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INVERTIS JOURNAL OF RENEWABLE ENERGY Volume 4 Patron Umesh Gautam
Chief Editor R.M. Mehra Sharda University
EDITORS
October-December 2014
No. 4
CONTENTS Exact analytical solutions of the parameters of different generation real solar cells using Lambert W-function : A Review Article Swati Sharma, Poonam Shokeen, Basant Saini, Sugandha Sharma Chetna, Jyoti Kashyap, Renu Guliani, Sandeep Sharma, Udaibir Manoj Khanna, Amit Jain and A. Kapoor
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Impact of Photovoltaic Appliances on Load Factor Ashwani Bawa, Sanjay Mahajan, Meenakshi Bawa and Sonika Virdi
195
Technical Note on Modeling and the Optimization of the Batteries for the Renewable Sources of Energy Kamal Nain Chopra
199
H2 Generation from Cu2O Quantum Dots (QDs) Sensitized Cu/p-CuI Photoelectrode R.D.A.A. Rajapaksha, C.A.N. Fernando and S.N.T. De Silva
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Ranjana Jha N S I T, Delhi University
Amita Gupta S S P L, D R D O, Delhi
Ediorial Assistance Sumit Kumar Gautam Owned, Published and Printed by Sanjeev Gautam, 60/10, Old Rajinder Nagar, New Delhi - 110 060 Printed at Alpha Printers, WZ-35/C, Naraina Ring Road, New Delhi - 110 028. Ph : 9810804196 Chief Editor : Prof. R.M. Mehra, School of Engineering & Technology, Sharda University, Knowledge Park 3 Greater Noida - 201 306, E-mail :
[email protected]
Exactof analytical solutions of Vol. generation solar ;cells Lambert W-function: A Review Article Invertis Journal Renewable Energy, 4, No.real 4, 2014 pp. using 155-194
Exact analytical solutions of the parameters of different generation real solar cells using Lambert W-function: A Review Article Swati Sharma1, Poonam Shokeen1, Basant Saini1, Sugandha Sharma1, Chetna1, Jyoti Kashyap1, Renu Guliani2, Sandeep Sharma3, Udaibir4, Manoj Khanna5, Amit Jain6 and A. Kapoor1 1 Photovoltaic and optoelectronics lab, Dept. of Electronic Science, UDSC 2 Dayal Singh College, University of Delhi, India 3 Dehradun Institute of Technology, Dehradun,India 4 Acharya Narendra Dev College, University of Delhi, India 5 Bhaskaracharya College of Applied Sciences, University of Delhi, India 6 Rajdhani College, University of Delhi, India E-mail:
[email protected] Abstract Present article reviews exact closed-form solution based on Lambert W-function to express the transcendental current-voltage characteristic of solar cell containing parasitic power consuming parameterslike series and shunt resistances. In this article different generation solar cells namely: inorganic solar cell, organic solar cell, dye-sensitized solar cell and hybrid solar cells are investigated. Equivalent circuits and explicit solutions of different cells are reviewed in detail to substantiate the relevance of renowned Lambert W-function in area of photovoltaic. Transient analysis using Lambert W-function is also reviewed to demonstrate the validity of W-function. Key words: Lambert W-function, current-voltage relationship, solar cell.
1.
Introduction
Equations where linear and exponential responses are combined appear in many areas of physics and engineering. Some examples are currentvoltage relationships of solar cells, photo detectors and diodes used as circuit elements. It is always desirable to express current as an explicit analytical function of the terminal voltage and vice versa. Such implementation would be computationally advantageous in device models that are to be used recurrently in circuit simulator programs, in problems of device parameter extraction, etc. Several attempts have been approached traditionally using iterative or analytical approximations [1-3], Lagrange's method of undetermined multipliers [4], approximation methods, least-squares numerical techniques [5] to achieve the explicit solutions containing only common elementary functions. A cautious exploration of
literature reveals that use of a function known as Lambert W-function [6, 7] commonly as ''W-function'' which is not frequently used in electronics problem is extremely important for solving such kind of problems. Solutions based on this function are exact and explicit and are easily differentiable. W-function originated from work of J.H. Lambert [8] on trinomial equation that he published in 1758 and was discussed by Euler [9] in 1779. W-function is defined by the solution of equation W exp(W)=x: Although rarely used, its properties are well documented [10-13] and several algorithms were published for calculating Wfunction. Some recent work includes exact analytical solution based on W-function for the case of a nonideal diode model comprised of a single exponential and a series parasitic resistance, bipolar transistor circuit analysis using W-function[14], and photorefractive two-wave mixing [15].
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Swati Sharma et al. Present article reviews the use of W-function to find the explicit solution for the current and voltage and use them to extract various parameters of different generation of solar cells. Comparisons are also made with the experimental data. 1.1 Introduction of Solar Cell
characteristics such as semiconductor material, application, and stage of development. 1.4.1 First Generation The first generation of solar cells is currently the most commercially widespread, as well as the oldest. These are conventional, silicon wafer solar cells. They use thinly sliced wafers of silicon as a semiconductor, that part of the solar cell that absorbs photons, exciting its electrons into creating employable electric current. Silicon solar cells currently control roughly 80% of the PV market due to their relatively high conversion efficiency. Because of the high investment cost, today's solar cells occupy only niche markets for power production although that is expected to change as second generation solar cells become cheaper and more efficient. 1.4.2 Second Generation
The ability to do work is called energy. Primary energy is the energy embodied in natural resources, such as coal, natural gas, and renewable sources. The final energy is the energy delivered to end-use facilities where it becomes usable energy that can provide services such as lighting, refrigeration, etc. The industrial revolution brought new machines and new energetic needs during the 19th century creating a major exploitation of coal to meet the increasing demand of energy. The early 20th century witnessed the first exploitation of petroleum, natural gas and later, nuclear energy as sources of energy to cover the ever increasing demand .The exploitation of non-renewable resources for energy leads to its depletion and has many disadvantages like pollution, global warming and environmental degradation. This necessitates exploring new energy sources which are abundant, inexpensive and environmentally friendly. Governments and industry across the globe are investing heavily in alternative renewable energy sources such as solar photovoltaic (PVs), wind, geothermal, hydroelectric, wave, tidal to meet the energy demands. 1.4 Generations of solar cells As solar photovoltaic (PV) technology solar should be omitted and no underline technology advances, solar cells, the electricity-producing heart of the solar panel, have been divided into four categories or generations. These represent a sort of timeline of solar innovations, separated by
Second generation photovoltaics include thin-film and building integrated (BIPV) technology. These solar cells are thinner, more flexible and more inexpensive than their first generation counterparts. Here the types of semiconductor materials broaden, from amorphous silicon (a-Si) to cadmium telluride (CdTe) and copper-indium-gallium-selenide (CIGS). Second generation, thin-film solar cells are expected to take over market dominance from first generation solar cells within five to ten years. So far their flexibility and diversity of available applications has not been enough to overtake conventional silicon solar cells, whose conversion efficiency often doubles that of thin-film products on the market. Laboratory testing has shown efficiencies equal to or greater than silicon solar cells, but manufacturers have not been able to transfer these numbers to market as of yet. 1.4.3 Third Generation Third generation solar cells are getting closer to market. They are still mostly in the laboratory/testing phase. Technologies include plastic (polymer) solar cells, photoelectrochemical solar cells and organic dyesensitized cells. Third generation solar cells promise incredibly cheap production costs and applications -to the point of solar paint and homemade solar cells. Barriers at this point include fast degradation of the cells and very low efficiencies. Nonetheless, third generation solar cells are expected to have a very promising future.
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Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article 1.4.4 Fourth Generation Fourth generation solar cells based on 'inorganicsin-organics', offer improved power conversion efficiency to current 3rd Generation (3G) Solar Cells, while maintaining their low cost base .Multi-junction concentrator cells belong to this category in which film layers are built up in such a way as to capture solar energy in different parts of the sun's spectrum. The main drawback is that today they carry a very high price tag. Multi-junction solar cells like the CSPs are really efficient when there is a very intense solar radiation. In other words, they are perfectly adapted for operations in desert zones. New Fourth generation Solar cells are being developed in the recently commenced Euro 11.6M European Union FP7 SMARTONICS programme [4] The SMARTONICS project aims to develop smart machines, tools and processes for large area production of Fourth generation solar cells engineered on the nanoscale, using roll-to-roll printing technology for high throughput and cost-efficient fabrication. 1.5 Semiconductor Solar Cell: In its simplest form solar cell is a simple p-n junction device. The p-n junction is effectively an interface between n and p type semiconductors [512]. The fundamental characteristics of a junction is presence of a strong electric field at the junction. The band diagram of such junction is shown in fig 1.1.
take place only when the energy of incident photons (h?) must equal or exceed the band gap Eg of the semiconductor. The ideal bandgap for solar cell materials should be between 1-2eV. The curve showing dependence of bandgap on it should be effciency instead of P of solar cell is given in fig 1.2.
Fig. 1.2 : Efficiency Vs Bandgap.
PV devices can be modelled as an ideal diode in parallel with light induced current generator ILwhose magnitude is a function of generation of electron-hole pair by absorption incoming light and collection efficiency for these charge carriers. The currentvoltage equation modified by light is given by : I = I0 [ exp(q V / k T) -1) - IL
(1.1)
Where: I = External current flow IO = Reverse saturation current q = fundamental charge (1.602 x 10-19C) k = Boltzmann's constant T = Temperature in Kelvin IL = Light induced current Thus the I-V curve is lowered by an amount proportional to light generated current as shown in fig. 1.3.
Fig. 1.1 : Semiconductor p-n junction in equilibrium (a) and under illumination (b).
In equilibrium, the electrochemical potentials on two sides of the junction are equal, and there is no net electric current. Under illumination electron-hole pairs are generated in the semiconductor and are subsequently separated by the electric field of the junction (fig. 1.1 b). The electron-hole generation will
When the device is short circuited (V=0), the terms from the diode equation cancels in Eq 1.1. However there is a short circuited current ISC through the device. Thus the I-V characteristics of fig. 1.3 cross the I-axis at negative values equal to IL. When there is an open circuit (I=0) across the device, V=VOC and is given by
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Swati Sharma et al.
Fig. 1.3 Characteristics of typical solar cell.
VOC = kTq ln[IL/ IO +1]
(1.2)
High values of V OC are achieved when (a) diffusion length of charge carrier is as high as possible (b) doping concentration ND and NA is as high as possible (c) crystal volume is as small as possible. From the I-V curve of solar cell it is evident that maximum power is delivered by the in the rectangular region of the curve. The points on the current and voltage axis corresponding to rectangular area are referred as optimum current (I mp) and optimum voltage (Vmp) respectively.The maximum power delivered in this area is given by Pm which is equal to product of optimum current and optimum voltage i.e, Pm =VmP Imp. It is clear that product VOC ISC is always less than Vmp Imp .the ratio VmP Imp / VOC ISC is called the fill factor (FF), and is figure of merit for solar cell design. FF measures the squareness of curve and is always between 0 and 1. For optimized solar cell it is between 0.6-0.75 while for ideal solar cell it is equal to 1. The optical properties of solar cell are key in achieving high performance. To generate as many carriers we need to (a) Reduce reflection from silicon (b) Reduce reflection from metal top surface (c) increasing absorption of light in semiconductors. They can be achieved by (a) using AR coating (b) Texturing and controlling the band gaps respectively.
resistances are due to sources at front side, because the fraction covered with the metal must be minimized on this side. In reality the series resistance is of distributed in nature but while modelling the cell it is taken as a lumped value Rs. this lumping approach is only valid when potential differences across the cell are very small. Shunt resistance is represented by Rsh. The cause of shunt resistance could be surface leakage along the edges of the cell, by diffusion spikes along dislocations or grain boundaries. They can also be due to fine metallic bridges along the microcracks, grain boundaries, or crystal defects such as stacking faults after the contact metallization has been applied. The new model of solar cell with series and shunt resistance is given in fig. 1.4.
Fig. 1.4 : Equivalent circuit of real solar cell.
This is also known as single diode model where dark current (ID) is described by a single exponential dependence modified by diode ideality factor n. Corresponding to above equivalent circuit new current voltage relation for real solar cell will be :
⎛ ⎛⎜ V + IRS ⎞⎟ ⎞ V + IRs nV − I o ⎜ e⎝ th ⎠ − 1 ⎟ I = IL − (1.3) ⎜ ⎟ Rsh ⎜ ⎟ ⎝ ⎠ Where symbols have their usual meanings. This equation is transcendental in nature and hence it is not possible to solve it for V in terms of I and vice versa. With addition of three new quantities namely series resistance (Rs), shunt resistance (Rsh) and diode ideality factor n , current-voltage characteristics of real solar cell modified accordingly. For an optimum real
Till now only ideal solar cells are considered not having any type of series or shunt resistance but real solar cells do have so [10-13, 14]. The series resistance arises due to (a) sheet resistance of the emitter type region (b) the resistance of the bulk type region (c) the contact resistance between metallisation and silicon at the front side and the back side and the resistance of metallisation itself. Almost all series 158
Fig. 1.5 : Efficiency curve
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article solar cell Rs should be less (mΩ), Rsh should be more (mΩ) and diode ideality factor between 1.0 and 2.0. Fill factor do depends on these new quantities. FF degrades with increasing value of Rs and improves with increasing value of Rsh. as :
η=
The efficiency of solar cell is define mathematically
POUT VOC ISC FF = PIN PIN
(1.4)
This relation implies that efficiency of solar cell increases with increase in VOC and ISC. There is another efficiency in relation to solar cell. i.e, Quantum Efficiency (QE). This is defined as ratio of photons incident to that of carriers collected. The curve showing QE vs wavelength is given in fig. 1.6.
equation W exp (W) = x. Alternatively it is defined to be the multivalued inverse of the function W → WEW . The history of Lambert W-function goes back to 18th century when Lambert and Euler published a paper [16-17]. In 1758, Lambert solved the trinomial equation x = q + xm by giving a series development for x in powers of q. later, he extended the series to give power of x as well [18-19]. In [17], Euler transformed Lambert's equation into the more symmetrical form xαxβ=(α - β)vxα+β
(2.1)
by substituting x-β for x and setting m = αβ and q = (α - β)v. Euler's version of Lambert's series solution was thus x = 1 + nv + 1/2 n (n + α + β) v2 + 1/6 n (n + α + 2β) (n + α + β) v3 + 1/24 n (n + α + 3β) (n + α + 2β) (n +α + β) v4 + etc. (2.2) After deriving the series, Euler looked at special cases, starting with α = β. To see what this means in the original trinomial equation, divide eq [ 2.2] by (α - β) and then let β → α to get Logx=vxa
(2.3)
Euler noticed that if we can solve eqn [ 2.3] for α =1, then we can solve it for any α = 1. to see this , multiply eqn [2.3] by α, simplify α log x to log xα, put z = xα and u = αv .we get
Fig. 1.6 : Quantum Efficiency Curve.
2.1 On Lambert W-function Mathematical functions do not by themselves uncover new physics-rather they assist the physicist by facilitating numerical and algebraic computations. A physicist therefore looks for many things in function before applying it to any problem. The first thing he looks for its general applicability. A book by Abramowitz and Stegun [15] contains is full of mathematical functions that can be used in physical problems most commonly. The second feature he looks for the function's ease for numerical evaluation and to pertinent algebraic properties. Such mathematical functions make the work of a physicist easy for evaluating physical problems or we can say physics and mathematics meets to explore a new world of interdisciplinary studies. One such function is Lambert W-function commonly known as W-function. The Lambert W is a transcendental function defined by solution of
log z = uz, which is just eqn[2.3] with α = 1. To solve this equation using [2.2] , Euler first put α = β = 1 and then rewrote [2.2 ] as a series for (xn -1 ) / n. Next he set n = 0 to obtain log x on the left hand side and a nice series on the right hand side:
Log Ax = v +
2! 2 32 3 4 3 4 54 5 v + v + v + v + ..... 2! 3! 4! 5!
(2.4)
This series which can been seen to converge for modv < 1/e, defines a function T(v) called the tree function. It equals -W (-v) where W(z) is defined to be the function satisfying W(z) e(w(z)=z
(2.5)
This is Lambert W-function. Several numerical approximations are available for W-functions [20-23].
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Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article Multiplying and dividing by Rs Rsh in exponential term we get
1+
blue and grey solar cell.
V + i( Rs + Rsh ) I 0 Rsh
(2.16)
Taking LCM of L.H.S and rearranging R.H.S ( Rsh I 0 + Rsh I ph − V − i( Rs + Rsh )
dependence modified by the diode ideality factor n the current-voltage relationship is given by: i = I ph
I0
−
⎛ V + iRs ⎞ ( Rs + Rsh ) = exp ⎜ ⎟ ⎝ nVth ⎠ ( Rs + Rsh )
Fig. 2.2 : Current and Voltage characteristics of
⎛ ⎛⎜ V + iRs ⎞⎟ ⎞ V + iRs nVth ⎠ ⎜ ⎝ − − Io e − 1⎟ ⎜ ⎟ Rsh ⎜ ⎟ ⎝ ⎠
I ph
Rsh I0 ⎛ ⎞ ⎛ − Rs (V + i( Rs + Rsh ) ⎞ VRsh = exp ⎜ ⎟ exp ⎜ ⎟ (2.17) ⎝ nVth ( Rs + Rsh ) ⎠ ⎝ nVth ( Rs + Rsh ) ⎠
(2.12)
where i and V are terminal current and voltage in amperes and volts respectively. I0 the junction reverse current (A), n the junction ideality factor and Vth the thermal voltage (kT/q), and Rs and Rsh are series and shunt resistance respectively. Eq [2.12] is transcendental in nature hence it is not possible to solve it for V in terms of I and vice versa. However explicit solutions can be obtained using properties of W-function. ⎛ V + iRs ⎞ ⎛ ⎞ ⎜ ⎟ V + iRs nVth ⎠ ⎜ ⎝ i = I ph − − I o exp − 1⎟ (2.13) ⎜ ⎟ Rsh ⎜ ⎟ ⎝ ⎠
Multiplying both sides by exp
Rs Rsh ( I ph + I0 ) nVth ( Rs + Rsh )
( Rsh ( I0 + Iph ) − V − i(Rs + Rsh )) Rsh I 0 ⎛ Rs Rsh ( I ph + I 0 ) ⎞ ⎛ −Rs (V + i( Rs + Rsh ) ⎞ exp ⎜ ⎟ exp ⎜ ⎟ ( ) + nV R R sh ⎠ ⎝ nVth ( Rs + Rsh ) ⎠ (2.18) ⎝ th s ⎛ Rs Rsh ( I ph + I0 ) ⎞⎛ ⎞ VRs = exp ⎜ ⎟⎜ ⎟ ⎝ nVth ( Rs + Rsh ) ⎠ ⎝ nVth ( Rs + Rsh ) ⎠ From the above eqn[ 2.18] we get:
Rearranging the terms in eq. (2.13) we get:
(
)
I ph ⎛ V + iRs ⎞ V + i (Rs + Rsh ) 1+ – = exp ⎜ ⎟ (2.14) I0 I0 Rsh ⎝ nVth ⎠
Rs Rsh ( I0 + I ph ) − V − i( Rs + Rsh )
⎛ R R (I + I Multiply both sides by exp ⎜ sh s ph 0 ⎜ nV ( R + R sh ⎝ th s rearranging
⎛ Rs [ Rsh ( I ph + I0 ) − (V + i( Rs + Rsh ))] ⎞ exp ⎜ ⎟ nVth ( Rs + Rsh ) ⎝ ⎠ ⎛ Rsh ( Rs I0 + Rs I ph + V ) ⎞ Rs I0 Rsh exp ⎜ ⎟ nVth ( Rs + Rsh ) nVth ( Rs + Rsh ) ⎝ ⎠
nVth ( Rs + Rsh )
⎞ ⎟⎟ and ⎠
( Rsh I0 + Rsh Iph − V − i( Rs + Rsh )) ⎛ Rs + Rsh ( I ph + I 0 ) ⎞ exp ⎜ ⎟ ⎝ nVth ( Rs + Rsh ) ⎠
⎛ ⎞ VRsh exp⎜ ⎟ + nV R R ( ) ⎝ th s sh ⎠
(2.19)
This is of the form: W(x) eW(x) = x
⎛ −R (V + i(Rs + Rsh )) ⎞ exp⎜ s ⎟ ⎝ nVth (Rs + Rsh ) ⎠
⎛ Rs Rsh (Iph + I0 )(V + i(Rs + Rsh )) ⎞ = +Rsh I0 exp⎜ ⎟ ⎜ ⎟ nVth (Rs + Rsh ) ⎝ ⎠
we get:
Wthere (2.15)
⎛ Rsh ( Rs I0 + Rs I ph + V ) ⎞ Rs I 0 Rsh exp ⎜ ⎟ nVth ( Rs + Rsh ) nVth ( Rs + Rsh ) ⎝ ⎠
(2.20)
⎛ Rs ⎡ Rsh ( I ph + I0 ) − (V + i( Rs + Rsh )) ⎤ ⎞ ⎣ ⎦⎟ W(x) = ⎜ ⎜ ⎟ nVth ( Rs + Rsh ) ⎝ ⎠
(2.21)
x=
161
Swati Sharma et al. W(x) is Lambert W-function and hence solution for equation can be written as : V = -i(Rs+Rsh)+Rsh Iph - nVthLambert W − Rsh (i − I ph − I 0 ) ⎞ ⎛ ⎜ I0 Rsh e ⎟ nVth ⎜ ⎟+ I R ⎜ ⎟ 0 sh nVth ⎜ ⎟ ⎝ ⎠
(2.22)
These are the exact analytical solutions of I and V where arguments of W-functions only contain corresponding variable and the model's parameters. This method of separating current and voltage from the current-voltage relationship of real solar cell was first time reported by Jain and Kapoor [34]. In addition to the apparent advantage another advantages lie in high computational speed and more accuracy.Due to its trivial properties this function is a useful tool in solving transcendental equation. Its significance in solving various physical problems is reported. The current-voltage relationship of real solar cell is also transcendental in nature and hence various numerical and analytical methods were used for extracting the solar cell parameters, as it was not possible to express current in terms of voltage explicitly and vice-versa. These numerical and analytical techniques are not accurate and consume much computational technique. The remedy to these problems came when the I-V equation of solar cell was written in the form : W(x) e W(x) = x
voltage (Voc), which is the output voltage when the load impedance is much greater than the device impedance or no load is connected at output terminal of solar cell; the short circuit current (Isc), which is the output current when the load impedance is much smaller than device impedance; and the "Fill Factor", the ratio of maximum output power to that of product of Voc and Isc. These three parameters determine the efficiency and the circuit conditions to be used with the cell or array of such cells. For an ideal solar cell having zero series resistance and infinite shunt resistance the current-voltage equation for the cell will be:
i = I ph
⎛ V ⎞ ⎛ ⎞ ⎜ ⎟ nV ⎜ − I 0 exp⎝ th ⎠ − 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
(2.24)
When Iph is equal to zero the current through such a cells or photodiode is determined only by the current through the ideal diode (represented by Shockley equation). For positive voltages current increases exponentially. Upon illumination the light generates a photocurrent Iph that is simply superimposed upon the normal rectifying IV characteristics of diode. [34 ]. The addition of Iph results in region in IV quadrant of IV characteristic of solar cell. Maximum voltage in IV quadrant develops when dark current just manages to cancel photocurrent Iph. This voltage is open circuit voltage Voc, I0 determines the height of characteristics.
(2.23)
where x and W(x) are defined in eqns [2.20] and [2.21]. Using this relation and properties of Lambert W function (W-function) current and voltage are separated from the transcendental relation of solar cell. These solutions are further used by Jain and Kapoor [35] to extract various model parameter of solar cell, study of organic solar cell [36] and calculations of solar cell array parameters [37]. 2.3 Solar Cell parameters extraction Solar cell behaviour can be examined easily through three main parameters: the open circuit
Fig. 2.3 Current and Voltage characteristics of blue and grey solar cell
2.3.1 Open Circuit Voltage At infinite load I=0 and V=Voc , eqn[2.24] changes to
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Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article
⎡⎛ I ph ⎞ ⎤ Voc = nVth Ιn ⎢⎜ ⎟ + 1⎥ ⎣⎢⎝ I0 ⎠ ⎥⎦
(2.25)
Voc, the open circuit voltage is directly related to the bandgap of the semiconductor through the energy barrier height at the junction. For a perfect junction n = 1 and Voc attains its highest value, while for larger n, I0 is larger in such a way that Voc is reduced. Effect of the Iph/ I0 ratio is relatively small. Typically Voc of silicon cells under solar conditions are around 0.55 Volts. As the eqn[2.25] describes the logarithmic nature of open circuit voltage it saturates as a function of light intensity.
i=
Rs I0 Rsh e
For real solar cell i.e., a cell having both parasitic series and shunt resistances the equation of the cell changes to:
Rsh ( Rs I ph + Rs I 0 + V )
nVth ( Rs + Rsh ) nVth ( Rs + Rsh ) − Rs Rsh ( I0 + I ph ) W
nVth
Rs + Rsh V = i( Rs + Rsh ) + I ph Rsh − nVth W Rsh (−i + I ph + I 0 ) ⎞ ⎛ ⎜ I 0 Rsh e ⎟ nVth ⎜ ⎟+ I R ⎜ ⎟ 0 sh nVth ⎜ ⎟ ⎝ ⎠
For a typical cell the logarithmic nature of Voc can be depicted diagrammatically as : The current I0 depends upon the bandgap of the material used and the temperature. Also decrease in I0 makes Voc increase with increase in bandgap or decrease in temperature.
V Rs + Rsh
(2.27)
The arguments of W-function in eqn [2.27] only contains corresponding variables and the model's parameter. The explicit solution of open circuit voltage Voc in terms of W-function can be evaluated by substituting I = 0 in eqn [2.27] Voc = I ph Rsh − nVth W Rsh ( I ph + I0 ) ⎞ ⎛ ⎜ I0 Rsh e ⎟ nVth ⎜ ⎟+ I R ⎜ ⎟ 0 sh nVth ⎜ ⎟ ⎝ ⎠
(2.28)
where W (x) is Lambert W (x). The dependence of Voc on I0 in this case is as shown in fig. 2.5 is same as for ideal solar cell without series and shunt resistance. Fig. 2.4 : Open Circuit Voltage Vs I[0].
i = I ph
⎛ ⎛⎜ V + iRs ⎞⎟ ⎞ V + iRs nV − − I0 ⎜ e⎝ th ⎠ − 1 ⎟ ⎜ ⎟ Rsh ⎜ ⎟ ⎝ ⎠
(2.26)
Where symbols have their usual meanings. This equation being transcendental in nature cannot be solved for V in terms of I and vice versa. However, explicit solution for current and voltage can be expressed using W-function as: 163
Fig. 2.5 : Voc Vs I0 for Ideal Cell.
Swati Sharma et al. Eqn. [2.28] suggests that open circuit voltage is independent of series resistance but depends on shunt resistance and ideal diode. Suppose the Rsh is not very high and the device is in dark. If a positive voltage is applied across the cell a voltage drop occurs across Rsh that is equal to voltage across the ideal diode. The current that can pass through the diode is determined by its IV characteristics [34]. The sum of currents through diode and Rsh yields the current through the solar cell for a applied voltage. Upon illumination, the current source generates the current Iph which is divided into current component part of which passes through the diode where a voltage drop is generated that is big enough to allow the rest of the Iph to go through the Rsh if output of the cell is open circuited. Its dependence on shunt resistance is shown in fig. 2.5.
Fig. 2.7 : Circuit Voltage Vs Ideality Factor.
or absence of electric fields on both sides of the junction, and the surface recombination velocity. The energy contained in sunlight is distributed over a wide range of wavelengths, and efficient conversion requires a wide spectral response. The bandgap dependence is due to absorption coefficient; wider bandgap materials absorb less sunlight and have smaller short circuit currents than narrow band gap material. Substituting V = 0 and I = Isc in eqn. 2.27 short circuit current is obtained as: Isc = Iph
(2.29)
Thus for a cell having zero series resistance and infinite shunt resistance short circuit current is equal to light generated current.
Fig. 2.6 : Open Circuit Voltage Vs Shunt Resistance.
From fig.2.6 it is clear that V oc increases exponentially for smaller values of shunt resistance than it becomes constant.Dependence of Voc on n is very strong as it is outside the W-function. n determines the shape of I-V curve and has therefore strong influence than Rsh or I0.
For a cell having finite series and shunt resistances the short circuit current is given by: ⎛ Rsh Rs ( I ph + I 0 ) ⎞ ⎛ ⎜ ⎟ ⎜ R I R e⎜⎝ nVth ( Rs + Rsh ) ⎟⎠ s 0 sh ⎜ W nVth ( Rs + Rsh ) ⎜ ⎜ I sc = − ⎝ Rs Rsh ( I ph + I0 ) + Rs + Rsh
2.3.2 Short Circuit Current: The short circuit current (Isc) is determined by the spectrum of the light source and the spectral response (electron-hole pairs collected per incident photon) of the cell. The spectral response in turn depends on the optical absorption coefficient, the junction depth, the width of the depletion region, the lifetimes and nobilities on both sides of the junction, the presence
⎞ ⎟ ⎟ nVth ⎟ ⎟ ⎠
(2.30)
In contrast to open circuit voltage, short circuit current depends both on series and shunt resistances. Variation of Isc with shunt resistance is shown in fig. 2.8. Initially Isc increases exponentially with increase in value of shunt resistance then it becomes constant.
164
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article Variation of Isc with shunt resistance is shown in fig. 2.9. Isc For small values of series resistance short circuit current remains constant, after that it starts decreasing rapidly. Variation of Isc with ideality factor is shown in fig. 2.10. Short circuit current increases in steps with increase in ideality factor.
Fig. 2.10 : Variation of Short Circuit Current with Diode Ideality factor.
Fig. 2.8 : Variation of Short Circuit Current Vs Shunt Resistance
Fig. 2.11 : Variation of Short Circuit Current with Temperature.
Fig. 2.9 : Variation of Series Resistance with Short Circuit Current
2.3.3 Series Resistance (RS) Series resistance is another very important factor that affect that functioning of the solar cell. The series
resistance arises due to (a) sheet resistance of the emitter type region (b) the resistance of the bulk type region (c) the contact resistance between metallisation and silicon at the front side and the back side and the resistance of metallisation itself. Almost all series resistances are due to sources at front side, because the fraction covered with the metal must be minimized on this side. In reality the series resistance is of distributed in nature but while modelling the cell it is taken as a lumped value Rs. this lumping approach is only valid when potential differences across the cell are very small [34, 37] for an ideal solar cell series resistance is zero but it is not true for solar cell. From the I-V characteristics shown in fig. 2.10 that for high series resistance the shape is approaching to triangular which is impractical.
165
Swati Sharma et al. Rsh = −
Voc − I ph + I 0
⎛ Voc ⎞ ⎜ ⎟ nV e⎝ th ⎠
(2.31) − I0
I-V characteristics of solar cell for different value of shunt resistance is as shown in fig. 2.11. Other than series and shunt resistance there are dynamic series and shunt resistance. The dynamic resistance Rso and Rsho at the open circuit voltage and short circuit current are given by:
⎛ ∂V ⎞ Rso = − ⎜ ⎟ v = Voc ⎝ ∂i ⎠
Fig. 2.12 : characteristics for different Series Resistance
2.3.4 Shunt resistance(Rsh) Shunt resistance is represented by Rsh. The cause of shunt resistance could be surface leakage along the edges of the cell, by diffusion spikes along dislocations or grain boundaries. They can also be due to fine netallic bridges along the microcracks, grain boundaries, or crystal defects such as stacking faults after the contact metallization has been applied. For an ideal solar cell shunt resistance should be infinite and for practical solar cell it should be very high. The mathematical expression for shunt resistance is :
⎛ Rsh ( I ph + I 0 ) ⎞ ⎞ ⎛ ⎜ ⎟ ⎜ I R e⎜⎝ nVth ⎟⎠ ⎟ sh 0 ⎟ Rsh W⎜ ⎜ ⎟ nVth ⎜ ⎟ ⎠ Rso = Rs + Rsh − ⎝ ⎛ Rsh ( I ph + I0 ) ⎞ ⎞ ⎛ ⎜ ⎟ ⎜ I R e⎜⎝ nVth ⎟⎠ ⎟ sh 0 ⎟ 1+ W ⎜ nVth ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
(2.32)
(2.33)
⎛ ∂V ⎞ Rsho = − ⎜ ⎟ i = Isc ⎝ ∂i ⎠
Rsho
⎛ Rsh ( I 0 + I ph − I sc ) ⎞ ⎞ ⎛ ⎜ ⎟⎟ nVth ⎜ I R e⎜⎝ ⎠ ⎟ sh 0 ⎟ Rsh W⎜ ⎜ ⎟ nVth ⎜ ⎟ ⎠ = Rs + Rsh − ⎝ ⎛ Rsh ( I 0 + I ph − Isc ) ⎞ ⎞ ⎛ ⎜ ⎟⎟ nVth ⎜ I R e⎜⎝ ⎠ ⎟ sh 0 ⎟ 1+W ⎜ nVth ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
(2.34)
Rso and Rsho are also the slopes of the I-V curve at open and short circuit conditions. 2.3.5 Fill factor The solar efficiency of the cell is defined as the ratio of the maximum power Pin to the solar power incident on the cell. A quantity called the fill factor (FF) is commonly introduced to relate Pm to the product VocIsc. Fig. 2.12b : characteristics for different Shunt Resistance
FF = Im Vm / VocIsc.= Pm/ VocIsc 166
(2.35)
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article FF is also the measure of squareness of the I-V curve of solar cell. For ideal solar cell it is equal to unity otherwise it is less than unity. Variation of FF with series resistance, shunt resistance are given in figs. 2.12, 2.13.
Fig. 2.15 : Characteristics of a diode for different ideality factors
2.3.6 Solar Cell Diode Ideality factor(n) Fig. 2.13 : Variation of Fill Factor with Series Resistance
The diode ideality factor has been introduced for a p-n junction solar cell after the consideration of the physical phenomenon that occurs in the diode. Most solar cells exhibit several exponential regions in the dark forward I-V characteristics which strongly suggest the presence of several current components (injection and recombination currents). Very seldom do the slopes of these exponentials (the value of n in qv/nkT) equal to 1 or 2. Transient Analysis of solar cell array and first and second order circuits using Lambert W-Function
Fig. 2.14 : Variation of Fill factor with Shunt Resistance
In real world applications a single cell is insufficient to be used as a power source. In order to be close to practical situations it is required to have expressions that take into account the current and voltages for a solar cell array, accounting contributions due to all the cells. PV generators are a collection of
Table 2.15 Comparison between experimental and theoretical data and relative accuracy Parameters
Voc(V) Isc(A) Rso(W) Rsho(W) Vm(V) Im(A) T(K)
Experimental Data of Charles et. al
Data using W-function
Accuracy
Blue solarCell
Grey Solar Cell
Blue Solar Cell
Grey Solar Cell
Blue Solar Cell
Grey Solar Cell
0.536 0.1023 0.45 1000 0.437 0.0925 300
0.524 0.561 0.162 25.9 0.390 0.481 307
0.53465 0.10229 0.44298 997.4018 0.43191 0.093396
0.52093 0.55931 0.16121 25.896 0.38473 0.48335
0.251 0.009 1.56 0.259 1.16 0.968
0.585 0.301 0.487 0.015 1.35 0.488
167
Swati Sharma et al. interconnected solar cells and other components. These cells can have any possible series- parallel connection to provide required terminal voltage and current ratings. Due to relatively high cost of solar cell array, the main objective of the system designer is to extract maximum available electrical power output at all insolation levels, for maximum utilization efficiency of the system. Since solar cells in array are never identical, which complicates the analysis of a large photovoltaic array operation under different load and environmental conditions. So efforts have been made to combine the cell parameters of the array into a single aggregate model to simplify calculations. In earlier works the array parameters were derived by different methods such as (a) incomplete analytical methods (b) complete analytical methods (c) simulation methods [38-41]. Some works deal with calculating series and shunt resistance of array with approximations [42]. This section of this chapter deals with calculation of various array parameters using Lambert W-function method to solve the current-voltage relationship of solar cell [43-44]. In recent years, some solar cell cross interconnection configurations have been proposed and tested to improve fault-tolerance [45-46]. These configurations are : 1.
2.
3.
Series-Parallel configuration- as shown in fig 3.1 (a) depicts constitution of parallel collection of strings having cells connected in series. Total-crossed-tied configuration - as shown in fig 3.1 (b) is obtained from simple SP array by connecting ties across each row of junctions. Bridge-linked configuration - as shown in fig 3.1(c) is obtained by connecting all cells in bridge rectifier fashion.
Fig. 3.1c : Bridge-linked configuration.
3.1.1 Explicit analytic solutions for a solar cell array The current-voltage relation of single solar cell in a photovoltaic array is given by:
⎛ i + I ph V − i Rs ⎞ V − i Rs In ⎜ − + 1⎟ = I0 Rsh nVth ⎝ I0 ⎠
(3.1)
where V and i are terminal voltage and current respectively; Iph is photocurrent; I0 is the diode reverse saturation current; Rs and Rsh are series and shunt resistance respectively; n is diode ideality factor ; Vth is the thermal voltage. In a series array consisting of N identical cells where the current through the array is equal to the current through individual cell it can be shown that: Ipha = Iph,
I0a = I0,
Rsa = N Rs,
Rsha =N Rsh,
Vtha =N Vth; And for parallel array : Ipha = N Iph,
I0a =N I0,
Rsa = Rs/N,
Rsha = Rsh /N,
Vtha = Vth; The explicit voltage equations for nth cell in an array as determined by solving equation 3.1is: Vn = Voc; Rsha = Rsh /N
(3.2)
where Vocn is given by :
Vocn = −LambertW
Fig. 3.1a : Series-Parallel
Fig. 3.1b : Total-crossed-
configuration.
tied configuration.
⎛ Iphn Rshn ⎞ ⎛ ⎜ ⎟ ⎜ Io Rsh e⎝ AVthn ⎠ n n ⎜ ⎜ AVthn ⎜ ⎝
168
⎞ ⎟ ⎟ nVthn + Rshn Iphn ⎟ ⎟ ⎠
(3.3)
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article Similarly the I-V equation of the array be of the same form as for the single cell in terms of voltage is: Vα = Vocα ( Rsα + Rshα )iα
(3.4)
Where Voca is given by: Vocα = −LambertW ⎛ Iphα Rshα ⎞ ⎛ ⎜ ⎟ ⎜ Io Rsh e⎝ AVthα ⎠ α ⎜ α ⎜ AVthα ⎜ ⎝
⎞ ⎟ ⎟ nVthα + Rshα Iphα ⎟ ⎟ ⎠
M
∑ V (i ) n
(3.5)
substituting Vn and ia in equation (3.63) different array parameters can be determined. Thus: N
∑V
Vtha =
(3.11)
thn
n =1
For identical cells:
Vtha = NVthn
(3.12)
Series resistance in array:
⎛ Vocα ⎞ ⎜ ⎟ AVthα ⎠ e⎝
(3.6)
Rsa =
N
∑R n= 1
(3.13)
sn
The explicit current equations for nth cell in an array as determined by solving equation 3.1 are:
Shunt resistance in array:
Vn in = Iscn + Rsn + Rshn
Rsha =
(3.7)
where Iscn (short circuit current for nth cell of the array) is:
I oα
−
Rsn
⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎟ ⎟ ⎟⎟ ⎠
Va Rsa + Rsha
∑R n =1
(3.14)
shn
⎛ 1 ⎞ ⎛ ⎞ ⎜⎜ ⎟⎟ ⎜ N ⎟ ⎝ ηn ⎠ = surd ⎜ Ion ,N ⎟ ⎜⎜ n=1 ⎟⎟ ⎝ ⎠
Ion = −
(3.8)
∏
Rshn inn + Rshn Iphn − Vn inn Rsn ⎛ −Vn + inn Rsn ⎞ ⎜⎜ ⎟⎟ Vth e⎝ n n ⎠ Rshn
(3.15)
and ηn =
Vthα N Vthn
where Photocurrent for open circuit array:
Expression for array current is:
I a = Isc a +
N
Reverse saturation current
Iscn = ⎛ ⎞ Rshn Rsn Iphn ⎛⎛ ⎜⎜ ⎟ ⎜⎜ AVthn ( Rsn + Rshn ) ⎟⎠ ⎝ Rs Io Rsh e n ⎜⎜ n n ⎜ ⎜ Rsn AVthn + Rshn AVthn LambertW ⎜ ⎜ ⎜⎝ ⎜ Rshn Rsn Iphn ⎜⎜ + ⎝ AVthn ( Rsn + Rshn )
(3.10)
a
n =1
where Iphoa is given by : Vocα + Ioα Rshα Rshα
When cells are connected in series current through the array is equal to minimum current through any cell while voltage across the array is equal to the sum of voltages across individual cells. Hence the voltage relation across the array will be:
Va (ia ) =
Two different photocurrents are introduced for array: Iphoa and Ipha. Iphoa is the photocurrent at open circuit voltage and current independent whereas Ipha is the array photocurrent at load and current dependent.
Iphoα =
3.1.2 Series Array:
(3.9)
169
⎛ 1 ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ N ⎟ ⎝ ηn ⎠ I ph oα = sud ⎜ Iphn ,N ⎟ ⎜⎜ n = 1 ⎟⎟ ⎝ ⎠
∏
(3.16)
Swati Sharma et al. where
second order circuits in this chapter. An assumption is made that:
Photocurrent for array:
(a) All the cells are identical in all respects. Ia
Ipha =
⎛ ⎛ ⎜ N ⎜ ⎛ Ia 1 − surd ⎜ ⎜1−⎜ ⎟ ⎜⎜ n =1 ⎜ ⎝ Iphn ⎠ ⎝ ⎝
∏
⎛ 1 ⎞ ⎜ ⎟ ⎞⎝ηn ⎠
⎞ ⎞ ⎟ ⎟ ⎟, N ⎟ ⎟ ⎟⎟ ⎠ ⎠
(3.17)
where surd(a, N) implies Nth root of a. It can be concluded from above equations that for series array having identical cells and so photocurrents, Iphoa = Ipha= Iph.
When cells are connected in parallel, current through array is equal to sum of currents through individual cells while voltage across the array is equal to voltage across any individual cells. (3.18)
and ia (V ) =
N
∑ i (V ) n
n= 1
A series-parallel array may be considered as an array consisting of M identical cells and having N strings of such arrays, where the current through the array is equal in each cell, thus the voltage will be the same as that of simple series array equation use eqn. [3.10] above. Expression for voltage of nth cell remains the same as in the case of a pure series array as above.
3.1.3 Parallel Array:
Va = Vn
(b) The array is a SP array, and that the entire array is dealt as a system constituting N strings in parallel and each string having M cells connected in series.
(3.19)
substituting in(V) in above equation and solving it results in: Series resistance for the array:
Voc a =
M
∑Voc
j
(3.23)
j =1
Consider a parallel array consisting of N identical cells where the voltage through the array is equal in each cell. The explicit current equations for nth cell in an array remains the same as in the case of a pure parallel array as above. Considering these series strings in parallel to each other thus keeping the voltage across the array constant and the current through the array is equal to sum of currents through parallel strings is evaluated with equation eqn.[3.19] above.
3.1.4 Series Parallel Array:
The equivalent model of solar cell consists of a diode, a series resistance and a shunt resistance in which shunt resistance paths are represented by Rsh. They are due to surface leakage along the edges of the cell, by diffusion spikes along dislocations or grain boundaries, or possibly by fine metallic bridges along microcracks, grain boundaries, or crystal defects such as stacking faults after contact metallization. Series resistance, represented by Rs, can arise from contact resistance to the front and back (particularly for high resistivity bases, 1-10 ohm-cm), the resistance of the base region itself, and sheet resistance of the thin diffused or grown surface layer. Although these are represented as lumped parameters for a single cell, when considering an array the net effect of these losses may be represented by a single lumped parameters Rsa and Rsha.
This is the array configuration that has been used to study the transient response of the first order and
Defining Rso = |dv/di| as the dynamic or incremental resistance of the PV element:
Rsa =
1
(3.20)
N
∑ Rs
n
n =1
Shunt resistance for the array:
Rsha =
1 N
1 Rsha n= 1
∑
(3.21)
N
Vtha in( Ioa ) Vthn in( Ion ) = Rsa Rsn n− 1
∑
(3.22)
170
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article
Rso =
3.1.5 Maximum Power:
Rs Rt 1− 1 + LambertW ⎛ R (ν + R ( I + I ) ) ⎞ ⎞ ⎛ s o ph ⎜ t ⎟ ⎜ ⎜⎜ ⎟⎟ ⎟ nV th ⎠⎟ ⎜ I 0 Rs Rt e⎝ ⎜ nVth ⎟ ⎜ ⎟ ⎝ ⎠
(3.24a)
where Rt = Rsh/(Rs+Rsh) For array Rsa = Rso * M / N
(3.24b)
A similar expression for Rsho can also be derived. For our analysis either Rso or Rsho is needed so Rso shall be used for all analyses to come.
Fig. 3.2 Mixed Array Power Vs Mixed Array Current.
Fig. 3.3 Mixed Array Power Vs Mixed Array Voltage..
A simple exact expression for calculating maximum power of an array could not be obtained due to constraints of maple software, a graphical representation of array power versus array current and array voltage is made in fig(3.2) and fig(3.3). The graphs are for an typical array with parameters Vtha = 0.73 V, Iph = 0.8 A (at an insolation of 1000W/m2), I0 = 0.5 mA, Rs = 0.05 Ω, Rsh = 105 Ω. The array consists of 18 parallel strings with 324 cells in series per string. 3.2 Transient Analysis - First Order Circuit First order circuits are those circuits that can be reduced to either RC or RL forms. The implication is that there may be more than one component of a particular type initially, provided that the circuit can be reduced to one equivalent value. It has been established by Zacharias et al [49] that for such circuits incremental method is the best approach to solve the first-order non-linear differential equation as compared to other existing techniques like 4th order Runge-Kutta method, linear and piecewise-linear models etc. The PV generator model considered by them does not account for current leakage losses in the form of shunt resistance for the PV generator. They have also expressed their concern over the nonexistence of an exact closed form solution to the transcendental current-voltage relation of a PV generator. An approach to overcome this difficulty has been proposed by Jain et al [50-53] where they have given an exact closed form solution to solve the transcendental current-voltage relation of a PV generator. We have developed a new approach based on incremental model of the PV generator [49] to study the effect of the losses in the PV generator due to series and shunt resistance. The use of Lambert W-function provides an exact closed form solution to solve the transcendental current-voltage relation of a PV generator. The describing differential equation, in each increment, is a linear equation, where the solution is presented in the form of a difference exponential equation. In this way, a greater degree of accuracy is attained to quantitatively estimate the transient behaviour, while at the same time, a qualitative description of the process can be visualized. This model also ensures the accuracy of a common integration program, even if a larger time step is used.
171
Swati Sharma et al. shown in fig 3.11. After the closure of the switch at t=0, as mentioned earlier in RL Load case the dynamic route along the i-v curve is determined starting from
Fig. 3.11 : PV Generator connected to a parallel RL Load circuit diagram.
Fig. 3.9 : PV Generator connected to a series RC Load dynamic route (heavy line) and the linear model
The solution for the capacitor voltage at t = tN+T is
(
)
vCN +1 = v N + RsaN iN + vCN − vN − RsaN iN e
where Bsan = 1 τ N
Bsan =
(
1
C R + RsaN
)
− Bsan T
(3.35) (3.36)
PV generator current is given by
iN + 1 =
vN + RsaN iN − vCN + 1 R + RsaN
(3.37)
Fig. 3.12 : PV Generator connected to a parallel RL Load dynamic route (heavy line) and the linear model.
A0 and terminating on A∞, as shown in fig 3.12 by a dark line. The circuit equation using Kirchoffs Voltage Law for the above circuit:
⎛1 1 L⎜ + ⎜ R Rsa ⎝ N
⎞ di vN + RsaN iN ⎟ LN + 1 + iLN +1 = ⎟ dt RsaN ⎠
(3.38)
leads to the solution for the inductor current iLN+1 at t = tN+T is iLN + 1 =
Fig. 3.10 : PV Generator current and voltage vs. time in a PV RC series circuit for (I) R = 3 Ohm and C=500microf (II) R = 14 Ohm and C=500 microf
vN + RsaN iN RsaN
(
)
+ iLN − vN − RsaN iN e
− Bsan T
(3.39)
where Bsan = 1/τ N
3.2.4 PV Generator connected to a Parallel RL Load Initial conditions are considered to be same as in the case of series RL Load. Consider a case where the PV generator is connected to a parallel R-L load via a switch S as
Bsan =
(
RRsaN
L R + RsaN
)
(3.40)
the PV generator current is given by
iN+1 =
174
v N + RsaN
( iN
+ iLN+1 )
R + RsaN
(3.41)
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article The circuit equation using Kirchoff's Voltage Law for the above circuit:
⎛1 CdvCN + 1 1 + vCN + 1 ⎜ + ⎜ dt ⎝ R RsaN
⎞ vN + RsaN iN ⎟= ⎟ RsaN ⎠
(3.42)
The solution for the capacitor voltage at t = tN+T is
ν CN +1 =
Fig. 3.13 : PV Generator current and voltage vs. time in a PV RL parallel circuit for (I) R = 14 Ohm and L = 0.3 H (II) R = 50 Ohm and L= 0.3 H.
(
R ν N + RsaN iN R + RsaN
)+
(
⎛ R ν N + RsaN iN ⎜ vCN − ⎜ R + RsaN ⎝
3.2.5 PV Generator connected to a Parallel RC Load Initial conditions are considered to be same as in the case of series RC Load.
where Bsan = 1 τ N
Bsan =
) ⎞⎟ e−B ⎟ ⎠
(3.43)
san T
R + RsaN CRRsaN
(3.44)
Consider a case where the PV generator is connected to a parallel R-C load via a switch S as shown in fig 3.14. After the closure of the switch at t=0, the system starts to operate at point Ao for an uncharged capacitor whereas for t= ∞, the capacitor behaves as an open circuit, so the operating point is A∞. Hence the dynamic route along i-v curve is determined, as shown by a dark line in fig 3.15.
Fig. 3.16 : PV Generator current and voltage vs. time in a PV RC parallel circuit for (I) R = 14 Ohm and C=500 microf (II) R = 50 Ohm and C=500 microf.
the PV generator current is given by Fig. 3.14 : PV Generator connected to a parallel RC Load circuit diagram.
iN + 1 =
vN + RsaN iN − vCN + 1 RsaN
(3.45)
3.2.6 Conclusion
Fig. 3.15 : PV Generator connected to a parallel RC Load dynamic route (heavy line) and the linear model.
In this chapter a comprehensive qualitative and quantitative analytical study of first order circuits powered by PV generator module using incremental method incorporating Lambert W-function has been completed. Various graphs have been plotted to describe the circuit behavior for a PV Generator current & voltage on a time line for RL series and parallel load circuits. Results obtained by using incremental model without incorporating shunt Resistances as reported by Zacharias et al. [49] (ZIM) 175
Swati Sharma et al. Table 3.1 Comparison of results between incremental model of Zacharias et al [49] (ZIM) and incremental model using LambertW function (LIM) for solutions in the PV-R-L series circuit for different time steps (R = 3W, L = 0.3H) Time (ms)
5 10 15 20 25 30 35 40 45 50 55 60 65
Step size = 0.1 ms
Step size = 1.0 ms
Step size = 2.5 ms
Step size = 5.0 ms
Z IM
LIM
Z IM
LIM
Z IM
LIM
Z IM
LIM
2.7966 5.3226 7.5787 9.5572 11.2357 12.5579 13.3757 13.5388 13.5409 13.5409 13.5409 13.5409 13.5409
1.637843 3.172524 4.610524 5.957922 7.22041 8.403316 9.511628 10.55001 11.52282 12.4341 13.28747 13.63772 13.63772
2.7967 5.3229 7.5792 9.5581 11.2370 12.5604 13.3802 13.5392 13.5409 13.5409 13.5409 13.5409 13.5409
1.637843 3.172524 4.610528 5.957931 7.220424 8.403337 9.511659 10.55006 11.52289 12.43419 13.28763 13.77709 13.77709
2.7974 5.3244 7.5818 9.5622 11.2437 12.5723 13.4041 13.5407 13.5409 13.5409 13.5409 13.5409 13.5409
1.637844 3.172527 4.610535 5.957944 7.220446 8.40337 9.511708 10.55013 11.52299 12.43434 13.28787 13.70251 13.70251
2.7997 5.3295 7.5905 9.5760 11.2658 12.6103 13.4806 13.5559 13.5425 13.5409 13.5409 13.5409 13.5409
1.637845 3.172531 4.610545 5.957964 7.220479 8.403421 9.511782 10.55023 11.52314 12.43456 13.28822 14.08708 14.09541
and incremental model using LambertW function (LIM) incorporating shunt resistance for solutions in the PV - R-L series circuit for different time steps (R=3ω, L= 0.3H) are tabulated in Table 3.1. Thus from the above study it may affirm that W function technique incorporating a finite shunt and series resistance overcomes the known inabilities due to the absence of shunt resistance in the PV generator, and hence provides good correspondence to existing experimental results. 3.3 Transient Analysis - Second Order Circuit Second order circuits are those circuits that have both capacitors and inductors as the circuit elements. The implication is that there may be more than one component of a particular type initially, provided that the circuit can be reduced to one equivalent value.
3.3.1 PV Generator connected to a Series RLC Load For a PV generator connected to series R-L-C circuit the governing differential equation can be written as: d 2 in + 1
din+ 1 in+ 1 (3.46) + =0 dt C dt This is a linear second order differential equation. When used with initial values for t = 0 (t = tN), it describes circuit behaviour in time interval 0 < t < T. The value of current iN+1 at any instant in this time interval comes from solution of this eqn, where iN+1 is at time t = T (t = tN+1 = tN + T ). L
+ ( R + Rsoa )
Thus the characteristic equation is Ls2 + (R+Rsoa)s+ 1/C = 0
(3.47)
The roots of the given equation are:
s1 =
s2 =
Fig. 3.17 : Series RLC Circuit connected across a PV Generator.
2
− R − Rsoa +
( R + Rsoa )2 −
4L C
2L −R − Rsoa −
( R + Rsoa )
2
4L − C
(3.48)
2L
Hence the solution to the equation 3.46 has the general form (3.49a) iN + 1 = A1 es1t + A2 e s2 t 176
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article also the voltage across the capacitor is given as R + Rsoa C >2 RRsoa L
(3.49b)
3.3.2 Overdamped case (R2 > 4L/C) In this case the roots of the characteristic equation are negative, real and unequal. Substituting the value of equation 3.48 in equation (3.49) results in
iN + 1 = A1 e
+ A2 e
iN + 1
⎛ ⎛ ⎜ ⎜ R + Rsoa − ⎜ −⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝ ⎛ ⎛ ⎜ ⎜ R + Rsoa + ⎜ −⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎝
( R + Rsoa )2 − 2L
( R + Rsoa )2 − 2L
4L C
vCN +1
⎞ ⎞ ⎟ ⎟ ⎟t ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠
4L ⎞ ⎞ ⎟ ⎟ C ⎟t ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠
(3.50a)
⎡⎛ 4L ⎞ ⎤ 2 ⎢⎜ R + Rsoa + ( R + Rsoa ) − ⎟ ⎥ C ⎠ ⎥ ⎢⎝ ⎢ ⎥ ⎛ ⎛ 2 4L ⎞ ⎞ ⎜ ⎜ R + Rsoa − ( R + Rsoa ) − ⎟ ⎟ ⎢ ⎥ C ⎟t ⎟ ⎜ −⎜ ⎢ ⎥ ⎜ ⎜ ⎟ ⎟ 2L ⎜ ⎜ ⎟ ⎟ ⎢ ⎥ ⎝ ⎠ ⎠ ⎥ 1 ⎢ A1 e⎝ = Edn − ⎢ ⎥ 2⎢ ⎛ 4L ⎞ ⎥ 2 + R + Rsoa − ( R + Rsoa ) − ⎟ ⎢ ⎜ C ⎠⎥ ⎢ ⎝ ⎥ ⎛ ⎛ ⎢ ⎥ 2 4L ⎞ ⎞ ⎜ ⎜ R + Rsoa + ( R + Rsoa ) − ⎟ ⎟ C ⎟t ⎟ ⎢ ⎥ ⎜ −⎜ ⎜ ⎜ ⎟ ⎟ 2L ⎢ ⎥ ⎟ ⎟ ⎢ A e⎜⎝ ⎜⎝ ⎥ ⎠ ⎠ ⎣ 2 ⎦
(3.50b)
Constants of integration A1 and A2 above can be evaluated with the help of initial conditions. These values are substituted in equation 3.50 to arrive at the expressions (as below) for current and voltage across capacitor at time T.
⎛ ⎛ ⎛ 2 4L ⎞ ⎞ ⎞ ⎜ ⎜ R + Rsoa − ( R + Rsoa ) − ⎟ ⎟ ⎜ C ⎟t ⎟ ⎟ ⎜ −⎜ ⎜ ⎜ ⎟ ⎟ ⎟ 2L ⎜ ⎟ ⎟ ⎜ ⎛ v − E + in ⎛ R + Rsoa + ( R + Rsoa ) 2 − 4L ⎞ ⎞ e⎜⎝ ⎜⎝ ⎠ ⎠ ⎟ ⎜ ⎟ ⎟⎟ dn ⎜ ⎜⎜ Cn ⎟ C 2 ⎝ ⎠ ⎠ ⎜⎝ ⎟ ⎛ ⎛ ⎜ 2 4L ⎞ ⎞ ⎟ ⎜ ⎜ R + Rsoa + ( R + Rsoa ) − ⎟ ⎟ ⎜ C ⎟t ⎟ ⎟ ⎜ −⎜ ⎜ ⎜ ⎟ ⎟⎟ ⎜ 2L ⎟ ⎟⎟ in ⎛ 4L ⎞ ⎞ ⎜⎝ ⎜⎝ ⎜ ⎛ 2 ⎠ ⎠ ⎜ + ⎜⎜ vCn − Edn + 2 ⎜ R + Rsoa − ( R + Rsoa ) − C ⎟ ⎟⎟ e ⎟ ⎝ ⎠⎠ ⎝ ⎝ ⎠ = 4L ( R + Rsoa )2 − C
(3.51a)
(3.51b)
177
Swati Sharma et al. where for i = in Edn and Rdn are calculated from
3.3.3 Underdamped case (R2 < 4L/C)
νn+1=(νn+Rsoa) - Rsoain+1 = Edn-Rsoain+1
(3.52a)
where Edn = νn +Rsoa
(3.52b)
From the foregoing analysis it is apparent that i(t) and vc(t) with t = nT may be calculated successively for all n, taking into account the initial conditions, so that the description of the response should be complete over the total time domain after closing the switch. Circuit current and capacitor voltage are plotted on a time line in Fig. 3.18 for an initially de-energized series RLC circuit for which the parameters are assigned such values that the roots of the characteristic equation are real and distinct.
In this case the roots of the characteristic equation are complex conjugate with negative real parts. Thus the roots of the given case are:
4L 2 − ( R + Rsoa ) C s1 = 2L 4L 2 −R − Rsoa − j − ( R + Rsoa ) C s2 = 2L − R − Rsoa + j
Substituting these values in equation 3.49, converting exponential functions with imaginary components to trigonometric functions and rearranging results in: in + 1 =
⎛ R + Rsoa ⎞ −⎜ t⎟ exp ⎝ 2 L ⎠
⎛ ⎛ 4L 2 ⎞ ⎞ − ( R + Rsoa ) ⎟ ⎟ ⎜ ⎜ ⎜ B1 cos ⎜ C t⎟ ⎟ ⎜ 2L ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎟ ⎜ ⎜ ⎟ ⎛ 4L 2 ⎞ − ( R + Rsoa ) ⎟ ⎟ ⎜ ⎜ ⎜ + B sin ⎜ C t ⎟⎟ ⎜ 2 2L ⎜ ⎟⎟ ⎜ ⎜ ⎟⎟ ⎝ −⎛⎜ R + Rsoa t ⎞⎟ ⎠⎠ ⎝ L 2 ⎠ vcn + 1 = Edn − exp ⎝
Fig. 3.18(a) : Overdamped behaviour of capacitor voltage for a RLC load connected in series across the PV Generator for R=10 Ohm L=10 mH C=1 mF.
⎛ 4L 2 − ( R + Rsoa ) ⎜ ( R + Rsoa ) B1 + B2 C ⎜ 2 ⎜ ⎜ ⎛ 4L 2 ⎞ ⎜ − ( R + Rsoa ) ⎟ ⎜ ⎜ ⎜ C t⎟ ⎜ cos ⎜ 2L ⎟ ⎜ ⎜ ⎟ ⎝ ⎠ ⎜ ⎜ ⎜ ( R + Rsoa ) B2 − B1 4L − ( R + Rsoa )2 ⎜ C ⎜+ 2 ⎜ ⎛ 4L ⎜ 2 ⎞ − ( R + Rsoa ) ⎟ ⎜ ⎜ C t⎟ ⎜ sin ⎜ 2L ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ ⎠ ⎝
Fig. 3.18(b) : Overdamped behaviour of current for a RLC load connected in series across the PV Generator for R=10 Ohm L=10 mH C=1 mF.
(3.53)
(3.54a)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 3.54b) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
where B 1 and B 2 are two integration constants determined from the initial conditions, that is the current in and the capacitor vcn at t = 0.
178
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article In order to strengthen the viewpoint of this study consider the following discussion: i.
The expressions apply in their present form only when R + Rsoa < 2 L C inequality is valid through the entire transient response. As a consequence of R s definition, R soa is always positive and monotonically increases 0 < I < Isc. Therefore, for the transient response to be underdamped over the total time domain, after switch closure at t = 0, R < 2 L C alone is a necessary condition but is not a sufficient condition to ensure underdamped response.
ii. Equation 3.50 (a) and (b) and fig 3.58 imply that the transition from initial state to steady state is accomplished through a damped oscillation characterized by a dynamic damped frequency 4L − ( R + Rsoa )2 C . Successive half cycles of ωd = 2L oscillation increase in width as time progresses. Also the presence of exponential factor in equation 3.50 (a) and (b) contributes to oscillation decrease in successive half cycles thus indicating the rise in R (or in turn current) with decrease in amplitude and vice-versa. This reveals that neither the damping rate nor the oscillation be confined to be a single time constant (or envelope) as is the case with a constant voltage source.
Fig. 3.19(b) : Underdamped behaviour of current for a RLC load connected in series across the PV Generator for R=1 Ohm L=10 mH C=1 mF.
Circuit current and capacitor voltage are plotted on a time line in Fig. 3.19 for an initially de-energized series RLC circuit for which parameters values ensure an underdamped transient response. 3.3.4 PV Generator connected to parallel RLC Load For a PV generator connected to parallel R-L-C circuit, as shown in fig 3.20, the governing differential equation can be written as: C
d 2 vn + 1 ⎛ 1 1 ⎞ dvn + 1 1 +⎜ + + =0 ⎟ 2 R R L dt ⎝ soa ⎠ dt
(3.55)
iii. Generally to block the flow of negative current a blocking diode is usually inserted between the PV source and the load. Thereby making the system oscillation free or free from negative current.
Fig. 3.20 : Parallel RLC equivalent circuit connected across a PV Generator.
Fig. 3.19(a) : Underdamped behaviour of capacitor voltage for a RLC load connected in series across the PV Generator for R=1 Ohm L=10 mH C=1 mF.
The above equation is a linear second order differential equation. When used with initial values for t =0 (t = tN), describes the circuit behaviour in time interval 0 ≤ t ≤ T. The value of voltage vn+1 at any instant in this time interval comes from the solution of this equation, where vn+1 is at time t = T (t = tN+1 = tN + T ). 179
Swati Sharma et al. Thus the characteristic equation is Cs2 + (1/R+1/Rsoa)s+ 1/L = 0
(3.56)
For this circuit, the roots of the characteristic equation can be written as under 2
− S1 =
2 L ( R + Rsoa ) ⎛ L ( R + Rsoa ) ⎞ 4LR2 Rsoa + ⎜ ⎟ − C C C ⎝ ⎠ 2LRRsoa 2
2 L ( R + Rsoa ) ⎛ L ( R + Rsoa ) ⎞ 4LR2 Rsoa − − ⎜ ⎟ − C C C ⎝ ⎠ S2 = 2LRRsoa
(3.57)
3.3.6 Underdamped case (R2 < 4L/C) The solution to the equation 3.57 has the general form
vn + 1 = e
It may be mentioned that PV generator operates at or near the short - circuit value for both overdamped and underdamped cases. Under this condition, it may be compared with a constant current source as in this region, Rsoa remains fairly constant. 3.3.5 Overdamped case (R2 > 4L/C) The solution to the equation 3.57 has the general form (3.58a) vn +1 = A1 es1t + A2 e s2 t Also the current across the inductor is given as
⎛ 1 1 ⎞ iLn +1 = IL f − ⎜ Cs1 + + ⎟ R Rsoa ⎠ ⎝ ⎛ 1 1 ⎞ S2 t A1 eS1 t − ⎜ Cs2 + + ⎟ A2 e R R ⎝ soa ⎠ Edn while I L f = Rsoa
can be calculated by solving eq's 3.58 (a) and (b). The transient response for the circuit is shown in fig 3.20. The sign of the radical in eq 3.57 determine if the response is overdamped or underdamped. The solution is overdamped if condition in R + Rsoa C otherwise, it is eq 3.57 is satisfied >2 RRsoa L underdamped.
(3.58b)
(3.59)
−
L( R + Rsoa ) C
T
⎛ 2 ⎞ ⎞ ⎞ ⎛ ⎛ 2 2 ⎜ ⎜ ⎜ 4LR Rsoa − ⎛ L ( R + Rsoa ) ⎞ ⎟ T ⎟ ⎟ B cos ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ 1 ⎜ ⎜⎜ C C ⎜ ⎝ ⎠ ⎟ ⎟ ⎟ (3.60a) ⎜ ⎝ ⎠ ⎠ ⎝ ⎜ ⎟ 2 ⎞ ⎞⎟ ⎛⎛ ⎜ 2 2 ⎜ + B2 sin ⎜ ⎜ 4LR Rsoa − ⎜⎛ L ( R + Rsoa ) ⎟⎞ ⎟ T ⎟ ⎟ ⎜ ⎜⎜ ⎟ C C ⎜ ⎝ ⎠ ⎟⎟ ⎟ ⎟⎟ ⎜⎜ ⎠ ⎠⎠ ⎝⎝ ⎝ Also the current flowing through the inductor is given as
iLn +1 = IL f + e
−
L( R + Rsoa ) T C
⎛ 2 ⎞ ⎞ ⎞ ⎛ ⎛ 2 2 ⎜ B cos ⎜ ⎜ 4LR Rsoa − ⎛ L ( R + Rsoa ) ⎞ ⎟ T ⎟ ⎟ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ 1 ⎜ ⎜⎜ C C ⎜ ⎝ ⎠ ⎟ ⎟ ⎟ (3.60b) ⎜ ⎝ ⎠ ⎠ ⎝ ⎜ ⎟ 2 ⎞ ⎞⎟ ⎛⎛ ⎜ 2 2 L R + Rsoa ) ⎞ ⎟ ⎟ ⎜ + B sin ⎜ ⎜ 4LR Rsoa − ⎜⎛ ( ⎟ ⎟T ⎟ ⎟ ⎜ ⎜⎜ ⎜⎜ 2 C C ⎜ ⎝ ⎠ ⎟ ⎟ ⎟⎟ ⎠ ⎠⎠ ⎝⎝ ⎝
Using equation 3.56 and 3.57, the value of inductor current for the overdamped and underdamped case
Fig. 3.22 : Underdamped behaviour of inductor current (I) and capacitor voltage (II) for a RLC load connected in parallel across the PV Generator for R=10 Ohm L=10 mH C=1 mF.
Fig. 3.21 : Overdamped behaviour of inductor current (I) and capacitor voltage (II) for a RLC load.
180
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article 3.3.7 Discussion and Conclusions In this chapter a comprehensive qualitative and quantitative analytical study of second order circuits powered by PV generator module based on incremental method using LambertW function has been completed. The conclusions reveal that transient performance of second-order circuits powered by a PV generator exhibit general characteristics of their input response. The incremental model of the PV generator can be considered to be a visual-circuitalmathematical tool offering many advantages over other numerical integration techniques. Although input experimental data used for this study is from Applebaum et al [55] it helps validate the output of the work significantly. However the above analysis can be directly applied to available solar panels for study without any change in expressions or method used by mere substitution of the respective parameter values. Thus from the above study it may affirm that W function technique incorporating a finite shunt and series resistance overcomes the known inabilities due to the absence of shunt resistance in the PV generator equivalent circuit, and hence provides good correspondence to existing experimental results [57]. Moreover, the performance characteristics presented provide a rough appreciation of the general behaviour expected for higher-order circuits. Finally the same approach may apply with equal validity for any circuit involving all three basic circuit elements in whatever combinations and to be of fundamental importance in the study of more realistic solar electrical systems. The analytical solutions provided here being explicit in nature may also serve as important inputs to other applications like determination of maximum power point for standalone PV systems. 4.
Application of Lambert-W function to Organic, DSSC and Hybrid Solar Cells
4.1 Organic Solar Cells: 4.1.1 Introduction Historically, researchers were limited to using crystalline silicon wafers or thin film deposition of other inorganic materials to develop solar cells and photovoltaic devices, a process that typically requires expensive manufacturing technologies. The development of low-cost methodologies to produce solar energy conversion devices is a much needed enabling technology for a variety of applications [58]. It has been shown that the purity of the compounds
has great influence on the electrical properties of the solar cell. Solar cells (IPV Cells)made from inorganic semiconductors have been studied since the 1950s and have been used as major material for solar cells .Recently several types of solar cells based on organic materials (OPV Cells) emerged as an alternative to IPV solar cells. OPV solar cells are comparitively cheap and can be made on flexible substratres [59-71]. The well known difference between the IPV and OPV solar cells in their mechanism of phtoconversion. In IPV cells light absorption directly leads to the creation of free electron-hole pairs while in OPV cells it results in mobile excited states (excitons) rather than free electron hole pairs. The process is given in fig 4.1 .
Fig. 4.1 : Conversion of incident photons to separated charges.
This is due to (a) The dielectric constants of organic phase is usually low compared to inorganic semiconductors, so the attractive Coulomb potential well around the incipient electron-hole pair extends over a greater volume than it does in inorganic semiconductors, and (b) The non-covalent electronic interactions between organic molecules are weak (resulting in a narrow band width) compared to the strong interatomic electronic interactions of covalently bonded inorganic semiconductor materials like silicon. Therefore, the electron's wave function is spatially restricted (small Bohr radius), allowing it to be localized in the potential well of its conjugate hole (and vice versa). This results in a tightly bound electron-hole pair (Frenkel exciton or mobile excited state) as the usual product of light absorption in organic semiconductors. It is a mobile, electrically neutral species which, to first order, is unaffected by electric fields. Excitons
181
Swati Sharma et al. dissociate primarily at hetero-interfaces resulting in electrons in one phase and holes, already separated from the electrons, in the other phase (Fig. 4.2). Band bending, Øbi, although it can be useful, is not required to separate the charge carriers. The interfacial exciton dissociation process leads to large concentration gradients of the two carrier types at the interface.
a mask (dry-processing) (b) Amount of organic materials are relatively small (100 cm thick films) and large scale production (chemistry) is easier than inorganic materials. (c) They can be tuned chemically in order to adjust separately bandgap, valance and conduction energies, charge transport as well as solubility and several other properties. (d) the vast variety of possible chemical structures and functionalities of organic materials (polymers, oligomers, dendriomers, organo-materials, dyes, pigments, liquid crystals etc.) favour an active research for alternative competitive materials with desired photovoltaic properties. 4.1.2 Analysis of Solar cell parameters of Organic Solar Cell The equivalent circuit diagram (ECD) for organic solar cell as suggested by Barbec [72] is given in fig 4.3. The load voltage across the cell ( V) can be determined by the equations:
Fig. 4.2 : Energy level diagram for an OPV Cell.
Various drawbacks associated with OPV cells are (a) Low motilities (b) spectral range of optical absorption is relatively narrow as compared to solar spectrum (c) when organic devices are exposed to the atmosphere, oxygen and other contaminants react with the device layers (d) organic materials are mechanically fragile and are easily attacked by chemicals used in photolithographic patterning. Therefore patterning often requires low resolution methods. Thus it can be concluded that organic devices will only be as reliable as their constituent organic materials are. With the breakthroughs in organic display research OPV solar cells becomes a hot topic for research today. Researchers from different fields like Material Science, Physics, Chemistry, Electrical Engineering etc. are taking interest in this area from different aspects. Blends of organic materials are devised to overcome various short-comings of individual materials and to cope with the problem of exciton dissociation and weak absorption in red. Several polymers are developed with increased light collection efficiency. Advantages of organic materials for photovoltaic solar cell application includes (a) they can be processed using spin coating or doctor blade techniques (wet-processing) or evaporation through
Fig. 4.3 Equivalent circuit diagram for organic solar cell.
V=IRsh Rsh + IRs
(4.1)
I= -IL +IRp +Id
(4.2)
⎛ V − IRs ⎞ I d = I0 e ⎜ ⎟ ⎝ nVth ⎠
(4.3)
Various blocks in the ECD of organic solar cell have different functions associated with them ; The current source generates current Iph upon illumination, which is equal to the number of dissociated exciton/s i.e, number of free electron hole pair immediately after the generation -before any recombination takes place. The series resistance Rs considers conductivity i.e, mobility of specific charge carriers in the respective
182
Exact analytical solutions of generation real solar cells using Lambert W-function: A Review Article transport medium, where the mobility is affected by space charges and traps of other barriers (hopping). The shunt resistance Rsh is due to recombination of charge carriers near the dissociation site (e.g. donor/ acceptor interface) and it may also include recombination farther away from the dissociation site (e.g. near electrode). The ideal diode represents the voltage dependent resistor that takes into account the asymmetry of conductivity. The diode is responsible for the nonlinear shape of the I-V curves. The diode characteristics are not necessary Shockley type. The cell can generate terminal voltage between 0 and Voc depending on the size of the load resistor. For obtaining voltage beyond this limit, an external voltage source is required. The voltage range between 0 and Voc can be simulated by the same voltage source so that applying an external voltage can scan entire ranges. Since current for the voltages beyond the ranges 0
