18.2.1 Electromagnetic transducers

Harvesters with electromagnetic transduction, such as the one in Fig. 18.1a, convert power via Faraday’s law, by the equations

$\mathit{f}\left(\mathit{t}\right)={\mathit{K}}_{\mathit{e}}\mathit{i}\left(\mathit{t}\right)\mathit{v}\left(\mathit{t}\right)={\mathit{K}}_{\mathit{e}}\dot{\mathit{x}}\left(\mathit{t}\right)$ [18.1]

[18.1]

where K_e is an electromagnetic conversion constant that is a function of the geometry of the coil, number of turns, and the magnetic field of the magnets. The specific functional dependency of K_e is typically linear in the field intensity and number of turns, while the dependency on the geometry depends on the configuration of the coil relative to the magnets.^11,15 The expressions in Equation [18.1] constitute the linearized version of what is always, to one degree or another, a nonlinear phenomenon exhibiting hysteretic and other losses.

The harvester dynamics are governed by the standard second-order differential equation

$\mathit{m}\ddot{\mathit{x}}\left(\mathit{t}\right)+\mathit{c}\dot{\mathit{x}}\left(\mathit{t}\right)+\mathit{kx}\left(\mathit{t}\right)=-\mathit{ma}\left(\mathit{t}\right)+{\mathit{K}}_{\mathit{e}}\mathit{i}\left(\mathit{t}\right)$ [18.2]

[18.2]

In the Laplace domain, voltage v is a consequence of dynamic system inputs a and i, as

$\widehat{\mathit{v}}\left(\mathit{s}\right)={\mathit{G}}_{\mathit{a}}\left(\mathit{s}\right)\widehat{\mathit{a}}\left(\mathit{s}\right)+{\mathit{G}}_{\mathit{i}}\left(\mathit{s}\right)\widehat{\mathit{i}}\left(\mathit{s}\right)$ [18.3]

[18.3]

where

${\mathit{G}}_{\mathit{a}}\left(\mathit{s}\right)=\frac{-{\mathit{K}}_{\mathit{e}}\mathit{ms}}{\mathit{k}+\mathit{cs}+\mathit{m}{\mathit{s}}^2}$ [18.4]

[18.4]

${\mathit{G}}_{\mathit{i}}\left(\mathit{s}\right)=\frac{{\mathit{K}}_{\mathit{e}}^2\mathit{s}}{\mathit{k}+\mathit{cs}+\mathit{m}{\mathit{s}}^2}$ [18.5]

[18.5]

18.2.2 Piezoelectric transducers

We assume a composite beam such as the one shown in Fig. 18.1b, in which piezoelectric material is bonded to a substrate material (such as aluminum). It is not necessarily the case that the cross-sectional properties of the beam are constant; indeed, there is some benefit to making the beam become progressively more slender as it extends from the root, as this makes strain deformations more uniform in the fundamental mode. It is also not necessarily optimal to bond the transducer patch over the entire extent of the beam, as is depicted in the figure. Rather, in many designs the patch extends from the root only a fraction of the beam length. This is advantageous because by placing the transducer only where the strain is high, the voltages produced can be higher. Finally, it is often the case that a proof mass is attached to the end of the beam. This can be used to tune the resonant frequency of the beam, as well as having other advantages.

In Reference 16, the electromechanical dynamics of a beam with multiple bimorph piezoelectric transducers were derived. Here, we merely provide an overview, specializing to the single-transducer case. Let ρ(x, y, z, t ) denote the density at location {x, y, z} in the substrate–piezo composite. Let S(x, y, z, t) ∈ ℜ⁶ and T(x, y, z, t) ∈ ℜ⁶ denote the Voigt (i.e., vector) form of the strain and stress tensors, and let E(x, y, z, t) ∈ ℜ³ and D(x, y, z, t) ∈ ℜ³ denote the electric field and displacement vectors. Then for small deformations and small fields, we can assume the linear, coupled constitutive law relating {S, E} to {T,D}, i.e.,

$\left[\begin{array}{c}\hfill \mathit{T}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \\ \hfill \mathit{D}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \mathit{c}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \\ \hfill \mathit{e}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \end{array}\begin{array}{c}\hfill -{\mathit{e}}^{\mathit{T}}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \\ \hfill \mathit{\epsilon }\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill \mathit{S}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \\ \hfill \mathit{E}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)\hfill \end{array}\right]$ [18.6]

[18.6]

where c is the modulus of elasticity at zero field, ε is the dielectric constant matrix at zero strain, and e is the coupling coefficient matrix. (Note that in the substrate material, e and ε are zero.)

Assuming the classical Bernoulli-Euler assumptions for the beam deformation, define d(x, t) as the transverse deflection of the beam centroid, and then it follows that

$\mathit{S}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)=\mathit{y}{\mathit{d}}^{″}\left(\mathit{x},\mathit{t}\right)\left[\begin{array}{c}\hfill {\widehat{\mathit{e}}}_{\mathit{x}}\hfill \\ \hfill 0\hfill \end{array}\right]$ [18.7]

[18.7]

A standard Galerkin approximation is typically used for d(x, t), i.e.,

$\mathit{d}\left(\mathit{x},\mathit{t}\right)={\displaystyle \sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{\varphi }}_{\mathit{i}}\left(\mathit{x}\right){\mathit{r}}_{\mathit{i}}\left(\mathit{t}\right)={\mathit{\varphi }}^{\mathit{T}}\left(\mathit{x}\right)\mathit{r}\left(\mathit{t}\right)}$ [18.8]

[18.8]

where r(t) is the vector of generalized displacements, and ϕ(x) is the vector of Galerkin shape functions for each generalized displacement. Electric field E is assumed to be constant inside each patch, zero in the substrate, and oriented in the ê_y direction everywhere; i.e.,

$\mathit{E}\left(\mathit{x},\mathit{y},\mathit{z},\mathit{t}\right)={\widehat{\mathit{e}}}_{\mathit{y}}\mathit{\psi }\left(\mathit{y}\right)\mathit{v}\left(\mathit{t}\right)$ [18.9]

[18.9]

where ê_y is the unit vector on the y direction, ψ(y) is a function of the patch thickness and beam geometry.

With these assumptions, the differential equations for the reduced-order system are found as

$\mathit{M}\ddot{\mathit{r}}\left(\mathit{t}\right)+\mathit{C}\dot{\mathit{r}}\left(\mathit{t}\right)+\mathit{Kr}\left(\mathit{t}\right)+\Theta v\left(\mathit{t}\right)=\mathit{Ga}\left(\mathit{t}\right)$ [18.10]

[18.10]

${\mathit{C}}_{\mathit{p}}\dot{\mathit{v}}\left(\mathit{t}\right)+{\mathit{R}}_{\mathit{p}}^{-1}\mathit{v}\left(\mathit{t}\right)-{\Theta }^{\mathit{T}}\dot{\mathit{r}}\left(\mathit{t}\right)=\mathit{i}\left(\mathit{t}\right)$ [18.11]

[18.11]

where {M, K, Θ, G, C_p} are found through the standard Rayleigh-Ritz projection. These terms are, respectively, the generalized mass and stiffness matrices, the coupling factor matrix, disturbance input matrix, and piezoca-pacitance. Generally, mechanical damping matrix C is found after the fact e.g., through a Rayleigh damping assumption. The dielectric leakage resistance R_p , is extremely high for typical designs, and is often assumed to be infinite.

As with the case of the electromagnetic harvester, the Laplace domain characterization of the above is

$\widehat{\mathit{v}}\left(\mathit{s}\right)={\mathit{G}}_{\mathit{a}}\left(\mathit{s}\right)\widehat{\mathit{a}}\left(\mathit{s}\right)+{\mathit{G}}_{\mathit{i}}\left(\mathit{s}\right)\widehat{\mathit{i}}\left(\mathit{s}\right)$ [18.12]

[18.12]

but where in this example the above transfer functions are

${\mathit{G}}_{\mathit{a}}\left(\mathit{s}\right)={\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]}^{\mathit{T}}{\left[\begin{array}{c}\hfill \mathit{sI}\hfill \\ \hfill {\mathit{M}}^{-1}\mathit{K}\hfill \\ \hfill 0\hfill \end{array}\begin{array}{c}\hfill -\mathit{I}\hfill \\ \hfill \mathit{sI}+{\mathit{M}}^{-1}\mathit{C}\hfill \\ \hfill -{\mathit{C}}_{\mathit{p}}^{-1}{\mathit{\Theta }}^{\mathit{T}}\hfill \end{array}\begin{array}{c}\hfill 0\hfill \\ \hfill {\mathit{M}}^{-1}\mathit{\Theta }\hfill \\ \hfill \mathit{sI}+{\mathit{C}}_{\mathit{p}}^{-1}{\mathit{R}}_{\mathit{p}}^{-1}\hfill \end{array}\right]}^{-1}\left[\begin{array}{c}\hfill 0\hfill \\ \hfill {\mathit{M}}^{-1}\mathit{G}\hfill \\ \hfill 0\hfill \end{array}\right]$ [18.13]

[18.13]

${\mathit{G}}_{\mathit{i}}\left(\mathit{s}\right)={\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]}^{\mathit{T}}{\left[\begin{array}{c}\hfill \mathit{sI}\hfill \\ \hfill {\mathit{M}}^{-1}\hfill \\ \hfill 0\hfill \end{array}\begin{array}{c}\hfill -\mathit{I}\hfill \\ \hfill \mathit{sI}+{\mathit{M}}^{-1}\mathit{C}\hfill \\ \hfill -{\mathit{C}}_{\mathit{p}}^{-1}{\Theta }^{\mathit{T}}\hfill \end{array}\begin{array}{c}\hfill 0\hfill \\ \hfill {\mathit{M}}^{-1}\Theta \hfill \\ \hfill \mathit{sI}+{\mathit{C}}_{\mathit{p}}^{-1}{\mathit{R}}_{\mathit{p}}^{-1}\hfill \end{array}\right]}^{-1}\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill {\mathit{C}}_{\mathit{p}}^{-1}\hfill \end{array}\right]$ [18.14]

[18.14]

18.2.3 General properties

Transfer function G_i(s) is the input impedance of the transducer. One can think of it as the driving-point impedance of an electric circuit that is equivalent to the true system from the vantage point of the harvesting electronics. Because harvesters are passive systems (i.e., they typically have no internal energy sources), this equivalent network will also be passive, i.e., it can be built from ideal resistors, inductors, capacitors, transformers, and gyrators. Because of this, one can always assume that G_i(s) is positive real, i.e.,

$\text{Re}\left\{{\mathit{G}}_{\mathit{i}}\left(\mathit{j\omega }\right)\right\}>0,\forall \mathit{\omega }\in \left[-\mathit{\infty },\mathit{\infty }\right]$ [18.15]

[18.15]

This property turns out to be important, in the determination of the maximum power generation for energy harvesters.

18.3.1 Classical impedance matching theory

Consider the case in which a(t) is a finite-energy signal, i.e., one for which ∫ ₀^∞a²(t)dt is finite, and let â(s) be its Laplace transform. Similarly, let $\widehat{\mathit{i}}\left(\mathit{s}\right)$ and $\widehat{\mathit{v}}\left(\mathit{s}\right)$ be the Laplace transforms for the corresponding current and voltage, which we also assume to have finite signal energy. The total amount of energy absorbed from the transducer by the electronics is

${\mathit{E}}_{\text{abs}}={\displaystyle {\int }_0^{\mathit{\infty }}-\mathit{v}\left(\mathit{t}\right)\mathit{i}\left(\mathit{t}\right)\mathit{dt}}$ [18.16]

[18.16]

By the Plancharel theorem, we have that this energy is equivalent to

${\mathit{E}}_{\text{abs}}=\frac{1}{2\mathit{\pi }}{\displaystyle {\int }_{-\mathit{\infty }}^{\mathit{\infty }}-\widehat{\mathit{v}}\left(-\mathit{j\omega }\right)\widehat{\mathit{i}}\left(\mathit{j\omega }\right)\mathit{d\omega }}$ [18.17]

[18.17]

Substituting Equation [18.12], we have

${\mathit{E}}_{\text{abs}}=\frac{1}{2\mathit{\pi }}{\displaystyle {\int }_{-\mathit{\infty }}^{\mathit{\infty }}\left({\mathit{G}}_{\mathit{a}}\left(-\mathit{j\omega }\right)\widehat{\mathit{a}}\left(-\mathit{j\omega }\right)+{\mathit{G}}_{\mathit{i}}\left(-\mathit{j\omega }\right)\widehat{\mathit{i}}\left(-\mathit{j\omega }\right)\right)\widehat{\mathit{i}}\left(\mathit{j\omega }\right)\mathit{d\omega }}$ [18.18]

[18.18]

Now, consider that at each ω we can find the $\widehat{\mathit{i}}\left(\mathit{j\omega }\right)$ maximizing E_abs. Performing this maximization, with the constraint that $\widehat{\mathit{i}}\left(-\mathit{j\omega }\right)$ is the complex conjugate of $\widehat{\mathit{i}}\left(\mathit{j\omega }\right)$, gives the optimal current as

$\widehat{\mathit{i}}\left(\mathit{j\omega }\right)=-\frac{{\mathit{G}}_{\mathit{a}}\left(\mathit{j\omega }\right)}{{\mathit{G}}_{\mathit{i}}\left(\mathit{j\omega }\right)+{\mathit{G}}_{\mathit{i}}\left(-\mathit{j\omega }\right)}\widehat{\mathit{a}}\left(\mathit{j\omega }\right)$ [18.19]

[18.19]

Correspondingly, the voltage $\widehat{\mathit{v}}\left(\mathit{j\omega }\right)$ with the optimal $\widehat{\mathit{i}}\left(\mathit{j\omega }\right)$ imposed, is

$\widehat{\mathit{v}}\left(\mathit{j\omega }\right)=\frac{{\mathit{G}}_{\mathit{i}}\left(-\mathit{j\omega }\right){\mathit{G}}_{\mathit{a}}\left(\mathit{j\omega }\right)}{{\mathit{G}}_{\mathit{i}}\left(\mathit{j\omega }\right)+{\mathit{G}}_{\mathit{i}}\left(-\mathit{j\omega }\right)}\widehat{\mathit{a}}\left(\mathit{j\omega }\right)$ [18.20]

[18.20]

and the optimized absorbed energy is

${\mathit{E}}_{\text{abs}}=\frac{1}{2\mathit{\pi }}{\displaystyle {\int }_{-\mathit{\infty }}^{\mathit{\infty }}\frac{{\mathit{G}}_{\mathit{i}}\left(\mathit{j\omega }\right){\left|{\mathit{G}}_{\mathit{a}}\left(\mathit{j\omega }\right)\widehat{\mathit{a}}\left(\mathit{j\omega }\right)\right|}^2}{{\left({\mathit{G}}_{\mathit{i}}\left(\mathit{j\omega }\right)+{\mathit{G}}_{\mathit{i}}\left(-\mathit{j\omega }\right)\right)}^2}\mathit{d\omega }}$ [18.21]

[18.21]

Recognizing that Im{G_i(jω)} = − Im{G_i(− jω)}and Re{G_i(jω)} = Re{G_i(− jω)} the above is equivalent to

${\mathit{E}}_{\text{abs}}=\frac{1}{2\mathit{\pi }}{\displaystyle {\int }_{-\mathit{\infty }}^{\mathit{\infty }}\frac{{\left|{\mathit{G}}_{\mathit{a}}\left(\mathit{j\omega }\right)\widehat{\mathit{a}}\left(\mathit{j\omega }\right)\right|}^2}{4\text{Re}\left\{{\mathit{G}}_{\mathit{i}}\left(\mathit{j\omega }\right)\right\}}\mathit{d\omega }}$ [18.22]

[18.22]

Note that the integrand above is real and positive at all frequencies, and constitutes the spectrum of optimized energy.

In the Laplace domain, the optimal $\widehat{\mathit{i}}\left(\mathit{s}\right)$ may be expressed as a feedback function of $\widehat{\mathit{v}}\left(\mathit{s}\right)$ as

$\widehat{\mathit{i}}\left(\mathit{s}\right)=-\mathit{Y}\left(\mathit{s}\right)\widehat{\mathit{v}}\left(\mathit{s}\right)$ [18.23]

[18.23]

where

$\mathit{Y}\left(\mathit{s}\right)={\mathit{G}}_{\mathit{i}}^{-1}\left(-\mathit{s}\right)$ [18.24]

[18.24]

is called the impedance matched feedback law. However, unlike our usual conventions for feedback, Y(s) above would need to be anticausal (i.e., anticipatory) in order for the closed-loop system to be stable. (In the time domain, this fact would be evident from the impulse response of Y(s) being left-handed, rather than right-handed.) As such, we can say that in general, in order for an VEH system to maximize the physically-available power, it must make power generation decisions based on future disturbance information.

18.3.2 Power availability and tuning for monochromatic disturbances

Now, consider the case in which a(t) is monochromatic, i.e.,

$\mathit{a}\left(\mathit{t}\right)={\mathit{a}}_{0}cos{\mathit{\omega }}_0\mathit{t}$ [18.25]

[18.25]

for time-invariant a₀ and ω₀. In this case, it turns out that an analogous optimization can be done, again arriving at the same matching condition in Equation [18.24]. However, in this case, â(jω) is zero at all frequencies except ω = ω₀, and it is therefore only necessary for a feedback law Y(s) to satisfy Equation [18.24] at s = jω₀. At all other frequencies, Y(jω) can be chosen arbitrarily, or to meet other design considerations. It is always possible to satisfy the matching condition Y(jω₀) = G_i^–1 (− jω₀) with a causal and stable feedback law, and often this feedback law is made equivalent to a simple two-component electric circuit that shunts the transducer, the admittance of which equals Y(s). Two such circuits are shown in Fig. 18.2. A feedback law equivalent to an inductor L in parallel with a resistor R would be

$\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)={\mathit{R}}^{-1}-\mathit{j}{\left({\mathit{\omega }}_0\mathit{L}\right)}^{-1}$ [18.26]

[18.26]

which, through choice of R and L, can be made any magnitude, and any phase in the range [− π/2,0]. Similarly, a feedback law equivalent to a capacitor C in parallel to a resistor R would be

$\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)={\mathit{R}}^{-1}+\mathit{j}{\mathit{\omega }}_0\mathit{C}$ [18.27]

[18.27]

which, through choice of R and C, can be made any magnitude, and any phase in the range [0,π/2]. Because G_i(s) is positive real, it is known that its phase at all frequencies is in the range [− π/2,π/2] and therefore, one of the above two cases always results in a matching circuit for any ω₀.

Suppose that the electronics are designed such that their dynamics relate i to v through some causal and stable transfer function, as in Equation [18.23], but not necessarily for the optimal Y(s). Then the closed-loop system is described in the frequency domain, by

$\widehat{\mathit{v}}\left(\mathit{j\omega }\right)={\mathit{H}}_{\mathit{a}}\left(\mathit{j\omega }\right)\widehat{\mathit{a}}\left(\mathit{j\omega }\right)$ [18.28]

[18.28]

$\widehat{\mathit{i}}\left(\mathit{j\omega }\right)=-\mathit{Y}\left(\mathit{j\omega }\right){\mathit{H}}_{\mathit{a}}\left(\mathit{j\omega }\right)\widehat{\mathit{a}}\left(\mathit{j\omega }\right)$ [18.29]

[18.29]

where

${\mathit{H}}_{\mathit{a}}\left(\mathit{j\omega }\right)=\frac{{\mathit{G}}_{\mathit{a}}\left(\mathit{j\omega }\right)}{1+{\mathit{G}}_{\mathit{i}}\left(\mathit{j\omega }\right)\mathit{Y}\left(\mathit{j\omega }\right)}$ [18.30]

[18.30]

For the case in which a(t) is characterized by Equation [18.25], the voltage and current in the time domain are

$\mathit{v}\left(\mathit{t}\right)={\mathit{a}}_0\left|{\mathit{H}}_{\mathit{a}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right|cos\left({\mathit{\omega }}_0\mathit{t}+{\mathit{\varphi }}_{\mathit{v}}\right),{\mathit{\varphi }}_{\mathit{v}}=\angle \left({\mathit{H}}_{\mathit{a}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right)$ [18.31]

[18.31]

$\mathit{i}\left(\mathit{t}\right)=-{\mathit{a}}_0\left|\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)\right|\left|{\mathit{H}}_{\mathit{a}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right|cos\left({\mathit{\omega }}_0\mathit{t}+{\mathit{\varphi }}_{\mathit{i}}\right),{\mathit{\varphi }}_{\mathit{i}}={\mathit{\varphi }}_{\mathit{v}}+\angle \left(\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)\right)$ [18.32]

[18.32]

The instantaneous absorbed power is P_abs (t) = − v(t)i(t). Substituting the above, and using some trigonometric identities, we have that

${\mathit{P}}_{\text{abs}}\left(\mathit{t}\right)=\mathit{P}\left[1+cos\left(2\left({\mathit{\omega }}_0\mathit{t}+{\mathit{\varphi }}_{\mathit{v}}\right)\right)\right]+\mathit{Q}sin\left(2\left({\mathit{\omega }}_0\mathit{t}+{\mathit{\varphi }}_{\mathit{v}}\right)\right)$ [18.33]

[18.33]

where P and Q are the so-called real and reactive energy flows, defined as

$\mathit{P}=\frac{1}{2}{\mathit{a}}_0^2{\left|{\mathit{H}}_{\mathit{a}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right|}^2\text{Re}\left\{\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)\right\}$ [18.34]

[18.34]

$\mathit{Q}=-\frac{1}{2}{\mathit{a}}_0^2{\left|{\mathit{H}}_{\mathit{a}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right|}^2\text{Im}\left\{\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)\right\}$ [18.35]

[18.35]

Note that on average, the reactive power oscillates about zero, while the real power has a mean value of P. This is the average power generation in monochromatic response.

Evaluating the value of P at the optimized Y(jω) = G_i^− 1(− jω) gives the physical limit on generated power, as

${\mathit{P}}_{\text{opt}}=\frac{{\left|{\mathit{G}}_{\mathit{a}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right|}^2}{8\text{Re}\left\{{\mathit{G}}_{\mathit{i}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right\}}{\mathit{a}}_0^2$ [18.36]

[18.36]

This is the fundamental limit on power generation for monochromatic disturbances. Meanwhile, the reactive power with the optimal feedback is

${\mathit{Q}}_{\text{opt}}=-\frac{{\left|{\mathit{G}}_{\mathit{a}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right|}^2\text{Im}\left\{{\mathit{G}}_{\mathit{i}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right\}}{8\text{Re}{\left\{{\mathit{G}}_{\mathit{i}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right\}}^2}{\mathit{a}}_0^2$ [18.37]

[18.37]

From Equation [18.33], we see that if | Q | is large, this implies that for each oscillatory cycle a significant amount of energy oscillates back and forth between the harvester and the electronics. If Q is zero, on the other hand, energy always flows from the harvester to the electronics. This is advantageous for a few reasons. Firstly, if it is only necessary for the electronics to facilitate energy flow in one direction, the processing circuitry can be simplified. Secondly, even if processing circuitry with bidirectional energy flow is used, the above analysis does not account for the conductive loss in the circuit, and this loss is minimized for a given P, by designing the system such that Q is zero. For this reason, it is often desirable to design the harvester to achieve Q = 0 in the development above by tuning the hardware parameters such that

$\text{Im}\left\{{\mathit{G}}_{\mathit{i}}\left(\mathit{j}{\mathit{\omega }}_0\right)\right\}=0$ [18.38]

[18.38]

In other words, the harvester is tuned such that v(t) and i(t) are in phase at the excitation frequency.^18,19

One way to accomplish this is through mechanical tuning design. For example, consider the SDOF electromagnetic harvester in Fig. 18.1a. In this case, we have that

${\mathit{P}}_{\text{opt}}=\frac{{\mathit{m}}^2{\mathit{a}}_0^2}{8\mathit{c}}$ [18.39]

[18.39]

${\mathit{Q}}_{\text{opt}}=\frac{{\mathit{m}}^2{\mathit{a}}_0^2\left(\mathit{m}{\mathit{\omega }}_0^2-\mathit{k}\right)}{8{\mathit{c}}^2{\mathit{\omega }}_0}$ [18.40]

[18.40]

Interestingly, P_opt does not depend on k at all, and thus the stiffness can be chosen so as to make Q_opt = 0. This is done by tuning the harvester so that ${\mathit{\omega }}_{\mathit{n}}=\sqrt{\mathit{k}/\mathit{m}}={\mathit{\omega }}_0$. So doing, the optimal Y(jω₀) is

$\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)=\frac{\mathit{c}}{{\mathit{K}}_{\mathit{e}}^2}$ [18.41]

[18.41]

Thinking back to the ‘equivalent circuit’ interpretation of the harvesting feedback law, we see that the tuned harvester is equivalent to a resistor (i.e., with no capacitor or inductor in parallel).

To increase P_opt for the SDOF electromagnetic harvester, either c must be decreased, or m must be increased. Indeed, P_opt can theoretically be made arbitrarily large by designing the harvester to have arbitrarily small damping. This is the reason why most energy harvesters are deliberately designed to have very low damping. However, in actuality there are often other constraints that limit power generation, requiring Y(s) to be different from its theoretically optimal value. For example, for the SDOF electromagnetic harvester, a displacement constraint |x(t)| ≤ x₀ may impose a tighter limit on power generation. Using Lagrange multipliers, it can be shown that with this constraint imposed the optimal feedback law is modified to

$\mathit{Y}\left(\mathit{j}{\mathit{\omega }}_0\right)=\frac{\mathit{c}+\mathit{\lambda }}{{\mathit{K}}_{\mathit{e}}^2}+\mathit{j}\frac{\mathit{k}-\mathit{m}{\mathit{\omega }}_0^2}{{\mathit{K}}_{\mathit{e}}^2{\mathit{\omega }}_0}$ [18.42]

[18.42]

where

$\mathit{\lambda }=max\left\{0,\frac{\mathit{m}{\mathit{a}}_0}{{\mathit{\omega }}_0{\mathit{x}}_0}-2\mathit{c}\right\}$ [18.43]

[18.43]

is an added term that enforces the closed-loop amplitude constraint |x(t)| ≤ x₀ The resultant constrained optimal power generation is

${\mathit{P}}_{\text{opt}}=\frac{{\mathit{m}}^2{\mathit{a}}_0^2\left(\mathit{c}+\mathit{\lambda }\right)}{2{\left(2\mathit{c}+\mathit{\lambda }\right)}^2}$ [18.44]

[18.44]

This expression tells us a lot about the harvesting potential of SDOF systems. Note that for small a₀ and nonzero c, the stroke limit is not saturated, and the available power is the ideal limit. For c → 0, the stroke limit is always saturated, and the available power is $\frac{1}{2}\mathit{m}{\mathit{\omega }}_0{\mathit{a}}_0{\mathit{x}}_0$. This is also an upper limit on the power generation, when c is finite.

Sometimes it is difficult or impossible to redesign the mechanical parameters of a VEH system to bring about Im{G_i(jω₀)} = 0. In many circumstances, an appealing alternative is to bring this condition about via insertion of the equivalent capacitor or inductor in Fig. 18.2 across the terminals of the transducer. So doing, the new G_i(jω₀) with this inserted component absorbed into the harvester model has zero phase at the excitation frequency, and consequently the optimized Y(jω₀) is equivalent to a resistor. Indeed, this approach is often much easier than mechanical tuning techniques. Its primary disadvantage is that sometimes the particular value of L or C necessary to bring about the matching condition is simply unrealistic. This is true, for example, in piezoelectric applications, in which the value of L can be in the thousands of henries. One clever workaround to address this issue is the SSHI circuit, discussed in Section 18.4.4.

18.3.3 Fundamental power generation limits for stochastic disturbances

When a(t) is stochastic, analogous causal limits can be found for the average power generation in stationary response. However, the mathematics associated with these limits is more complicated, and involves optimal control theory.²⁰ Nonetheless, there is one very simple but conservative limit that can be stated here, which is useful in sizing harvesters. Suppose that a(t) is extremely broadband, and can justifiably be modeled as white noise with spectral intensity Φ_a. Then it is a fact that the optimal average power generation is bounded from above by

${\mathit{P}}_{\text{opt}}=\frac{1}{2}\mathit{m}{\Phi }_{\mathit{a}}$ [18.45]

[18.45]

where m is the total mass of the harvester. It turns out that this limit holds irrespective of the harvester type; i.e., it can be a piezoelectric beam, an SDOF electromagnetic system, or even a harvester with significant nonlinearities such as an electrostatic transduction system. Furthermore, it is true whether the feedback law is linear or nonlinear, and is insurpassable irrespective of the circuitry used.

This simple limit illustrates a rather uncompromising truth about harvesting energy from broadband stochastic disturbances. The only way to raise the limit is to add mass. However, it is also worth noting that most harvesters, when excited by broadband disturbances, would fall well short of this upper limit, due to internal dissipation in the harvester as well as dissipation in the circuitry. Design of stochastic VEH technologies that come anywhere near reaching the above limit is generally quite challenging.

18.4.1 Resistive loads

By far, the most convenient situation in energy harvesting would be where the power generated by the harvester is consumed by a resistive load, with a resistance that can be chosen freely. In this case, the designer would first electrically tune the harvester, if it is not already tuned mechanically, and then simply choose the load resistance to be the optimal R derived from the previous section.

However, the reality of an energy harvesting application is usually much more complicated than this scenario. To start with, it is hardly ever the case that the subsystems for which power is being generated consume this power like a simple resistance, and even less common that the resistance may be treated as a design parameter. For sensing applications, harvested energy is usually delivered to a storage subsystem, consisting of a supercapacitor, rechargeable battery, or some combination of both. Typically, sensing systems operating on harvested energy will spend the majority of the time in a sleep mode, where they consume very little power, during which time the storage bus is trickle charged. Then, periodically, the sensor will come online, conduct measurement, computation, and broadcasting tasks, before returning to sleep mode. As such, the sensing system consumes energy in short bursts of activity, during which time the electronics must maintain a constant voltage on the power bus.

In this context, our energy harvesting analysis from the previous section really pertains to the behavior of the system during the ‘sleep mode’ periods, when the primary objective is to replenish energy storage. During this time, the electronics must facilitate energy transferral from an oscillatory voltage (i.e., e) to the storage device, which typically will possess a positive voltage V_S that rises as a slowly varying parameter as the stored energy increases.

18.4.2 The diode bridge rectifier

The simplest (and by far the most prevalent) recharge circuit is a simple full-bridge diode rectifier. This circuit is shown in Fig. 18.4a. Full-bridge rectifiers conduct current when the transducer voltage magnitude |v(t)| exceeds the bus voltage V_S plus the diode conduction voltage V_d, and disconnects the transducer from storage otherwise. Assuming V_S is a slowly varying parameter, the rectifier imposes a nonlinear relationship between voltage and current, governed by

$\left|\begin{array}{c}\hfill \mathit{v}\left(\mathit{t}\right)\hfill \\ \hfill \mathit{v}\left(\mathit{t}\right)\hfill \end{array}\right|\begin{array}{c}\hfill <{\mathit{V}}_{\mathit{s}}+{\mathit{V}}_{\mathit{d}}\hfill \\ \hfill ={\mathit{V}}_{\mathit{s}}+{\mathit{V}}_{\mathit{d}}\hfill \end{array}\begin{array}{c}\hfill :\hfill \\ \hfill :\hfill \end{array}\begin{array}{c}\hfill \mathit{i}\left(\mathit{t}\right)=0\hfill \\ \hfill \mathit{i}\left(\mathit{t}\right)\ne 0\hfill \end{array}$ [18.47]

[18.47]

As such, the rectifier imposes an amplitude saturation on v(t), and draws the current i(t) necessary to ensure this condition. This is illustrated in Fig. 18.4b.

In using a diode bridge rectifier, one would ideally use a capacitor or inductor to tune the harvester. Then, using the method of harmonic balance, an approximately equivalent input resistance R_eq (i.e., a resistance which absorbs the same energy per cycle) can be found for the electronics, which is shown in Fig. 18.4c. The specific expression for R_eq is complicated, but the general behavior is that R_eq for a tuned harvester depends on the ratios (V_s + V_d)/e₀ and R₀ = Z(jω₀ ). As shown in Fig. 18.4d, a fraction of this equivalent resistance (R_d) accounts for the power dissipated by the current as it is transmitted through the rectifier. The remainder of the resistance (R) represents the power absorbed by the storage system. For a tuned harvester, power transferral is optimized with the matching condition

$\mathit{R}={\mathit{R}}_0+{\mathit{R}}_{\mathit{d}}$ [18.48]

[18.48]

The problem with the diode bridge rectifier is that R_eq varies over time, as V_s rises, and cannot be controlled. For example, consider the scenario in which the recharge process is initiated at V_s ≈ 0, and assume a₀ is sufficiently large such that e₀ > V_s + V_d (otherwise, the rectifier will not conduct at all). Then over time, as V_s rises due to the recharge operation, R_eq will increase from a low initial value at V_s ≈ 0, to infinity when V_s = e₀ – V_d. This has two specific drawbacks:

These drawbacks can limit the usage of the diode bridge rectifier as a recharge circuit. However, for a given application, it makes sense to first assess the typical magnitudes of e₀, and the desired recharge rate, and to see if the diode bridge rectifier will be sufficient. It may also be worth considering a redesign of the harvester, expressly to better cope with these limitations. This is because, even though its performance is limited, the full-bridge rectifier does have the distinct advantage of simplicity. In particular, it has no transistors, and does not require control. As such, it is definitely the preferred option for applications where its limitations can be tolerated.

18.4.3 Pulse-width-modulation (PWM)-controlled DC/DC converters

Various techniques have been proposed to overcome the aforementioned drawbacks of the full-bridge rectifier. The central issue is the desire for a recharge circuit with an input impedance that looks like the optimal equivalent resistance R derived in the previous section. Then, by placing a passive reactive component (i.e., an inductor or capacitor) in parallel with this recharge circuit, the input admittance of electronics will look equivalent to one of the circuits in Fig. 18.2, resulting in optimal absorption.

One way to accomplish this is to insert a switch-mode DC/DC converter between the bridge rectifier and the storage bus.^21,22 One example of such an implementation, shown in Fig. 18.5a, consists of a MOSFET and diode, together with an inductor, and two capacitors for current filtering. It is controlled by switching the MOSFET on and off, in a PWM cycle. When the MOSFET is switched on, it conducts current, while the diode does not. Alternatively, when the MOSFET is switched off, it appears as an open circuit, and the inductor current flows through the diode. The idea is that by switching the MOSFET on and off periodically at frequency f_s = 1/t_s, and with duty ratio d, the converter pumps energy from the transducer to the storage bus, using the inductor as an intermediary. Standard procedure is to operate the circuit at a switching frequency which is much higher than the frequency of excitation.

The particular converter shown is called a ‘buck-boost’ converter, because it can transfer power from V_R to V_S, irrespective of which voltage is the larger. This converter is particularly useful for energy harvesting because, when it is operated in the discontinuous conduction regime (i.e., when the inductor current drops to zero before the end of each PWM cycle), its input impedance is approximately linear and resistive at frequencies well below f_s²³. More specifically, it turns out that at the excitation frequency, (ω₀ the input impedance from the point of view of transducer can be expressed approximately as

${\mathit{R}}_{\text{eq}}\approx \frac{\mathit{r}}{{\mathit{d}}^2}$ [18.49]

[18.49]

where r is a constant that depends primarily on f_s and L, but is highly insensitive to V_S. As such, the input resistance of the converter can be specified indirectly through proper choice of d, and stays approximately constant as the storage system recharges.²⁴

The use of a DC/DC converter allows for power to be absorbed from the harvester in a manner consistent with the theory in the previous section, i.e., like a resistance. Consequently, the use of such converters in parallel with a tuned inductor or capacitor effectively realizes the linear input admittances in Fig. 18.2. The conductive losses in the MOSFET and diodes can also be taken into account in a manner similar to the way they were for the diode bridge rectifier in the previous subsection, to optimize R_eq.

However, there are also disadvantages to using DC/DC converters. Most significantly, every time the MOSFET is switched on, its gate capacitance must be charged, and this requires a small amount of energy. Generally, it is not easy to recover this energy when the MOSFET is switched off again. Thus, the gate drive circuit must draw power off of the power bus to run the switching circuit, at a rate that increases with f_s. Consequently, f_s cannot be too high or else the gating losses become prohibitively high. On the other hand, f_s must be kept well above the bandwidth of the harvester in order for the system to operate as intended. As P_opt is made smaller, this tradeoff becomes tighter and tighter, and there is ultimately a critical power level below which a given converter is no longer viable. However, the power level where this occurs depends heavily on the specifics of the hardware used, and significant progress is still being made to scale down DC/DC converters to ever-lower power levels for VEH systems.^25,26

18.4.4 Synchronized switching circuits

Piezoelectric harvesters possess a significant internal capacitance, and consequently a supplemental inductor L must be attached to the harvester terminals to electrically tune it such that Im{G_i(jω₀)} = 0. Often, the magnitude of L can be prohibitively large, with values in the hundreds or even thousands of henries. This makes passive electrical tuning of the harvester impossible. There are several approaches that could be used to address this issue. One is to use a power-electronic converter with bidirectional energy flow capability, in which case both the real and reactive energy flows are facilitated by the converter.²⁷ This approach is theoretically advantageous, but the converter circuitry (and its control) are more complicated and exhibit higher parasitic losses. Another approach is to realize the inductor artificially, using a gyrator circuit. Such circuits use operational amplifiers and capacitors to create the illusion of an inductor. However, it is difficult to make such circuits acceptably efficient at small power scales. A third option, which has received significant attention for energy harvesting applications, involves the use of a synchronized switching circuit.

Figure 18.6 illustrates the insertion of a synchronized switch between a piezoelectric transducer and the DC/DC converter circuit discussed in the previous subsection. This particular circuit is called a parallel synchronized switch harvesting inductor (SSHI) circuit, which was originally proposed for these types of applications in Reference 28. Typically, the inductor used in this circuit is very small, many orders of magnitude lower than that necessary to electrically tune the harvester. The switch in the circuit is realized with a MOSFET and diode. Unlike the PWM-controlled switch used in the DC/DC converter, the switch in the SSHI switch is only triggered twice in a vibratory period of the harvester. Each time, it induces a near-instantaneous polarity reversal in v, thus causing it to have zero crossings at desired times. Through proper switch timing, the phase of the fundamental harmonic of v(t) can be forced to be the same as it would with passive tuning techniques.

Synchronized switching circuits have been shown to be highly effective at greatly enhancing the power generation from monochromatically-excited piezoelectric harvesters.^29–33 Especially near resonance, the optimal switch trigger times are nearly coincident with the displacement peaks of the harvester, and consequently, the switching actions can be instigated using a displacement sensor and a simple peak-detection circuit. In contrast to the PWM-controlled switches, the gating losses associated with the SSHI circuits are negligible, because they are only triggered twice in a vibratory cycle. Meanwhile, they have two drawbacks that can in some circumstances reduce their efficiency. First, they conduct current in a short-period pulse each time the switch is triggered, and this can lead to significant conductive dissipation. Second, they introduce significant harmonics into the periodic response, and much of the energy transferred to these higher frequencies is not recovered. More detailed modeling of the losses in SSHI systems can be used to compensate for these losses.³²

18.5.1 Frequency-robust monochromatic energy harvesting

Consider again the optimal power generation limit for a monochromatically-excited SDOF harvester, from Equation [18.39]. As we have discussed, this limit is raised by making c as small as possible, resulting in the general practice of making harvesters very lightly-damped. However, when passive techniques are used to tune the harvester (either mechanically or electrically) this also makes the harvester’s power generation capability extremely narrowband, which can have undesirable ramifications if the frequency ω₀ is uncertain. One remedy for this is to deliberately introduce conservative nonlinearities (such as nonlinear stiffnesses) into the electromechanical dynamics of the harvester. If done properly, this can have the effect of creating high-amplitude open-circuit voltages over a wider range of ω₀ values. This has led to a flurry of recent activity on nonlinear energy harvesting, as recently summarized in Reference 34.

Rather than engineer mechanical nonlinearities into a harvester to maintain high power levels as ω₀ is varied, it can be alternatively done by adapting Y(jω) such that the condition Y(jω₀) = G_i^–1 (–jω₀) is maintained as ω₀ slowly varies. This has been investigated recently in Reference 35. Because the reactive power is actively regulated by the electronics, the rectifier and DC/DC converter shown in Fig. 18.5 cannot be used, and must be replaced with a bidirectional converter. This is challenging because bidirectional converters require more elaborate switching control, and generally exhibit higher parasitic losses.

For piezoelectric applications employing SSHI circuits, adaptive tuning can be achieved merely by adapting the phase angle at which the switch is triggered. This has recently been examined in Reference 36.

18.5.2 Broadband stochastic energy harvesting

In many applications, the disturbance a cannot be presumed to be monochromatic, and is much more appropriately modeled as a stochastic process, possibly with a low quality factor. For such cases, determination of the optimal energy harvesting circuit is more challenging because the system must harvest energy from a continuous band of frequencies simultaneously. It should be noted that this problem is fundamentally different from the aforementioned problem in which the disturbance is assumed to be monochromatic but with uncertain frequency. Indeed, it may be the case that a system optimized for one of the two problems performs poorly when applied to the other. A number of recent studies^37,38 have investigated stochastic analogies of the development presented here, in which an equivalent circuit for Y(s) is assumed to be a simple R – C and R – L shunt, and a is assumed to be white noise.

However, it has also been shown in Reference 20 that for white-noise-excited harvesters, the optimal power generation is achievable only with active control, i.e., with a bidirectional converter such as a standard H-bridge drive. Furthermore, the determination of the optimal power extraction can be framed as a feedback optimization problem, with the optimal feedback being determined via the solution to an associated linear-quadratic-Gaussian (LQG) control problem. In Reference 39, this analysis is generalized to the case in which a(t) is colored noise of arbitrary quality factor.

18.5.3 More efficient power electronics

It is almost beyond debate that in the near future, the innovations that will have the largest impact on the state-of-the-art in VEH systems will be those that enhance the efficiency of recharge circuits. This is especially true for power levels below about 100 μW. Efficiency improvements may be made by finding ways to reduce the dissipation of energy as it flows from the transducer terminals to storage, especially in the semiconductor components of the circuit. For example, one simple and well-known technique for reducing conductive losses is to replace the rectifier diodes with MOSFETs which are actively switched to behave like diodes. Although some power is required to actively switch these MOSFETs, they dissipate much less energy than diodes at low current levels, and as such, the net energy consumption is reduced. Another well-known approach to reduce losses is to use DC/DC converters that exhibit soft switching, i.e., MOSFETs that are only switched on or off when their current is zero. This reduces the transition losses associated with switching, actions. There are also opportunities to reduce losses in the gate drive circuits for MOSFETs, through the recapture and reuse of the gating energy. This enables the effective use of DC/DC converters at much lower power levels.

Beyond these ideas, which may be viewed as incremental advancements on existing electronics topologies, it is also likely that in very low-power regimes, custom solid-state converter topologies and transmission techniques will be required. For example, energy harvesting circuits at MEMS scale can be built using CMOS-switched capacitors and without inductors (i.e., charge pumps), which can greatly reduce conduction losses. There has been significant advancement in this area over the last ten years, which is summarized in a recent survey by Chao.¹⁴ These results suggest that the next decade will see a dramatic expansion in the spectrum of power scales over which VEH systems are viable.