Basic Model and Governing Equation of Solar Cells used
in Power and Control Applications
Afshin Izadian, Senior Member, IEEE, Arash Pourtaherian, and Sarasadat Motahari
Abstract—This paper provides an overview of modeling of a group of
commercially available solar cells to ease the study of solar powered
electric systems. The models solar cells can be accurately used to predict
the behavior of the system operation under different conditions.
Index Terms—Solar cell equivalent circuits, Silicon based solar cells,
Thin-films, Dye-sensitized, Organic solar cells.
I. INTRODUCTION
HE structure, material, and fabrication process of solar cells have
been significantly improved since their introduction. Much of
this effort has been focused on efficiency enhancement of energy
conversion. The typical efficiency of current PV products for space
applications is around 35%-38%. In average, non-concentrating
terrestrial cells have efficiencies around 14%-15% for wafer-based
Si, and around 9-10% for thin films. Energy conversion efficiencies
are directly related to the type of material and structure of the cells
[1]. The most common types of solar cells are made in silicon in form
of single crystal, poly crystal, and amorphous. Other materials such
as cadmium telluride, copper indium, and gallium arsenide have been
used to obtain desired characteristics. Thin films and dye-sensitized
solar cells use organic materials to lower the cost of the cells.
Because of various materials, the structure of solar cells and therefore
their behavior is different. To accurately express and predict the
behavior of these cells, mathematical governing equations and
equivalent circuits are required.
In this paper, the mathematical modeling and equivalent circuit of a
group of solar cells that are commonly used in industrial applications
are studied. Various circuit elements and their physical effects are
illustrated to help in better understanding of the cell behavior under
various operating conditions and when connected to power
converters.
T
II. CURRENT SOURCE
Solar cells generate current in a large range independent from the
load; thus, these cells are modeled as current sources. They are
designed with built-in asymmetries to capture the photo-excited and
released electrons and send them through an external circuit to build
electric currents. The current generated from the cell is directly
dependent on the illumination area and the probability that one
photon can release one electron in the device. Solar cells in zero
illumination behave more like a diode. Therefore, their current in
dark condition is a function of the cell’s voltage. In illumination,
however, based on the light intensity, the cell generates a current that
affects the diode characteristics at the terminal. In open circuit, the
voltage is created based on the recombination of carriers in solar cell.
The open circuit voltage is defined as the voltage at which the short
circuit current and the forward bias diffusion currents become equal
with opposite polarities. Open circuit voltage has also been defined as
the separation of Fermi energies at the equilibrium of electron-hole
generation [1]. The short circuit current of the cell increases as the
thickness of the cell increases. However, the open circuit voltage
decreases as the thickness increases. Since thicker material absorb
more photons and increase recombination, the optimum thickness is
where rate of additional recombination and absorbed photons
equalize [1].
In different cell types, several effects such as recombination (in the
space charged region, at the surface, or in bulk) and diffusion may
also cause a drop in the cell generated current. Applications and
operating conditions may also change the cell’s voltages and current
generations. For instance, to model the partial shading behavior, a
voltage-controlled current source is required to model the mismatches
between cells.
III. BASIC MODEL OF SOLAR CELLS
As mentioned earlier, a solar cell can be modeled as a current
source. In its basic form, the current generated from the photocurrent
source is directly conducted to the terminals. A diode is connected
across the terminals to model the I-V curve normally generated from
the cells. To obtain the model of illuminated solar cells, one
technique is to use dark models. The Shockley equation [2] relates
the current and voltage of the cell in zero-illumination condition by

 eV   ,
(1)
I D = I 0  exp 
 − 1
 kT  

where e is the charge of an electron, k is the Boltzmann constant, T is
the temperature in ºK, and I0 is the inverse saturation current of the
diode. The term kT is equal to the thermal voltage Vt. When the cell
e
is illuminated by a light source, the I-V curve is offset from the origin
by the photo-generated current IL [1]. Therefore, its behavior can be
modeled [3-9] as
  eV   ,
(2)
I = IL − I0 exp
 − 1
  mkT  
where m is the ideality factor which is defined as how closely a diode
follows the ideal diode equations. This is mainly caused by
recombination effect and can vary significantly between different
types of cells and from 1 and 5. In the majority of practical cases, m
is between 1 and 2 for high and low voltages, respectively [3]. In
typical applications, a value of 1.3 can be considered initially, until
more estimations determine the exact values [10]. The circuit
elements of this model can be obtained as a photocurrent source IL to
be the reverse current with amplitude linearly proportional to the
solar irradiance G (W m-2) [2, 11]. The dark current diode across the
source takes the current ID from the generated current IL [4] to deliver
the current I in the terminals. The equivalent circuit of the ideal diode
is shown in Figure 1.
The paper was submitted on September 10, 2011.
A. Izadian, A. Pourtaherian, and S. Motahari are with the Purdue School of
Engineering and Technology, Indianapolis, IN, 46202, USA. E-mail:
izadian@ieee.org.
Fig. 1. A single diode equivalent circuit for the ideal solar cell.
978-1-4673-0803-8/12/$31.00 ©2012 IEEE
1483
The current source IL is a temperature-dependent value. It can be
obtained by
(3)
I L = I L( T1 ) (1 + K 0 (T − T1 )) ,
where
I L(T1 ) = GI SC (T1 ,nom ) / G( nom )
,
K 0 = I SC (T ) − I SC (T1 ) / (T − T1 ) ,
(
)
(4)
(5)
and I SC(T ) is the short circuit current at temperature T. The reverse
circuit elements. These effects are mostly shown in the photocurrent
generation process; therefore, an extra current source, as required in
the thin films, can be used to show the current loss.
In advanced fault diagnosis of solar cells, a dynamic model that
requires modeling junction capacitors is used. In this case, the model
representation will have a capacitor across the diode in the basic
model of solar cells. Next sections of this paper will illustrate these
elements, their operations, and some values in various types of cells.
saturation current from its value at temperature T1 can be obtained [4]
by
3
T
I 0 = I 0(T1 ) 
 T1
 eV
m
 exp  − g
 mk


1 1
 −
 T T1
 ,
 


(4)
where Vg is the band-gap voltage (1.12eV for crystalline silicon and
~1.75eV for amorphous silicon). The saturation current at T1, I o (T1 )
can be calculated from the open-circuit voltage and short-circuit
current [4] by
I SC (T1 )
.
(5)
I0 =
 eV OC (T1 )

exp 
− 1 
mkT
1


This basic model is useful to achieve I-V curves at different
temperatures. However, the behavior of the cell at extreme short
circuits and open circuits cannot be thoroughly investigated from this
model. In addition, this model does not provide the details of power
loss and contact characteristics of the cell. For instance, as the model
implies, under short-circuit conditions, the diode should not conduct
any current, but in reality, the diode conducts a small current [2].
Therefore, a more accurate circuit model is needed to express these
effects and obtain detailed governing equations. A model with series
and parallel resistors can illustrate some of these effects. A list of all
circuit elements and physical effects that are used in solar cells is
provided in Table I.
TABLE I.
CIRCUIT ELEMENTS, PHYSICAL EFFECTS, AND THEIR APPLICATION
Circuit
Element
Physical Effect
• Electron excitation and
extraction by light photons
• All cells
Ideal Diode
• Dark current
• Diffusion current from base
and emitter
• All cells
Series Resistor
• Contact resistors
• Layer resistors
Parallel Resistor
• Recombination modeling
Avalanche
Current Source
• Generation-recombination in
the junction space-charge
region
• Voltage-controlled current
source
• Reverse bias in shading
A. Modeling of Voltage Drop with Series Resistance
The series resistance represents the ohmic voltage drop at the
contacts and through the layers of materials. Therefore, the value of
the series resistance can vary under different illuminations and in
various types of cells. This resistance relates the maximum power
generation of the cell with the open circuit voltage [2]. Figure 2
illustrates a model that can be used for power generation analysis in a
network of solar cells [2], [12-16].
Fig. 2. A single diode and a series resistance model equivalent circuit.
The effects of series resistance on the current generation and
terminal voltage is described by

 e
(V − IRs ) − 1 ,
I = I L − I 0 exp
mkT

 

MPPT
Power loss
High power
Fault diagnosis network
analysis
• High power
•
•
•
•
• Polycrystalline
• Cascade models
• Partial shading
• Network analysis
• Mismatch of solar cells
Recombination
Current Source
• Electron recombination in
strong electric fields
• Thin film
Capacitors
• Capacitance of cell layers
• High frequency
• Dye sensitized
• Surface recombination
• Most cells
• Carrier trapping
• Most cells
Embedded in
Photocurrent
Source
Embedded in
Photocurrent
Source
The process of photocurrent and terminal-current generation is
affected by many parasitic sources. To model these parasitic sources,
series and parallel resistances are introduced as a practical technique.
Series resistance at the terminal of the cell is used to model the
voltage drop. The parallel resistance is used to model the current
leakage in the device proportional to the terminal voltage [3]. The
current leakage is distributed throughout the device and may not
always be represented as a lumped element.
Application
Photocurrent
Source
Double Diode
IV. MODELING OF CURRENT AND VOLTAGE DROPS
As the table illustrates, each of the circuit elements represents a
physical effect. Among all the physical effects, the surface
recombination and carrier trapping effects are less represented by the
(6)
where RS is the series resistance. This equation illustrates the
principle of reverse photocurrent generation. Several studies have
been done to determine the best approximation of the internal series
resistance [17-25]. Wolf [17], and Iles [18] showed the series
resistance as Rs=rb+re/2. Where, re≡RES2 is the geometrical
resistance in Ω.cm2, RE is the sheet resistance of the layer, and S is
the length. rb≡ρBWB is the total resistance of the base, where ρB is the
resistivity of the base, and WB is the base width. A more accurate
expression of this resistance has been provided [27] as follows:
1) The series resistance will change in small current densities at
which IRs≤ Vt ≈ mkT/e [26]. When rb»re, the resistance can be
calculated by Rs=rb+re/3 [27].
2) RS varies by illumination conditions. Its maximum reaches when
the cell is saturated, Rs=rb+re/3, and its minimum reaches when the
cell is an open circuit, RS=(rbre)2 [27]. In dark conditions, RS varies
within the range of rb+re/3≥ RSdark≥(rbre)1/2 . The base resistance is a
non-zero value rb ≠0, which sets a lower bound on RS [27].
Doping the boundary layers in silicon-based solar cells can achieve
either an n+-p-p+ or a p+-n-n+ structure and can reduce the series
resistance. In most commercially available solar cells (e.g. the
passivated emitter solar cell PESC), this parameter is nearly
independent of current density, irradiance, and illumination
conditions. In modern solar cells, such as back-surface fields, SiN
surface passivations, selective emitters, shallow emitters, backcontact cells, HIT cells, buried-contact cells, and floating-junction
1484
passivation of surfaces, the resistance of the bulk of the material does
not contribute significantly to the series resistance.
low as 1.15. As the material and structure of solar cells improve, their
photocurrent densities increase, and their ideality factors decrease.
C. Application of models with Series and Parallel Resistance
B. Modeling of Current Loss with Parallel Resistor
The current leakage around the edge of the cell, the diffusion path
along dislocations, and the small internal short circuits can be
represented by a parallel resistor in parallel to the dark current diode
[3]. In crystalline silicon solar cells, this resistance has almost no
effect under normal operating conditions. A parallel resistor has
effects on the device in low voltage applications and can be
eliminated in high current densities [17]. Figure 3 illustrates the
equivalent circuit of the solar cell with parallel and series resistors.
Circuits with parallel and series resistors are being used in many
applications including studies considering high power applications
[30], in the development of Maximum Power Point Tracking
Techniques [31-44], and in fault diagnosis [45, 46]. This model can
also be used to provide details on the effects of partial shading on PV
arrays [47-51], in which some of the cells may operate in reverse
bias, sinking the generated power of other cells [31]. Series resistance
Rs is primarily related to the reduction of the maximum output
current Isc, and Rsh is related to the reduction of the maximum
available output voltage Voc [7].
V. PARTIAL SHADING
Fig. 3. A single diode with series and parallel resistance equivalent circuits.
The I-V characteristics of solar cells with series and parallel
resistors can be represented by
  e
(7)
I = I L − I 0 exp
(V + IRs ) − 1 − V + IRs .
mkT
Rsh

 

The circuit parameters Rs, Rsh, I0, and m at a certain illumination
and temperature can be obtained from the parameter values of Voc, Isc,
Vm, Im, Rso, and Rsho, where Vm is the voltage at the maximum power
point, Im is the current at the maximum power point, and Rso and Rsho
are defined [28],[29] as
 dV 
,
(8)
Rso = −

 dI V =VOC
and RshO = − dV 
.
 dI  I =ISC
, (10)
Vm + RSO Im − VOC
 



VOC 
Vm
Im
+
VT ln I SC −
− Im  − ln I SC −

Rsho
Rsh  I SC − (VOC Rsho ) 
 



V 
 eV 
I s =  I SC − OC  exp − OC  ,
R sh 
 mkT 

R s = R so −
mkT
 eV OC  ,
exp  −

eI s
 mkT 


V ,
 eV  
I = I L − I0  exp
 − 1 M(V ) −
R sh

 mkT  

M(V ) =


R 
 eI R  
(13)
I L = I SC  1 + s  + I s  exp SC s  − 1  .
Rsh 
 mkT  


A suitable initial guess for solving the Newton-Raphson or the
Phang equations can be set for a 3" mono-crystalline solar cell under
1 sun (normally AM1 = 100 mW/cm2) illumination as IL = 1 A, Rs =
0.05 Ω, Rsh = 500 Ω, I0 = 10-7 A, m = 1.3, Voc = 0.545 V, Isc = 1 A,
Rso = 0.0838 Ω, Rsho= 497 Ω, Vmax =0.415 V, Imax = 0.916 A, T= 27
°C, and Fill Factor as FF=0.698. A low fill factor may indicate a
short across the diode, a high internal resistance, inadequate doping,
or a high interface resistance. For a 20 µm thick cell, with light
trapping, at exposure AM 1.5 the Isc=413 A/m2, Imax=401 A/m2,
Voc=0.707 v, Vmax=0.702 V, FF=0.89 [1].
These values in a 6" square-shaped poly-silicon cell are Voc = 0.6V,
and Isc = 8 A, and the ideality factor at the higher voltages can be as
1
(1 − (V
Vb
))
,
(15)
n
where Vb is the breakdown voltage in reverse bias, and n is the Miller
constant. Figure 4 shows the electric equivalent circuit of a partially
shaded solar cell.
Fig. 4. A solar cell equivalent circuit considering the avalanche effect [52].
The avalanche breakdown effect can also be modeled on the path of
shunt resistor. The leakage current source E(V) is added into the
shunt resistance and is expressed by
(11)
(12)
(14)
where M(V) is the multiplication factor as a voltage controlled current
source, and represents the avalanche effect [52, 53]. This factor can
be calculated by
(9)
The lumped circuit parameters can be obtained by solving the
governing equations of the cell with the Newton-Raphson technique
[28], or by using the Phang equations [29]. These equations represent
an analytical approach to calculate the circuit parameters obtained by
the following:
m=
Reverse bias occurs when PV modules are partially shaded. In this
case, the shaded cell behaves like a load for the circuit [31]. A
mismatch may occur where the current of the series string becomes
higher than that of the shaded cell's short-circuit current and harm the
cell. To model the mismatch, the avalanche breakdown effect can be
represented as a dependent current source [52].
The mathematical model of the solar cell in reverse bias is
expressed as
 V + IR s
E(V ) = 1 + a 1 −
Vb

−n
 .


(18)
This model can express the mismatch between solar cell
interconnections to form PV modules very accurately [54]. The
current can be calculated by
−n
 V + Rs I   , (19)
  e(V + Rs I )    V + Rs I 
 1 + a1 −

I = I L − I0  exp
 − 1 − 


Vb  
  mkT    Rsh 


where a is a constant. The corresponding equivalent circuit is shown
in Figure 5.
Fig. 5. Bishop’s equivalent circuit.
1485
Lorenzo has shown the reverse-bias saturation current and the
open-circuit voltage as a function of temperature [3] by
 E go 
(16)
 ,
I 0 = KT 3 exp −
 kT 
VOC (T ) =
E go
e
−
kT  KT 3  .

ln
e  I L 
(17)
where K and Ego (the band gap at 0º K) are approximately constant
with respect to temperature.
VI. MODEL OF THIN FILM SOLAR CELLS
Thin film solar cells like other types of cells generate photocurrents
in two steps: photogeneration (or release of electrons) and charge
separation (or current creation) [34]. A built-in voltage separates
charges across the p-n junctions in a layer called the space charge
region. This is more sensible in thin film solar cells such as
hydrogenated amorphous silicon (a-Si:H) [34] and hydrogenated
microcrystalline silicon (mc-Si:H) [55]. The aging of solar cells
degrades the space charge layer and provides a recombination center
that sinks some of the generated current and, therefore, reduces the
power of the solar cell.
An additional recombination loss is represented by the current Irec
to show the counteraction effect on the photogenerated current. The
current source
is added into the equivalent circuit, as shown in
the dashed lines in Figure 6 [56]. The expression for the
recombination current loss Irec within the space charge layer equals
the generation current IL multiplied by the ratio of the cell thickness
di over the effective drift length
(µτ )eff E
(Schubweg) in the space
charge layer [57] as
I rec = I L
di
(µτ )eff [Vbi − (V − IRs )] di
(µτ )eff
where
=2
,
µ noτ no .µ poτ po
,
µ noτ no + µ poτ po
(18)
Fig. 7. An equivalent circuit model of a DSSC.
The model is a combination of the photocurrent source, a diode to
represent the dark current behavior, series and parallel resistors, and
capacitors. The diode behaves more like a variable resistor, R2. Series
resistance, RS, represents the effects of impedances related to charge
transport at the electrode, photosensitized material, and electrolyte.
R3 is the Warburg impedance, and Rh is defined as a resistance in the
high-frequency range of about 106 Hz. C1 and C2 are capacitance
elements of R1 and R2, respectively. Since dye-sensitized material can
be made out of non-toxic and biocompatible material, they can be
used in health care products as well as domestic applications such as
paint pigmentation [76].
VIII. POLYCRYSTALLINE SOLAR CELLS
Polycrystalline solar cells require more details in their equivalent
circuits to represent accurate I-V characteristics. For accurate
modeling, the diode should have a variable ideality factor m [62], or
be replaced with two parallel diodes with different ideality factors
m=1 and 2 [2, 11, 63]. Figure 8 illustrates the double diode
equivalent circuit of the polycrystalline solar cells. The first diode
with current ID1 models the diffusion current from the base and the
emitter, and the second diode models the generation-recombination
current ID2 in the junction space-charge region [64]. This
recombination current is negligible in wafer-based silicon cells
except for low-cost mass-produced cells. For this reason, the use of
the double diode model is very limited except for polycrystalline
devices [65, 66].
(19)
µ no and τ po are the band mobility of the free carriers. This
expression has been obtained by neglecting diffusion currents and is
valid only for thin cells, strong fields in the space charge layer, and
low defect densities in the space charge layer [57].
Fig. 6. An equivalent circuit for photovoltaic thin film solar cells and modules. The
current sink (dashed lines) takes into account the current losses due to recombination in
the space charge layer of the device [57].
VII. DYE-SENSITIZED SOLAR CELLS
Dye-sensitized solar cells are in the same family of thin film solar
cells. These cells are semiconductors formed between a
photosensitized material and an electrolyte to generate current from
light. Because of these semiconductors, the cell has large capacitors
notable to be included in its model as shown in Figure 7. The cell can
be modeled through electrochemical impedance spectroscopy [58].
Some models consider distributed elements similar to an electrical
transmission line [59], while others point out a concentrated model
[60, 61]. The equivalent circuit model of a dye-sensitized solar cell
with capacitance is shown in Figure 7.
Fig. 8. A double-diode equivalent circuit for the ideal solar cell.
The I-V characteristics of double-diode equivalent circuits can be
obtained by
  eV  
  eV  
I = I L − I D1 − I D2 = I L − I01  exp  − 1 − I02 exp
 − 1 , (20)
  kT  
  mkT  
where m is the ideality factor of diode D2 ,and I01 and I02 are inverse
saturation currents of diodes D1 and D2, respectively [64, 67]. The
current source IL models the photocurrent source, I01 represents the
first diode saturation current density resulting from the thermal
generation of minority carriers, and I02 represents the second diode
saturation current density resulting from the generation
recombination [68].
A more accurate mathematical expression of the double-diode
model for a single solar cell is identified as two-exponential
transcendental models [69]. This can also be represented as a doublediode model [63], [66], [70], [71], [77], and two resistors, Rs and Rsh
to determine the ohmic losses and the shunt leakages, respectively
[64, 68]. The model is represented by
  e(V + IRs )  
  e(V + IRs )   V + IRs
. (21)
I = IL − I01exp
 −1 − I02exp
 −1 −
kT




  mkT   Rsh
1486
Figure 9 illustrates the equivalent circuit of the is shown in.
R sh and Rs resistances are functions of externally applied voltage
and are non-linear in nature. The parallel resistance can be modeled
V
as a constant value as R sh ≅ int . The series resistance is a variable
IL
Fig. 9. A double-diode lumped model equivalent circuit. The shunt resistance represents
the loss of internal solar cells due to charge recombination, and the series resistance
represents the charge extraction process.
Ho and Morgan [72] used the double-diode model to represent the
SPICE modeling of cascade solar cells [73, 74]. This model is used
when the series resistance of two diodes cannot be separated. Several
studies have identified the parameters of double-diode models [54,
75]. For instance, Veissid [75] used the standard deviation method to
obtain the parameters. This method models the reverse bias currents
as
 eV 
(22)
I 01 = N 1T 3 exp − G  ,
 kT 
3
 eV 
(23)
I02 = N2T 2 exp − G  ,
 2kT 
where N1 and N2 are constant values. Polycrystalline cell model
elements have been identified by Gow and Manning [63].
that depends on the magnitude of the current across it and can be
represented [4] as R s = Vint − V . The magnitude of internal voltage is
I net
Vint = (I L − I − f (V ))Rsh , and the net current flowing through the series
resistance is I net = I L − I − f (V ) .
X. SUMMARY
Commercially available solar cells are manufactured in various
material types and are used in a variety of applications. These cells
have many effects in common, and some specific characteristics that
need to be modeled and thoroughly understood. This paper illustrated
a class of solar cells, their common and specific effects, and provided
circuit equivalents and governing equations to express their behavior
in electric power generation applications.
REFERENCES
[1].
[2].
IX. ORGANIC SOLAR CELLS
The equivalent circuit of organic cells in the absence of extrinsic
series and parallel resistances can be modeled by some diodes to
replace these resistors. The proposed model is shown in Figure 10.
[3].
[4].
[5].
[6].
[7].
Fig. 10. An equivalent circuit model for an organic solar cell.
Diode Drec represents the effect of recombination, and diode Dext
represents the extraction of free carriers. Diode Ddark represents the
dark current. These diodes are forward-biased in the presence of light
and are reverse-biased in dark. The cell characteristics in illumination
can be expressed by
I = I L − I rec − f (V ) ,
(24)
Irec = f rec (Vint ) ,
(25)
I SC = I L − f rec (Vint ) = f ext (Vint ) ,
(26)
I L = f rec (Vint ) + f (VOC ) ,
(27)
f (VOC ) = f ext (Vint − VOC ) ,
(28)
where f (V ) represents the I-V characteristics of the cell, f rec
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solar cells. Ideal diodes D1 and D2 are utilized to model the forward
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