Piezoelectric fans and their application In electronics cooling

Piezoelectric fans seem to represent an example of research and development that has culminated in a product that is deceptively simple. Although piezoelectric technology is capable of producing rotary motion, the fans operate quite differently from rotary fans, as they generate airflow with vibrating cantilevers instead of spinning blades.

Piezoelectric, as derived from Greek root words, means pressure and electricity. There are certain substances, both naturally occurring and man-made, which will produce an electric charge from a change in dimension and vice-versa. Such a device is known as a piezoelectric transducer (PZT), which is the prime mover of a piezoelectric fan. When electric power, such as AC voltage at 60 Hz is applied, it causes the PZT to flex back and forth, also at 60 Hz.



The magnitude of this motion is very tiny, so to amplify it, a flexible shim or cantilever, such as a sheet of Mylar, is attached and tuned to resonate at the electrical input frequency. Since piezoelectric fans must vibrate, they must use a pulsating or alternating current (AC) power source. Standard 120 V, 60 Hz electricity, just as it is delivered from the power company, is ideal for this application, since it requires no conversion.

[If direct current (DC), such as in battery-operated devices, is the power source, then an inverter circuit must be employed to produce an AC output. An inverter may be embodied in a small circuit board and is commercially available with frequency ranges from 50 to 450Hz.]

Driving the fan at resonance minimizes the power consumption of the fan while providing maximum tip deflection. The cantilever is tuned to resonate at a particular frequency by adjusting its length or thickness. The PZT itself also has a resonant frequency, so the simplistic concept of adjusting only the cantilever dimensions to suit any frequency may still not yield optimum performance. (Conceivably, tuning the electrical input frequency to match existing cantilever dimensions may work, though with the same caveat, that the resonant frequencies of all the components must match, within reason.

Applications for piezoelectric fans are just in their infancy and could really thrive through the imagination of designers. This article, which originally appeared in the April 2011 issue of Qpedia (published by Advanced Thermal Solutions, Inc. and used with permission here) explores the principles, construction, implementation, and installation of piezoelectric fans.


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Performing efficient arctangent approximation

Fast and accurate methods for computing the arctangent of a complex number x = I + jQ have been the subject of extensive study because estimating the angle θ of a complex value has so many applications in the field of signal processing. The angle of x is defined as θ = tan-1(Q/I).

Practitioners interested in computing high speed (minimum computations) arctangents typically use look-up tables where the value Q/I specifies a memory address in programmable read-only memory (PROM) containing an approximation of angle θ.

Those folks interested in enhanced precision implement compute-intensive high-order algebraic polynomials, where Chebyshev polynomials seem to be more popular than Taylor series, to approximate angle θ.

(Unfortunately, because it is such a non-linear function, the arctangent is resistant to accurate reasonable-length polynomial approximations. So we end up choosing the least undesirable method for computing arctangents.)

Here’s another contender in the arctangent approximation race that uses neither look-up tables nor high-order polynomials. We can estimate the angle θ in radians, of x = I + jQ using the following approximation



where –1 ≥Q/I ≤1. That is, θ is in the range –45 to +45 degrees (–π/4 ≥θ≤+π/4 radians). Equation (13–107) has surprisingly good performance, particularly for a 90 degrees (π/2 radians) angle range.

Figure 13–59 below shows the maximum error is 0.26 degrees using Eq. (13–107) when the true angle θ is within the angular range of –45 to +45 degrees. A nice feature of this θ computation is that it can be written as:



eliminating Eq. (13–107)’s Q/I division operation, at the expense of two additional multiplies.



Figure 13–59. Estimated angle theta error in degrees.

Another attribute of Eq. (13–108) is that a single multiply can be eliminated with binary right shifts. The product 0.28125Q2 is equal to (1/4+1/32)Q2, so we can implement the product by adding Q2 shifted right by two bits to Q2 shifted right by five bits.

This arctangent scheme may be useful in a digital receiver application where I2 and Q2 have been previously computed in conjunction with an AM (amplitude modulation) demodulation process or envelope detection associated with automatic gain control (AGC).

We can extend the angle range over which our approximation operates. If we break up a circle into eight 45 degree octants, with the first octant being 0 to 45 degrees, we can compute the arctangent of a complex number residing in any octant. We do this by using the rotational symmetry properties of the arc tangent:



These properties allow us to create Table 13-6 below.




Table 13–6 Octant Location versus Arctangent Expressions

So we have to check the signs of Q and I, and see if | Q | > | I |, to determine the octant location, and then use the appropriate approximation in Table 13–6. The maximum angle approximation error is 0.26 degrees for all octants.

When θ is in the 5th octant, the above algorithm will yield a θ’ that’s more positive than +π radians. If we need to keep the θ’ estimate in the range of –π to +π, we can rotate any θ residing in the 5th quadrant +π/4 radians (45 degrees), by multiplying (I +jQ) by (1 +j), placing it in the 6th octant.

That multiplication yields new real and imaginary parts defined as



Then the 5th octant θ’ is estimated using I’ and Q’ with



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