Solar Cells in Concentrating Systems and Their High ... · PDF fileM. A. Mosalam Shaltout et...

Tarn Yates, Senior Thesis, Physics Department UCSC Summer 2003

Solar Cells in Concentrating Systems

and Their High temperature Limitations

A Thesis Submitted in Partial Satisfaction of the Requirements for the Degree of Bachelor of

Sciences in Physics at the

University of California, Santa Cruz

by

Tarn A. Yates

September 3, 2003

________________________ _______________________ Ali Shakouri Clemens Heusch Technical Advisor Supervisor of Thesis, 2002-2003

________________________ David Dorfan

Chair, Department of Physics


2

Table of Contents

Abstract 1

Introduction 2

Theory i. The p-n junction. 6 ii. The p-n junction under an applied bias. 7 iii. Solar cell principles. 9

Summary of Related Papers

i. Thermally Affected Parameters of the Current-Voltage Characteristics of Silicon Photocell. E. Radziemska, E. Klugmann 13

ii. The Effect of Temperature on the Power Drop in Crystalline Silicon Solar Cells. 20 iii. Temperature Dependence of the Spectral and Efficiency

Behavior of Si Solar Cell Under Low Concentrated Solar Radiation.

M. A. Mosalam Shaltout et al. 23

Derivation of Power Conversion Formula 27 Verification of Power Conversion Formula 34 Experiment 36

i. Apparatus. 36 ii. Data Collection. 37 iii. Data. 37 iv. Data Analysis. 39 v. Conclusions. 42

Conclusion 44 References 46 Acknowledgements 47


3

Abstract A mathematical model of power conversion vs. illumination for a solar cell is

presented. Heating of the solar cell under illumination and temperature dependent

properties are taken into account. This model is designed for use in the construction of

solar concentrating devices and takes into account reflection losses, efficiency, loss of

efficiency due to heat, and thermal resistance of the cell. Experimental results on the

thermal resistance of a cell are also presented.


4

Introduction

Current means of energy production all have environmental drawbacks. These

drawbacks include the damming of rivers, the creation of nuclear waste, and the pollution

associated with fossil fuels. Compounded with these problems is the dependence on other

nations for oil that can lead to global conflict, and the fact that the supply of fossil fuels is

quickly being depleted. It is apparent that an alternative source of energy must be

developed. A potential solution to this problem can be found in the production of solar

electricity. It has been calculated that approximately one kilowatt per square meter falls

on the earth during the day [1]. Putting this energy to use would diminish the demands

put on the environment and reduce the reliance on other sources of energy.

Photovoltaic technology was developed in 1954 [2], but it has not caught on as a

widely used source for the production of electricity. This is because the cost of solar

panels compared to the amount of power they produce makes their purchase

uneconomical for most buyers. Significant gains have been made in the cost per watt

ratio since the 1970’s reducing the price from $70 per Watt to under $4 per Watt today

[2]. However, this ratio needs to be reduced further before photovoltaic technology

becomes a viable resource.

At present, there are two types of photovoltaic cells that dominate the market. The

most widely used are silicon cells, which constitute 86% of the current market [4]. Single

crystalline cell technology is well developed because it uses the same manufacturing

techniques that are used in the electronics semiconductor industry. In this process, a

single silicon crystal is grown and then sliced into wafers. This produces identical cells


5

that are up to 15 percent efficient [2]. However, the process is very time consuming and

expensive. A less expensive technique is the production of polycrystalline silicon cells. In

this process molten silicon is poured into molds and allowed to cool, then sliced into

wafers similar to those in the single crystal method. Many small silicon crystals are

formed in the mold instead of one large one as in the single crystalline cell. This results in

the efficiency of the polycrystalline cell being lower, between 11 and 14 percent [2].

Thin film semiconductors make up the remaining 14% of the PV market [3]. In

this process, a thin film of semiconductor material, most often amorphous silicon, is

deposited on an inexpensive substrate such as glass. This is a fast process and uses much

less material because the silicon layer is only one micron thick [2]. Thin film has no

crystalline structure, which results in efficiencies of only six to seven percent. The low

cost of production makes up for the poor efficiency, and thin films may be the future of

the photovoltaic market.

More efficient cells have been developed, but due to their high price, they are

used mainly in research or in space. The most promising high-efficiency cell is the multi-

junction cell. Multi-junction cells are several layers of photovoltaic cells stacked on top

of one another. Each successive layer has a lower band gap energy, allowing for the

absorption of a wider range of the spectrum. Boeing holds the current record for the

efficiency of a multi-junction cell at 34.2 percent [4]. This is more than twice the

efficiency of cells currently on the market, and efforts are being made to increase the

efficiency up to 40 percent [4]. It would not be practical to build a solar panel out of

multi-junction cells because they are so expensive. However, a solar concentrating

system utilizing multi-junction cells could be cost effective.


6

Solar concentrators have been in use since as early as 212 BC, when it was

reported that Archimedes used mirrors to concentrate sunlight onto invading Roman

ships, setting them on fire [5]. Whether or not this is true, it is evident that the principles

behind solar concentration have been known for some time. Concentrators make it

possible to focus solar radiation falling on a large area onto a very small area. This

increases the intensity of the sunlight, leading to greater power falling on the area of

focus. The solar intensity or concentration ratio, denoted by the letter C, is determined by

the ratio of the area of incident sunlight to the area that it is focused on [6]. Concentration

ratios in the thousands have been achieved [5].

The use of concentrators creates the potential for the production of less expensive

solar panels using high efficiency mono-crystalline or multi-junction cells. In general the

materials used to build concentrators are less expensive than photovoltaic cells. The

concentrator takes up most of the area of a concentrator system, and only a small amount

of photovoltaic material is needed. Concentrator panels could reduce the cost per watt

ratio to the point where solar power is an economical alternative.

At present there are many different types of concentrators that could be used in a

photovoltaic system. These concentrators can be divided into two different groups:

reflecting concentrators and lenses. In the former group the most common concentrators

are flat plate mirrors, spherical and parabolic mirrors, and cylindrical trough collectors.

The most prominent lens is the Fresnel lens. It was developed in 1822 for use in

lighthouses and can achieve high concentration ratios [7]. Newer lenses such as Aspheric

lenses and TIR (Transmission, total Internal reflection, Refraction) lenses can be used

together to achieve concentration ratios of over 300 while being only 2 cm thick [8].


7

The use of concentrators is not free of problems. Most concentrators rely on being

directly focused on the sun; any deviation causes a severe drop in the concentration ratio.

Some of the collected power must be used to run a tracking system. Also, though it seems

that the power converted by a solar cell would increase indefinitely with increasing

illumination, that is not the case. As the intensity of illumination increases, the solar cell

heats up. It is a well-documented fact that the efficiency of solar cells decreases as the

temperature of the cell increases. The loss in efficiency is about 10% for every 25 K

increase in temperature [9], although the exact loss in efficiency depends on the specific

cell.

This paper concerns the presentation of a mathematical model of power

conversion vs. illumination. The model takes into account the thermal resistance,

efficiency, and loss of efficiency due to an increase in temperature of the cell. It also

considers reflection losses. This model could be useful in the designing of solar

concentrator systems. It shows that there is a maximum power conversion point beyond

which any increase in illumination causes a decrease in converted power. The model also

reveals the importance of using cells with low thermal resistance in concentrator systems.

Basic solar cell theory will be reviewed, including a discussion of p-n junctions, a

derivation of the diode current, and the production of electron-hole pairs leading to the

photocurrent. This is followed by a summary of papers related to temperature effects on

solar cells. The results of an experiment to test the thermal resistance of a solar cell are

presented in the final section.


8

Theory

The p-n Junction

In order to understand how a solar cell works, it is necessary to understand p-n

junctions. A p-n junction is formed when an n-type semiconductor is put together with a

p-type semiconductor. See Figure 1 The n-type semiconductor is doped with donor atoms

that have more electrons than the surrounding material, and the p-type semiconductor is

doped with acceptor atoms that have fewer electrons than the surrounding material.

Atoms in the p-type semiconductor with fewer electrons than the surrounding material

are said to have holes. These holes are thought of as positive entities much like electrons

in that they can move throughout the material and contribute to the current. When a hole

and an electron meet, they essentially annihilate each other; this is called recombination.

Figure 1: p-n Junction (electrons are depicted as filled circles, holes are depicted as empty circles, negatively charged donor atoms

in the p-side and positively charged acceptor atoms in the n-side are also shown.)

Putting an n-type semiconductor together with a p-type semiconductor creates an

electron/hole concentration gradient. This concentration gradient causes a diffusion

current with electrons diffusing to the p-side and holes diffusing to the n-side. The area in


9

which this diffusion takes place is called the depletion region. When electrons from the n-

side diffuse to the p-side they meet with holes and recombine leaving negatively charged

donor atoms on the p-side. The holes from the p-side diffusing to the n-side create

positively charged donors on the n-side. The ionized donors create an internal electric

field in the depletion region [10], [11]. See Figure 2. The Electric field in the region

works as a barrier preventing more electrons from diffusing from the n-side to the p-side.

Only those electrons with a high enough energy to overcome the field can make the

transition. In equilibrium there is no net current so the diffusion current and the drift

current (due to the internal electric field) cancel each other.

Figure 2: Creation of Internal Electric Field

The p-n Junction Under an Applied Bias

When a potential is applied across a p-n junction, it can either increase or

decrease the internal electric field. If the negative side of the potential is connected to the

p-side, then the electric field is increased. This is called a reverse bias. Alternately, a

forward bias, where the positive side of the potential is connected to the p-side, results in

the reduction of the internal field. When the internal field is decreased by a forward bias

the number of electrons on the n-side that have enough energy to cross the depletion


10

region to the p-side increases by a factor of exp ( )TkeV . Here e is the charge of the

electron, V the applied voltage, k the Boltzmann coefficient, and T absolute temperature.

The resulting electron current from the n-side to the p-side is 0eI exp ( )kTeV . There is a

small electron current 0eI from the p-side to the n-side. This is due to the very few

electrons (minority carriers) in the p-side. Thus the total electron current is:

( )10 −= TkeVee eII 1

The same relation holds for the hole current from the p-side to the n-side. A

forward bias results in a hole current:

( )10 −= TkeVhh eII 2

where 0hI is the equilibrium hole current. Putting equations 1 and 2 together gives the

total current, also known as the diode current:

)1(0 −=+= TkeVhe eIIII 3

where

000 he III += 4

0I is sometimes called the dark current [12].


11

Solar Cell Principles

When photons of a high enough energy are incident on a semiconductor, they

create an electron-hole pair. This can be understood by looking at the energy band

diagram of a semiconductor. Figure 3 shows the three distinct energy bands of electrons

in a semiconductor. Valence band states are fully occupied by electrons and the first

empty band (conduction band) is separated by a band gap. Electrons in the valence band

can not be involved in conduction. This is due to the Pauli exclusion principle, since there

are no low lying empty energy states for the electrons in the valence band to move to

under and electric field. When electrons acquire a sufficient amount of energy, they can

enter the conduction band. When a photon with an energy greater than the band gap is

incident on a semiconductor, it gives an electron in the valence band enough energy to

move to the conduction band. Both the electron in the conduction band and the hole that

has been created in the valence band can be involved in the conduction of a current under

an electric field [12].

Figure 3: Band Diagram and Electron-Hole Pair Production

A solar cell can be constructed by putting a very thin, heavily doped n-type layer

on top of a thicker p-type layer. As can be seen in Figure 4, the depletion region is mostly

on the p-side. Light is absorbed through the n-type layer. Because the n layer is so thin,


12

most photons penetrate into the depletion region, or the p-side before creating an

electron-hole pair. When an electron-hole pair is created in the depletion region the

electric field moves the electron into the n-side and the hole into the p-side. This gives

the previously neutral n-side a negative charge and the previously neutral p-side a

positive charge. When a load is connected to the cell, the electron can travel through the

circuit, do work, and recombine with the hole.

Figure 4: Electron-Hole Pair Behavior in Solar Cell

If the light penetrates into the neutral p-side, then there is no electric field to

separate the electron-hole pair. Instead the electron and the hole diffuse at random

through the material and recombine if they meet. The average time between pair

production and recombination for an electron is eτ . In this time, the electron diffuses a

mean distance of eee DL τ2= where eD is the diffusion coefficient in the p-side. If the

electron-hole pair is created within eL of the depletion region, then the electron can


13

diffuse to the depletion region and be moved by the electric field over to the n-side. For

this reason it is important for the diffusion length eL to be as long as possible. The same

process takes place for electron-hole pairs created in the n-side. In silicon, the diffusion

length is longer for electrons than it is for holes. This is why the thin top region is n-type,

and the thicker region is p-type.

When an illuminated solar cell is short-circuited, a current flows through the

circuit in the opposite direction of the diode current. This current is a result of electron-

hole production in the solar cell and is called the photocurrent phI . The photocurrent is

directly proportional to the intensity of light. If the cell is in a circuit with some

resistance, then there is a voltage across the junction. This voltage acts like a forward bias

and results in a diode current through the cell. The total current is then [11]:

( )10 −−= kTeVph eIII 5

A typical IV curve for a solar cell can be seen in Figure 5.

The Fill Factor(ff) of a solar cell is a measure of the quality of the cell. It is

defined as:

OCSC

mppmpp

VIVI

ff = 6

where mppI and mppV are the current and the voltage at the maximum power point on the

IV curve, SCI is the short circuit current, and OCV is the open circuit voltage.


14

Figure 5:

A typical IV curve for a solar cell under illumination. MPP is the maximum power point of the cell.

The efficiency of a cell is the ratio of the maximum converted power CP to the input

power IP :

I

C

PP

=η 7

Converted power is the product of the current and voltage at the maximum power point

so equation 7 can be rewritten as [10]:

I

OCSC

PVIff ⋅

=η 8


15

Summary of Related Papers

Paper 1) Thermally Affected Parameters of the Current-Voltage Characteristics of

Silicon Photocell (2002) E. Radziemska, E. Klugmann Energy Conversion &

Management 43 1889-1900

This paper presents a comprehensive explanation of the temperature effects on

silicon solar cells. It gives experimental results showing a slight increase in short circuit

current with increased temperature and a large decrease in open circuit voltage with

increased temperature. Radziemska and Klugmann show that the product OCSCVI

degrades by 0.8 % for every 1 K increase in temperature. They assert that temperature

losses account for 7.6 % of PV conversion losses in a working power plant. This paper

introduces the following main temperature effects.

Series Resistance

A solar cell in a circuit with some load resistance can be represented by the

equivalent circuit shown in Figure 6. The voltage across the load is:

sLphL rIVV −= 9

where LI is the current through the load.


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Figure 6:

Electrical model of a solar cell. jr is the junction resistance, sr is the series resistance, and LR is the load

resistance.

When the temperature of the cell increases, it causes an increase in the series

resistance sr . This is due to the fact that the mobility of charge carriers, µ , is inversely

proportional to the temperature:

23−

∝ Tµ 10

As the temperature rises, there is increased carrier scattering on lattice vibrations

(phonons) and impurities. This decrease in mobility causes a decrease in the conductivity

and an increase in the series resistance. In the normal operating range of a solar cell, 300

to 380 K, this increase in very small.

jr LRphV LV

sr


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Short Circuit Current

An increase in the temperature of a solar cell results in a slight increase in the

short circuit current. The short circuit current from a solar cell is the photocurrent. At a

given wavelength of light the photocurrent is:

λληλ eNI ph =)( 11

where λN is the number of photons at that wavelength, and λη is the efficiency of the

solar cell at that wavelength, also known as the spectrum response. The power of

irradiation is:

λλλhcNP = 12

Combining this with equation 11 gives:

hcP

eI phλ

ηλ λλ=)( 13

In order to find the total photocurrent equation 13 must be integrated over all

wavelengths that are absorbed by the solar cell. A photon must have an energy greater


18

than the bandgap of the solar cell in order to contribute to the photocurrent. This

introduces a maximum photon wavelength 1λ . The total photocurrent is then:

λληλ

λλ dPhceI ph ∫=

1

0 14

As the temperature of a solar cell increases, the bandgap gE decreases. To a linear

approximation, the bandgap follows the equation:

)300()300()( KTdTdE

KETE ggg −+= 15

For silicon 4103.2 −×−=dTdEg eV/K.

This decrease in the bandgap allows photons with longer wavelengths, lower

energy, to be absorbed by the solar cell. For example, an increase in temperature of 80 K

raises the limiting wavelength by 19 nm. As a result, the short circuit current of the cell is

increased.

This is offset by the fact that increasing temperature decreases the built in voltage

of a p-n junction. The built in voltage for a p-n junction is:

20 lni

DA

nNN

ekTV = 16


19

where AN and DN are the number of acceptor and donor atoms respectively that the

junction has been doped with, and in is the intrinsic carrier concentration of the junction.

In silicon, in follows the equation:

)2exp(1087.3 2316 kTET goi −×=η 17

In a silicon cell where 1610=AN 3−cm and 1510=DN 3−cm , an increase in temperature

from 300 to 380 K causes the built in voltage to decrease from 0.66 V to 0.35 V. This

decrease in the built in voltage allows thermally agitated charge carries to cross over the

p-n junction in both directions. The voltage is also much less effective at separating

electron-hole pairs. Due to these two factors, there is less build-up of excess charge,

negative on the n-side, positive on the p-side, and the short circuit current is decreased.

Open Circuit Voltage

An increase in temperature reduces the open circuit voltage. As the bandgap

decreases with increasing temperature, more electrons are able to move into the

conduction band. The extra electrons in the conduction band and the holes in the valence

band lead to an increase in the dark current. It is simple to show that an increase in the

dark current results in a decrease in the open circuit voltage.

The output current of a solar cell is:

( )10 −−= kTeVph eIII


20

Under open circuit conditions, the output current is zero, so the photocurrent and diode

current must be equal:

( )10 −= kTeVph eII 18

This can be rearranged to give:

+== 1ln

0II

ektVV ph

OC 19

As can be seen from equation 19, an increase in dark current with increasing temperature

leads to a lower open circuit voltage. Dark current is related to the temperature by this

proportionality:

∝

kTE

I gexp0 20

Experimental Results

These theories were tested on a Siemens type 5, photovoltaic cell. The cell was

placed on a thick copper plate to control the temperature and illuminated with a solar

simulating halogen lamp. Data was taken on the open circuit voltage and short circuit


21

current at various temperatures. Figures 7 and 8 are the graphs that resulted from this

research. From these graphs, it was found that the open circuit voltage decreased by 0.38

% 1−K , and the short circuit current increased by 0.033 % 1−K . The paper reported that

the product OCSCVI decreased by 0.8% 1−K [13].

Figure 7: Temperature dependence of the open circuit voltage for the Siemens Solar cell.

Figure 8: Short circuit current vs. temperature of the Siemens solar cell.


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Paper 2) The Effect of Temperature on the Power Drop in Crystalline Silicon Solar Cells

(2003) E. Radziemska Renewable Energy 28 1-12

This paper presents data showing a decrease in maximum output power of a

silicon solar cell of 0.65% 1−K , a decrease in maximum output power of a silicon solar

module of 0.66% 1−K , and a decrease in module efficiency of 0.08% 1−K .


Data was recorded from a single-crystalline silicon solar cell illuminated by a

halogen lamp. A thick copper plate, which was heated by an electric heater, was used as a

base for the cell to ensure a stable temperature. Four MOSFETs were used to monitor the

temperature of the cell. Data was also taken from an ASE-100DGL-SM solar module

illuminated by the sun. A water cooling system was used on the module, and the

temperature was measured using a copper-constantan thermocouple.

Figure 9 is a graph of the output power vs. voltage for the single-crystalline

silicon cell at different temperatures. A dramatic drop in maximum output power can be

seen with each increase in temperature. Figure 10 is a graph showing how the maximum

power degraded with temperature. It was determined from the graph that the maximum

power decreased by 0.65% 1−K .


23

Figure 9: Output power vs. voltage of a single-crystalline silicon solar cell at various temperatures.

Figure 10: Temperature dependence of the maximum output power.

Graphs of both the current vs. voltage and power vs. voltage for the solar module at

different temperatures can be seen in Figures 11 and 12.

It is apparent that the module suffers from the same loss of output power with

increased temperature that the cells do. Parameters collected from the data presented in

Figures 11 and 12 can be found in Table 1. These numbers can be used to show that the


24

maximum power output drops by 0.66% 1−K and that the efficiency decreases by

0.08% 1−K .

Figure 11: Current vs. voltage and output power vs. voltage of the solar module at 25 C.

Figure 12: Current vs. voltage and output power vs. voltage of the solar module at 60 C.


25

)(VU OC )(AI ph )(WPm FF (%)η

CT °= 25 42.18 2.545 79.60 0.724 13.3

CT °= 60 34.75 2.555 61.28 0.690 10.3

Table 1

Radziemska suggests that the temperature of the module could be lowered by

decreasing the heat produced by:

-non-active absorption of photons, which do not generate pairs,

-recombination of electron-hole pairs,

-photocurrent (Joule’s heat generated during the current flow in the series

resistance of the p-n junction) and parasitic currents [14].

Paper 3) The Temperature Dependence of the Spectral and Efficiency Behavior of Si

Solar Cell Under Low Concentrated Solar Radiation (2000) M.A. Mosalam Shaltout,

M.M. El-Nicklawy, A.F. Hassan, U.A. Rahoma, M. Sabry Renewable Energy 21 445-458

This paper presents data on the behavior of solar cell efficiency and maximum

power output at different temperatures under various levels of illumination. Data was

taken at light concentrations between 1 and 5 suns. These concentrations were chosen to

mimic low cost static concentrators, such as parabolic and v-trough concentrators. The


26

authors believe these concentrators to be very promising due to the fact that they do not

need tracking systems and can be used with low cost, one-sun solar cells.


The data was taken on a monocrystalline solar cell illuminated by a variable

intensity halogen lamp. Cell temperature was controlled by a large serpentine of brass

that was cooled by a water circulator. The following graphs (Figures 13 and 14) show

how the maximum power of the cell varied with temperature and illumination. It is

interesting to note in Figure 13 that at higher illuminations, the temperature had a much

larger influence on the maximum power than at lower illuminations. Also, that at 90 C

there is only an 8 % difference between the maximum power at 4010 2/ mW and at 2812

2/ mW . In Figure 14, when the cell is at 85 C, a large increase in the illumination causes

a relatively small increase in the maximum power.

Figure 15 is a graph of the cell efficiency vs. back cell temperatures. It shows that

an increase in temperature always leads to a decrease in cell efficiency. The change in

cell efficiency with temperature is more dramatic as the illumination level goes up.

Mosalam Shaltout et al. all concluded that one-sun solar cells could be used in solar

concentrating devices with low concentration ratios. Beyond a certain temperature, there

is no use for increased illumination. They believe that the optimum concentration ratio

for one-sun solar cells is around 4 times or 3000 W/m 2 [15].


27

Figure 13: Decrease of cell maximum power with temperature at different illuminations.

Figure 14: Maximum power variation of the cell with illumination at three cell temperatures.


28

Figure 15: Efficiency behavior of the cell with temperature at five illuminations.


29

Derivation of Power Conversion Formula

The following model was created for this paper, in an attempt to understand the

effects that increased temperature has on concentrating systems.

The increase in a cell’s temperature under illumination is a function of the thermal

resistance of the cell TR and the amount of power that is involved in heating the cell HP .

The relation is:

TH RPT =∆ 21

where T∆ is the change in temperature. There are three factors that govern the amount of

power that is involved in heating the cell. These factors are the power incident on the cell

IP , the percentage of light that is reflected off the surface of the cell α , and the

efficiency of the cell Eη . The power transmitted past the surface coating of the cell is:

)1( α−= IT PP 22

where TP is the transmitted power. This is the power that can be converted into

electricity by the cell. It is also the power that goes into heating the cell. Transmitted

power that is not converted into electricity is absorbed as heat. The equation governing

this relation is:

)1( ETH PP η−= 23


30

This relation shows that the higher the efficiency of the cell is, the less power there will

be to heat the cell. Putting equations 22 and 23 together gives:

)1)(1( EIH PP ηα −−= 24

Combining this expression with the relation for the increase in temperature results in:

)1)(1( ETI RPT ηα −−=∆ 25

With this known, it is then possible to determine how the efficiency of the cell

changes with illumination. Solar cell efficiency follows the equation:

TTNNE ∆−= µηηη 26

where Eη is defined more precisely as the efficiency of the cell under increased

illumination, Nη is the efficiency of the cell under normal illumination (one sun) at a

given temperature, and Tµ is the percentage decrease in efficiency with increased

temperature. This relation is made more complex by the fact that T∆ is itself a function

of the efficiency. Plugging equation 25 into equation 26 gives:

)1)(1( ETITNNE RP ηαµηηη −−−=

Pulling Nη out in front and multiplying out results in:


31

( ) ( )[ ]ITTEITTNE PRPR αµηαµηη −+−−= 111

This can be rearranged and solved for Eη :

ITTN

ITTE

PR

PR

)1(1)1(1

αµη

αµη

−−

−−= 27

Once this has been found, it is then possible to create a relation for the power

converted based on a given illumination. As mentioned above, the power available to the

cell to convert into electricity is TP . Multiplying this by the efficiency gives the

converted power:

)1( αηη −== IETEC PPP

or

ITTN

ITTIC

PR

PRPP)1(1

])1(1)[1(

αµη

αµα

−−

−−−= 28

where CP is the converted power. A graph of this function using standard values for

,,, TT Rµα and Nη can be seen in Figure 16. The intercepts are at 0=IP and

)1(1

αµ −=

TTI R

P .


32

Figure 16 illustrates how the increasing temperature of the solar cell effects the

converted power. Up to a certain point, an increase in illumination causes an increase in

converted power. However, there is a point at which the efficiency becomes so poor, due

to the rise in temperature, that any increase in illumination causes a decrease in converted

power. The point at which this occurs represents the maximum power that a cell can

convert.

Figure 16: Power Converted vs. Power Incident

( 95.1,004.,12.,06. ==== TTN Rµηα wKcm /2 )

)/( 2cmwPC

)/( 2cmwPI


33

Finding the Maximum Power Conversion Point

The maximum power point mpcP can be found by taking the derivative of equation

28 with respect to incident power, setting it equal to zero, and solving for IP . This yields

a quadratic, which, when solved gives two answers for the maximum power point:

( )[ ]2111)1(

1N

TTNmpc R

P ηαµη

−±−

= 29

This equation can be simplified by using the expansion ( ) ...81

2111 221

NNN ηηη −−≈− .

Plugging the expansion into equation 29 gives:

( )

−−±

−≈ 2

81

2111

11

NNTTN

mpc RP ηη

αµη 30

The answer in which the addition sign is used shows the maximum power point to the

right of the positive intercept. This answer is non-realistic. The correct answer is the one

in which the minus sign is used:

( )

+

−≈ ...

81

21

11

NTT

mpc RP η

αµ 31


34

It can be seen that the ratio of the maximum power point to the positive intercept is just

slightly greater than one half.

Varying Efficiency and Thermal Resistance

It is interesting to look at the effect that a change in either the efficiency or the

thermal resistance has on the converted power. This is illustrated in Figure 17. Curve 1 is

a graph of equation 28 using the same values as in Figure 16. Curve 2 was created by

doubling the efficiency used to create Curve 1 and holding everything else constant.

Curve 3 was created by reducing the thermal resistance used in Curve 1 by one half and

again keeping everything else constant.

As would be expected an increase in efficiency greatly increases the power

converted at a given illumination. More interesting is the fact that a decrease in thermal

resistance can also cause a large gain in power at increased illuminations. And, if

working with very high levels of illumination, decreasing the thermal resistance is more

important than increasing the efficiency of the cell. These results could be very useful in

constructing a solar concentrating panel.


35

Figure 17:

Change in CP curve by varying Nη and TR . For Curve 1 the parameters used were:

95.1,004.,12.,06. ==== TTN Rµηα wKcm /2 . For Curve 2 all the parameters remained

constant except for the efficiency, which doubled to 24.=Nη . Curve 3 was produced using the same

parameters as Curve1 except that the thermal resistance was cut in half to 975.=TR .

)/( 2cmwPC

)/( 2cmwPI


36

Verification of Power Conversion Formula

The power conversion vs. illumination model can be verified using data from a

working concentrator system. The following data was taken from a parabolic dish

concentrator that is part of a power plant at White Cliffs, Australia. Light from the dish

was focused on the receiver at a concentration of about 340 times. The receiver consisted

of 384 series-connected HEDA312 silicon solar cells from SunPower Corporation. A

water cooling system was used to keep the receiver from overheating. Pictures of the

concentrating dish and the receiver can be seen in Figures 18 and 19.

Figure 18: Parabolic concentrator power plant.


37

Figure 19: The receiver consisting of 384 series connected cells.

Power Incident on Receiver IP 25.26 2/ cmW

Module Efficiency Nη 26.5 %

Temperature Coefficient Tµ 0.0038 1−K

Reflection α 13.78 %

Thermal Resistance TR 0.451 WKcm /2

Area of Receiver A 576 2cm

Table 2: Data from parabolic concentrator at White Cliffs.

Plugging these numbers into equation 28 predicts that the receiver should convert

3232.5 watts. This is very close to the recorded value for converted power of 3448 watts

[16]. It is a difference of 6.25 %.


38

Experiment

At this point, with the resources available, it is not possible to do a test of the

power conversion equation (equation 28) presented in the previous section. Instead the

following experiment was run in order to test the thermal resistance of a small solar

panel. This experiment was conducted because manipulation of the power conversion

equation pointed out the importance of the thermal resistance. Measurements of the

panels fill factor and efficiency are also presented.

Apparatus

The solar panel used in this experiment was a Radio Shack Sola 6 volt, 50 mA

panel. It consisted of twelve 0.5 volt silicon cells wired in series. The cells were housed

in a black plastic container with a clear plastic anti-reflection cover. Positive and negative

wires protruded from one end of the panel. The wires were connected to a 0-1 Ωk

variable resistor. A brass plate was used as a base for the solar panel. There were Type E,

Omega Engineering, INC. thermocouples mounted on the inside bottom of the solar panel

and on the brass plate. The two Constantan ends of the thermocouples were soldered

together so that the thermocouples measured the temperature difference between the solar

panel and the brass plate. GB Instruments GDT-11 multimeters were used to measure the

value of the resistance, the voltage across the resistor, the current through the resistor,

and the voltage difference between the two ends of the thermocouples. A sun meter was


39

used to measure the intensity of the incident sunlight. Also used were a Jameco solderless

breadboard, a screwdriver, thermal grease, and black plumbers tape.

Data Collection

Data collection began by setting the solar panel mounted on the brass plate and

the light meter out in the sun. No data was recorded until the temperature difference

between the solar panel and the plate was stable. When the temperature was stable, the

intensity of the sunlight was measured, and the open circuit voltage and short circuit

current of the solar panel was recorded. The panel was then connected to the variable

resistor, which was set at a low resistance. Multimeters were connected in both parallel

and series with the resistor to measure the voltage drop and the current. The resistance

was then increased in small increments with measurements of the current and voltage

taken at each increment. At each resistance the intensity of the light was recorded to

ensure that it not vary too drastically.

Data

)(kfcL )(mVV leThermocoup∆

)(ΩR )(mAI Iσ Volts Vσ

95.0 -1.0 ∞≈ 6.04 0.042

95.0 -1.0 0≈ 55.5 0.418

94.6 -0.9 12.3 55.3 0.415 .65 0.005


40

94.4 -0.9 32.1 53.7 0.403 1.73 0.012

94.5 -0.9 40.4 51.7 0.388 1.89 0.013

94.5 -0.9 51.0 48.8 0.366 2.50 0.018

94.6 -0.9 58.9 46.4 0.348 2.82 0.020

94.4 -0.9 68.3 43.5 0.326 3.10 0.022

94.2 -0.9 79.6 41.0 0.308 3.41 0.024

94.8 -0.9 85.4 39.5 0.296 3.52 0.025

94.5 -0.9 91.2 38.2 0.287 3.64 0.025

94.5 -0.9 101.2 36.2 0.272 3.81 0.027

94.5 -0.9 110.5 34.3 0.257 3.96 0.028

94.4 -0.9 120.0 32.6 0.245 4.08 0.029

94.4 -0.9 138.3 29.7 .0223 4.26 0.030

94.2 -0.9 165.0 26.2 0.197 4.48 0.031

94.0 -0.9 206 22.3 0.167 4.73 0.033

93.9 -0.9 303 16.4 0.123 5.09 0.036

94.0 -0.9 488 10.9 0.082 5.42 0.038

93.9 -0.9 990 5.7 0.043 5.62 0.039

Table 3

In this table L is the intensity of the light; the unit of measurement is foot-

candles. The error in the intensity is 05.0=Lσ kfc . leThermocoupV∆ is the voltage generated

by the thermocouple due to the temperature difference between the solar panel and the


41

brass plate. There was an error in the temperature data of 05.0=Bσ mV . The area of the

solar panel is: 0.0062 00003.0± 2m

Data Analysis

In order to find the fill factor, efficiency, and thermal resistance of the panel, the

data for the intensity of the light had to be converted into incident power in watts, and the

amount of power incident on the panel had to be calculated. Also the power delivered to

the resistor had to be found. The conversion for foot-candles is:

1 01610281.=fc 2mwatts . Converted power is simply the product of the current and

the voltage. These values can be found in Table 4.

)/( 2mwattsP )(wattsPI PIσ )(wattsPC PCσ

1529.8 9.48 0.046 0 0

1529.8 9.48 0.046 0 0

1523.3 9.44 0.046 0.0359 0.0004

1520.1 9.42 0.046 0.0929 0.0009

1521.7 9.43 0.046 0.0977 0.0010

1521.7 9.43 0.046 0.1220 0.0013

1523.3 9.44 0.046 0.1308 0.0014

1520.1 9.42 0.046 0.1349 0.0014

1516.8 9.40 0.046 0.1398 0.0014

1526.5 9.46 0.046 0.1390 0.0014


42

1521.7 9.43 0.046 0.1390 0.0014

1521.7 9.43 0.046 0.1379 0.0014

1521.7 9.43 0.046 0.1358 0.0014

1520.1 9.42 0.046 0.1330 0.0014

1520.1 9.42 0.046 0.1265 0.0013

1516.8 9.40 0.046 0.1174 0.0012

1513.6 9.38 0.046 0.1055 0.0011

1512.1 9.37 0.046 0.0835 0.0009

1513.6 9.38 0.046 0.0591 0.0006

1512.1 9.37 0.046 0.0320 0.0003

Table 4

The error on P is .81 2/ mwatts . IP was the power incident on the solar panel,

and CP was the power that the solar panel converted.

In order to find the fill factor and the efficiency, it is necessary to know the

voltage and the current at the maximum power point. The maximum power point can be

found by plotting the converted power vs. the voltage. See Figure 20. From this graph,

the voltage at the maximum power point can found. The current at the maximum power

point can then be found from a graph of the current vs. voltage. See Figure 21. The

current at the maximum power point is 0.41=mppI 308.0± mA, and the voltage is

41.3=mppV 024.0± V. Using these numbers, the short circuit current, the open circuit

voltage, and equation 6 gives a fill factor of =ff 0.419 006.0± . The efficiency can then


43

be found using equation 8, the fill factor, and the incident power at the maximum power

point =IP 9.40 watts. This yields an efficiency of 0149.0=η 0003.0± .

The thermal resistance of the panel can be found using equation 21. A Type E

thermocouple table shows that a -0.9mV voltage difference corresponds to a T∆ of K1 .

The error on T∆ is as large as 0.5 K . This gives a thermal resistance of

0.000659 0003.± 30 WKm /2 or 6.59 30.3± WKcm /2 .

Figure 20: power vs. voltage


44

Figure 21: current vs. voltage

Conclusions

The value calculated for the fill factor of 0.419 is very low compared to expected

results of 0.75-0.85. Also, the efficiency of the panel 1.49 % is very low. A typical

amorphous silicon solar cell has an efficiency between 6 % and 9%. The low efficiency

of this panel is a result of the fact that only about half of the area of the panel is covered

in solar cells. This automatically cuts the efficiency of the cell in half. Another factor

leading to the low efficiency is the reflection of incident light off the plastic cover.


45

The experiment to calculate the thermal resistance was very rough, and more

precise equipment would be needed in order to get an accurate result. The largest

inaccuracy involved the measuring of the temperature with the thermocouples. The

multimeter used to measure the voltage difference only made measurements down to a

tenth of a millivolt, this made the T∆ data very vague. Subsequent experiments should

use multimeters capable of measuring microvolts.

The data for the incident power was also inexact. This is because no estimation

was made of how much light the clear plastic cover reflected. It is reasonable to include

the loss of power due to reflection in the measurement of the efficiency of a panel so that

value should not have been affected. However, the thermal resistance value is lower than

it would be if light reflection had been taken into consideration.

Experiments should be run on a wide range of solar cell packages to test their

thermal resistance. In these experiments, it should be possible to estimate reflection

losses by placing the solar panel’s plastic cover over the light meter. Also, various heat

sinks should be used to see if there is any effect on the thermal resistance.


46

Conclusion

Concentrating systems could make solar technology an economical source of

energy production. A concentrating system utilizes less photovoltaic material and is

therefore less expensive to produce than a conventional solar panel. However,

concentrating systems suffer from a loss of power due to the effect that increased

temperature has on a solar cell. As the temperature of a solar cell increases the open

circuit voltage, efficiency, and output power of the cell decrease.

With this in mind, a mathematical model of power conversion vs. illumination has

been created. This model shows that beyond a certain illumination, the power converted

by a cell decreases. The model can be used to calculate the illumination at which a

maximum level of power is converted. Manipulation of the power conversion model

shows that the thermal resistance of a cell can greatly affect the amount of power a cell

converts at increased levels of illumination. Great attention should be paid to decreasing

the thermal resistance of solar cells used in concentrating systems.

As a result of these findings, an experiment has been carried out to determine the

thermal resistance of a typical solar panel. The experiment suffered from a lack of

appropriate equipment; however, a rough measurement of the thermal resistance was

made. The cell’s thermal resistance, along with a measured value for the efficiency of the

cell, and some typical values for efficiency loss with increased temperature, and

reflection losses can be plugged into the power conversion model. See Figure 22. The

model predicts that a Radio Shack Sola 6 volt, 50 mA panel will produce a maximum

power of 0.142 2/ cmw at an illumination of 20.25 2/ cmw .


47

Figure 22: Power conversion curve for a Radio Shack Sola 6 volt, 50 mA panel.

( 59.6,004.,0149.,06. ==== TTN Rµηα wKcm /2 )

Future research should focus on a confirmation of the power conversion model

using solar simulators capable of producing the adequate concentration ratios. If verified,

the presented model would allow an optimal match to be made between concentrating

system and solar cells. This would ensure the most efficient use of the available material,

and bring solar technology closer to being an economical source of energy production.

)/( 2cmwPC

)/( 2cmwPI


48

References

1. Welford, W.T Winston, R. (1998) The Optics of Nonimaging Concentrators. (Academic Press,

New York)

2. Perihelion (2002) http://www.imakenews.com/solarlight/e_article000032551.cfm (Solar Electric

Light Fund) 4. Hettelsater, D. (2002) Photovoltaic Technology Overview. University of California, Santa Cruz. 5. Hathwar, M. (2001) Photovoltaic Technology.

http://mhathwar.tripod.com/thesis/photovoltaic_technology.html 6. Cheremisinoff, P. Regino, T. (1978) Principles and Applications of Solar Energy. (Ann Arbor

Science Publishers, Inc., Ann Arbor, Michigan) 7. Hsieh, J. (1986) Solar Energy Engineering. (Prentice-Hall, Inc., Englewood Cliffs, New Jersey) 8. Anicin, B.A. Babovic, V.M. Davidovic, D.M. (1989) Fresnel Lenses. American Journal of Physics

57. 4, 312-316 9. Mulligan, W. Terao, A. Daroczi, S. Chao Pujol, O. Cudzinovic, M. A Flat-Plate Concentrator:

Micro-Concentrator Design Overview. (SunPower Corporation, Sunnyvale, California) 10. Radziemska, E. Kleugmann, E. (2002) Thermally Affected Parameters of the Current-Voltage

Characteristics of Silicon Photocell. Energy Conversion and Management 43. 1889-1900 11. Hettelsater, D. (2002) Solar Cell Lab. University of California, Santa Cruz 12. Kasap, S.O. (2002) Principles of Electrical Engineering Materials and Devices, Second Edition

(Mc Graw Hill) 13. Hook, J.R., Hall, H.E. (2000) Solid State Physics, Second Edditon. (John Wiley & Sons) 14. Radziemska, E. (2003) The Effect of Temperature on the Power Drop in Crystalline Silicon Solar

Cells. Renewable Energy 28. 1-12 15. Mosolam Shaltout, M.A., El-Nicklawy, M.M., Hassan, A.F., Rahoma, U.A., Sabry, M. (2000) The

Temperature Dependence of the Spectral and Efficiency Behavior of Si Solar Cell Under Low Concentrated Solar Radiation. Renewable Energy 21. 445-458

16. Verlinden, P.J., Terao, A., Smith, D.D., McIntosh, K., Swanson, R.M., Ganakas, G., Lasich, J.B.

Will We Have a 20%-Efficient (PTC) Photovoltaic System? (SunPower Corporation, Sunnyvale, CA)


49

Acknowledgements

I am deeply grateful to professor Ali Shakouri for his guidance and patience with

me throughout the duration of this project. His continued faith in me, as I struggled to

complete this project, was extremely helpful. The direction and assistance of Clemens

Heusch was also very valuable. Finally, I must thank my mother, who simply would not

allow her son to wallow on in procrastination. Her help and motivation are the reason this

project was completed.

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