IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 3, MARCH 1979 165

Application of the Superposition Principle to Solar-Cell Analysis

Abstract-The principle of superposition is used to derive from funda- mentals the widely used shifting approximation that the current-voltage characteristic of an illuminated solar cell is the dark current-voltage characteristic shifted by the short-circuit photocurrent. Thus the derivation requires the linearity of the boundary-value problems that underlie the electrical characteristics. This focus on linearity defines the conditions that must hold if the shifting approximation is to apply with good accuracy. In this regard, if considerable photocurrent and considerable dark thermal recombination current both occur within the junction space-charge region, then the shifting approxima- tion is invalid. From a rigorous standpoint, it is invalid also if low- injection concentrations of holes and electrons are not maintained throughout the quasi-neutral regions. The presence of sizable series resistance also invalidates the shifting approximation. Methods of analysis are presented to treat these cases for which shifting is not strictly valid. These methods are based on an understanding of the physics of cell operation. This understanding is supported by labo- ratory experiments and by exact computer solution of the relevant boundary-value problems. For the case of high injection in the base region, the method of analysis employed accurately yields the depen- dence of the open-circuit voltage on the short-circuit current (or the illumination level).

LIST OF SYMBOLS Electron and hole diffusivities. Excess electron and hole densities. Electric field. Dielectric permittivity. Optical generation rates of electrons and holes. Steady-state net generation current source

Solar-cell current density, current. Dark diode current density, current (subscript 0

Electron and hole current densities. Short-circuit current density, current. Uncompensated photocurrent density. Boltzmann's constant. Electron and hole mobilities (b = p n / ~ p ) , Electron and hole densities. Ionized acceptor and donor impurity concentra-

(A/cm3).

denotes saturation current).

tions.

Manuscript received March 1977; revised May 26, 1978. This work was supported by the United States Department of Energy and by the National Aeronautics and Space Administration.

F. A. Lindholm and J . G. Fossum are with the Electrical Engineering Department, University of Florida, Gainesville, FL 32611.

E. L. Burgess is with Sandia Laboratories, Albuquerque, NM 87185.

ni Intrinsic carrier density.

RN, Rp Electron and hole steady-state recombination 4 Electron charge.

rates for thermal processes. RS Series resistance. P Space-charge density. T Absolute temperature. V Solar- cell terminal forward voltage. VB Quasi-neutral base region voltage. VZ Intrinsic Fermi potential (or electrostatic

VJ Junction space-charge region voltage. potential).

VN, Vp Electron and hole quasi-Fermi potentials.

C I. INTRODUCTION

ONVENTIONALLY, the current in an illuminated solar cell is described by the dark (nonilluminated) current-

voltage characteristic ID(V) shifted by the short-circuit current Isc :

I( V ) = -ID( V ) I- Isc. (1) This description has served as a basis for many past theoretical estimates of various measures of solar-cell performance. The separation of I ( V ) into the components of (1) is useful because it reduces the problem of characterizing the illuminated cell connected to an arbitrary load to two simpler problems: that of describing the current I ~ c of the illuminated cell connected to zero load (short circuit), and that of describing the current- voltage dependence ID(V) of the nonilluminated cell under applied forward voltage.

As we show in this paper, however, the fundamental reasons justifying (l), which we term the shifting approximation, hold only under restricted conditions. The use of the shifting ap- proximation beyond its range of validity can lead to sizable errors in predicting the electrical characteristics of solar cells and thus to misleading conclusions about design.

The paper has several purposes:

a) to define the range of validity of (1) from fundamental consideration of the relevant boundary-value problems;

b) to note instances of practical importance for which the use of (1) will yield erroneous conclusions concerning cell performance;

c) t o illustrate, by laboratory experimentation and by exact computer solution of the pertinent boundary-value problems, the errors that can result from the use of (1);

0018-9383/79/0300-0165$00.75 0 1979 IEEE

3 66 IEEE 'TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 3, MARCH 1979

and to develop approximate analytic expressions F o x cell performance, based on the device physics, that: hold when (1) fails to apply.

Certain important limits on the validity of (1) were recopized by Wolf [ l ] . But no rigorous derivation has yet beer1 done that fully exposes these limits. This paper is meant to I: rovide such a derivation. The derivation is based on the prin,cfple of superposition of differential-equation theory, which apltlies if and only if the relevant boundary-value problems are 1inc;tr.

Following this derivation, we consider several examrdes of cell operation for which superposition and the shifting approxi- mation of (1) do not apply. In these examples, we treat the effects of: a) optical generation and net thermal recor-tbina- tion in the junction space-charge region; b) high4njectic;ta and moderate-injection concentrations of holes and electron;; and c) series resistance. The treatment of high injection relaxes to cells in concentrated sunlight or to cells in normal sunlight that have a light base doping concentration. Exact corrI:mter solutions of the boundary-value problems underlyinl: cell operation together with laboratory experimentation are used to support the theoretical developments.

11. APPLICABILITY OF THE SUPERPOSITION PRINCIPLE TO SOLAR CELLS

If a system is linear, its response to several excitations 1.9 the sum of the responses to each excitation applied alone. This is the principle of superposition, which is a basic theorlern of linear differential equations [2] and which is widely used i:n wch fields as electrical circuit theory [3] and control systems 1'41.

We now apply the principle of superposition to a p-n junc- tion solar cell. The fundamental equations describing this system in dc steady state are [5], [6]

JN = t qD,VN + qpnNE (2)

= -q&NVVN 6:2a)

Jp = -qDpVP + qppPE (3)

= -qppPVVp i3a)

O = + V . J N + I S S = V . J N + q ( G o - R s s ) (4)

O = - V . J p t I s ~ = - V . J p + q ( G O - R s S ) (5)

V E = -V2V1 (P/E) = (q/e)(P - N + NDD - NAA) (6) J = JN + Jp. (7)

The List of Symbols contains the definitions of the syrrlI:jols in (2)-(7). In [6] , the functional dependence of R J : ~ : is discussed for various special cases such as Auger and Shockley- Read-Hall recombinations.

In general, this set of equations is nonlinear. Thus super- position does not apply unless approximations are made that remove the nonlinearity from the system description.

The first approximation to be made follows the wiiiely used Shockley method of analysis [5] which divides the cell into three coupled systems: the quasi-neutral emitter (QidE) and quasi-neutral base (QNB) regions, and the junction spi~ce- charge region (SCR) that separates them. The total cur m t density J drawn from the cell results from the recombination-

Fig. 1. Onedimensional model of a p-n junction solar cell.

1

L I r J BACK CONTACT

Fig. 2. Equivalent circuit of the solar cell illustrating that the Kirchhoff current law holds regardless of the applicability of superposition.

generation processes in these three regions and at the front and back surfaces. To demonstrate this for the one-dimensional n+-p junction solar cell of Fig. 1, consider

where the integral is the net generation current from the bulk material of the solar cell and the other two terms are net generation currents at the front and back surfaces. If the terms are grouped as

we see that the current density J is the net generation current density JQNE from the quasi-neutral emitter, including both bulk and surface components, its counterpart JQNB from the quasi-neutral base, and the net generation current J ~ C R from the junction space-charge region; that is

Equation (1 1) is analogous to Kirchhoff's current law of electric circuit theory [3] , as is illustrated in the equivalent- circuit representation of Fig. 2. Equation (1 1) and Fig. 2 are valid, in general, regardless of whether or not superposition can be applied to the system describing the solar cell.

LINDHOLM et al.: SUPERPOSITION PRINCIPLE IN SOLAR-CELL ANALYSIS 167

We now examine the applicability of superposition to each of the regions of the cell. Consider first the quasi-neutral emitter and base regions. If low-injection conditions prevail in the quasi-neutral regions and the effects of series resistance are negligible, then the expressions for the minority-carrier current [SI and the net steady-state thermal recombination rate Rss [6], [7] both become linear in the minority-carrier concentration. Thus the continuity equation for minority carriers (4) is linear, and the principle of superposition applies. The linearity of the system holds in general despite the presence of built-in electric fields arising from impurity concentration gradients [ l ] . It holds also despite heavy-doping effects in the emitter (such as energy-band-edge distortion) because the models commonly used to describe these effects [6] introduce parameters, such as the effective intrinsic density or the effective energy band gap, that are independent of the depen- dent variables, P and N.

To illustrate the use of superposition, consider the p-type base region of the solar cell of Fig. 1. For this system the independent excitations are the optical generation rate Go'(x) and the excess minority-carrier (electron) concentration AN(0) at the plane x = 0, corresponding to the edge of the space-charge region in the base. The response is the minority- carrier base current JQNB of (1 1). From (10) and (1 1)

JQNB = 4 I,,, (GO - ~ s s ) dx +JN(wB) . (12) The integral in (12) is linearly related to the excess minority- carrier concentration AN and to the optical generation rate Go, and JN( W,) is linearly related to AN and Go if the surface recombination velocity S, at the back face is constant. Thus superposition yields

JQNB =JQNB 1 c8=o +JQNB 1 G ~ f O . (1 3) The first term of (13) is a dark current - J ~ N B . The second term is a photocurrent J ~ N ~ . Therefore

JQNB = -JGNB + J ~ N B (14)

which can be represented by the equivalent circuit of Fig. 3. This decomposition of JQNB focuses attention on the dark- current component, which is given by [ 5 ]

JGNB = J $ N B O ( ~ 9 V W - 1) (1 5 )

for the assumed conditions of low injection and negligible series resistance. In (15), J&BO denotes the saturation current (pre-exponential factor) for the dark recombination current from the quasi-neutral base, which can be determined by a graphical procedure [SI, [ 9 ] from the measured I-V characteristics.

Analogous results describe the current JQNE from the quasi-neutral emitter. Thus if superposition applies, as it does for low-injection conditions, the components of current from the two quasi-neutral regions have the form of (1); it con- sists of a dark-current characteristic shifted by a photocurrent.

We now consider the applicability of superposition to the junction space-charge region. In this region, the electric field E depends on the hole and electron concentrations, N and P.

AN(O)#O U ( 0 ) = 0

4 (BACK CONTACT1 Fig. 3. Equivalent circuit for the quasi-neutral base if superposition

applies. A similar equivalent circuit represents the quasi-neutral emitter.

(FRONT1

P

l i J 6

@ACU CONTACT1

Fig. 4. Equivalent circuit for the entire solar cell.

This introduces nonlinearity into the current-density expres- sions, ( 2 ) and (3). Furthermore, the net thermal recombination rates, RN and Rp, are nonlinearly related to N and P. There- fore, the continuity equations for holes and electrons, (4) and (S) , are both nonlinear, and superposition does not apply to the junction space-charge region.

The results of this section are illustrated by the equivalent- network representation of the solar cell in Fig. 4. The dashed lines that connect the current generators representing the quasi-neutral regions are closed or open depending on whether superposition is applicable or not in the quasi-neutral regions. The breaks in connection in the space-charge region represen- tation indicate that superposition is never valid in the junction space-charge region.

Although the principle of superposition never applies to the junction space-charge region, the inapplicability does not necessarily invalidate the resolution of the solar-cell current- voltage characteristics indicated in (1). Using Fig. 4, we now consider special cases that illustrate this.

A. Cases for which the Shifting Approximation of (1) Applies Case A : Here the quasi-neutral regions dominate in deter-

mining both the photocurrent and the dark current. Low injection prevails throughout the quasi-neutral emitter and base, and series resistance is negligible. Thus the dashed lines in Fig. 4 are short circuits, indicating that superposition

168 IEEE TRJLNSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 3, MARCH 1979

applies. From Fig. 4

2 J&R - J ~ N O [exp (qV/kT) - 11 I: 16) which is consistent with (1).

Case B: Here the conditions are those stated for c8aia A except that the space-charge region now also contrilutes significantly to the photocurrent. From Fig. 4

Case C: Here the dark current of the cell is domir.ated by the space-charge region and the photocurrent is domir.ated by the quasi-neutral regions. Series resistance is negliijble. Thus from Fig. 4 and the theory of dark p-n junction diodes [7], we have

J = J ~ N R - Jgco [exp (qV/m,,kT) - 13 I: 18) where msc is the reciprocal slope factor that characterizes Shockley-Read-Hall recombination in the junction ,,I (1 )ace- charge region (1 < msc < 2).

Case D: Here the conditions are those stated in C I ~ I ; E : A except that the space-charge and quasi-neutral regions 30th contribute significantly to the dark current. Thus from Fig. 4 and [7]

J = JgNR - {JgNO [exp (qV/kT) - 11

+ Jfco [exp (4Vlmsc k T ) - 1 1 1 (: 19) again in agreement with (1).

has the simple form Discussion: For cases A, B , and C, the open-circuit vo iage

where m = 1 for cases A and B, and m = m,, for case C. 'The open-circuit voltage for case D is the solution for V of (19) when J = 0, but no simple explicit expression for Voc exi,s ::;.

The cases just described have practical importance. For example, cases A and B can describe a p-n junction solar cell made on a single-crystal silicon base having light or rlod- erate doping concentration. Cases C and D can describe single-crystal silicon cells having highly doped base regions (PBASE - 0.1 i2 . cm [8] ) or polycrystalline silicon cells.

Experimental Results: To demonstrate the validity of [18) and (19) for cases C and D, we measured the illuminated and dark I-V characteristics of a shallow diffused junction n-1:Nn-p silicon solar cell having a highly doped base region ( N ~ A = I O1 cm-'). For this cell, the emitter and the junction srme- charge regions are too thin to contribute significantly to the photocurrent, which thus originates predominately in the quasi-neutral base region. At voltages V near V& for me- sun illumination, the space-charge and quasi-neutral regions both contribute to the dark current (case D). For the current range resulting from an illumination level of l/lOO sun, however, the recombination in the space-charge region ; w n e determines the dark current (case C). Our measured Slata showed excellent agreement with (1) for both levels 01' il- lumination, which is consistent with our predictions basecl on the principle of superposition.

B. Cases for which the Shifting Approximation of (1) Does Not Apply

Any of the following conditions will invalidate the super- position principle and thus (1):

a) the junction space-charge region contributes importantly to both the photocurrent and the dark current; or

b) the carrier concentrations in the quasi-neutral regions exceed low-injection levels; or

c) the series resistance contributes significantly to the cell current-voltage characteristics; or

d) the material parameters, such as the carrier recombina- tion liftetime, depend on the illumination level; or

e) the volume of the regions producing the photocurrent changes appreciably as the cell is loaded (and the terminal voltage V changes).

The invalidity of (1) for CdS cells has been discussed and shown experimentally [ 101 . The invalidity of (1) for amor- phous silicon (a-Si) cells [ l l ] and the consequent poor fill factor probably arises, in part, from the decrease in volume of the junction space-charge region that accompanies loading of the cell. Because the diffusivity and hence the diffusion length in the quasi-neutral regions of a-Si cells are small, the photocarriers generated in the junction space-charge region dominate in determining the photocurrent I . Thus the invalidity of (1) for a-Si cells owes its origin in part to condition e) listed above and, in part, to condition a). In single-crystal silicon, condition b) can produce condition d) because the carrier recombination (Auger or Shockley- Read-Hall) lifetime is injection-level dependent [6]. Such effects can occur at AM0 illumination for base resistivities, as low as 10 i2 . cm and for much lower base resistivities if the cell is illuminated by concentrated sunlight. Condition d) can also result if the extrinsic portion of the solar spec- trum changes the occupancy of the recombination centers, thus causing the carrier lifetime in the dark to differ from that under illumination [6].

Other instances of the invalidity of (1) for single-crystal silicon cells will be discussed in the following sections, which treat the solar-cell characteristics for cases b) and c).

111. HIGH INJECTION I N THE QUASI-NEUTRAL BASE One important example of high injection occurs in the

quasi-neutral base region of a solar cell exposed to concen- trated sunlight [ 121 . Despite the inapplicability of super- position to this cell from a strict standpoint, an analytic description based on superposition can be developed for the case of high-level injection for which P --N > > N A ~ in the base region. For this case, as will be discussed, superposition yields

J - J d O ) = J N ( 0 ) AN(O)=O

= -Jo + Jupc. (21) In (21), JD is the dark current-voltage characteristic of the cell. The term J ~ p c in (21) is the photocurrent in a highly illuminated cell to which a reverse bias is applied to insure AN(0) = 0; JUpc is essentially the maximum current that can

LINDHOLM e l al.: SUPERPOSITION PRINCIPLE IN SOLAR-CELL ANALYSIS 169

v,, IVI O'*i 0.75 UL

Fig. 5

P n J b + l q UPC -

0.55 - - -

I I I I I1111 I I I I I 1 1 1 1 1 I I 1 10 50

Jupc IAkm'l

Superposition-based theoretical dependence of VOC on Jupc showing invalidity of (1) for high injection but showing agreement with values (points) obtained from computer solution of the differential equations.

be drawn from the solar cell for a given level of illumination. We thus call it the uncompensated photocuwent. In general,

The result of superposition shown in (21) is an approxima- tion. A strict equality in (21) would require strict linearity in the continuity equations, (4) and (9, for N and P and in the relations, (2) and (3), between JN and N and Jp and P.

To explore this linearity, it is instructive to consider the ambipolar transport equation [ 131 , [ 141

JUPC > Jsc.

dAN d2AN O=Go - Rss + p a E z +Da -

dx

which is derived by combining (4) and ( 5 ) and by using the condition from quasi-neutrality that AN = AP. In (22)

In general, (22) is nonlinear in AN. We now consider a solar cell in which the base region is

homogeneously doped (NAA = constant) and in which Rss 0 in the quasi-neutral base, which is a good approximation for a well-designed solar cell in which high injection occurs for illumination levels ranging from 25 to 100 suns [ 121 . Then, in both the low- and high-injection limits, (22) reduces to the form

dZAN O=Go+D,-

dx

where D, = D, for low injection and D, = DH = 2 D,D,/ (Dn t D,) for high injection. For low injection, the response JN becomes

which is also linear in AN. For high injection, comparison of (24) with (4) implies that

if Go = 0 in the quasi-neutral base, except near the space- charge region [ 1 2 ] . In (26), C is a space-independent con- stant; C --l for the condition, J = JN(O), assumed in (21).

Thus for silicon, for which D, - D,, in both the low- and high-injection conditions, the ambipolar transport equation for AN and the current equation for JN are both linear in AN, they have exactly the same form, and they have coefficients that are the same order of magnitude. It follows, therefore, that the quasi-neutral base region can be described approxi- mately by a single set of linear equations even though the injection condition across the base region varies from high- to lowlevel injection; this variation in injection level is implied by the boundary condition in (21) that AN(0) = 0 and by the possible boundary condition AN(WB) = 0 (corresponding to an ohmic contact or high surface recombination velocity at the back face). Because the quasi-neutral base region can be described by a single set of linear equations, the superposition principle can be applied and (21) results.

Using (21) and including effects of high injection in the base region, one can derive analytical expressions for the open- circuit voltage VOC of highly illuminated silicon solar cells [12]. In Fig. 5, values of Voc calculated from these ex- pressions for 1042 * cm n+-p cells [ Voc proportional to (2/b t l ) (kT/q) In (Jvpc/JDo)] and 4-Q . cm p*-n cells [ VOC proportional to (2b/b t l ) (kT/q) In ( J u p c / J ~ o ) ] are plotted against J u p c ; (b = Pa//+). These dependencies of VOC on J u p c (or, to a first approximation, on JSC) are com- pared at several levels of illumination with values of Voc calculated from exact numerical solutions of (2) through (7). These results are contrasted with the classical relation, derived from (1) for low-injection conditions

VOC = (kT/q) In (JSClJDO) (27) which is a special case of (20). At low-injection levels, Ju,:

170 IEEE TRAIiSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 3 , MARCH 1979

I

1.2 - 0 - 0.3 Q-cm lSOlO5A-lll 1

Fig. 6. Experimental dependence of VOC on Jupc normalized to one- sun values. V O C , ~ and Jupc, x designate the open-circuit vaJtage and the uncompensated photocurrent at X suns. Note the agreenimt with the theoretical results of Fig. 5.

Jsc; otherwise, Jupc >J,c, because a considerable potential difference can develop across the quasi-neutral base [ 121. The inapplicability of (27) is apparent in Fig. 5 . The method of analysis, based on the superposition principle that underlies (21), gives the needed correction and provides the theoret ml dependence of VOC on Jsc (or .Iupc),

These theoretical results are supported by experimental d&ta (Fig. 6) plotted in normalized fashion against the measured Jupc for three types of n+-p solar cells: a cell having a bflse resistivity of 0.3 n * cm, and two cells having a base resis- tivity of 10 8 * cm, one made with a back-surface field (%SF) and the other without a BSF. At high levels of illuminat on (large Jupc), high injection occurs in the base of the 1042 ::an cells at the open-circuit condition. For the 10-8 - cm d l without a BSF, the measured dependence of VOC on JL;VC agrees with the theoretical predictions of Fig. 5 and demm- strates experimentally that (27) is invalid. The VOC depw- dence observed for the BSF cell again demonstrates ..he inapplicability of (27) . The nonlinearities associated with high injection in the base region of this cell present ;m analytical problem different from that describing a non-B!iiF cell. Basic corrections of classical theory required by these nonlinearities will be treated in a later publication.

High injection does not occur in the 0.342 cm cell, and i t s VOC versus Jupc characteristic agrees with the classical res& of (27) .

IV. SERIES RESISTANCE In a p-n junction solar cell, series resistance Rs can originate

in the metal contacts, in the metal-semiconductor interface, in the emitter, and in the base. The presence of R s invalida..::s the direct use of superposition. To illustrate this, considx, as the system under study, a p-type quasi-neutral base regic 11. Though the electron continuity equation (4) remains linear for low injection, the presence of R s results in a nonlinear bour (1- ary condition relating AN and JN

W O ) a [exp ( q V / W exp (41Rs/kT> - 11 ( 2 8) where I depends on JN(0). The nonlinearity of (28) precludes

INTRINSIC STRUCTURE

P

(b) Fig. 7. (a) Entire solar cell in which AN(0) depends nonlinearly

on I . (b) Intrinsic solar cell in which AN(0) is proportional to [exp W I s l k T ) - 11.

the use of superposition in which AN(O), defined by (28), is regarded as an independent excitation.

A solar cell with significant Rs can be treated analytically by separating Rs from the intrinsic system, as in Fig. 7. The intrinsic system is defined so that

A N O ) a [exp (4V ls lW - 11 . (29) Since AN(0) thus defined depends only on the voltage VIS across the system, it can be regarded as an independent excita- tion. Thus the intrinsic system of Fig. 7 can be treated as described in Section 11, where we indicated the conditions for which superposition applies.

Use of a superposition requires that the excitation AN(0) be set to zero, which defines the photocurrent J u x . As for the high-injection condition described in Section 111, here also a short circuit applied across the cell terminals fails to make AN(0) = 0 because

V = VIS - I R s . (30)

From (29) we see that AN(0) = 0 implies V, = 0, and from (30) that a reverse bias of about IupcRs must, therefore, be applied to the solar cell to experimentally determine Jum. We note again that Jupc > J ~ c , where J ~ c = J( V = 0).

For low-injection conditions, Rs is approximately constant, independent of I in an illuminated cell, and can be determined experimentally [ 151 , [ 161 .

A similar method of analysis will extend the applicability of the superposition principle to solar cells whose characteristics are affected by shunt resistance.

V. DISCUSSION The basis underlying this paper is the superposition principle.

Its use here provides insight about the shifting approximation of (1) that the characteristic of an illuminated solar cell is the dark current-voltage characteristic shifted by the short- circuit current. Interpreted from the standpoint of the super- position principle, the current I ( V ) in (1) is the sum of the responses to two excitations applied independently, one at a time. One excitation is the hole-electron optical generation rate resulting from the sunlight. The other is the excess carrier concentration at each edge of the junction space- charge region, which is related to the voltage difference

LINDHOLM et al.: SUPERPOSITION PRINCIPLE IN SOLAR-CELL ANALYSIS 171

across the cell terminals. The two responses are, therefore, the short-circuit current and the dark-current characteristic.

The optical generation rate and the excess carrier concentra- tion at the space-charge-region edges are chosen as the ex- citations because they are the independent variables in the boundary-value problem describing the cell. These variables are linearly related to the current and to the excess carrier concentrations within the cell under certain well-defined conditions which, therefore, are necessary for superposition to apply. These conditions are:

a) the junction space-charge region may contribute impor- tantly to either the photocurrent or the dark current, but not to both;

b) the carrier concentrations in the quasi-neutral regions must stay within low-injection levels;

c) the series resistance (and shunt resistance) must contri- bute negligiljly to the cell current-voltage characteristics;

d) the material parameters, such as the minority-carrier lifetime, must be essentially independent of the illumina- tion level; and

e) the volume of the regions that contribute appreciably to the photocurrent must stay essentially constant as the cell is loaded.

We have described how a superposition-based theory can be developed even if conditions b) or c) are violated. The viola- tion of condition b) discussed here relates to an application of much practical importance: the performance of silicon p-n junction cells in concentrated sunlight. Although, in a strict sense, the superposition principle is invalid for this application, we used superposition in developing the cor- rection. This use led to a theory for the dependence of VOC on ISC that agreed well both with laboratory experimenta- tion and with the exact computer solution of the underlying differential equations (2)-(7).

This is the first application of the superposition principle of differential-equation theory to the analysis of solar cells, although the principle has been used previously to describe the saturated operation of the bipolar transistor [ 171 , [ 181 . We believe that this principle offers insight about the range of validity of (l), the shifting approximation, and about the considerable simplifications that (1) yields. Beyond this, the treatment discussed here of a cell in concentrated sunlight has shown that superposition provides a basis from which to develop expressions useful for design when superposition, from a strict standpoint, fails to hold. We anticipate that the superposition principle also will prove useful in applications other than those discussed in detail here. Other treatments [19], [20] of the shifting approximation and superposition are now appearing, prompted by an early conference version [21] of our present paper.

ACKNOWLEDGMENT The authors wish to thank Prof. C . T. Sah, Prof. M. Wolf,

Prof. R. J. Van Overstraeten, and Dr. B. Williams for helpful discussions. We also thank the anonymous reviewers who indicated several places in the manuscript that needed major revisions.

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(31 D. F. Tuttle, Network Synthesis. New York: Wiley, 1958, p. 19. H. H. Skilling, Electrical Engineering Circuits. New York: Wiley, 1965, p. 353.

[4] J. L. Bower and P. M. Schultheiss, Introduction to the Design of Servomechanisms. New York: Wiley, 1958, pp. 11, 100.

[5] W. Shockley, The theory of p-n junctions in semiconductors and p-n junction transistors, Bell Syst. Tech. J . , vol. 28, pp. 435- 489, July 1949.

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- , The silicon solar cell design handbook, in Rec. 9th IEEE

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