.05 --.<

w_j26.00 _:_(1.

25"t_ F

0 20OI

T_ME(sEc)

I40O

I5O0

J6O0

161

SENSITIVITYANALYSISUSINGDYNAMICMODEL

MFOV TOTAL SIMULATION

REDUCED CONDUCTANCE BETWEEN NODES 1 AND 2

ACTIVE CAVITY SENSOR WIRE RESISTANCE CHANGED BY .5 _ (NOMINAL 2319.1 _)

twLUD-._5 --=I

I--bJ

.18 --IJ.u.O .10

0I

IO0

ACT CAV - REF CAVACT CAV-CU HEAT _gNIK

I I I2.0O 30O 4OO

"riME(sEc)

I ISO')

I

i-

162

_GIN._ PXGE

OE'_OR QUALITY

163

CONCLUSIONS

• DYNAMICMODELSHELPTHE UNDERSTANDINGOF THE EP_E INSTRUMENTS

• SIMPLECOUNTCONVERSIONALGORITHMSHAVEBEENDEVELOPED

STEADYSTATESIMULATIOIIAND GROUNDCALIBRATIONRESULTSARE IN

GENERALAGREEMENT

• THE ERRORSDUE TO COUNTCONVERSIONARE BEIMGANALYZEDBY SIMULATION

• INITIALSENSITIVITYAND ERRORANALYSESPROVIDECONSTRUCTIVERESULTS

OVERALLAPPROACHSEEMSWELL-SUITEDTO ASSESSINSTRUMENTPERFORMANCE,

ACCURACYAND VALIDATION

_lNo, PAGeet,aNIINO'I'

165

APPENDIX B

El{BE SCANNER SIMULATIONS

.¢

_cm_o P_o_SL_Ir_ rR4m_•

167

These graphical displays were produced during development of the simula-

tion model. Please note that the plot of "OUTPUT (CNTS)" versus "TIME (SEC)"

in the graph on pages 169, 173, 174, 177 and ig0 does not display quantita-

tively absolute values for counts generated from the simulation model due to

uncertain gain/offset parameters in the electronics.

168

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183

APPENDIX C

ERBE NON-SCANNER THERMAL MODEL

B,.

PRBCED_ PAGE BLANK NOq' FILMED

185

_J

NON-SCANNER MODEL

The energy balance equation

n T. - Tkr J

Mk Tk -- j-g-i _j+ _ _k + Qsource

(i)

The second term in the right-hand side of tile above equation is the

radiation interaction of the kth element to the other elements in an en-

closure. This variables are defined by the following approach.

For the radiant interchange in an enclosure having both specular and

diffuse surfaces, the surface properties are defined as the followings:

The reflectivity is

d s

Ps = Ps + Ps

d s

Ps = Pl + Pl

where subscripts, s and l, signify short and long waves and superscripts,

d and s, denote diffuse and specular.

For diffuse and gray approximation, the following realtionships are

obtained

+_ = iPs + ks s

Pl c._ _[ = I.

From now, the subscripts for short and longwaves are omitted for brevity.

The radiosoty at the kth surface is

d _ d + TkEkMk = E k_T + PkEk(2)

The incident radiant flux at the kth surface is

- d

Ek = _ Fk-jM jJ

(3)

186

Combhlin>_the above two equations yields

Md Z(6." d- -i (( + T.E.)j = . 31- 0jFj-i) k_i I I1

From (3) and (4),

Ek = Z(_iOi + TiEi) _ Fk-j Pj-ii ]

where

_- (_.. d- -iPj-i 31- pjFj-i)

= OT4.

(4)

(5)

The radiant engergy balance equation is the difference between the

incident energy and the radiosoty. That is,

_k = Ek - Mk

s= Ek - Ek _k - Ok Ek - Ok Ek - _k Ek

Substituting (5) into (6)we have

(6)

_k = Z [(i - pk ) c i Z Fk-j P" - (k _ki ] _"• j-i Ii j

+ Z [(i - pk ) T i _ Fk-j Pj-ii 3

- Tk Ski] E i (7)

LetHk_ i = Z Fk-j Pj-i

J

Then, for the cavity a, the equation (5) can be

expanded as

EIO = Hlo-i el _I

+ [HIo-8 E8 _+ HIO-9 (9] _8

a _I0+ HI0-10 _I0

a _ (A)+ HIO_I 0 TIO EIO

b - cavity __

i0

187

ii

b-cavity

EFov

El0

For the cavity, b

~ b f_10El0 -- HI0-5 _5 _5 + HIO-6 e6 _6 + HI0-10 ci0

• bb +

+ HI0_I0 TI0 El0 HI0-11 _ii EFOV(B)

From (A) and (B),

EIO = G _i HI0-1 _i

a n5+ G _5 TI0 _0-5 HI0-10

a

+ G c6 TI0 HIO_6 HI0_I 0 _6

+ G[HI0-8 _8 + HI0-9 _9 ] _8

a a b

+ G _Io[HIo_I0 + TI0 HI0_I 0 HI0_I0] _QI0

a

+ G TI0 TII HI0_I 0 HI0_I 1 EFO v

2 a .b )-i.where G = (i - TI0 HI0_I 0 HI0_I 0

~

EIO = G b nli TI0 HIO-10 HI0-1

+ G 45 HI0_5 _i

+ G _6 HI0-6 _6

b [I[10-8 _8 + _ ] d8+ G TI0 HI0_I 0 HI0-9 9

b Ha+ G _I0 HI0-10[I + TIO i0-I0 ] _i0

b

+ G TII HI0_I I EFO V

188

$i = _[(I - pi ) Hij Ej - _ij ej] _j + Z[(I. - pi ) Hij T.j - _ij Tj]Ejj J

In cavity a:

a 2 G b -i] _i$i = Ei[(i - of)HI_ I + (i - OI) TI0 HI_f0 HI0_I0 HI0_ I

+ (i - OI) (5 TIO G HI_f0 HI0_5 _5

+ (i - PI) (6 TI0 G HI_I0 HI0_6 _"6

2 b

+ (i - OI) {((8 HI-8 + e9 HI-9) + TI0 G HI_I0 HI0_10 ((8 B10-8

+ HI0-9 E9 )} _8b a

+ (i - pz) (to _-zo [I + Tzo c _zo_zo(l + Zlo _1o_zo)] _zo

+ (1 - O1 ) "_i0 "_11 G 1-11_10 I-Ii0_11 EFOV

A 8 A 9

a : ¢8 + _ ¢9$8 A(8+9 ) A(8+9 )

2 b nl¢8 = (i - p8)[_ 1 H8_ I + eI _i0 G H8_10 Hl0_l 0 Hl0_ I]

+ (i - P8 ) e5 %10 G H8_I0 H10_5 _5

+ (i - p8) e6 "rlO G HS_I0 1-110_6 _Q6

2 b

+ {(8[(i - p8 ) H8_ 8 - i] + (]-- p8) TI0 G H8-].0 HI.0-10 (HlO-8 (

+ H10_ 9 (9) + (i- 8) e9 H8-9} _8

b (i + a+ (I .- P8 ) el0 H8_I0 [i + TI0 G HIO_I 0 TIO HI0_I0)] RI0

+ (i- 08) %10 TII G HS_I0 HI0_I I EFO V

189

2 b¢9 = (i - 09) el[H9_I + TI0 G H9_I0 HI0_I0 HI0_I ] _i

+ (i - 09 ) _5 _i0 G H9_I0 HI0_5 _5

+ (I - p9 ) (6 TI0 G H9_I0 HI0_6 _6

+ { (i - p9 )

2 b

9 H9-8 + c9[(I - 09) H9-9 - i] + (i - 09 ) TI0 G H9_I0 HI0_I 0

(HI0_ 8 E8 + HIO_ 9 _9)}f_8

b (i + a+ (I - 09 ) el0 H9_I0[I + TI0 G HI0_I 0 TI0 HI0-10)] _i0

+ (i - 09 ) TIO TII G H9_IO HI0_I 1 EFO v

In Cavity b:

b_5 = (I - 05 ) _I TI0 G H5_I0 H10_l _i

2 a

+ e5 {[(I - 05 ) H5_ 5 - i] + (i - 05) TI0 G.H5_I0 HI0_5 HI0_I 0} _5

2 a

+ (I - 05 ) E6[Hs_ 6 + TI0 G H5_I0 HI0_6 HI0_I 0] _6

+ (i - 05 ) TI0 G H5_I0 (e8 HI0_8 + e9 HI0_9) _8

+ (i 05 ) el0 H5_I0[I + TI0 G a _b- HI0_I 0 .(i + TI0 HI0_I0)] _QI0

2 G a+ (i - 05 ) TI0 TII H5_I0 HI0_I 0 HI0_I I EFO v

b¢6 = el(l - 06) TI0 G H5_I0 HI0_I _i

2 G H_0-10] n5+ (i - 06 ) e6[H6_5 + TI0 H6_I0 HI0_5

2 a

+ E6{[(I - p6 ) H6_ 6 - i] + (i - 06) TID G H6_I0 HI0_5 HI0_10} _6

+ (i - 06) TIO G H6_I0 (_8 HIO-8 + _9 H'O._I-9 ) fl'8

a b

+ (i - 06 ) el0 H5_I0 [i + TI0 G HI0_I 0 (i + TI0 HI0_I 0)] _i0

2 G a+ (i - 06 ) TI0 TII H5_I0 HI0_I 0 HI0_I I EFO v

190

b G[(i ) b - I] _i_i0 -- _I _i0 - 010 HI0-10 HI0-1

2 G[(i 010) b - i] a+ (5 HI0-5 {(i - 010) + TIO HI0-10 H]O-10} _5

2 G[ (i Ol0 ) b - i] a+ _6 HI0-6 { (i - 010 ) + TI0 - HI0_10 HI0-10} R6

b -I] (_ +_+ TI0 G[(I - 010 ) HI0_10 8 _0-8 9 _0-9 ) _8

b -i] {i+ a a b+ (I0 [(I - 010) HI0-10 TI0 G[HI0-10 + TI0 HI0-10 HIO-IO]} _I0

2 G [(i 010 ) b a+ TI0 TII - HI0_I 0 - i] HI0_I 0 HI0_I 1 EFO V

_Oa a Al0b h

_i0 filter - Al0(a+b ) _i0 + AlO(a+b ) _i0

Cavity C:

¢i=.c(i[(i. - Pl ) HI_ I - 1] R1 + (I - pl ) (2 H1-2 _2

+ (I - pl ) (4 HI-4 _4

C

_2 = (I - 02) _i H2-1 _i + E2[(I - 02) H2-2 - i] n2

+ (I - 02) _4 H2-4 _4

C

_4 = (i - 04 ) (i H4-1 nl + (i - 04 ) E2 H4_ 2 _2

+ c4[(i - 04) H4_ 4 - i] _4 "

c c

where _ and 0 are e and 0

Cavity d=

d

_2 = _2 [(l - 02) H2-2 - i] _2 + (i - 02) _3 H2-3 _3

+ (I - 02 ) (7 H2-7 _7

191

r_ •

_3d ffi (1 - 03 ) (2 H3-2 _2 + c3 [(1 - P3 ) H3_ 3 -

+ (1 - P3 ) e7 H3-7 _7

d_7

I] n 3

= (1 - 07 ) ¢2 H7-2 _2 + (1 - 07 ) _3 H7-3 _3

+ c7[( 1 - p7 ) H7_7 - 1] _7

Cavity e:

_le ffi _l[(1 - pl) HI_1 - l] _]

+ (I - P I) _3 HI-3 _3

+ (I - Pl ) ¢2 HI-2 _2

+zeffi (i- 02) _I_2-i_{ +_2[(I- o2)

+ (1 - P2 ) _3 H2-3 _3

e$3 _(1 - 03 ) (1 H3-1 _Q1 + (1 P 3 )

2 H3-2 _2

+ e3[( 1 _ p3 ) H3_ 3 -11 _Q3

Cavity f :

s f ffic3[(i - 03 ) H3_ 3 - i] _3 + (l - P3 ) c 7 H3-7 _73

_b7f = (1 - 07 ) _3 H7-3 _3 + c7[(l - 07) H7-7 - 1] _7

The energY balance equation (i) is written for each node as the

followings :

T2-T I c c e e a a QHI}TI i { + AI $i + AI $I + AI $i +

= _ii RI-2

192

- T4-T2 T3-T2 T7-T2 e e,-7-i{TI-T2 + -- + -- + -- + A2 _2 + Q2 }

T2 = n 2 R2_ I R2- 4 R2- 3 R2- 7

i T2-T3 e e d d f f

T3 = M 3 { _ + A3 _3 + A3 _3 + A3 _3 + QH3 }

i T2-T4 T5-T4 T8-T4 c c= -- + -- + _ + A4 _4 + Q4 }

T4 M4 { R4-2 R4-5 R4-8

- T6-T5 TI2-T5 b _}i {T4 T5 +__+ __+

T5 = M5 R5_ 4 R5_ 6 R5-12 A5

i Ts-T6 b b

= -- + _6 }T6 M6 { R6-5 A6

i T2-T7 d d f f

= M7 --R2-7 + A7 + + Q7}-T7 { _7 A7 _7

I T4-T8 T5-T8 TI0-T8 a _a}9= __+ __+ __+ a+ aT8 M8 { R8-4 R8-5 R8-10 A8 _b8 A9

• i {T4-TI0 + T8-TI0 + a a + _0 b.... ¢lo }TIO MIO R10_4 RIO_8 AI0 _lO

Where M = V X C. In addition there are three more equations which were

derived from the electronics model. These equations are already discussed

in Section II.

In the above equations, the _s can be splitted as

_k = (_k)shortwave + (_k)longwave"

Substituting _k into the above equations and rearranging the constants

yeild a following type of equation

-- . . T4 + Qk + BkTk Z Akj Tj + Z Dkj 3 BQk EFOV3 J

193

where

%j1

kRk_j

Dkj = constants including all radiative parameters

Bk = constants related with the irradiance at FOV limlter of

radiometers.

The above constants are described in the following pages.

i 1

{AI_ I = MIRI_ 2 ' AI_ 2 = _RI_ 2

i

A2_ I - M2R2_I 'i i

A2_ 3 - M2R2_3 ' A2_ 4 - M2R2_4

=__(--_--L + ! + ! +_)A2_ 2 _z R2_ I R2- 4 R2- 3 R-27

i

M2R2_ 7

I I

{A3_ 2 - , A3_ 3 =M3R3_ 2 M3R3- 2

i

1A4_ 2 = M4R4_2

i

A4- 5 = M4R4_ 5

=_z---(--L+ ! +_ l )A4_ 4 M4 R4_ 2 R4- 5 R4- 8

i

A4_ 8 -M4R4_ 8

i

A5_ = M5R5_4 '.i

_A5-6 MsR5_------_ '

= - _!--- ( i + I + I.__.____)A5- 5 M5 R5- 4 Rs- 6 R5-12

A5-12

i

M5R5-12

194

{A6_5 = i , A6_6 = iM6R6_ 5

{A7_ 2 _ iM7R7_ 2

IAs_ -. IMsRs_ 4

iAs_10_ iMsR8_I0

A7_ 7 =M7RT_ 2

i

A8_ 5 = MsRs_5 ' A8_ 5 -z (._k_.z+ __.i_z + z__k_)M 8 R8_ 4 R8_ 5 R8_I0

i i

{AI0_4 = MIoRI0_4 ' AI0_ 8 = MIoRI0_8 ' AI0_I0 =I (-k +__i )

M10 R10- 4 RI0- 8

{AI a a a a 2 a b a= _ _ Ill_f0 HI0_I0 HI0-1DI_ l _ _1[(1 0I) HI_ 1 + (1 Ol ) ri0 G - i]

C c c c - i] + e e e e - I]}+ A 1 _i[( 1 - 01) HI_ 1 A1 _1[( 1 - 01) HI_ 1

= _ {A I (i - 0I) Ec c e e e eDI-2 M 1 2 HI-2 + AI (i - 01) e2 HI_ 2}

O e e e e

= _ HI_ 3DI-3 _ii AI (i 01) e3

a a b

DI-5 _II 5 TI0 HI-10 HI0-5}

= _ c (i - 01) e4 cDI-4 MI AI HI-4

a a a a Hb= - HI-10 10-6DI_ 6 _ A I (i 0 1 ) _6 Tl0 G

195

o a a a a + a a 2 a " b=- HI_I0 HI0-10DI_ 8 M1 A I (i - pl ) {(e 8 HI_ 8 9 HI-9) + TI0 G

a a a a(_8 HI0-8 + e9 HI0-9)}

= o a a a G b a

DI-10 _ii AI el0 HI-10[I + TI0 HI0-10(1 + TI0 HI0-10) ]

i a a G a b

B1 = M-_ AI (I - 01) TI0 TII HI_I0 HI0_11

O C C c c + e e e e

D2_ 1 = _2-2[A2 (i - p2 ) eI H2_ 1 A 2 (i - 02) cI H2_ I]

= O___{A2[(I P2 ) c - I] c + A2d d d d - I]D2-2 M 2 - H2-2 2 2 [(I - 02) H2_ 2

e e e e+ A2 _2[(i - 02) H2_ 2 - I]}

Ad d d d e e e e= _ { (i - P2 ) _3 H2-3 A2 H2-3}D2_ 3 M2 + (i - p2 ) _3

= _ {A 2 (i - p2)e4 H2-4 }D2-4 M 2

d d d d

= _ H2_ 7D2-7 _22 A2 (I 02) e7

(7 e e e e

D3-1 = M--3A3 (i - 03) E1 H3_ 1

o d3 d d d e e e e= _ H3_ 2 A3 H3_ 2 }D3_ 2 _ {A (i 03) e2 + (I- 03) E2

= __ d d[(l- 0_ ) d _ i] + e e eD3_ 3 M3 {A e3 H3_ 3 A 3 e3[(l - 03)

f f f f

+ A 3 e3[(l - 03) H3_ 3 - i]}

196

c) Ad d (I - 0d) d + f (i- 0_) c f fD3-7 = _33 { c7 H3-7 A3 7 H3-7}

= _ C C C C

D4_ I M4 A4 (i - 04) _i H4-1

---- O" C C C C

D4-2 _44 A4 (i - 04) e2 H4-2

c c c c c -i]D4_ 4 = M4 A4 E4 [(I - 04) H4_ 4

O {Ab (I_ f)b)eI G b aD5-I-- H5-I0HI0-I}

2 b b aO b b b) Hb - i] + (I - 0b) TI0 G H5_I0 HIO_5 HI0_I0]}

D5-5 = M'-5{A5[_5[ (i - 05 5-5

c b b b b 2 b b= - [Hs_ 6 TI0 HS-10 HI0-6 Hal0_10]

D5-6 _55 A5 (I 05) e5 + G

c b b) G b a a + eD5-8 = M-55A5 (i - 05 TI0 H5-10 (_8 HI0-8 9 HI0-9)

o b b b b [I + G a bD5-10 = M--5A5(I - 05) el0 HS-IO TI0 HI0-10(I + TI0 HI0-10)]

1 b b) 2 b aB5 = _55 A5 (i - 05 TII[TIO G HS_I0 HI0_I 0 HI0_I I + H5_II]

=c_ b a b G b a

D6-1 _66 A6 E1 (i - 06) TI0 H5_IO HI0_I

o b b) b b 2 G b b a ]D6_ 5 = _66 A6 (i - 06 e6[H6_ 5 + TI0 H6_I0 HI0_5 HI0_I 0

2 G b aD6_6 = M__A6_ b E6b {[(i - 0_) H6_6b _ i] + (i - 0_) TI0 Hl0_ 5 HI0_I0}

197

---- ab b G b (a a +

D6_ 8 _ A 6 (i - 06 ) TI0 H6_I0 8 HI0-8 9 HI0-9)

b b b b G a Hb

D6_I0 =_6 A6 (i - 06) el0 H6-10[I + TI0 HI0_10 (i - TI0 I0_i0 )]

i b b 2 G b a

B 6 = _ A 6 (i - 06) TII[H6_II + TI0 H6_I0 HI0_I 0 HI0_I 0]

__O d (i - 07 ) d dD7-2 M_7 A7 H7-2

=_ Ad d d d + f f f fDT_ 3 _ { (I - 07) _3 H7-3 A 7 (I - 07) c3 HT_ 3}

_ {AT d d d -i] + f efT[(l Pf7) f - i]}D7-7 = M_ c7[(i - p7 ) HT_ 7 A 7 - HT_ 7

D8_ I

a 2 a b ao {A8 (i 08 ) a a +_ G

= M_ Aa - [el HS-I i TI0 H8-10 HI0-10 HI0-1](8+9)

a

A 9 a a + 2 G a b a ]} A+ (i - 09) El[H9_ I TI0 H9-10 HI0-10 HI0-1 (8+9)

Aa(8+9)

b a b a a b a b

= _ {A8 (i - 08) _5 G + (i- 09) _5 GM- TI0 H8-10 HI0-5 A 9 TI0 H9-10 Hlo- 5}D8_ 5--8

a b + a (i- 09) Eb a b° " o-o oD8-6 = M_

D8- 8 O a a a a - i) + (i 0_) 2 G a b a=-- {A8[_8((I - 08) H8_ 8 - TI0 H8-10.HI0-10 (HIo_ 8M 8

aa+ a a a a a a +E8 Hi0-9 _9 ) + (I - 08) _9 H8-9] + A [(I - 09) _9 H9-8 9

((i- 09) H9_ 9 - i)2 G b a

+ (i - 09) TI0 H9_I0 HI0_I0(HI0_8a + a a

_8 HID-9 c9)]}

198

{A8 (i - p_) a a b aDS-Z0 M8 10 H8-10[1 + TZ0 G (i + ]=-- HI0_10 rl0 HI0_10)

a a [i + G b a+ A9 (i - p_) (i0 H9_I0 TI0 HI0_I 0 (i + rl0 HI0_I0)]}

1 a

B8 = _88 {AS (i - p8 ) TI0 rllG a b

}{8-10 HI0-11 + A9 (i - p9 ) TI0 TII G a9-10b }

HI0-11

C a a a a 2 a a b

=_ 010) TI0 010 ) HI0_10 HI0_10]DI0-1 M10 {AI0 'i HI0-1[(1 - + G((I - - I)

a G[ (i b b - i] a1 rl0 - Pl0 ) HI0-10 HI0_ I}

G.q_ a b G[(I a a bDZ0-S = MZ0 {At0 '5 TZ0 - 0Z0) HZ0_Zo - Z U_0_5

b b b 2 b b+ AI0 c5 HI0_5[(1 - + G (iPI0 ) rl0 - Pl0 ) HI0_10

a

z) azo_zo] }

O b a a bD10-6 M10 {Ac0 '6 TI0 G[ (1 Pl0) - i: _ - HZ0-10 "10-6

b b b b 2 G((I b b a+ AI0 (6 HI0-6[(1 - P 0 ) + TI0 Pl0 ) 50_10 i ]}- _ HI0_10

c a a a a a a 2 a aM10 {Al0(e 8 HI0_8 + L9 HI0_9) [(i - + G((IPlo) "qo - Ozo) a_o_zo -z)

b b G[(I b b - i]( a a a aHI0-10] + AI0 TI0 - PI0) HI0-10 8 H10-8 + _9 HI0-9)}

O a

:-- Pzo) _zo-lo - z] [z + Tzo G (1 + ]DI0_I0 MI0 {A10 el0[(l _ a a b aHI0-10 TI0 HI0_I0)

b b b Hb+ AI0 (i0[(i - Of 0 ) i0-i0 - i] [i + TI0 G a bHI0_I 0 (i + TI0 HI0_I0)]}

199

l_!- a G[(I a a - i] b

BI0 = Ml0 {AI0 TI0 _Ii - 010 ) lIl0-10 HI0-11

b G[(Z pzo) b - z] a b+ AI0 TI0 TII - HI0_10 HI0-10 HI0-11

b (i b b b+ AZO - OiO) zlZ HZO-Zl}

b a G[(I a a - I]1 T1 HI0-11 {AI0 Zl0 - 010 ) HI0-10

BI0 = MI0.

b (T_0 G a [(i b b - i] + i - b+ AI0 HI0-10 - 010 ) HI0-10 010 )}

_gOED_G PAGE PL#2_K NOT FDLM_hb

201

APPENDIX D

ERBE SCANNER THERMAL MODEL EQUATIONS

_I-¢,]_EDING PAGE BLANK NOT FILMF._

203

SCANNER MODEL

The scanner model was developed by the same methods which were used

for the non-scanner model development. The Equations (i) through (7) in

the Appendix C are used to generate the constants of the thermal model

Equation (24) in the Section 2.1.3.

To define the coefficients in the themal model, the enclosure coupling

technique was used as another alternative approach. The following pages

contain those coefficients defined by using the enclosure coupling method.

The attached figure shows the scanner model which has node description

and enclosure descriptions. The enclosure has the number which is circled

around. The numbers describing enclosure surfaces are underlined. In the

illustration of the coefficients, the word that explains the procedure of

derivation was omitted because they were already shown in Appendix C.

b ,jJ

204

V V

/._.1LI.J

C)

LI.I

2O5

i i

AI, i = MIRI_ 2 ' AI, 2 = MIRI_ 2

AI,3 AI,4 = AI, 5 = AI, 6 = AI, 7 = AI, 8 = AI, 9 = AI,10 = AI,II = AI,12 =

AI,I 3 = AI,14 =0

i , A2,2 _ (_2)(__i +____i +---_i +---_i +---_i + I-¢A2, I = MIR2_I = R2_I R2- 3 R2-4 R2- 7 R2- 8 R2-13

i , A2, 4 iA2, 3 = M2R2_3 = M2R2_4

A2, 5 = A2, 6 = 0

l I i

A2, 7 = M2R2_7 ' A2, 8 = M2R2_ 8 ' A2,13 = M2R2_I3

_A2, 9 = A2,10 = A2,11 = A2,12 = A2,14 = 0

i i

= 0 , A 3 = + M3R3_2 = M3R3_2

3,4 A3,5 = A3,6 = A3,7 = A3,8 = A3,9 = A3,10 = A3,11 = A3,12 = A3,13 = A3,14 = 0

• 1 =0

A4, I = 0 , A4, 2 = M4R4_2 , A4, 3

: _ (i (__!__i+.___!__z+ ___!_i)A4,4 _4"4) R4_ 2 R4-6 R4- 8

1 = 0A, : , A4, 7_,6 M4R4_ 6

A4"' = 0' 5

' A4 8 .=9

_4,9 = A4,10 = A4,11 = A4,12 = A4,13 = A4,14 = 0

M4R4-8

206

A5,I = A5,2 = AS,3 = AS,4 = 0

i i= , A5,6 =A5,5 MsRs_6 MsRs-6

A5,7 = A5,8 = A5,9 = AS,10 = AS,If = A5,12 = A5,13 = A5,14 = 0

_A6, I = A6,2 = A6,3 = 0

i I- , A6, 5 = M6R6_5A6,4 M6R6_ 4

A6,6 R6_ 4 R6_ 5 R6_ 8

i

A6,7 =" 0 ' A6,8 " M6R6_ 8

_A6, 9 = A6,10 = A6,11 = A6,12 = A6_13 = A6,14 = 0

• = 0 A7,2 = 1A7, I , M7R7_ 2

A7, 3 = AT, 4 = A7, 5 = A7, 6 = 0

A7,7 = _ (M_)( i_.___+ i ) , = iR7_ 2 R7- 8 ' A7, 8 M7R7_------_

_7,9 = AT,10 = A7,11 = A7,12 = A7,13 = A7,14 = 0

207

i i_'A = 0 A8 -- 0 A8 =8,1 ' ,2 = MsR8_2 ' A8,3 ' ,4 M8R8_4

i iAS,5 = 0 , AS,6 = MsRs_6 , AS,7 = MsRs_------_

=_ i___+ i__+!_+A8,8

R8_ 2 R8- 4 R8_ 6 R8_ 7 R8-10 R8-11

i i

- , A 8 -A8, 9 = 0 , AS,10 MsRs_I 0 ,ii MsRs_II

_A8,12 = A8,13 - A8,14 = 0

A9,1 = A9,2 = A9, 3 = A9, 4 = A9, 5 = A9, 6 = A9, 7 = A9, 8 = 0

i , A9,10 = 0 , A9,11 = 0 , A9,12A9, 9 = M9R9_I2

i

M9R9_I2

A9,13 = A9,14 = 0

"AI0, I = AI0,2 = AI0,3 = AI0,4 = AI0,5 = AI0,6 = AI0,7 = 0

i

AI0,8 - MIoRI0_8 _ , AI0_9 = 0

AI0_I 0 = _ (_i)(i+

M1o RI0-8

i i--+ )RI0-11 RI0-14

I =0 =0

_A10_I 1 - M10R10_I 1 A10-12 A10,13 A10,14

1

MIoRI0-1.4

208

All, I = All, 2 = All, 3 = AII,4 = AII,5 = AII,6 = AII,7 = 0

= i =0 i

AII,8 MIIRII_8 ' AII,9 ' AII,I 0 - MIIRII_I0' AII,I 2

_0

= AII,I 4 = 0 , AII,I I = - (.i)( l___!___'+ ___!___l)_ii,13 KII-8 RII-10

AI2,1 = A12,2 ffiA12,3 ffiA12,4 = A12,5 = A12,6 = A12,7 ffiA12,8 = 0

= i , AI2,10 = AI2,11 = 0 A12,12 _ iA12,9 MI2RI2_9 ' MI2RI2_9

AI2,13 ffi0 , AI2,14 = 0

i

AI3,1 ffi0 , A13,2 - MI3RI3_2 , A13,3 = A13,4 = A13,5 = A13,6

"0

A13,7 = A13,8 ffiA13,9 ffiAI3,10 = AI3,11 = AI3,12 = 0

i

_13,13 - MI3RI3_2 ' A13,14 = 0

AI4-1 = A14,2 = A14,3 = A14,4 = A14,5 = A14,6 = A14,7 = A14,8 = A14,9 ffi0

_ i _0 =0 =0

AI4 ,i0 MI4RI4_I0 ' AI4 ,ii ' AI4 ,12 ' AI4 ,13

Al4ml 4 -MI4RI4-10

209

For the total channel, the filter glasses are removed, Accordingly,

the (DI-_--) related with node number of filters have to set zero.

i-3

BQ = [0, 0, 0, 0, 0, 0, 0, M_' 0, 0, 0, 0, 0, 0]

Scanner:

Let

(Hkji)A = (_.k-j PJ-i)A '3

where subscript A denotes the enclosure A.

Then let

2 -i

G 1 = [i- TI3 (HI3jI3) D (H13m13) F]

2 GI )FSI = (H17mlT) F + TI3 (Hl7ml 3 (HI3jI3)D (HI3mI7)F

2 -i

G2 = [I- TI7 SI(HI7jlT) C]

2 SIS2 = (HI6jI6) C + TI7 G2(HI7jI6) C (HI6jI7)C

2 $2 )B]-IG 3 = [I - TI6 (Hl6ml 6

2

S3 = (Hl4ml4) B + TI6 S2 G3(HI4mI6) B (H16m14) B

2 -i

G4 = [I- TI4 S3 (HI4jI4) A]

210

XI = FI,I_ I + FI,2_ 2 + FI,4_ 4 + FI,5_ 5 + FI,6_ 6 + FI,8_ 8 + FI,9_ 9

+ FI,13_13

where

FI,I

FI,2

FI,4

= TI3 _i GI G2(HITmI3)F (HI3jl)D

2 GI )F= _2 G2 {(HI7m2)F + TI3 GI(HITmI3)F (HI3J2)D + TI3 (H17m13

(HI3jI3) D (H13m2) F}

2

= _4 G2 {(HI7m4)F + TI3 GI(HI7mI3)F (HI3mI3)D (HI3m4)F}

FI,5 = TIT _5 Sl G2(HITJ5)C

FI,6 = TI7 c6 Sl G2(HITj6)C

FI,8 = TI7 _8 Sl G2(HI7j8)C

FI, 9 = TI7 c 9 SI G2(HI7j9) C

FI,13 = 2 GI_13 G2 {(HITmI3)F + TI3 GI(HI7mI3)F (HI3jI3)D + TI3 (HITmI3)F

(HI3jI3) D (H13m13) F}

X 2 = F2,1_ I + F2,2_ 2 + F2,4_ 4 + F2,5_ 5 + F2,6_6 + F2,8_ 8 + F2,9_ 9 + F2,11_II

F2,12_12 + F2,13_13 + F2,14_14

211

where

= (HI6jlT) c (HITmI3)F (HI3jl) DF2,1 TI3 TIT GI G2 G3

F2,2 = TIT 2 G2 G3(HI6jI7)C {(HlTm2) F + TI3 GI(HI7mI3) F (HI3j2) D 2 GI+ TI3

(H17m13) F (HI3jI3)D (H13m2) F}

2 GIF2,4 = TI7 _4 G2 G3(HI6jI7)C {(HITm4)F + TI3 (HI7mI3)F (HI3mI3)D (HI3m4)F}

2 Sl 5)C}F2,5 = e5 G3 {(HI6j5)C + _17 Gz(HI6JI7)C (HI7j

F2,6 = _6 G3 {(HI6j6)C

F2,8 = (8 G3 {(HI6J8)C

F2, 9 = c9 G 3 {(HI6j9) C

2 SI (H1 6)C}+ TIT G2 (HI6JI7)C 7j

2 Sl 8)C}+ TIT G2(HI6jlT) C (HI7 j

2 SI G2 )C }+ TI6 S2(HI6m9)B + _17 (HI6jlT)C (HITj9 "

F2,11 = TI6 _ii S2 G3(HI6mlI)B

F2,12 = TI6 c12 S2 G3(HI6mI2) B

F2,132 GI= TIT El3 G2 G3(HI6jI7) C (H17m13) F {i + TI3 GI(HI3jI3) D + TI3

(HI3jI3) D (H13m13) F}

F2,14 = TI6 14 $2 G3(HI6mI4)B

212

v

El4 = UI,I_ I + UI,2_ 2 + UI,4R 3 + UI,5_ 5 + UI,6_ 6 + UI,8_ 8 + UI,9_ 9 + Ul,10_10

+ UI,II_II + UI,12_12 + UI,13_13 + UI,14_14 + UI,15E

Ul, I ffiTI6 G4(HI4mI6) B F2,1

UI, 2 = TI6 G4(HI4mI6) B F2,2

UI, 4 ffiTI6 G4(HI4mI6) B F2, 4

Ul,5 = _16 G4(HI4mI6)B F2,5

Ul, 6 ffi TI6 G4 (H14m16) B F2,6

ffi (H14m16) B F2,8U1,8 T16 G4

Ul,9 -= G4{YI6(HI4mI6) B F2, 9 + _9(HI4m9)B}

Ul,10 = TI0 S3 G 4 _Io(HI4jI0)A

Ul,ll = G4 {Cll(Hl4mll) B + TI6(HI4mI6) B F2,11}

Ul,12 = G4 {el2(Hl4ml2)B + TI4 c12

/

Ul,13 ffiTI6 G4(HI4mI6) B F2,13

S3(HI4jI2) A + TI6(HI4mI6)B F2,12}

Ul,14 = G4 {c 14(HI4mI4)B + T14 c14 S3(HI4jI4) A + TI6(HI4mI6) B F2,14}

Ul,15 ffiTI5 G4(HI4jI5) A

2i3

El4 = U2,1_ I + U2,2_ 2 + U2,4_ 4 + U2,5_ 5 + U2,6_ 6 + U2,8_ 8 + U2,9_ 9 + U2,10_I0

+ U2,11_II + U2,12_12 + U2,13_13 + U2,14_14 + U2,15 E

U2,1 = TI4 TI6 G4(HI4jI4) A (H14m16) B F2,1

U2, 2 = TI4 TI6 G4(HI4hI4) A (H14m16) B F2,2

U2,4 = TI4 YI6 G4(HI4JI4)A (HI4mI6)B F2,4

U2, 5 = TI4 TI6 G4(HI4jI4) A (H14m16) B F2,5

U2,6 = TI4 Y16 G4(HI4JI4)A (HI4mI6)B F2,6

U2,8 = TI4 z16 G4(HI4JI4)A (HI4mI6)B F2,8

U2, 9 = G4(HI4jI4) A {TI4 TI6(HI4mI6) B F2, 9 + T14 c9(H14m9) B}

= 2 S3 G4U2,10 _I0(HI4jI0)A {i + TI4 (HI4jI4) A}

U2,11 = G4(HI4jI4) A {TI4 TI6(HI4mI6) B F2,11 + TI4 Cil(Hl4mll) B}

U2,12

2

= TI4 TI6 G4(HI4jI4) A (H14m16) B F2,12 + _I2(HI4jI2)A[I + TI4 S3 G4

(HI4jI4)A] + TI4 _12 G4(HI4jI4)A (HI4mI2)B

U2,13 = TI4 YI6 G4(HI4jI4)A (HI4mI6)B F2,13

214

U2,14 -- (HI4JI4)A {TI4 el4 G4(HI4mI4)B + TI4 TI6 G4(HI4mI6)B F2,14 + _14

2 S3 G4 }[I + TI4 (HI4jI4) A]

U2,15 = TI5(HI4jI5) A {i + TI4 G4(HI4jI4) A}

A

El6 = U3,1f_ I + U3,2f22 + U3,4D,4 + U3,5_ 5 + U3,6f_ 6 + U3,Sf_8 + U3,9_ 9 + U3,10_I0

+ U3,11_II + U3,12_12 + U3,13_13 + U3,14_14 + U3,15E

U3,1 : TI4 TI6 S2 G3(HI6mI4) B U2,1 + TI3 TIT G1 G 2 G 3 _I(HI6jI7)C (H17m13) F

(HI3jl) D

U3,2 = TI4 TI6 $2 G3(HI6mI4)B U2,2 + TI3 TI7 _2 GI G2 G3(HI6jI7)C (HI7mI3)F

2

(HI3jI3)D + TI7 _2 G2 G3(HI6JI7)C (HITm2)F + TI3 TI7 c2 GI G2 G3

(HI6JI7) C (HITmI3)F (HI3JI3) D * (H13m2) F

U3,4 = TI4 TI6 $2 G3(HI6mI4)B U2,4 + TI7 _4 G2 G3(HI6jI7)C (Hl7m4)F

2

+ TI3 TI7 _4 GI G2 G3(HI6jI7)C (HI7nI3)F (HI3jI3)D (HI3m4)F

2 SI G2U3, 5 = TI4 TI6 S2 G3(HI6mI4) B U2,5 + G 3 c5(H16j5) C + TIT e 5 (HI6jI7) C

(HI7j5) C

2

U3,6 = TI4 TI6 G3(HI6mI4)B U2,6 + _6 G3(HI6j6)C + TI7 _6 Sl G2 G3 (HI6jI7)C

(HITj6) C

215

2U3,8 = TI4 r16 S2 G3(HI6mI4)BU2,8 + _8 G3(HI6jS)C + TIT c8 SI G2 G3(HI6JI7)C

(HI7j8)C

U3,9 = TI4 TI6 S2 G3(HI6mI4)B U2,9 + 2 _8 SI G29 G3(HI6j9)C + _17 G3(HI6JI7)C

(HITj9)C + YI6 _9 $2 G3(HI7j9)C

U3,10 = TI4 TI0 S2 G3(HI6mI4)B U2,10

U3,11 = YI4 TI6 S2 G3(HI6mI4)BU2,11 + TI6 Ell $2 G3(HI7JlI)C

= (H16m14 ,12 TI6 c12 G3(HI7jI2)CU3,12 TI4 TI6 $2 G3 )B U2 + $2

U3,13 = TI4 YI6 S2 G3(HI6mI4)B U2,13 + TI7 _13 G3(HI6JI7)C (HI7mI3)F

+ TI3 TI7 _13 GI G2 G3(HI6JlT)C (HI3jI3)D (HI7mI3)F[I + _13(H13 ml3)F]

= (H16m14) B U2,14 TI6 c14 G3(HITjI4) CU3,14 TI4 TI6 S2 G3 + S2

= (Hl6mi4) B U2,15U3,15 TI4 YI6 $2 G3

v

El6 = U 4,1_I + U4,2f_2 + U4,4_4 + U4,5_5 + U 4,6_Q6 + U4,8_Q8 + U4,9f_9 + U 4,10_-I0

+ U4,11_II + U4,12_12 + U4,13_13 + U4,14_14 + U4,15E

2 S2 G 3 + TU4, I = T14(Hl6mi4)B[l + TI6 (H16m16) B] U2,1 16 TI7 G3(HI6mI6)B

FI(HI6jI7)C ,i

216

2 S2 G3 U2 + G3U4,2 = TI4(HI6mI4)B [I + TI6 (HI6mI6)B] ,'2 TI6 TI7 (HI6mI6)B

(HI6j 17)C FI,2

2 $2 )B] U2 +U4,4 = TI4(HI6mI4)B [I + TI6 G3(HI6mI6 ,4 TI6 TI7 G3(HI6mI6)B

(HI6jI7) C FI,4

2 $2 B] U2 +U4,5 = TI4(HI6mI4)B[I + TI6 G3(HI6mI6) ,5 TI6 TIT G3(HI6mI6)B

(HI6j 17)C FI,5 + TI6 _5 G3(H]_6mI6)B(HI6j5)C

2 $2 B] +U4,8 = TI4(HI6mI4)B [I + TI6 G3(HI6mI6) U2,8 _16 TIT G3(HI6mI6)B

(HI6jI7)C FI,8 + TI6 _8 G3(HI6mI6)B (HI6j8)C

2 $2 G3 + G3U4,9 = TI4(HI6mI4)B[I + TI6 (H16m16)B] U2,9 TI6 TI7 (H16m16)B

(HI6jI7)C FI,9 + _9(HI6m9)B+ TI6 _9 G3(HI6mI6)B (HI6j9)C

2+ TI6 9 $2 G3(HI6mI6)B (HI6m9)B

2 $2 G3 U2U4,10 = TI4(HI6mI4)B[I + TI6 (HI6mI6)B] ,i0

U4,11 2 $2 )B] U2= TI4(HI6mI4)B[I + TI6 G3(HI6mI6 ,ii

2 $2+ TI6 ell G3(HI6mI6)B (Hl6mll) B

+ ell(Hl6mll)B

U4,12 = TI4(HI6mI4)B[I + T2 S2 G3 U216 (HI6mI6)B] ,12

2 $2 G3 )B+ TI6 C12 (H16m16 (Hl6mi2) B

+ _I2(HI6mI2)B

217

2 S2 G3 B] + r1.14,13= TI4(HI6mI4)B[I + TI6 (Hl6ml6) U2,13 16 TI7 C3(HI6mI6)B

(HI6jI7)C FI,13

22 $2 B] U2, + T _ S2 G3U4,14 = _I4(HI6mI4)B [I + r16 G3(HI6mI6) 14 16 14 (HI6mI6)B

(H16m14)B + (14(HI6mI4)B

2U4,15 = TI4(HI6mI4)B[I + TI6 S2 G3(HI6mI6)B] U2,15

El7 = U5,1_I + U5,2_2 + U5,4_4 + U5,5_5 + U5,6_6 + U5,8_8 + U5,9_9 + U5,10_I0

+ U5,11_II U5,12_12 + U5,13_13 + U5,14_14 + U 5,15E

= _ (HI7jI6)C ,iU5,1 i TI3 G1 G2(HI7mI3)F (HI3jl)D + TI6 TI7 SI G2 U4

= _ c (HI3jI3) DU5,2 2 G2(HI7m2)F+ 2 TI3 G1 G2(HI7mI3)F[(HI3j2)D + (H13m2)F

TI3] + TI6 TI7 SI G2(HI7jI6) C U4,2

U5,4 = _4 G2(HI7m4)F+ _2

4 TI3 G1 G2(HI7mI3)F (HI3jI3)D (H13m4)F + TI6 TI7

SI G2(HI7jI6) C U4,4

= _ (HI7jI6 ,5U5,5 5 TI7 SI G2(HI7j5)C + TI6 TI7 SI G2 )C U4

U5,6 = (6 TI7 SI G2(HITj6)C + TI6 TI7 SI G2(HITjI6)C U4,6

U5,8 = _8 TI7 Sl G2(HI7j8)C + TI6 TI7 Sl G2(HI7jI6)C U4,8

218

U5,9 = _9 TI7 Sl G2(HI7j9)C + "[16 "[17 SI G2(HI7jI6)C U4,9

U5,10 = TI6 TI7 SI G2(HITjI6) C U4,10

U5,11 = _16 TI7 SI G2(HI7jI6)C U4,11

U5,12 = TI6 TI7 S1 G2(HI7jI6) C U4,12

= (H17m13)FU5,13 el3 G2 + _13 TI3 GI G2(HI7mI3)F (HI3jI3)D[I + TI3(HI3mI3 )F]

+ TI6 TI7 SI G2(HI7jI6) C U4,13

U5,14 = TI6 TI7 SI G2(HI7jI6) C U4,14

U5,15 = TI6 TI7 SI G2(HITjI6) C U4,15

A

El7 --U6,1f21 + U6,2f_2 + U6,4f_4 + U6,5_ 5 + U6,6f_ 6 + U6,8f_ 8 + U6,9_ 9 + U6,10f_10

+ U6,11_II + U6,12_12 + U6,13_13 + U6,14_14 + U6,15E

U6,1 = _i TI3 TI7 GI G2(HI7jI7)C (HI7mI3)F (HI3jl)D + _16 G2(HI7jI6)C U4,1

U6,2 = _2 TI7 G2(HI7jI7)C (HI7m2)F + e2 TI3 TI7 GI G2(HI7j 17)C (HI7mI3)F

(HI7jI6 U4,2[(HI3j2) D + TI3(HI3jI3)D (HI3m2)F] + TI6 G2 )C

2

U6,4 = _4 TI7 G2(HI7jI7)C (HI7m4(F + _4 TI3 TI7 GI G2(HI7jI7)C (H17m13) F

(HI3jI3) D (H13m4) F + TI6 G2(HI7jI6) C U4,4

219

U6,5 = _5 G2(HI7j5)C + TI6 G2(HI7jI6)C U4,5

U6,6 -- E6 G2(HI7j6) c + TI6 G2(HI7jI6) C U4,6

= _ G2(HI + G2 ) U4U6,8 8 7j8)C TI6 (HI7jI6 C ,8

U6,9 = _9 G2(HI7j9)C + TI6 G2(HI7jI6)C U4,9

U6,10 --TI6 G2(HI7jI6) C U4,10

U6,11 -- TI6 G2(HI7jI6) C U4,11

U6,12 = TI6 G2(HI7jI6) C U4,12

U6,13 = _13 TI3 TI7 GI G2(HI7jI7)C (HI7mI3)F (HI3jI3)D[I + TI3(HI3mI3)F]

+ _13 TI7 G2(HI7jI7)C (HI7nI3)F + TI6 G2(HI7jI6)C U4,13

U6,14 = TI6 G2(HI7jI6) C U4,14

U6,15 = TI6 G2(HI7jI6) C U4,15

El3 = U7,1_ 1 + U7,2_ 2 + U7,4_ 4 + U7.5_ 5 + U7,6_ 6 + U7,8oQ 8 + U7,9_ 9 + U7,10_10

+ U7,11_II + U7,12_12 + U7,13_13 + U7,14_14 + U7,15E

U7,1 = _i GI(HI3jl)D + TI3 TI7 GI(HI3jI3)D (HI3mI7)F U6,1

220

U7,2 = _2 GI(HI3j2)D + ¢2 TI3 GI(HI3jI3)D (HI3m2)F + "[13 TI7 GI(HI3jI3)D

(H13m17) F U6,2

U7,4 = _4 TI3 GI(HI3jI3)D (HI3m4)F + YI3 TI7 GI(HI3jI3)D (HI3mlT)F U6,4

U7, 5 = TI3 TI7 GI(HI3jI3) D (H13m17) F U6,5

U7,6 = "[13 TI7 GI(HI3jI3)D (HI3mI7)F U6,6

U7, 8 = TI3 TIT GI(HI3jI3)D (H13m17) F U6,8

U7, 9 = TI3 TI7 G1 (HI3jI3) D (H13mlT) F U6,9

U7,10 = TI3 TI7 GI(HI3jI3) D (HI3mI7)F U6,10

U7,11 : TI3 TI7 GI(H13jI3) D (14k3mlT) F U6,11

U7,12 = TI3 TI7 GI(HI3jI3) D (H13m17) F U6,12

U7,13

U •

7,14

el3 GI(HI3jI3)D + _13 TI3 GI(HI3jI3)D (HI3mI3)F + TI3 TIT GI

(HI3jI3) D (H13m17) F U6,13

TI3 TI7 GI(HI3jI3) D (HI3mI7)F U6,14

U7,15 = TI3 TI7 GI(HI3jI3) D (H13m17) F U6,15

221

El3 = U8,1_I + U8,2_2 + U8,4_4 + U8,5_5 + U8,6_6 + U8,8_8 + U8,9_9 + U8,10_I0

+ U8,11_II + U8,12_12+ U8,13_13 + U8_1414 + U8,15E

U8,1 = _i TI3 GI!HI3mI3)F (HI3jl)D + TI7 GI(HI3mlT)F U6,1

U8,2 = E2 TI3 GI(HI3mI3)F (H13i2)D + 2 G2(Hlqm2_F+ TI7 GI(HI3mlT)F U6,2

U8,4 = c4 GI(HI3m4)F + TI7 GI(HI3mlT) F U6,4

U8,5 = TI7 GI(HI3mI7)F U6,5

U8,6 = TI7. GI(HI3mI7)F U6,6

= (H13m17 ,8U8,8 TI7 GI )F U6

U8,9 = TI7 GI(HI3mI7)F U6,9

U8,10 = TI7 GI(HI3mI7)F U6,10

= (H13m17)F U6,11U8,11 TI7 GI

U8,12 = TI7 G!(HI3mI7)F U6,12

= _ _ (H13m13)F TI7 GI(HI3mI7)F U6,13U8,13 13 TI3 GI(HI3mI3(F (HI3jI3)D + 13 G1 +

U8,14 = r17 GI(HI3mI7)F U6,14

U8,15 = r17 GI(HI3mI7)F U6,15

222

OAI_- (-_) [(i-_i)

DI,I_i (HIJl)D

- E + (I-Q I) T13 (HljI3)D US,I]S.L1

oAl + (1-pI) (Hlj 2]DI,2 = (__i)[(I_01) _2(HIj2)D "[13 13)D US, S,L

=0DI,3

_A I

DI, 4 = (-_i)[(I-0 I) TI3(HIjI3)D US,4]S,L

,_(_AI _I3(HIjI3) D U 8,5]S,L

DI, 5 = _--_;[(i-0 I)

°A I U 8

DI, 6 = (-_I)[(I-Pl) TI3(HIjI3)D ,6]S,L

=0DI,7

oAf _I3(HIjI3) D U8 _8]S9L

DI, 8 = (-MI)[(I-O I)

oA I

DI, 9 = (-_i)[(I-0 I) _I3(HIjI3)D US,9]S,L

___AI_I3(HIjI3)D U8,10]S,L

DI,10 = t-_l)[(l-o I)

aA 1

nl,ll = (-_i)[(I-0 I) TI3(HIjI3)D US,II]S,L

°At U8 ]S.= (__i)[(I-01) _I3(HIjI3)D ,12 LDI,12

(_AI + (I-0 I)

DI,13 = (-_i)[(I-0 I) _I3(HIjI3)D UI,13

<_AI

DI,14 = (--_I)[(I-01) _I3(HIjI3)D U8'I4]S'L

13(HIjI3)D]S,L

223

D2,1 = (_2) {A2D(I-02)[eI(H2jl)D + TI3(H2jI3) D U8, I] + A2F(I-P2)[TI3

(H2m13) F UT,I + TI7(H2mI7)F U6,1]}S, L

D2, 2 -- (_2) {A2E[(I'o 2) _2(H2j2)E- e2 ] + A2D[(I-o 2) _2(H2j2 ) -c 2

+ (1-02) TI3(H2jI3)D U8,2] + A2F|(I-o2) c2(H2m2)F - _2 + (1-02)

+ (1-02) T13 *(H2m13) F U7, 2 + (1-02 ) TI7(H2mI7) F U6,2]}S, L

D2, 3 = (_2)[A2E(I-02) e3(H2j3)E]S,L

D2, 4 = (_2) {A2D(J-o 2) TI3(H2jI3) D U8, 4 # A2F[(I-02)_4 (H2m4)F + (1-02) TI3

(H2m13) F U7,4 + (1-02) TIT(H2mI7)F U6,4]}S, L

D2,5 = (_2){A2D (1-02) TI3 (H2jI3) D U8,5 + A2F(I-P2) [TI3(H2mI3)F U7, 5 + TIT

(H2m17) F U6,5]}S,L

D2, 6 = (_2) {A2D(I-o 2) TI3(H2jI3) D U8,6 + A2F(I-02)[TI3(H2mI3)F U7, 6 + TI7

(H2m17) F U6,6]}S, L

D2, 7 = (_2)[A2E(I-O 2) e7(H2j7)E]S,L

D2, 8 = ( ) {A2D(I-P2 ) TI3(H2jI3) D U8,8 + A2F(I-02)[YI3(H2mI3) F UT, 8 + TI7

(H2m17) F U6,8]}S, L

224

oD2,9 = (_-_2){A2D(I-o 2) Ti3(H2jI3) D U8,9 + A2F(I-O2)[TI3(H2mI3) F U7,9

+ TI7(H2mI7)F U6,9]}S,L

D2,10 = (_2) {A2D(I-P2) TI3(H2jI3) D U8,10 + A2F(I-o2)[TI3(H2mI3) F U7,10

+ TIT(H2mI7)F U6,10]}S,L

D2,11 = (-_2) {A2D(I-P2) TI3(H2jI3)D U8,11 + A2F(I-P2)[TI3(H2mI3) F UT,ll

+ TI7(H2mlT)F U6,11]}S,L

D2,12 = (_22) {AzD(I-P2) TI3(HzJ 13)D U8,12 + A2F(I-P2)[TI3(H2mI3)F U7,12

+ TI7(H2mI7) F U6,12]}S,L

D2,13 = (-_2) {A2D(I-P2)[eI3(H2jI3)D + TI3(H2jI3)D'U8,13] + A2F(I-P2)[_I3

(H2m13) F + TI3(H2mI3) F U7,13 + 17(H2mI7)F U6,13]}S,L

D2,14 = (_2) {A2D(I-o 2) TI3(H2jI3) D U8,14 + A2F(!-O2)[TI3(H2mI3) F U7,14

+ TI7(H2mI7) F U6,14]}S,L

D3, I = 0

D3, 2 = (_._3)A3[(I-D 3) _2(H3j2)E]S,L

225

D3,3

D3,4 = D3, 5

A3[(I-03) , 3(H3j3)E - _3]S,L

=0= D3, 6

D3,7

D3,8

A3[(1-03 ) ¢7(H3j7)E]S,L

= D3,10 = D3,11 = D3,12 = D3,13 = D3,14 = 0

D4,1

D4,2

oA 4

oA 4

--

{(I-04)[TI3(H4mI3)F U7,1 + TI7(H4mI7) F U6,1]}S,L

+ TI7(H4mI7) F U6,2]}S,L{ (1-04) [E2(H4mZ)F + _I3(H4mI3)F U 7,2

D4,3

D4,4

--0

oA 4

(--_4) { (I-04) _ 4(H4m4)F

(H4m17) F U6,4}S,L

- _4 + (1-04) TI3 (H4m13) F U7,4 + (l-p 4) TI7

D4,5

D4,6

oA 4 + _ U 6 5 ]}

(__4) {(I_04)[TI3(H4mI3)F U7, 5 17(H4mI7)F , S,L

(_4){ U7 U6,6]}(i-04) [TI3 (H4m13) F ,6 + TI7 (H4m17) F S,L

=0D4,7

D4,8

oA 4-- { (1-04 ) [_I3(H4mI3)F U7,8

= (M4)

+ TI7(H4,17) F U6,S]}S,L

226

D4,9 =

D4,10

D4,11

D4,12 =

D4,13 =

D4,14 =

OA4 U7(___44){ 41-04) [_13(H4mI3)F ,9

+ TI7(H4mI7)F U6,9]}S,L

41-04) [_13(H4m13)F ,i0(_4){ u7

(_A4

(__4) { (1-P4) [_I3(H4mI3)F U7 ,ii

(1-P4) [_13(H4m13 ) U7,12(_4){ .F

[_I3(H4mI3)F + _I3(H4mI3)F U7,13

(_A4 U 7

(__4) { (!-p4) [_Z3(H4mI3).F ,14

+ rl7(H4ml7)F U6,10]}S,L

+ _I7(H4mI7)F U6,11]}S,L

+ TI7(H4mI7)F U6,12]}S,L

+ TI7(H4mI7)F U6,13] (I-P4)}S,L

+ _17(ll4ml7)F U6,14]}S,L

D5,1

D5,2

_- { (l-P5) [TI6 (l_15j16) c U4,1

°A 5 U 4

= (__5) { (i-05) [T16 (H5j 16) ¢ ,2

= _lT(H5j17)c U5,1]}S,L

+ _17(H5j17)c U5,2]}S,L

D5,3 = 0

+ _IT(H5jI7) c U(_A5 U 4

D5,4 = (___5){(l-05)[_16(H5j16)c ,4

D5,5

(_A5 U 4

= (___5){ (i-05) [T16 (H5j 16) c ,5

+ _17(H5]17 )

(H5j5) c - _5]}S,L

oA 5 + ,6

D5, 6 = (-_5){(I-P5)[_6(H5j6)c TI6(H5jI6)c U4

5,4]}S,L

c u5,5] + [(t-Ps) _5

+ _17(H5j17)c U5,6]}B,L

227

=0D5,7

°A5 cD5, 8 -- (--_5){(I-05)[Es(H5j8) + _16(H5j16)c U4,8

+ TI7 (H5jI7) c U5 ,8]}S,L

D5,9= (i-05) [_9 (H5j 9) c r16 (H5jl6)c U4,9

+ _17(H5j17)c U5,9]}S,L

_- °A5 +D5,10 (--_5){(I-05)[TI6(H5j 16)c U4,10 TIT(H5jI7)c U5,10]}S,L

D5,11

oA 5

= (--_5){(i-05) [_16(H5j16)c U4,11+ TI7(H5jI7) c U5,11]}S,L

oA 5 +

D5,12 = (-_5){(I-05)[TI6(H5jI6)c U4,12 TI7(H5jI7)c U5,12]}S,L

D5,13 (O--AM_){ c= . (1-05) [TI6(H5jI6) U4,13 + TI7(H5jI7) c U5,13]}S,L

DS, 14

_A 5

= (--_5){(I-05)[II6(HDjI6)c U4,14+ TI7(H5jI7) c U 5,14])S,L

D6,1°A6 U 4

= (--_6)((i-06) [TI6 (H6j 16) c ,i+ TI7(H6jI7) c U5,1]}S,L

$6,2

oA 6

= (-_6){(I-06)[TI6(H6jI6)c U4,2+ TI7(H6jI6) c U5,2]}S,L

=0D6,3

D6,4= (1-06)[TI6(H6jI6) c U4,4 + %17(H6jlT)c U5,4]}S,L

228

D6,5 (o_) { c= (i-06)[_5(H6j5)c + TI6(H6jI6)c U4,5 + TI7(H6j17) U5,5]}S,L

oA6 c -

(-_6){(I-06)[¢6(H6j6) c + TI6(H6jI6) c U4,6 + _17(}{6j17 ) U5,6 C6}S,L

=0D6, 7

aA6 + +

D6, 8 = (--_6){(I-06)[_8(H6j8)c _I6(H6jI6)c U4,8 TiT(H6jI7) U5,8]}S,L

D6,9 (°_--66){ +_- (1-06) [E9(H6jg) e Tl6(H6jl6)c U4,9+ _17(H6j17)c U5,9]}S,L

aA 6 +

D6,10 = (7-_6){(I-06)[TI6(H6hI6) e U4,10 _17(H6j17)c U5,10]}S,L

D6,11

oA 6

= (-_6)( (1-06 ) [TI6(H6jI6) c U4,11

+ TI7(H6jI7)c U5,11]}S,L

oA6 cD6,12 = (-_6){(I-06)[TI6(H6jI6) U4,12 + TI7(H6jlT) c U5,12]}S,L

D6,13 (_6){= (1-06) [_16(H6j16)c U4,13+ YI7(H6jI7)c U5,13]}S,L

D6,14

oA 6

= (-_6){(I-P6) [TI6(H6jI6)c U4,14+ TI7(H6jI7) c U5,15]]S,L

=0D7,1

_A 7

D7,2 = (-_7)E{(1-07) _2(H7j2)E}S'L

229

aA 7

D7, 3 = (-_7)E{(I-07) 3(H7j3 )E}S,L

= 0= D 7

D7, 4 = D7, 5 ,6

oA 7

D7,7 = (-_7) f (l-P7)E

D7, 8 = D7, 9 = D7,10

- _7,}7(H7j7)E

= D 7= D7 ,ii ,12

S,L

= D7,13 = D7,14

=0

D8,1

D8,2

: (1_08) [TI6 (HsjI6) c ,i(_s){ u4

oA 8

: (__8) { (1_08) [TI6(H8jI6) c U4,2

+ "[17(Hsj17)c U5,1]}S,L

+ TI7(HsjI7)c U5,2]}S,L

D8,3

D8,4

D8,5

D8,6

=0

: (l_P8) [-[16 (H8jI6)c ,4(_s){ u4

oA 8 +

: (-_S) { (L-_8) [_ 5(HSj 5)c 16(Hsj 16)

8j17) u 41}s+ TI7( H c 5, ,L

U 4c ,5

oA8 (H8j 6) + TI6(HSjI6)c U4,6

(-_8) { (I-P 8) [¢ 6 c

+ TI7(Hsjl7)c U5,5]}S,L

+ TI7(HBjI7)c U5,6]}S,L

=0D8,7

_A 8 + .

D8,8 = (__8){(l_P8.)ie8(H8j8) c TI6(HSjI6)c U4,8

+ TI7 (H8jI7)c U5,8] - _8}$'L

230

oA8D8,9 = (--_8)

{ (l-f) 8) [e9(I18j9)¢ + T16(liSj16) c U4,9 + _17(H8j17)c U5,9]}S,L

OA8 + cDS,10 = (--_8){(l-08)[T16(li8j16)¢ U4,10 _I7(H8jI7 ) U5,10]}S,L

D8,11 (_s){= (i-08) ['[16(H8j16)c U4 ,ii+ _lT(Hsj17)c U5,11]}S,L

D8,12 - (_8){ (1-08 ) [_16(H8j16)c U4,12

+ "[17(H8jlT)c U5,12]}S,L

gAs c +D8,13 = (-_8){(I-08)[_I6(H8jI6) U4,13 "_I7(H8JlT)c U5,13]}S,L

D8,14

OA 8

-- (--_8){ (i-08) [Tl6(liSjl6)c U4,14

+ _17 (H8j17) c U5,14]}S,L

D9,1

D9,2

+ T16(l'19m16)B U3,1] + A9c(i-09) c

= (_9){A9B(I-P9)B[_I4(H9mI4)B U2,1

+ "[17(H9jlT)c U5,1]}S,L[TI6(H9jI6)c U4,1

-- (_9){A9B(I-P9)B[_I4(H9mI4)B U2,2 + Tl6(li9ml6) B U3,2] + A9c(I-P9) c

+ TI7(H9jI7)c U5,2]}S,L[_16 (H9jI6)c U4,2

D9,3

D9,4

--0

+ "[16(H9mI6)B U3,4] + A9c(I-P9) c

(_9){A9B(I-o9B[TI4 (H9mI4)B U2,4

+ _[17(H9j17)c U5,4]}S,L['[16(H9j 16) c U4,4

231

D9,5 (_9){A9B(I-O9)B[TI4(H9mI4)B U2,5 + TI6(H9mI6)B U3,5] + A9¢ (l-P9) c

[E5(H9j5)c + TI6(H9jI6) c U4,5 + TI7(H9jI7)c U5,5]}S,L

D9,6 (M9__)o{A9B(I-P9)B[TI4(H9mI4) B U2,6 + TI6(H9mI6)B U3,6] + A9c(l-P9)c

[E6(H9j6)c + TI6(H9jI6) c U4,6 ÷ TI7(H9jlT) c U5,6]}S,L

D9,7 = 0

D9,8 (_9){A9B(!-P9)B[TI4(H9mI4)B U2,8 + TI6(H9mI6)B U3,8] + A9c(l-P9)c

[_8(}{9j6)c + T16(H9jl6)c U4,8 + TI7(H9jI7) c U5,8]}S,L

D9,9 = (%){A9B(I-09)B[_9(H9m9)B + TI4(H9mI4)B U4,9 + TI6(H9jI6)c U5,9] - A9B_9

+ A9c(i-O9)c[C9(H9j 9) + c U4 + U5 ] -c TI6 (H9jI6) ,9 TI7 (H9jI7)c ,9 A9c 9}S,L

D9,10 (_9){A9B(I-P9)B[TI4(H9mI4)B U2,10 + TI6(H9mI6)B U3,10]

(H9jI6) c U4,10 + TI7(H9jI7) c U5,10]}S,L

+ A9c[TI6

D9,11 = (%){A9B(I-O9)B[ClI(H9mlI)B = TI4(H9mI4)B U2,11 + TI6(H9mI6)B U3,11]

+ A9c(l-09)c[_16(H9jl6)c U4,11 + TI7(H9jI7) c U5,11]}S,L

D9,12 (_9) {A9B(1-09) B[e12(H9m12)+ U2B TI4 (H9m14)B ,12 + TI6(H9mI6)B U3,12]

+ A9c(I-O9)c[TI6(H9jI6) c U4,12 + TI7(H9jI7) c U5,12]}S,L

232

D9,13

D9,14

(_9){A9B(I-09)B[TI4(H9mI4)B U2,3 + "c16(H9m16) m

[TI6(H9jI6)c U4,13 + TIT(H9jI7) c U5,13]}S,L

U3,13 ! + A9c(I-09) c

+ TI6(FI9mI6) B U3,14]

= (_9){A9B(I-09)B[_ 14(H9mI4)B + TI4(H9mI4) B U2,14

+ A9c(I-09)c[TI6(H9jI6) c U4,14 + TI7(H9jI7) c U5,14]}S,L

DI0,1

DI0,2

oAI0

--{(i-010)= ( MZ0 )

oaf0

: (Tzo){(z-Ozo)

TI4 (HI0jI4)A UI,I}S,L

TI4(HIoJ14 )A UI,2 }S,L

=0DI0,3

DI0,4

DI0,5

DIO,6

oAI0

: (Tzo){(z-Ozo)

OAI0

-- {(I-010): ( MI0 )

_AI0

: (--_i0){(1-010)

TI4 (HI0jI4)A UI,4}S,L

"[14 (HI0jI4)A UI, 5} S,L

TI4 (HI0j I4)A UI,6}S,L

=0DI0,7

DI0,8

DI0,9

°Al0

: (TlO){ (1-olO)

OAl0

: (_i0 ){ (i-010)

TI4 (HI0j 14) A UI,8}S,L

TI4(HI0jI4)A UI,9}S,L

233

(_AI0 _ +DI0,10 = (-_I0)[(I-PI0) _I0(HI0jI0)A el0 (I-P10) TI4(HI0jI4)A UI,10}S,L

oAI0DI0,11 = (-_I0){(I-PI0) _I4(HIojI4)A UI,II}

(_AI0{(i-01 O) TI4(HI0jI4)A UI }SDI0,12 = (.MIo) [eI2(HI0jI2)A + ,12] ,L

°AloDI0,13 = (-_i0){(i-010) TI4(HIojI4)A UI,13}S,L

(_AI0DI0,14 = (-_i0){ cI4(HIOjI4)A

+ _I4(HIojI4)A UI,14}

DII,I_AII { (i-011) U2

= (MII--) [TI4(HIImI4)B ,i+ TI6(HIImI6) B U3,1]}S,L

oAII +DII,2 = (--_II){(I-PlI)[_I4(HIImI4)B U2,2 YI6(HIImI6)B U3,2]}S,L

DII,3 = 0

oAII += (Hllml4) BDII,4 (--_II){(I-oII)[TI4 U2,4 _I6(HIImI6)B U3,4]}S,L

°AllDII,5 = (-_II){(I-PlI)[TI4(HIImI4)B U2,5 + 16(HIImI6)B U3,5]}S,L

DII,6°All U2-- { (l-Pil) IT14(Hllml4) B ,6

= ( MII )+ TI6(HIImI6) B U3,6]}S,L

=0DII,7

DII,8OAII U 2-- {(I-PlI)[TI4(HIImI4)B ,8

= ( M11)+ _I6(HIImI6)B U3,8]}S,L

234

°All + +DII,9 " (-_II){E9(HIIm9)B YI4(HIImI4)B U2,9 TI6(HIImI6)B U3,9]}S,L

OAll= (IMI-){_I4(HIImI4)B U2'I0DII,10

+ TI6(HIImI6) B U3,10]}S,L

°All= (--Mll){(l-Pll) ell(Hllmll)B - _II + TI4(HIImI4)B U2,11DII, ii

+ TI6(HIImI6) B U3,11}S,L

°All= -- { (I-P11) [_12(HIImI2)B + TI4(HIImI4) B U2,12

Dli, 12 ( Mil )

+ TI6(HIImI6) B U3,12]}S,L

DII,13

DII,14

°All-- {(i-01 I)[_I4(HIImI4)B U2,13 + _[16(HIImI6)B U3,13]}S,L

= ( MII )

OAII

= (--_II){(I-011)[_I4(HIImI4)B + TI4(HIImI4)B U2,14

+ TI6(HIImI6) B U3,14]}S,L

DI2,1

D12,2

o {AI2A(I_oI2) A _14 (HI2j 14 )= (MI2)

+ TI6(HI2mI6) B U3,1]}S,L

A UI,I

= (M_2){AI2A(I-PI2) A TI4

+ TI6(HI2mI6) B U3,2]}S,L

(HI2j 14)A UI,2

+ AI2B(1-OI2)B[_I4(HI2mI4)B U2,1

+ AI2B(I-012)B[TI4(HI2mI4) B U2,2

=0D12,3

235

D12,4 = (M_2){AI2A(I-012) A TI4(HI2JI4)A UI,

+ TI6(HI2mI6)B U3,4]}S,L

4 + AI2B (I-012) B[TI4 (HI2mI4)B U2'4

D12,5 = (M_2) AI2A(I-012)A _I4(HI2JI4)A UI,5

+ _I6(HI2mI6)B U3,5]}S,L

+ AI2 B (1-012) B[TI4 (H12m14) B U2,5

D12,6 = (M_2){AI2A(I-012) A rI4(HI2jI4)A UI

_+ TI6(HI2mI6) B U3,6]}S,L

,6 + AI2B(I-012)B[TI4(HI2mI4)B U2,6

D12,7 = 0

D12,8 = (M_2){AI2A(I-oI2) A TI4(HI2JI4)A UI,8

+ _I6(HI2mI6)B U3,S]}S,L

+ AI2B(I-oI2)B[TI4 (HI2mI4)B U2,8

D12,9

DI2,10

(_2) {AI2A(I-012) ATI4 (HI2JI4)A UI,9 + AI2B(I-012) B [E9 (H12m9) B

+ TI4(HI2mI4) B U2,9 + TI6(HI2mI6)B U3,9]}S,L

-- (M_2){AI2A(I-012)A[_I0(HI2jI0) A + TI2(HI2jI4) A UI,10] + AI2B(I-oi2) B

[TI4(HI2mI4) B U2,10 + TI6(HI2mI6)B U3,10]}S,L

DI2,11 = (M_2){AI2A(I-012) A TI4(HI2jI4)A UI,II + AI2B(I-012)B[¢II(HI2mlI) B

+TI4(HI2mI4)B U2,11 + TI6(HI2mI6) B U3,11]}S,L

236

D12,12 = (M_2){AI2A(I-oI2)A[_I2(HI2jI2) A + TI4(HI2jI4)A Ul,12] - AI2A El2

++ AI2B(I-PI2)B[_I2(HI2mI2) B _I4(HI2jI4)B U2,12 + TI6(HI2mI6B

U3,12] - AI2B eI2}S,L

D12,13 = (M--_2){AI2A(I-PI2)A TI4(HI2jI4)A UI,13

+ TI6(HI2mI6) B U3,13]}S,L

+ AI2B(I-PI2)B[TI4(HI2mI4) B U2,13

DI2,14 (M_2){AI2A(I-012)[_I2(HI2JI4)A + YI4(HI2jI4)A Ul,14 + AI2B(I-oI2) B

[_I4(HI2mI4)B + TI4(HI2mI4) B U2,4 + TI6(HI2mI6) B U3,14]}S,L

DI3,1 = (_3){AI3F(I-PI3)F[TI7(HI3mlT)F U6,1 + TI3(HI3mI3) F U7,1] - AI3 F U7,1

+ AI3D(I-oI3)D[_I(HI3jl) D + TI3(HI3jI3) D U8,1] - AI3 D TI3 U8,1}S,L

D13,2 = (M---_3){AI3F(I-013)F[TI7(HI3mlT) F U6,2 + TI3(HI3mI3) F U7,2

+ Ez(HI3m2) F] - AI3 F TI3 U7,2 + AI3D(I-013)D[C2(HI3j2) D

+ TI3(HI3jI3) D U8,2] - AI3 D TI3 U8,2}S,L

D13,3 = 0

D14,4 -- (M-_3){AI3F(I-PI3)F[TI7(HI3mI7) F U6,4 + TI3(HI3mI3) F + e4(Hl3m4)F]

_13 U7,4 + AI3D[(I-PI3)D TI3(HI3jI3)D - TI3] U8,4}S,L

- AI3 F

237

D13,5 (M-_3){AI3F(I-013)F[TI7(HI3mI7)F U6,5 + TI3(HI3mI3) F U7,5] - AI3F

%13U7,5 + AI3D[(I-oI3)D TI3(HI3j 13)D - %13] U8,5}S,L

D13,6 (M-_3){AI3F(I-PI3)F[TI7(HI3mI7)F U6,6 + %I3(HI3mI3)F U7,6] - AI3F

%13U7,6 + AI3D[(I-013)D %I3(HI3j 13)D - %13] U8,6}S'L

D13,7 = 0

D13,8 U6,8 + 8](M--_3){AI3F(I-PI3)F[%I7(HI3mI7)F %I3(HI3mI3)F U7,

%13U7,8 + AI3D[(I-QI3)D TI3(HI3j 13)D - %13] U8,8}S,L

- AI3F

D13,9 = (¢?{AI3F(I-013)F[TI7(HI3mI7) F U6,9 + YI3(HI3mI3)F U7,9] - AI3F

%13U7,9 + Ai3D[(I-PI3)D %I3(HI3j 13)D - TI3]U8,9}S,L

DI3,10 (M--_3){AI3F(I-oI3)F[YI7(HI3mI7)F U6,10 + %I3(HI3mI3)F U7,10] - AI3F

%13 U7,10 + AI3D[(I-PI3)D %I3(HI3j 13)D - %13] U8,Z0}S'L

DI3,11 (M_3){AI3F(I-PI3)F[TIT(HI3mI7)F U6,11 + %I3(HI3mI3)F U7,11]

%13 U7,11 + AI3D[(I-013)D %I3(HI3j 13)D - TI3] U8,11}S'L

- AI3 F

D13,12 (M-_3){AI3F(I-PI3)F[TI7(HI3mI7)F U6,12 + %I3(HI3mI3)F U7,12]

Y13 U7,12 + AI3D[(I-013)D YI3(HI3j 13)D - TI3] U8,12}S'L

- AI3 F

238

D13,13 = (M_3){AI3F(I-oI3)F[TI7(HI3mI7)F U6,13 + TI3(HI3mI3)F U7,13 + _13

(H13m13)F] - AI3F(eI3 + r13 U7,13) + AI3D(I-oI3)D[_I3(HI3jI3) D

+ TI3(HI3jI3)D U8,13] - AI3D (el3 + TI3 U8,13)}S,L

DI3,14 = (M_3){AI3F(I-oI3)F[TIT(HI3mI7) F U6,14 + TI3(HI3mI3)F U7,14] - AI3F

TI3 U7,14 + AI3D[(I-013) D 13(HI3jI3)D - TI3] U8,14}S,L

DI4,1 = (#4){AI4A TI4[(I-014)(HI4jI4)A - i] Ul, I + AI4B[(I-014) TI4

(H14m14) B - r14] U2, I + AI4 B r16(l-014)(H14m16)B U3,1}S, L

D14,2 = (M--_4){AI4A rl4[(l-Pl4)(Hl4jl4)A - i] UI, 2 + AI4B[(I-014) TI4

(H14m14) B - TI4] U2, 2 + AI4 B T16(I-014 )(H14m16) B U3,2}S,L

D14,3 = 0

D14,4 = (M--_4){AI4A TI4[(I-oI4)(HI4jI4) A - i] UI, 4 + AI4 B T14[(I-014)

(H14m14) B - i] U2, 4 + AI4 B TI6(I-014)(HI4mI6) B U3,4}S, L

D14,5 = (M_4){AI4 A TI4[(1-014)(HI4jI4) A - l] Ul, 5 + AI4 B T14[(1-014)

(H14m14) B - l]U2, 5 + AI4 B TI6(I-014)(HI4mI6) B U3,5}S, L

239

D14,6 (M_4){AI4A TI4[(1-014)(HI4jI4) A - I] UI, 6 + AI4B T14[(1-014)

(H14m14)B - i] U2,6 + AI4B TI6(I-014)(HI4mI6)B U3,6}S,L

D14,7 = 0

D14,8 (M_4){AI4A -[14[ (1-014)(H14h14)A - i] UI, 8 + AI4B TI4[ (I-P14)

(H14m14)B - i] U2,8 + AI4B TI6(I-014)(HI4mI6)B U3,8}S,L

DI4,9 (M_4){AI4A _I4[(I-oI4)(HI4j 14)A - i] UI, 9 + AI4B(I-014) B[E9(H14m9)B

+ TI4(HI4mI4) B U2,9 + TI6(HI4mI6)B U3,9] - AI4B TI4 U2,9}S,L

DI4, i0 ffi (M_4){AI4A(I-014)[_Io(HI4jI0)A + TI4(HI4jI4) A UI,IO]

+ AI4B(I-PI4)[TI4(HI4mI4) B U2,10 + TI6(HI4mI6)B U3,10]

- AI4A TI4 UI,IO

- AI4B TI4 U2,10}S,L

DI4,11 = (M_4){AI4A TI4[(I-0i4)(HI4jI4)A - i] UI,II + AI4B(I-014)[ClI(HI4mlI) B

+ TI4(HI4mI4) B U2,11 + TI6(HI4mI6) B U3,11] - AI4B TI4 U2,11}S,L

D14,12 = (M_4){AI4A(I-Pi4)[eI2(HI4jI2) A + TI4(HI4jI4)A Ul,12] - AI4A TI4 UI,2

+ AI4B(I-014)[cI2(HI4mI2) B + TI4(HI4mI4)B U2,12 + TI6(HI4mI6)B U3,12]

- AI4B TI4 U2,12}S,L

D14,13 = (M_4){AI4A TI4[(I-oI4)(HI4jI4) A - i] UI,13 + AI4B(I-oI4)[TI4(HI4mI4) B

U2,13 - TI6(HI4mI6)B U3,13] - AI4B TI4 U2,13}S,L

240

DI4,14 = (M-_4){AI4A[(I-014)(HI4jI4) A - i](c14 + TI4 UI,14) + AI4B(I-014)

[_ )B + ) U2 + U314(H14m14 TI4(HI4mI4 B ,14 TI6(HI4mI6)B ,14] - AI4B

(_14 + TI4 U2,14)}S,L

(BI)s, L = (M_)[AI(I-0 I) TI3(HIjI3) D U8,15]S, L

(B2)s, L = (M_)[A2D(I-02) TI3(H2jI3) D U8,15 + A2F(I-02){TI3(H2mI3) F U7,15

+ TI7(H2mI7)F U6,15}]S, L

(B4)s,L =(M_)[A4(I-o4){_I3(H4mI3)F U7,15 + TI7(H4mlT)F U6,15}] S,L

(B5)s, L = (M_)[A5(I-05) {_16(H5j16)c U4,15 + _17(H5j17)c U5,15}]S,L

(B6)s, L = (M_)[A6(I-o6){TI6(H6jI6) c U4,15 + TI7(H6jI7) c U5,15}]S,L

(B8)s, L = (M_)[As(I-O8){TI6(H8jI6) c U4,15 + TI7(H8jI7) c U5,15}]S, L

(B9)s, L = (M_)[A9B(I-o9)(TI4(H9mI4) B U2,15 + TI6(H9mI6)B U3,15} + A9c(l-09)

{TI6(H9jI6)c U4,15 + TI7(H9jI7)c U5,15}] S,L

(BI0)S, L = (M-_0)[AI0(I-010){TI4(HI0jI4) A UI,15 + TI5(HI0jI5)A}]S, L

(BII)S, L = (MI-_)[AII(I-011){TI4(HIImI4) B U2,15 + TI6(HIImI6) B U3,15}]S,L

241

(BI2)S, L = (M_2)[AI2A(I-012){TI4(HI2jI4) A UI,15 + TI5(HI2jI5) A} + AI2B(I-012)

{TI4(HI2mI4)B U2,15 + rl6(Hl2ml6)B U3,15}]S, L

(BI3)S, L = (..i)[AI3 D TI3 { - I}U8, + { )m13 (1-013) (HI3jI3)D 15 AI3F TI3 (i-013

(H13m13)F - i}U7,15 + AI3F(I-PI3) TI7(HI3mI7) F U6,15]S,L

(BI4)S, L (MI_)[AI4A YI4{ ) )A I}UI, += (1-014 (HI4jI4 - 5 AI4A(I-oI4) YI5

(HI4jI5) A + AI4 B TI4((I-014)(HI4m14) B - I}U2,15 + AI4 B TI6

(l-P14) (H14m16) B U3,15 ]S, L

For the total channel,

(Bi)s, L B.1

TI3 = TI4 = i ,

:" c = 0 .13 14

Scanner Model Equations

Total Channel

" AijT j 4 + BQiQ + B.E + b + eT.1 = + Dij Tj I

SW/LW Channels

Ti = A._jT.3 +D..T4_3 j + BQ iQ + (Bi)sEs + (Bi)LEL + b + e

242

Scanner (using Enclosure Coupling Method)

E

o

:i

i2

_3

_4

_5

6where m = 6

For each cavity (or enclosure in the scanner model)

EE_I i

ETI E ETI E

. -- _I

E E

L _

v

i

Es= EI!

E°f

i i i i 1 _ 1 i

where E1 = El5 , E2 = El0 , E3 = EI__ , E4 = El2

2 2 2 2 2 2 2 2 2

E1 = El4 , E 2 = Ell , E3 = Ek6 , E4 = E9 , E5 = El2

3 3 _ 3 3 ,3 3 3 3 3 3 3

E1 = EIO ' E2 = E8 ' E3 = '6 ' E4 = E5 ' E5 = El7 ' E6 = E9

4 4 E_ 4 4 _4 E4 4E1 = El7 ' _ = E4 ' E3 = '2 ' 4 = El3

5 5 , E_ 5 5 5E1 = El3 _ =oE 2 , E3 = E 1

6 6 6 6 6 6

E1 = E3 , E2 = E2 , E3 = E 7

243

E1 100000_ I E2= 000

L ooooj000

E2=0010

0000

0000

0000

0000

E1 +0000000000

i00000 I000000

000000J

E3

_010_

;_= °°°° I

iOOOO

_0000

0000

0000liO 00il E4

E2+ 00

O O00

[o°°00

Ioooll= 000000000000

_o_E3+ 00

U °

E5

= 00

00

_4 _6:o9

Thus D matrix is formed by the above Es.

244

. °

0010

0000

0000

I0000

0000

0000

0000

I0000

_0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

O000i

00001

00001

0000

0000

0000

0000

0000

0000

OlO0

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

000o

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

tO00

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

oooo

0000

0000

000'0

0000

O0

O0

O0

O0

000000

000000

000000

000000

000000_

000000

000000

000000

000000

001000

000000

I00000

000000

000000

000000

O0000t

0ooooo

000000

O0

O0

O0

000

000

000

000

000

000

000

000

000

000

000

000

000

000

I00

000

ooo

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

ooo

000

o

sl

m

10000 ..........

00000 .......... 0

00000 .......... 0

00000 .......... 0

o.°,.°°,°.°...,

00000 .......... 01

245

0

x14

0

T14

0

1

0

0

1

0

0

0

1

0

1

0

0

x13

x13

0

0

0

0

246

TD= O0 0 0

O0 0 0

00 0 Oi

00 0 0

00 TI40

O0 0 0

O0 0 0

O0 0 0

'00 0 0

00 0 0

O0 0 0

00 0 0

O0 0 0

00 0 0

00 0 0

O0 0 0

00 0 0

O0 0 0

00 0 0

O0 0 0

00 0 0

00 0 0

O0 0 0

O0 0 0

00 0 0

0 0000

0 0000

TI40000

0 0000

0 0000

0 0000

0 0000

0 0000

0 0000

0 0100

0 0000

0 0000

0 0000

0 0000

0 0000

0 0000

0 0000

0 0000

0 0000

0 0000)

0 0000

0 0000

0 0000

0 0000

0 0000

000000

000000

000000

000000

i000000

000000

I.00000)

000000

000000

0000_0

000000

000000

000000

000000

000000

000010

000000

000000

000000

000000

000000

000000

000000

000000

000000

000

000

000

000

000 0

000 0

000 0

000 0

000 0

000 0

000 0

000 0

000 0

i00 0

000 0

000 0

000 0

000 0

000 0

000"r13

lO00 0

_000 0

000 0

000 0

000 0

0

0

0

0

0

0

0

0

0

0

0

0

0

'l"13

0

0

0

0

0

0

0 O0

0 O0

0 00

0 00

0 00000

0 00000

0 00000

0 00000

0 00000

00000

00000

00000

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001000

00000

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00

00

O0

247

A

1

IAI| AI2 AI3 At4

A21 A22 A23 A24

A31 A32 A33 A34

A41 A42 A43 A44

4

All A12 AI3 A14 AI5

A21 A22 A23 A24 A25

A31 A32 A33 A34 A35

IA41 A42 A43 A44 A45

A51 A52 A53 A54 A55

!All AI2 A13 A14 AI5 A16

A21 A22 A23 A24 A25 A26

A31 A32 A33 A34 A35 A36

A4! A42 A43 A44 A45 A46

A51 A52 A53 A54 A55 A56

A6! A62 A63 A64 A65 A66

All A12 A13 A14

A21 A22 _23 A24

A31 A32 A33 A34

A4I A42 A43 A44

All AI2 AI3

A21 A22 A23

A31 A32 A33

6

AI] AI2 AI31

A21 A22 A231

A31 A32 A33J

A = (I - dE)-I E

/_'_EDING PAGE BLANK NOT FILMED

249

APPENDIX E

ERBE SOLAR MONITOR THERMAL MODEL

_bl2qG PAGE BLANK NOT FILMI_)

251

SOLAR MONITOR MODEL

As described in the non-scanner and scanner modeling efforts, the same

notations are used for constants and variables in developing the energy

balance equation of the solar monitor.

To avoid the burden to write the same equations as used in developing

the other models, the same variable name in the equation such as Rij is

here simply used.

A general form of the equation for the solar monitor which is divided

into 19 nodes is presented here as

T = AT + DT 4 +BQQ + BE + b.

Now the computational algorithm for the constants of the above equation

has been developed by using the enclosure coupling technique. It is avail-

able and can be used for all 8 non-scanner, scanner and solar monitor models.

The following constants, Aij, represent the coefficients of themal

conduction terms in the energy balance equations•

All = - I/(RI3MI )

AI2 = 0

AI3 = I/(RI3M I)

A21 = 0

A22 = - I/(R23M_)

A23 = I/(R23M 2)

252

C

!

CII C12 Ct3 C14

C21 C22 C23 C24

C31 C32 C33 C34

C41 C42 C43 C44

CII C12 C13 C14 C15 i

C2I C22 C23 C24 C25

C31 C32 C33 C34 C35

C41 C42 C43 C44 C45

CSI C52 C53 C54 C55

CI 1 C12 C13 C14 C15 C16

IC2] C22 C23 C24 C25 C26,

C31 C32 C33 C34 C35 C36

C4I C42 C43 C44 C45 C461

C51 C52 C53 C54 C55 C56

C6! C62 C63 C64 C65 C66

4 5 6

CI! CI2 el3 C14

iC2l C22 C23 C24

C31 C32 C33 C34

C41 C42 C43 C44

CI1 C12 C]3

C21 C22 C23

C31 C32 C33

Cll C12 C13

C21 C22 C23

C31 C32 C33

C = (I - 0dE) -I E T"

= [(I -0- TD)(I - CD) I- A - _]_ + [(i -0- TD)(I - CD)-

C-T] F S

253

A31 = i/(R3_3)

A32 = l/(R32M3 )

= _(i__ + _: + _i )/._A33 R31 R32 K34 J

A34 ffi1/(R34H3 )

A43 ffiI/(R43M4 )

--!__)/M4: -( i_ + _ + i + R4A44 R43 R45 R47

A45 ffiI/(R45M4 )

A47 ffiI/(R47M4 )

A4_I 9 ffii/(R4_I9M4 )

A54 = I/(R54M5 )

= _(i_!_ + Z----)/M5A55 _54 R56

A56 = I/(R56M 5)

A65 = I/(R65M6 )

= -(i____+ l___)/M 6A66 R65 R67

A67 : I/(R67M6 )

254

W

®

II'

.4-,--

I"!

!

II

!III

III!

II!

SOLAR MONITOR MODEL

i. Active Cavity

2. Reference Cavity

3. Button, Duplex Cone

Cavities

4. Heat Sink

5. Aperture, Primary

6. Shield, Aperture

7. Body (Rear Barrel)

8. Body (Front Barrel)

9. Aperture, Secondary

(or FOV Limiter)

i0. Motor Housing

11. Chopper Blade

12. Chopper Pendulum

13. Motor, Chopper

14. FOV Limiter

15. Front Top Housing Cover

16. Rear Top Housing Cover

17. Front Housing

18. Rear Housing

19. Holder, Heat Sink

20. Fictitious Surface

p. Temperature Probe

E. Total Incident Radiatio_

E.. Irradiance in enclosure a

E. Irradiance in enclosure b

Rate of Incident Radiation

From a Neighboring Surface

Cavities Within The

Sensor, i.e., Enclosures

or Spaces Between Sensor

Nodes

255

A74 = I/(R74M 7)

A76 = I/(R76M 7)

= _(i_.i__+ _ + i )IM7A77 R74 R76 R78

A78 = 1/(R78M 7)

A87 -- 1/(Rs7M 8)

ASS- I(i--!--+ i___+ I____)/MsR87 R89 RS-10

A89 = I/(R87M8 )

As_j0= l/(R8_IOM8)

A9_ 8 = i/(Rg_sM 9)

I

A9_ 9 = R9_8M9

Ai0_8 = I/(Ri0_8Mi0)

AI0_I0 = - i/(RI0_8MI0 )

AI0_I2 = i/(RI0_I2MI0 )

AI0_I 3 1 I/(RI0_I3MI0 )

- I/(RI0_I2MI0) - i/(RIo_I3MI0 )

AII_I 2 = I/(RII_I2MII )

AII_I I = - I/(RII_I2MII)

256

AI2_I I = I/(RI2_IIMI2 )

AI2_I 2 = -(I/RI2_I I + I/RI2_I3)/MI2

A12_13 _ i/(RI2_IIMI2 )

A13_12 -- i/(RI3_I2MI2)

AI3_I 3 = -(I/RI3_I 2 + I/RI3_I0)/MI2

A 3_Io --l!(R13_1o is)

A14-14

A14-15

A14-17

= -(I/RI4_I 5 + I/RI4_I7)/MI4

= I/(RI4_I5MI4 )

= i/(RI4_ITMI4 )

A15_15 = - I/(RIs_I6MI5) - I/(RI5_I4RI5 )

A15_16 = i/(RI5_I6MI5)

A15_14 = i/(RI5_I4MI5 )

A16_15 -- 1/(RI6_I5MI6 )

AI6_I 6 = -(I/RI6_I 5 + I/RI6_I9)/MI6

A16_19 = I/(RI6_I9MI6 )

257

AI7_I 4 = I/(RI7_I4MI7)

Al7_l 7 = -(I/RI7_I 4 + I/RI7_I8)/MI7

AI7_I 8 = I/(RI7_I8MI7 )

AIS_I7 = I/(RIs_I7MI8)

AI8_I 8 = -(I/RI8_I 7 + I/RI8_I9)/MI8

_8-19 = I/(RI8-19MI8)

A19_4= I/(RI9_4MI9)

AIg_I 6 = I/(19_16M19)

AI9_I 8 = I/(RI9_I8MI9 )

AI9_I 9 = -(I/R19_4 + I/RI9_I 6 + I/RI9_I8)/MI9

For the radiative interaction term, considering the specular reflection,

d s (1)p=p +p

E +p+T =i (2)

d _ d + Tk_kM k = (kOT + PkEk(3)

=. M dEk Z Ek_ j 33

(4)

The energy balance equation is now

_k = Ek = Mk(5)

258

where

s + dMk = kEk Mk •

Thus

d s _ Yk_k<k = Ek = ¢k_k - OkEk - okEk(6)

where

flk = °T4

From (3) and (4),

n dE ,-I, fi + r._.)M_ = _ (_ij - Oj j_l _ ( i i i l

i=l

(7)

From (7) through (4)

Ek = _ Ek- j _ Pj-i (e + -j i_i TiE i)

n

[(I_ ifli + _ "m )i i Ek-j Pj-i± j--1

(8)

Equation (6) becomes

_k = (! - Ok )Ek - _k_qk - _kEk

= I [(1 - Ok)_ i ! Ek-jPj-i - 'k_ki ] _ii ]

+ _[(i - pk)'ri _ Ek_jPj_ i - rk_ki ] Eii O

(9)

Let

Hkj i j Ek-j Pj-i

=Hnm £ L En_mPm-£

m

259

From the above drawing, the enclosures, a, b and c, are found. For

each enclosure, the incoming radiance to a certain surface can be described

by Equation (8).

Choose the enclosure b and c and couple them together, then one can

define the transmitted radiance through the filtering surface.

First,

enclosure b

enclosure c

elements 6,7,8,9,11,20

elements 1,5,20

Index

i,j,k

_,m_n

From Equation (8)

l 3 i j

where in this equation i, j and k are dummy index

or

Ek = [ e o Hk j + _ T E Hkji i_i i _ ii i

For the enclosure c, Equation (8) becomes

A

E20 = (I _£H2om£) c + T20 E20(H2om20)c£

(i0)

where £, m and n = i, 5 and 20.

260

Likewise, for the enclosure b,

A

E20 = (I ¢i_iH20ji)b + T2oE20(H20j20)b + TIIEII(H20jlI)bi

(ii)

where i and j = 6, 7, 8, 9, ii and 20

Combining (i0) and (ii) to obtain E20 and E20 as a function of Eli, we

have

E20 = GI[( _El_fH2omZ) c + T20(H20m20)c (! c i_iH2oji)b

I

+ TII T20(H20m20)c (H20jll)b Eli ](12)

and

*%

E20 = GI[(_ _i_i H2oji) n + T20(H20j20)b (_ E£_£H20m£)c1

+ TII(H20jlI)b Eli ](13)

where

2 c]-IGI = [i - T20(H20j20)b (g20m20)

The radiation energy balances in the enclosures b and c are

: [[(i - p k) _ -_klb i i Hkji k_ki ] _i

+ I[(1 - pk)T i Hkj ii

- Tk6ki ] (Ei=20 or Ei=ll)(14)

261

and

= I[(1 - On) eIHnml - en6n£] _££

A

+ I[(i - On) TlHnm l - Tn_n£ ] El£

(15)

Second,

enclosure d

enclosure b

elements 11,14,21

elements 6,7,8,9,11,20

Index

l,m,n

i,j,k

From Equation (8)

D

Eli = (I _i_iHllji)b + TII(HIIjlI)bEII + T20(HIIj20)bE20i

(16)

and

A

Eli = (_" _£Hllml)a + Tll(Hllmll)a Eli + T£

21(Hllm21)a E(17)

Substituting (12) into (16), Equation (16) becomes

A

Eli = ([ _i_iHllji)b + T20 GI(HIIj20) b (£_ _£H20m£)ci

2 GI c .+ T20 (HIIj20)D (H20m20) (_. _i_iH20ji)bl

2 GI (H2 J ] -+[TII(HIIjlI)b + TII _20 (Hllj20)b (H20m20)c - ll)b Eli(18)

Substituting (18) into (17)

Eli = G2[(_ _£_£ Hnml)a+ T21(HIIm21)aE + Tll(Hllmll)a (_. Ei_iHllji)b

l

+ TII T20 GI(HIImlI) a (Hllj20)b (_ _£_£ H20m£) c

2 GI c .+ TII T20 (Hllmll) a (HIIj20) b (H20m20) (_ Ei_iH20ji) b]I

(19)

262

where2 -i

G2 -- [i -TII SI(HIImlI) a]

2 GISI = (HIIjlI) b + T20 (HIIj20) b (H20m20)c (H2ojlI) b

On the other hand, substitution (17) into (18)

A

Ell -- G2[(_._i_iHllji) b + T20 GI(HIIj20) b (_ _I_£ H20ml) c11.

2

+ T20 GI(HIIj20) b (H20m20) c (_ ci_iH20ji)b

+rll SI( _ _l_l Hllm£) a + TIIT21 SI(HIIm21) a E](2O)

For the enclosure a: nodes: 11,14,21

10 for open position

= [(i - 011 ) Cll(_llmll) a - Cll]_ll + (i- 014 ) _14(_im14)a _14

+ [I - 01 I) T21(Hllm21)a E + [(i - 011 ) S(Hllmll)a. - TII]EII

0 for closed position

0 for open position

I - -- E= (i - 0i4 ) l(Hl4mll)a _iI + [(! 014 ) Gl4(Hl4ml4)a 14 ] _14

A

+ (i - 014 ) Tll(Hl4mll)a Ell + (i - 014 ) T21(Hl4m21)aE

263

For the enclosure b: nodes: 6,7,8,9,11,20

_6 b = [(i - 06) _6(H6j6)b - _6] _6 + (i - 06) E7(H6j7)b _7

0 for open position

+ (i - 06) _8(H6j8)b _8 + (i - 06) c9(H6j9) b _9 + (i - 06) _ll(H6jll)b _ii

_0 for closed position ~+ (i - 06) _II(H6jlI)D Eli + (i - 06) T20(H6j20)B E20

_7Ib = (i - 07) _6(H7j6)b _6 +[(i - 07) c

+ (i - 07) _8(H7j8)b _8 + (L - 07)

7(H7j7)b - _7] fZ7

0 for open position

9(H7jg)b _9 + (i - 07) ]II(H7jlI)B _ii

/ -+ (i - 07 ) ll(H7jll)b Ell + (i - 07) T20(HTj20) b E20

0 for closed position

= (1 - o8) 6(H8j6)b _6 + (i - 08) 7(H8j7)b _7 + [(i - 08) ¢8

(H8js)b - _8 ] _8

0 for open position

+ (! - 08 ) _9(H8j9)b _9 ÷ (i - 08) _ll(H8jll)b _ii

+ (! - 08)/ll(Hsjll)b Eli + (i - 08) T20(H7j20)b E20

0 for closed position

264

I = (i - 09) E6(H9j6)b _6 + (i - 09) E7(H9jT)b _7 + (L - 09) _8(H9j8)b _8_9 b

0 for open position

+ [(i - 09) (9(H9j9)b - (9 ] _9 + (i - 09) /ll(H9jll)b _ii

7 -+ (i - 09) ll(H9jll)b Eli + (i - 07) T20(H9j20) b E20

0 for closed position

b: (i - Pll ) (6(H11j6)b f_6 + (1 - Pll ) E

(HllJ 8)b _8

7 (Hllj7)b _7 + (1 - Pll )

+(i - Pll ) f9(Hllj9)b _9 + [(i - PlI)SI(HlljlI) b - (11 ] _ii

0 for open position

+ (L - Pll ) T20(HIIj20) b E20 + [(1 -Pll ) 1(Hzlj11)b - "[II ] Ell

0 for closed position

@20 b = (i - P20 ) E6(H20j6)b _6 + (I - P20 ) E7(H20j7) b _7

O_r open position

+ (I- 020 ) _8(H20j8)D _8 + (i - P20 ) _9(H20j9)b _9 + (I - P20)_l

(H2ojll)b _ii

+ (i - P20) TII(H2ojlI) b Eli + [(i - P20 ) _20(H20j20)b - T20] E20

265

Enclosure c: Nodes: 1,5,20

= [(i - 01) _l(Hljl)c - -el] _I + (L 01) 5(Hlj5)c _5

A

(1 - 01) r20(Hlj20)c E20

_51 c = (i - 05 ) _l(H5jl)c _i + [(i - 05 ) e5(H5h5)c -

+ (1 - 05 ) "r20(tt5j20) c E20

I - _5_20 = (i - 020 ) EI(H2ojl) c _i + (I 020 ) f5(H2oh5)c

c

A

+ [(i - 020 ) T20(H20j20) - T20] E20

For the enclosure d: Nodes: i,j,k = 1,3,4,5

_I Id = [(I-01) ei(Hlil)d" -_I] _I + (i - 01) _3(Hlj3)d _3

+ (i - 01) e4(Hlj4)d _4 + (i - 01) _5(Hij5)d _5

I : (I - 03) _l(H3jl)d f_l + [(i - 03) _3(H3j3)d - E3] f_3_3 d

+ (i - 03 ) _4(H3j4)d _4 + (i - 03) E3(H3j5) d _Q5

4)4 d : (i - 04 ) EI(H4jl) d _i + (I - 04) _3(H4j3)d f_3

+ [(i - 04) ¢4(H4j4)d - E4] _Q4 + (i - 04) _5(H4j5)d _5

266

_5Id = (1-05) Cl(H5jl)d _i + (1-05) _3 (H5j3)d _3

+ (1-05) E4 (H5j4)d _.4+[(1-05) _5 (H5j5)d - _5] _5

For the enclosure e: Nodes: 4,16,18,19

= [(1-04) c4 (H4j4) e - E4] _4 + (1-04) _16 (H4jl6)e _16

+ (1-04) f18 (H4jl8)e f_18+ (I-P4) el9 (H4jl9)e _219

_161e = (1-016) c4 (Hl6j4)e _4 + [(1-016) _16 (Hl6jl6)e - _16] _16

+ (1-016) _18 (Hl6jl8)e _18 + (1-016) _19 (Hl6jl9)e _19

= (I-P18 16 (HI8jI6 n16_18 e (I-P18) £4 (Hl8j4)e _4 + ) _ )e

+ [(I-PI8) _18 (Hl8jl8)e - _18 ] _18 + (I-P18) _19 (Hl8jl9)e _19

_191e = (1-019) _4 (Hl9j4)e _4 + (!-0i9) _16 (Hl9jl6)e _16

+ ) )e + [ (1-019) _(I-P19 ¢18 (H19j18 _218 19 (H19j19)e - _19 ] 219

For the enclosure f: Nodes: 4,7,8,14,15,16,17,18

_4 If = [(1-04) e4 (H4j4)f - _4 ] _4 + (I-P4) _7 (H4j7)f _7

+ (1-P4) _.8 (H4j8)f _8 + (1-04) ci4 (H4jl4).f _14 + (1-04) _15 (H4jl5)-f _15

+ (1-04) _16 (H4jl6)f _16 + (1-P4) _17 (H4jl7)f _17 + (1-04) El8 (H4jl8)f _18

267

-- (I-P7) '4(H7j4)f _4 + [(I-P 7) _

(H7j8)j _28

7 (H7j7)f - _7 ] _7 + (I-P 7) c 8

+ (1-07) (14 (H7jl4)f _14 + (I-0 7) _15 (H7jl5)f _215 + (1-0 7) _16

(H7jl6)f f_16

+ (l-P7) (17 (H7jI7) f f217 + (l-P7) el8 (H7jI8) f _218

¢81 --"(I-P8) E4 Q4 + (f (H8j4)f (I-P 8)

(Hsj 8)f - (8 ] f_8

7 (H8j 7) f n7 + [ (1-ps) (s

+ (I-P8) (14 (Hsjl4)f _14 + (I-P 8) (15 (H8jI5) f _15 + (I-P 8) (16

(HsjI6) f n16

+ (I-P8) (17 (Hsjl7)f _217 + (I-P 8) (18 (Hsjl8)f f_18

l:_14 f (I-P14) _4 (Hl4jl4)f f_4 + (I-P14) _7 7) _7 + ) _(HI4 j f (1 -pl 4

(m14js)f_8

+ [(I-PI4) (14 (Hl4jl4)f (14] fZl4 +- (l-Pl 4) f

(I-P14) (16 (Hl4jl6)f f_16

15 (Hl4jl5)f _15 +

+ (1-P14) (17 (H14ji7)f _17+ (1-P14) _18 (H14j18)f _18

268

= (1-015) 4 (Hl5j4)f f24 + ) ((1-015

(H15j8) f f_8

7 (Hl5j7)f _7 + .((l-P15) 8

+ (1-015) (14 (Hl5jl4)f _214 + [(1-015) _15 (Hl5Jl5)f - _15 ] _15 +

(1-015) _16 (Hl5jl6)f f_16

+ (1-015) _17 (Hl5jl7)f f_17 + (i -015 ) _18 (Hl5jl8)f P'18

= (1-016) 4 (Hl6j4)f f_4 + (1-016) (7 7) _7 +e (HI6 j f (1-016) E

(HI6j8) f f28

+ (I-P16) e14 (H16j14)f _14 + (I-P16) (15 (Hl6jl5)f _15 + [(1-P16) (

(Hl6jl6)f - _16 ] f_16

+ (1-016) (17 (Hl6jl7)f f217 + (I-PI7) el8 (HI6jI8) f _18

16

-- (I-P17) e4 (HlTj4)f f_4 + (I-P17) e

(HI7j S) f _8

7 (Hl7j7)f _7 + ) ((I-P17

+ (I-P17) 14 (Hl7jl4)f DI4 + (I-PI7) _15 (Hl7jl5)f _15 + ) (( (l-Pl 7

(H17j16)f _16

16

+ [(1-P17)c17 (Hl7jlT)f - _17 ] _17 + (I-P17) el8 (HI7jI8) f ni8

269

_181f = (I-P18) _4 (Hl8j4)f _f + (1-018) c7 (HI8j7) f _7 + (1-018) ¢8

(Hlsjl8)f _8

+ (l-Pi8) _14 (Hl8jl4)f _14 + (I-@18) '15 (Hl8jl5)f n15 + (1-018)

el6 (Hl8jl6)f _16

+ )_(I-P18 17 (Hl8jl7)f _17 + [(1-018) _18 (Hl8jl8)f - _18 ] ill8

For the enclosure g: Nodes: 8,9,14

= [(1-P8) E8 (H8j8)g - _8 ] _8 + (l-Q8) E

+ (1-08) ¢14 (H8jl4)g _14

9 (H8j9)f _9

_9 Ig = (1-P9) E8 (H9j8)g _8 + [(i-09) _

+ (1-09) _14 (H9jl4)g _14

9 (H9j9)g-_9 ] _9

_14 Ig = (1-014) _8 (Iil4j8)g _8 + (1-014) ¢

+ [(I-P14) el4 (Hl4jl4)g - _14 ] _14

9 (Hl4j9)g n9

For the enclosure h: Nodes: 8,9,14,17 - i,j,k

_8 lh = [(I-P8) _8 (H8j8)h - E8] _8 + (1-08) _9 (H8j9)h _9

+ (I-P8) _14 (H8jl4)h _QI4 + (i-08) _17 (H8jl7)h QI7

27O

-( 9] _9

= (I-P 9) (8 (I{9_8)h _8 + t(I-P9) _9 (B9J 9)h

+ (I-P 9) _14 (H9314)h _14 +(l-Q9) _17 (_9317)h _17

ffi(l-Pi4) B (I_1418)h _B + (l-Of4) E9 (B14]9)h _9 n17) (-17 (_14JlT)h

+ [(1-P14 ) (14 (1414_14)h- _14] _14 + (1-P14

qblTlh (I-P17) (8 (_II7_B)B _IB + (i-017) _9 (H17_g)h %39

+ (l-Pl7) ¢14 (BlT_14)h _14 + [(1-p17) _17 (HlT_17)h 17 ] _17

For the enclOSUre l: NodeS:

¢2I_ L(i-o2) 2 (Bziz) i

+ (1_o2) 44 (_2_4)i _4

2,3,4 - i'_'k

¢2 ] _2 + (1-02) _3 (B233) i_3

= (1-P 3) _ 2 (_3_ 2) t e2 + [(l-P3)

¢3/i

3 (B3_3) - (3 ] _3

+ L(i-p_) 4 (B434) i

The following are the constants for the radiation transfer terms in

the energy balance equation.

_- 2GI [i- T20 (H20j20)b (H20m20)c]-I

2 GISI = (HIIjlI) b + T20 (Hllj20)b (H20m20)c(H20jll)b

2 sl ]-iG 2 = [i - rll (Hllmll) a

2 GIUk = G 2 ek [(Hllhk) b + T20 (Hllj20)b (H20m20)c (H20jk) b]

2 GI G2Vk = GI _k T20 [(H20j20)b (H20mk)c + TII (H20jll)b (Hllmll)a (HIIJ 20)b

(H20mk) c ]

2 2 GI G2Yk = GI _k [(H20mk)c + Ill T20 (H20m20)c (H20jlI)B (Hllmll)a (HIIJ 20)b

(H20mk) c ]

2

W k --G I [(H20jk) b ek + TII (Hllmll) a (H20jll)b U k]

DII = (_i) {Alc[(l-0 I) eI (Hljl) c -e l] + Alc (i-0 I) T20 (HIj20) c V I

+ AId[(l-01) _i (Hljl)d - el ]}

DI2 = 0

DI3 = (_i) Aid (i-01) _3 (Hlj3)d

DI4 = (_i) Aid (I-01) _4 (Hlj4)d

272

DI5 = (_i) {Alc[(l-Pl) _5 (Hlj5)c + (i-01) T20 (Hlj20)c V5] + Aid (i-01)

_5 (HljD)d}

DI6 : (_i) Alc (Z-01) T20 (HIj20) c W6

DI7 = (_i) Alc (I-0 I) T20 (HIj20) c W7

DIS = (_i--) Alc (I-0 I) T20 (HIj20) c W8

DI9 = (_I) Alc (i-0 I) T20 (HIj20) c W9

DI_I0 = 0

DI-II = (_I) Alc (i-01) T20 (Hlj20)c [WII + G1 G2 TII ell (H20jll)b (Hllmll)a]

DI_I2 = DI_I3 = 0

DI_I4 = (_i) Alc (i-0 I) T20 (HIj20) c TII

DI_I5 = DI_I6 = DI_I7 = DI_I8 = DI_I9 = 0

14 GI G2 (H20jll)b (Hllml4)c

= (Alc) aBI M I TII r20 T21 (i-0 I) G I G 2 (HIj20) c (H20jlI) b (Hllm2 I)

D2_ I = 0

D2_ 2 = (_2) A2i[(i-02)- c 2 (H2j2) i - c2]

273

D2-3 ffi (_i) A2i (1-02) _3 (H2j3)i

D2-4 = (_2) Azi (I-P2) _4 (H2j4)i

D2_5 = D2_ 6 ffiD2_ 7 = D2_ 8 = D2_ 9 = D2_I0 = D2_II ffiD2_12 = D2_13 ffiD2_14

= D2_15 = D2_16 ffiD2_17 = D2_18 = D2_19 = 0

D3_ 1 : (_3) A3d (1-P 3) eI (H3jl) d

D3_ 2 : (_3) A3d (1-0 3 ) E2 (H3j2) i

D3-3 ffi (_3) {.A3d[(1-P3) _3 (H3j3)d- c3] + A3i[(1-P3) _3 (H3j3)i- _3 ]}

D3-4 ffi (_3) {A3d (1-03) c4 (H3j4)d + A3i (1-P3) _4 (H3j4)i}

D3-5 = (_3) A3d (1-03) _5 (H3j5)d

D3_ 6 = D3_ 7 ffi D3_ 8 = D3_ 9 = D3_10 ffi D3_ll ffi D3_12 = D3_13 = D3_14 " D3_15

= D3_16 = D3_17 = D3_18 "= D3_19 = 0

ffi eI (H4j 1D4-1 (_4) A4d (1-04) )d

D4_ 2 = ([_4) A4i (1-04 ) (2 (H4j 2) i

D4-3 = (_4) {A4d (1-04) c3 (H4j3)d + A4i (1-04) _3 (H4h3)i}

274

D4_4 -- (_4) {A4d[(l-o 4) 4 (H4j4)d - _4] + A4e[(l-04) _4 (H4j4)e - _4]

+ A4f[(l-04) c4 (H4j4)f -E4] + A4i[(i-04) _4 (H4j4)i- _4]}

D4-5 =(_44) {A4d (1-04) _5 (H4j5)d}

D4_6 = 0

D4-7 = (_4) A4f (i-P4) e7 (H4j7)f

D4-8 _ (_4) A4f (1-04) ¢8 (H4h8)f

D4_ 9 =" D4_IO = D4=11 = D4_12 = D4_13 = 0

D4-14 = (_4) A4f (1-04) _14 (H4jl4)f

D4-15 = (_4) A4f (1-04) _15 (H4jlS)f

D4_16 = (_4) {A4e (1-04 ) E16 (H4j16) + A4f (1-04) Cl 6 (H4jl6)f }

D4-17 = (_4) A4f (I-04) _17 (H4jlT)f

D4-18 = (_4) {A4e (i-04) f18 (H4jlS)e + A4f (1-D 4 ) _18 (H4jl8)f}

D4-19 = (_4) A4e (1-04) ¢19 (H4jl9)e

D5_ I = (_5) {A5c (1-05)[_ 1 (H5jl) c +

(H5j i)d }

r20 (H5j 20)c Vz] + A5d (I-P5) el

275

D5_2 = 0

D5_3 = (_5) A5d (i-o 5) c3 (H5j3)d

D5_4 -- (_5) A5d (i-05) c4 (HDj4)d

D5-5 = (_5) {ADd[(l-05) _5 (HDh5)c- _5

+ A5d[(1-05) _5 (H5j5)d - _5]d }

D5_6 -- (_5) A5c (1-05 ) T20 (H5j20) c W6

D5_7 = (_5) A5c (1-05 ) T20 (H5j20) c W7

D5_8 = (_5) A5c (1-05 ) T20 (H5j20) c W8

D5_9 = (_5) A5c (1-05 ) T20 (H5j20) c W9

D5_I0 = 0

+ (1-05) T20 (H5j20) c V5]

D5-11 = (_5) A5c (1-05) T20 (H5j20)c[WII + GI G2 TII _ii (H20jll)b

(Hllmll) a _14]

D5_12= D5_13= 0

D5_14= (_) A5c TII T20 GI G2"(l-05)(H5j20)c (H20jll)b (Hllml4)a _14

D5_I5 = D5_16= D5_17= D5_18= D5_19= 0

276

B5 (A5c) (1-05) GI G2 a= M5 TII T20 _21 (H5j20)c (H20jll)b (Hllm21)

D6-1-- (M6--)A6b (I-P6)[_211 T20 _i GI G2 (H6jll)b (Hllmll)a (Hllj20)b

(H20ml)c + T20 (Hgj20)b YI ]

D6_2 -- 0 , D6_3 -- 0 , D6_4 = 0

c (1_P6)[ 2D6-5-- (M-_)A6b TII "[20

+ T20 (H6j20)b Y5]

(5 GI G2 (H6jll)b (Hllmll)a (HIIJ20) c

D6-6 = (-_6) A6b{(I-P6) (6 (H6j6)b - (

2+ T20 (H6j20) b (H20m20) c W6]}

6 + (I-P6)['r121 (H6jll)b (Hllm11)a U6

_6 2 )a U7D6_ 7 = ( ) A6b (1-06 ) {E 7 (H6j7) b + TII (H6jll)b (Hllml I

2

+ T20 (H6j20) b (H20m20) c W 7}

_6 2D6L8 = ( ) A6b (1-06) {(8 (H6j8)b + TII (H6jll)b

2

+ T20 (H6j20) b (H20m20) c W8}

(Hllmll) a U 8

_6 2D6-9 = ( ) A6b (1-06) {E9 (H6j9)b + TII (H6jll)b (Hllmll)a U9

2

+ T20 (H6j20) b (H20m20) c W9}

D6_I0 = 0

277

= _ (1-P6) {( )b +D6-11 (M6) A6b ii (H6jll TZl (H6jll)b (Hllmll)a

2(TII Ull + _ii G2) + T20 (H6j20)b (H20m20)c[Wll + GI G2 +ii TII

(H20jlI) b (Hllmll) a}

D6_12= D6_13= 0

_6 2 GI G2D6-14 = ( ) A6b (I-P6)[TII el4 G2 (H6jll)b (Hllm14) + TII T20 (14

(H6j20)b (H20m20)c (H20jlI) b (Hllmll) a]

D5_I5 = D6_16= D6_17= D6_18= D6_19= 0

A_.

B6 = (M--_)(1-P6) TII T21 G2[(H6jlI)b (Hllm21)a

(H20jlI)b (Hllm21)a]

2

+ GI T20 (H6j20)b (H20m20) c

D7_ I = (_7) A7b (I-P7){T_I T20 GI G2 (H7jlI) b (Hllmll)a (Hllj20)b (H20ml) c e1

+ _20 (H7j20)b YI }

DT_ 2 = D7_ 3 = 0

D7_ 4 = (_i) A7f (1-P 7) E4 (H7j4) f.

D7-5 = (_7) A7b (1-07) T20 {T_I _5 GI G2 (H7jll)b (Hllmll)a (Hllj20)b

(H20m5) c + (H7j20) b Y5 }

D7-6 = (_7) A7b (1-07) {(6 (H7j6)b2 (H7j U6 + 2+ TII ll)b (Hllmll) a T20

(HTj20) b (H20m20) c W6}

278

D7_7 = (_7) {A7b[(1-07) c7 (H7j7)b - _7 + (1-07)[T_I (H7jll)b (Hllmll)a U7

2 (H7j )c W7]] + [ _7 - ]}+ T20 20)b (H20m20 A7f (1-07) (H7jT) f E7

(_7 2D7_8 -- ) {A7b (l-P7)[c 8 (H7j8)b + TII (H7jlI) b (Hllmll) a Us

2+ T20 (H7j20)b (H20m20)cW8] + A7f (l-P7) _8 (H7j8)f}

D7_9 -- (_7) {A7b (l-P7)[_ 9 (H7j9)b

2+ T20 (H7j20)b (H20m20)c W9]}

2+ TII (H7jlI) b (Hllmll) a U9

DT_I0 = 0

D7_11-- (_7) A7b

(Hl-ml i )a

(I-P7) {eli (H7jll)b + TII (H7jll)b (TII UII + ell G2)

2+ T20 (HTj20)b (H20m20)c[Wll + G1 G2 ell TII (H20jlI) b

(Hllmll)a]}

D7_12= D7_13= 0

_7) + 2 GID7-14 = ( {ATb (1-07) TII G2 _14[(H7jll)b (Hllml4)a _20 (H7j20)b

(H20m20)c(H20jll)b (Hllml4)a] + A7f (1-07) _14 (H7jl4)f}

D7_15= (_7) A7f (1-07 )

D7_16= (_7) A7f (1-07)

15 (H7jl5)f

16 (H7jl6)f

279

D7-17 = (_7) A7f (1-07) _17 (H7jl7)f

D7_18 = (_7) A7f (1-07 ) el8 (HTjI8) f

D7_19 = 0

= (ATb) G2 {T21 aB7 M_-- (1-07) TII (H7jll)b (Hllm21)

2

+ T20 T21 G I (H7j20) b

(H20m20) c (H20jll)b (Hllm21) a}

D8_ I = (_8) A8b (I-08){T_I T20 eI G I G 2 (H8jlI) b (Hllmll) a (HIIj20) b (H20ml) c

+ T20 (H8j20) b YI }

DS_ 2 = DS_ 3 = 0

D8-4 = (_8) A8f (1-08) _4 (H8j4)f

D8-5-- (_8) A8b (I-08){T211 T20 _5 GI G2 (H8jll)b (Hllmll)a (Hllj20)b

(II20m5) c + T20 (H8j20) b Y5 }

2 )a U6D8_ 6 = (_8) A8b (1-08 ) {E6 (H8j6) b + TII (H8jlI) b (Hllmll

2

+ T20 (H8j20)b (H20m20)c W 6}

D8_ 7 = (_8) {Asb (I-08)[c7 (H8j7)b + T211 (H8jll)b

2

(Hllmll) a U7 + T20

(H8j20)b (H20m20) W7] + A8f (1-08) _7 (H8j7)f}

28O

{A8b[ (l-P8) _ (H8j - E8 + (1-08) 2D8-8 = (M--8) 8 8)b [_ii (H8jll)b

2 )c W8]] + [(1-08) _8 c+ T20 (H8j20)b (}120m20 A8f (H8js) f -

[(1-°8) _8 (H8j8)g

(Hllmll) a U8

- _8 ] + A8h[(1-08 )

8] + A8g

_8 (Hsjs)h - _8 ]}

2 2

= _ {A8b (I-08)[E9 (H8j9)b + YII (H8jll)b (Hllmll)a U9 + T20D8- 9 (M8)

(H8j20) b (H20m20)c W9] + A8g (1-08) _9 (H8j9)g + A8h (1-08) _

(H8j9) h}

D8_I0 -- (%) x 0 -- 0

D8-11 = (_88) A8b (I-08) {¢ii (H8jll)b + _Ii (H8jll)b (llllmll)a (TII UII

2

+ ell G2) + T20 (H8j20)b (H20m20)c[Wll + TII ell G I G 2 (H20jlI) b

(Hllmll)a]}

D8_12 = D8_13 = 0

2 GID8-14 = (_8) {A8b (i-08) TII _14 G2[(H8jlI)B (Hllml4)a - T20 (H8j 20)b

(H20m20) c (H20jll)b (Hllml4)a] + A8f (1-08) _14 (H8jl4)f

+ A8g (1-08 ) ,_14 (H8jI4) +g A8h (I-P 8) 14 (H8jl4)h}

D8-15 = (_8) A8f (1-08) _15 (Hsjl5)f

D8_16 -- (_8) A8f (i- 8) _16 (Hsjl6)f

281

D8-17 = (_8)[A8f (1-08) _17 (H8jl7)f + A8h (1-08) _17 (H9jl7)h]

D8_19 = 0

B8 (A8b) (1-08) G2 { a + 2 G1 a= M8 TII T21 (Hllm2 I) T20 (H8j20) b (H20m20) b (Hllm2 I) }

2

D9_ I = (%) A9b (1-09 ) T20{TII GI G 2 (H9jlI) b (Hllmll) a (HIIj20) b (H20ml)

c (H9j20) bi + YI }

D9_ 2 = 0 , D9_ 3 = 0 , D9_ 4 -- 0

2 GI G2 (H9jD9- 5 = ( ) A9b (I-P9) T20{_II _5 ll)b (Hllmll)a (HllJ20)b

(H20m5) c + (H9j20) b Y5 }

2 2

D9-6 = (_9) A9b (1-09) {_6 (H9j6)b + Tll (H9jll)b (Hllmll)a U6 + T20

(H9j20) b (H20m20)c W 6}

D9_7 = (_9) A9b (1-09) {E2 U7 + 27 (H9jT)b + TII (H9jll)b (Hllmll)a T20

(H9j20) b (H20m20)c W 7}

= _ {A9b (i-09)[_D9- 8 (M9)

2 2

8 (H9j8)b + TII (H9jll)b (Hllmll)a U8 + T20

(H9j20)b (H20m20)c W8 + A9g (1-09) _8 (H9j8)g + A9h (1-09) _8

(H9j 8) h}

282

= (H9j 9)b (Hgjll)b (Hllmll)D9_ 9 (_99) {A9D[(I-09) c9 - _9 + (I-09)[T211 a U9

2 _ -_9] ++ T20 (H9j20) b (H20m20) c W9]] + A9g[(l-09) 9 (H9j9)g A9h

[(1-09) _9 - _9]}(Hgj9) h

D9_I0 = 0

D9-11 = (_99) A9b (1-09) {_Ii (H9jll)b + TII (H9jll)b (Hllmll)a (Yll UII +

2

(iI G2) + T20 (H9j 20)b (H20m20)c[WII + _Ii TII GI G2 (H20Jll)b

(Hllmll)a]}

D9_12 = D9_13 = 0

-_9 2 GID9-14 = ( ) {A9b TII _14 G2[(H9JlI)b (Hllml4)a + Y20 (H9j20)b (H20m20)c

(H20jlI) b (Hllml4) a] + A9g (1-09) _14 (H9jl4)g + A9h (1-09) _14

(H9jI4) h}

D9_I 5 = D9_16 = 0

D9-17 = (%) A9h (1-09) _17 (H9jl9)h

D9-18 = D9-19 = 0

: (Asb) (1-09) c2{B9 M 9 TII T21 (H9jll)b (Hllm21)a

(H20jlI) b (Hllm21)a}

2 G1+ T20 (H9j20)b (H20m20)c

283

DI0_I = DI0_2 = D10_3 = D10_4 =

DI0_I 0 = DI0_I 1 = DI0_I 2 = DI0_I 3

DI0_I 7 = DI0_I 8 -- DI0_I 9 = 0

D11-6

DII-7 = (M_I)[AIIb {(i-011) e7 (Hllj7)b

DIO_5 = DI0_6 = DIO_7 = DIO_8 = DIO_9 =

= DI0_I 4 = DI0_I 5 = DI0_I 6 --

= 2 G I G 2 1 [ - i]DII-I (M_I) [AIIb{TII T20 ¢ (i-011) (HIIjlI) b (Hllmll) a (HIIj20)

(H20ml)c + (i-011) T20 (Hllj20)b Y1 } + Alla TII T20 _i G1 G2[(I-oII)

(Hllmll) a - i] (Hllj20) a (H20ml)c ]

DII_2 = DII_3 = DII_4 = 0

DII_ 5 z (M_I)[AIID{TII2 "['20GI G2 _5[(i-011 ) (Hlljll)n - l](Hllmll) a (HIIj20) b

(H20mS)c + (i-011) T20 (Hllj20)b Y5 } +" Alla _ii Y20 _5 G1 G5

[(l-Pl I) (H_ ) - i]llmll a (Hllj20)b (H20m5)c]

= (M_I)[AIIb {(I-PlI) _6 (Hllj6)b + T_I[(I-oII) (Hlljll)b -i]

2

(Hllmll)a U6 + r20 (i-011) (Hllj20)b (H20m20)c W6} +All a TII

[(I-011) (Hllmll) a - i] U6]

2

+ YII [(I-011) (HIIjlI) 5 - I] U7

2

+ T20 (I-011) (HIIj20) b (H20m20) c W9} + All a TII[(I-011)

(Hllmll) a - i] U7]

284

= 2

DZi-8 (M_I)[Alib {(1-011) _8 (Hlij8)b + Tli[(l-Pll) (Hlljil) b - i]

2

(Hllmll) a U8 + T20 (Z-Pll) (HIIj20) b (H20m20) c W8}+ Ali a TII

[(I-o11) (Hllmll) a - 1] US]

DII-9 ( ) [Aiib { (l-Oil) _ 9 (Hiij9)b + _il [ (i-Pii) (Hiljli)b - i]

2

(Hllmll) a U9 + T20 (1-011) (H11j20) b (H20m20) e W9}+ All a Tll

[ (1-011) (Hllmll) a - i] U9]

D : 011-i0

DII-11 = (M#I)[Allb {(1-011) Eli (H11jI1)b - _11 + %11[(1-011) (Hlljll) b -i]

2

(Hllmll) a (TII Ull + eli G 2) + T20 (1-0il) (HIIj20) b (Hl120m20) c

[WII + ii TII GI G2 )a]} + { )E (H20jII) b (Hllml I Alla (i_011 ell

(Hllmll)a - _ii + %11[(i-011 ) (Hllmll) a - l][Ull + TII ell G2

S1 )a]}](Hllml I

DII_I 2 : DII_I 3 : 0

DII_I4 = (M_I)[AIIb{%11 c14[(1-011) (HlljlI) b _ 1]G2 (Hllml4)a + (1-011) TII

2

T20 el4 G1 G2 (Hllj20)b (H20m20)c(H20jll) b (Hllml4) a} + Ali a (I-011)

El4 SI G2[(I-011) (Hllmll)a}]

285

DII-15 = DII-16 = Dli_17 = DII_I 8 DII_I 9 = 0

BII = (M_I)[AIIb{TII r21 G2[(I-DII) (HIIjlI) 5 - I] (Hllm21) a + (l_Pll) T11

2

T20 T21 G1 G2 (HIIj20) 5 (H20m20) c (H20jlI) b (Hllm2k)} + All a {(l-Pll)

2

Y21 (Hllm21)a + TII T21 S1 G2[(I-Ol I) (Hllmll) a - i] (Hllm21)a} ]

DI2_ 1 DI2_ 2 = D12_3 = D12_4 = D12_5 = D12_6 = D12_7 = D12_8 = D12_9

DI2_I 0 = DI2_I 1 = DI2_I 2 = DI2_I 3 = DI2_I 4 = DI2_I 5 = DI2_I 6 =

D12-17 = D12_18 = D12_19 = 0

DI3_I = D13_2 = D13_3 = D13_4 = D13_5 = D13_6 = D13_7 = D13_8 = D13_9 "

DI3_I 0 = DI3_II = D13_12 = DI3_I 3 = DI3_I 4 = DI3_I 5 = DI3_I 6

D13_17 = D13_18 = DI3_I 9 = 0

DI4-1 = D14-2 = D14-3 = D14-4 = DI4_ 5 = D14_6 = D14_7 = 0

D14-8 = (M_4)[AI4g (1-014) E8 )g + _8(HI4j8 AI4 h (1-014) (HI4jS) h ]

D14-9 = (M_4) [Al4g (l-P14) _9 (Hl4j9)g + AI4 h (l-P14) e9 (Hl4jg)h]

DI4-10 = DI4_II = D14_12 = DI4_I 3 = 0

DI4_I 4 = (M_ 4) {Al4g

D14_15 = DI4_I 6 = 0

14[(I-P14) (Hl4jl4)g - i] + AI4 h _14[(i_P14) (H14j14) h _ i]}

DI4_I7 = (M_4) Al4h (1-014) el7 (l_l14jlT)h

D14_18= D14_19= 0

DIS_ I = DI5_ 2 = DIS_ 3 = 0

D14_4 = (M_ 5) AI5 f (1-015) _4 (Hl5j4)f

DI5_ 5 -_ D15_6 = 0

D15_7 = (M_5) Al5f (1-015) E 7 (HlSj7)f

D15_8 = (M_5) AI5 f (1-015) _8 (Hl5j8)f

D15-9 = DIS-10 = DIS_If = D15_12 = D15_13 = 0

DI5_I 4 = (M_5) AI5 f (1-015) _14 (H15314)f

DIS_I 5 = (M_5) A15f[(l-015) _15 (Hl5jl5)f- _15]

_) (1-015) (HISjI6) fD15_16 = (Mi 5 AI5 f _16

DI5_I 7 = (M_ 5) Al5f (1-015) c17

DI5_I 8 = (M#5) AI5 f (1-015) _18

D15_19 = 0

(HI5jI7) f

(HI5jis)f

DI6_ I = DI6_ 2 = DI6_ 3 = 0

287

: (M_. + Al6f(l-Pl6) _4 (Hl6j4)f}D16-4 ) {AI6e(I-PI6) (4 (Hl6j4)e

D16_5 : D16_6 : 0

D16-7 : (M_6) A16f(I-PI6) (7 (Hl6jT)f

D16-8 : (M_6) Al6f(l-Pl6) (8 (Hl6j8)f

DI6_ 9 : DI6_I0 : D16_11 : DI6_I 2 : DI6_I 3 = 0

D16-14 : (M_6) A16f(l-P16) (14 (Hl6jl4)f

D16-15 : (#6) AI6f(I-PI6) _15 (Hl6jl5)f

D16_16 : (M--_6) {A16e[(l-P16)(HZ6j16) e - i] + AI6 f (16[(I-o16)(HI6jI6) f _ 1]}

D16-17 : (M_6) AI6f(I-PI6) (17 (Hl6jl7)f

D16-18 : (M_6) {AI6e(I-PI6)_18 (HI6jI8) + A16f(1-PI6)_18 (HI6j 18)f}

D16-19 : (M_6) Al6f(l-Pl6) _19 (Hl6jl9)e

DI7_I = D17_2 : D17_3 = 0

D17-4 : (M--_7)Al7f(l-Pl7) (4 (Hl7j4)f

D17_5 = D17_6 = 0

D17-7 : (M--_7) Al7f(l-Pl7) _7 (H17j7)f

288

D17-8 = (M--_7){AlTf(l-Pl7) c8 (HlTjS)f + Al7h(l-Pl7) _8 (HlTjS)h}

D17-9-- (M--_7)A17h(l-017) _9 (Hl7j9)h

DI7_I0 = DI7_I I = DIT_I2 = DIT=I3 = 0

DI7_I4 = (M-_7){Ai7f(l-Pl7) 14 (Hl7jl4)f + Al7h(l-Pl7) _14 (HlTjl4)h}

D17-15 = (M_7) AI7f(I-PI7) (15 (Hl7jl5)f

D17-16 = (¢7) AI7f(I-PI7) _16 (Hl7jl6)f

DI7_I7 -- (M--_7){AI7 f (17[(l-P17)(H17j17) f - i] + AI7h _I7[(I-oI7)(HI7jI7) h - I]}

DIT_I8 = D17_19= 0

DIS_I = D18_2= D18_3= 0

D18-4 = (M--_8){Al8e(l-Pl8) e4 (Hl8j4)e + Alsf(l-PlS) E4 (Hl8j4)h}

D18_5= D18_6= 0

D18_7= (M--_8)AI8f(I-PI8) E7 (HI8j7) f

D18-8 = (M-_8)AI8f(I-PI8) _8 (Hl8jS)f

D18_9= DI8_I0 = DI8_I 1 = DI8_I2 = DIS_I3 = 0

D18-14 = (M--_8)AI8f(I-PI8) _14 (Hl8jl4)f

289

DI8_I5 = (M_--_)Alsf(l-Pl8) c15 (HI8jI5) f18

D18-16 = (M-_8) {AI8e(I-PI8) _16 (HIsjI6) + AI8f(I-PI8) _16 (Hl8jl6)f}

D18-17 = (M_8) Al8f(l-Pl8) _17 (Hl8jl7)f

DI8_I 8 = (M_ 8) {AI8 e _18 [(I-pls)(HI8jI8) e - i] + AI8 f _18 [(l-p18)(H18j18) f -I]}

M_8 •DI8_I 9 = ( ) AI8e(I'-PI8) El9 (HI8jI9) e

DI9_I = D19_2 = D19_3 = 0

D19-4 = (M--_9)Al9e(l-Pl9) _4 (Hl9j4)e

D19_5 = D19_6 = D19_7 = D19_8 = D19_9 -- DI9_I 0 = DI9_I 1 = DI9_I 2 -- DI9_I 3 "

DIg_I 4 = DIg_I 5 = 0

DI9_I 6 - (M--_9)AI9e(I-PI9) El6 (HI9jI6) e

DI9_I 7 = 0

D19-18 = (M--_8)Al9e(l-Pl9) _18 (HI9jI8)

D19_19 = (M--_8)AI9 e EI9[(1-PI9)(HI9jI9) e - i]

290

Standard Bibliographic Page

I. Report No.

NASA CR-1782922. Government Accession No.

4.Title_dSubtitle Development of Response Models for the

Earth Radiation Budget Experiment (ERBE) Sensors:

Partl - Dynamic Models and Computer Simulations for th

EP_E Nonscanner, Scanner and Solar Monitor Sensors

7. Author(s)

Nesim Halyo, Sang H. Choi,

Dan A. Chrisman, Jr., Richard W. Sannns

9. Per_rming Org_ization Name and Address

Information & Control Systems, Incorporated

28 Research Drive

Hampton, VA 23666

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Langley Research CenterR_mnf on VA 2_665

15. Supplementary Notes

3. Recipient's Catalog No.

5. Report Date

March 20, 1987_B. Performing Organization Code

8. Performing Organisation Report No.

FR 687103

10. Work Unit No.

665-45-30-01

11. Contract or Grant No.

NASI-16130

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

Langley Technical Monitor: Robert J. Keynton

Final Report

16. A_tract

Dynamic models and computer simulations were developed for the radiometric

sensors utilized in the Earth Radiation Budget Experiment (ERBE). The models were

developed to understand sensor performance, improve measurement accuracy by updatin@

model parameters and provide the constants needed for the count conversion algor-

ithms. Model simulations were compared with the sensor's actual responses demon-

strated in the ground and inflight calibrations. The models consider thermal and

radiative exchange effects, surface specularity, spectral dependence of a filter,

radiative interactions among an enclosure's nodes, partial specular and diffuse

enclosure surface characteristics and steady-state and transient sensor responses.

Relatively few sensor nodes were chosen for the models since there is an accuracy

tradeoff between increasing the number of nodes and approximating parameters such

as the sensor's size, material properties, geometry, enclosure surface character-

istics, etc. Given that the temperature gradients within a node and between nodes

are small enough, approximating with only a few nodes does not jeopardize the

accuracy required to perform the parameter estimates and error analyses.

17.Key Words(Suggestedby Authors(s))

Dynamic Models, Computer Simulations,

Radiometric Sensors, Count Conversion

Algorithms, Ground and Inflight Calibra-

tions, Thermal and Radiative Exchange

Effects, Surface Specularity,

Spectral Dependence, Enclosure Nodes

18. Distribution St_ement

Unclassified - Unlimited

Subject Category 35

19. Security Classif.(of this report)

Unclassified 20. Secu_ty Classif.(ofthispage) 21. No. ofPages122. P_ceUnclassified 305 AI4

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