Power Optimization of TENGs via Load Capacitance Sizing

Abstract

A power optimization strategy is described for triboelectric energy harvesting systems by optimizing the load capacitor size within a full wave rectifier (FWR). In AC harvesters such as triboelectric nanogenerators (TENGs) with an FWR, the average power delivered to the rectifier increases in each cycle, ultimately reaching a steady state determined by system parameters. Through cycle-level analysis of input voltage, current, and rectifier turn-on time during mechanical motion, an optimal load capacitance is identified that maximizes power delivery while minimizing transient time to reach this maximum power. This approach achieves peak power delivery via capacitor sizing alone, eliminating the need for additional circuitry. Experimental results using a vertical contact-separation triboelectric nanogenerator with internal capacitance varying from 24pF to 96pF demonstrate that the optimal rectifier capacitance of 390pF achieves maximum power delivery (900nW at 1.7Hz and 2.7μW at 5Hz) within the second cycle, while suboptimal capacitances either fail to reach peak power or delay it to the seventh cycle or later. Sensitivity analysis reveals that the method exhibits high robustness, with capacitances within ±30% of the optimal value can still maintain ≥90% of peak power, providing flexibility when implementing or selecting the capacitor size.

Graphical Abstract

I. Introduction

RECENT advances in portable ultra-low power microelectronic devices, particularly for the Internet of Things (IoT) and wireless sensor networks (WSNs) in implantable and wearable applications, have created significant challenges in securing sustainable power sources [1]–[4]. Conventional batteries at these power levels either lack sufficient power density for prolonged operation or are too bulky to meet design constraints [5]–[7]. By scavenging ambient energy, batteries can be recharged continuously, enabling more sustainable operation. Various ambient energy sources have been explored to power miniaturized devices. RF energy harvesters offer perpetual operation, but suffer from low power density and distance-dependent performance [8], [9]. Solar energy harvesters provide high power density under sunlight, but are typically bulky for equivalent power output [10], [11]. Among these sources, kinetic energy is particularly attractive due to its ubiquity [12], [13]. Piezoelectric energy harvesters (PEHs) can convert human motion to electricity, but they use fragile, costly materials and exhibit low efficiency with irregular human movements [14], [15]. In contrast, triboelectric nanogenerators (TENGs), first developed by Wang and co-workers in 2012 [16], convert mechanical energy through contact electrification and electrostatic induction. TENGs offer low-cost production, ecological safety, biocompatibility with human contact [17], and versatile fabrication using diverse materials and designs [18], [19], making them highly applicable for wearable and implantable devices.

However, the harvested power often cannot directly satisfy the requirements due to the need for a relatively stable DC voltage/current. A power management unit (PMU) serves as the interface between the harvester and the load, converting the harvested energy into a stable DC voltage [20]. Fig. 1 illustrates a typical energy harvesting system, consisting of an energy harvesting device, a PMU, and a load. The PMU should not only regulate and store the harvested energy, but also intelligently manage the power delivery to the load.

Fig. 1:

Typical energy harvesting system including the energy harvesting device, power management unit (PMU) and load.

In AC-based harvesters, a rectifier is used first to convert the AC output to DC, which is then regulated to the required voltage level via a DC-DC converter. The resulting DC energy is stored in supercapacitors or batteries to power various electronic loads. However, since supercapacitors typically have low impedance, the rectified energy is initially stored in an intermediate capacitor at the output of the rectifier before being transferred to the battery to avoid any mismatch [21], [22].

Depending on the available energy source and the characteristics of the harvester, the amount of harvested power is typically limited. In addition, a portion of this harvested power is lost during transfer to the PMU due to mismatches between the harvester and the PMU. Various optimization strategies, collectively referred to as power control methods in Fig. 1, have been devised to reduce these mismatches. One such method is the perturb and observe (P&O) maximum power point tracking (MPPT) approach [23], [24], which utilizes complex analog circuitry to dynamically track power and set it to its maximum available level. For example, Moon et al. introduced a novel P&O MPPT method utilizing a mixed-signal architecture to reduce the power consumption of conventional analog circuitry [25]. A simpler alternative replaces P&O’s extensive analog circuitry with fractional open-circuit voltage (FOCV) MPPT [26], monitoring the optimal rectifier output voltage to maximize power transfer. This method achieves passive power optimization with lower overhead. An additional enhancement involves adding control circuitry for a dynamic FOCV-MPPT system [27], [28]. In this approach, the control circuitry samples the open-circuit voltage at specific time intervals and updates the optimum voltage accordingly. Furthermore, parallel and series synchronous switched harvesting on inductor (P-SSHI) techniques have also been adopted to improve PMU efficiency [29]. However, these methods generally require additional control circuitry, which increases the overall power consumption. Furthermore, they rely on steady-state operation of the circuit, often requiring a relatively large post-rectifier capacitor to store energy until the circuit reaches steady state. Once steady state is reached, the control method is applied to achieve maximum power delivery [30].

Alternative approaches specific to TENGs have focused on addressing the variable capacitance characteristics inherent to triboelectric harvesters. Zargari et al. presented a power optimization method that adds a fixed capacitance in series with the harvester and rectifier to mitigate the detrimental effects of variable capacitance on power delivery [31]. Similarly, a capacitor optimization method was proposed demonstrating that a TENG with a bridge rectifier under periodic mechanical excitation exhibits charging behavior analogous to a DC voltage source in series with internal resistance [32].

However, these methods generally require additional control circuitry, which increases the overall power consumption, or rely on steady-state operation of the circuit, often requiring a relatively large post-rectifier capacitor to store energy until the circuit reaches steady state [30]. Additionally, some approaches have limited applicability, such as the model in [32], which assumes that the load capacitance significantly exceeds the internal capacitance of the harvester, restricting its use to later operational cycles.

By performing a cycle-level analysis of the rectifier signals, this paper demonstrates that the selection of the intermediate capacitance size at the output of the rectifier plays an important role in transferring the maximum available power of the harvester.

The primary contributions of this paper are as follows:

• An analytical approach is introduced to optimize the rectifier circuit based on its transient behavior, enabling the extraction of maximum power within the first few cycles, without waiting for steady-state operation.

• The turn-on time of the diodes within the FWR is introduced as an important circuit parameter that is independent of the characteristics of the harvester (such as force and frequency). Through this parameter, our proposed model provides comprehensive insight into voltage and current behavior, resulting in power optimization that does not depend upon harvester characteristics such as force and frequency.

• Experimental validation of the model is provided using a vertical contact-separation triboelectric generator.

The rest of the article is organized as follows: Section II provides a background on TENGs and describes an equivalent electrical model. In Section III, the impact of load capacitance on the transfer of power to the PMU is described. The behavior of the rectifier circuit on a per cycle basis is analytically investigated in Section IV. The proposed method to determine the optimal rectifier capacitance is described in Section V. Finally, Section VI validates the proposed model by providing experimental results based on the TENG of [33], including comparison results with existing methods, and Section VII concludes the article.

II. Triboelectric Nanogenerator (TENG) Electrical Model

This work utilizes a TENG-based self-powered pressure sensor designed for large-scale pressure detection in knee implants, capable of converting various pressure levels into electrical power for an embedded load sensor [33]. The device employs a cuboid array structure comprising aluminum and cuboid-patterned silicone rubber layers in vertical contact mode, housed within a knee implant package. This vertical contact-separation configuration, illustrated in Fig. 2(a), represents a widely adopted approach in the TENG design [34]. In this setup, two plates of conductive electrodes, each with an area $S$, are used. One plate is fixed and coated with a dielectric (insulating) material of thickness $t_d$, while the other moves vertically in a periodic cycle. When the plates come into contact, an exchange of charge occurs, creating opposing charge with density $\sigma$ on both the moving electrode and the dielectric surfaces. This charge remains trapped because at least one of the materials is an insulator.

Fig. 2:

A contact-separation TENG: (a) the cross-section of the device illustrating the stationary and moving electrodes, and the dielectric material in between the electrodes, (b) the equivalent electrical model consisting of a voltage source in series with a variable capacitor.

Assuming that the generated charge, $-\sigma S$, are uniformly distributed on the dielectric surface, an electric field $E_d=\sigma /\left({ϵ}_0{ϵ}_d\right)$ is created between the two surfaces, where ${ϵ}_0$ is the permittivity of the free space and ${ϵ}_d$ is the dielectric constant of the material. In this scenario, the system can be represented as a fixed capacitance $C_d=\left({ϵ}_0{ϵ}_{d}S\right)/\left(t_d\right)$. Once the electrode begins to separate from the dielectric, an air gap is introduced. Considering the time-varying displacement $x(t)$ of the moving electrode, this new gap has a different permittivity, creating an electric field $E_a=\sigma /{ϵ}_0$ and can be modeled as a time-varying capacitance $C_a(t)={ϵ}_{0}S/x(t)$. In general, the electrical model of TENG includes a capacitance $C_{\text{EH}}(t)$, which is formed by the series combination of $C_d$ and $C_a(t)$, in series with the voltage source $V_{\text{oc}}(t)=\frac{\sigma x(t)}{{ϵ}_0}$ [35], as depicted in Fig. 2(b).

Considering the commonly used motion profile for TENG characterization [36], [37], the displacement $x(t)$ is modeled as a sinusoidal motion,

$$x(t)=\frac{x_{\max }}{2}(1-cos\omega t),$$ (1)

where $\omega$ is the angular frequency of the motion and $x_{\max }$ represents the maximum separation distance between plates. A more general derivation, valid for non-sinusoidal displacement $x(t)$ is provided in Appendix. Based on this motion pattern, the open-circuit voltage $\left(V_{\text{oc}}(t)\right)$ and the equivalent time-varying capacitance of the TENG $\left(C_{\text{EH}}(t)\right)$ over one period are shown in Fig. 3 and determined by the following,

$$V_{\text{oc}}(t)=\frac{\sigma x(t)}{{ϵ}_0}=V_m(1-cos\omega t),$$ (2)

$$C_{\text{EH}}(t)=\frac{S{ϵ}_0}{\frac{t_d}{{ϵ}_d}+\frac{x_{\max }}{2}(1-cos\omega t)},$$ (3)

where $V_m=\left(\sigma x_{\max }\right)/\left(2{ϵ}_0\right)$ is the peak open-circuit voltage.

Fig. 3:

The behavior of the triboelectric harvester illustrating the movement of the electrode, $x(t)$, open circuit voltage, $V_{\text{oc}}(t)$, and the variable capacitor, $C_{\text{EH}}(t)$.

According to Eq. 3, $C_{\text{EH}}(t)$ in the electrical model of TENG varies between $C_{\max }$ in full contact mode at the beginning of each cycle and $C_{\min }$ when the plates are fully separated at each half cycle. Thus, $C_{\max }$ and $C_{\min }$ are,

$$C_{\max }=\frac{S{ϵ}_0}{t_d/{ϵ}_d},$$ (4)

$$C_{\min }=\frac{S{ϵ}_0}{x_{\max }+t_d/{ϵ}_d}.$$ (5)

The ratio of these two values is defined as $m$,

$$m=\frac{C_{\max }}{C_{\min }}=1+\frac{x_{\max }}{t_d/{ϵ}_d}.$$ (6)

Using $m$, the time-varying capacitance of the harvester can be rewritten as,

$$C_{\text{EH}}(t)=C_d\Vert C_a(t)=\frac{2C_{\max }}{1+m+(1-m)cos\omega t}.$$ (7)

The electrical model presented in this work is based on the contact-separation mode, which is a widely adopted configuration for TENGs due to its ability to mimic body movements, making it particularly suitable for wearable and implantable devices that harvest energy from human motion [33], [34], [38]–[40]. The proposed power optimization method can be extended to other TENG modes sharing similar electrical models. For instance, the sliding mode and freestanding mode also use a voltage source in series with a variable capacitor, though the capacitance and open-circuit voltage equations differ due to different charge transfer mechanisms and electromechanical coupling, with variable internal capacitance for sliding mode TENGs [41] and time-independent internal capacitance for freestanding mode TENGs [42], [43].

III. Impact of Load Capacitor on Rectifier Transient Power

When a TENG device is interfaced with a rectifier for AC-to-DC conversion, only a portion of the power produced by the harvester in each cycle is transferred to the rectifier, depending on the rectifier configuration and load. For an FWR, as shown in Fig. 4, the delivered power, $P_{\text{in}}(t)$, is determined by the product of the current from the harvester, $I_{\text{EH}}(t)$, and the input voltage of the rectifier, $V_{\text{in}}(t)$. Note that a purely capacitive load is employed at the rectifier, as subsequent DC-DC converters in low-power applications typically operate in Discontinuous Conduction Mode (DCM) to improve overall efficiency [26], [29], [44]. Consequently, for most of the operating cycle, the converters remain disconnected from the rectifier, leaving the intermediate capacitor $C_{\text{rect}}$ as the primary load.

Fig. 4:

A full wave rectifier (FWR) interfaced with a TENG device modeled as a voltage source in series with a variable capacitor.

To evaluate the power transferred from the harvester to the rectifier, it is helpful to focus on the average power delivered per cycle. Starting from the first cycle (e.g., $n=1$), the average power delivered increases steadily until it reaches maximum power $P_{\text{in,av,max}}$, at cycle $n_{\text{opt}}$, where the average input voltage $V_{\text{in},\text{av}}$ overlaps with the average harvester current $I_{\text{EH},\text{av}}$. This optimal cycle, $n_{\text{opt}}$, depends on the load impedance and the internal capacitance of the harvester. For example, the cycle-averaged input power of the rectifier is simulated for various values of $C_{\text{rect}}$, as illustrated in Fig. 5. The harvester in this example produces an open-circuit peak voltage of 100V at a frequency of 1Hz with an internal capacitance ranging from 18pF to 72pF. As shown in this figure, at low $C_{\text{rect}}$ values (e.g., 100pF), the rectifier power never reaches its maximum average value of 72nW. Increasing $C_{\text{rect}}$ to 330pF enables the rectifier to reach this peak power in the second cycle (i.e., $n=2$). However, increasing $C_{\text{rect}}$ to 1nF delays the cycle in which the maximum power is reached to $n=5$. This simulation demonstrates that the size of the rectifier capacitance plays an important role in power transferred from the harvester to the rectifier.

Fig. 5:

Cycle-level average input power of the rectifier for different values of $C_{\text{rect}}$.

In this paper, based on the characteristics of the harvester, an analytical method is proposed to determine the optimal rectifier capacitance so that (1) the optimal cycle $\left(n_{\text{opt}}\right)$ in which the power is maximized is attained and (2) $n_{\text{opt}}=2$, meaning that the maximum average power is delivered by the second cycle to minimize the transient time.

IV. Cycle-Level Analysis of FWR

To capture the cycle-level power behavior of the FWR circuit in Fig. 4, the input voltage and input current of the rectifier should be modeled during the first two cycles ($n=1$ and $n=2$), as described in the following subsections.

A. Input Voltage of the Rectifier

Referring to the schematic in Fig. 4, the transient behavior of $V_{\text{oc}}(t)$ and $V_{\text{in}}(t)$ is illustrated in Fig. 6. Assume that at $t=0$, both $C_{\text{EH}}(t)$ and $C_{\text{rect}}$ are discharged, and $V_{\text{oc}}(t)$ begins to rise. Once $V_{\text{oc}}(t)$ reaches twice the forward voltage drop across a diode (e.g., $2V_D$), $D_1$ and $D_4$ in Fig. 4 conduct, and the harvester current, $I_{\text{EH}}$, charges both $C_{\text{EH}}(t)$ and $C_{\text{rect}}$ until $V_{\text{oc}}(t)$ reaches its peak at half the cycle, $T/2$ ($T$ is the period of the harvester signal). During this interval, $V_{\text{in}}(t)$ also increases, while $C_{\text{rect}}$ continues to charge. At the end of this first half-cycle, the diodes $D_1$ and $D_4$ are turned off, and $C_{\text{rect}}$ maintains its charge (neglecting the diode leakage current).

Fig. 6:

Behavior of FWR: (a) voltage waveforms within the FWR circuit of Fig. 4 for the first two cycles, (b) rectifier input voltage $V_{\text{in}}(t)$. Sep and ct refer, respectively, to separation and contact times of the electrodes.

In the second half cycle, $D_2$ and $D_3$ turn on when $V_{\text{oc}}(t)$ reaches $2V_D+V_{\text{in}}\left(\frac{T}{2}\right)$, thereby charging $C_{\text{rect}}$ to a higher voltage $V_{\text{in}}(t)$ by the end of the cycle. This pattern repeats in subsequent cycles. Consequently, each half-cycle can be divided into ON and OFF states, as illustrated in Fig. 6(a). In this figure, the OFF state corresponds to the initial portion of the half-cycle during which all diodes are off, and the ON state represents the latter portion when one pair of diodes (either $D_1$ and $D_4$ or $D_2$ and $D_3$) is on. In cycle $n$, the first half cycle is referred to as the odd half cycle, and the turn-on time of $D_1$ and $D_4$ in this half cycle is $t_{\text{o},\text{n}}$. The even half-cycle is the second half-cycle, with $t_{\text{e},\text{n}}$ denoting when $D_2$ and $D_3$ turn on. These turn-on times for the first two cycles are illustrated in Fig. 6(b).

It is important to note that while both $V_{\text{oc}}(t)$ and $V_c(t)$ (the voltage across the internal capacitor $C_{\text{EH}}$) remain positive with respect to the ground, $V_{\text{in}}(t)$ represents the differential voltage $V_p-V_n$ across the rectifier input terminals and alternates between positive and negative values. During the first (odd) half-cycle when $D_1$ and $D_4$ conduct, $V_n$ is effectively connected to the ground, making $V_{\text{in}}=V_p$ positive. During the second (even) half-cycle when $D_2$ and $D_3$ conduct, $V_p$ is connected to the ground, making $V_{\text{in}}=-V_n$ negative. This alternating polarity ensures unidirectional current flow through $C_{\text{rect}}$, while maintaining the positive nature of both $V_{\text{oc}}(t)$ and $V_c(t)$.

Referring to Fig. 6(b), the rectifier input voltage $V_{\text{in}}(t)$ at cycle $n$ can be expressed as the sum of a fraction of the open circuit voltage and a DC component,

$$V_{\text{in},\text{n}}(t)=\alpha (t)\times V_{\text{oc}}(t)+V_{\text{DC},\text{n}},$$ (8)

where $V_{\text{DC},\text{n}}$ is the DC voltage at cycle $n$ and has distinct values for odd half-cycles $\left(V_{\text{DC},\text{o},\text{n}}\right)$ and even half-cycles $\left(V_{\text{DC},\text{e},\text{n}}\right)$, during the ON and OFF periods of the circuit. Referring to Fig. 4, $\alpha (t)$ is defined as the ratio of the equivalent input impedance observed from the rectifier input (between $V_p$ and $V_n$) to the equivalent input impedance observed from the open-circuit voltage of the harvester (between $V_k$ and $V_n$). During the ON state $\alpha (t)$ is determined as,

$${\alpha }_{\text{on}}(t)=\frac{C_{\text{EH}}(t)}{C_{\text{EH}}(t)+C_{\text{rect}}}=\frac{2C_{\max }}{2C_{\max }+(1+m+(1-m)cos\omega t)C_{\text{rect}}}.$$ (9)

At each half-cycle, when the plates of the TENG are fully separated, $\alpha$ becomes,

$${\alpha }_{\text{sep}}=\frac{C_{\text{max}}}{C_{\text{max}}+m{C}_{\text{rect}}}.$$ (10)

Applying Kirchhoff’s Voltage Law in Fig. 4, the voltage across the internal capacitor of the harvester during the ON state in each cycle, $V_{\text{c},\text{n}}(t)$, is determined as,

$$V_{\text{c},\text{n}}(t)=V_{\text{oc}}(t)-V_{\text{in},\text{n}}(t)=\left(1-{\alpha }_{\text{on}}(t)\right)\times V_{\text{oc}}(t)-V_{\text{DC},\text{n}}.$$ (11)

When aiming for maximum power transfer by the second cycle $\left(n=2\right)$, we focus on the first (i.e., odd) half-cycle since the average power in this half-cycle is more sensitive to turn-on time variations. Additionally, we assume that, during the odd half-cycle of the first cycle, the diodes switch on immediately once the cycle begins. Using Eq. 8, Eq. 9 and Eq. 11, $V_{\text{in}}$ can be determined separately at the beginning of each cycle, each mid-cycle, and at the start of each ON-state, as indicated by Eq. 12.

Based on Eq. 12, we can determine voltages $V_1$, $V_2$ and $V_3$ in Fig. 6(b), which, as discussed earlier, helps to predict the required voltage and current essential for power calculation in odd half-cycle of cycle $n=2$. According to this equation, $V_1$ is,

$$V_1=V_{\text{in}}\left(\frac{T}{2}\right)=2{\alpha }_{\text{sep}}{V}_m.$$ (13)

When diodes turn on at $t=t_{\text{o},\text{n}}$, as shown in Fig. 6(a), $V_{in}$ is

$$V_{\text{in}}\left(t_{\text{o},\text{n}}\right)=V_c\left(t_{\text{o},\text{n}}\right)=\frac{1}{2}{V}_{\text{oc}}\left(t_{\text{o},\text{n}}\right).$$ (14)

Using this equation, $V_2$ in Fig. 6(b) is,

$$V_2=V_{\text{in}}\left(t_{\text{o},2}\right)=\frac{V_m}{2}\left(1-cos\omega t_{\text{o},2}\right).$$ (15)

Equating this result with Eq. 8 at $t=t_{\text{o},1}$ gives,

$$V_{\text{DC},\text{o},2}=\left(\frac{1}{2}-\alpha \left(t_{\text{o},2}\right)\right)\left(1-\cos \omega t_{\text{o},2}\right)V_m.$$ (16)

Thus, $V_3$ in Fig. 6(b) is obtained by replacing Eq. 16 in Eq. 8 at $t=\frac{3T}{2}$,

$$V_3=V_{\text{in}}\left(\frac{3T}{2}\right)=V_m\left[2{\alpha }_{\text{sep}}+\left(\frac{1}{2}-\alpha \left(t_{\text{o},2}\right)\right)\left(1-cos\omega t_{\text{o},2}\right)\right].$$ (17)

The average voltage during the odd half-cycle of the second cycle $\left(n=2\right)$ can then be determined as,

$$V_{\text{in},\text{av},\text{o},2}=\frac{V_2+V_3}{2}.$$ (18)

B. Input Current of the Rectifier

In the next step, we analyze the input current of the rectifier, which is equal to the current flowing through $C_{\text{EH}}(t)$. To determine the average current in a half-cycle, we calculate the amount of charge transferred during that half-cycle. Since the odd half-cycle of the second cycle $\left(n=2\right)$ is used for maximizing power, the average current during this half-cycle is expressed as,

$$I_{\text{EH},\text{av},\text{o},1}=\frac{Q\left(\frac{3T}{2}\right)-Q\left(t_{\text{o},2}\right)}{\frac{T}{2}},$$ (19)

$$Q\left(t_{\text{o},2}\right)=-C_{\max }{V}_c\left(t_{\text{o},2}\right),$$ (20)

$$Q\left(\frac{3T}{2}\right)=C_{\min }{V}_c\left(\frac{3T}{2}\right),$$ (21)

where $V_c\left(t_{\text{o},2}\right)$ is obtained from Eq. 14 and Eq. 15, and $V_c\left(\frac{3T}{2}\right)$ is extracted by substituting Eq. 17 in Eq. 11. Thus, the average current during the half-cycle is determined as,

$$I_{\text{EH},\text{av},\text{o},2}=\frac{2C_{\max }}{mT}\left[2V_m-V_3-m{V}_2\right].$$ (22)

V. Optimal Load Capacitance

A small load capacitance is typically desirable for the rectifier to minimize transient time and attain a steady-state DC voltage. However, with a small load capacitance, the charge transferred to $C_{\text{rect}}$ can elevate $V_{\text{in}}$ above a critical threshold, causing the intersection point between the input voltage and the current (where the maximum power occurs) to be missed (see Fig. 5). As a result, the cycle with $P_{\text{in,av,max}}$ is lost. Therefore, an optimal load capacitance, $C_{\text{rect,opt}}$, is needed to ensure maximum power transfer while minimizing transient time. In this paper, $C_{\text{rect,opt}}$ is determined using a two-step approach so that $P_{\text{in,av,max}}$ is achieved in $n=2$. First, the optimal turn-on time of rectifier diodes in the first half-cycle is determined based on the intersection of its average input voltage (Eq. 18) and current (Eq. 22) per cycle, as derived in the previous section. Then, by relating this turn-on time to the load capacitance, $C_{\text{rect,opt}}$ is determined to ensure a minimal transient time within a single harvester motion cycle. These two steps are described in the following subsections.

$$\begin{gathered} V_{\text{in}}(0)=0\approx V_{\text{in}}\left(t_{\text{o},1}\right) \\ {V}_{\text{in}}\left(\frac{T}{2}\right)=2{\alpha }_{\text{sep}}{V}_m \\ {V}_{\text{in}}\left(t_{\text{e},1}\right)=-V_m\left[2\left(1-{\alpha }_{\text{sep}}\right)-\left(1-cos\omega t_{\text{e},1}\right)\right] \\ {V}_{\text{in}}(T)=-V_m\left[2\left(1-{\alpha }_{\text{sep}}\right)-\left(1-{\alpha }_{\text{on}}\left(t_{\text{e},1}\right)\right)\left(1-cos\omega t_{\text{e},1}\right)\right] \\ {V}_{\text{in}}\left(t_{\text{o},2}\right)=V_m\left[-2\left(1-{\alpha }_{\text{sep}}\right)+\left(1-{\alpha }_{\text{on}}\left(t_{\text{e},1}\right)\right)\left(1-cos\omega t_{\text{e},1}\right)+\left(1-cos\omega t_{\text{o},2}\right)\right] \\ {V}_{\text{in}}\left(\frac{3T}{2}\right)=V_m\left[-2\left(1-2{\alpha }_{\text{sep}}\right)+\left(1-{\alpha }_{\text{on}}\left(t_{\text{e},1}\right)\right)\left(1-\cos \omega t_{\text{e},1}\right)+\left(1-{\alpha }_{\text{on}}\left(t_{\text{o},2}\right)\right)\left(1-cos\omega t_{\text{o},2}\right)\right] \end{gathered}$$ (12)

A. Optimal turn-on time of diodes

The average input power of the rectifier during the odd half-cycle of $n=2$ is determined as the product of its average input voltage and current, derived from Eq. 18 and Eq. 22, respectively. The turn-on time of $D_1$ and $D_4$ in this half-cycle affects $V_2$ and $V_3$ in Fig. 6(b), and hence the average voltage and current, as shown in Fig. 7.

Fig. 7:

Waveforms of (a) harvester current $I_{\text{EH}}(t)$, and (b) rectifier input voltage $V_{\text{in}}(t)$ for different ratios of $m$ (with the same $C_{\max }$), during the first two cycles.

By substituting $V_2$ from Eq. 15 and $V_3$ from Eq. 17 and assuming $m\approx 1$ (which represents a TENG with a fixed internal capacitance of $C_{\text{EH}}$, where $x_{\max }$ in Eq. 3 is negligible compared to $t_d$), these average voltage and current values can be expressed as functions of the diode turn-on time $t_{\text{o},2}$ in the corresponding half-cycle,

$${\left.V_{\text{in},\text{av},\text{o},2}\right|}_{m\approx 1}=V_m\left\{\frac{C_{\text{EH}}+0.5\left(1-cos\omega t_{\text{o},2}\right)C_{\text{rect}}}{C_{\text{EH}}+C_{\text{rect}}}\right\},$$ (23)

$${\left.I_{\text{EH},\text{av},\text{o},2}\right|}_{m\approx 1}=\frac{2V_m{C}_{\text{EH}}}{T}\left\{\frac{\left(1+cos\omega t_{\text{o},2}\right)C_{\text{rect}}}{C_{\text{EH}}+C_{\text{rect}}}\right\}.$$ (24)

Based on these equations, there is an optimal diode turn-on time at which the average voltage and current overlap, maximizing the average input power. This optimum time is determined by,

$$\frac{\partial P_{\text{in,av,o,2}}{|}_{m\approx 1}}{\partial t_{\text{o},2}}=\frac{\partial }{\partial t_{\text{o},2}}(V_{\text{in,av,o,2}}{|}_{m\approx 1}\times I_{\text{EH,av,o,2}}{|}_{m\approx 1})=0$$ (25)

.

Substituting the average voltage and current values from Eq. 23 and Eq. 24, we can determine the optimal turn-on time of diodes during the odd half-cycle for cycle $n=2$ as,

$${\left.cos\omega t_{\text{o},2,\text{opt}}\right|}_{m\approx 1}=\frac{C_{\text{EH}}}{C_{\text{rect}}}.$$ (26)

Note that even though we begin the analytic derivation with the simplified case of $m\approx 1$ for mathematical tractability, this analysis is subsequently extended to higher values of $m$ to account for significant internal capacitance variability, as demonstrated in the following subsection and experimental validation. This extension to higher values of $m$ is achieved by determining the intersection point of the average input voltage (Eq. 18) and average input current (Eq. 22), assuming that $C_{\text{rect}}$ is much larger than $C_{\min }$. The analysis yields a general expression for the optimal turn-on time of the diodes during the odd half-cycle of $n=2$, given by,

$$cos\omega t_{\text{o},2,\text{opt}}\approx \frac{m-1}{m+1},$$ (27)

$$t_{\text{o},2,\text{opt}}\approx \frac{1}{\omega }co{s}^{-1}\left(\frac{m-1}{m+1}\right).$$ (28)

The FWR circuit in Fig. 4 was simulated to determine the optimum turn-on time (as a fraction of the entire cycle). This analysis was performed with different values of $V_m$ and $f$ to represent different harvester forces and frequencies. Since a TENG device characteristically produces low-frequency, high open-circuit voltages, these simulations considered peak voltages up to 100V and frequencies up to 5Hz (due to the typical TENG operating range between 1Hz to 5Hz [38]). For each configuration, the parameter m was varied from 1 to 10 to assess its impact on the optimum diode turn-on time.

The results are shown in Fig. 8. The analytic result obtained by Eq. 28 is also included in the figure. According to this figure, as $m$ increases, the optimal turn-on time decreases. For example, at $m=2$, the optimum turn-on time occurs at 16.9% of the cycle, whereas for $m=10$, the optimum turn-on time is reduced to 8.6% of the cycle. Thus, for a harvester where the difference between $C_{max}$ and $C_{min}$ is greater, rectifier diodes should be turned on earlier in the cycle to ensure maximum power transfer. Importantly, the characteristics of the harvester (operational frequency and maximum voltage output) have minimal impact on the optimal turn-on time. This result is important as it establishes turn-on time of the diodes as a parameter independent of the harvester characteristics, making it a feasible option for dynamic power optimization, as targeted in this paper. This figure also shows the accuracy of Eq. 28 with respect to the simulation results. At $m=2$, the analytic approximation estimates the optimum turn-on time with an error of 4.7%, which decreases to 1.1% for $m=10$. As shown in Section VI, this error has a negligible impact on the power delivered to the load.

Fig. 8:

Simulation results illustrating the optimal turn-on time of rectifier diodes (normalized to cycle time) as a function of $m$ for a harvester with different applied forces and frequencies. The analytically approximated optimal turn-on time based on Eq. 28 is also indicated.

B. Capacitance-dependent turn-on time of diodes

In this subsection, the relationship between turn-on time and rectifier capacitance is examined, aiming to identify the rectifier capacitance that achieves the optimum turn-on time described in the preceding subsection. Based on Fig. 6(b) and assuming that the positive and negative peaks of $V_{\text{in}}(t)$ in Fig. 6(b) have the same magnitudes during the OFF state, Eq. 12 can be rewritten as,

$$V_{\text{in}}\left(t_{\text{e},1}\right)-V_{\text{in}}(T)=-V_{\text{in}}\left(\frac{T}{2}\right)+V_{\text{in}}\left(t_{\text{o},2}\right),$$ (29)

$$V_{\text{in}}\left(t_{\text{o},2}\right)=-V_{\text{in}}(T),$$ (30)

resulting in,

$$\cos \omega t_{\text{e},1}=4{\alpha }_{\text{sep}}-1,$$ (31)

$$\cos \omega t_{\text{o},2}=1-4{\alpha }_{\text{sep}}-2{\alpha }_{\text{on}}\left(t_{\text{o},1}\right)-4\left(1-2{\alpha }_{\text{sep}}\right){\alpha }_{\text{on}}\left(t_{\text{e},1}\right).$$ (32)

Since diodes turn on immediately after harvester starts operating (due to typical $C_{\text{EH}}$ values that are sufficiently small), $\cos \omega t_{\text{o},1}\approx 1$. By replacing Eq. 31 into Eq. 9 we obtain ${\alpha }_{\text{on}}\left(t_{\text{o},1}\right)$ and ${\alpha }_{\text{on}}\left(t_{\text{e},1}\right)$. From these terms, the diode turn-on time during the odd half-cycle of $n=2$ can be expressed as a function of $C_{\text{max}}$ and $C_{\text{rect}}$,

$$\begin{array}{l}cos\omega t_{\text{o},2}=\\ \frac{C_{\max }^3+(m-6)C_{\text{rect}}{C}_{\max }^2+m(2-7m)C_{\text{rect}}^2{C}_{\max }+m^3{C}_{\text{rect}}^3}{C_{\max }^3+(m+2)C_{\text{rect}}{C}_{\max }^2+m(2+m)C_{\text{rect}}^2{C}_{\max }+m^3{C}_{\text{rect}}^3}.\end{array}$$ (33)

C. Optimal ${\text{C}}_{rect}$

To maximize $P_{\text{in},\text{av}}$ during the odd half-cycle $\left(n=2\right)$, diodes $D_1$ and $D_4$ should satisfy the optimal turn-on time. The corresponding optimal $C_{\text{rect}}$ can be determined by comparing the capacitance-dependent $\cos \omega t_{\text{o},2}$ and its optimal value derived in Subsections A and B of Section V. As illustrated in Fig. 9, according to the simulation results, for $m\approx 1$, the average input voltage and current overlap at $C_{\text{rect}}=5.52\times C_{\max }$. This value is sufficiently close to the analytically obtained value of $5.81\times C_{\max }$ (with an error of approximately 5%), determined by equating Eq. 33 and Eq. 26 with $m\approx 1$. The relationship between optimal $C_{\text{rect}}$ and m for higher values of $m$ is obtained by equating Eq. 33 and Eq. 27, resulting in a more general analytical solution for $C_{\text{rect,opt}}$ as,

$$C_{\text{rect},\text{opt}}\approx 5.81\times \frac{m+2}{2m+1}\times C_{\text{max}}.$$ (34)

Fig. 9:

Simulation results illustrating the average input voltage and current of the rectifier in cycle $n=2$ as a function of rectifier capacitance. For the harvester, $m\approx 1$.

According to the average power extracted from the circuit with the optimal load capacitance, Eq. 34 is rewritten, using Eq. 27, to show the dependence of the optimal load capacitance on the turn-on time of diodes during the odd half-cycle of $n=2$, producing

$$C_{\text{rect},\text{opt}}\approx 5.81\times \frac{3-cos\omega t_{\text{o},2,\text{opt}}}{3+cos\omega t_{\text{o},2,\text{opt}}}\times C_{\max }.$$ (35)

For $C_{\text{rect}}$ values lower than the optimum value, the power delivered to the rectifier never reaches the peak power. However, if the rectifier capacitance is larger than the optimum value, the maximum power is delayed to later cycles, causing unnecessary delay.

VI. Validation of the Proposed Model

A. Measurement results

To validate the theoretical predictions, experimental measurements were performed using a fabricated TENG system under controlled laboratory conditions. The TENG operates on a vertical contact-separation mode and was fabricated with layers of aluminum and cuboid-patterned silicone rubber [33]. This TENG is incorporated into a pressure sensor for extensive pressure detection in a durable and reliable knee implant, intended to provide insight into knee functionality after surgery. The energy harvester converts a wide range of pressures into electrical power to drive a load sensor within the implant. According to the characteristics of the TENG listed in Table I, the maximum harvester capacitance is $C_{\max }\approx 96\text{pF}$ and the minimum harvester capacitance is $C_{\min }\approx 24\text{pF}$, resulting in a ratio $m=4$. According to our model (Eq. 34 and Eq. 35), for a harvester with $m=4$ and $C_{\max }$ of 96pF, the optimal $C_{\text{rect}}$ that maximizes power transfer while minimizing the cycle time (i.e., n=2) is approximately 374pF.

TABLE I:

Characteristics of the TENG based on [33].

Parameter	Symbol	Value
Effective contact area	$S$	4.5 cm2
Charge density	$\sigma$	39–64 μCm−2
Maximum time-variant gap	$x_{\max }$	0.124 mm
Silicone rubber thickness	$t_{\text{d}}$	0.207 mm
Dielectric constant of silicone rubber	${ϵ}_{\text{d}}$	5
Permittivity of air	${ϵ}_0$	8.85 × 10−12

Fig. 10 presents the experimental setup for measurements, with displacement control on a universal compression testing machine (MTS 858 Mini Bionix II) to apply controlled forces to the harvester package, creating low frequency sinusoidal movement. Package-harvester assemblies are placed within the MTS machine. The applied force is adjusted to generate an open-circuit voltage with a peak of 250V. Two operating frequencies are considered for the harvester. In the first scenario, the frequency of the MTS is maintained at 1.7Hz, while in the second scenario, it is increased to 5Hz.

Fig. 10:

Illustration of the experimental setup for measurements utilizing a contact-separation TENG [33].

All of the experiments were conducted in a temperature-controlled laboratory environment at 22±2°C with relative humidity maintained at 45±5%. The test setup was isolated from external vibrations using rubber isolation pads and shielded from electromagnetic interference using grounded metal enclosures. Each measurement was repeated three times with 100 data acquisition periods at 600Hz. The frequency accuracy of the MTS was ±0.01 Hz with an amplitude control precision of ±2%.

Using D14007 diodes in the FWR with a forward voltage drop of 1V, the average input power of the rectifier circuit is measured in both scenarios for three load capacitances: 100pF, 390pF (discrete capacitor closest to the optimal predicted capacitor) and 10nF. The average power is calculated by multiplying $V_{\text{in}}$ and $I_{\text{EH}}$ for each load capacitance, with typical waveforms illustrated in Fig. 11. As shown in the figure, the load capacitance significantly affects both voltage buildup and current flow patterns. For capacitances smaller than the optimal value (100pF), the voltage increases sharply with large incremental steps in each cycle, while the current exhibits low-amplitude pulses. This combination of sharp voltage buildup and low current amplitude results in low average power. At optimal capacitance (390pF), the incremental voltage steps decrease while the peak current increases significantly, creating optimal conditions where the multiplication of these averaged values reaches maximum power extraction. For capacitances much larger than the optimal value (10nF), the incremental voltage steps become sufficiently small per cycle while current peaks are high, but the low voltage buildup dominates, resulting in reduced average power despite the higher current amplitudes.

Fig. 11:

Measurement results for the voltage and current waveforms of the harvester with $V_m=250\text{V}$ and frequency of 1Hz for three different load capacitances.

To enhance measurement accuracy and reliability, the voltage $\left(V_{\text{in}}\right)$ and current $\left(I_{\text{EH}}\right)$ waveforms were processed using digital filtering techniques implemented in MATLAB. A two-stage filtering approach was employed: (1) a wavelet denoising filter (Daubechies-4, 5 decomposition levels) to remove both low-frequency drift and high-frequency noise while preserving transient signal characteristics, and (2) a 60 Hz notch filter $(Q=30)$ to eliminate power line interference common in laboratory environments. The wavelet filter was specifically selected for its superior time-frequency localization capability, which is particularly effective for TENG signals containing both slow voltage buildup patterns and sharp current pulse features. This analysis approach ensures that the critical pulse timing and amplitude characteristics essential for accurate power calculations remain intact while achieving superior noise suppression. Power spectral density analysis validates the filter effectiveness, demonstrating that the approach preserves more than 97% of signal power in the operational frequency band (0–10 Hz) while achieving significant suppression of high-frequency noise components for both voltage and current measurements. Importantly, since the majority of noise components are concentrated in higher frequency ranges well outside the operational band, the filtering validates the accuracy of the measurements, confirming that the voltage-current relationships essential for power calculations represent true TENG behavior.

The per-cycle average power comparison for frequencies of 1.7 Hz and 5 Hz is shown in Fig. 12-a and b, respectively, calculated using the filtered waveforms to ensure measurement reliability. According to the measurement results in Fig. 12(a), the maximum available power for the first scenario $(f=1.7\text{Hz})$ is approximately 900nW, which is reached in the second cycle $\left(n=2\right)$ when $C_{\text{rect}}=390\text{pF}$, which confirms the proposed analytic model. If $C_{\text{rect}}$ is larger (10nF), the circuit requires eight cycles to reach this power, unnecessarily delaying the transfer of the maximum power. Alternatively, when $C_{\text{rect}}$ is smaller (100pF), the maximum available power is never delivered to the rectifier. These experimental results confirm the behavior of the circuit and simulation results described in Section III.

Fig. 12:

Measurement results of the average delivered power from harvester to the rectifier with different rectifier capacitances when $V_m=250\text{V}$ and (a) $f=1.7\text{Hz}$ and (b) $f=5\text{Hz}$. According to the proposed model, $C_{rect}=390\text{pF}$ corresponds to the optimal scenario.

In the second scenario, the operating frequency of the harvester is increased to 5Hz. As illustrated in Fig. 12(b), the optimal rectifier capacitance remains unchanged at $C_{\text{rect}}=390\text{pF}$, and the maximum power is still achieved in cycle $n=2$. This frequency-independent behavior confirms that the optimal capacitance and the number of cycles required to reach maximum power transfer are determined solely by the sizing of the capacitor, not by the harvester frequency. The maximum available power in this scenario increases to 2.7μW due to the higher frequency, but the timing (cycle $n=2$ for 390pF and cycle $n=8$ for 10nF) remains identical to the case where $f=1.7\text{Hz}$. This experimental validation confirms the theoretical prediction of Eq. 35 and Fig. 8 that the optimal turn-on time depends only on the capacitance ratio $m$.

1. Sensitivity analysis: To assess the robustness of the proposed method, a sensitivity analysis was performed around the optimal load capacitance. For a harvester with an open-circuit voltage of 100 V and $m=2$ with $C_{\max }=134$ pF, the optimal load capacitance is 622 pF, as calculated from Eq. 34. The simulation results demonstrate that capacitances within ±35% of $C_{\text{rect,opt}}$ (420 pF to 880 pF) maintain ≥ 94% of peak power.

Experimental validation using a different harvester configuration (250 V open-circuit voltage, 5 Hz frequency, $m=4$, $C_{\max }=96$ pF) confirms this robustness, showing that capacitances within ±30% of the optimal value can maintain ≥ 90% of the peak power. This wide range of tolerance provides flexibility for the selection of components, accommodating practical constraints such as discrete component availability (for PCB design) and space limitations (for on-chip implementations). The inherent robustness stems from the fact that small capacitance changes primarily affect diode turn-on timing within cycles, while much larger deviations are required to shift the optimal cycle number.

B. Comparison with FOCV-MPPT Method

Both the proposed model for optimizing rectifier capacitance and the established FOCV-MPPT technique seek to maximize power delivery from the rectifier output, albeit by employing different control variables. In this section, a concise analysis of the FOCV methodology is provided, followed by a comparative evaluation to corroborate the accuracy of the proposed model.

In the existing FOCV-MPPT technique, an optimal rectifier output voltage is determined, thereby yielding the maximum extractable power at the rectifier output. This optimal voltage is given by [29],

$$V_{\text{rect},\text{opt}}\approx \frac{V_m}{2(1+m)},$$ (36)

where $V_m$ is the peak amplitude of the signal generated by the harvester. According to this equation, the optimum voltage varies as the harvester output changes, which requires additional sensing circuitry. Alternatively, in our proposed method, the optimum turn-on time and optimum capacitance (Eq. 28 and Eq. 35) do not depend on the characteristics of the harvester.

As detailed in [40], the optimal rectifier output voltage for the FOCV-MPPT method is generated using a low dropout regulator (LDO) and a reference voltage generator. A voltage comparator then continuously monitors the rectifier output voltage relative to this optimal value. The maximum power is achieved when $V_{\text{rect}}$ reaches its optimal level [40]. In contrast, the proposed model relies on the diode turn-on time as the primary parameter for maximum power detection. Therefore, unlike the FOCV-MPPT method, which focuses on tracking the optimal output voltage of the rectifier, the proposed approach determines an optimal diode turn-on time precisely synchronized to the second cycle $\left(n=2\right)$, thus maximizing power extraction during this interval.

To validate the equivalence of the proposed method and the traditional FOCV-MPPT method, two harvester configurations for $m\approx 1$ and $m=4$ were simulated using the rectifier circuit shown in Fig. 4 over a range of $C_{\text{rect}}$ values. The cycle-level output power, the turn-on time of the diodes, and the average rectifier output voltage (during the odd half-cycle of each cycle) are illustrated in Fig. 13(a)-(f). Based on $C_{\text{rect},\text{opt}}$ derived from Eq. 34, three different load capacitances were selected for the simulations: one approximating $C_{\text{rect},\text{opt}}$ (red waveforms), one below $C_{\text{rect},\text{opt}}$ (black waveforms), and one above $C_{\text{rect},\text{opt}}$ (blue waveforms).

Fig. 13:

Simulation results of the rectifier for three different capacitance $\left(C_{\text{rect}}\right)$ values for a harvester with $V_m=100\text{V}$, $f=1\text{Hz}$ and $C_{\max }=72\text{pF}$. The red curve $\text{(}{C}_{\text{rect}}=400\text{pF)}$ corresponds to the optimum capacitance determined by the proposed model. The cycle-level average input power of the rectifier during each odd half-cycle for (a) $m\approx 1$ and (b) $m=4$, the turn-on time of diodes during the odd half-cycle for (c) $m\approx 1$ and (d) $m=4$ and the cycle-level average output voltage of the rectifier during each odd half-cycle for (e) $m\approx 1$ and (f) $m=4$. In (c) and (d), the optimum turn-on time based on Eq. 28 is also illustrated, which intersects with the simulated optimum turn-on time time at $n=2$, when $C_{\text{rect}}=400\text{pF}$. In (e) and (f), the optimum rectifier voltage based on FOCV-MPPT method is also illustrated, which is sufficiently close with the simulated rectifier voltage at $n=2$, when $C_{\text{rect}}=400\text{pF}$.

As illustrated in Fig. 13 (a) and (b), the maximum power output is achieved at $n=2$ when the load capacitance is chosen according to the proposed method. At lower capacitance values, the power maximum is not reached because the diodes are activated after the intersection point of the average input voltage and current (where the maximum power occurs). Alternatively, higher $C_{\text{rect}}$ values postpone the peak power to subsequent cycles. Fig. 13 (c) and (d) illustrate the corresponding diode turn-on times during the cycle for each load capacitance. With optimal load capacitance, diodes turn on at 25% of the cycle for $m\approx 1$ and 10.6% of the cycle for $m=4$, achieving maximum power delivery. According to Eq. 36 of the FOCV-MPPT method, for $V_m=100\text{V}$, the optimal rectifier output voltage is 50V for $m\approx 1$ and 20V for $m=4$, as highlighted in Fig. 13(e) and (f). According to the average rectifier output voltage during the odd half-cycle, with $C_{\text{rect},\text{opt}}$ (calculated according to the proposed model), $V_{\text{rect}}$ approaches these optimal values at approximately $n=2$ (with deviations denoted as $e_1$ and $e_4$ in the figures). However, when nonoptimal $C_{\text{rect}}$ values are used, the system either fails to reach the optimal output voltage or delays the optimal voltage to subsequent cycles. This analysis demonstrates that the proposed method to determine rectifier capacitance achieves peak power transfer at $n=2$ without requiring additional circuitry needed for the existing FOCVMPPT method.

Table II lists the power consumed by the existing FOCVMPPT method used for various harvesters. This overhead power ranges from 275nW to 4μW for triboelectric harvesters and up to 5μW for piezoelectric harvesters.

TABLE II:

Comparison of the proposed power optimization method with the existing FOCV-MPPT method.

Reference	JSSC’20 [26]	JSSC’15 [27]	IEEE Access’21 [29]	TPEL’25 [25]	This work
Harvester	Triboelectric	Piezoelcetric	Triboelectric	Triboelectric	Triboelectric
Power optimization method	FOCV-MPPT	Adaptive FOCV-MPPT	FOCV-MPPT	P&O-MPPT	Load capacitance sizing
Technology	0.18μm BCDMOS	0.35μm BCDMOS	Discrete	0.18μm BCDMOS	Discrete
Operation frequency (Hz)	1−5Hz	90kHz	10Hz	5Hz	1−5Hz
Power consumed by MPPT system	257nW	5μW	1 to 4μW (MPPT + SSHI)	786nW	None

While the proposed method offers advantages in terms of simplicity and reduced power overhead, the primary trade-off with respect to traditional MPPT methods is the static nature of the proposed optimization determined by the harvester’s internal capacitance ratio $m$. Dynamic MPPT methods can adapt to varying environmental conditions and harvester characteristics at the expense of higher power overhead. The proposed method can track changes in the frequency and force of the harvester, but cannot track changes in m, which is uncommon as it would require mechanical changes in the harvester.

C. Utility of the proposed method and model

Given a triboelectric harvester with a specific capacitance ratio $m$, the proposed method determines the optimal rectifier capacitance that maximizes power delivery during the second cycle. The method offers several key advantages over traditional MPPT approaches, making it particularly suitable for practical energy harvesting implementations. Simplicity and Independence: The proposed model is independent of harvester voltage and frequency (see Fig. 8 and Fig. 13), as the optimum turn-on time depends only on the fixed parameter $m$. This eliminates the need for additional tracking circuitry required by dynamic FOCV-MPPT methods [25], [27], reducing system complexity and power overhead. Overall System Efficiency: The proposed optimization approach improves the overall efficiency of the PMU by simplifying the requirements on control circuitry. The subsequent DC-DC converter operates in DCM to prevent loading effects on rectifier performance [28], [39], [40]. The control circuitry can then be reduced to a zero current detector for frequency tracking and a two-bit counter for half-cycle counting. This triggers charge transfer in the optimal cycle $n=2$. This method eliminates the complex voltage sensing and comparison circuits required by FOCV and P&O methods. Integration with Energy Storage Capacitor: The optimized pF-scale rectifier capacitance serves as an impedance matching interface between the high-impedance TENG harvester and downstream storage systems. A DC-DC converter bridges the gap between the small $C_{\text{rect}}$ at the rectifier output and the much larger storage capacitor. The storage capacitor is typically in the μF/mF range at the DC-DC converter output. The DC-DC converter operates in DCM to periodically transfer accumulated charge when a sufficient voltage is reached. This two-stage approach prevents the larger storage capacitor from loading the rectifier while enabling rapid voltage build-up for efficient DC-DC operation [26], [27]. Parameter Estimation and Adaptability: For practical parameter determination, our previous work [45] demonstrates an online method to estimate $C_{\max }$ and $C_{\min }$ using load resistance measurements and voltage derivative analysis, enabling real-time determination of the capacitance ratio $m$ without harvester disassembly. Future work can integrate comprehensive real-time parameter adaptation, including continuous monitoring of internal capacitance variations and open-circuit voltage changes, enabling dynamic recalculation as harvester characteristics evolve with aging or environmental conditions. Broader Applications: The proposed model provides comprehensive insight into cycle-level, transient behavior of voltage and current waveforms for rectifier circuits in both triboelectric and piezoelectric energy harvesting systems, making it applicable across diverse energy harvesting technologies. In implantable medical devices, particularly knee joint implants, the proposed technique can effectively harness energy from regular flexion-extension cycles during daily activities. The method’s capability to efficiently convert low-frequency biomechanical motions (typically 0.5–5 Hz during walking) into electrical energy, without requiring additional physical circuitry, makes it well-suited for powering embedded sensors that monitor joint health parameters such as applied forces and movement frequency. Similarly, for long-term wearable applications, the approach addresses key challenges, including variable activity patterns and the need for consistent power delivery during low-activity periods. For remote IoT sensor deployments, where battery replacement is often impractical, the proposed method enables autonomous operation essential for applications such as environmental monitoring, structural health assessment, and precision agriculture. The optimization method’s ability to maintain efficient energy capture in varying operating conditions ensures reliable sensor operation for extended periods.

VII. Conclusion

This paper introduces a power optimization method for triboelectric energy harvesting systems by sizing the rectifier capacitance. By modeling the cycle-level rectifier voltage and current waveforms, we establish a relationship between the rectifier capacitance and the rectifier turn-on time. The proposed optimal capacitance maximizes power transfer while minimizing transient time, thereby negating the need for additional optimization circuitry. Experimental results from a vertical contact-separation TENG device validate the accuracy of this approach in achieving optimal power delivery. The static nature of this optimization method makes it particularly suitable for biomedical implants, wearable devices, and remote IoT sensors where circuit complexity must be minimized. Future research directions include exploring multi-mode TENG integration to capture energy from various motion patterns simultaneously, investigating hybrid control architectures that combine static optimization with adaptive mechanisms for enhanced dynamic performance, and integrating machine learning techniques for intelligent, self-optimizing systems capable of predictive parameter adjustment.

Biographies

Maryam Hosseini received the M.Sc. degree in Electrical Engineering from Urmia University, WA, Iran, in 2014. She is currently pursuing the Ph.D. degree in Electrical Engineering at Stony Brook University, NY, USA.

Her current research interests include analog/digital/mixed-signal circuits, energy harvesting, power management, and circuit/system design for biomedical sensor applications.

Mahmood Chahari received the M.Sc. degree in mechanical engineering from the Sharif University of Technology, Tehran, Iran, in 2016. His current research interests include robotics and control, MEMS sensors, and energy harvesting.

Milutin Stanacevic (Senior Member, IEEE) received the B.S. degree in electrical engineering from the University of Belgrade, Belgrade, Serbia, in 1999, and the M.S. and Ph.D. degrees in electrical and computer engineering from Johns Hopkins University, Baltimore, MD, USA, in 2001 and 2005, respectively.

In 2005, he joined the Faculty of the Department of Electrical and Computer Engineering, Stony Brook University, Stony Brook, NY, USA, where he is currently a Professor. His current research interests include mixed-signal VLSI circuit design for RF energy harvesting in implantable devices and tag networks, ultra-low power biomedical instrumentation, and acoustic source separation. Dr. Stanacevic was a recipient of the National Science Foundation CAREER Award and the IEEE Region 1 Technological Innovation Award. He was an Associate Editor of the IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITSAND SYSTEMS. He also serves on several technical committees for the IEEE Circuits and Systems Society.

Shahrzad Towfighian received the B.S. degree from the Amirkabir University of Technology, Tehran, Iran, in 2001, the M.S. degree from Ryerson University, Toronto, ON, Canada, in 2006, and the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada,in 2011.

She is a Professor and the Director of graduate studies at the Department of Mechanical Engineering at Binghamton University, NY, USA. Her research interests are micro-electromechanical systems and energy harvesting for bio-medical devices. She focuses on creating theoretical and experimental frameworks to explain the underlying mechanism of electromechanical systems. Using these frameworks, she seeks innovative methods to improve functionality of devices for various applications.

Dr. Towfighian is an Associate editor of the Journal of Vibration and Acoustics. She has been the recipient of several National Science Foundation and the National Institute of Health awards. She received a SUNY Chancellor’s Award in Scholarship and Creative Activities in 2024.

Ryan Willing received the Ph.D. degree in mechanical engineering with Queen’s University, Kingston, ON, Canada, in 2010.

He is an Associate Professor with the Department of Mechanical and Materials Engineering, Western University and a Biomedical Engineering faculty member. Prior to joining Western University, he was an Assistant Professor with Mechanical Engineering, State University of New York at Binghamton. He has been involved in biomechanical engineering research for more than ten years, with a special focus on human joint biomechanics and implant design. His research employs a combination of state of the art computer modeling and experimental analysis. He is associated with the Lawson Health Research Institute, and a member of Western’s Bone and Joint Institute. He is also a member of the Canadian and American Societies for Mechanical Engineering (CSME, ASME) and Orthopaedic Research (CORS and ORS).

Emre Salman (Senior Member, IEEE) received the B.S. degree in microelectronics engineering from Sabanci University, Istanbul, Turkey, in 2004, and the M.S. and Ph.D. degrees in electrical engineering from the University of Rochester, Rochester, NY, USA, in 2006 and 2009, respectively.

Since 2010, he has been with the Department of Electrical and Computer Engineering, Stony Brook University, NY, USA, where he is currently a Professor. His broad research interests include analysis, modeling, and design methodologies for integrated circuits and VLSI systems with applications to low power and secure computing, the Internet of Things with energy harvesting, and implantable devices.

Dr. Salman was a recipient of the National Science Foundation Faculty Early Career Development Award in 2013, the Outstanding Young Engineer Award from IEEE Long Island, NY, USA, in 2014, and the Technological Innovation Award from IEEE Region 1 in 2018. He is a Distinguished Lecturer of the IEEE Circuits and Systems (CAS) Society and previously served as the Chair for the VLSI Systems and Applications Technical Committee of the IEEE CAS Society.

Appendix I

The Proposed Model for a Non Sinusoidal Movement Pattern

The proposed optimization method can be extended to arbitrary periodic displacement patterns beyond sinusoidal motion. Although the fundamental optimization principle remains valid for any periodic plate motion $x(t)$, the specific optimal timing varies with different motion profiles. This section presents a generalized mathematical framework for a periodic displacement $x(t)$ with period $T$ and maximum displacement $x_{\max }$. For a general periodic motion $x(t)$, the fundamental parameters $V_m$, $C_{\max }$, $C_{\min }$, $m$, and ${\alpha }_{\text{sep}}$ remain the same as Eq. 2, Eq. 4, Eq. 5, Eq. 6, and Eq. 10, respectively. However, the time-dependent $\alpha (t)$ is generalized as

$$\alpha (t)=\frac{C_{\max }}{C_{\max }+C_{\text{rect}}\left(1+\frac{x(t)}{d}\right)},$$ (A.1)

Following the same cycle-level analysis methodology, the rectifier input voltages during the optimal cycle are updated as functions of the displacement x(t)

$$V_2=V_{\text{in}}\left(t_{\text{o},2}\right)=V_m\times \frac{x\left(t_{\text{o},2}\right)}{x_{\max }},$$ (A.2)

$$V_3=V_{\text{in}}\left(\frac{3T}{2}\right)=V_m\times \left[2{\alpha }_{\text{sep}}+\left(1-2\alpha \left(t_{\text{o},2}\right)\right)\frac{x\left(t_{\text{o},2}\right)}{x_{\max }}\right],$$ (A.3)

Substituting these generalized voltage expressions into the average power calculation and maintaining the assumption that $C_{\text{rect}}\gg C_{\min }$, the average power during the odd half-cycle of cycle n=2 is determined by

$$P_{\text{in},\text{av},\text{o},2}\approx \frac{2V_m^2{C}_{\min }}{T}\times \left[\frac{x\left(t_{\text{o},2}\right)}{x_{\max }}\right]\times \left[1-(m+1)\frac{x\left(t_{\text{o},2}\right)}{x_{\max }}\right],$$ (A.4)

Maximizing this power with respect to the displacement yields the optimal electrode displacement as

$$\frac{\partial P_{\text{in},\text{av},\text{o},2}}{\partial x\left(t_{\text{o},2}\right)}=0,x\left(t_{\text{o},2,\text{opt}}\right)=\frac{1}{m+1}\times x_{\max }.$$ (A.5)

This optimal displacement is independent of frequency and open-circuit voltage, depending only on the capacitance ratio m. For any specific motion profile x(t), this relationship can be used to determine the corresponding optimal turn-on time.

The relationship between displacement and rectifier capacitance follows the same derivation methodology as Section V-B, yielding,

$$x\left(t_{\text{o},2}\right)=\frac{4C_{\max }{C}_{\text{rect}}\left(C_{\max }+m^2{C}_{\text{rect}}\right)\times x_{\max }}{\left(C_{\max }+m{C}_{\text{rect}}\right)\left(C_{{\max }^2}+m^2{C}_{\text{rect}}^2+2C_{\max }{C}_{\text{rect}}\right)}.$$ (A.6)

Equating the optimal displacement from Eq. A.5 with Eq. A.6, the optimal rectifier capacitance for arbitrary periodic motion is determined as

$$C_{\text{rect},\text{opt}}\approx \frac{7.6m+12}{1.9m+1}\times C_{\max }.$$ (A.7)

Comparing this result with the specific case of sinusoidal movement (Eq. 34), Fig. A.1 demonstrates that both expressions yield similar optimal capacitance values, with a difference of 16% at low $m$ values and 31% at higher values of $m$. This deviation indicates that motion-specific optimization can provide more accurate results for highly variable internal capacitances (i.e. higher values of $m$). The sinusoidal model, however, remains a practical approximation for most applications, particularly when $m$ is smaller than 5.

Fig. A.1:

Comparison of optimal rectifier capacitance predictions for sinusoidal motion (Eq. 34) versus general periodic motion (Eq. A.7)