The Sandia PV Array Performance Model (SAPM) defines five points on the IV curve.  These points are shown in the figure below.

The SAPM defines the primary points ($I_{sc}$, $I_{mp}$, $V_{oc}$, and $V_{mp}$) with the following equations:

$I_{sc}=I_{sc0}\times&space;f_{1}&space;\left&space;(&space;\frac{E_{b}f_{2}+f_{d}E_{d}}{E_{0}}&space;\right&space;)\times&space;\left&space;(&space;1+\alpha&space;_{Isc}\left&space;(&space;T_{c}-T_{0}&space;\right&space;)&space;\right&space;)$           (eq. 1)

$I_{mp}=I_{mp0}\left&space;(&space;C_{0}E_{e}+C_{1}{E_{e}}^{2}&space;\right&space;)\left&space;(&space;1+\alpha&space;_{Imp}\left&space;(&space;T_{c}-T_{0}&space;\right&space;)\right&space;)$           (eq. 2)

$V_{oc}=V_{oc0}+N_{s}\delta&space;\ln&space;\left&space;(&space;E_{e}&space;\right&space;)+\beta&space;_{Voc}\left&space;(&space;T_{c}-T_{0}&space;\right&space;)$          (eq. 3)

$V_{mp}=V_{mp0}+C_{2}N_{s}\delta&space;\ln&space;\left&space;(&space;E_{e}&space;\right&space;)+C_{3}N_{s}\left&space;(&space;\delta&space;\ln&space;\left&space;(&space;E_{e}&space;\right&space;)&space;\right&space;)^{2}+\beta&space;_{Vmp}\left&space;(&space;T_{c}&space;-T_{0}\right&space;)$            (eq. 4)

## Functions:

$f_{1}$ is a 4th order polynomial function of absolute air mass, $AM_{a}$, and is called the air mass modifier.  It is defined as:

$f_{1}\left&space;(&space;AM_{a}&space;\right&space;)=a_{0}+a_{1}AM_{a}+a_{2}\left&space;(&space;AM_{a}&space;\right&space;)^{2}+a_{3}\left&space;(&space;AM_{a}&space;\right&space;)^{3}+a_{4}\left&space;(&space;AM_{a}&space;\right&space;)^{4}$

where $a$ is the vector of coefficients that are determined from module testing.

$f_{2}$ is a 5th order polynomial function of angle of incidence, $AOI$, and is called the angle of incidence modifier. it is defined as:

$f_{2}\left&space;(&space;AOI&space;\right&space;)=b_{0}+b_{1}AOI+b_{2}\left&space;(&space;AOI&space;\right&space;)^{2}+b_{3}\left&space;(&space;AOI&space;\right&space;)^{3}+b_{4}\left&space;(&space;AOI&space;\right&space;)^{4}+b_{5}\left&space;(&space;AOI&space;\right&space;)^{5}$

where $b$ is the vector of coefficients that are determined from module testing.

$E_{e}$ is the “effective irradiance”.  It is defined as:

$E_{e}=\frac{I_{sc}}{I_{sc0}\left&space;\{&space;1+\alpha&space;_{Isc}\left&space;(&space;T_{c}-T_{0}&space;\right&space;)&space;\right&space;\}}$, where:

• $I_{sc0}$ is the short circuit current at reference conditions.
• $I_{sc}$ can be calculated from (eq. 1) above.

$\delta$ is a function of $T_{c}$ defined as:   $\delta&space;=\frac{n\times&space;k\left&space;(&space;T_{c}+273.15&space;\right&space;)}{q}$, where:

• $n$ is an empirically determined ‘diode factor’,
• $k$ is Boltzmann’s constant ($1.38066\times&space;10^{-23}J/K$),
• $q$ is the elementary charge constant ($1.60218\times&space;10^{-19}coulomb$)

$\beta&space;_{Voc}$ is a function of effective irradiance,$E_{e}$ , defined as: $\beta&space;_{Voc}=\beta&space;_{Voc0}+m_{\beta&space;Voc}\left&space;(&space;1-E_{e}&space;\right&space;)$, where:

• $\beta&space;_{Voc0}$ is the temperature coefficient for module open circuit voltage at irradiance conditions of $1000&space;W/m^{2}$ .
• $m_{\beta&space;Voc}$ is a coefficient describing the irradiance dependence for the open circuit voltage temperature coefficient (typically equals zero)

## Parameters:

• $E_{b}$ is beam irradiance on the plane of array
• $E_{d}$ is the diffuse irradiance on the plane of array
• $E_{0}$ is reference solar irradiance ($1000&space;W/m^{2}$)
• $T_{c}$ is cell temperature ($^{\circ}C$)
• $T_{0}$ is reference cell temperature ($25^{\circ}C$)
• $f_{d}$ is the fraction of the diffuse light that is used by the module.  For typical flat plate modules $f_{d}$ is usually assumed to be equal to 1.  For concentrators the value can be smaller than 1.
• $\alpha_{Isc}$ is the normalized temperature coefficient for short circuit current.  Units are $1/^{\circ}C$
• $\alpha&space;_{Imp}$ is the normalized temperature coefficient for maximum power current.  Units are $1/^{\circ}C$
• $N_{s}$ is the number of cells in series
• $C$ is a vector of coefficients determined by module testing using a method developed at Sandia.