350 |
12 Filters and Resonators |
12.6Numerical Method for Modeling of Pipelines with Side Branches
Most acoustic filters can be modeled as a system of pipelines with side branches as shown in the figure below. The main pipe is the vertical pipe, and the side branch is the horizontal pipes in series as shown in the figure below:
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Hint: Doing this example by hand can be difficult because (1) the formula for acoustic impedance is in complex number format and (2) power transmission coefficients need to be calculated at multiple frequencies. You can use the MATLAB functions provided in Computer Code Section (Sect. 11.7) or any suitable programming language for this project.
12.7Project
Use the MATLAB code provided in the Computer Program Section (Section 11.7) to calculate and plot the power transmission coefficient of the three basic filter designs:
(a) Low-pass filter as shown in the sketch below:
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12 |
Filters and Resonators |
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Pi þ Pr ¼ P1 ¼ P2 ¼ P3 |
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Prove that: |
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Exercise 12.2
A side branch, length L, and a cross-section area Sis have an acoustic impedance at its termination given as Zos = ρoc/Sis. The cross-section area of the main pipe is Si (¼S2) as shown below.
The acoustic impedance of Z1 at the intersection is:
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Zos þ jZis tan ðkLÞ |
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jZos tan ðkLÞ þ Zis |
Treat Z1 and Z2 as given values to solve the exercise:
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1 = 1 + 1
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Intersection as a point
(a)Calculate the power reflection coefficient (Rw)from the cross-section at the intersection.
12.8 Homework Exercises |
353 |
(b)Calculate the power transmission coefficient (Tw2) through the intersection of the main pipe.
(c)Calculate the power transmission coefficient (Tw1) through the side branch.
(Answers):
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ZoþZi |
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ð ZoþZiÞZ1 |
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