The relation about FOM and power extraction improvement

I saw some paper,their comparison is FOM or Power Extraction Improvement(called PEI here),and they are related,PEI=FOM*100,when FOM*100,the PEI value will usually bigger than 100%,take 126% or 425% for example.I want to ask,why will most authors multiply 100 by FOM?why don't they just multiply 10?

The FOM's definition is $\frac{P_{o}}{CV^{2}f }$ ,$P_o$ is the output power,and the $C$ is $C_{pz}$,the capacitor of the PZT;$V$=$V_{PZ(OC)}$,it is the voltage that PZT produce;$f$=$f_{vib}$,it is the PZT vibration frequency.

For first picture,i just want to show that the PEI are really bigger than 100% often;For second picture,i just want to let you know what is the meaning of $P_{o}$

• Seems suspicious that “this work” has both a peak efficiency of 95.4% and a PEI of 681% suggests the average power is very low due to extraction of crest factor off resonance rather RMS integration. So they let the kinetic energy build up then extract bigger peaks but still don’t compare RMS power of every competitor – Tony Stewart EE75 Jun 28 '19 at 22:23

I'm not familiar with the nomenclature in this area, but I've got an idea where things are coming from. The journey begins with a question of "How much (electrical) power should I expect from my piezo generator?

A piezo generator is roughly a capacitor. If I excite the piezo generator at a given frequency, I will see an AC voltage $V_{PZ_{OC}}$. I might imagine that my system could be modeled as a current source in parallel with a capacitor, making imaginary power. My circuit might look something like the following:

simulate this circuit – Schematic created using CircuitLab

In this case, I would say the complex power could be characterized as: $$|P|= \dfrac{V^2}{Z}=V_{PZ_{OC}}^2(\dfrac{1}{2\pi fC})^{-1}=2\pi fCV_{PZ_{OC}}^2$$

This is the magnitude of the energy required to create an open-circuit voltage across an ideal capacitor. If this was a perfectly adequate description of a piezo generator, then you probably could never recover more energy than this value. If nothing else, this does provide a useful baseline expectation of system performance that should help reduce the impact of physical size differences.

Your figure of merit (FOM) is basically a ratio of recovered power to this "ideal baseline" case. The multiplication by 100 reflects the conversion from a ratio to a percent. $$FOM=\dfrac{P_{measured}}{P_{baseline}}=\dfrac{P_{measured}}{2\pi fCV_{PZ_{OC}}^2}$$

In your paper, they advertised a FOM of 4.93 at resonance, meaning their measured power output was 4.93 times higher than the baseline value. Saying it another way, their FOM was 493%

I'm not sure where the $2\pi$ got off to in the FOM

• Why is that $Z$=$\frac{1}{2 \pi f C}$? – XM551 Apr 4 '18 at 4:54
• @XM551 It's the impedance of the capacitance presented by the piezo element. – W5VO Apr 4 '18 at 5:21
• but isn't it $Z=-j \frac{1}{2 \pi f C}$? – XM551 Apr 4 '18 at 5:27
• @XM551 Sure, and I probably put an |abs| in there somewhere to get rid of it. The point of this FOM is to compare magnitudes of a baseline power to measured power. – W5VO Apr 4 '18 at 5:48