Bore reconstruction from input impedance measurements

1
Bore reconstruction from input impedance
measurements

• Jean-Pierre Dalmont
• Laboratoire dAcoustique, Université du
  Maine (Le Mans), UMR CNRS 6613

2
Aim

• To determine the bore profile of a pipe with a
  non invasive technique
• Propose an alternative to the pulse reflectometry

3
Pipe response characterization

• Impedance measurement
• Measure pressure P and volume flow U at the input
  in the frequency domain
• Z(f)P(f)/U(f)

• Pulse reflectometry
• Measure the reflection of a pulse at the input in
  the time domain
• r(t)pref/pinc

Microphone
Source
r(t)fft-1(Z-Zc)/(ZZc) (Zc the characteristic
impedance)
4
Process

• Four steps
• Measure the reduced input impedance
• Z(f)
• Complete the input impedance for low frequencies
• Calculate the reflection function
• r(t)fft-1(Zr-Zc)/(ZrZc)
• Apply the Layer Peeling algorithm
• S(x) lt r(t) where S(x) is the cross section
  at abcissa x

5
What are the advantages of input impedance
measurements compared to the pulse reflectometry ?
First step Input impedance measurements

• Advantages
• Input impedance measurements can now be very
  accurate.
• Signal to noise ratio can be much better.
• No long tube needed.
• Disadvantages
• An inverse Fourier transform is needed which may
  result in ripple in the reflection function r(t)
• Spatial resolution is limited by the frequency
  range of the measured impedance

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Measurement setup (principle)
First step Input impedance measurements

• Source is a piezoelectric buzzer
• Cavities are as small as possible

gives P
gives U
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Measurement setup
First step Input impedance measurements
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Measurement setup
First step Input impedance measurements
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Result impedance of a trumpet
First step Input impedance measurements
____ Theory ____ Experiment
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Result impedance of a trumpet bell
First step Input impedance measurements
Amplitude
Phase
Frequency (Hz)
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Result impedance of a trumpet bell
Second step complete the input impedance for low
frequencies
Amplitude
Phase
Frequency (Hz)
Input impedance for low frequencies has to be
extrapolated
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Result impedance of a trumpet bell
Second step Complete the input impedance for
low frequencies
Amplitude
Phase
Frequency (Hz)
Input impedance for low frequencies is
extrapolated
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Third step calculate the reflection function
Reflection function at the input of the trumpet
bell
Some ripple
Time (s)
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Third step calculate the reflection function
Reflection function at the input of the trumpet
bell with virtual extra length added
Less ripple
Virtual extra length
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The Layer Peeling algorithm
Fourth step calculate the reflection function

• A non zero value of the reflection function
  indicate a change of the cross section
• R(S0-S)/(S0S)
• To the discrete time correspond a discrete space
  along the bore (DxcDt/2 and Dt1/(2fmax))
• The following procedure is repeated, starting at
  t(0)0 and x(0)0
• At time t(i) the reflection function r(t,i) gives
  the cross section S(i)
• The reflection function r(t,i1) for a pipe
  starting at abcissa x(i1) is calculated
• Errors grow exponentially gt the algorithm dont
  like strong discontinuities

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Fourth step calculate the reflection function
Reconstructed bore of trumpet bell
____ Reconstruction ____ Exact geometry
Virtual extra length
Looks nice!
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Fourth step calculate the reflection function
Reconstructed bore of trumpet bell
____ Reconstruction ____ Exact geometry
Not so nice!
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Fourth step calculate the reflection function
Reconstructed bore of trumpet bell
____ Reconstruction ____ Exact geometry
Not so nice!
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Fourth step calculate the reflection function
Reconstructed bore of trumpet bell
Reconstruction from measurement not worst than
from simulation (except at beginning)
20
Fourth step calculate the reflection function
Relative error on bore of trumpet bell
____ Experiment ____ Simulation
Reconstruction from measurement not worst than
from simulation (except at beginning)
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Conclusion

• Bore reconstruction from impedance is an
  efficient technique (probably as good as pulse
  reflectometry)
• Accuracy is in the range -5 for a bell
• Can the accuracy be improved ?
• Probably not simply !
• Is an accuracy of - 5 sufficient ?
• Probably not for copies, but this can be a
  useful complementary measurement in some cases
  difficult to investigate