The current density of an ideal p-n junction under illumination can be described by:
J(V)=Jph−J0(eqVkT−1)
where Jph is the photo-current density, J0 the saturation-current density, q the elementary charge, V the voltage, k the Boltzmann's constant, and T the temperature.
A crystalline silicon solar cell generates a photo-current density of Jph=40mA/cm2 at T=300K. The saturation-current density is J0=1.95∗10−10mA/cm2.
Assuming that the solar cell behaves as an ideal p-n junction, calculate the open-circuit voltage Voc (in V).
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J0=1.95∗10−10mA/cm2=1.95*10^-9A/m^2
Jph=40mA/cm2=400A/m^2
Voc=kT/q*ln(Jph/J0+1)
T=300K
k=1.35*10^-23J/K
q=1.6*10^-19C
J(V)=Jph−J0(eqVkT−1)
The open-circuit voltage is the voltage at which the net current is zero where J(Voc)=0
Solving for Voc we have
Voc=0.0253*ln((400+1.95*10^-9)/(1.95*10^-9))
Voc=0.659 V
To calculate the open-circuit voltage (Voc), we need to find the value of V for which J(V) is equal to zero.
The equation for J(V) is given as:
J(V) = Jph - J0(eq(V/kT) - 1)
Since we want to find the open-circuit voltage, we set J(V) equal to zero:
0 = Jph - J0(eq(Voc/kT) - 1)
Rearranging the equation, we get:
Jph = J0(eq(Voc/kT) - 1)
Now we can substitute the given values:
Jph = 40 mA/cm^2
J0 = 1.95 x 10^(-10) mA/cm^2
k = Boltzmann's constant = 1.38 x 10^(-23) J/K
T = 300 K
Plugging in these values, the equation becomes:
40 = 1.95 x 10^(-10)(e^(Voc/ (1.38 x 10^(-23) x 300)) - 1)
To solve for Voc, we need to isolate it on one side of the equation. Let's go through the steps step by step:
1. Divide both sides of the equation by J0:
40/J0 = (e^(Voc/ (kT)) - 1)
2. Add 1 to both sides of the equation:
1 + (40/J0) = e^(Voc/ (kT))
3. Take the natural logarithm (ln) of both sides:
ln(1 + (40/J0)) = (Voc/ (kT))
4. Multiply both sides by (kT):
Voc = (kT) * ln(1 + (40/J0))
Now we can substitute the given values of k and T to calculate Voc:
k = 1.38 x 10^(-23) J/K
T = 300 K
Voc = (1.38 x 10^(-23) J/K * 300 K) * ln(1 + (40 / (1.95 x 10^(-10) mA/cm^2)))
Calculating this expression will give us the open-circuit voltage (Voc) of the crystalline silicon solar cell.