Table 3 VALUES OF Amin ATINTERPOLATION POINTS

•vo J'o -oxi Joro•YoJi ~o•voJ'o-i

22-2 V19 5 V190 V120 V15-7 V

J'o-t

•Y2 J'o - 0

• v o> 'o ; -2

8-5 V8-9 V

17-2 V150 V90 V

nomial of second degree in three variables:

P 2 ( x y z ) = 2 2 - 2 + 2 - 8 8 ( z - z 0 ) - O - 3 2 ( z - z o ) ( z - z { )

+ 76-2(v - Vo) + 0-67()' - yofe ~ -7o)

+ 35-7(y-.vo)(v-y1) + 0-71(.v-.vo)

- 0052(.v - .vo)(z - z0) - 3-74(.v - xo)(y - y0)

The validity of this polynomial as a model for Aquestioned in two different situations, namely:

was

(a) if the polynomial is evaluated at points far from the inter-polation points within the tolerance region

(b) if Amin is predicted in the sweep generator of another TVreceiver. In this case all components apart from x, y, z wouldbe randomly different within nominal tolerances.

To validate case («), measurements were performed with theoriginal circuit, circuit no. 1, at three points far from the inter-polation region.

To validate case (h), Amin was measured in another TV recei-ver, circuit no. 2, at five interpolation points identical to thoseselected for the original circuit.

A comparison between experimental values of Amin and pre-dicted values with the polynomial showed a difTerence smallerthan 1 V in all cases considered above. This accuracy wasconsidered fairly good to describe the value of Amin in thiscircuit.

frequency

0 20

0 10

0 17 20AminV

Fig. 3 Histogram of Amln

A Monte Carlo simulation was performed using the poly-nomial, and the histogram of Amin obtained is shown in Fig. 3.Since values of Amin greater than 17 V imply parametric fail-ures of the circuit, it can be seen that 7% of the circuits wouldfail.

Conclusions: The output voltage of a series regulator and thesynchronising pulse requirement of a TV sweep generatorwere adequately modelled for statistical simulation byNewton's interpolation polynomials.

A circuit simulation program or experimental measure-ments are required at a few interpolation points, but theMonte Carlo run can be done with the polynomial alone,thereby saving considerable computer time.

Since the theory of multivariate approximating functions iscurrently of great interest, doubtless better approaches can befound.

R. AMADORR. L6PEZLahoratorio de MicroelectrbnicaInstitute Superior Politecnico J. A. EcheverriaA part ado 8016Cuidad Httbana 8, Cuba

28th March 1984

References

1 BALABAN, p., and GOLEMBESKI, J.: 'Statistical analysis for practicecircuit design", IEEE Trans., 1975, CAS-22

2 BALABANIAN, N.: 'Synthesis of active RC networks'. Revista Com-unicaciones, 1970, Havana, Cuba (in Spanish)

3 ISAACKSON, E., and KELLER, H. B.: 'Analysis of numerical methods'(Wiley, 1966)

ACCURATE ANALYTICAL METHOD FORTHE EXTRACTION OF SOLAR CELL MODELPARAMETERS

Indexing term: Solar cells

Analytical expressions are derived for the rapid extraction ofsolar cell single diode model parameters from experimentaldata. The resulting parameter values are shown to have lessthan 5% error for most solar cells.

The determination of solar cell model parameters from experi-mental data is important in the design and evaluation of solarcells. Whilst a number of methods have been suggested formeasuring the series resistance of a solar cell,1"4 other par-ameters, which are also important, have not received the sameamount of attention, and no rapid and accurate methods ofextracting these other parameters have so far been proposed.

The single diode lumped parameter equivalent circuit of asolar cell is given in Fig. 1. At a given illumination, thecurrent/voltage relationship is given by

' = ' * -!R£ ~ / s exp

nVT(1)

It has been shown5 that the circuit parameters Rs, Rsh, Is andn at a particular temperature and illumination can be com-puted from the values Voc, Isc, Vm, /„„ Rso and Rsho measuredfrom the IJV characteristic, where Voc = open-circuit voltage,/sc = short-circuit current, Vm = voltage at the maximumpower point, /„, = current at the maximum power point,

" - - ' I T ) (2)

(3)

It has also been shown5 that the lumped circuit parameterscan be evaluated from the following nonlinear equations:

//exp -^ - p̂ ^f) - lj\ + £) + £ = 0 (4)V nVr nVT J \ RJ Rsh

(5)

(6)1

expRsh Rsh0-Rs nVT nVT

= 0

- /s expnVr

= 0 (7)

Kennerud5 and Charles et al.6 have shown that the four par-ameters n, /„., Rs and Rsh may be determined by using theNewton-Raphson method of solving the nonlinear simulta-neous eqns. 4-7. However, this method requires extensive

406 ELECTRONICS LETTERS 10th May 1984 Vol. 20 No. 10

computation and also good initial guesses for the iterations toconverge. In many cases, it was found that there are difficultiesin determining these guesses in order to solve the equations.7

Thus there is a need for analytical expressions that will enablethe direct determination of Iph, n, Is, Rs and Rsh.

Consider the following typitical parameters for a 3" solarcell under AMI illumination: Iph = 10 A, Rs = 005 Q, Rsh =500 ft, / , = 10~7 A, n = 1-3, Voc = 0-545 V, /,c = 10 A, R\o =00838 ft, Rsho = 497 Q, Vm = 0-415 V,/m = 0-916 A, T = 27°Cand FF = 0-698.

[869/11

Fig. 1 Single diode model for solar cells

By substituting these values into eqn. 4, it is evident thatexp (VoJnV;-) is much greater than exp (IscRJnVT). The term/, exp (Isc RJnVr) may therefore be neglected provided theillumination is such that Iph is less than 10 A (i.e. equivalent toillumination of less than 10 suns) for less than 10% error.Also, since Rs <? Rsh, I + (RJRsh) may be written as 1. For thetypical values cited above, in eqn. 5 the term (lJnVT) xexp (V0C/nVr) is much greater than i/Rsh, and in eqn. 6 theterm (IJnVr) exp (IscRJnVr) is much less than 10% of theother terms provided Iph is less than 6 A (i.e. illumination lessthan 6 suns at AMI).

After ignoring the terms that may be neglected we obtainthe following:

(Rso - *») " V exp •%- ~ l = 0n Vr n VT

is exp -T7 + -^-—- - /,„ - / , expn VT Rsh n Vr

^ = 0

(8)

(9)

(10)

)

(11)

From these equations an analytical expression for n in termsof the measured parameters is

n =

/„ Rs and Iph may then be found from:

nVr

(12)

(13)

(14)

(15)

Thus, all the parameters of the single diode model can befound from eqns. 10, 12, 13, 14 and 15.

Table 1 COMPARISON OF PARAMETERS EXTRACTEDBY ANALYTICAL EXPRESSIONS ANDITERATIVE METHOD FOR TWOEXPERIMENTAL CELLS

Experimentaldata

Vuc, V

/ w , AI/,, V/„„ ARso, QRslw> fl7, K

Results

V An!s, LIARs, mQRv/l, fi

Blue cell

0-536010230-437009250-45

1000300

Iterative Analytical

01023 010231-5017 1-501901034 01036

68-51 68-2610031 10000

Grey cell

0-5240-5610-3900-4810162

25-9307

Iterative Analytical

0-5627 0-56101-7168 1-72255-326 5-514

78-52 77-692603 25-9

The errors present in the analytical expressions in eqns. 10,12-14 were computed. These errors introduced by the assump-tions incorporated in the derivation of the analytical expres-sions were computed as follows. A theoretical I/Vcharacteristic was computed from the model using base valuesof Iph= 1 A, /s = 10~7 A and n = 1-3 and different com-binations of Rs and Rsh. Rs was varied in the range I mQ to1 Q and Rsh was varied in the range 10 Q to 10 kfl Theparameters were recovered from the characteristic using theanalytical expressions and compared with the actual valuesused in generating the characteristic. The differences wereexpressed in terms of percentage errors and plotted in a gridof Rs and Rsh values. The results are presented in Fig. 2, wherethe i% and 5% error contours for parameters n, Is, Rs and Rsh

are drawn in an Rs — Rsh grid. The results in Fig. 2 show thatthe analytical expressions for all four parameters are accurateto within 1% provided Rs is in the range 1 mQ to 150 mQ andRsh is in the range 30 Q to 3000 Q. For 5% errors, the range ofvalidity is even wider, and would include most solar cells.

1

0-1

<2to

rr

001

0001

-

.-

///CO/v ,

• \ /

\ /

- \ /

\

; \\57, 1

\ ,

I \

%

//

1 "j

l c

\

//

\

\i * \ .

57

17.

i

>

\

17o and

M7.

5 7.

57.

10 100 1000 10k

Fig. 2 Error contours of extracted parameters

Is errorRs errorRsh error

The model parameters of two cells of differing quality wereextracted using these analytical expressions and comparedwith parameters extracted using the iterative method.5 Theexperimental data (taken from Charles et a/.6) and the resultsare set out in Table 1. It can be seen that the results obtainedby the two methods are virtually identical for the high-quality'blue' cell, and differ by no more than 4% for the lower-quality

ELECTRONICS LETTERS 10th May 1984 Vol. 20 No. 10 407

'grey' cell. These results together with the error contours effec-tively demonstrate that the analytical expressions enable therapid and accurate determination of lumped circuit par-ameters for most solar cells.

J. C. H. PHANG 22nd March 1984D. S. H. CHANJ. R. PHILLIPS

Department of Electrical EngineeringNational University of SingaporeKent Ridge, Singapore 0511

References

1 WOLF, M., and RAUSCHENBACH, H.: 'Series resistance effects on solarcell measurements', Adv. Energy Conversion, 1963, 3, pp. 455-479

2 PHANG, J. c. H., and CHAN, D. S. H.: 'Comments on the experimentaldetermination of series resistance in solar cells', IEEE Trans., ED(to be published)

3 RAJKANAN, K., and SHHWCHUN, J.: 'A better approach to the evalu-ation of the series resistance of solar cells', Solid State Electron.,1979, 22, pp. 193-197

4 ARAUJO, G. L., and SANCHEZ, E.: 'A new method for experimentaldetermination of the series resistance of a solar cell', IEEE Trans.,1982, ED-29, pp. 1511-1513

5 KENNERUD, K. L.: 'Analysis of performance degradation in CdSsolar cells', ibid., 1969, AES-5, pp. 912-917

6 CHARLES, J. P., ABDELKRIM, M., MUOY, Y. H., a n d MIALHE, P.: 'A

practical method of analysis of the current-voltage characteristicsof solar cells', Sol. Cells, 1981, 4, pp. 169-178

7 BRAUNSTEIN, A., BANY, J., and APPELBAUM, j . : 'Determination of

solar equation parameters from empirical data', Energy Conver-sion, 1977, 17, pp. 1-6

IN-THE-LENS SECONDARY ELECTRONANALYSER FOR IC INTERNAL VOLTAGEMEASUREMENTS WITH ELECTRON BEAMS

Indexing terms: Integrated circuits, Electron microscopy

A new analyser scheme for voltage measurements on inte-grated circuits is described. The analyser is built.into theobjective lens of a scanning electron microscope. It featuressmall working distances, high voltage resolution capability,high transmission, and reduced sensitivity to measurementerrorsv

Quantitative voltage measurements on integrated circuits(ICs) can be performed using a secondary electron (SE)analyser and a finely focused electron beam which acts as acontactless, nonloading, easily positionable probe.

This analyser has to fulfil three functions: (i) provide anextraction field to eliminate the measurement accuracy degra-dation due to the influences of microfields on the IC surface;(ii) provide a retarding field that analyses the SE energy; (iii)provide a deflection unit that deflects the SEs towards a detec-tor.

In a conventional scanning electron microscope (SEM),which generates and controls the electron probe, the energyanalyser is located between the bottom of a 'pinhole' objectivelens and the IC. In this arrangement the analyser limits theminimum working distance (i.e. distance between the bottomof the objective lens and IC surface) to about 12-25 mm. Thisin turn compromises the probe-forming capability of the elec-tron optical system, as the shorter the working distance, thehigher the probe current density, and the better the spatialresolution of the arrangement. The voltage resolution of theanalyser, on the other hand, improves as the spacing betweenthe extraction and analyser grids increases, which can only beaccomplished by lengthening the analyser. These conflictingrequirements necessitate a compromise in terms of spatial andvoltage resolution to be struck.

Using an ISI model SS-60 SEM, which employs an open-bore objective lens, with SE detection above the lens, a newanalyser scheme has been developed that overcomes the trade-offs between spatial and voltage resolution.

Fig. 1 shows a schematic diagram of the new analyserscheme, compared to the below-the-lens arrangement useduntil now. Primary electrons (PE) pass through a series ofgrids or apertures and generate SEs at the IC surface, which islocated typically 3 mm below the objective lens. This shortworking distance allows the achievement of 3-5 times smallerbeam probes, containing the same current as is possible at'25 mm working distance.

det.

PE

U

in -lensanalyser

below-lens _ _analyser ma

Fig. 1 Comparison between below- and in-lens analysers

The SEs are accelerated by the extraction field and enter theretarding region. Planar equipotential lines parallel to the gridplanes are assured by a resistive coating on the inner wall ofthe analyser. All SEs with energies larger than the retardingvoltage enter the deflection region and are detected by the SEdetector.

Applying a ramp signal to the retarding grid yields theintegral SE spectral distribution, or 'S-curve', which is shiftedwhen voltages are applied to the specimen. The voltage mea-surement is performed by analysing this S-curve shift. High-accuracy measurements are possible only if the S-curve shaperemains constant with applied specimen voltage. This is thecase for the in-the-lens analyser as demonstrated in Fig. 2.

Fig. 2 S-curves taken at Vexlr — 15 kV and specimen voltages of +4 V{left curve), 0 V (centre curve) and —4 V (right curve)

The ramp signal applied to the retarding grid varies from + 12 to- 2 0 V

The voltage resolution AV of SE analysers is given by

AV = , AE/E, Vret, /V(£)](A///pe)1/2

where n is the acceptable signal/noise ratio on the detectedsignal, and C is a factor which depends on the transmission T,the analyser energy resolution AE/E, the retarding voltage Vre,and the spectrum of the SEs, given by N(E). A /= systembandwidth and Ipe = primary electron beam current. Thefactor C on an aluminium surface was determined to be6 x 10"9 V(As)1/2 at Vextr = 500 V, and 8 x 10"9 V(As)1/2 atKxtr— 1500 V. The factor C for the in-the-lens analyser ismore than 20 times smaller than achieved by below-the-lensanalysers1 and reduces the measurement (integration) time fora given voltage resolution by a factor of about 5.

The energy resolution AE/E was measured by simulatingSEs with a low energy electron gun with low energy spread of

408 ELECTRONICS LETTERS 10th May 1984 Vol. 20 No. 10