Electrical modelling

The Hybrid piezo-pyro effect is almost close to the algebraic sum of the pure piezo and pure pyro effects. Therefore, we can say the total charges induced by the hybrid effect

Q
= Q _{t} + Q _{M
}
1

Pyroelectric effect induced charge, voltage and energy

The pyroelectric effect induced charges can be calculated using Eq. 1.

Q_{t
=} ρΔTA
2

where ΔT is the temperature change and A is the surface area of the harvester. Assuming that during one cycle of cooling or heating process, pyroelectric effect induced charge is QT. In a random n th cycle, the charge will be distributed between the internal and external capacitors Eq 2

ΔQ
+ Q_{p }(n+1) +Q_{E}(n-1) = Q_{P} (n) + Q_{E}(n)
3

where ΔQ is the charge induced in the nth cycle; QP and QE are the charges stored in external capacitor (CE) and internal capacitor (Cp), respectively, typically, CE Cp.

Q_{P
}(n) = V(n) C_{P
}
4

Q_{E
}(n) = V(n) C_{E}

Q_{P
}(n-1) = V (n-1) C_{P}

Q_{E
}(n-1) = V (n-1) C_{E}

where
V is the voltage across CE. It should be noted that the sign in third
equation in Eq. 3 depends on the signs of temperature change ΔT.
That is, if both the n^{th} and n1^{th} cycles are
cooling or heating process, the current generated will be in the same
direction; however, if the n^{th} cycle is cooling (heating)
but the n-1^{th} cycle is heating (cooling), then the current
generated in these two cycles will be opposite and a negative sign
will be used. By substituting Eq. 4 in Eq.3, we can get (Batra et
al., 2011, Cuadras et al., 2010)

V_{n}
= +()V_{n-1 }=
+ () V_{n-1}
5

Assume
V_{1}=0 and with CE >>Cp, Eq. 4 reduces to

V_{n}
= _{ } =
6

where
Q_{n} is the accumulated charges on CE after n^{th}
thermal cycle.

The energy stored in CE is thus,

E_{n}
= C_{e }(^{
} )^{2
}^{7}

Piezoelectric effect induced charge, voltage and energy

The dynamic motion of the cantilever can be described using the mass-spring-damper equation

m_{t}
z ̈+ bz + k_{1}Z
+ k_{2}
Z^{3 }= A
cos(ωt )
8

where
m_{t} , b, k_{1}, k_{2} are total effective
mass ,damping coefficient, linear stiffness, and nonlinear stiffness
respectively. z(t) is the tip displacement. The first derivative of
the z with respect to the time t represents the velocity, and the
second derivative denotes the acceleration. A is the amplitude of the
external periodic driving force, and ω is the driving frequency.
When ω= ω0 (ω0 being the natural frequency of the cantilever; ω =
here we only analyse the fundamental resonance), the deformation
of the cantilever reaches to the maximum. Here we set the driving
frequency same as the nature frequency of the cantilever to ensure an
optimal match between the excitation and the beam. Under the external
driving, the cantilever vibrates, the deflected cantilever can be
treated as an arc. The curvature 1/r (r is the radius) of this arc is
expressed with respect to the tip deflection z

= 9

The axial strain on the surface of the piezoelectric layer along the x-axis can be obtained in terms of curvature and tip deflection

є_{x(t)
} =
10

The axial stress σ in the piezoelectric layer is derived through the stress-strain relation, that is

σx (t)= Yε_{x}(t)
11

The piezoelectric constitutive equations relate four field variables stress components Σ, strain components S, electric field components E, and the electric displacement components D, which can be described as [] = | | [], We shall focus on the generated electric displacement D from the mechanical stress Σ. Assuming no external electric field is applied to the cantilever, D= dΣ, d being the piezoelectric constant. The electrical charges generated from a tip deflection z(t) can therefore be obtained.

Q_{M
=} ѡLd_{33}σx
(t) = (t)
12

By substituting the Eq 12 and Eq 6 in Eq 1 we will get the total charge induced by the piezo pyro effect,

Q
= Q _{t}
+ Q _{M }=
ρA∑∣ΔT∣
+ ѡLd_{33}σx
(t) _{
}

Similarly, by Eq 6 and Eq 7 we can calculate the voltage and the energy.