Capacitors are made of dielectric material with permittivity and dielectric loss. Permittivity and dielectric loss depend on frequency and temperature. In this case, frequency is super low, and temperature is stable.

A polypropylene capacitor was chosen for an integrator because of the high resistivity. Most polymer is a dielectric material. Dielectric material polarizes by the external electric field. There are several models of the polarizations that are electron cloud, ion of the atom and ion, and dipole orientation.

In this case, permittivity is increased as the movement of dipoles that follow the direction of the external electric field until aligning the orientation.

Dielectric relaxation is explained as a procedure of mechanical orientation. During the procedure of orientation, dipoles have resistance from the molecules around them, and orientation takes time to complete.

For a high-frequency electric field environment, the dipole cannot follow the change of the electric field as it is faster than the orientation process. Also, for low-frequency electric fields, dipole follows the change in the electric field without delay.

The electronic model of a capacitor with dielectric absorption is shown below. The model consists of multiple RC time constants in parallel. Due to parasitic capacitors and resistors with a very long time constant, the capacitor behaves as if the capacitor memorized the previous voltage.

Dielectric absorption is standardized by IEC / EN 60384-1. According to the procedure of measurement, charge a capacitor at DC voltage for sixty minutes, followed by discharge for ten seconds. Then measure the voltage recovery for fifteen minutes, which indicates dielectric absorption voltage. The percentage of the voltage before and after is the level of the absorption. Polypropylene film capacitor has dielectric absorption of 0.05 to 0.1%.

Measuring ultra-low currents in the femtoampere range requires enough time for dielectric absorption and relaxation.

Figure 4-16 shows multiple cycles of integration measurements with a varying start voltage.

Using the data in Figure 4-17 to calculate I_B, we can see that the initial I_B measurements vary significantly from the settled measurements. This effect is an additional error from the I_B starts from different points and gradually aligns together as time goes by, which is Vcal goes up in this case. Though the relaxation process lasts long, we suppose the dielectric absorption was almost negligible at Vcal = 0.4V as the curves are aligned. With that, the slope of I_B versus Vcal for Vcal is from 0.4V to 0.6V, indicating resistance. From the slope, the resistance is more likely as 16.2 x 10¹⁵Ω.

At this point, we have only three parameters to consider. These are dielectric absorption, capacitor resistance, and I_B as shown in the drawing.

Estimate an error for the measured slope depending on what Vcal we took derivative. A calibration uses the resistance 16.2PΩ to provides 6aA leakage when the voltage across the capacitor is 100mV.

For example, if we measure the slope of the buffer (gain 10x) output is 10uV/sec, and Vcal is 1V which is the voltage across the integration capacitor of 100 [pF], I_B is calculated as

10 x 10^-6 / 10 x 100 x 10^-12 = 100aA (without calibration)

10 x 10^-6 / 10 x 100 x 10^-12 + 0.1 / 16.2 x 10^-15 = 106aA (with calibration).

We can add or subtract leakage current to or from the measured number depending on the signs of Vcal and I_B.