Charging capacitors using diodes at different temperatures. II Numerical studies

Abstract

This study is presented in a series of two papers. The first paper is an analytical study. This is the second paper, and here we numerically study the thermal energy harvesting capability of two electronic circuits. The first circuit consists of a diode and capacitor in series. We solve the time-dependent Fokker-Planck equation and show the capacitor initially charges and then discharges to zero. The peak charge on the capacitor increases with temperature, capacitance, and diode quality. The second circuit has two current loops with one small capacitor, two storage capacitors, and two diodes wired in opposition. When the diodes are held at different temperatures we observe a non-zero steady-state charge is accumulated on both storage capacitors. The magnitude of the stored charges are nearly equal but the signs are opposite. When resistors are used in place of diodes there is no transient and no steady-state charge buildup. Numerical studies for the time-independent Fokker-Planck equation are presented and confirm the steady state charges.

Figures (7)

• Figure 1: Single-loop circuit with a capacitor, diode, and a DC bias voltage. Numerical results use the following parameters $u_0=0.025$, $R=1$, $C=4$, $k_BT=1$, and $V=0$, unless otherwise stated in the figure. (a) Schematic of the circuit. Average charge on the capacitor in time is shown with alternative parameter values for (b) temperature, (c) capacitance, (d) resistance, (e) bias voltage, and (f) diode parameter.
• Figure 2: Multiple diodes in series or parallel with a capacitor. Numerical results use the following parameters: $u_0=0.025$, $R=1$, $C=4$, $k_BT=1$, and $V=0$. (a) Schematic of the series and parallel circuits. Average charge on the capacitor in time is shown for various numbers of diodes connected in (b) series and (c) parallel.
• Figure 3: Two-loop circuit dynamics using the following parameters: $k_BT_1=10$, $k_BT_2=1$, $u_0=1$, $R=0.1$, $C_0=4$, and $C_1=C_2=100$. (a) Circuit schematic. Average charge in time on capacitor (b) $C_1$ and (c) $C_2$. Variance of the charge distribution in time on capacitor (d) $C_1$ and (e) $C_2$. Time evolution of charge probability distribution on capacitor (f) $C_1$ and (g) $C_2$.
• Figure 4: Two-loop circuit dynamics using the following parameters: $k_BT_1=1$, $k_BT_2=100$, $u_0=1$, $R=0.1$, $C_0=4$, and $C_1=C_2=100$. Average charge in time on capacitor (a) $C_1$ and (b) $C_2$. Variance of the charge distribution on capacitor (c) $C_1$ and (d) $C_2$. Time evolution of the charge probability distribution on capacitor (e) $C_1$ and (f) $C_2$.
• Figure 5: Numerical results for the time-dependent Fokker Planck equation while varying $k_BT_2$. Parameter values are $u_0 = 0.2$, $R=0.1$, $C_0=4$, $C_1=C_2=100$, and $k_BT_1=1$. Average charge in time on capacitor (a) $C_1$, and (b) $C_2$. Average sum and difference scaled charges in time (c) $\xi$ and (d) $\eta$.