The JPL Low-Cost Solar Array Project is sponsored by the U.S.^^lc^. Department of .... interface in solar cell modules is an important requirement in the Low Cost .... voltaic current-voltage (IV) profile which measures photovolaic performance. ..... tD y .^ s. W c. N r'. C7. O. J. O. N to. O. O r. 00. O. 0 to > o^. O. N. CO. O i. aL. V).
NOTICE
THIS DOCUMENT HAS BEEN REPRODUCED FROM MICROFICHE. ALTHOUGH IT IS RECOGNIZED THAT CERTAIN PORTIONS ARE ILLEGIBLE, IT IS BEING RELEASED IN THE INTEREST OF MAKING AVAILABLE AS MUCH INFORMATION AS POSSIBLE
JPL NO. 9950-535 DOS/JPL 954739-4 t
Annual Report
SC5106,104AR
Period Covered; January 1, 1980 through December 31, 1980 STUDY PROGRAM FOR ENCAPSULATION MATERIALS INTERFACE FOR LOW — COST SOLAR ARRAY To
Jet Propulsion Laboratory California Institute of Technology
345^^
for the ««
Encapsulation Task of the Low-Cost
Solar Array Project
JUN 1.0 8.1
^CEIaED. ;SAS $Tl FACIUi `n1c, ACCESS I)'M
The JPL Low-Cost Solar Array Project is sponsored by the U.S.^^lc^ Department of Energy and forms part of the Solar Photovolta`ifLConversinn Program to initiate a major effort toward the develop-
ment of low-cost solar arrays. This work was performed for the Jet Propulsion Laboratory, California Institute of Technology by agreement between NASA and DOE. This report contains 'information proposed by tle Science Center, Rockwell International, under JPL Subcontra, t 954739. This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Department of Energy, nor any of their employees,
nor any of their contractors, subcontractors, or -their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represent". that its use would not infringe privately-owned rig' ts. February, 1981 D. H. Knolble, F. B. Mansfeld, M. Kendig, C. Leung
Science Center Rockwel International
l
1049 Camino Dos trios Thousand Oaks, Callrllbrnia 91360 (NASA-C-154328) STUDY PROGRAM FOR ENCAPSULATION MATiitIALS INTERFACE FOR LOW-COST SOLAR.. ARRAY Annual Report, l .Ja.n. - 31 Dec. 1980 ( Rockwell . :International Science Center) 81 p HC A05/MF A01.CSCL 10A G3 /44 —rte
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Rockwell International % Science Center SC5106.104AR
TABLE OF CONTENTS
Page ACKNOWLEDGEMENTS ....................................................
1
ABSTRACT ...........................................................
2
1.0
OBJECTIVES .......................................................,.
3
2.0
SUMMARY ............................................................. 2.1 Atmospheric Corrosion Model Verification ...............0000....
4 4
2.2 AC Impedance Monitoring and Analysis ........................... 2.3 Hydrothermal Stress Analysis of Solar Arrays ................... 2.4 Criteria for Encapsulant Bonding ...............................
5 5 6
3 .0
INTRODUCTTON ................................................,.......
7
4.0
ANALYTICAL MODELS ................................................... 4.1 Solar Array Current-Voltage (I-V) Model ........................ 4 .2 AC Impedance Mod el ,............,. ...........................0...
10
5.0
10 16
EXPERIMENTAL ........................................................
24
5.1 5.2 5.3 5.4
1"iead Site Atmospheric Corrosion Monitoring ......... ............ Hydrothermal Stress Analysis ................................... Hydrothermal Cell Cracking and Corrosion ....................... Interfacial Bonding for Corrosion Protection.. ... .10 ... ..,,.....
24 33 46 52
6.0
CON%^JLUSIONS .........................................................
55
7.0
RECOMMENDATIONS .....................................................
a7
B.O
NEW TECHNOLOGY .............,,,.................,.................:....
58
9 .0
REFERENCES .......................................................0..
59
APPENDIX 1:
Mathematical Relations for Current-Voltage (I-V)
Response in Single Solar Cells .............................
61
APPENDIX 11: Electrical Impedance of Solar Cells and Low
Cost Solar Arrays ....... .. ... ....,,.....................,...
e
e
li
69
1
WNW Rockwell international Science Center SC5106,104AR
,
ACKNOWLEDGEMENTS
The authors wish to express their appreciation to the many individuals who contributed to this research and this report. At the Jet Propulsion Laboratory the helpful discussions and supply of test materials by M. Sarbol ouki , H. Maxwell, J. Repar and A. Gupta were mast helpful. Professor Charles Rogers of Case Westf.rn University and Professor Wolfgang Knauss of Cal ifornia Institute of Technology provided progress reports and direct advice to this program. The technical cooperation provided by Steve Forman and his colleagues at the Lincoln Laboratories, M.1.T. were important to the successful completion of the Mead, Nebraska corrosion monitoring experiment. Jovan Moacanin is technical monitor of this subcontract for JPL and Cliff Coulbert of JPL is Manager of the Encapsulation Task of the Low Cost Sol ar Array Project. Their helpful technical guidance is greatful ly acknowl edged.
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ABSTRACT
The early validation of a 20 year service integrity for the bonded interface in solar cell modules is an important requirement in the Low Cost Solar Array (!.SA) project. The first annual report, (Science Center Report No SC5106.22AR) out-, lines and implements a physical/chemical evaluation program for solar cell encapsul ants. The results of computer controlled ultrasonic and optical /ell ipsometric mapping for interfacial defect characterization in solar modules is summarized in the second annual
report (SC5106.49AR).
va.l idation of an atmospheric corrosion model
The development and
and test pl . % for LSA outdoor
service at the Mead, Nebraska test site is presented in the third annual report (S 5106.86AR) . In the present fourth phase of study detailed in this annual report emphasis is placed on the development of AC impedance cis a nondestructive evaluation (NDO methodology for solar arrays and the further development of corrosion models and materials selection criteria for corrosion resistant interfaces.
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international 01% Rockwell Science Center SCS106.104AR . 1.0 OBJECTIVES
The general objective of this phase of the program is to broaden the present corrosion model and atmospheric corrosion studies developed for the Mead, ;iebraska test site. The two major objectives included in this present study are stated as follows:
1)
Continue further development of atmospheric corrosion monitors and AC impedance measurements as nondestructive evaluation (NDE) tools for LSA module performance and life prediction.
2)
Initiate development of materials selection criteria and tests for corrosion protection and environment resistant interfaces as required for validation of corrosion models.
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2.0 SUMMARY
2.1
Atmos heric Corrosion Model Verification
a.
A 13 month experimental study of climatology and atmospheric corrosion monitoring (ACM) of the Mead, Nebraska LSA test site has been compl.,ted.
l).
During this 13 month period the ACM units recorded the corrosion protection function of an encapsulant system
(a
reactive primer
GE-SS4155 on Zn/Cu plates encapul ated by Syl Bard 184).
C.
A non-encapsulated ACM unit correlated Mead climatology with corrosion rates as defined by an atmospheric corrosion model.
d.
Both the intermediate and final results of this study verify the prediction that atmosphr,-ric corrosion rate is the product of the moisture condensation probability (P c) and the maximum ionic diffusion current (1 L ) at the corrosion surface or interface.
e.
Encapsulant corrosion protection is specifically related to the suppression of I L at the potential corrosion interface.
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2.2
AC Impedance Monitoring and Analysis
a
A predictive model for solar array current-voltage (I »V ) response has been developed and correlated to AC impedance.
b.
A new and simpltified methodology for conducting AC impedance measurement and spectrun analysis has been developed and verified for nondestructive evaluation of LSA performance.
c.
A preliminary design for an AC impedance measurement, spectrum analysis, and performance optimizing control for a solar array branch circuit has been outlined.
2.3
Hydrothermal Stress Analysis of Solar Arrays
a.
A computer model for hydrothermal stress analysis (HTSA) has been successfully applied to evaluate the combined effects of
temperature-humidity cycling on solar cell cracking mechanism in LSA modules using a fiber board substrate.
b.
The protective effect of moisture diffusion barrier coatings of
ethylene vinyl acetate (EVA) and pol ytri fl uorochl oroethyl ene .
(KEL-F) is predicted to delay but not change the solar cell cracking process.
.
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2.4
Criteria for Encapsul ant Bonding
4
a,
A new criterion for encapsulant bonding is being developed and specifically directed at supressing the micrecorrosion process in the presence of internal defects such as micro-cracks.
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30 INTRODUCTION
In view of the promising early resins of the atmospheric corrosion model developed for the 1 ow cost solar array (LSA) at the Mead, Nebraska test
site the present program was expanded relative to a general LSA life prediction program as shown in Fig. 1. The logic flow diagram of Fig. 1 describes the materials imputs to an LSA system in the upper block. The next lower block describes the environmental stresses known to cause LSA performance degradation. The specific mechanisms of degradation and consequent degradation effects ire detailed in the subsequent decendi q g blocks of Fig. 1 with debonding and corrosion near the bottom of the figure and directly connected with photovoltaic rel I
U
l sty ai,d durability. On the 'right of
r"• ig. 1
are
categorized the specialized measurementand analysis programs which correlate with each degradation mechanism and effect. The prior annual report (SC5106,86AR) details the results of the corrosion program whose scope is defined by the inner bracket on the right of Fig. 1. In this prior program consistent interrelations have identified the corrosion chemistry, changes in the AC impedance spectrum, and the. photovoltaic current-voltage (IV) profile which measures photovolaic performance. In the present program the corrosion studies were expanded to include development of a 1 i fe prediction model and 1 i fe prediction test plan which included the debonding and corrosion process. This report details the progress of this 0
effort under a Task I which now encompasses corrosion modeling, life prediction in terms of photovoltaic performance and nondestructive evaluation of
7 C3047A/ES
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MODULE MATERIALS MEASUREMENT AND ANALYSIS PROGRAM UV RADIATION.
HYDRO y 4NRMAL CYCLING ECHANICAL LOADING L UV DAMAGE - f4," UV*VISIBLE SPECTRUM IR SPECTRUM CHEMICAL ANALYSIS
INCREASE ? BANCE OPTICAL AMR
SWELL GEL LICUIO CHROMATOGRAPHY
CROSSLINKING
SCISSION
MIXED THERMAL TRANSITIONS
:.EIGHT L0
ESS CON SNTRA STRESS CONC,NTRATION
ADHESIVE TACK OWT RETENTION
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I
STRES&STRAIN RESPONSE
CAVITY FORMATION
FAILURE ENVELOPE
TASK
FAILURE STATISTICS DEFECT ANALYSIS
PROBABILISTIC PREDICTIONS CRAZE
SURFACE ENERGETICS BONDING CHEMISTRY
CRACKING
FAILURE ANALYSIS
DEBONOING I—
,AUGER PROFILING PRIOR
AC 1MPEOANCF, SPECTRUM
CURRENT•VOLT.AGE PROFILE PROGRAM CORROSION MODEL
CORROSION
CORROSION TEST PLAN LIFE P`►,EDICTION MOOit:L
PHOTOVOLTAIC RELIABILITY--OURASILITY
LIFE PREDICTION TEST PLAN MATERIALS-PROCESS OPTIONS
it
Fig. 1 General LSA life pred
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p4rformance of modules and solar array branch circuits. Tbi s phase of the study is closely coupled to extensive studies in photovoltaic design optimization already completed by JnL. In order to provide for new materials selections in terms of
advancing LSA technology a new study program in
materi .,l selection criteria
for corrosion resistant bonding in the presences of microdefects has been initiated.
This study is defined as Task 2 in the right bracket of Fig.
This task implements an already developed computer model (7) for
I.
hydrothermal
stress analysis (''HTSA) to define potential mechanisms of LSA microfail ure and to foc ► s the mateeials studies and bonding criteria to the specific failure mechanisms. In addition, a new physical/chemical
bonding
criteria and test
protocol has been initiated for minimizing ionic conduction at encapsulant interfaces in LSA.
9 G3047A/ES
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A.0 ANALYTICAL MODELS
Two analytical models have been developed within this program to translate the results of corrosion analysis
performance and durability. The
into predictions of photovoltaic
First of these models develops the current,.
voltage relations for a singlo solar cell performance. By introduction of parallel
interconnection the
performance and
AC impedance response of a branch circuit
can be calculated.
The second
sunning relations for series and
model
is developed for a simplified computational methodology to analyse AC impedance spectrum. The combination of these models provides a potential new basis for nondestructive evaluation (NDE) and feedback control of solar arrays.
4.1
Sol ar
Array Current
Voltage (I -V) Model
Appendix I provides a detailed derivation of an
analog model for
solar cell response which permits direct calculation of the I -V fill factor, maximi,mi power point and other performance oroperties
relating to solar cell
response. The equation, diagram and nomenclature of Fig. 2 give a brief synopsis of this I-V response model. Based upon controlled variations of isolation S and temperature T meters of p his curves in
the curves of Fig. 3 show how the design para-
14 response model can he
the tipper right
evaluated experimentally. The 14
portion of Fig. 3
are takon frori a published report
and represent typical silicon module response. %4 ) The calibration experiments of Fig. 3 define all of the solar cell design
paramee ters with the exception of
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AC impedance is viewed as a method for augmenting power output measurement to provide highly sensitive and interpretable data on environmental aging effects. The interpretation and± control methodology outlined above would integrate nondestructive evaluation (NOE) with a corrective action program. to locate and minimize environmental aging effects on array performance.
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5.0 EXPERIMENTAL
This section discusses
and interrelates the results of natural
sion studies at the Mead test site and
computer
corro-
modeling and accelerated aging
at the Science Center.
5.1
Mead Site Atmospheric Corrosinn Monitoring As reported in the third annual report (SC5106.86AR), a new atmos-
pheric corrosion model,
as outlined
in Fig. 11 was developed and implemented
in corrosion studies at the Mead test site. As shown in Fig. 11, the corrosion rate is predicted to be the product of the surface condensation probability of water
vapor PC
and the
diffusion
controlled corrosion current I L-
Th i s model combines surface physical chemistry and electrochemisty and is verified by the direct correlation between relative humidity and the logarithm of measured corrosion current as shown in the upper right view of Fig. 11. The corrosion monitor experiment was initiated at Mead, Nebraska on July 12, 1979 and continued for over a year until terminated in October 1980. A pnotographic view of the Mead, Nebraska L,SA test site is shown in the upper view of Fig. 12. The
lower view
of Fig. 12 shows two atmospheric corrosion
moni tore (ACM) installed on t'ne right rear portion of the array.
The monitors
consist of the edges of alternating plates of zn and Cu seperated by a 50 um polyester (DuPont-Mylar) dielectric film. Moisture condensation on the exposed surface closes the circuit between the plates and current flows.(10) The solid state circuitry of the ACM converts 24 C3047A/ES
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logarithm of the current
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portion of array. 26
01% Rockwell international Science Center SC5106,104AR
(microamperes) to a proportional DC voltage which is transmicted to the
recordi>>g station located in the power control unit (PCU) located behind the right end of the array (see upper view Fig. 12). Specific details of the
corrosion and climatology recordings are discussed in the third annual report
(SC5106.86AR). During the 13 month deployment at the Mead, Nebraska test site the ACM units recorded, at 10 minute intervals, the corrosion protection function of an encapsulant system consisting of a reactive primer, General Electric
GE-SS4155, on Zn/Cu plates encapsulated by 2nm of Dow Sylgard 184 silicone. A non- encap Fe > Ni - Pb > Sn. although kinetic aspects of passivation and alloying influences upon chemical potential can produce significant changes in this ordering of actual lability, the selective migration of Fe and 50 C3047A/ES
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Cr from the Pb/Sn electrode which exhibits positive polarity under illumination argues for an electrochemical process leading to the
dissolution of Fe
yid Cr from the upper electrode. An additional NTSA computer run was conducted in which moisture exposure was suppressed at 0% R. H. and only the thermal cycle of lower Fig. 19 was imposed on the 4 layer solar cell model. The purely thermoelastic stresses are shown by this ccxnputation to be noncritical and noncunui Live.
5.4
In terfacial Bonding . for Corrosion Protection A good deal of evidence has been accumulated in the experimental
studies of this program that environmentally resistant irate;*face bonding of encapsul ant to all solar cell and metallization surfaces is essential for corrosion protection. This problem is even more acute when individual cells display microcracks as discussed in the preceeding sections. The atmospheric corrosion model discussed in earlier sections and summarized in Fig. 11 defines explicit factors for reduction of corrosion rate by suppression of ionic diffusion mechanisms. In order to extend the theoretical range of this corrosion model, an additional acid-base criteria for interface bond stability, as shown in Fig. 26, will be explored. This criteria was originally developed and verified by Bolger and b.::hael s (13) and has been i ncorpor ted into a general adhesion theory by Kaelble. (14) It is also evident that E. Plueddemann(15)
52 C3047A/ES
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ha, recently exploited this system of
silane adhesion graphical
definition
promoters for LSA. The
method for
in selection and testing of
diagram of Fig. 26 provides a compact
selecting either acidic (right margin) or basic (left
margin) organic radicals to achieve
respective high AA or eQ with metal oxides
(see upper margins) of differing isoelectric points for surfaces (IEPS).
Interfaces formed
according to the combined criteria Fig. 26 will be tested by
atmospheric corrosion monitors and environmental corrosion simulation as already developed in the prior program. Additionally, a new ion migration test method, recently described by Mittal and Lussow (16)
small scale screening test
for interface stability
a
53 C3047A/ES
will be explored as a
and corrosion
resistance.
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6.0 CONCLUSIONS
Atmospheric corrosion monitors (ACM) have been returned to the Science Center following 13 months deployment at the Mead, Nebraska test s i te. During this period the ACM units recorded the corrosion protection function of an encapsul ant system and correlated Mead climatology with corrosion rates of a nonencapsulated ACM. The fundamental assumptions of a new atmospheric corrosion model were verified in this study. This corrosion model predicts that corrosion rate is the product of a condensation probability (P c ) and the maximuti ionic diffusion current (I L ). Encapsulant corrosion protection is specifically related to its effi-,iency in suppresssing
I L at the potential corrosion interface. AC impedance measurements combined with impedance spectrum analysis and feed back control to a branch circuit appear as a direct means of remotely locating degraded .LSA modules and modifying series to parallel S/P interconnects between modules to achieve maximums LSA power efficiency. A hydrothennal stress analysis (HTSA) computer model has been successfully applied to evaluate the combined effects of temperature-humidity cycling on development of intetnal stresses and solar
cell
cracking in solar
cell modules using a fiberboard substrate. The effect of protective polymer coatings of ethylene vinyl acetate (EVA) and polytri fl uoro-chl oroethyl ene If
(Kel-F) is to delay but not change the failure process.
55 C3047A/ES
oi% Rockwell International Science Conter SC5106,104AR A new materials selection criterion for encapsulant bonding is being developed which incorporates both ionic conduction and electrochemical mechanisms of corrosion. The new selection criteria and test, methodology specifically directed at presence of internal
suppressing the micro-corrosion process in the
defects such as micro- cracks.
56 C3047A/ES
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International 01% Rockwell Science Center SC5106.104AR
7.0 RECOMMENDATIONS
The following specific recommendations in the area of corrosion and encapsul ant life prediction are presented from this study:
1.
AC impedance monitoring combined with spectrum analysis and feed back control is suggested for field trails at an LSA test site.
2.
Develop materials selection criteria and tests for corrosion and environment resistant interfaces with 20-year life capability.
3,
Develop and validate a more general reliability and 1 i fe prediction model for LSA which includas corrosion, interfacial integrity and hydrothermal stress analysis (HTSA) as specialized subjects and directly describes photovoltaic current-voltage (I-V) response.
57 C3047A/ES
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Rockwell International Science Center
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8.0 NEW TECHNOLOGY
This study has developed and demonstrated new corrosion models and test methods. AC impedance measurements and analysis has been advanced to a
potential field deployable state for nondestructive evaluation (NDE) and remote management of LSA branch circuits. Hydrothermal stress analysis (HTSA) has been demonstrated as a method for predicting solar cell life time and failure mechanisms. Atmospheric corrosion monitors (ACM) are effective tools
for evaluating atmospheric corrosion resistance of encapsulant bonding materials and evaluating climatology influence on corrosion rates.
58 C3047A/CS
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9.0 REFERENCES
1.
R. G. Ross, Jr., "Photovoltaic Design Optimization for Terrestrial Applications, paper presented at the 13th IEEE Photovoltaic Specialists Conference, Washington, D.C. June 5-8, 1978.
2.
R. G. Ross, Jr., "Flat Plate Photovoltaic Design Optimization," presented at the 14th IEEE Photovoltaic Specialists Conference, San Diego, California, Jan. 7-10, 1980.
3.
C. Gonzales and R. Weaver, "Crrcuit Design Considerations for Photovoltaic modules and Systems," Ibid.
4.
R. G. Ross, Jr., C C. Gonzalez, "Reference Conditions for Reporting Terrestrial Photovoltaic Performance," presented at the AS/ISES 1980 Annual Meeting, Phoenix, Arizona, June 2-6, 1980.
5.
D. Schwartz, "Module Performance in a Varying Test Environment," Proceedings of the 11th Project Integration Meeting, JPL Report No. 5101-109 (DOE/JPL-1012-126), October 1978
December 1978, pp. 3-36
to 3-42. 6.
Proceedings of Workshop on "Flat Plate Photovoltaic Module and Array Circuit Design Optimization," Engineering Area, LSA Project, Jet Propulsion Laboratory, March 31
7.
April 1, 1980.
C. L. Leung and D. H. Kaelble, "Moisture Diffusion and Microdamage in Composites," ACS Symposia Series No. 132, American Chemical Society
4
(1980), pp. 419
434•
59 C3047A/ES
.A
Rockwell International 01% Science Center SC5106.104AR
8.
N. G. Mc Crum, B. E. Read and G. Williams, "Anel astic and Dielectric Effects in Polymeriz Solids," Wiley, New York, (1967), Chap. 4.
9.
L. B. Bucciarelli, "Power Loss in Photovoltaic Arrays Due to Mismatch in Cell Characteristics," Solar Energy, 23 (1979), p. 277.
10.
F. Manfeld and J. V. Kenkel , "Electrochemical Measurements of Time-ofWetness and Atmospheric Corrosion Rates," Corrosion, 33(1), (1977), p . 13.
11.
D. Kaelble, F. Mansfeld, S. Jeanjaquet and M. Kendig, "Atmospheric Corrosion Model and Monitor for Low Cost Solar Arrays," Proc. Corrosion 81, Toronto, Canada, April 6-10 (1981).
12.
E. Laue and A. Gupta, "Reactor for Simulation and Acceleration of Solar Ultraviolet Damage," JPL Report No. 5101-135 (DOE/JPL 1012-31) for Low-Cost Solar Array Project, Sept. 21, 1979.
13.
J. C. Bolger and A. S. Michaels, "In Interface Conversion (Editor: P. Weiss) Elsevier, Amsterdam (1968), Chap. I.
14.
0. H. Kaelble, Physical Chemistry of Adhesion, Wiley-Intersecion, New York (1971), Chap. 5-5.
15.
E. P. Pl ueddemann, "Chemical Bonding Technology for Terrestrial Solar Cell Modules," JPL Report No. 5101-132 for , Low-Cost Solar Array Project, Sept. 1, 1979.
16.
K. L. Mittal and R. n, Lussow, "Adhesion Measurement of Polymeric Films and Coatings with Special Reference to Photoresist Materials," in Adhesion and Adsorption of Polymers, (Editor L. H. Lee) Polymer Science and Technology Series Vol. 12B, Plenum Press, New York, 1979, pp. 503-520.
f,
50 C3047A/ES i M
-77—
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Rockwell international % Scionce Conter 5C5106.104AR
APPENDIX I I#
MATHEMATICAL RELATIONS FOR CURRENT-VOLTAGE (I-V) RESPONSE IN SINGLE SOLAR CELLS
Introduction
It is often useful to devel op an anal og model of physical response in order that the interrelations between variables are more fully understood.
The model for I-V response presented here is mathematically simple and yet resembles many aspects of real physical response of single solar cells. Of particular interest in this discussion is the analytical description of the
fill factor
(F),
and the maximum power point Wm of the I-V curve which
describes cell power generating efficiency. Table 1 summarizes the symbols
and nomenclature used in this discussion.
General Relations
Let us assume that short circuit current
Io
is proportional to
insolation S:
Io =
+K1S
(1)
ID
f,
and that the open circuit voltage Vo decreases with increased temperature T and insolation S as follows: 61 C3047A/ES
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V Q = V D - K 2 (t + K 3 S)
(2)
where K 1 , K 2 , and K 3 are positive constants. By introduction of a shape factor K, which is defined as follows:
K
^ V0
Zn
RS
SN
(B)
where R S and R SH are respective series and shunt resistance the IV curve can
be expressed as follows:
I = Io
X-1
Cl - exp K(V
X - 1 - exp(
(4)
Vo)]
KV o )
(5)
Determining R S and RBy differentiation of Eq. (4) for constant I o
,
Vo
,
and K we obtain
the following relation;
W
KI oa -1 [exp K(V - V o)]
(6)
At the open circuit condition, where V - V o and I - 0, Eq. (6) becomes:
dI
-KIoX-1
(7)
Rl 62
C3047AJES
International Oilo Rockwell Science Center SC5106 104AR
A At the short circuit condition, where V = 0 and I
x 10 ,
Eq. (6) becomes
dI = -KI o A -1 [exp(- KV o )] = - ;^-^-SN
(0)
S
The limiting slopes of the I-V curve are defined by Eq.
(7) to determine R S at
V = V o and R S + R SH at I = Io.
Oeten nining the IV Fill Factor F The fill factor of an IV curve is defined by the following integral relation:
Vo
(9)
j IdV F=,1V '0 D 0 By substituting Eq. (4) into Eq. (0) we obtain: ^-1 Vo F = V f [ 1 - exp K (V - Vo ) dV 0 0
V T V-1 V V ' j o[exp K(V - Vo )] dV J 0d im o o 00
(10)
Integrating Eq. (10) by parts we obtain the following relation;
1
F = a-1 - KV - [1 - exp(- KV o )] 0 Substituting Eq. (5) into Eq. (11) provides the following relation: E
63 C3047A/ES i
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Fa
(12)
as shown by Eq. (12) when KV o a 0 the fill factor F x 0,50 and when KVo achieves high values KV o > 50 the fill factor approaches unity.
Determining the Maximum mower Voltage
The power output W of the solar cell is deterjained by the following product
W = IV
(13)
The maximum power point, with respect to voltage V, is obtained by differentiating Eq. (13) as follows;
dW - dI W-I+V3V
(14)
By solving for maximum power where d14/dV i 0 and in substitution of Eq. (4) and Eq. (6) into Eq. (14) we obtain the following expression:
aV
I = 10 [1
VI exp K (V - V o )] _ IO K exp K (V - V o ) = 0
(15)
By simplification and rearrangement, Eq. (15) defines the voltage V m where maximum power Wm is produced from the I-V curve by the following relation:
64 C3047A/ES
International 01% Rockwell Science Center SC5106.104AR . in (KV m + 1) + KV m = KV o (16) A
Solutions to Eq. (16) are obtained by introducing values of KVm in the left side and solving for KV o . The subsequent determination of ratio of Vm/Vo versus KV o is obtained by dividing the input KV,, on the left side of Eq. (16) by the output KV o on the right side of Eq. (16) to obtain the dimensionless ratio V1WVo.
Determining the Maximum power Output For a given I-V curve
the maximum power output Wm is expressed by the
rel ation:
(17)
Wm=ImVm
where I m and V m are the respective current and voltage at the maximum power point. The relation
for Im is provided by substitution of V = V. in Eq. (4)
to provide the following
Im =
I0X -1
relation:
CL - exp K (V n - V o )] i
By substituting Eq. (18) into Eq. (17)
(18)
provides the following relation for
maximum power:
W in = l ox -1 Vm [l - exp K(Vm - Vo)l
65 C3047A/ES
(1g)
01% Rockwell International Scisnce Cente
r
5C6106 .104AR
Alternatively, the maximum power W. can be reduced by the product I O V o to "
define a maximum power output ratio P am as
follows
1 Pm ^ ' 1.r K Vm^
0
00
CI - ex K(Vrn - V o )^
(2Q)
Shunt to Series Resistance Ratio By rearranging Eq. (3) the ratio of shunt to series resistance fie,
given by the following relation:
RSH
exp (KV o )
1
(21)
S Sample Computations for IV Properties The procedure for conducting abbreviated calculations for IV response proceeds as shown in Table 2. The calculation proceeds from left to right across each line of Table 2 and from top to bottom through the list of equations in lower Table 2. For convenience, this calculation simply inputs
tin assigned value of KV m into Eq. (16) and computes a value of
KVo
in the two
left columns of Table 2. The dimensionless ratio V m /V o is defined and the reciprocal value a"
1
calculated as shown in the third and fourth columns of
Table 2. The remaining properties F, Pm, and RSH/RS in the right column of
Table 2 are readily calculated to complete the calculations. /
It
is evident that in full calculations both Eq. (1.) and Eq. (2)
would be utilized with input of two environment conditions T and S and six
66
C3047A/ES i
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Rockwell International
Sclence center SC5106.104AR
0
solar cell constants I D , VD , K, K 1 , K 2 , K3 with computation of Io and Vo . The next computation inputs given value of KVo into Eq. (16) and successive values of KY. substituted to satisfy the equality with a determined value of KVm as output. Successive steps of the full calculation follow those of the abbreviated procedure of 'fable 2.
Tabl e 1 List of Principal Current-Voltage; (I-V) Design and Performance Parameters Maning
Symbol
D esign Parameters Kl
Coefficient of light developed current
K2
Thermal coefficient of voltage
KS
Thermal coefficient of light
ID VD
Dark current Dark voltage
RS
Series resistance
R SH
Shunt resistance Environment Parameters
T
Ambient temperature
S
Ambient insolation Performance Variables
Y
io
Short circuit current
Vo
Open circuit voltage
A, X
Current-voltage (I-V) response factors
F
14 fill factor
Vm
Voltage at the maximum power point
Wm
Wattage at the maximum power point
Im
Current at the maximum power point
Pm
Proportion of ideal power output
67 0304 7AIES i
Rockwell International Science Center SC5106,104-AR A
Tabl e c Sample Computations of Solar Cell IV Response .
KVo
Vm/Vo
A-1
.001
1.9995E-3
.5001
5.006E2
.50017
.2501
.002
.002 .005 .01 .02 .05
3.9978E-3 9.9875E-3 1-995E-2 3.903E-2 9.789E-2
.5003 .5006 .5012 .5025 .5061
2.506E2 1.006E2 5.0627E1 2.6125E1 1.0724E1
.4991 .5007 .5016 .5033 .5082
.2503 .2506 .2512 .2525 .2562
.004 .0100 .0201 .0406 .1038
.10
1.963E-1
.5120
5.6366
.5162
.2623
.2157
.20
3.823E-1
.5231
3.1474
.5318
.2744
.4657
.50
9.055E-1
.5522
1.6789
.5744
.3090
1.473
1.00 2.00 5.00 10 20 50 100
1.693 3.099 6.792 1.2398E1 2.3045E1 5.3932E1 1.0462E2
.5906 .6455 .7362 .8066 .8679 .9271 .9559
1.2254 1:0472 1.0011 1.0000 1.0000 1.0000 1.0000
.6347 67245 .8539 .9193 .9566 .9815 .9904
.3618 .4506 .6142 .7333 .8266 .9089 .9465
4.436 2.117E1 8.895E2 2.423E5 1-019E10 2.6442E23 2.728E45
200
2.0530E2
.9742
1.0000
.9951
.9693
CO
500
5.0622E2
.9877
1.0000
.9980
.9857
00
1.00691E3
.9932
1.0000
.999
.9932
KVm
1000
F
Pm
RSH/RS
KVo = An (KVm + 1) + KV m IV Shape Factor (Eq. 16)
F
Vrvv o = KV m/KVo Max. Power Voltage Ratio X -1 1/El - exp(-KV o )] Function of Shape Factor (Eq. 5) = a -1 - (KV^)- 1 IV Fill Factor (Eq. 12) Pm = (V m/V o )1. El - exp K(V m - V o l)] Max. Power Ratio (Eq. 20) RSH/RS = exp (KV o ) - 1 Shunt/Series Resistance (Eq. 21)
1
.
68
C3047A/ES
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Rockwell International Science Center
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APPENDIX II
Analysis for Frequency Dependence of AC Impedance
Introduction AC (alternating current) impedance measurements have been applied experimentally to I.SA (low cost solar array) single solar cells or to modules of 50 to 75 cells electrically connected in series. The AC impedance measurement is conducted by scanning the frequency spectrum from 1 I-1z through 1 MHz using low amplitude AC voltage (i10 W) and constant temperature. The frequency response For AC impedance of a single solar cell appears to fit the classical Debye typel single relaxation mechariism. When multiple solar ::el 1 s are examined, the frequency response for AC impedance is more complex but appears to be well described by a symmetrically broadened frequency distribution of Debye type relaxation processes. The empirical distribution function developed by Cole and Cole provides a set of general equations for broadened AC impedance response which reduce to the Debye single relaxation as a special case. In this discussion, the analysis renters on the Cole-Cole distribution function as a means for evaluating performance parameters of both single solar cells and multiple cells in modules or branch circuits. The AC impedance measurement represents a readily conducted nondestructive test which when combined with the dielectric analysis discussed here becomes a method for conducting nondestructive evaluation (NOE) of LSA performance. A list of the principal parameters of this analysis is presented in Table I.
69 C3047A/ES
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Rockwell International Science Center
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11
The Cole-Cole Distribution Function The characteristic relaxation time T o for t ,o `hree element circuit 0,iown in Fig. 5 is defined by the following relation:
(1)
Iro - C RSH
The storage or in-phase impedance z' is defined as a function of test frequency w
(radians/ s) by the following rel ation-,
R SH Cl + (wT) a cos(Oar/2)] Rs
-
1
+ (WT)
20
+ 2( wx)
s
(cos Off/2)
The loss or out-of-phase impedance z" is defined as a function of test frequency by the following relation:
^r,, i (oT) a si n(Oir /2)
(3)
1 + (wry ° + 2(wr) cos(Oli /2)
The distribution quality factor R is unity, a ; 1.0, for the Debye single
relaxation and defines a range 0