acoustic impedance measurements acoustic impedance measurements presented by: brendan sullivan june...

Acoustic Impedance Acoustic Impedance MeasurementsMeasurements

Presented by:Brendan Sullivan

June 23, 2008

Agenda for TodayAgenda for Today What acoustic impedance is and why we are interested in

it

Physical interpretations of acoustic impedance Notes on an instrument Electrical circuits

How to measure acoustic impedance First, in General Mainly, in a trumpet Phase Sensitive

Results No general theory, but some interesting data

Future Plans

What is Acoustic Impedance?What is Acoustic Impedance?

P(x) U(x)

Z(x) =

Air Pressure

Longitudinal Particle Velocity

Specific Acoustic Impedance

Units are Acoustical Ohms (Pa-s/m), or Ώ for short.

What What ReallyReally is Acoustic is Acoustic Impedance?Impedance?

Take a look at this typical impedance spectrum:

Image modified from J. Backus, J. Acoust. Soc. Am. 54, 470 (54)

Blue lines (maxima) are accessible frequencies

Red lines (minima) are inaccessible frequencies

The first peak is the fundamental

Subsequent peaks are harmonicsHarmonics decrease in

amplitude – just as in the overtones of an instrument

Ohms? Impedance? This sounds Ohms? Impedance? This sounds like a circuit...like a circuit...

...because it is!

Any acoustical system creates an acoustical circuitParts of the acoustical system behave exactly like the

components of a circuit

Image modified from J. Backus, J. Acoust. Soc. Am. 54, 470 (54)

Zi – Mouthpiece input impedance

Z – Mouthpiece output Impedance

L – The inductance, or the area between the cup and tube

R, C – Values determined by geometry of mouthpiece

The Circuit Components:

How Do We Measure Impedance?How Do We Measure Impedance?

P(x) U(x)

Z(x) =

Pressure Microphone

Time-Integrated DifferentialPressure Microphone

Two quantities to measure: pressure (P) and particle velocity (U)

For pressure, we use a pressure microphone

For particle velocity, we use a (time-integrated) differential

pressure microphone

How the Microphones WorkHow the Microphones WorkElectret Condenser Microphone (P-mic)Electret Condenser Microphone (P-mic)

d

V = E dPressure (sound) waves press against front plate, changing d,

thereby inducing a voltageAssuming elastic particle-plate collisions, conservation of

momentum ensures induced voltage is linear in pressure

Condenser microphone schematic

How the Microphones WorkHow the Microphones WorkFix this: Differential Pressure Microphone Fix this: Differential Pressure Microphone

(DPM)(DPM)

Measures the pressure immediately to the right and left of a particular location

Numerically integrates to find the pressure at that location

Differential pressure microphone schematic

Placing the Microphones in a Placing the Microphones in a TrumpetTrumpet

A trumpet bell - notice the large, accessible geometry

The openness of the trumpet bell makes mounting the exit microphones easy

Microphones can be secured outside the trumpet and simply placed in

Wiring can also be done externally

Schematic of the bell: the mics easily fit in the bell and can be wired/secured externally

Placing the Microphones in a Placing the Microphones in a TrumpetTrumpet

Mouthpiece is much narrower than the bell

Harder to use microphones

Drill tiny holes in mouthpiece to run wires/brackets through

As tiny as possible so as not to change the instrument

Can't just run directly out of the mouthpiece because the path is blocked by a transducer...

Schematic of the mouthpiece notice that the wires run through small holes

in the mouthpiece

Exciting the TrumpetExciting the Trumpet

Schematic of the mouthpiece The transducer has a position that

goes as x(t) = A sin(ω t)

A player's lips resonate at a specific frequency

Excites the instrument with nearly monochromatic sound wave

Using a function generator, drive the transducer at a specific frequency

Much like a piston

Closely recreates an actual player

Some aspects still not reproducible yet, i.e., humidity

Adding Complexion to the Adding Complexion to the Measurement:Measurement: Lock-in Amplifiers Lock-in Amplifiers

We want this to be a phase-sensitive measurement We can do this using a lock-in amplifier

How lock-in amplifiers work: Pick out any components of the desired frequency; in this case, the function generator's frequency Resolve vector into real (in phase) and imaginary (perfectly out of phase) parts Record the real and imaginary values separately

A phasor diagram:The lock-in amplifier will pick outthe blue vector and resolve it intoits real (red) and imaginary (green) components.

An overview of the setup: each microphone is connectedto a lock-in amplifier which is recorded on a computer.The spectrum is obtained by sweeping a frequency range.

Above: A picture of the trumpet with measurements being taken. The four closed boxes are the microphones and the open box is the piezo driver

Left: A picture of the measurement setup.

Results: An OverviewResults: An Overview

First time a phase-sensitive measurement of this sort has been made

No general theory can explain all the data Even for non-phase sensitive, theory is

inaccurate

Imaginary component very small compared to real component Like a correction factor

Pressure vs. FrequencyPressure vs. Frequency

Magnitude of output is much less than real (output is even amplified 10x)

Output component switches sign each harmonic

Output part generally increasing, real part increases then decreases

Higher notes seem louder

A plot of input (blue) and output (pink) pressure versus frequency

Pressure Phase vs. FrequencyPressure Phase vs. Frequency

A plot of output (blue) and input (pink) phase difference versus frequency

Output is mostly noise below ~250 Hz

Distinct Patterns Output like tan(φ) Input has defined peaks and troughs Period increases with frequency

Indicative that something cyclical is happening with phase difference

Pressure in the Complex PlanePressure in the Complex Plane Different way to

look at the last plot – the elliptical nature of the plots indicates the repeating phase shift

Bigger loops correspond to higher frequencies

No 'deeper' interpretation of this data

No general theory, yet

A parametric plot of output (blue) and input (pink) pressure in the complex plane

Complex Acoustic ImpedanceComplex Acoustic Impedance

A plot of output (blue) and input (pink) impedance versus frequency

Distinct peaks and troughs on input we noted earlier

Output is nearly linear (three separate lines, perhaps)

Relates to structure of musical notes, but we won't go into that Can only access the output frequencies at input peaks

How the Notes Line UpHow the Notes Line Up

Each data point is the frequency of output at an input impedance peak (e.g., C4 = Middle C = 261.626 Hz)

Very small deviations from “accepted” notes Since measurement errors on experiment were ~ 5%, these

notes clearly coincide with accepted notes

Looking AheadLooking Ahead This summer, same experiment for an Oboe and

Clarinet Much smaller instruments make it harder These instruments use reeds, not metal mouthpieces

Data may help with a more general theory

Above: Clarinet mouthpieceLeft: Oboe reed and top of mouthpiece

RecapRecap Acoustic Impedance is defined as pressure over particle

velocity

Relates to the accessible sounds an object can make

Measured using a DPM and U-mic

No general theory yet, though some interesting data

Questions?Questions?

Special Thanks to David Pignotti, Special Thanks to David Pignotti,

Professor Errede, and all of you!Professor Errede, and all of you!