IEEE Circuits and Systems Magazine - Q3 2018 - 32

The Direct Energy Transfer or Standard Energy Transfer
approach was the first attempt to implement
a self-powered energy harvester.

To optimize the harvested energy over a wide frequency band, the complete impedance matching of the
piezoelectric impedance must be achieved. To analytically evaluate the power loss due to incomplete impedance matching, the power extraction efficiency (7) is
expressed by considering the vibrational frequency
~. The two power expressions are obtained considering, in the first case, the complete matching expression
Pcomplete, and, in the second case, just the modulus of the
matching impedance Ppartial . A similar analysis has been
already conducted by Brufau-Penella and Puig-Vidal in
[59], who achieved a different result for the power extraction efficiency.

The Thevenin equivalent circuit representation, comprised by the electromechanical impedance Z m and the
load impedance Z L, is shown in Fig. 4.
Assuming that the input excitation force can be modeled as a sinusoidal voltage source with an amplitude
Vm, and that the model itself has only an electronic nature, the average power on the load P can be expressed
considering the input voltage Vm, the transducer impedance Z m, the load impedance Z L and R L the load resistance [59]:
P=

Vm
Zm + Z L

2

RL
2

(8)

Considering the matching impedance as the load value, Z L = Z *m = R m - jX m, and substituting it in Eq. (8) the
resulting complete power expression can be derived as
Vm

Pcomplete =

Zm
ZL

(dBm)

1
0.8
0.6
0.4
0.2
0
45

Power Expressions
Pcomplete
Ppartial
50

55
60
65
Vibrational Frequency (Hz)
(a)

70

Power Extraction Efficiency

2

Vm
Rm
=
2
8R m

(9)

Rm
2
2
Vm
R
= m 2
2
2
4 R m + R m R 2m + X m
+ Xm

Ppartial =

Vm
Zm + Zm

2

(10)

Therefore, the power extraction efficiency expression
can be further deduced to be:
50

55
60
65
Vibrational Frequency (Hz)
(b)

70

Figure 5. numerical estimation of (a) the complete and partial power expression and (b) the power extraction efficiency.
32

2

In [55] and [59] a similar expression can be found.
The power expression in [55] is independent from the
vibrational frequency, while in Eq. (9) the value of the
resistance R m, and consequently of the average power,
can be re-expressed considering the vibrational frequency. Although the value of R m and consequently of
Pcomplete, according to the estimations, stays almost constant when considering different vibrational frequencies, as shown in Fig. 5(a).
The partial power expression, defined as the average power supplied to the load when its value is
equal to the modulus of piezoelectric impedance,
Z L = Z m = R 2m + X 2m , can be expressed as:

Figure 4. Thevenin equivalent circuit representation.

10
5
0
-5
-10
-15
-20
-25
45

Vm
Z m + Z *m

Ieee cIRcuITs anD sysTems magazIne

h=

=

Ppartial
Pcompleted
2
m

R + Rm

2R 2m
2
Rm
+ X 2m + X 2m

(11)

Combining expressions (3), (4), and (11), it is possible to analytically express h as function of the vibrational frequency.
THIRD quaRTeR 2018



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