"""Collection of functions related to calculating current collection in solar cells
"""
import numpy as np
from scipy.integrate import simpson
from scipy.interpolate import interp1d
from scipy.constants import elementary_charge
import photovoltaic as pv
[docs]
def collection_probability(x, thickness, s, l, d):
"""
Returns the collection probability (unit 0 to 1) at a distance x (cm) from the junction.
See [1]_.
Parameters
----------
x : array-like
array of x positions from a junction to a surface (typically [cm]).
thickness : numeric
Layer thickness [cm].
s : numeric
Surface recombination velocity [cm/s].
l : numeric
Minority carrier diffusion length [cm].
d : numeric
Minority carrier diffusivity [cm^2/Vs].
Returns
-------
cp : array-like
Collection probability along x.
References
----------
.. [1] https://www.pveducation.org/pvcdrom/solar-cell-operation/collection-probability
"""
cosh_xl = np.cosh(x / l)
cosh_wl = np.cosh(thickness / l)
sinh_xl = np.sinh(x / l)
sinh_wl = np.sinh(thickness / l)
num = s * l / d * cosh_wl + sinh_wl
den = s * l / d * sinh_wl + cosh_wl
cp = cosh_xl - (num / den) * sinh_xl
return cp
[docs]
def calculate_jsc_from_tau_cp(
tau,
wafer_thickness,
d_base,
s_rear,
generation,
depth,
w_emitter=0.36,
l_emitter=15,
d_emitter=5,
s_emitter=1e4,
xp=0.00000024,
):
"""
Returns cell Jsc given lifetime and cell parameters
Jsc is calculated via integrating collection probability and optical generation profiles
through the wafer depth. Includes contribution of the emitter, depletion region (assumes
CP = 1 in the depletion region), and base.
Parameters
----------
tau : numeric
Carrier lifetime [us].
wafer_thickness : numeric
Wafer thickness [um].
d_base : numeric
Minority carrier diffusivity [cm^2/Vs].
s_rear : numeric
Rear surface recombination velocity [cm/s].
generation : array-like
array of generation current G(z), e.g. from PVlighthouse OPAL2 [cm^3/s] [1]_.
depth : array-like
array of wafer depth 0-z [um].
w_emitter : numeric, default 0.36
Emitter thickness [um].
l_emitter : numeric, default 15
Minority carrier diffusion length of the emitter [um].
d_emitter : numeric, default 5
Minority carrier diffusivity in the emitter [cm^2/s].
s_emitter : numeric, default 1e4
Front surface recombination velocity [cm/s].
xp : numeric, default 0.00000024
width of the depletion region [m]. Treated as fixed width, as it is very small compared
to the bulk, so injection-dependent variations will have very small effects.
Returns
-------
jsc : numeric
Short circuit current of the solar cell [mA/cm^2].
Notes
-----
Default emitter parameters w, l, d, and s are supplied, but users may wish to experiment to
find the most suitable parameters to model different cell types. Default values are adapted
from [2]_ and [3]_.
Pros of this approach
can account for e.g. lighttrapping via the generation profile
Cons of this approach
requires a generation profile
Future work:
include default generation and depth profile.
References
----------
.. [1] K. R. McIntosh and S. C. Baker-Finch, “OPAL 2: Rapid optical simulation of silicon solar
cells,” in 2012 38th IEEE Photovoltaic Specialists Conference, IEEE, 2012, pp. 000265–000271.
doi: 10.1109/PVSC.2012.6317616.
.. [2] A. Fell et al., “Input Parameters for the Simulation of Silicon Solar Cells in 2014,”
IEEE Journal of Photovoltaics, vol. 5, no. 4, pp. 1250–1263, Jul. 2015, doi:
10.1109/JPHOTOV.2015.2430016.
.. [3] W. J. Yang, Z. Q. Ma, X. Tang, C. B. Feng, W. G. Zhao, and P. P. Shi, “Internal quantum
efficiency for solar cells,” Solar Energy, vol. 82, no. 2, pp. 106–110, Feb. 2008, doi:
10.1016/j.solener.2007.07.010.
"""
q = elementary_charge
# Unit conversions:
w_depletion = xp * 100 # depletion region width, m to cm
w_emitter = w_emitter * 1e-4 # um to cm
l_emitter = l_emitter * 1e-4
w_base = wafer_thickness * 1e-4 - w_emitter # width of the base in cm
tau = tau * 1e-6 # us to s
s_base = s_rear # cm/s
depth = depth * 1e-4 # um to cm
# 1. calc diffusion length
l_base = np.sqrt(tau * d_base) # diffusion length in cm
# 2. set up depth profile and cell regions
index_emitter = np.searchsorted(depth, w_emitter)
emitter_depth = depth[0:index_emitter]
emitter_depth = np.flip((-emitter_depth + np.max(emitter_depth)).to_numpy())
index_depletion = np.searchsorted(depth, w_emitter + w_depletion)
depletion_depth = depth[index_emitter:index_depletion]
base_depth = depth[index_depletion:]
# 3. calc c.p. for emitter, depletion region, and base
cp_emitter = collection_probability(
emitter_depth, thickness=w_emitter, s=s_emitter, l=l_emitter, d=d_emitter
)
cp_emitter = np.flip(cp_emitter)
emitter_depth = np.flip((-emitter_depth + np.max(emitter_depth)))
cp_depletion = np.ones(len(depletion_depth))
cp_base = collection_probability(base_depth, w_base, s_base, l_base, d_base)
depth_array = np.concatenate(
np.array([emitter_depth, depletion_depth, base_depth], dtype=object), axis=0
)
collection_array = np.concatenate(
np.array([cp_emitter, cp_depletion, cp_base], dtype=object), axis=0
)
# 4. Reinterpolate to match the generation depth profile
f = interp1d(depth_array, collection_array, kind="cubic")
collection_array_interp = f(depth)
# 5. integrate
jsc = simpson(collection_array_interp * generation, depth) * q
return jsc * 1000
[docs]
def calculate_jsc_from_tau_iqe(
tau,
wafer_thickness,
d_base,
s_rear,
spectrum,
absorption,
wavelengths,
w_emitter=0.36,
l_emitter=15,
d_emitter=5,
s_emitter=1e4,
xp=0.00000024,
):
"""
Returns cell Jsc given lifetime and cell parameters
Calculates Jsc via calculating cell internal quantum efficiency (IQE) and absorption for Si
Parameters
----------
tau : numeric
Carrier lifetime [us].
wafer_thickness : numeric
Wafer thickness [um].
d_base : numeric
Minority carrier diffusivity [cm^2/Vs].
s_rear : numeric
Rear surface recombination velocity [cm/s].
spectrum : array-like
photon flux density solar spectrum.
absorption : array-like
absorption coefficient of Si, e.g. loaded from 'photovoltaic' library [1]_.
wavelengths : array-like
wavelength series (nm) for absorption coefficient and solar spectrum.
w_emitter : numeric, default 0.36
Emitter thickness [um].
l_emitter : numeric, default 0.5
Diffusion length of the emitter [um].
d_emitter : numeric, default 4
Minority carrier diffusivity in the emitter [cm^2/s].
s_emitter : numeric, default 4.86e5
Front surface recombination velocity [cm/s].
xp : numeric, default 0.00000024
width of the depletion region [m]. Treated as fixed width, as it is very small compared
to the bulk, so injection-dependent variations will have very small effects.
Returns
-------
jsc : numeric
Short circuit current of the solar cell [mA/cm^2]
Notes
-----
Default emitter parameters w, l, d, and s are supplied, but users may wish to experiment to
find the most suitable parameters to model different cell types. Default values are adapted
from [2]_ and [3]_.
Pros of this approach:
requires only fundamental inputs: Si absorption and spectrum
Cons:
does not account for any anti-reflection or light trapping
To do:
accept user inputs for spectrum and absorption, otherwise use photovoltaic library
References
----------
.. [1] “photovoltaic.” https://github.com/pvedu/photovoltaic
.. [2] A. Fell et al., “Input Parameters for the Simulation of Silicon Solar Cells in 2014,”
IEEE Journal of Photovoltaics, vol. 5, no. 4, pp. 1250–1263, Jul. 2015, doi:
10.1109/JPHOTOV.2015.2430016.
.. [3] W. J. Yang, Z. Q. Ma, X. Tang, C. B. Feng, W. G. Zhao, and P. P. Shi, “Internal quantum
efficiency for solar cells,” Solar Energy, vol. 82, no. 2, pp. 106–110, Feb. 2008, doi:
10.1016/j.solener.2007.07.010.
"""
# Unit conversions:
w_depletion = xp * 100 # depletion region width, m to cm
w_emitter = w_emitter * 1e-4
l_emitter = l_emitter * 1e-4
w_base = wafer_thickness * 1e-4 - w_emitter # width of the base in cm
tau = tau * 1e-6 # us to s
s_base = s_rear # cm/s
# 1. calculate diffusion length
l_base = np.sqrt(tau * d_base) # diffusion length in cm
# 2. calculate iqe
_, _, _, qet = pv.cell.iqe(
absorption,
w_depletion,
s_emitter,
l_emitter,
d_emitter,
w_emitter,
s_base,
w_base,
l_base,
d_base,
)
# 3. integrate iqe with spectrum
jsc = simpson(qet * spectrum, wavelengths) / 10
return jsc
[docs]
def generation_current(generation, depth):
"""
Returns cell generation current given generation profile
Parameters
----------
generation : array-like
array of generation current G(z), e.g. from PVlighthouse OPAL2 [cm^3/s] [1]_.
depth : array-like
array of wafer depth 0-z [um].
Returns
-------
j_gen : numeric
generation current [mA/cm^2]
"""
q = elementary_charge
j_gen = simpson(generation, depth) * q / 10
return j_gen
