An analysis of the coupling effect for a hybrid piezoelectric and electromagnetic energy harvester

Considering a hybrid energy harvester integrated with piezoelectric (PE) and electromagnetic (EM) conversion mechanisms, as shown in figure 1, the magnets are supported by a double-end fixed beam, piezoelectric layers polarised in the beam thickness direction are attached on the top surface of the beam, and the coil is under the magnets. In [20, 25], when the harvester structure vibrates at its resonant frequency, the model of the PE-only or EM-only energy harvester can be simplified as an SDOF (single degree of freedom) system, so the hybrid energy harvester can be modeled as a mass–spring–damper–PE-element–EM-element system, as shown in figure 2. This consists of the PE harvesting element, the EM harvesting element and the mechanical structure, and the PE and EM elements connect with their respective load resistors. When the acceleration is applied to the harvesting system, an effective mass m_e is bounded on a spring of effective stiffness K, a damper of coefficient c_m , a PE element and an EM element. In this approach, the coupled PE and EM elements alter the vibration response of the harvesting system, which in turn alters the power output obtained from each individual energy harvesting technique. The effective circuits of the PE and EM elements are shown in figures 3 and 4, where θ and k_em are the PE and EM transfer factors, respectively [31, 32], R_p and R_m are the load resistors of the PE and EM elements, C_p is the capacitance of the piezoelectric layer, and R_coil and L_coil are the resistance and the inductance of the coil, respectively. These parameters are dependent on the material constants and the design of the energy harvester, which can be derived by standard model analysis [31, 32].

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Let ${{z}_{i}}(i=0,1)$ be the displacement of the magnet, V_p be the output voltage of the PE harvesting element and I_em be the output current of the EM harvesting element. The governing equations of the piezoelectric-only energy harvester can be described by [33]

$$\begin{eqnarray}&&{{m}_{e1}}{{\ddot{z}}_{1}}(t)+{{c}_{m1}}{{\dot{z}}_{1}}(t)+{{K}_{1}}{{z}_{1}}(t)+\theta {{V}_{p1}}(t)=-{{m}_{e1}}a(t)\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\frac{{{V}_{p1}}(t)}{{{R}_{p}}}+{{C}_{p}}{{\dot{V}}_{p1}}(t)-\theta \frac{d{{z}_{1}}(t)}{dt}=0\end{eqnarray} \tag{ 2 }$$

where $a(t)$ is the applied acceleration.

Therefore, using an analogy method, the governing electromechanical equation of the hybrid piezoelectric and electromagnetic energy harvester can be expressed as

$$\begin{eqnarray} &&\qquad \qquad \ \ \ {{m}_{e}}\ddot{z}(t)+{{c}_{m}}\dot{z}(t)+Kz(t)+{{k}_{em}}{{I}_{em}}(t) \\ &&\qquad \qquad \qquad \qquad +\theta {{V}_{p}}(t)=-{{m}_{e}}a(t) \\ \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{L}_{\text{coil}}}\frac{d{{I}_{em}}(t)}{dt}+\left( {{R}_{\text{coil}}}+{{R}_{m}} \right){{I}_{em}}(t)+{{k}_{em}}\frac{dz(t)}{dt}=0\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\frac{{{V}_{p}}(t)}{{{R}_{p}}}+{{C}_{p}}{{\dot{V}}_{p}}(t)-\theta \frac{dz(t)}{dt}=0.\end{eqnarray} \tag{ 5 }$$

Equations (4) and (5) are derived from figures 4 and 3 by KCL (Kirchhoff's current law) and KVL (Kirchhoff's voltage law), respectively [34].

According to references, [26, 31, 35] the inductance of the coil can be neglected in the low vibrating frequency (lower than 1 kHz) because the impedance is mainly determined by the resistance of coil. Hence, equations (4) and (5) can be expressed in the frequency domain:

$$\begin{eqnarray} {{I}_{em}}(\omega ) & = & \frac{{{k}_{em}}}{{{R}_{\text{coil}}}+{{R}_{m}}}z(\omega )j\omega , \\ {{V}_{p}}(\omega ) & = & \frac{\theta {{R}_{p}}}{1+j\omega {{C}_{p}}{{R}_{p}}}z(\omega )j\omega . \\ \end{eqnarray} \tag{ 6 }$$

So the transfer function of the harvesting system giving the displacement as a function of the applied acceleration is given by:

$$\begin{eqnarray} &&\frac{z(s)}{a(s)}=-{{\mu }_{1}}{{m}_{e}}\left[ {{m}_{e}}{{s}^{2}}+\left( {{c}_{m}}+\frac{k_{em}^{2}}{{{R}_{\text{coil}}}+{{R}_{m}}} \right)s \right. \\ &&\qquad \quad {{\left. +K+\theta \frac{s\theta {{R}_{p}}}{1+s{{C}_{p}}{{R}_{p}}} \right]}^{-1}} \\ \end{eqnarray} \tag{ 7 }$$

where ${{\mu }_{1}}$ is the correct factor for the SDOF model of the harvester [32]. Suppose the source of the forcing function comes from the vibration of the base of the structure, and this gives ${{m}_{e}}{{a}_{M}}$. Then the amplitude of magnet at any frequency $\omega$ is

$$\begin{eqnarray} &&{{z}_{M}}={{\mu }_{1}}{{m}_{e}}{{a}_{M}}\left\{ {{\left[ -{{m}_{e}}{{\omega }^{2}}+k+\frac{{{\omega }^{2}}R_{p}^{2}{{\theta }^{2}}{{C}_{p}}}{1+{{\left( \omega {{C}_{p}}{{R}_{p}} \right)}^{2}}} \right]}^{2}} \right. \\ &&\qquad \ \,+{{\left\{ \omega \left[ {{c}_{m}}+\frac{k_{em}^{2}}{{{R}_{\text{coil}}}+{{R}_{m}}}+\frac{{{\theta }^{2}}{{R}_{p}}}{1+{{\left( \omega {{C}_{p}}{{R}_{p}} \right)}^{2}}} \right] \right\}}^{2}}{{\}}^{-1/2}} \\ \end{eqnarray} \tag{ 8 }$$

where ${{a}_{M}}$ is the magnitude of the acceleration. As illustrated in equation (8), the vibration characteristics of the hybrid energy harvester are affected by the coupling effect of the PE harvest element and the EM harvest element. In particular, the coupling effect of the PE element changes the effective stiffness and damping coefficient of the harvesting system, while that of the EM element just influences the effective damping coefficient. Thus, the effective stiffness and damping coefficient of the hybrid energy harvester at any excitation frequency $\omega$ are shown, respectively, as

$$\begin{eqnarray}&&{{K}_{ef}}=K+\frac{{{\omega }^{2}}R_{p}^{2}{{\theta }^{2}}{{C}_{p}}}{1+{{\left( \omega {{C}_{p}}{{R}_{p}} \right)}^{2}}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{{c}_{ef}}={{c}_{m}}+\frac{k_{em}^{2}}{{{R}_{\text{coil}}}+{{R}_{m}}}+\frac{{{\theta }^{2}}{{R}_{p}}}{1+{{\left( \omega {{C}_{p}}{{R}_{p}} \right)}^{2}}}.\end{eqnarray} \tag{ 10a }$$

Similarly, equation (6) is rearranged as

$$\begin{eqnarray} \left| {{I}_{em}} \right| & = & \frac{\omega {{k}_{em}}}{{{R}_{m}}+{{R}_{\text{coil}}}}{{z}_{M}}, \\ {{\left| {{V}_{p}} \right|}^{2}} & = & \frac{{{\omega }^{2}}R_{p}^{2}{{\theta }^{2}}}{1+{{\left( \omega {{C}_{p}}{{R}_{p}} \right)}^{2}}}z_{M}^{2}. \\ \end{eqnarray} \tag{ 10b }$$

Thus, the output power of the PE and EM harvesting elements can be calculated, respectively, by

$$\begin{eqnarray} {{P}_{em}} & = & \frac{{{R}_{m}}k_{em}^{2}{{\omega }^{2}}}{2{{({{R}_{m}}+{{R}_{\text{coil}}})}^{2}}}z_{M}^{2}, \\ {{P}_{p}} & = & \frac{{{R}_{P}}{{\omega }^{2}}{{\theta }^{2}}}{2+2{{\left( \omega {{C}_{p}}{{R}_{p}} \right)}^{2}}}z_{M}^{2}. \\ \end{eqnarray} \tag{ 11 }$$

In equation (11), it can be concluded that the power of the PE element for the hybrid energy harvester is not only decided by the piezoelectric properties' parameters, but is also affected by the electromagnetic properties' parameters because of the coupling effect. Conversely, the power of the EM element is also influenced by the PE properties' parameters.

Lastly, the total power of the harvesting system is

$$\begin{eqnarray} {{P}_{e}} & = & {{P}_{em}}+{{P}_{p}} \\ {} & = & \left[ \frac{{{R}_{m}}k_{em}^{2}{{\omega }^{2}}}{2{{({{R}_{m}}+{{R}_{\text{coil}}})}^{2}}}+\frac{{{R}_{P}}{{\omega }^{2}}{{\theta }^{2}}}{2+2{{\left( \omega {{C}_{p}}{{R}_{p}} \right)}^{2}}} \right]z_{M}^{2}. \\ \end{eqnarray} \tag{ 12 }$$

Suppose that

${{\eta }_{p}}=\frac{{{\theta }^{2}}}{K{{C}_{p}}}$ as the piezoelectric coupling coefficient;

${{\eta }_{m}}=\frac{{{\omega }_{n}}k_{em}^{2}}{K{{R}_{\text{coil}}}}$ as the electromagnetic coupling coefficient;

$\lambda =\frac{\omega }{{{\omega }_{n}}}$ as the normalized frequency;

${{\zeta }_{m}}=\frac{{{c}_{m}}}{2{{m}_{e}}{{\omega }_{n}}}$ as the mechanical damping ratio;

${{r}_{p}}={{R}_{p}}{{C}_{p}}\omega$ as the normalized load resistance of the PE harvesting element.

${{r}_{m}}=\frac{{{R}_{m}}}{{{R}_{\text{coil}}}}$ as the normalized load resistance of the EM harvesting element.

The normalized vibration magnitude and output power of the hybrid energy harvester are then described by

$$\begin{eqnarray} {\bar{z}} & = & {{z}_{M}}/\left( {{m}_{e}}{{a}_{M}}/K \right) \\ {} & = & {{\mu }_{1}}\left[ {{\left( -{{\lambda }^{2}}+1+\frac{r_{p}^{2}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right. \\ {} & {} & {{\left. +{{\left( 2{{\zeta }_{m}}\lambda +\frac{1}{1+{{r}_{m}}}\cdot \lambda {{\eta }_{m}}+\frac{{{r}_{p}}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right]}^{-1/2}} \\ \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} {\bar{P}} & = & {{P}_{e}}/\left[ \frac{{{\left( {{m}_{e}}{{a}_{M}} \right)}^{2}}}{{{\omega }_{n}}{{m}_{e}}} \right] \\ {} & = & \left[ \frac{{{r}_{p}}}{2\left( 1+r_{p}^{2} \right)}\lambda {{\eta }_{p}}+\frac{{{r}_{m}}}{2{{\left( {{r}_{m}}+1 \right)}^{2}}}{{\eta }_{m}}{{\lambda }^{2}} \right] \\ {} & {} & \mu _{1}^{2}\left[ {{\left( -{{\lambda }^{2}}+1+\frac{r_{p}^{2}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right. \\ {} & {} & {{\left. +{{\left( 2{{\zeta }_{m}}\lambda +\frac{1}{1+{{r}_{m}}}\cdot \lambda {{\eta }_{m}}+\frac{{{r}_{p}}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right]}^{-1}}. \\ \end{eqnarray} \tag{ 14 }$$

As shown in equations (13) and (14), the normalized displacement and power mainly depend on the vibration frequency $\lambda$, the mechanical damping ratio ${{\zeta }_{m}}$, the electromechanical coupling coefficients ${{\eta }_{p}}$ and ${{\eta }_{m}}$, and the normalized loads ${{r}_{p}}$ and ${{r}_{m}}$. Thus, the scheme to optimize the power is either by adjusting the coupling coefficient and damping ratio by optimal structure design or selecting the optimal load and exciting at a suitable vibration frequency.

When ${{\eta }_{m}}=0$, the normalized displacement and power of the piezoelectric-only energy harvester are

$$\begin{eqnarray} {{{\bar{z}}}_{op}} & = & {{\mu }_{1}}\left[ {{\left( -{{\lambda }^{2}}+1+\frac{r_{p}^{2}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right. \\ {} & {} & {{\left. +{{\left( 2{{\zeta }_{m}}\lambda +\frac{{{r}_{p}}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right]}^{-1/2}} \\ \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} &&{{P}_{op}}=\frac{{{r}_{p}}}{2\left( 1+r_{p}^{2} \right)}\ \lambda {{\eta }_{p}} \\ &&\qquad \times \frac{\mu _{1}^{2}}{{{\left( -{{\lambda }^{2}}+1+\frac{r_{p}^{2}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}}+{{\left( 2{{\zeta }_{m}}\lambda +\frac{{{r}_{p}}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}}}. \\ \end{eqnarray} \tag{ 16 }$$

When ${{\eta }_{p}}=0$, the normalized displacement and power of electromagnetic-only energy harvester are

$$\begin{eqnarray}&&{{\bar{z}}_{om}}={{\mu }_{1}}{{\left[ {{\left( -{{\lambda }^{2}}+1 \right)}^{2}}+{{\left( 2{{\zeta }_{m}}\lambda +\frac{1}{1+{{r}_{m}}}\cdot \lambda \cdot \beta {{\eta }_{m}} \right)}^{2}} \right]}^{-1/2}}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} {{P}_{om}} & = & \left[ \frac{{{r}_{m}}}{2{{({{r}_{m}}+1)}^{2}}}\beta {{\eta }_{m}}{{\lambda }^{2}} \right] \\ {} & {} & \times \frac{\mu _{1}^{2}}{{{(-{{\lambda }^{2}}+1)}^{2}}+{{\left( 2{{\zeta }_{m}}\lambda +\frac{1}{1+{{r}_{m}}}\cdot \lambda \cdot \beta {{\eta }_{m}} \right)}^{2}}} \\ \end{eqnarray} \tag{ 18 }$$

where $\beta$ is the correct factor of electromagnetic coupling coefficient when the electromagnetic-only energy harvester is working.

Compared with the output characteristics of the two separated devices in equations (15)–(18), the performance of the hybrid energy harvester is affected by the coupling among the PE element, the EM element and the mechanical element simultaneously.

According to equations (9) and (10), the effective stiffness and damping coefficient of the hybrid energy harvester cannot remain constant like a conventional second-order vibration system because they are related to the excited frequency, the load resistor and the coupling coefficient. Therefore, the resonant frequency of the hybrid energy harvester is no longer equal to the natural frequency that is designed aiming at environment frequency.

From references [31, 32], the vibration energy harvesters output the maximum power at the resonant frequency. In order to move the resonant frequency into the expected frequency range, the shift values of the resonant frequency relative to the natural frequency need to be studied for different coupling effects and loads for the hybrid energy harvester.

When the harvester is excited at resonance, the value of displacement is also maximal. Therefore, the resonant frequency can be calculated by the derivative of the expression of displacement. Then, suppose $\frac{d{{z}_{M}}}{d\lambda }=0$. Because ${{m}_{e}}{{a}_{M}}\ne 0$ and

$$\begin{eqnarray} &&\left[ {{\left( -{{\lambda }^{2}}+1+\frac{r_{p}^{2}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right. \\ &&\left. \quad +{{\left( 2{{\zeta }_{m}}\lambda +\frac{1}{1+{{r}_{m}}}\cdot \lambda \cdot {{\eta }_{m}}+\frac{{{r}_{p}}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)}^{2}} \right]>0 \\ \end{eqnarray} \tag{ 19 }$$

the solution of $\frac{d{{z}_{M}}}{d\lambda }=0$ can be derived, such that

$$\begin{eqnarray} &&\left( -{{\lambda }^{2}}+1+\frac{r_{p}^{2}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right)\left( -2\lambda \right) \\ &&\quad +\left( 2{{\zeta }_{m}}\lambda +\frac{1}{1+{{r}_{m}}}\cdot \lambda \cdot {{\eta }_{m}}+\frac{{{r}_{p}}}{1+r_{p}^{2}}\cdot {{\eta }_{p}} \right) \\ &&\quad \times \left( 2{{\zeta }_{m}}+\frac{1}{1+{{r}_{m}}}{{\eta }_{m}} \right)=0 \\ \end{eqnarray} \tag{ 20 }$$

and simplified to

$$\begin{eqnarray}&&{{\lambda }^{3}}+p\lambda +q=0\end{eqnarray} \tag{ 21 }$$

which is a Cartan equation, and $p$, $q$ are given by

$$\begin{eqnarray} p & = & 2\zeta _{m}^{2}+\frac{2{{\zeta }_{m}}{{\eta }_{m}}}{1+{{r}_{m}}} \\ {} & {} & +\frac{\eta _{m}^{2}}{2{{\left( 1+{{r}_{m}} \right)}^{2}}}-1-\frac{{{r}_{p}}{{\eta }_{p}}}{1+r_{p}^{2}} \\ \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&q=\frac{{{r}_{p}}{{\zeta }_{m}}{{\eta }_{p}}}{1+r_{p}^{2}}+\frac{{{r}_{p}}{{\eta }_{p}}{{\eta }_{m}}}{2\left( 1+{{r}_{m}} \right)\left( 1+r_{p}^{2} \right)}.\end{eqnarray} \tag{ 23 }$$

Equation (21) can be solved according to the value of the Cartan discriminant

$$\begin{eqnarray}&&\Delta ={{\left( \frac{q}{2} \right)}^{2}}+{{\left( \frac{p}{3} \right)}^{2}}\end{eqnarray} \tag{ 24 }$$

and the solution is

$$\begin{eqnarray*}\begin{array}{*{35}{l}} {{\lambda }_{1}} & = & \sqrt[3]{-\frac{q}{2}+\sqrt{{{\left( \frac{q}{2} \right)}^{2}}+{{\left( \frac{p}{3} \right)}^{3}}}} \\ {} & {} & +\sqrt[3]{-\frac{q}{2}-\sqrt{{{\left( \frac{q}{2} \right)}^{2}}+{{\left( \frac{p}{3} \right)}^{3}}}} \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{*{35}{l}} {{\lambda }_{2}} & = & m\cdot \sqrt[3]{-\frac{q}{2}+\sqrt{{{\left( \frac{q}{2} \right)}^{2}}+{{\left( \frac{p}{3} \right)}^{3}}}} \\ {} & {} & +{{m}^{2}}\cdot \sqrt[3]{-\frac{q}{2}-\sqrt{{{\left( \frac{q}{2} \right)}^{2}}+{{\left( \frac{p}{3} \right)}^{3}}}} \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray} {{\lambda }_{3}} & = & {{m}^{2}}\cdot \sqrt[3]{-\frac{q}{2}+\sqrt{{{\left( \frac{q}{2} \right)}^{2}}+{{\left( \frac{p}{3} \right)}^{3}}}} \\ {} & {} & +m\cdot \sqrt[3]{-\frac{q}{2}-\sqrt{{{\left( \frac{q}{2} \right)}^{2}}+{{\left( \frac{p}{3} \right)}^{3}}}} \\ \end{eqnarray} \tag{ 25 }$$

where $m=\frac{-1+i\sqrt{3}}{2}$.

Combining equations (22), (23) and (25), it can be concluded that the resonant frequency of the hybrid energy harvester is affected by structural characteristics (the damping ratio ${{\zeta }_{m}}$ and coupling coefficients ${{\eta }_{p}},{{\eta }_{m}}$) and the normalized loads ${{r}_{m}},{{r}_{p}}$. Therefore, the resonant frequency of the hybrid energy harvester shifts when the coupling coefficient and the load resistor vary, and we will discuss the effect of these factors on the resonant frequency in section 4. In addition, substituting equation (25) into equations (13) and (14), the normalized displacement and power can be calculated at resonance.

Suppose that $\left( {{\eta }_{m}}+{{\eta }_{p}} \right)/{{\zeta }_{m}}\ll 2$ for weak coupling. In this case, the vibration characteristics are almost independent of the feedback effect of the electric element, and the mechanical energy is mainly dissipated by damping.

In table 1, most of the micro-harvesters belong to the weak coupling group. Thus, the performance of the hybrid energy harvester in weak coupling is investigated to give methods of optimizing power for the micro-hybrid energy harvesters. Assume the mechanical damping ratio ${{\xi }_{m}}=0.026$ and coupling coefficient $({{\eta }_{m}}+{{\eta }_{p}})=0.0046$ in the following study, so $({{\eta }_{m}}+{{\eta }_{p}})/{{\zeta }_{m}}=0.177$ corresponds to a weak coupling. In order to analyze the effect of each coupling coefficient of the hybrid energy harvester on the performance, two different cases are considered: ${{\eta }_{m}}<{{\eta }_{p}}$ and ${{\eta }_{m}}>{{\eta }_{p}}$. When ${{\eta }_{m}}<{{\eta }_{p}}$, suppose that ${{\eta }_{m}}=0.0016$ and ${{\eta }_{p}}=0.003$. Conversely, when ${{\eta }_{m}}>{{\eta }_{p}}$, suppose that ${{\eta }_{p}}=0.0016$ and ${{\eta }_{m}}=0.003$. For these two cases, figure 5 demonstrates the normalized resonant frequency at an arbitrary load resistance for the hybrid energy harvester.

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From the analysis results in figure 5, although the shift of the resonant frequency for weak coupling is relatively small, a definite relation exists between the resonant frequency and the load resistance of the PE and EM harvesting elements, which is that the resonant frequency increases with an increased load resistance, and which keeps constant after reaching the maximum value. At the same time, no matter whether ${{\eta }_{m}}<{{\eta }_{p}}$ or ${{\eta }_{m}}>{{\eta }_{p}}$, the normalized resonant frequency mainly depends on the normalized PE load resistance instead of the EM load. Thus, for weak coupling, the resonant frequency is almost equal to the natural frequency of the hybrid energy harvester.

The normalized displacement, power and conversion efficiency are plotted against the normalized load resistance of the hybrid energy harvester in figure 6, based on the analytical solution at resonance, and it can be concluded that the normalized displacement increases as the EM load resistance increases until it attains a maximum, and it exibits a minimum value at ${{r}_{p}}=1$. On the other hand, the normalized power and conversion efficiency reach a peak value when ${{r}_{m}}=1$ and ${{r}_{p}}=1$ simultaneously, which means the optimal PE load resistance is 1/ωC_p and the optimal EM load resistance is ${{R}_{\text{coil}}}$ for the weak coupling effect. Therefore, the internal resistance of the hybrid energy harvester is not affected by the coupling effect at the weak coupling.

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In the optimal load resistance, the normalized displacement, power and conversion efficiency against the excitation frequency are numerically calculated and compared with only the PE or EM energy harvester, as illustrated in figure 7. It is clear from analysis results that the power and conversion efficiencies of the hybrid energy harvester are bigger than those of the PE-only or EM-only energy harvesters, although the values of displacement are almost equal. Moreover, the hybrid energy harvesting techniques can enhance the bandwidth at the same condition, which is a conclusion also reached by Lallart and Inman [29]. However, the energy conversion efficiency is less than 10% at the weak coupling effect, so most of the mechanical energy is lost because of the mechanical damping ratio and internal resistance.

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In addition, the power and conversion efficiency of the hybrid energy harvester at η _m  < η _p is bigger than the situation of η_m  > η_p when the total coefficient stays at one value. Thus, for the hybrid energy harvester, although the total coupling coefficient is the same, the power is decided by each value of the PE and EM coupling coefficient. Moreover, for weak coupling, the hybrid energy harvesting technique should be adopted instead of single the harvesting mechanism.

We define the medium coupling as (η_m  + η_p )/ζ_m  ≈ 2, and table 1 reveals that most of the meso-harvesters belong to this case. At this time, the feedback effect of the electric-element and mechanical damping has the equivalent effect on the harvesting system.

Suppose η_m  + η_p  = 0.05 and ζ_m  = 0.026, then (η_m  + η_p )/ζ_m  = 1.923, which are typical parameters for PE and EM power generators designed as the meso-structure [36]. According to the analysis results in table 1, the effects of the coupling coefficient on the performance for the hybrid energy harvester are classified into three cases: (1) EM medium coupling and PE weak coupling; (2) PE medium coupling and EM weak coupling; and (3) EM medium coupling and PE medium coupling. Here we assume that η_m  = 0.045 and η_p  = 0.005 for case 1; η_m  = 0.005 and η_p  = 0.045 for case 2; and η_m  = 0.025 and η_p  = 0.025 for case 3. Then, the resonant frequency is numerically calculated for medium coupling when the load resistance of the PE and EM elements vary, and results are shown in figures 8(a), (e) and 9(a), which reveal that although the shift of the resonant frequency for medium coupling is much bigger than that for weak coupling, the change law is similar. Therefore, the conclusion can be reached that the shift of the resonant frequency is proportional to the coupling coefficient for the hybrid energy harvester. However, in contrast with two others of medium coupling, the extent of the resonant frequency shift is more obvious when the harvester is in the state of EM weak coupling and PE medium coupling.

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When the hybrid energy harvester is excited at the resonant frequency, the normalized displacement, power and conversion efficiency versus load resistance are as shown in figures 8(b)–(d), (f)–(h) and 9(b)–(d), so the performance of the harvester is mainly decided by the PE coupling coefficient for EM weak coupling and PE medium coupling. Conversely, they are mainly related to the EM coupling coefficient for EM medium coupling and PE weak coupling. Therefore, it can be said that the coupling coefficient directly affects the vibration characteristics and performance of the hybrid energy harvester. The load resistance is optimal while the power reaches a maximum value. When η_m  = 0.045 and η_p  = 0.005, the optimal load resistances are r_m  = 2.1 and r_p  = 1 simultaneously for the hybrid energy harvesting; while the optimal normalized load is r_p  = 1 and r_m  = 4.1 at η_m  = 0.005 and η_p  = 0.045, as illustrated in figures 8(c), (g) and 9(c). At the same time, when η_m  = 0.025 and η_p  = 0.025, the optimal load resistances are r_m  = 2.9 and r_p  = 1 for the hybrid energy harvesting while the optimal loads are r_m  = 1.6 and r_p  = 1 for the EM-only and PE-only energy harvesters, respectively. Therefore, it can be concluded that the internal resistance of the EM harvest element increases as the coupling effect of the hybrid energy harvester is enhanced. However, the optimal PE load resistance is almost unchanged at the medium coupling effect. In addition, the optimal EM load is influenced by the PE coupling efficient, and the greater the PE coupling coefficient, the bigger the optimal load resistance of the EM element. Furthermore, the optimal power of the hybrid energy harvester allows the sensitivity to EM load resistance to decrease.

When the hybrid energy harvester is connected with an optimal load, the normalized displacement, power and conversion efficiency against the normalized frequency are plotted in figures 10 and 11, which demonstrate that the shift of resonant frequency mainly relates to the PE coupling coefficient instead of the EM coupling efficient. Compared with the weak coupling, the conversion efficiency of the hybrid energy harvester is much bigger for medium coupling, which means that much more energy is converted into electric energy instead of being lost by damping. Moreover, the conversion efficiency of the harvester at EM weak coupling and PE medium coupling is bigger than any other two cases of medium coupling; in particular, the efficiency is up to 30%.

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In addition, the power of the hybrid energy harvester is not increased compared with the PE-only energy harvester at EM weak coupling and PE medium coupling, or of the EM-only energy harvester at EM medium coupling and PE weak coupling. The former situation should take into account the internal loss of the electromagnetic transducer element, and the latter may be caused by the stiffness variation after combing the PE harvesting element. Furthermore, compared with only one conversion technique, the power of the hybrid energy harvester is much bigger, and the bandwidth is much wider for EM medium coupling and PE medium coupling, which is the same result as the experimental test in [26, 29].

Therefore, as the electromechanical coupling coefficient increases, the total power will not definitely increase after combining the PE and EM energy harvester, but depends on the cases of each value of the PE and EM coupling coefficients, which need to be considered in designing the hybrid energy harvester. At the same time, according to analysis results, the output of the hybrid energy harvester is mainly decided by the PE element for PE medium coupling and EM weak coupling; similarly, the harvester output is mainly related to the EM element for EM medium coupling and PE weak coupling. Therefore, compared with the case of EM medium coupling and PE medium coupling, the current of the hybrid energy harvest is still relatively small for EM weak coupling and PE medium coupling, and the voltage is also not increased for EM medium coupling and PE weak coupling. For these two cases, the performance of the hybrid energy harvester cannot achieve both a big current and voltage output. Hence, the hybrid energy harvester designed with EM medium coupling and PE medium coupling could enhance the output power and conversion efficiency.

Define the strong coupling as (η_m  + η_p )/ζ_m  ≫⃒ 2, which is a special case for the energy harvester, but may be achieved by an SSHI (synchronized switch harvesting on inductor) technique for boosting the power [20, 21]. Suppose that η_m  = η_p  = 0.1 and ζ_m  = 0.026, then (η_m  + η_p )/ζ_m  = 7.69. In this case, the feedback effect of the electric element on the harvesting system is much greater than the mechanical damping, so the coupling coefficient plays a major role in energy transfer, and the most of mechanical energy will be converted into electric energy by the harvester.

Using equation (25), the resonant frequency of the hybrid energy harvester is numerically calculated with a different load resistance for strong coupling. The results, given by figure 12(a), indicate that the resonant frequency shift of strong coupling is much greater than that of weak coupling and medium coupling. Similarly, the resonant frequency increases with the increase of the load resistance of the PE element and the EM element, and the normalized frequency is mainly related to the PE coupling coefficient compared to the EM coupling coefficient.

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When the harvesting system is excited at the resonant frequency, the corresponding normalized displacement, power and conversion efficiency for strong coupling are as shown in figures 12(b)–(d), which shows that the variation law of the normalized resonant frequency against the normalized load resistance is the same as the conclusions with weak coupling and medium coupling. However, when η_m  = η_p  = 0.1, the normalized load resistance corresponding to the maximum power is r_m  = 8 and r_p  = 1 for the hybrid energy harvester, while the optimal load resistance is 3R_coil and 1/ωC_p for the EM-only and PE-only energy harvester, respectively. Hence, it can be concluded that the internal resistance difference between the hybrid energy harvester and the EM-only harvester increases as the coupling coefficient is enhanced.

With the optimal load, the normalized displacement, power and conversion efficiency versus the normalized frequency are numerically calculated, and the results are plotted in figure 13, which reveals that the resonant frequency shift of the strong coupling is much greater than that of the weak coupling and medium coupling. In addition, the total power and conversion efficiency further increase when the coupling effect rises from weak coupling to strong coupling. Moreover, the efficiency can be up to 50% when the hybrid energy harvester is of the strong coupling type, which means that the bigger the coupling effect, the more the electric energy output.

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From figure 13, the total power of the hybrid energy harvester is bigger than that of the EM-only harvester, but is almost equal to that of the PE-only harvester. Furthermore, the conversion efficiency has the same conclusion with the output power, and the harvesting bandwidth is just equal to the PE-only energy harvester. According to equation (36), the perfomance of the hybrid energy harvester in strong coupling is mainly determined by the coupling coefficient compared with the damping ratio, which means that the electric output feedback of the piezoelectric and electromagnetic elements for strong coupling are much bigger than those of weak and medium coupling. Therefore, the effective stiffness and damping are much bigger in strong coupling. In addition, the internal resistance of the hybrid harvester increases as the coupling coefficient strengthens. It can be determined that there is much more power loss for strong coupling because of the effective electric damping and internal resistance, and the power output of the electromagnetic element in the hybrid device may be consumed because of the strong coupling effect. Thus, it can be concluded that the total power and conversion efficiency of the hybrid energy harvester are no longer raised compared with the PE-only harvester when the coupling coefficient is up to strong coupling.