A variation of the voltage applied to a p-n junction is linked to a variation of charge, which yields a capacitance. Two different capacitances have to be taken into account: the transition (or space charge) capacitance C_t and the diffusion capacitance C_d. C_d and C_t are parallel, so that the cell capacitance is expressed as:

C = C d + C t (1)

In dynamic regime, the junction schematic representation is given in Figure 1.

Therefore, the effect of the transition and diffusion capacitance on the diode is sketched in Figure 2.

In several papers [12] [18], the junction or transition capacitance, which is due to the charges stored in the depletion region, is expressed as:

C t = C 0 1 − V d Φ (2)

In Equation (2), Φ is the diffusion potential; V_d = U is the voltage over the diode; C₀ is the voltage for U = 0. The voltage V = Φ − U is the potential between the limits of the depletion region. The transition capacitance dominates for small voltages.

The diffusion capacitance C_d is due to the charges stored outside the depletion region, in the neutral regions, when the solar cell is under illumination. In some works [12], it is expressed as:

C d = τ I d n V T (3)

In Equation (3), τ is the mean carrier lifetime. I_d is the current conducted by the forward biased diode. V_T is the thermal voltage and is expressed as:

V T = K T q (4)

C_d dominates for forward biased p-n junctions.

In several works, the Negative Capacitance (NC) is studied. The admittance Y(ω), for a frequency ω, is expressed as [15]:

Y ( ω ) = δ I ( ω ) δ V ( ω ) (5)

for δI and δ V ~ e i ω t .

A small δV voltage impulse yields a transient harmonic current δI. The capacitance is expressed as:

C ( ω ) = 1 ω Im [ Y ( ω ) ] (6)

The conductance g(ω) is expressed as:

g ( ω ) = Re [ Y ( ω ) ] (7)

In all cases, the capacitance is linked to the voltage. In several works [19] [20], the capacitance is linked to the excess minority carrier’s density under illumination. Let a p-n junction under illumination be considered. In the p base, the excess minority carriers density generated δ(x) is gotten from the continuity equation. The resolution of this equation yields the photocurrent density J_ph, the photovoltage V_ph and the capacitance. The photovoltage V_ph is expressed as:

V p h = V T ln [ 1 + N b δ ( 0 ) n i 2 ] (8)

where δ ( 0 ) = δ ( x = 0 ) ; N_b is the base doping density; n_i is the intrinsic carriers density. The transition capacitance C_t and the diffusion capacitance C_d are expressed as [19] [20]:

C t = q V T n i 2 N b (9)

C d = q V T δ ( 0 ) (10)

Hence, from Equation (1), one gets:

C = q V T n i 2 N b + q V T δ ( 0 ) (11)

Moreover, from Equation (8), for V = V_ph, one gets:

1 + N b δ ( 0 ) n i 2 = e V V T (12)

Consequently, one gets:

δ ( 0 ) = n i 2 N b ( e V V T − 1 ) (13)

Then the capacitance, given by Equation (11), becomes:

C = q V T n i 2 N b + q V T n i 2 N b ( e V V T − 1 ) (14)

Finally, one gets:

C = q V T n i 2 N b e V V T (15)

Equation (15) shows explicitly, for the first time, the voltage dependence of the capacitance.

Very often, the single diode dynamic model is used for the dynamic study of solar cells. In some of the studies, only the forward-bias conducting diode D_f, the parallel capacitance C_p (which is voltage dependant), the shunt resistance R_sh and the series resistance R_s are taken into account, as shown in Figure 3 [10] [11].

In such model, the impedance Z of the circuit is given by:

Z = Re ( Z ) + j Im ( Z ) (16)

where:

Re ( Z ) = R s + R s h 1 + ω 2 C p 2 R s h 2 (17)

Im ( Z ) = − ω C p R s h 2 1 + ω 2 C p 2 R s h 2 (18)

In Equations (17) and (18), ω is the frequency. In other works [12], a more extended model is used, which, in addition, includes a series inductance L_s and a reverse-bias diode D_r, with a breakdown voltage V_bd, as shown in Figure 4.

For such model, the impedance Z is expressed as:

Z = ( R s + R s h α + 1 ) + j ( L s ω − ω C p R s h 2 α + 1 ) (19)

where α = ω 2 C p 2 R s h 2 .

The study of the variation of Z with the frequency ω is a mean to determine the cell parameters such as R_sh, R_s, C_p and L_s.

In the present paper, by taking into account Figure 1 and Figure 2, the dynamic resistance R_d is included in the model, without the reverse-bias diode and the series inductance, as shown in Figure 5.

The introduction of R_d in the model is justified by the following calculations. Let the capacitance current be considered. The diode voltage V_d is expressed as:

V d = V + R s I (20)

The charge Q is:

Q = C V d (21)

Therefore, the current i is given by:

i = ∂ Q ∂ t = ∂ ( C V d ) ∂ t = C ∂ V d ∂ t + V d ∂ C ∂ t (22)

i = C ∂ ( V + R s I ) ∂ t + ( V + R s I ) ∂ C ∂ t (23)

Then the current-voltage characteristic is given by [10]:

I = I p h − I s [ exp ( V + R s I n V T ) − 1 ] − V + R s I R s h − ( V + R s I ) ∂ C ∂ t − C ∂ ( V + R s I ) ∂ t (24)

In Equation (24), I_ph is the photocurrent, I_s is the diode saturation current and n is the diode ideality factor.

In the present paper, R_d is introduced as indicated below:

For a voltage u over the capacitance, C is expressed as:

C = δ Q δ u = δ i t δ u = t δ u δ i (25)

Since δ u = R d δ i

One gets:

∂ C ∂ t = 1 R d (26)

Moreover:

∂ ( V + R s I ) ∂ t = ∂ V ∂ t + R s ∂ I ∂ t = ∂ V ∂ t + R s ∂ I ∂ V ∂ V ∂ t = ∂ V ∂ t + g R s ∂ V ∂ t (27)

In Equation (27), g is the conductance. Hence:

∂ ( V + R s I ) ∂ t = ∂ V ∂ t ( 1 + g R s ) (28)

Therefore, Equation (24) becomes:

I = I p h − I s [ exp ( V + R s I n V T ) − 1 ] − V + R s I R s h − V + R s I R d − C ∂ V ∂ t ( 1 + g R s ) (29)

The solution of the nonlinear Equation (29) yields the output of the solar cell under illumination:

If the term g R s ≈ 0 , Equation (29) becomes:

I = I p h − I s [ exp ( V + R s I n V T ) − 1 ] − V + R s I R s h − V + R s I R d − C ∂ V ∂ t (30)

Let a solar cell be considered with the following parameters: V o c = 0.5925   V ; n i = 1.45 × 10 10     cm − 3 ; N b = 10 17     cm − 3 ; q = 1.602 × 10 − 19     C ; T = 299.16   K . A plotting of logC against V is sketched on Figure 6. It is a straight line with slope

a = 1 ln 10 1 V T = 16847 and intercept b = log C t = − 1.8839 . For the chosen solar

cell, the voltage increases quickly from 0.013 pF (for the short-circuit) to 125.3 μF (for open voltage V o c = 0.5925   V ).

Hence; for any value of the voltage, the capacitance can be calculated and that is useful for the evaluation of the current as shown by Equation (29) and the impedance.

The evaluation of the dynamic resistance R_d was made by using the following reduced equation:

I = I p h − I s [ exp ( V + R s I n V T ) − 1 ] − V + R s I R s h − V + R s I R d (31)

The calculation was made for the following PV cell described in [10]:

I p h = 1.035   A ; I s = 1.05 × 10 − 10     A ; n = 1.2 ; R s = 0.08   Ω ; R s h = 3.15   Ω

At the maximum power point ( V m = 0.5   V ; I m = 0.8   A ), it is found: R d = 23.53   Ω .

The present work gives, for the first time, an estimation of the dynamic resistance.

This estimation of R_d at the maximum power point shows that, like R_sh, R d ≫ R s .

Equation (31) shows that the variation ΔI_m of the current, due to the introduction of R_d in the solar cell characteristics, at the maximum power point is:

Δ I m = V m + R s I m R d = 0.024   A (32)

Hence the relative variation of I_m is Δ I m I m = 0.03 . It shows that R_d leads to a 3% reduction of the current, at the Maximum Power Point (MPP).

It should be noted that R_d is intensity dependent, as shown by Equations (30) and (31). For the short-circuit (V = 0), it is higher, so that its influence on the short-circuit intensity I_sc is very weak. The short-circuit intensity I_sc is linked to the photocurrent, I_ph by the following relation:

I s c ≈ I p h 1 + R s R s h (33)

As for the open voltage V_oc, it can be expressed by the semi-empiric relation [21]:

V o c ≈ V T ln ( 1 1.5 J p h 10 8 [ mA cm 2 ] ) + E g q (34)

In Equation (34), J_ph is the photocurrent density, E_g is the band gap and q is the electron charge. Hence the introduction of R_d which leads to a very small decrease of I_sc results in a weak decrease of V_oc. But as shown above, its influence on I_m at the maximum power point is not negligible. Consequently, the dynamic resistance R_d has an influence on the maximum output power P m = I m V m , while its influence on I_sc and V_oc is negligible.