Fresnel with AR coating

iam_fresnel_ar.py

"""
Approximately duplicates plot on this pvsyst help here:
https://www.pvsyst.com/help/iam_loss.htm
"""

import numpy as np
import matplotlib.pyplot as plt
plt.ion()


# constants
n_glass = 1.56
n_air = 1.0
theta_inc = np.linspace(0, 88, 100)


def snell(theta_1, n1, n2):
    """Snell's equation"""
    sintheta_2 = n1/n2 * np.sin(np.radians(theta_1))
    return sintheta_2, np.degrees(np.arcsin(sintheta_2))


def refl_s(theta_1, theta_2, n1, n2):
    """Fresnel's equation"""
    n1_costheta_1 = n1*np.cos(np.radians(theta_1))
    n2_costheta_2 = n2*np.cos(np.radians(theta_2))
    return np.abs((n1_costheta_1 - n2_costheta_2)/(n1_costheta_1 + n2_costheta_2))**2


def refl_p(theta_1, theta_2, n1, n2):
    """Fresnel's equation"""
    n1_costheta_2 = n1*np.cos(np.radians(theta_2))
    n2_costheta_1 = n2*np.cos(np.radians(theta_1))
    return np.abs((n1_costheta_2 - n2_costheta_1)/(n1_costheta_2 + n2_costheta_1))**2


def refl_eff(rs, rp):
    """effective reflectivity"""
    return (rs+rp)/2


def trans(refl):
    """transmissivity"""
    return 1-refl


def refl0(n1, n2):
    """reflectivity at normal incidence"""
    return np.abs((n1-n2)/(n1+n2))**2


def fresnel(theta_inc, n1=n_air, n2=n_glass):
    """calculate IAM using Fresnel's Law"""
    _, theta_tr = snell(theta_inc, n1, n2)
    rs = refl_s(theta_inc, theta_tr, n1, n2)
    rp = refl_p(theta_inc, theta_tr, n1, n2)
    reff = refl_eff(rs, rp)
    r0 = refl0(n1, n2)
    return trans(reff)/trans(r0)


def ashrae(theta_inc, b0=0.05):
    """ASHRAE equation"""
    return 1 - b0*(1/np.cos(np.radians(theta_inc)) - 1)


def fresnel_ar(theta_inc, n_ar, n1=n_air, n2=n_glass):
    """calculate IAM using Fresnel's law with AR"""
    # use fresnel() for n2=n_ar
    _, theta_ar = snell(theta_inc, n1, n_ar)
    rs_ar1 = refl_s(theta_inc, theta_ar, n1, n_ar)
    rp_ar1 = refl_p(theta_inc, theta_ar, n1, n_ar)
    r0_ar1 = refl0(n1, n_ar)
    # repeat with fresnel() with n1=n_ar
    _, theta_tr = snell(theta_ar, n_ar, n2)
    rs = refl_s(theta_ar, theta_tr, n_ar, n2)
    rp = refl_p(theta_ar, theta_tr, n_ar, n2)
    # note that combined reflectivity is product of transmissivity!
    # so... rho12 = 1 - (1-rho1)(1-rho2) 
    reff = refl_eff(1-(1-rs_ar1)*(1-rs), 1-(1-rp_ar1)*(1-rp))
    r0 = 1-(1-refl0(n_ar, n2))*(1-r0_ar1)
    return trans(reff)/trans(r0)


# plot Fresnel for normal glass and ASHRAE
plt.plot(theta_inc, fresnel(theta_inc))
plt.plot(theta_inc, ashrae(theta_inc))

# calculate IAM for AR with n=1.1 and plot
iam_ar11 = fresnel_ar(theta_inc, n_ar=1.1)
plt.plot(theta_inc, iam_ar11)

# repeat for AR with n=1.2
iam_ar12 = fresnel_ar(theta_inc, n_ar=1.2)
plt.plot(theta_inc, iam_ar12)

# make plot pretty
plt.legend(['Fresnel, normal glass', 'ASHRAE, $b_0=0.05$', 'Fresnel $n_{AR}=1.1$', 'Fresnel $n_{AR}=1.2$'])
plt.title("IAM correction, Fresnel vs. ASHRAE, using basic eqn's")
plt.ylabel('IAM')
plt.xlabel(r'incidence angle $\theta_{inc} [\degree]$')
plt.grid()
plt.ylim([0.55,1.05])