Enhancement of surge-induced synchronized switch harvesting on inductor strategy

When the current in an inductor varies quickly, the inductor generates a high voltage called a surge voltage. In SSHI, the switch is turned off when the current in the inductor is zero. Therefore, a surge voltage is not generated. Even when a surge voltage appears, the circuit in figure 1 cannot use it to increase the stored energy in ${C_{\text{h}}}$. Therefore, to exploit the surge voltage, the SSHI system should be modified in two aspects.

First, the circuit is modified such that the surge voltage enhances energy storage. Figure 3 depicts two such circuits. No components have been added to the original SSHI circuit. A rectifier can be connected to the main circuit in three ways: (a) across the piezo element (circuit P in figure 1), (b) across the inductor (circuit I in figure 3(a)), or (c) across the switch (circuit S in figure 3(b)). The SSHI approach uses circuit P, whereas the conventional and high-frequency S³HI approaches [19, 20] use circuit I. The last circuit, S, has not been investigated using the S³HI approach. In addition, when the switch in circuit S remains open, the circuit appears similar to that used in the standard FBR method. Therefore, given a large vibration amplitude, circuit S might harvest some energy without the need for switching control, thereby resulting in an auto-reboot of a system undergoing shutdown owing to complete battery discharge. Therefore, it is important to investigate how circuit S performs in accordance with the S³HI strategy, albeit the realization of the auto-reboot feature is a topic for future research. Because the only difference between the three circuits is the rectifier connection, their behaviors can be described by a single equation if no current flows through the rectifier.

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Second, the switching pattern for voltage inversion is modified to generate surge voltage. Figure 4 shows four switching patterns. Pattern S keeps the switch turned on for period $\pi /{\omega _{\text{e}}}$ synchronized with half the period of the electric vibration, as described in section 2, and thus it does not generate surge voltage. Pattern E keeps the switch turned on for period ${\tau _{\text{T}}}$, which is shorter than $\pi /{\omega _{\text{e}}}$. This pattern generates a surge voltage at the end of the voltage inversion. Pattern H turns the switch on and off at a high frequency during the period ${\tau _{\text{T}}}$. We propose pattern B, which maintains the switch in the ON state for period ${\tau _{\text{N}}}$, turns it OFF for a short period ${\tau _{\text{F}}}$, and finally maintains it in the ON state for period ${\tau _{{\text{N}}2}}$. This switching pattern generates a surge voltage close to the beginning of the voltage inversion. Voltage inversion decreases the energy stored in the piezoelectric element to ${\gamma ^2}$ times (i.e. less than 2/3 even when $\gamma$ is as large as 0.8). Pattern B aims to transfer the energy before it dissipates.

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We analyzed the combinations of both circuits (I and S) in figure 3 with every switching pattern, E, H, and B, in figure 4. Note that the conventional SSHI combines circuit P and switching pattern S. The combinations of circuit I with switching pattern E (S³HI) [19] and circuit I with pattern H (high-frequency S³HI) [20] have already been studied. Note that we refer to all the combinations exploiting surge voltage as S³HI and indicate the circuit and switching pattern to identify the combination (e.g. IE indicates S³HI combining circuit I with switching pattern E). The circuit S and switching pattern B constitute our proposed configuration in this study.

In the remainder of this section, we analyze the typical operation of the different methods in the S³HI family. Approximate performance estimates are also derived, except for the cumbersome combinations IH and SH. Figure 5 shows the typical behaviors of voltage ${V_{\text{p}}}$ and currents ${\dot Q_{\text{p}}}$ and ${\dot Q_{\text{h}}}$ in long and short time scales. ${V_{{\text{pn}}}}$ in figure 5(a) denotes the value of ${V_{\text{p}}}$ upon completion of voltage inversion, and its value according to the combination is indicated in the figure legend

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IE uses circuit I and switching pattern E. Similar to SSHI, IE turns the switch on when ${V_{\text{p}}}$ reaches its maximum or minimum and turns it off after period ${\tau _{\text{T}}}$. In IE, ${\tau _{\text{T}}} = \beta \pi /{\omega _{\text{e}}}$ for $0.5 < \beta < 1$ to generate surge voltage.

We use the assumptions from section 2 unless otherwise stated. If no current flows through the bridge rectifier of circuit I, the electric behavior of circuit I is described by equation (7). Therefore, as described in section 2, the values of ${Q_{\text{p}}}$, ${\dot Q_{\text{p}}}$, and ${V_{\text{p}}}$ at $t = {\tau _{\text{T}}}$ can be derived from equation (7) as follows:

$$\begin{align}{Q_{{\text{p}}1}} & = {C_{\text{p}}}{V_{{\text{p}}0}}{\text{exp}}\left( { - \zeta \beta \pi /{\omega _{\text{e}}}} \right)\left\{ {\left( {\zeta /{\omega _{\text{e}}}} \right)\sin \left( {\beta \pi } \right) + \cos \left( {\beta \pi } \right)} \right\} \nonumber\\[3pt] & \quad - {C_{\text{p}}}{V_{{\text{oc}}}},\end{align} \tag{ 14 }$$

$$\begin{equation}{\dot Q_{{\text{p}}1}} = - {C_{\text{p}}}{V_{{\text{p}}0}}{\text{exp}}\left( { - \zeta \beta \pi /{\omega _{\text{e}}}} \right)\left\{ {\left( {\zeta /{\omega _{\text{e}}}} \right) + {\omega _{\text{e}}}} \right\}\sin \left( {\beta \pi } \right),\end{equation} \tag{ 15 }$$

$$\begin{equation}{V_{{\text{p}}1}} = - {\gamma _\beta }{V_{{\text{p}}0}},\end{equation} \tag{ 16 }$$

where ${\gamma _\beta } = {\text{exp}}\left( { - \zeta \beta \pi /{\omega _{\text{e}}}} \right)\left\{ {\left( {\zeta /{\omega _e}} \right)\sin \left( {\beta \pi } \right) + \cos \left( {\beta \pi } \right)} \right\}$. The switch is turned off at ${\tau _{\text{T}}}$. Then, it is clear from figure 3(a) that subsequently ${\dot Q_{\text{p}}} = 0$ and ${V_{\text{p}}} = {V_{{\text{p}}1}}$ hold. However, the current in the inductor cannot vary instantly. Therefore, current ${\dot Q_{\text{h}}} = \left| {{{\dot Q}_{{\text{p}}1}}} \right|$ starts to flow through the inductor, rectifier, and ${C_{\text{h}}}$, charging ${C_{\text{h}}}$ as shown in figure 5(b). Thus, the induced surge voltage overcomes voltage barrier $2{V_{\text{d}}} + {V_{\text{h}}}$.

The subsequent behavior of ${Q_{\text{h}}}$ is described as

$$\begin{equation}L{\ddot Q_{\text{h}}} + {R_{{\text{T}}1}}{\dot Q_{\text{h}}} + {V_{{\text{B}}1}} = 0,\end{equation} \tag{ 17 }$$

where ${R_{{\text{T}}1}} = {R_{\text{h}}} + {R_{\text{L}}}$, ${V_{{\text{B}}1}} = {V_{\text{h}}} + 2{V_{\text{d}}}$, and ${R_{\text{h}}}$ is the total series resistance of two diodes and the storage capacitor. We assume that ${C_{\text{h}}}$ is sufficiently large, and ${V_{\text{h}}}$ can be considered constant. After some manipulations of equation (17), we observe that ${\dot Q_{\text{h}}}$ ceases to flow at $t = {\tau _{\text{T}}} + {\tau _{{\text{z}}1}}$, and the increment of charge in ${C_{\text{h}}}$ by this moment is given by

$$\begin{equation}{{\Delta }}{Q_{\text{h}}} = \left( {L/{R_{{\text{T}}1}}} \right)\left\{ {{{\dot Q}_{{\text{p}}1}} - ({V_{{\text{B}}1}}/{R_{{\text{T}}1}}){\text{ln}}\left( {1 + {{\dot Q}_{{\text{p}}1}}{R_{{\text{T}}1}}/{V_{{\text{B}}1}}} \right)} \right\},\end{equation} \tag{ 18 }$$

where ${\tau _{{\text{z}}1}} = \left( {L/{R_{{\text{T}}1}}} \right){\text{ln}}\left( {1 + {{\dot Q}_{{\text{p}}1}}{R_{{\text{T}}1}}/{V_{{\text{B}}1}}} \right)$. In the next half period of structural vibration ($\pi /{\omega _{\text{s}}}$) the switch is kept open, and thus ${Q_{\text{p}}}$ remains constant. In contrast, ${V_{\text{p}}}$ varies from ${V_{{\text{p}}1}}$ to ${V_{{\text{p}}1}} - 2{V_{{\text{oc}}}}$ as ${V_{\text{a}}}$ varies from${V_{{\text{oc}}}}$ to −${V_{{\text{oc}}}}$. As the system operates in steady state, ${V_{{\text{p}}1}} - 2{V_{{\text{oc}}}} = - {V_{{\text{p}}0}}$, as shown in figure 5(a). Therefore, from equation (16), we obtain

$$\begin{equation}{V_{{\text{p}}0}} = 2{V_{{\text{oc}}}}/\left( {1 - {\gamma _\beta }} \right).\end{equation} \tag{ 19 }$$

In the next half period of structural vibration, ${C_{\text{h}}}$ is additionally charged by the amount described in equation (18). Therefore, the energy stored in ${C_{\text{h}}}$ per cycle of structural vibration is given by

$$\begin{equation}{{\Delta }}E = 2\left( {\frac{{{V_{\text{h}}}L}}{{{R_{{\text{T}}1}}}}} \right)\left\{ {{{\dot Q}_{{\text{p}}1}} - ({V_{{\text{B}}1}}/{R_{{\text{T}}1}}){\text{ln}}\left( {1 + {{\dot Q}_{{\text{p}}1}}{R_{{\text{T}}1}}/{V_{{\text{B}}1}}} \right)} \right\},\end{equation} \tag{ 20 }$$

where ${{\Delta }}E$ is obtained by substituting equations (15) and (19) into equation (20). Note that the above analysis is limited to the typical case in which ${V_{{\text{p}}0}}$ given by equation (19) satisfies equation (4).

IB uses circuit I and switching pattern B for the energy in the piezoelectric element to be harvested at the beginning of voltage inversion. We use the assumptions from section 3.2 unless otherwise stated. If the switch is turned on at $t = 0$ and turned off at $t = {\tau _{\text{N}}} = \beta \pi /{\omega _{\text{e}}}$, the charge given by equation (18) is additionally stored in storage capacitor ${C_{\text{h}}}$ during subsequent period ${\tau _{z1}}$, and the values of ${Q_{\text{p}}}$, ${\dot Q_{\text{p}}}$, and ${V_{\text{p}}}$ at $t = {\tau _{\text{N}}} + {\tau _{\text{F}}}$ are respectively given by equation (14), ${\dot Q_{\text{p}}} = 0$, and equation (16) assuming that ${\tau _{{\text{z}}1}} \leqslant {\tau _{\text{F}}}$. For IB, we assume that $0 < \beta < 0.5$, which typically results in ${V_{{\text{p}}1}} > 0$, as shown in figure 5(c). IB then turns the switch on again at $t = {\tau _{\text{N}}} + {\tau _{\text{F}}}$ to complete voltage inversion. Then, as described in section 2, voltage inversion is completed at $t = {\tau _{\text{N}}} + {\tau _{\text{F}}} + \pi /{\omega _{\text{e}}}$.

IB typically turns the switch off at this moment. As in section 2, ${V_{\text{p}}}$ at this moment can be derived from equation (11) as follows:

$$\begin{equation}{V_{{\text{p}}2}} = - \gamma {V_{{\text{p}}1}} = \gamma {\gamma _\beta }{V_{{\text{p}}0}}.\end{equation} \tag{ 21 }$$

As in section 3.2, it is clear that ${V_{{\text{p}}2}} - 2{V_{{\text{oc}}}} = - {V_{{\text{p}}0}}$, and thus ${V_{{\text{p}}0}}$ is given by

$$\begin{equation}{V_{{\text{p}}0}} = 2{V_{{\text{oc}}}}/\left( {1 + \gamma {\gamma _\beta }} \right).\end{equation} \tag{ 22 }$$

The amount of energy stored per cycle of structural vibration is given by equation (20) and can be obtained from equations (15) and (22).

SE uses circuit S and switching pattern E. We use the assumptions from section 3.2 unless otherwise stated. If no current flows through the bridge rectifier, the behavior of circuit S can be described by equation (7). Therefore, the values of ${Q_{\text{p}}}$, ${\dot Q_{\text{p}}}$, and ${V_{\text{p}}}$ at $t = {\tau _{\text{T}}} = \beta \pi /{\omega _{\text{e}}}$ (i.e. ${Q_{{\text{p}}1}}$, ${\dot Q_{{\text{p}}1}}$, and ${V_{{\text{p}}1}}$) are respectively given by equations (14)–(16). SE turns the switch off at this moment. As current ${\dot Q_{\text{p}}}$ cannot vary instantly, it starts to flow through the piezoelectric element, inductor, diode, storage capacitor, and next diode, charging the storage capacitor as shown in figure 5(e). Hence, ${\dot Q_{\text{h}}} = - {\dot Q_{{\text{p}}1}}$, and the subsequent behavior of ${Q_{\text{p}}}$ is described as

$$\begin{equation}L{\ddot Q_{\text{p}}} + {R_{{\text{T}}2}}{\dot Q_{\text{p}}} + {Q_{\text{p}}}C_{\text{p}}^{ - 1} + {V_{{\text{B}}2}} = 0,\end{equation} \tag{ 23 }$$

where ${V_{{\text{B}}2}} = {V_{{\text{oc}}}} - {V_{\text{h}}} - 2{V_{\text{d}}}$ and ${R_{{\text{T}}2}} = {R_{\text{p}}} + {R_{\text{L}}} + {R_{\text{h}}}$. After some manipulations of equation (23), we can see that ${\dot Q_{\text{p}}} = 0$ at $t = {\tau _{\text{T}}} + {\tau _{{\text{z}}2}}$, as shown in figure 5(e), completing voltage inversion. In addition, ${Q_{\text{p}}}$ at this moment is given by

$$\begin{align}{Q_{{\text{p}}11}} = {e^{ - {\zeta _2}{\tau _{{\text{z}}2}}}}& \left\{ {\left( {{{\dot Q}_{{\text{p}}1}} + {\zeta _2}{Q_{{\text{pp}}}}} \right)\omega _{{\text{e}}2}^{ - 1}\sin \left( {{\omega _{{\text{e}}2}}{\tau _{{\text{z}}2}}} \right)}\right. \nonumber\\[3pt] & \quad \left.{ + {Q_{{\text{pp}}}}{\text{cos}}\left( {{\omega _{{\text{e}}2}}{\tau _{{\text{z}}2}}} \right)} \right\} - {C_{\text{p}}}{V_{{\text{B}}2}},\end{align} \tag{ 24 }$$

with

$$\begin{equation}{\tau _{{\text{z}}2}} = {\omega _{{\text{e}}2}}^{ - 1}{\text{ta}}{{\text{n}}^{ - 1}}\left\{ {{{\dot Q}_{{\text{p}}1}}/\left[ {{\zeta _2}\omega _{{\text{e}}2}^{ - 1}\left( {{{\dot Q}_{{\text{p}}1}} + {\zeta _2}{Q_{{\text{pp}}}}} \right) + {\omega _{{\text{e}}2}}{Q_{{\text{pp}}}}} \right]} \right\},\end{equation} \tag{ 25 }$$

where ${Q_{{\text{pp}}}} = {Q_{{\text{p}}1}} + {C_{\text{p}}}{V_{{\text{B}}2}}$, ${\zeta _2} = {R_{{\text{T}}2}}/\left( {2L} \right)$, and ${\omega _{{\text{e}}2}} = {\left\{ {{{\left( {L{C_{\text{p}}}} \right)}^{ - 1}} - \zeta _2^2} \right\}^{1/2}}$. Moreover, ${V_{\text{p}}}$ at this moment is given by

$$\begin{equation}{V_{{\text{p}}11}} = {Q_{{\text{p}}11}}/{C_{\text{p}}} + {V_{{\text{oc}}}}.\end{equation} \tag{ 26 }$$

The additionally stored charge in the storage capacitor within this period ${\tau _{{\text{z}}2}}$ is equal to the corresponding decrease in ${Q_{\text{p}}}$:

$$\begin{equation}\Delta {Q_{\text{h}}} = {Q_{{\text{p}}1}} - {Q_{{\text{p}}11}}.\end{equation} \tag{ 27 }$$

As shown in figure 5(a), after voltage inversion, ${V_{\text{p}}}$ varies from ${V_{{\text{p}}11}}$ to ${V_{{\text{p}}11}} - 2{V_{{\text{oc}}}}$ as ${V_{\text{a}}}$ varies from ${V_{{\text{oc}}}}$ to $- {V_{{\text{oc}}}}$. If ${V_{{\text{p}}11}} - 2{V_{{\text{oc}}}} < - 2{V_{\text{d}}} - {V_{\text{h}}}$, current ${\dot Q_{\text{p}}}$ starts to flow when ${V_{\text{p}}}$ reaches $2{V_{{\text{oc}}}} - 2{V_{\text{d}}} - {V_{\text{h}}}$ while varying toward ${V_{{\text{p}}11}} - 2{V_{{\text{oc}}}}$. However, we omit this case because it is rare, as analyzed below. Hence, we assume that ${V_{{\text{p}}11}} - 2{V_{{\text{oc}}}} > - 2{V_{\text{d}}} - {V_{\text{h}}}$ and focus on the typical case. As the system is in steady state,

$$\begin{equation}{V_{{\text{p}}11}} - 2{V_{{\text{oc}}}} = - {V_{{\text{p}}0}}.\end{equation} \tag{ 28 }$$

Unlike IE and IB, we need to solve algebraic equation (28) and the above equations to obtain the value of ${V_{{\text{p}}0}}$ for SE. Once we determine ${V_{{\text{p}}0}}$, $\Delta {Q_{\text{h}}}$ can be obtained by substituting equations (14) and (24) into equation (27). In the next half period of structural vibration, ${C_{\text{h}}}$ is additionally charged by the amount described in equation (27), and the energy stored per period of structural vibration is given by

$$\begin{equation}\Delta E = 2{V_{\text{h}}}\Delta {Q_{\text{h}}}.\end{equation} \tag{ 29 }$$

SB drives the switch in the proposed circuit S according to the proposed switching pattern B shown in figure 4. We use the assumptions from section 3.4 unless otherwise stated. Again, if the switch is turned on at $t = 0$ and turned off at $t = {\tau _{\text{N}}} = \beta \pi /{\omega _e}$, the charge given by equation (27) is additionally stored in capacitor ${C_{\text{h}}}$ during the next period ${\tau _{z2}}$, and ${Q_{\text{p}}}$, ${\dot Q_{\text{p}}}$, and ${V_{\text{p}}}$ at $t = {\tau _{\text{N}}} + {\tau _{\text{F}}}$ are respectively given by equation (24), ${\dot Q_{\text{p}}} = 0$, and equation (26), provided that ${\tau _{z2}} < {\tau _{\text{F}}}$.

For SB, we assume that $0 < \beta < 0.5$, and therefore ${V_{{\text{p}}11}}$ is positive as shown in figure 5(f). SB turns the switch on again at this moment and turns it off at $t = {\tau _{\text{N}}} + {\tau _{\text{F}}} + \pi /{\omega _{\text{e}}}$. Subsequently, ${\dot Q_{\text{p}}}$ becomes zero, and this completes the voltage inversion, as shown in figure 5(f). The value of ${V_{\text{p}}}$ at this moment is given based on equation (11) by

$$\begin{equation}{V_{{\text{p}}21}} = - \gamma {V_{{\text{p}}11}}.\end{equation} \tag{ 30 }$$

We assume that ${V_{{\text{p}}21}} - 2{V_{{\text{oc}}}} \geqslant - {V_{\text{h}}} - 2{V_{\text{d}}}$ to focus on the typical case. Then, as described in section 3.4, it is clear that

$$\begin{equation}{V_{{\text{p}}21}} - 2{V_{{\text{oc}}}} = - {V_{{\text{p}}0}}\end{equation} \tag{ 31 }$$

in the steady state. Therefore, ${V_{{\text{p}}0}}$ can be obtained by solving equation (31). The amount of energy stored per period of structural vibration is obtained from equation (29) along with equations (14), (24), and (27), and ${V_{{\text{p}}0}}$ can be obtained accordingly.

IH turns the switch of circuit I on at $t = 0$, off at $t = {\tau _{\text{N}}}$, on at $t = {\tau _{\text{N}}} + {\tau _{\text{F}}}$, off at $t = 2{\tau _{\text{N}}} + {\tau _{\text{F}}}$, and so on. Typically, voltage inversion is expected to complete when the switch is turned off at $t = \left( {{N_{\text{F}}} + 1} \right){\tau _{\text{N}}} + {N_{\text{F}}}{\tau _{\text{F}}}$, where ${N_{\text{F}}}$ is the number of periods ${\tau _{\text{F}}}$.

SH controls the switch of circuit S like IH. The typical behaviors of ${V_{\text{p}}}$, ${\dot Q_{\text{p}}}$, and ${\dot Q_{\text{h}}}$ are illustrated in figures 5(d) and (g) for IH and SH, respectively. The energy stored per cycle of structural vibration can be derived by using the equations derived in the previous sections. However, they are not shown here for brevity.

As the considered system includes diodes, it is convenient to describe the governing equations for the six states listed in table 1. Let ${i_{\text{r}}}$ denote the current flowing through the bridge rectifier as shown in figures 1 and 3. To write the equations compactly, we introduce variable S, whose value is shown in table 1 according to the switch state and polarity of ${i_{\text{r}}}$. Table 1 lists the applicable governing equations per state:

$$\begin{equation}{\dot Q_{\text{p}}} + S{\dot Q_{\text{h}}} = 0,\end{equation} \tag{ 32 }$$

$$\begin{equation}{R_{\text{p}}}{\dot Q_{\text{p}}} - S\left( {2{V_{\text{d}}} + {R_{\text{h}}}{{\dot Q}_{\text{h}}} + C_{\text{h}}^{ - 1}{Q_{\text{h}}}} \right) + {V_{\text{a}}} + C_{\text{p}}^{ - 1}{Q_{\text{p}}} = 0,\end{equation} \tag{ 33 }$$

$$\begin{equation}{\dot Q_{\text{h}}} = 0,\end{equation} \tag{ 34 }$$

$$\begin{align}L\left( {{\ddot Q}_{\text{p}}}\right. & \left. + S{{\ddot Q}_{\text{h}}} \right) + \left( {{R_{\text{p}}} + {R_{\text{s}}} + {R_{\text{L}}}} \right){\dot Q_{\text{p}}} + \left( {{R_{\text{L}}} + {R_{\text{s}}}} \right)S{\dot Q_{\text{h}}} \nonumber\\[3pt] & + {V_a} + C_{\text{p}}^{ - 1}{Q_{\text{p}}} = 0,\end{align} \tag{ 35 }$$

$$\begin{equation}{\dot Q_{\text{p}}} = 0,\end{equation} \tag{ 36 }$$

$$\begin{equation}L{\ddot Q_{\text{h}}} + \left( {{R_{\text{L}}} + {R_{\text{h}}}} \right){\dot Q_{\text{h}}} + 2{V_{\text{d}}} + C_{\text{h}}^{ - 1}{Q_{\text{h}}} = 0,\end{equation} \tag{ 37 }$$

$$\begin{equation}{ }\left( {{R_{\text{p}}} + {R_{\text{s}}}} \right){\dot Q_{\text{p}}} - S\left( {2{V_{\text{d}}} + {R_{\text{h}}}{{\dot Q}_{\text{h}}} + C_{\text{h}}^{ - 1}{Q_{\text{h}}}} \right) + {V_{\text{a}}} + C_{\text{p}}^{ - 1}{Q_{\text{p}}} = 0,\end{equation} \tag{ 38 }$$

$$\begin{align}L\left( {{\ddot Q}_{\text{p}}}\right. & \left. + S{{\ddot Q}_{\text{h}}} \right) + \left( {{R_{\text{p}}} + {R_{\text{s}}} + {R_{\text{L}}}} \right){\dot Q_{\text{p}}} + {R_{\text{L}}}S{\dot Q_{\text{h}}} + {V_{\text{a}}} \nonumber\\[3pt] & + C_p^{ - 1}{Q_{\text{p}}} = 0,\end{align} \tag{ 39 }$$

$$\begin{align}L{\ddot Q_{\text{p}}} & + \left( {{R_{\text{p}}} + {R_{\text{L}}}} \right){\dot Q_{\text{p}}} - S\left( {2{V_{\text{d}}} + {R_{\text{h}}}{{\dot Q}_h} + C_{\text{h}}^{ - 1}{Q_{\text{h}}}} \right) + {V_{\text{a}}} \nonumber\\[3pt] & + C_{\text{p}}^{ - 1}{Q_{\text{p}}} = 0,\end{align} \tag{ 40 }$$

$$\begin{equation}{R_{\text{s}}}\left( {{{\dot Q}_{\text{h}}} + S{{\dot Q}_{\text{p}}}} \right) + {R_{\text{h}}}{\dot Q_{\text{h}}} + 2{V_{\text{d}}} + C_{\text{h}}^{ - 1}{Q_{\text{h}}} = 0,\end{equation} \tag{ 41 }$$

$$\begin{equation}L{\ddot Q_{\text{p}}} + \left( {{R_{\text{p}}} + {R_{\text{L}}} + {R_{\text{s}}}} \right){\dot Q_{\text{p}}} + S{R_{\text{s}}}{\dot Q_{\text{h}}} + {V_{\text{a}}} + C_{\text{p}}^{ - 1}{Q_{\text{p}}} = 0.\end{equation} \tag{ 42 }$$

Table 1. Switch states, values of S, and applicable governing equations per case considered in numerical simulations.

			Governing equations
Switch state	Polarity of ${i_{\text{r}}}$	S	Circuit P	Circuit I	Circuit S
Off	${i_{\text{r}}} > 0$	S = 1	(32), (33)	(36), (37)	(32), (40)
	${i_{\text{r}}} < 0$	S = −1
	${i_{\text{r}}} = 0$	S = 0	(32), (34)	(34), (36)	(32), (34)
On	${i_{\text{r}}} > 0$	S = 1	(33), (35)	(38), (39)	(41), (42)
	${i_{\text{r}}} < 0$	S = −1
	${i_{\text{r}}} = 0$	S = 0	(34), (35)	(34), (39)	(34), (42)

The simulation program was scripted in Fortran wherein the applicable governing equations were selected in accordance with table 1 based on the switch and current ${i_{\text{r}}}$ states. The equations were numerically integrated to evaluate ${Q_{\text{p}}},{ }{\dot Q_{\text{p}}},{ }{Q_{\text{h}}}$, and ${\dot Q_{\text{h}}}$. The states of the switch and ${i_{\text{r}}}$ were checked at each successive time instant, and if any state demonstrated a change, the corresponding governing equation to be integrated was changed to its applicable form. When the governing equations were changed, the current flowing through the inductor and the charge of the capacitors were maintained at a constant value. Updated values of other variables were calculated using the governing equations prior to proceeding to the next time step. All simulations were performed under double precision on a PC (Panasonic CF-LX3; Osaka, Japan). The magnitude of the time-step increment equaled 7.64 ×10⁻⁹ s.

To evaluate the approximate equations and numerical simulations, we obtained the corresponding energy harvesting performances of SSHI, IE, SE, IB, and SB using the same input data. The circuit parameters are listed in table 2. The parameter values were determined by measuring those of actual components from the corresponding experimental setup. In addition, the switching timing listed in table 3 was used to obtain the performances. The timing values were obtained from the values in table 4, which lists near-optimal experimental values. Because we assumed that ${C_{\text{p}}} \ll {C_{\text{h}}}$ when an approximate result is obtained, the value of ${C_{\text{h}}}/{C_{\text{p}}}$ was set to 10 000 during simulations performed in this study. We also analyzed excitation frequencies of 100 and 25 Hz.

Table 2. Parameters of energy harvesting circuit.

Parameter	Value	Unit
${C_{\text{p}}}$	1.41	μF
${C_{\text{h}}}$	4.7	μF
L	10.5	mH
${R_{\text{p}}}$	4.5	Ω
${R_{\text{s}}}$	0.5	Ω
${R_{\text{L}}}$	5.5	Ω
${R_{\text{d}}}$	10	Ω
${V_{\text{d}}}$	0.65	V
${C_1}$	100	nF
${C_2}$	200	nF
${R_1}$	420	Ω
${R_2}$	340	Ω

Table 3. Switching timing to obtain simulation and approximate results.

Method	SSHI	IE and SE	IB and SB
Parameter	${\tau _{\text{T}}}{\omega _{\text{e}}}/\pi$	${\tau _{\text{T}}}{\omega _{\text{e}}}/\pi$	${\tau _{\text{N}}}{\omega _{\text{e}}}/\pi$	${\tau _{\text{F}}}{\omega _{\text{e}}}/\pi$	${\tau _{\text{T}}}{\omega _{\text{e}}}/\pi$
Value	1	0.781	0.206	0.126	1.310

Table 4. Near-optimal switching times obtained from experiments at ${V_{\text{h}}}/{V_{{\text{oc}}}} = 10$.

Method	${\tau _{\text{N}}}{\omega _{\text{e}}}/\pi$	${\tau _{\text{F}}}{\omega _{\text{e}}}/\pi$	${\tau _{\text{T}}}{\omega _{\text{e}}}/\pi$	$\Delta {\text{E}}/{C_{\text{p}}}V_{{\text{oc}}}^2$	Additional parameters
IE	—	—	0.781	5.12	—
SE	—	—	0.794	5.23	—
IH	0.264	0.0096	1.045	5.45	${N_{\text{F}}}\,$ = 3.82, duty = 0.964
SH	0.264	0.0096	1.045	5.58	${N_{\text{F}}}{ }$ = 3.82, duty = 0.964
IB	0.206	0.126	1.310	6.60	${\tau _{{\text{N}}2}}{\omega _{\text{e}}}/\pi \,$ = 0.979, duty = 0.904
SB	0.182	0.107	1.241	6.62	${\tau _{{\text{N}}2}}{\omega _{\text{e}}}/\pi \,$ = 0.955, duty = 0.916

Note: Duty is defined in the footnote of table 5.

Table 5. Optimal switching timing at ${V_{\text{h}}}/{V_{{\text{oc}}}} = 10$, corresponding $\Delta {\text{E}}/{C_{\text{p}}}V_{{\text{oc}}}^2$, and effect of precise optimization.

Method	${N_{\text{F}}}$	${\tau _{\text{T}}}{\omega _{\text{e}}}/\pi$	Duty	$\Delta {\text{E}}/{C_{\text{p}}}V_{{\text{oc}}}^2$	Increment
IE	—	0.807	—	5.20	0.08
SE	—	0.779	—	5.27	0.04
IB	—	1.245	0.952	6.73	0.13
SB	—	1.229	0.939	6.90	0.28
IH	2	1.020	0.970	5.76	0.31
	3	1.020	0.970	5.79	0.34
	4	1.010	0.970	5.80	0.35
SH	2	0.985	0.968	5.87	0.29
	3	0.985	0.970	5.89	0.31
	4	0.980	0.970	5.90	0.32

Note: For IB and SB, ${\text{duty}} = \left( {{\tau _{\text{N}}} + {\tau _{{\text{N}}2}}} \right)/{\tau _{\text{T}}}$; for IH and SH, ${\text{duty}} = {\tau _{\text{N}}}/\left( {{\tau _{\text{N}}} + {\tau _{\text{F}}}} \right)$. The increment in $\Delta {\text{E}}/{C_{\text{p}}}V_{{\text{oc}}}^2$ (last column) is due to precise optimization

Figure 6 depicts the results obtained from numerical simulations and approximations. For the sake of clarity, the results obtained for IE and SB are not shown. The values of the normalized energy harvested per structural-vibration cycle (i.e. ${{\Delta }}E/\left( {{C_{\text{p}}}V_{{\text{oc}}}^2} \right)$) are plotted against those of V_h/V_oc. As can be observed, the assumptions considered to derive the approximate equation are satisfied in all cases. When considering IE and SB as well, the maximum difference between the values of ${{\Delta }}E/\left( {{C_{\text{p}}}V_{{\text{oc}}}^2} \right)$ obtained using the approximate equation and numerical simulation equals 0.67 at a 100 Hz excitation frequency, whereas it remains below 0.1 at a 25 Hz excitation frequency. This is consistent with the assumption ${\omega _{\text{e}}} \gg {\omega _{\text{s}}}$ considered to derive the approximate equation. Therefore, the approximate equations demonstrate good agreement with the numerical-simulation results.

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To verify the experimental results reported in section 5, we obtained the corresponding results from numerical simulations. The circuit parameters listed in table 2 were used, except for ${V_{\text{d}}}$ and ${R_{\text{d}}}$ for SSHI, which were set as described in section 5.2. In addition, the experimental switching timing listed in table 4 was used, and ${C_{\text{h}}}/{C_{\text{p}}}$ was set to 10 000. First, ${C_{\text{h}}}$ was charged up to voltage ${V_{\text{h}}}$ to set an initial condition, and integration continued over 50 cycles of mechanical vibration. After confirming that the system reached the steady state, the stored charge per cycle was obtained by integrating ${\dot Q_{\text{h}}}$.

Figure 7 depicts the mechanical setup for vibration-energy harvesting. As can be seen in figure 7(a), a conical metal fitting adheres to each end of a stack piezoelectric element (PICMA P-885.51; PI Ceramic, Lederhose, Germany). Additionally, the semiconductor strain gauges (KSP-1-350-E4; Kyowa, Tokyo, Japan) adhere to the side surfaces of the element, thereby evaluating the axial strain therein. Subsequently, the element was clamped between the dints on the support structure and cantilevered plate. Thus, the element was compressively preloaded, as depicted in figures 7(b) and (c). The support structure comprised a rod with adjustable length. An 8 kg mass was mounted on the plate, which was sinusoidally excited at 100 Hz frequency by an electromagnetic shaker (WF1974; Akashi Seisakusho, Tokyo, Japan), whereas the resonant frequency was 144 Hz.

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Figure 8 shows a diagram of the electrical setup for the experiments. Channel 1 of the function generator (WF1974; NF corporation, Kanagawa, Japan) provided a 100 Hz sinusoidal signal, which was amplified and fed to the shaker to induce mechanical vibration. The switch shown in figures 1 and 3 was implemented using two field effect transistors (FETs; 2SK4017; Toshiba, Tokyo, Japan). Channel 2 of the function generator provided a 200 Hz square wave, whose amplitude, duty, and phase delay with respect to channel 1 were controllable. To drive the FET switch, the output from channel 2 was directly used when applying switching pattern S or E (figure 4). For switching pattern B, the output from channel 2 was modified using RC delay circuits and logic components and fed to the switch. For switching pattern H, the output from channel 2 was modified by using the output from a clock generator (TL494IN; Texas Instruments, Dallas, TX, USA), which provided a square signal. The frequency and duty of the output were adjustable by modifying the resistance and capacitance of the clock generator (not shown in figure 8). Therefore, all the circuits, P, I and S (figures 1 and 3), and all the switching patterns, S, E, H, and B (figure 4), could be implemented by adjusting the function generator, variable resistances of the RC delay circuits, capacitor, and variable resistance connected to the clock generator, and three snap switches shown in figure 8. The inductor in the circuit was a model ELC18B103L (Panasonic, Osaka, Japan), and the diodes composing the rectifier were of model 1N4148TR (Vishay Intertechnology, Malvern, PA, USA). The various voltages were measured typically via unit-gain buffers using an Analog–Digital converter board (LPC-320724; Interface, Hiroshima, Japan) connected to a desktop PC. The unit-gain buffers used OPA445AP (Texas Instruments, Dallas, TX, USA) as the operational amplifier. A signal conditioner (CDV-230 C, Kyowa, Tokyo, Japan) was used for strain measurements. Figure 9 shows a picture of the circuit, where the jumper lines for voltage measurement at various points were removed. The phase difference between the outputs from channel 2 of the function generator and clock generator was not controllable.

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The parameters of the components for the experiments were measured using an LCR meter at a frequency of 1.3 kHz, which is close to ${\omega _{\text{e}}}/\left( {2\pi } \right) = 1.31\,{\text{kHz}}$. The relation between the applied voltage and current of the diode was measured as shown in figure 10. In this experiment of S³HI, the current through the diodes was approximated to 5 mA from preliminary experiments and simulations. Therefore, the values of ${V_{\text{d}}}$ and ${R_{\text{d}}}$ were estimated from the tangent around 5 mA, as shown in figure 10. The identified parameters are listed in table 2, which provides the important parameters of this system, including $\gamma = 0.826$, ${\omega _{\text{e}}} = 8.20{ } \times { }{10^3}\,{{\text{s}}^{ - 1}}$, and ${V_{\text{d}}}/{V_{{\text{oc}}}} = 3.25$ at ${V_{{\text{oc}}}} = 0.2\,{\text{V}}$. On the other hand, the current through the diodes was estimated to be below 10 $\mu {\text{A}}$ when applying SSHI. Therefore, ${V_{\text{d}}}$ and ${R_{\text{d}}}$ for SSHI were 0.45 V and $630\,{{\Omega }}$, respectively, as obtained from figure 10. These values were used only when we compared the experimental and simulation results.

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Because S³HI facilitates energy harvesting from low-amplitude vibrations, the output amplitude from channel 1 was adjusted such that the amplitude of open-circuit voltage of the piezoelectric element ${V_{{\text{oc}}}}$ equaled 0.2 V. This value of ${V_{{\text{oc}}}}$ is equivalent to a strain of 1.74 μ for an 18 mm long piezoelectric element. The value of ${V_{{\text{oc}}}}$ was manually adjusted by monitoring the temporal evolution of ${V_{\text{p}}}$ and/or strain values obtained from the strain gauges attached to the piezoelectric element. For each method in the S³HI family, load resistor ${R_{{\text{load}}}}$, which makes the value of ${V_{\text{h}}}$ to be approximately 2 V, was first connected to the storage capacitor as shown in figure 8. Then, the switching timing was manually adjusted to maximize ${V_{\text{h}}}$. The obtained and measured switching timings are listed in table 4. Typical values of ${C_1},\,{C_2},\,{R_1}$ and ${R_2}$ in figure 8 obtained via tuning are listed in table 2. As these values were manually obtained during experiments, they can be considered near-optimal.

Subsequently, by maintaining this condition for each method, the load resistance was varied with different resistors. By measuring ${V_{\text{h}}}$, the energy stored in a period of structural vibration was obtained as ${{\Delta }}E = 2V_{\text{h}}^2\pi /\left( {{R_{{\text{load}}}}{\omega _{\text{s}}}} \right)$. To characterize the operation of each method, we measured various voltages, including ${V_{\text{p}}}$, voltage across the inductor, and voltage across the switch, through unity-gain buffers.

Figure 11 shows the experimental and simulation results of energy harvested per vibration cycle obtained for each evaluated method. The normalized performance, ${{\Delta }}E/\left( {{C_{\text{p}}}V_{{\text{oc}}}^2} \right)$, is shown according to the normalized voltage of the storage capacitor, ${V_{\text{h}}}/{V_{{\text{oc}}}}$. The experimental results are mostly consistent with the simulation results. The performance depends on the switching pattern but is almost independent of the circuit topology (i.e. circuit I or S). These results may not correspond to the maximum performance of each method because the switching timing was roughly optimized to ${V_{\text{h}}}/{V_{{\text{oc}}}} \approx 10$.

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When IE or SE is applied, figure 11 shows that the experimental and simulation results suitably agree. Figure 12 shows a close-up view of the experimental and simulation evolution of the input voltage to the bridge rectifier of circuit I (i.e. voltage across the inductor) for IE. The curves virtually coincide. When the switch is turned off after keeping it on for period ${\tau _{\text{N}}}$, a high voltage is induced, and the storage capacitor is charged at the end of voltage inversion, overcoming the voltage barrier of ${V_{\text{h}}} + 2{V_{\text{d}}} = 3.34\,{\text{V}}$.

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The experimental and simulation results differ by the high-frequency vibration in the experimental curve. This vibration may be generated by the stray capacitance and the inductor. The vibration frequency of 112 kHz and inductance of 10.5 mH result in a stray capacitance of 0.192 nF. The vibration amplitude suggests an energy dissipation of approximately 7.01 × 10⁻¹⁰ J, being around 0.15% of the mechanical vibration converted into electrical energy. Therefore, the dissipation is negligible. Figure 13 shows the evolution of ${V_{\text{p}}}$ and ${V_{\text{h}}}$ in a longer time scale. The value of ${V_{\text{h}}}$ jumps at every voltage inversion, showing that some energy is added as intended in the proposed design.

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Figure 11 reveals that the discrepancy between the experimental and simulation results of the performance estimation for IH and SH is relatively large when $10 < {V_{\text{h}}}/{V_{{\text{oc}}}} < 20$, possibly due to the estimation error of ${\tau _{\text{N}}}$. As ${\tau _{\text{N}}}$ is shorter than 4 μs, it is difficult to obtain an accurate measure from data sampled at 1.54 MHz. If we consider a shorter ${\tau _{\text{N}}}$, the discrepancy decreases.

Figure 14 shows the evolution of the experimentally obtained input voltage to the bridge rectifier during voltage inversion for IH. The switch is turned off four times within period ${\tau _{\text{T}}}$, generating a high voltage and charging the storage capacitor four times during one voltage inversion process.

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Figure 11 shows that IB and SB outperform the other S³HI methods. Figure 15 shows the experimental and theoretical evolution of the input voltage in the bridge rectifier of circuit S (i.e. voltage across the switch) for SB. As is the case for IE, both curves coincide except for the high-frequency vibration that appears in the experimental results whenever the current ceases to flow. Such high-frequency vibration may also be related to the stray capacitance, and its effect on harvesting is negligible.

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Figure 16 shows the evolution of the experimentally obtained ${V_{\text{P}}}$ and ${V_{\text{h}}}$ in long and short time scales. Figures 15 and 16 show that a high surge voltage is induced, and the storage capacitor is charged near the beginning of voltage inversion, overcoming the barrier of ${V_{\text{h}}} + 2{V_{\text{d}}} = 3.77{\text{ V}}$ as designed. The voltage pulse at the end of period ${\tau _{{\text{N}}2}}$ indicates that ${\tau _{{\text{N}}2}}$ is slightly shorter or longer than $\pi /{\omega _{\text{e}}}$.

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The preceding sections of this paper discussed the investigation and comparison of the performance of the different S³HI methods using near-optimal switching times obtained via experiments. This section discusses the corresponding investigations and comparisons performed after precise optimization of the switching time. We determined two sets of optimal switching timings through simulations to maximize $\Delta E/{C_{\text{p}}}V_{{\text{oc}}}^2$ at ${V_{\text{h}}}/{V_{{\text{oc}}}} = 10$ and ${V_{\text{h}}}/{V_{{\text{oc}}}} = 30$. We fixed ${V_{{\text{oc}}}}$ to 0.2 V like in the experiments. For IB and SB, the optimal value of ${\tau _{{\text{N}}2}}{\omega _e}/\pi$ is approximately 0.99–0.98 in many cases. However, even when the other two parameters are optimized while maintaining ${\tau _{{\text{N}}2}}{\omega _{\text{e}}}/\pi$ at 1.0, $\Delta E/{C_{\text{p}}}V_{{\text{oc}}}^2$ decreases only by 0.1% compared with the optimal value. Therefore, we fixed ${\tau _{{\text{N}}2}}{\omega _{\text{e}}}/\pi$ to 1.0 and considered the other two parameters as design parameters. In IH and SH, ${N_{\text{F}}}$ was considered to be an integer, and the optimal values of the other two independent parameters were searched for ${N_{\text{F}}} =$2, 3, and 4. Table 5 lists the obtained optimal timings, maximum values of $\Delta E/{C_{\text{p}}}V_{{\text{oc}}}^2$ at ${V_{\text{h}}}/{V_{{\text{oc}}}} = 10$, and increments of $\Delta E/{C_{\text{p}}}V_{{\text{oc}}}^2$ due to the precise optimization.

By using the optimal switching timing, we performed various numerical simulations. Figure 17 shows the performances of the SSHI and S³HI methods at various values of ${V_{\text{h}}}/{V_{{\text{oc}}}}$. For IH and SH, $\Delta E/{C_{\text{p}}}V_{{\text{oc}}}^2$ is almost independent of ${N_{\text{F}}}$, as listed in table 5. Therefore, figure 17 only shows the case for ${N_{\text{F}}} = 3$. As shown in table 5 and by comparing figures 17 and 11, the performance improvement due to the precise optimization of the switching timing is negligible. Hence, the methods in the S³HI family are relatively robust to variations in switching timing, with IE and SE being the most robust, IB and SB being moderately robust, and IH and SH being the least robust methods.

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Figure 17 shows that SSHI can harvest energy in a limited range of ${V_{\text{h}}}/{V_{{\text{oc}}}}$, whereas any method in the S³HI family achieves harvesting in a wider range beyond the values shown in the figure and can harvest more energy. Even for cases in which SSHI can harvest, its performance (i.e. $\Delta E/{C_{\text{p}}}V_{{\text{oc}}}^2$) is much lower than that of the methods in the S³HI family. In addition, the curve showing the relation between $\Delta E/{C_{\text{p}}}V_{{\text{oc}}}^2$ and ${V_{\text{h}}}/{V_{{\text{oc}}}}$ depends on the value of ${V_{\text{h}}}/{V_{{\text{oc}}}}$ at which switching timing is optimized. This dependency is strong in IH and SH and weak in IE and SE. Regarding both peak performance and average performance, SB and IB provide the highest values. Hence, the proposed switching pattern B substantially enhances the performance of S³HI. Figure 17 reveals that when $4 < {V_{\text{h}}}/{V_{{\text{oc}}}}$, the switching pattern B demonstrates a harvesting performance that has increased by 11%–31% and 15%–450% compared to patterns E and H, respectively, depending on the ${V_{\text{h}}}/{V_{{\text{oc}}}}$ value.

As observed, SE, SH, and SB demonstrate slightly better performance compared to the IE, IH, and IB, respectively, at ${V_{\text{h}}}/{V_{{\text{oc}}}}$ values corresponding to optimum switching times, as indicated by the value of $\Delta {\text{E}}/{C_{\text{p}}}V_{{\text{oc}}}^2$ listed in table 5. However, the observed differences in table 5 and figure 17 are marginal. This demonstrates that circuit S is a good alternative to circuit I.

We have exclusively focused on small-amplitude vibrations and large FVD (i.e. ${V_{\text{d}}}/{V_{{\text{oc}}}} = 3.25$) in the simulations and experiments reported thus far in this paper, because this is the main operating condition considered in this study. However, we have also investigated the performance and characteristics of the different methods in the S³HI family under conditions of large-amplitude vibration and/or small FVD (i.e. ${V_{\text{d}}}/{V_{{\text{oc}}}} = 0.1$). More specifically, we set ${V_{{\text{oc}}}} = 6.5\,{\text{V}}$, and the other parameters were set as listed in tables 2 and 5. Hence, the switching timing was not optimized for ${V_{\text{d}}}/{V_{{\text{oc}}}} = 0.1$.

Figure 18 shows the harvesting performance of SSHI, SE, SH, and SB for ${V_{\text{d}}}/{V_{{\text{oc}}}} = 0.1$. Similar to figure 17, figure 18 reveals that SSHI can harvest energy only when ${V_{\text{h}}}/{V_{{\text{oc}}}}$ is lower than a certain value, whereas all the methods in the S³HI family achieve harvesting over a wide range of ${V_{\text{h}}}/{V_{{\text{oc}}}}$ values beyond that shown in the figure. Therefore, S³HI can store significantly more energy than SSHI when ${V_{\text{d}}}/a = 0.1$. Unlike the case wherein ${V_{\text{d}}}/{V_{{\text{oc}}}} = 3.25$, SSHI provides the highest performance when ${V_{\text{h}}}/{V_{{\text{oc}}}} < 7$. Otherwise, SB provides the highest performance. If the switching timing is optimized for ${V_{\text{d}}}/{V_{{\text{oc}}}} = 0.1$, the performance of each method in the S³HI family increases. However, the performance increase at ${V_{\text{h}}}/{V_{{\text{oc}}}} = 10$ and ${V_{\text{h}}}/{V_{{\text{oc}}}} = 30$ due to this optimization is below 0.12 for SE and 0.75 for SB. In practice, this increase is considered negligible. Hence, the timing parameters determined at ${V_{{\text{oc}}}} = 0.2\,{\text{V}}$ for SE and SB can be used in practice even when ${V_{{\text{oc}}}}$ assumes a large value of 6.5 V. This demonstrates the robustness of the proposed strategy to variations in the vibration amplitude.

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As discussed above, we have demonstrated that the performance of the proposed circuit S is equivalent to that of circuit I. Nevertheless, the use of circuit S might provide additional advantages when vibration amplitude is large or ${V_{\text{d}}}/{V_{{\text{oc}}}}$ is small. Figure 18 demonstrates the performance of the SS method, which is an energy-harvesting method based on the combination of circuit S and switching-pattern S. Therefore, when circuit S is used, SS and SB become interchangeable via a simple change in the switching pattern from S to B and vice versa. Figure 18 reveals that the performances of SS and SSHI are identical and that they outperform other methods when ${V_{\text{h}}}/{V_{{\text{oc}}}} < 7.$ When ${V_{\text{h}}}/{V_{{\text{oc}}}}\, >$ 7, SB outperforms other methods. This fact indicates that one can always attain the best performance by using circuit S and adaptively selecting the preferred switching pattern between B and S depending on the ${V_{\text{h}}}/{V_{{\text{oc}}}}\,$value.

Figure 19 compares the performances of passive FBR, SSHI, and SB (a typical member of the S³HI family). The FBR circuit was implemented by neglecting the switch and inductor of the circuit P shown in figure 1. Thus, the performance of FBR was obtained using the simulation program described in section 4.2, keeping the switch open. The parameter values listed in table 1, which were used in all the numerical simulations in this study, were used in this simulation as well. The value of ${V_{\text{d}}}$ was 0.65 V. The switching timings for SB were ${\tau _{\text{T}}}{\omega _{\text{e}}}/\pi = 1.25$ and duty = 0.97, which are optimal at ${V_{\text{h}}}/{V_{{\text{oc}}}}$ = 30 and ${V_{\text{d}}}/{V_{{\text{oc}}}}$ = 3.25. The value of FVD was ${V_{\text{d}}}$ = 0.65 V, and the vibration frequency was 100 Hz. Figure 19(a) shows the case of small-amplitude vibration, which is the target condition of this study. The figure shows that the FBR harvests no energy in this case, irrespective of the storage-capacitor voltage. However, SB achieves a high harvesting rate even when ${V_{\text{h}}}/{V_{{\text{oc}}}}$ is large and the vibration amplitude is small. This figure depicts well that SB performs effectively from the perspective of our objective. Figure 19(b) shows the case of large-amplitude vibration; in this case, FBR can harvest some energy only when ${V_{\text{h}}}/{V_{{\text{oc}}}}$ is relatively small. SB realizes a high harvesting rate when ${V_{\text{h}}}/{V_{{\text{oc}}}}$ is large, in this case as well.

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Table 6 summarizes figure 19. Power enhancement is defined as ${{\text{max}}{\Delta}}E/{{\text{max}}{\Delta}}{E_{{\text{FBR}}}}$ in this study; max and ${{\Delta }}{E_{{\text{FBR}}}}$ are defined in the footnote of table 6. When the amplitude of open-circuit voltage ${V_{{\text{oc}}}}$ is as small as ${V_{\text{d}}}/V_{{\text{oc}}}^{} = 3.25$, power enhancement is infinity because ${{\text{max}}{\Delta}}{E_{{\text{FBR}}}} = 0$. ${V_{{\text{h max}}}}$ denotes the maximum value of ${V_{\text{h}}}$ at which $0 < {{\Delta }}E$ holds, and ${E_{{\text{max}}}}$ denotes the upper limit of storable energy. Because no energy is additionally harvested when ${V_{{\text{h max}}}} < {V_{\text{h}}}$, the upper limit is given by ${E_{{\text{max}}}} = {C_{\text{h}}}V_{{\text{h max}}}^2/2$.

Table 6. Comparison of performances of SBR, SSHI, and S³HI-SB.

Publication	$\displaystyle{\frac{{{V_{\text{d}}}}}{{{V_{{\text{oc}}}}}}}$	$\gamma$	Method	$\displaystyle{\frac{{{{\text{max}}{\Delta}}E}}{{{C_{\text{p}}}V_{{\text{oc}}}^2}}}$	Power enhancement	$\displaystyle{\frac{{{V_{{\text{h max}}}}}}{{{V_{{\text{oc}}}}}}}$	$\displaystyle{\frac{{{E_{{\text{max}}}}}}{{{C_{\text{h}}}V_{{\text{oc}}}^2}}}$
This work	3.25	0.826	S3HI-SB	11.4	$\infty$	>1000	>5.0 × 105
			SSHI	2.08	$\infty$	4.8	11.5
			FBR	0	$-$	0	$0$
	0.1	0.826	S3HI-SB	11.5	18.0	>1000	>5.0 × 105
			SSHI	10.9	17.1	11.5	66.1
			FBR	0.639	1	0.80	$0.32$
[3]	0	—	SECE	4.0	4.0	$\infty$	$\infty$
[6]	0	0.822	Series SSHI	10	10	1.0	0.50
[7]	0	0.74	Hybrid SSHI 1)	7.7	7.7	30	450
[10]	0	0.8	ESSE 2)	8.0	8.0	$\infty$	$\infty$
[14]	0	0.7	SICE 3)	6.4	6.4	$\infty$	$\infty$

Note: Power enhancement: ${{\text{max}}{\Delta}}E/{{\text{max}}{\Delta}}{E_{{\text{FBR}}}}$, max: maximum value with respect to V_h/${V_{{\text{oc}}}}$, ${{\Delta }}E$: energy harvested per vibration cycle, ${{\Delta }}{E_{{\text{FBR}}}}$: value of ${{\Delta }}E$ of FBR. 1) Transformer ratio $m = 30$, 2) efficiency of buck-boost converter ${\gamma _{\text{c}}}$ = 0.9, 3) energy transfer efficiency $\eta = 0.85$.

Table 6 shows the performance of certain previously proposed methods as well. When the value of ${V_{\text{h}}}$ was not included in a study, it was calculated from the corresponding values of load resistance and harvested power in each study. Unfortunately, values of parameters such as ${V_{\text{d}}}/V_{{\text{oc}}}^{}$ and γ are different in each case. However, table 6 shows that ${E_{{\text{max}}}}$ is very large when S3HI-SB, SECE, ESSE, or SICE is applied. In principle, ${{\text{max}}{\Delta}}E/{C_{\text{p}}}V_{{\text{oc}}}^2$ decreases when ${V_{\text{d}}}/V_{{\text{oc}}}^{}$ increases. However, table 6 shows that when S3HI-SB is applied, value of ${{\text{max}}{\Delta}}E/{C_{\text{p}}}V_{{\text{oc}}}^2$ is larger than those of prior methods even though ${V_{\text{d}}}/V_{{\text{oc}}}^{} = 0$ in them. The objective of this study was not only to obtain a large value for ${{\Delta }}E/{C_{\text{p}}}V_{{\text{oc}}}^2$, but also to increase the upper limit of the stored energy. Therefore, table 6 demonstrates that our objective is achieved well by, e.g. SB of the S³HI family.