3.11 Harmonic Vibration
Consider the finite piezoelectric semiconductor rod shown in Fig. 3.17. The left end
is fixed. The right end is under the action of a time-harmonic extensional force
[8]. Both ends are electroded with ohmic contacts. There is no interface nonlinearity,
and the interface resistance between the crystal and electrodes is negligible at the two
ends. The two end electrodes are connected by an output circuit whose impedance is
Z in time-harmonic motions. Such a rod can operate as an energy harvester for
converting mechanical energy to electrical energy. As an energy harvester, the
screening of the effective polarization charge by the mobile charge in a semiconductor is undesirable. It reduces the piezoelectrically generated output voltage by the
mechanical load.
The governing equations are from Eqs. (1.1) 1 with i ¼ 3, (1.10), (3.2), (1.9) with
i ¼ 3, and (3.6):
∂T 3
∂x 3
¼ ρ € u 3 ,
∂D 3
∂x 3
¼ q Δp À Δn
ð
Þ ,
ð3:131Þ
q
∂
∂t
Δp
ð Þ ¼ À
∂J
p
3
∂x 3
,
q
∂
∂t
Δn
ð Þ ¼
∂J
n
3
∂x 3
:
ð3:132Þ
T 3 ¼ c 33 S 3 À e 33 E 3 ,
D 3 ¼ e 33 S 3 þ ε 33 E 3 ,
ð3:133Þ
J
p
3 ¼ qp 0 μ
p
33 E 3 À qD
p
33
∂ Δp
ð Þ
∂x 3
,
J
n
3 ¼ qn 0 μ
n
33 E 3 þ qD
n
33
∂ Δn
ð Þ
∂x 3
,
ð3:134Þ
Fig. 3.17 A piezoelectric
semiconductor rod under a
time-harmonic force
3.11 Harmonic Vibration
61