Shading Effects on CdS/Cu₂S Solar Cells & Array Design

Energy Conversion. Vol. 14, pp. 61-71. Pergamon Pres.?, 1975. Printed in Great Britain
Effect of Shading on CdS/Cu,S Solar Cells
and Optimal Solar Array Design*
MOHAMED SAYEDj-§ and LARRY PARTAINS
(Received July 1973)
Abstract-The effects of shading on solar arrays in general has been analyzed using the specific Z-V characteristics of a CdS/Cu,S
solar cell. The major differences in the amount of power lost with various array configurations subjected to identical shading
conditions has been determined. The conditions under which localized heating or ‘hot spots’ occur has been treated and the criteria
for their avoidance has been precisely defined. The effects of protective diodes in reducing power loss and in preventing ‘hot spots’
has been included. Genera1 design rules to assist in obtaining optimal solar structures have been formulated and presented.
1. rntroduction
Loss of output power from an array of solar cells is
unavoidable if a portion of the array is shaded from
incident light. However, the amount of power lost
depends on the configuration used to connect the cells
[l-71. Some connection schemes with particular portions
shaded result in reverse voltage across individual cells.
These cells then absorb rather than deliver power and
severe localized heating can result [7, 81. Such ‘hot
spots’ can cause individual cel1 failure [9] and can
result in power loss from large sections of an array [7].
The addition of shunt diodes shorts out these reverse
voltages preventing ‘hot spots’ [2-71. Instead of reverse
voltages, recent studies have shown that other combina-
tions of 41s and shading produce reverse currents
through individual cells [ 101. The resulting power
absorption is avoided by plating diodes in series with
the cells to black the reverse current. These reverse
current levels are low and no significant heating or
‘hot spot’ formation has been attributed to them.
In the references cited above, power loss studies have
been largely confined to particular cases. Few attempts
have been made to define rules for optimal arrays in
general. This previous work has dealt specifically with
silicon solar 41s intended for space applications where
considerations of weight, size, efficiency, but not tost,
are strong constraints. When tost becomes a major
concern, as in large scale terrestrial solar energy con-
version, materials other than silicon become attractive
alternatives. In particular, thin film CdS/&S hetero-
junction cells have the potential for reducing power
conversion costs by several orders of magnitude [l 1, 121
even though their efficiency is somewhat less than
that of silicon (5-8 per cent [l 1-131 compared to 11 per
cent [3]).
* Supported by a grant from the National Science Foundation.
t Present address: Microwave Division, Hewlett-Packard
Company, Palo Alto, California 94304, U.S.A.
$ Correspondence should be directed to this author. Electrical
Engineering Department, University of Delaware, Newark,
Delaware 19711, U.S.A.
5 Institute of Energy Conversion, University of Delaware,
Newark, Delaware 19722, U.S.A.
This paper investigates the effects of shading various
configurations in genera1 using CdS/Cu,S cel1 charac-
teristics. Since the I-V properties of al1 solar 41s are
similar, implications for al1 cel1 types are obtained.
In particular, we have studied the precise shading
conditions under which reverse voltage and current
occur. Confìgurations less susceptible to ‘hot spots’
and excessive power loss have been found. We have
quantified the shading effects for the various array
contìgurations so that direct comparisons can be made
for series and parallel connections with and without
protective diodes. Finally, we have formulated genera1
design guidelines to aid in obtaining optimal solar array
structures.
2. Analysis of Shading
The effects of shading depend on the I-V charac-
teristics of the solar cells, the type of biasing (e.g. maxi-
mum power, minimum load current fluctuation or
minimum load voltage fluctuation) and load used
(resistive, inductive, constant or variable) and the
connection scheme.
A. Measurement of cel1 characteristics
The I-V curves of a 3 x 3 in., CdS/Cu,S solar cel1
(No. 1-1: 6-5) manufactured by Clevite-Gould, were
measured under three conditions. First the cel1 was
exposed to a light intensity equivalent to the mean maxi-
mum solar value of 100 mW/cms. Next an opaque
cover was placed over one half the cel1 surface so that
only the uncovered portion was exposed to light. Finally
the I-V characteristics were measured in the dark. The
light source was a calibrated solar simulator of intensity
air mass one. Air mass one is the mean maximum
solar intensity at sea leve1 in contrast to air mass zero
which is the mean value in space. This simulator ap-
proximated the spectra1 distribution of sunlight and
was calibrated (& 2 per cent) with a standard cel1
flown from NASA’s Lewis Research Center. The resulting
reverse voltage characteristics are shown in Fig. 1.
Figure 2 gives the forward characteristics. Only in the
fourth quadrant is the I-Y product negative indicating
61

62
VOLTAGE (VOLTS 1
Flg. 1. The reverse voltage I-V clauam of a CdS/Ca,S
solar cell ader fully shaded, half shaded and fally lllmalaated
COIldltlOUS.
VOLTAGE ( VOLTS)
i i
-800
//’ \ I I I
Fig. 2. The forward voltage I-V clmactetics of a C!dS/Ca,S
BolarcellImdertilllyshaded,halfshadedaadfallyillaaliaated
cmditbm. Dashed liaea iadicate the locas of points of amtaat
power. The load line iadica~ the re&tive load for maximam
pOWW.
power generation. In the fìrst and third quadrants, the
product is positive and power is absorbed by the cell.
For full illtination, the short circuit current (&)
was 700 mA and the open circuit voltage (Vac) was
0.47 V. These curves are the basis for al1 the calculations
that follow.
It is of interest to note that no detectable cel1 damage
resulted from forcing up to 1.10 A of third quadrant
current through the cel1 for short periods of time.
By this we mean that the fourth quadrant characteristic
remained unchanged within the measurement accuracy
(_t 2 per cent). Thi
‘s is in contrast to results found with
silicon cells. Jett and Miller [7] reported that breakdown
and irreversible damage occurred to silicon devices at
shaded cell, third quadrant current levels approximately
equal to the full illumination, short circuit current
value.
B. Bìasing for maximum power
One of the simplest methods of determining the
maximum power available from a cel1 is by plotting
curves of constant power (IY = constant) on the cel1
characteristic [13]. This series of hyperbolas are shown
as the dashed lines in Fig. 2. For clarity only a few are
shown. The intersection of the cel1 curve with the highest
power hyperbola quickly gives the maximum power
(Pm) to be 220 mW at a current (Im> of 600 mA and a
voltage (Vm) of 0.36 V. The resistive load (&) that
allows this maximum power to be delivered is given by
Vm/Im and is equal to 0.6 Q. Since the characteristic
is not square, V,& is less than VoJsC. The ratio
(V,&/Vo&) is a measure of cel1 quality called the
fill factor F [14]. The squarer the characteristic, the
higher the fill factor. In the present case the F value is
67 per cent.
An arbitrary number N of identical cells can be
connected in series. If each cel1 is biased at its maximum
power point, a total output voltage of NV, is obtained
but the output current remains Im. Thus the load re-
sistance is NR, and the maximum power output is
NPm. If the N cells are not identical, the composite
I-V characteristic can be obtained by adding the voltage
across each cel1 for each current value [7]
V&O = E vt(G, (1)
i=l
where V{(I) is the voltage across the ith cel1 which is a
single valued function of 1. Knowing the composite
I-V characteristic, the load resistance for maximum
power and the maximum power available can be deter-
mined, as in the case of a single cell.
Alternately M identical cells biased for maximum
power can be connected in parallel. From the duality
theorem the output voltage is V,, the load current is
MIm, the load resistance is RmIM and the maximum
available power is MP,. Also if the M cells are not
identical, the composite I-V characteristic can be
obtained by adding the current of each cel1 for each
voltage value
MV) = E la(V), (2)
i=l
where la(v) is the current from the ith cel1 which is
single valued function of V. The maximum power
parameters again can be obtained from this composite
curve as with a single cell.
C. Efects of shding
In realistic applications of solar cells, the load could
be expected to change from time to time. Such random
variations are difficult to treat analytically. Since it is
desirable to obtain as much power as possible from the
cells, we have chosen to treat the case of solar arrays
connected to constant resistive loads equal to that
needed for maximum power output when there is no
shading. Correctly designed systems should not have
loads fluctuating signitìcantly from this value.
(1) Series conjìgurution. For N identical cells con-
nected in series and one cel1 completely shaded, Kirch-

63
2
,..’
I
10 20 x) 40 50 so 70 80 90 100
SH4DIffi 1 PERCENT)
Fig. 3. The ratio of the power los.9 to the unsbded power output
a3 8 fmction of slmdiag for the simple series conmction witb
and without ideal shunt diodes and lor the theoretical minimum
power loss case.
hoff’s voltage law requires
(N - l)(Vr - I&&) + (VS - IR??&) = 0, (3)
where VZ and VS are the voltages across the illuminated
and shaded cells respectively. Figures 1 and 2 were used
to obtain plots of (VZ - Iz&) VS IZ and of ( Va - 18Rm)
VS Is, where Ir and Is are the current of the illuminated
and shaded cells respectively. These were used to obtain
trial and error solutions to Equation (3) so that
Ir = Is = 1 for N values between 2 and 50.
Since N = 2 represents 50 per cent shading, N = 3
represents 33.3 per cent shading, etc. these results have
been plotted in Fig. 3 (solid curve) as the ratio PL/Po VS
per cent shading. Here PL is the power lost with shading
and PO is the initial unshaded power output. The power
values were obtained from
and
Po = NV,,& (4)
PL = NV,& - [(N - 1)Vz + Vs]l. (5)
Results for shading between 50 and 100 per cent were
determined by considering 10 cells in series and 6-9
cells shaded to give the 60-90 per cent data points.
For this, equation (3) was modified for multiple cel1
shading and solved by trial and error. Note that power
loss is essentially 100 per cent if shading is 12.5 per cent
or more because shaded cel1 current (Fig. 1) is negligible
unless a voltage greater than - 2.0 V is applied. This
required eight or more illuminated cells (12.5 per cent
or less shading).
As shading decreases to 2.8 per cent the power loss
drops to 35 per cent and the load current rises to ap-
prox. 480 mA. At this value the shaded cel1 drops
- 4.75 V so that the shaded cel1 absorbs 2.28 W instead
of delivering the 0.22 W available when it was unshaded.
This power dissipation heats the shaded cel1 and produces
the so-called ‘hot spot’ [8]. Note that the shape of the
reverse characteristic strongly affects this power loss [5].
Cells with lower reverse currents result in greater power
loss for a given current value.
As N approaches itinity (shading approaches 0 per
cent), (VS - IRm) < (N - l)( VZ - IRm) and Equation
(3) requires that 1 be equal to V./R, = Im. At this
(bl
Fig. 4. Array co~tions for (a) the simple series amay witb
shunt diodes and @) the simple parallel army witb series diodes.
value the reverse voltage across the shaded cel1 is
- 5.1 V. This means that the maximum power dissipated
by the shaded cel1 in a series string under worst conditions
is 3.06 W.
This reverse voltage and power absorption can be
completely eliminated if an ideal diode is placed in
parallel with each series 1~11, as shown in Fig. 4(a)
[2, 31. Shunt diodes short out any reverse voltage and
shunt current around shaded cells. The positive voltage
of illuminated cells reverse bias their diodes so that
unshaded cells are unatfected. Kirchhoff’s law for this
case is the same as Equation (3) except that now V. is
zero so that
(N- 1)
Vz-I$$] =o,
must be satisfied. This was solved directly by plotting
the load line R = NRm/(N - 1) on the I-V charac-
teristic (Fig. 2) and obtaining VZ and 1 from the inter-
section of the curves. The results are shown by the
dashed line curve in Fig. 3 as the ratio PLIPo VS per-
centage shading where Po is the same as Equation (4)
but
PL = NV,,& - (N - 1)VzI. (9
As can be seen, shunt diodes drastically reduce power
loss for low percentage shading. With 50 per cent shading
the loss is 65 per cent compared to 100 per cent without
diodes. For 10 per cent shading, PL/Po is 11 per cent
rather than 96 per cent without diodes, a factor of 8.8
reduction in lost power. As percentage shading decreases
further, this reduction factor grows smaller and eventu-
ally approaches one at zero shading. One might conclude
that diodes are unnecessary for large arrays where
per cent shading should remain smal1 but the diodes
prevent ‘hot spots’. As percentage shading decreases
the ‘hot spot’ heating grows worse eventually approaching
a constant value asymptotically (of 3.06 W per shaded
cel1 in our case) for very large arrays with very small
percentage shading.
Real diodes of course have some forward bias voltage

64
MOHAMED SAYED and LARRY PARTAIN
drop so that the actual shunt diode reduction in power
loss is not as great as shown in Fig. 3 for ideal diodes.
This real voltage drop is nevertheless much less than the
- 5.0 V of the example discussed for cells without
diodes. Comparing power absorption at 580 mA, a
diode with 0.2 V drop would dissipate 96 mW rather
than the 2.28 W occurring with the unshunted cell, a
factor of 24 reduction even with real diodes. Since
germanium diodes have approximately a factor of three
less forward voltage drop than silicon (0.2 compared
to 0.7 V typically), germanium should provide the best
shunt diode protection.
A calculation of the theoretically minimum power
loss possible when one cel1 is shaded provides a useful
standard for evaluating various connection schemes.
Power loss to the shaded cel1 is avoided by shunting it
with an ideal diode. To obtain the ideal minimization,
the other unshaded cells should be maintained at their
maximum power points. This is accomplished by reducing
the load resistance to a value of (N - l)R,. This then
gives the lowest power as, using Equation (5)
PL = NV&, - (N - l)VmZ, = VmIm, (8)
so that
PL 1
_ =_,
Po N
(9)
under the best possible circumstances. This is shown as
the dotted line in Fig. 3. As percentage shading drops
from 50 to 10 per cent, the minimum power loss de-
creases from 50 to 10 per cent compared to the 65-11 per
cent decrease with shunted cells. This means that a
variable load could produce an additional factor of
1.3 reduction in power loss for 50 per cent shading
and a factor of 1.1 at 10 per cent shading. This ratio
of real to theoretical minimum loss becomes less than
1.01 for shading below 6 per cent. Hence a variable
load cannot reduce power loss appreciably for low
shading percentages.
(2) Parallel conjîguration. With M identical cells
connected in parallel, Section 2B gives the load resistance,
voltage and current for maximum power output with
no shading. With one cel1 shaded Kirchhoff’s law again
requires
- (M-l)(II-;)+(I,-;)=O* (l”)
Notice that Equation (10) can be obtained from Equation
(3) by replacing N by M, Vl by ZI and I by V. This is
essentially the well-known duality theorem. Plots of
(11 - VZ/&) VS VZ and of (Is - V,I&J VS VS were
obtained from Figs. 1 and 2 and trial and error solutions
to Equation (10) were found so that VZ = VS = V for
M values between 2 and 50. These results are plotted
as the solid curve in Fig. 5 for shading levels between
0 and 50 per cent. The values for 50-100 per cent shading
were obtained for 10 cells in parallel by modifying
Equation (10) to apply for shading 6-9 cells.
The power loss varies from 69 to 12 per cent as per-
centage shading decreases from 60 to 10 per cent. This
contrasts sharply with the series case which had 100 per
cent power loss for 50 per cent shading that only de-
creased to 96 per cent at 10 per cent shading (Fig. 3).
Obviously, simple parallel connection results in much
less power loss with shading than simple series con-
nection.
There is some power absorption by a shaded cel1 in
parallel with other illuminated ones. The unshaded
ones maintain a positive voltage across the shaded
cel1 (except for P configuration ‘hot spot’ formation
which is discussed later). Figure 2 shows that such
positive voltage results in some current flow in the
shaded cel1 and biases it in the power absorbing fitst
quadrant. This partially shorts out the photocurrent
generated in the illuminated cells. In contrast to the
series case, the shaded cel1 voltage and current both
remain low. With VS around 0.3 V, Is is less than 25 mA
(Fig. 2) so that less than 7.5 mW is absorbed compared
to absorption as high as 2.28 W calculated for the
series case. Resultant heating is insignificant in com-
parison. The forward characteristics do not strongly
affect this power loss unless the cel1 becomes a low
resistance short circuit.
As M approaches infinity, percentage shading ap-
proaches zero and the operating current and voltage
StWJBNG ( PERCENT)
Fig. 5. The ratio of the power loss to the uushaded power output as a
function of shading for the simple parallel connection with and without
ideal series diodes and for the theoretical miaimmn power loss case.

Effect of Shding on CdS/CuaS Solar Cells and Optimd Solar Army Design
65
approach their values before shading (similar to N + co
for series cordìguration). For this worst condition case,
the shaded cell dissipates 12.6 mw, a factor of 17.5 less
than its unshaded power production. This contrasts
sharply with the series case where shaded power loss
for the worst case was over an order of magnitude larger
than the unshaded power production.
This smal1 loss can be eliminated by the addition of
ideal diodes in series with each cel1 and oriented so
that only forward current can flow as seen in Fig. 4(b).
(As is traditional, a forward photocurrent is plotted as
a negative value in Figs. 1 and 2.) Application of Kirch-
hoff’s voltage law gives the same result as Equation (10)
except that now I8 is zero. This can be rewritten as
(11)
This was solved directly by plotting the load line
R=R,(M- l)/M on the 1-V characteristic for the
various M values. As before this expression was modified
for multiple cel1 shading to obtain values for shading
between 50 and 100 per cent. The diode reduction in
power loss was so small that the results, either with or
without diodes, are shown by the same solid line in
Fig. 5. This is due to the shaded ccll’s smal1 current
value (less than 35 mA) compared to the illuminated
cel1 current (over 600 mA). The cel1 voltages for both
cases are comparable.
The power loss that occurs in real series diodes makes
their use impractical. For a forward cc11 current of
600 mA through an illuminated cel1 and diode, a forward
voltage drop of 0.2 V across the diode would drop
120 mW at each diode. This is approximately half the
power generated by the cell. Such 50 per cent power loss
with no shading at al1 is of course unacceptable.
The theoretical minimum power loss for M cells in
parallel would be obtained with ideal diodes in series
with each cel1 and a variable load that would change
from R,,,/M to R,,,/(M - 1) when one cel1 is shaded.
This would maintain maximum power output from the
remaining cclls and give the theoretical minimum power
loss ratio as
PL 1
_=_*
Po M (12)
This minimum loss is shown as the dashed line in
Fig. 5.
Connecting cells in parallal is equivalent to increasing
the surface area of a single cell. Thus the results of
Fig. 5 apply to partial shading of a single cel1 as wel1
as parallel cells. In particular, Fig. 5 should correctly
describe power loss due to shading by the conducting
grid placed over the front of solar cells. Actual measure-
ment on a CdS/Cu,S cel1 with the grid shaded area
reduced from 15 to 5 per cent showed a 20 per cent
increase in output power [15]. This agrees with the
solid curve of Fig. 5 to within 3.5 per cent.
(3) Combination series and parallel arrays. The large
decrease of power loss with shading for simple parallel
strings of cells rather than series strings make the former
WIDTH-hl (bl
Fig. 6. Large array cxmnection schemes for (a) the basically
parallel or P array witb shunt diodw and (b) the basically series
or s array witb shunt diodes.
configuration attractive. However, the voltage output
for this arrangement (approximately 0.36 V) is too
low for practica1 applications. Series connection must
be used to obtain useful voltage output levels. If enough
cells are available to provide a given voltage, the question
arises as to how to best connect in additional cclls.
Two possibilities immediately present themselves and
are shown in Fig. 6. The primarily parallel arrangement
of Fig. 6(a) is called the P coníìguration, and the more
series layout of Fig. 6(b) is termed the S cotiguration.
A series of specific examples have been taken to
analyze the characteristics of these two connection
schemes. Arrays of N cells high (2 I N I 10) and
M cells wide (2 I M 2 10) are the basis of this analysis.
The N cel1 height is connected in series to give a desired
voltage and the M cel1 width is connected in either the
P and S coníìgurations to give a desired current. The
amount of power lost when one cell was shaded was
calculated for a constant resistive load. By extending
the results of Section 2B, the load resistance for maximum
power output is NR,/M for both coníigurations with
no shading. This load value was used for each case
considered which gave an unshaded output power of
MNP,,,.
(a) P conzguration. For the P configuration, shading
a single cel1 lowers the current output from the parallel
section in which it is located. Since this section is in
series with the other (N - 1) non-shaded sections, the
total current is decreased. Kirchhoff’s voltage law
requires that
(N - 1)Vr + Ve -
MII(NR,/M) = 0, (13)
where VS is the voltage across the shaded cel1 and the
(M - 1) fully illuminated cells in parallel with it. The