development of a power dissipation model for the power ... #272...qualification, back-up and...

NO. REV. NO.

~

Development of a Power Dissipation Model for the Power Conditioning Unit (PCU)

ATM 783

1 PAGE OF

23

{ tams Division DATE 15 July 1968

This memorandum summarizes the effort to develop a mathematical model of the power dissipation within the PCU and to adapt that model to assist in the central station power/thermal analytical studies that have been required during ALSEP system development.

')

Prepared by: ~~ H. W. Wilson

/rl ! . / / Approved by: ''>.--<(". { .J :,. ".1 J

o. Neau

NO. REV. NO.

Development of a Power Dissipation Model for the Power Conditioning Unit (PCU}

ATM 783

PAGE 2 OF 23

Aerospace r~"~--\~ems Division

DATE 15 July 1968

1. Introduction and Summary

Throughout the ALSEP Program., ATM 449 (Power Budget} has provided a catalog of power availability and power demand associated with the various equipment items of the system. This type of power budgeting has bePn essential to system power management and to the forecasting of system performance in terms of:

- permissible experiment power demands - thermal support power availability - maintenance of voltage regulation under all operational modes - thermal control of central station electronics.

This budget process was simplified by the fact that all units except the Power Conversion Unit (PCU} have., at most., two or three operational modes in which their power demands differ appreciably. Hence, although the number of operational modes is relatively large, the associated power demands at the output of the PCU are normally predictable. The power dissipation (W} associated with the PCU requires a knowledge of this system power demand as well as the power coming in to the system from the RTG, and is determined by summing the electrical dissipations associated with the two PCU functions of power conversion and voltage regulation, viz. :

where

W = prt + p c

P rt = Reserve power dissipated within PCU

P = Voltage conversion losses. c

( 1}

It is the purpose of this memorandum to discuss the history of development of the model used for the prediction of PCU power dissipation and to discuss the application of this model during ALSEP system test and lunar surface operation.

NO. REV. NO.


ATM 783

PAGE 3 OF 23

Aerospace /,-~"1rtems Division DATE 15 July 1968

2. Mathematical Modelling of PCU Dissipation

The distribution of power in the ALSEP system can be described in terms of the subdivisions shown in Figure 1 to be

P. = p + p + p 1n 0 c r

where

pin = Power into PCU /"\ ,,F:'-

l\ L \

Po = Power out of PC U f ~

p = Power used in voltage conversion c

pr = Power dissipated in shunt regulator

(2)

The last two items are determined by the performance of the PCU. Hence, it is necessary to be able to establish the power usage of the PCU in order to evaluate the balance of power throughout the system. A first approximation of the PCU conversion loss was made during engineering model testing of that unit. At that time it was assumed that the power consumed in the PCU for voltage conversion and associated functions varied linearly with PCU power output in the following manner:

P = C+KP c 0 (3)

where

C = 5. 0 watts

K = 0. 08

The reserve power dissipated within the PCU is simply stated as: ··:.,·' ' ' ''· :" '

(4)

where

P r = Total reserve power

P rr = Reserve power external to PCU

.,.,., ._ ..,. ._. ,_., ..,.. . ....._., ... tli?AZI' ~

I Pow£R Ccn.tvee<.-sto~ lh"',.. (Pcu) -1

I ""Po...ve R

""Pel

Co>-•vE..({ "tE ~~-

1. ------·

~-

1 I I • '?o , • ----- ----, )

I

I I I I

' • i

.. R~~\)\..1\iOR' "k'c·:v:;, "\C' v

1.

f I

! ~y.H~~ ~---r C:,NvE:~, f.: i< I I

' I z. . . _ J I ----- I

I

' '

I .fu.l

I -·-------,

r C i: t-.I'T~Al 'STAll 0 t-1

LoA:O ·~---r~~~=~~~~J. lRI:\"-l':.l~·n~~1

---- J

I I

~ ; -----,---·-··~

-~-

l(FC,.u U.\\ 0 1<.

l<.es I'S\O R.. 2

I _L

......., ----· C.~N"'t~U\\.. C.S'ifl.."\" \0 N EL.ECTR()!-1..\C s BA. '(

Figure 1: Simplified Diagram of Power Distribution

I . \::xT~T<'N ~\\. .. I LoA~

T J

' ~ "

1:1> Ill 1-3 ~ ~

I ~-.] 0 00 ..... w N w

. .. .


NO .

ATM 783

PAGE 5

REV. NO.

OF 23

DATE 15 July 1968

Since

or

then

where

2 prr = I · Rext r

(:;J 2 p = .R rr ext

p 2 p r = rr pv

Ir = Current in shunt regulator

Rext = Shunt regulator circuit resistance external to PCU

Ein = Voltage at PCU input

= 2

E. ln

R ext

Substituting (5) into (4) yields

= p r -::-)

By definition., the sum of equations (3) and (6), represent the total electrical dissipation within the PCU and can be written as

+ C + K P 0

( 5)

(6)

(7)

: •, I ~

NO. REV. NO.


ATM 783

PAGE 6 OF 23

DATE 15 July 1968

When the system parameters other than reserve power (P ) and output r

power (P 0

) are available, this model can be expressed in alternate, but equivalent, forms such as,

2 p p )

w = p s c (8) (8) s p

v

where

ps = p. Po 1n

and c

p o) ( ~v (Pin - C - ( l+K) -Pin+ C + (l+K) Po) w = +C+KP (9)

p c 0

v

This last equation was investigated for the effect of changes in Pin' P 0 and Pv by use of an analog computer. The results of this analog modelling effort were published in ATM 67 5, and are reproduced in Figure 2.

To improve the flexibility and accuracy of these calculations, a digital computer program was written which provided a tabulated, printed output of the calculation of W for selected values of the primary parameters~ This program was extended to provide plots of W vs P

0 in several forms

(see Figures 3 and 4) to aid in the analysis of various power distribution problems.

As a result of data obtained from tests performed on both the qualification, back-up and prototype Power Conversion Units, the need for modifying this preliminary model of PCU dissipation became evident. These tests were reported in ATM 753 and revealed, among other things, that P c is actually a nonlinear function of P

0. Figures 5 and 6 are repro

duced from that report. Since a change in the dissipation model was necessary to reflect this new information, it was decided to also change the parameters used in the model so that it would be useful in the analysis of test and operational data. The only power parameters telemetered in the housekeeping data are:

R. T. G.

0 lJ

~ u T

\6 2.5 "Recau l..A.ToR.

Pv

ATM-783

Figure 2

0 0

;l I.

~~

~~ I

8 ~

8 :!!

~81 t-nl a:-::; (f)

c;s ~2 w 0..

8 ..;

g .;

g nl

g cb.oo

FIGURE 3 PCU OISIPATION VS PCU OUTPUT !IN WATTSJ

I I I I I I !'1 Input Power (watts)

55 Watt Regulator

·-+-----+---· ·-+--- ·--+-------- --·--· -··--+--·-··-·---+-- ----+-··- -------· -- ·--+- ----+---· ·--+-·--·----~--·----! s.oo 10.00 1s.oo ro.oo . 2s.oo ~.oo . 3s.on 110.00 ijS.on so.oo 55.oo 60.00 &S.oo 10.00 75.00 eo.oo Pcu· OUT IWRlTSI

. '

1:J)> Ill f-3

~ ~ I

00-..)

0 00 ....,w N w

\ .

ifj t-

! 4:

~ // -t-·".

~ ~

~-· ;

.__, t-

: \·~

Ll

a._

3 ~ ~

oo·112~or;:;;-.j__-r-·· ..

oo·zz oo·oz

~ oo·gt

oo·M

oo·g

AT

M-7

83

P

ag

e 9 o

f Z3

oo·~

20

15

!! '.;! ~ I .: .2 '.;! .2-"' .. iS ~

10

... " :s ::> u p..

5

·\

l -0

'· 0

Figure 5 DISSIPATION WITHIN PCU Vs PCU OUTPUT POWER

33 WATT REGULATOR

//,/// 1/

5 1 0 15 20 25 30 35 40 45

Total Load on All PCU Outputs (Watts)

,, "\ '\

50 55 60

i Page I~

~ Input power 74 Watts

65

:_ -~- -~

20

"' ~ 15 ;3':

~ .... !-< < p..

~ E ..1 10 < z lk: "l !-< := ::> u p..

5

0

0 5 b

Figure 6

DISSIPATION WITH IN POWER

~-, PC. 1 .. ·T-· i

~- -~· """'

~

l ·1

I

~~, ~· . .,. \.\ "••" \Z'w '\.~\, '; "' .. ! . I

·:- ··--!

~

\ \

\ \\ " \

.. \ .S

~ ~\ ....----.--\·:...:.. :

Conversio . ~C.!- ••. ~__,..-'-~--! n Loss ~·-·--- • '· · ,

I -------~-~ -----· ··-·------ ...

----··----- PC ' ' --, ____ .-.·-·" . . '· .

r-·- -- ---- ··---·----;----- ---+---~---+---~

15 20 25 30 35 40 45 50 55 60 65

TOTAL LOAD ON ALL PCU OUTPUTS (Watts)

! i

_)

NO. REV. NO.

: :- . ATM 783

Input Voltage:

Input Current:

Reserve Current:

Development of a Power Dissipation Model for the Power Conditioning Unit {PCU) PAGE

DATE

This model was thus modified to express the test-derived values of PCU dissipation in terms of these input parameters.

The amount of reserve power dissipated inside the PCU, (i.e., Prt} was already derivable from these parameters using equation {4). The relation between PCU conversion loss and the input power parameters was derived by empirical curve-fitting techniques. The details of this effort are given in Appendix A. The revised PCU dissipation model now has the form

+ (

{ 1 0)

12 c1 OF 23

lr" Sri u~ Cull!~ E t,J "'" e.:res- 1 r-J Pu'i

\?:e)lr~4 .. ~~ rc.u1 A-=- :s. e<a~ <~,) ~-; D.O~'S (PC.ut) C.-:< 0007SC! (PC.I.P)

-:.. ~& t!VPt.JT" Pw~- ~~t::"1Z\1G Pw~ where

: Ein (Iin - Ir)

A digital computer program has been written for this relation which provides tabulated and/or analog plotted presentations of the PCU dissipation as a function of Pi~ . Typical plots are presented in Figure 7.

Au ..:.....;_,_L-

f,v !";. 9o v f e.'Yf C/. -2. ..lL

/,,v 3. 7 7 A

0 0

::1 N

0 0

N N

0 0

0 N

0 0 .., -0 0

Ul -0 0

::1 -0 0

::z:~

0 0

0 -0 0

.,;

0 0

.,;

0 0

;

0 0

N

0 c. ·~

9l.oo 5.00 10.00 15.00 20.00

PIN PRIME VS. W

25.00

1", .. = 75 ....

"Re,T • 4. 2 7 .A.

FIGURE 7

30.00 35.00 40.00 45.00 PIN PRIME

50.00

ATM 78? Page l xf 23

55.00 60.00 65.00 70.00 15.00 80.00

NO. REV. NO.

: . I


ATM 783

PAGE 14 OF Aerospace .::!-~-- Divi. . /"""''l!l!!:eDu:nnS 81011 DATE

3. Application of the Revised Model

When substituting specific values for the parameters in this model it is, of course. important to properly interpret the definition of each parameter. Figure 1 illustrates the assumed distribution of power and emphasizes the assignment of all PCU power to "conversion" losses or "reserve power" dissipation. When considering the real PCU (Figure 8) it is necessary to be more specific in the definition of the P and P t c r terms in the mathematical model. As stated above, the P term in the

c revised model was molded to suit the results of a specific test designed to measure the PCU conversion losses. To make these measurements under a range of load conditions it was necessary to establish the meaning of "conversion loss" so that the test data would be consistent. For purposes of this test the operating point of the regulator servo was set to "cut-off". This point was set by varying the voltage of the power source in such a manner as to yield a maximum voltage drop across the regulator transistor. This establishes a unique condition of power system operation where the regulator is imposing minimum load on the power source and the supply voltages are on the verge of dropping due to excessive load. Hence the measurements of power consumed within the PCU under this condition (and herein designated P c) represent the total PCU dissipation when reserve current is at a minimum.

When the above test measurements were made the data was correlated in terms of input power using a measurement of Ein made at the PCU connector. When values of Ein" obtained during system test or operation (from the System Test Set or from the telemetered data) are substituted into this math model, it must be recognized that these voltage readings may require some correction to represent the voltage at the input connector. The resolution of the telemetered input power parameters introduces an additional uncertainty into the results obtained by this model. To illustrate, the resolution of these parameters at a typical power operating point, in terms of the possible change in the parameter without a change in octal representation, is as follows:

Input Voltage Resolution:

Input Current Resolution:

Reserve Current Resolution:

o.o8v@l6v

0. 0275A ~ 4. 25A

O.Ol72A

23

N017S: J i.A.Ir «.,....,....""'""' WED/JC?I.~,'(.I,q27, ,If~/

,.r~ 1..1~ AR41 T4 """" L ,XC~Pr A.t IVO'T7D;

AU. liE~ AAI JN ~ AoWJ ARE y,.,.. rtltlfTT AU. Coi'AII(YTClo't,f ......,. IN ll!lt:,/II~ADJ

~ A" ,IGNM.S •t> TO D-14. <f ••• ..-NA.L.S QO 10 .ACTIVE UJMIIC.

~ CAA«'I7'l:lJilt.S:: '0~ C4AACffQII/U Atfllll C 3R/4.,0S.I(P

.O/Jn> CAo"'Wcln;w,s AA6 CK.If/16Jf/0.1Kiill.

• CONfll/ltCNI!NT3 A.I(1T USID: "-'~.lf.J~ A'7t:.,f~77

" .

-

r -~· ... !1'1"'!! 1()1/1.

··~~ ~ l~O ... ...

!"~~f.ll.l~ ll"'! ·=l -~

~

Figure 2 - PCU Electrical Schematic, Complete

I

... l ~q~ ,tio .. ..

II ~ -4 "'" :1 ,; I!! 1£.:: l-1 =fi: __,._ I !I I:: ~1M I;:

'"" ·-

II lFf-F

;:'f ~- f~

& .. f*: -..... .....:: ~ ;::..

-·4!- .. ~ _.:;: -111 >K -_l!ff -.... :iL.'

,:. f:Jl ... 7,,.,_ --·-

~ CN ~~·4"1!1"

I r4"t- . ~~~

-"'~-~ . .. _ .. ~ ··-··,·-~ . ··-· ·--~'~ ,._#_,g_ W<n

.. ---'~'·-l~'t --·-I~ ·-· -• jL,~ "",_"'n-1!4. --

""'

[\../'

" . ..... ..

--.. , ..

..

..

1:!~ Ill t-3 ~ ~

I ~-'-J tnoo 0 w H-.

N (J1

t.t,\

:;D


NO.

ATM 783

16

REV. NO.

PAGE OF

DATE

This uncertainty is provided solely by the digitizing process and does not include any of the measurement errors imposed by voltage and temperature changes. These measurement uncertainties, if multiplied together in the most adverse combinations to determine power, yield the following typical resolution products:

6 Input Power = +(16. 08 X 4. 2775) -(15. 92 X 4. 2225)

= 68.7822 67.2221

=

~Reserve Power = +(16. 08 x 0. 3275) - (15. 92 x 0. 2725)

= 5. 2662 4. 3382

= o. 928(~5 w)

NO. REV. NO.


ATM-783

PAGE 17 OF 23

Aerospace /-~~.-tems Division DATE 15 July 1968

APPENDIX A

Curve-Fitting to PCU Conversion Loss Test Results

On 9 May 1968, a test was performed on the Qualification Back-uE.,_ Model of the PCU (Part No. 2330000-3, Serial No. 2} to determine the

~e"ctrical power""'consumed by the PCU when the regulator is inactive. The procedure for this test was the first 62 pages of the Thermal Mapping Test Procedure for the PCU. The results of this test are plotted in Figure 9 against input power, Pin· Using the measurement data shown in Table I, rather than these hand-fitted curves, a series of equations of the form

Pen = An + Bn (PirU + Cn (Pi:r;_) 2

where Pi~ = Pin - Pr

=Pin {when Pr = 0}

n: relevant power converter {1 or 2)

were fitted numerically using simple curvilinear regression techniques.* This curve -fitting process yielded the following relations which are the ones being used to compute the PCU conversion loss, Pc• in the revised model:

Pel = 3. 893 + 0. 0205 Pi~ + 0. 000759 {Pi~ ) 2

Pc2 = 3. 97 + o. 0234 Pi~ + o. 000605 (Pi~ ) 2

The following paragraphs detail the method for determining the coefficients of these relations.

*REFERENCE: Ezekiel and Fox, "Methods of Correlation and Regression Analysis", Third Edition, Wiley.

p c

watts

Input Power (with reserve power = 0) --- watts

.10 30 40 50 60 70 80 90 10(\ 108 +- I f • • ' ' ' · j_ -- 1 ' i · ' r ' - t ' · ' ' : ' • ' I I • • • w J. •-t i 1 + • ' [ ffij'- w· "Jill ·v ili . , , ; ' ·i - - t -., : , 1 1 1 _, : , l 1 - ; , , • : · t . ; - i t: , r 1 n t, F : r 1 : ·t t t1 :1 1; --~ --.:'~-- .. -t- ··--:----f_- • - -f- _-. .;. --· \ . _--- ~- t .. L . .:. -r:-' + ~-- -l I'~- I:-;~ +.:b.-t -·-~-.: _·l= ---,~ :J t·'·!, H•

I t : ' j ---1- .. , .. ·- --1-- 't ___ -1 ·I I -I- ! II j r-- --1 1-j I 1.1 'I I: _I II_~.-''~ ''t'l- "'J_t:·'-~ ·'llf!,·llntt·• 11· 1'' 1 1 1 - t + - r - ,_ ----- •· 't • + _; · , 1 1 • · :1 · i t , - • ' -- 't· -. · -• 'li + - ..;~ 9 t I I j - '- _:_t: -- . ! - - - - ::.1:....:.L ~ -: - ' • . \ . . - I I ·' 1-~ .L ;- - -, - j.J. ·~ ' tTjj9

1 · · 1 ' : • ' 1 -r - · I 1 1· .• 1- ·---- 1- -" 1- .t .• ,_ - · -~ ·- • 1 1· ' 1· r - 1 ~ . - i = -1-1 : H- nl _i- .r ,;; _ I _ j 1 , l 1 u JHJ 1- l t~ ~- -_r~_-_:- -~--~--~~- IT-~- -l-r:1 __ , i·t-_r--:--t·_-J:t _t.:f·:t-L~_= r=r:·----~-t------~~r--~ ~rr~::~- -1·-- -~ ·-:·:uJ=t-~--~ ·:--r-1 +---~-1.c- ···, ===t- =-r ., -H-·~c~ ·-. : ti. , -i ' 1 ·f~' \:' +i-r ti. -t -L · Hi =: 1 itt:; r+= ~ [1:±:; · li·r'=tt ,1

1::

8 j_ ':·_ ::-~-~--- ~---i---~~~: .. ! :: __ ' -- :---~-- r_! _~ __ 1_-_-!- j" . _-----~- : t~--,_-_·-~~~- ____ :-~--~-~-1-~~~- ~t~-~:t:i~:fg ~ -~_-·-:·:f· t_+ ill:-:-::. ~-~~-4Js 1 1 I 1 -. '"·-~-- , f 1. , -~ -r 1 1·· HI• 1+1-- .. Jil . 'T t:-t:-±- .... - tt=rt: tt.,..,,_,_l:t El Hil' !L iLJJ+bm11ttJ,

i I :- ·'+· n.··· / f f 1-- ·'lEtr·· . l L_j:·j.. ·F -·"!··- bf_1;i, f:t:7t::=iH-·ft+tUl= ii Wil:8JI+B.W

6~-r~-r++~~~~++rH~H+~~~~~~~~R4*PR+H+~+H~~~~~*ffi~~~

Figure 9: PCU Conversion Loss Test Results.

lo(j> Ill t-1 ~ ~

I ~-J oooo 0 I..V ..., N w

.. -~

Power Converter 1 Pin Pc

15.009 4.336 21. 148 4.645 27.044 4.955 33.201 5.348 39.921 5. 975

~c,,_4,5 0 9 3 9 6.430 52.437 7. 197 58.637 7.766 64.719 8.408 71. 964 9. 184

TABLE I:

ATM 783 of 23 Date: -------

Power Converter 2 Pin Pc

14.997 4.406 21. 004 4.647 26.930 4.984 32.980 5.322 39.271 5.700 45.343 6.224 51. 530 6.769 57.419 7. 197 63.760 7.904 70.519 8.562

PCU CONVERSION LOSS TEST DATA

i X ;

15.009 21. 148

;

i27. 044 \33.201 39.921 45.939 ~52.437 :58. 637 64.719 71.964

! ! ~~=430.02 I )

Assuming a quadratic curve fit for the data and using a simple curvilinear regression technique, then

p 1 = a + b (Pin') +

In the following table Y = P 1, X = Pin' U = x2

{P· ')2 1n

y x2

4.336 225.27 4. 645 447.24 4.955 731.38 5.348 1102.31 5.975 1593.69 6.430 2110.39 7. 197 2749.64 7.766 3438.30 8.408 4188.55 9. 184 5178.82

Tabulation for P 1 Equation TABLE II

xu u2

3, 381. 0 5. 07{104) 9,458.2 20. 00(104)

53. 5 (104) 19,779.4 36,597.8 1. 22{106) 63, 621. 7 2. 54{106) 96,949.2 4. 45(106)

144,182.9 7. 56{106) 201,611.6 11. 82(106) 271,078.8 17.54(106} 372,688.6 26. 82(106)

XY UY

65.08 976.77 98.23 2,077.43

134.00 3, 624.00 177.56 5,895.52 238.53 9,522.30 295.39 13,569.81 377.39 19, 789. 16 455. 16 26, 701. 84 544. 16 35,217.33 660.92 47,562.28 I

I

i

2Y=64.27 L;U=21, 765.60 L:xu=l,219,349 ,L;U2=72. 73{10 6) ~XY=3,046.63 ~UY=164,936

Determine the following auxiliary relations in order to find the parameters a, b, & c;

2x _ 430.02 = 43 Mx = -- 10 n ~=2;Y =

n

~ 2 " 2 2 ~x = L X - nMx = 3, 2 7 5. 6

I;xu = L XU - nMxMu2 = 283, 669

I;u2 = L u2 - nMU2 = 25. 43(106}

L:xy = [xy - nMxMy = 282

Luy = L UY - nMuMy = 25, 085

64. 27 = 6. 427 10

21,765.6 = z, 176 Mu = 10

······ .. en ~ /I

I ~ ;:

> t ..... .. t:J (1) .... (1)

li s ..... ::s Ill .... ..... 0 ::s 0 '""' tr1

,.c s::: Ill .... ..... 0 ::s 0' li

"'d () .....

10 ,.. -1 m

..... U1

1:.-j

g. '< ..... ~ 0' 00

t:J (1)

'""'< 0 (1)

li 0 .... "'Cj ;; s

(1)

"'d::s 0 .... ~ 0 (1) '""' li Ill

()"'d 0 0 g.~ ..... (1) .... li ..... 0 t:J ::s ..... ,.... rn ::S rn

()Q .....

c::~ e. o: .... 0 -::s "'d~ ()0 C::p,. -(1)

......

, ,.. a m

N 0

0 "11

N uv

> ~ t-3 •

~ t -.I 00 uv

,., m ::. z ?

Aerospace Sy~ems Division

,r -.,_


NU. t<t: V. I'IU.

ATM-783

PAGE 21 OF 23

DATE 15 July 1968

b and c can now be found by solving the following equations simultaneously.

Then

( L x 2 )b + ( L xu) c = £ xy

(Lxu)b + (L u2 ) c = L;uy

c 1= . 000759

b.= . 0205 I

and the following relationship is used to determine a;

a.= 3. 893 I

Therefore,

Pc t 3. 893 + 0. 0205 Pin' + 0. 000759 (Pin')2

X

14.997 21. 004 26.930 32.980 39.271 45.343 51. 530 57.419 63.760 70.519

~

L LX=423.74

Using the same as surnptions given in Appendix A-1, then

y u

4.406 225 4.647 441 4.984 725.22 5.322 1,087.7 5.700 1,542.2 6.224 2,056.0 6.769 2,655.3 7. 197 3,296.9 7.904 4,065.5 8.562 4,972.9

LY=61. 73 LU=21,067. 7

p ~2 = a . .+ b:(Pin') + c{Pin' )2

Tabulation for Fez Equation TABLE III

xu uz

3, 375.0 5. 06{104) 9,262.8 19. 45{104)

19,530.2 52. 59(104) 35,872.3 1. 183{ 106) 60, 563. 7 2. 378{106) 93,225.2 4. 227{106)

136, 827. 6 7. 050{106) 189,304.7 10. 869{106) 259,216.3 16. 528{106) 350,683.9 24. 730{106)

I XU=!, 157,861.7 L u 2=67. 74{lo6)

XY UY

66.09 991. 35 97.60 2,049.33

134.22 3, 614.50 175.52 5,788.74 223.84 8,790.54 282.21 12,796.54 348.8 17,973.73 413.24 23,727.79 503.96 32, 133.71 603.78 42,577.97

L XY=2,849.26 LUY=l50,444 --- -- - ---------------------- ------------------------ L________ _____ --------- - L_

I:x = 4 2.374 Mx = --n--

Lx2 =l:x2 -n~ = 3,112.2

Lxu = LXU - nMxMu = 265, 842

l: u2 = L u2 - nMu = 23. 35{106

)

Lxy = LXY - nMxMy = 233.8

Luy = LUY - nMuMy = 20,391

~= ~y=6.173 Mu = LU = 2, 106. 8 n

'f)~ l ~ ;a a-! 21 s 1:

> t:J ~ ro .. ~ <

o ro t:J 11 0 ro ~o-o ..,. ;T t:l ro ro P 11 'tJ ro s 0 g. 5' ~ 0 Ill ro ~ c-. 11 Ill 0 () hoi ;:l 0 v 0 ;:l 0 ~ p..~

,_.. ro M ::r. 11

..0 0 H ~ ;:l v Ill ........ ~ ::s til ,_.. ()Q til 0 .... ;:l c:: "0 ~ ;:l Ill 0 .... c-. 11 ~ 0

'tJ -;a;:l

b. () ~ N C::O

-P.. ro t-'

R,.

c ., > ~ ,.. ,.. 1-3 _. a !7 m m i==oo

1-' N I \J1 N -.J

00 t..r I.N g.

'<: 0 XI

'""' I"" ~ ...0 • 0' z 00 N p

l..V


(L x 2)b + ( L.xu)c = L xy = 233. 8

(Lxu)b + ( := u2)c = [uy = 20,391

Solving Simultaneously,

and

a 7.= My - hMx - eMu= 3. 97

Then

2 Pc2 = 3. 97 + o. 0234 Pin' + . 000605 (Pin')

NO. REV. NO.

ATM-783

PAGE 23 OF 23

DATE 15 July 1968

development of a power dissipation model for the power ... #272...qualification, back-up and...

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