(Uwe Hansen, Dept. of Physics, Indiana State Univ., Terre Haute, IN 47809, and P.L. Hoekje, Dept. of Physics and Astronomy, Baldwin-Wallace College, Berea, OH 44017) Presented at Session 3aED Education in Acoustics: Demos 2000 of the 136th Meeting of the Acoustical Society of America
Inexpensive impedance head for classroom use
The input impedance is defined as Zin(f) = p / u, where p
is the pressure amplitude response to the applied excitation volume velocity
u, which is oscillating at frequency f. To measure the
input impedance of tubes and musical wind instruments easily and inexpensively
in the classroom and for demonstrations, consider an adaptation of a design
introduced by Benade and Ibisi for an impedance head(1),(2) that uses low cost components yet is suitable
for research work.
As a sound source, this design uses a piezoelectric "buzzer"
disk, which has a high acoustical impedance and is therefore not strongly
affected by the air column's own resonances. Buzzers with housings such
as Radio Shack 273-073 or 273-064 are easy to find and have leads attached,
but they must be disassembled in order to remove the brass disk with the
piezoelectric element. Or, the disk itself can be obtained from electronics
supply houses such as Mouser or Digi-Key.
The piezodisk displacement is proportional to the voltage applied between
the brass disk and the silvered top of the electrode; the disk's center
moves approximately 10-7 meters/volt.(3)
The disk's useful operating frequency range is from DC up to a little below
its first resonance frequency, which is often between 2 kHz and 6 kHz and
may have a Q between 10 and 100. Below this frequency, the velocity amplitude
of the disk is proportional to frequency for a given voltage amplitude,
and this needs to be compensated by a -6dB/octave equalization either in
the applied voltage or in the measured response. The piezodisk can be driven
directly from the output of any audio device, including either the "line
out" or the "speaker out" of a computer sound card. Because
it also has high electrical impedance, it won't draw very much current,
but voltages of more than 30V should be avoided in order to avoid re-polarizing
the crystal and changing its sensitivity(4).
To measure the pressure response, use an electret microphone element,
which has flat frequency response and can be powered by low voltages. A
suitable one will have two terminals, such as the RS 270-090. Most computers
and sound cards with microphone inputs have built-in facility for powering
a microphone; there are a few different schemes for implementing this, but
the most common one simply has the signal and power delivered on the tip
of the 1/8" mini-plug.(5) Both the AC-coupling
on the sound card input and an internal leak around the sides of the electret
diaphragm reduce the response at very low frequencies, below about 10 Hz.
The piezodisk and microphone are inexpensive enough to allow custom-building of impedance heads for a wide variety of instruments, or to allow a class to build several impedance heads that they can use in their own measurements on common water pipe. A good general-purpose head can be built from a coupler for 3/4" CPVC plumbing pipe (technically, this is called 3/4" CTS, for Copper Tubing Size), as detailed in Figure 1. Cut off a short piece of 3/4" tubing about 0.1" less than half the length of the coupler, and use the appropriate solvent pipe cement to glue this in place so that it is just below flush with the end of the coupler. When this is dry, a hole may be drilled to fit the microphone(6). A 5-minute epoxy may be used to attach the piezodisk and the microphone, but be sure not to get any epoxy on the felt that covers the small microphone aperture.
Sound editing and analysis software
With the boom in computer multimedia, computers with sound recording
and playback capabilities have become common, and software for editing waveforms
and for spectral analysis have become common and inexpensive. Some of this
inexpensive software is listed further below. For this demonstration, two
tools called CoolEdit and WavPrism will be used.
The general approach is to have the computer both provide the excitation
signal to the piezodisk as well as to record the microphone response. To
do this requires that the sound card be able to operate in "full-duplex
mode", which is true of most recent models. If not, the excitation
signal could be recorded onto a tape recorder and played back that way.
CoolEdit has sound generation facilities, including particularly the ability
to generate swept tones. However, though self-contained, this is not an
especially satisfying way to proceed. Better is to use a "Schroeder
chirp(7)", which sounds like a swept
sine, but in fact is composed by summing individual sinusoids that are running
continuously. The phases have been adjusted in a way that effectively minimizes
the ratio of (peak amplitude)/(RMS amplitude)(8).
Each sinusoid has a frequency at the center of its corresponding frequency
bin when analyzed using an FFT, for example, with Cool Edit, or WavPrism(9). Therefore, it is best to use a "rectangular"
analysis window.
To start, CoolEdit should be opened, and then a chirp file loaded. For demonstrations or exploration, CoolEdit can be placed in Loop mode and the sound played continuously. Figure 2 shows a CoolEdit screen in which 5 seconds of a repeating chirp have been loaded. This particular chirp has been equalized by -6dB/octave in order to compensate for the piezodisk response, so the amplitude varies through each chirp. Equalization can be done in CoolEdit.
To monitor the response, a second window may be opened. For recording
purposes, a new "instance" of CoolEdit may be started while the
original window is still playing the chirp. Or, for demonstrations, a real-time
analyzer such as WavPrism may be started. Most inexpensive sound cards require
that the sampling frequencies of both input and output be the same. In any
case, the same number of points should be used for the FFT analysis as are
used in each cycle of the repeating chirp, and a "rectangular"
window used, if available. Figure 3 shows a WavPrism screen for the same
chirp as in Figure 2, fed directly from the output back to the input. At
the top is the waveform, with the chirp nature clearly apparent. At the
bottom is the spectrum, illustrating the band of frequencies present, and
the -6dB/octave equalization.
Figures 5 and 6 show the response of a tube attached to the impedance head, when the piezodisk is driven by the same chirp. The MS-Win Volume Control should be used to control levels and to ensure that only the microphone signal is connected via the internal mixer to the input. Figure 5 illustrates the characteristic 1-3-5... sequence of the closed-open tube resonances, while in Figure 6 the resonances are in the sequence 2-4-6, equivalent to 1-2-3 an octave higher. Note that the spectrum is only driven between 100 and 3000 Hz; outside of this range, only noise is detected.
When a trombone bell is added to the end of the tube, all of the resonances move to lower frequencies, and become more closely spaced, because the tube is longer. This is shown in Figure 6. In fact, the waves penetrate poorly into the bell at low frequencies, but reach all the way down to the end at the "cutoff frequency" of about 1000 Hz. So, the resonances accordingly become more closely spaced at higher frequencies. These high frequency waves also penetrate beyond the bell, so they can't reinforce the standing wave, and therefore the resonance peaks are not as strong.
Lesson ideas!
Inexpensive software tools for sound editing and analysis
For computers running MS-Windows:
For Macintosh computers:
Notes
1. M.I. Ibisi and A.H. Benade, "Impedance and impulse response measurements using low cost components," J. Acoust. Soc. Am. Suppl. 1 63, S63 (1982).
2. A.H. Benade and M.I. Ibisi, "Survey of impedance methods and a new piezo-disk-driven impedance head for air columns," J. Acoust. Soc. Am. 81, 1152-1167 (1987).
3. As an optics demonstration, a small chip of a quality mirror can be waxed onto the disk and used as one arm of a Michelson interferometer and moved by applying a low frequency oscillating voltage.
4. Physically, however, these thin crystals will typically survive a few hundred volts.
5. Contact phoekje@bw.edu for more information about other configurations or about 3-terminal microphones.
6. Be sure to use appropriate ventilation; if any sanding or filing is planned, a breathing filter also should be used, as CPVC dust does not decompose in the lungs.
7. M. Schroeder, "Synthesis of low-peak-factor signals and binary sequences with low autocorrelation," IEEE Trans. Inf. Theory 16, 85-89 (1970).
8. A more general approach is described by Andrew Horner and James Beauchamp, "A genetic algorithm-based method for synthesis of low peak amplitude signals," J. Acoust. Soc. Am. 99, 433-443 (1996).
9. Contact phoekje@bw.edu for more information or for a sample file. Some additional materials may be available at the website http://www.bw.edu/~phoekje.