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A quick and simple method for estimating the transmission or insertion loss of an acoustic filter
AppliedAcoustics15(1982) 347 354
A QUICK AND TRANSMISSION
SIMPLE METHOD OR INSERTION FILTER
FOR ESTIMATING THE LOSS OF AN ACOUSTIC
ST1G SODERQVIST
1FM Akustikbyrdn, Warfvinges Vtig 26, S-112 51 Stockholm (Sweden) (Received: 10 November, 1981)
SUMMARY
A method is presented by which the dominating term in the expression for TL or IL can be set up without using the complete equation system. This dominating term is the one that contains the highest power of Ac/A p (chamber area divided by pipe area). The approximation is valid everywhere except at the filter resonances where sine and cosine factors contained in the term tend to zero. In practice, however, this is not a very serious limitation. The approximation is also invalid at low frequencies where the length of chambers and pipes is smaller than about 12"5: per cent of the wavelength. I f this frequency region is of interest a lumped-element description of the filter can be used.
INTRODUCTION
An acoustic filter is made up of a number of impedance mismatches (for instance, area changes) where sound energy is reflected back towards the source. In the process of calculation one often neglects the effects of constant flow, at least in ventilation systems where the Mach number is small. This also means that resistive components due to pressure drops are neglected. Let us limit our interest to the low frequency region where only plane waves can propagate in the system. We will also limit our interest to impedance mismatches in the form of area changes. Principally, it is very easy to achieve a mathematical description of the standing waves in the system. However, the practical calculations are often tedious. Every area change introduces two new unknown variables with two equations, one describing the continuity of pressure and the other describing the continuity of volume velocity. 347 Applied Acoustics 0003-682X/82/0015-0347/$02-75 c~ Applied Science Publishers Ltd, England, 1982 Printed in Great Britain
348
STIG SODERQVIST
As an example consider the transmission loss of the two-chamber arrangement shown in Fig. 1. Transmission loss is defined by: TL = l O log ( W,/ W,)
(1)
where W i is the incoming sound power and W, is the sound power delivered to a reflection-free load. Areo A3 Area AQ["
Are° A1
JArea AO
l
J
Area AQ
Fig. 1. Two-chamberacoustic filter. It takes an equation system of eight equations to calculate the TL of the arrangement shown in Fig. 1. Of course, it is an even greater task to calculate the insertion loss (IL) which describes the actual noise reduction when the chambers are placed in a given system. In this case not only the filter, but the whole system, must be included in the calculations. With modern computer techniques it is also possible to obtain exact solutions for very complicated systems. However, for the noise control engineer it is often convenient to have access to a simple method for rough estimations without the necessity of going back to the computer. The method presented here is based on finding, with simple reasoning, the term that dominates the T L or IL in the greatest part of the frequency range. It also indicates where the system resonances are to be found. In what follows p stands for pressure and u for volume velocity. These are continuous across the area changes.
EXPANSION CHAMBER
Consider the expansion chamber shown in Fig. 2. In the greatest part of the frequency region its terminating impedance, Z (pressure divided by volume velocity)
I ~--
Area A Length b u-=--
I
p ~,'-Z
Fig. 2. Expansionchamber.
TRANSMISSION OR INSERTION LOSS OF AN ACOUSTIC FILTER
349
has a high value. We compare the total volume velocity, u, at the inlet with total pressure, p, at the outlet. The result is:
u = p A (jsinkb + ~-A-coskb)
(2)
where: pc = w a v e impedance, k = co/c = w a v e number, ~o = a n g u l a r frequency, c = speed of sound and j = x / - 1. The trick is to omit the last term and thus suppose that: A u =p--jsin
pc
kb
(3)
As can be seen from eqn. (2), there are two cases where this approximation does not hold. The first is when sin kb is close to zero, i.e. when kb ~ n. ft. These are the resonances of the chamber. The second case is when Z has its resonances so that it is no longer much greater than pc/A. In the low frequency region where kb < 0.8 it is seen from eqn. (3) that the chamber can be treated as a lumped element, i.e. a spring with impedance: Zc=
pc 2 joAb
(4)
INTERCONNECTING PIPE
In a pipe connecting two chambers (Fig. 3) we compare the inlet pressure, p, with the outlet volume velocity, u. The result is:
p=uP~(jsinkb+p~coskb)
(5)
Omitting the last term leads to the approximate expression:
pc
p = u ~ - .jsin
Area
kb
A
P Length b
Fig. 3.
Interconnecting pipe.
(6)
350
STIG SODERQVIST
Similar to the case with the expansion chamber, the approximation is invalid at the pipe resonances where sin kb is close to zero. It is also invalid in the regions where Z is not much smaller than pc/A. This happens, for instance, at the resonances of the downstream chamber. In the flow frequency region where kb < 0.8 the pipe can be treated as a lumped mass with impedance: Zp-
j~pb A
(7)
Equations (6) and (7) can also be used for a pipe between a constant pressure source and an expansion c h a m b e r or for a pipe between a chamber and free space (end pipe).
PIPE BETWEEN CONSTANT-VELOCITY SOURCE AND EXPANSION CHAMBER
For a pipe between a constant velocity source and an expansion c h a m b e r it is suitable to c o m p a r e the inlet volume velocity, uo, with the outlet volume velocity, u. F r o m Fig. 4 one obtains:
uo=u(coskb+pfAJsinkb )
(8)
Areo A i.~u 0 Length b
Fig. 4. Pipe between constant velocity source and expansion chamber.
Thus, we can use the approximation: u 0 = u cos
kb
(9)
everywhere except for the regions where c o s k b is close to zero or where Z is not much smaller than pc/A. In the low frequency region the lumped element expression (eqn. (7)) can also be used. However, a mass impedance connected to a constant velocity source does not influence its performance.
TRANSMISSION OR INSERTION LOSS OF AN ACOUSTIC FILTER
351
EXPANSION CHAMBER WITH INTERNAL PIPES
If one or both of the connecting pipes are drawn into the expansion chamber the calculations are somewhat more complicated. This arrangement is shown in Fig. 5. I f A 1 ,~A and A 2,~A the result is: A 1 u = p pc cos k x cos k y
sinkb+~-cosKto-y)cosky
(10)
where A, A 1, A2, b, x and y are defined in Fig. 5. The last term can be omitted if we
Area A ; Length b I Area A1
Area A2 u..
P
]
*-z
I,
r I
l= Fig. 5.
X
._
.~
Y
_~
Expansion chamber with internal pipes.
exclude the frequency regions where Z is small (resonances of the connected system) and where sin kb is small (resonances of the chamber). We then have: A sin kb u =P~Jcoskxcosky
(11)
which shows that this configuration is favourable compared with the simple chamber with x - y = 0. In the low frequency region where k b = 0.8 the cosine terms must be close to 1, so that eqn. (4) is valid also in this case. EXAMPLES
In order to show the step-by-step calculation process we estimate the transmission loss of the two-chamber arrangement shown in Fig. 1. -4 3 . U3 = P 3 ~ . j s l n k b 3
Pl
(12)
= u3 pc .jsin k b 2 = - P 3 A3 sin k b 2 sin k b 3
Ao
u l = p l A lpc .jsinkbl
(13)
Ao A I A 3 sin k b l sin kb z sin k b 3 = - J P• 3 ~opC
(14)
352
STIG SODERQVIST
We assume total reflexion at the inlet chamber 1. Thus, the incoming pressure:
ulpc
(15)
P i - 2A ° so that:
A A3
p~ = - j 2 ~ AoAoX sin kb t sin kb 2 sin kb 3
(16)
Equation (16) is equal to the term in the complete solution that dominates in the greater part of the frequency region. If, also, low frequencies (kb < 0.8) are of interest we can, in this region, use the lumped element model of Fig. 6. For the incoming pressure we obtain: P~ = 2
(17)
where Z~ is the impedance seen when looking into the filter. The calculation is
Ul
Z2
Zi
1
o
A0
T
Z1 = pc2 j w A 1 b1
T Z2 : Jwob2 A0
' Z3_ pc2 -JwA3b 3
Fig. 6. Lumped element circuit for Fig. I at low frequencies.
somewhat more tedious than eqns (12) to (16) but still much simpler than the exact solution. The result can be written as follows:
p~=P3-[2-(kb2) 2V'-V2+V3 +jkbz V' + V2+ V3V2
j(kb2) 3VxV3q~-z2j
(18)
where Vv= b~A,. is the volume of element v. From eqns (16) and (18) the transmission loss is calculated as: TL = 10.log P'- 2 P3 Calculations have been made for bl = bz = b and shows the result.
(19)
A1/A o = A3/A o = 10. Figure 7
TRANSMISSION OR INSERTION LOSS OF AN ACOUSTIC FILTER
353
TLdB 60 b 1 =b2=b3=b
50
A1/A 0 = A3/A 0 = I0
/.0 30
20 10 0 0
t -
-
Fig. 7.
2
~3
/
/*
5 kb radians
= complete solution = approximate solution from equation(16) . . . . . . . . . . . . . approximate low-frequency solution from equdion(18) -
-
~
-
-
Calculated transmission loss for the two-chamber acoustic filter of Fig. 1.
As another example consider the insertion loss of an expansion chamber connected to a constant velocity source and terminated by an end pipe as shown in Fig. 8. Without a chamber, i.e. with//1 = Ao, we have: u° U 2 o - c o s k ( b ° + bl + b2)
(20)
whilst the approximate solution with a chamber is: --
U 0
(21)
U2 - A I
- - c o s k b o sin k b 1 sin k b 2 Ao
The insertion loss is: IL = 10. log u2-~° 2
(22)
lU2I
In the low frequency region we use a discrete element representation. Area A 1 Area A0
J
I AreaA0
L.no,,
Fig. 8.
.
u2
Constant velocity source with expansion chamber.
354
STIG S()DERQVIST IL dB
bo/b1:0,3
b 2/
bl=0,/.
AlIA 0 =I0
40
~
30
20
/i
~"
""
10 0 -
10
-20
-0
, 1
3
2
- -
= complete solution
------
= approximate
The approximate
4
5
kb,
radiQns
sotution from equations (20}-[21)
~ow-frequency solution coincides with the complete
solution.
Fig. 9.
Calculated insertion loss for the arrangement of Fig. 8.
In Fig. 9 this approximate solution is compared with the complete solution for bo/b 1 = 0 . 3 ; b2/b I = 0 . 4 and AI/A o = 10.
CONCLUSIONS
The method presented here permits the extraction, by simple means, of the term that dominates the greatest part of the frequency region. This is the term containing the highest power of Acha,nb¢r/Apipe. The higher this figure, the better will be the approximation. However, at the system resonances this term no longer dominates because it contains sine or cosine factors that tend to zero. Thus, the approximation is not valid at the resonances. In practice this is not a very serious limitation. In complicated systems the loss close to resonances is determined by the balance between two or more terms of about equal size. Small changes of dimensions or sound velocity are sufficient to change this balance radically and these figures are usually not known with sufficient precision. It is also in the resonance regions that resistive losses due to pressure drops may influence the result.
A QUICK AND TRANSMISSION
SIMPLE METHOD OR INSERTION FILTER
FOR ESTIMATING THE LOSS OF AN ACOUSTIC
ST1G SODERQVIST
1FM Akustikbyrdn, Warfvinges Vtig 26, S-112 51 Stockholm (Sweden) (Received: 10 November, 1981)
SUMMARY
A method is presented by which the dominating term in the expression for TL or IL can be set up without using the complete equation system. This dominating term is the one that contains the highest power of Ac/A p (chamber area divided by pipe area). The approximation is valid everywhere except at the filter resonances where sine and cosine factors contained in the term tend to zero. In practice, however, this is not a very serious limitation. The approximation is also invalid at low frequencies where the length of chambers and pipes is smaller than about 12"5: per cent of the wavelength. I f this frequency region is of interest a lumped-element description of the filter can be used.
INTRODUCTION
An acoustic filter is made up of a number of impedance mismatches (for instance, area changes) where sound energy is reflected back towards the source. In the process of calculation one often neglects the effects of constant flow, at least in ventilation systems where the Mach number is small. This also means that resistive components due to pressure drops are neglected. Let us limit our interest to the low frequency region where only plane waves can propagate in the system. We will also limit our interest to impedance mismatches in the form of area changes. Principally, it is very easy to achieve a mathematical description of the standing waves in the system. However, the practical calculations are often tedious. Every area change introduces two new unknown variables with two equations, one describing the continuity of pressure and the other describing the continuity of volume velocity. 347 Applied Acoustics 0003-682X/82/0015-0347/$02-75 c~ Applied Science Publishers Ltd, England, 1982 Printed in Great Britain
348
STIG SODERQVIST
As an example consider the transmission loss of the two-chamber arrangement shown in Fig. 1. Transmission loss is defined by: TL = l O log ( W,/ W,)
(1)
where W i is the incoming sound power and W, is the sound power delivered to a reflection-free load. Areo A3 Area AQ["
Are° A1
JArea AO
l
J
Area AQ
Fig. 1. Two-chamberacoustic filter. It takes an equation system of eight equations to calculate the TL of the arrangement shown in Fig. 1. Of course, it is an even greater task to calculate the insertion loss (IL) which describes the actual noise reduction when the chambers are placed in a given system. In this case not only the filter, but the whole system, must be included in the calculations. With modern computer techniques it is also possible to obtain exact solutions for very complicated systems. However, for the noise control engineer it is often convenient to have access to a simple method for rough estimations without the necessity of going back to the computer. The method presented here is based on finding, with simple reasoning, the term that dominates the T L or IL in the greatest part of the frequency range. It also indicates where the system resonances are to be found. In what follows p stands for pressure and u for volume velocity. These are continuous across the area changes.
EXPANSION CHAMBER
Consider the expansion chamber shown in Fig. 2. In the greatest part of the frequency region its terminating impedance, Z (pressure divided by volume velocity)
I ~--
Area A Length b u-=--
I
p ~,'-Z
Fig. 2. Expansionchamber.
TRANSMISSION OR INSERTION LOSS OF AN ACOUSTIC FILTER
349
has a high value. We compare the total volume velocity, u, at the inlet with total pressure, p, at the outlet. The result is:
u = p A (jsinkb + ~-A-coskb)
(2)
where: pc = w a v e impedance, k = co/c = w a v e number, ~o = a n g u l a r frequency, c = speed of sound and j = x / - 1. The trick is to omit the last term and thus suppose that: A u =p--jsin
pc
kb
(3)
As can be seen from eqn. (2), there are two cases where this approximation does not hold. The first is when sin kb is close to zero, i.e. when kb ~ n. ft. These are the resonances of the chamber. The second case is when Z has its resonances so that it is no longer much greater than pc/A. In the low frequency region where kb < 0.8 it is seen from eqn. (3) that the chamber can be treated as a lumped element, i.e. a spring with impedance: Zc=
pc 2 joAb
(4)
INTERCONNECTING PIPE
In a pipe connecting two chambers (Fig. 3) we compare the inlet pressure, p, with the outlet volume velocity, u. The result is:
p=uP~(jsinkb+p~coskb)
(5)
Omitting the last term leads to the approximate expression:
pc
p = u ~ - .jsin
Area
kb
A
P Length b
Fig. 3.
Interconnecting pipe.
(6)
350
STIG SODERQVIST
Similar to the case with the expansion chamber, the approximation is invalid at the pipe resonances where sin kb is close to zero. It is also invalid in the regions where Z is not much smaller than pc/A. This happens, for instance, at the resonances of the downstream chamber. In the flow frequency region where kb < 0.8 the pipe can be treated as a lumped mass with impedance: Zp-
j~pb A
(7)
Equations (6) and (7) can also be used for a pipe between a constant pressure source and an expansion c h a m b e r or for a pipe between a chamber and free space (end pipe).
PIPE BETWEEN CONSTANT-VELOCITY SOURCE AND EXPANSION CHAMBER
For a pipe between a constant velocity source and an expansion c h a m b e r it is suitable to c o m p a r e the inlet volume velocity, uo, with the outlet volume velocity, u. F r o m Fig. 4 one obtains:
uo=u(coskb+pfAJsinkb )
(8)
Areo A i.~u 0 Length b
Fig. 4. Pipe between constant velocity source and expansion chamber.
Thus, we can use the approximation: u 0 = u cos
kb
(9)
everywhere except for the regions where c o s k b is close to zero or where Z is not much smaller than pc/A. In the low frequency region the lumped element expression (eqn. (7)) can also be used. However, a mass impedance connected to a constant velocity source does not influence its performance.
TRANSMISSION OR INSERTION LOSS OF AN ACOUSTIC FILTER
351
EXPANSION CHAMBER WITH INTERNAL PIPES
If one or both of the connecting pipes are drawn into the expansion chamber the calculations are somewhat more complicated. This arrangement is shown in Fig. 5. I f A 1 ,~A and A 2,~A the result is: A 1 u = p pc cos k x cos k y
sinkb+~-cosKto-y)cosky
(10)
where A, A 1, A2, b, x and y are defined in Fig. 5. The last term can be omitted if we
Area A ; Length b I Area A1
Area A2 u..
P
]
*-z
I,
r I
l= Fig. 5.
X
._
.~
Y
_~
Expansion chamber with internal pipes.
exclude the frequency regions where Z is small (resonances of the connected system) and where sin kb is small (resonances of the chamber). We then have: A sin kb u =P~Jcoskxcosky
(11)
which shows that this configuration is favourable compared with the simple chamber with x - y = 0. In the low frequency region where k b = 0.8 the cosine terms must be close to 1, so that eqn. (4) is valid also in this case. EXAMPLES
In order to show the step-by-step calculation process we estimate the transmission loss of the two-chamber arrangement shown in Fig. 1. -4 3 . U3 = P 3 ~ . j s l n k b 3
Pl
(12)
= u3 pc .jsin k b 2 = - P 3 A3 sin k b 2 sin k b 3
Ao
u l = p l A lpc .jsinkbl
(13)
Ao A I A 3 sin k b l sin kb z sin k b 3 = - J P• 3 ~opC
(14)
352
STIG SODERQVIST
We assume total reflexion at the inlet chamber 1. Thus, the incoming pressure:
ulpc
(15)
P i - 2A ° so that:
A A3
p~ = - j 2 ~ AoAoX sin kb t sin kb 2 sin kb 3
(16)
Equation (16) is equal to the term in the complete solution that dominates in the greater part of the frequency region. If, also, low frequencies (kb < 0.8) are of interest we can, in this region, use the lumped element model of Fig. 6. For the incoming pressure we obtain: P~ = 2
(17)
where Z~ is the impedance seen when looking into the filter. The calculation is
Ul
Z2
Zi
1
o
A0
T
Z1 = pc2 j w A 1 b1
T Z2 : Jwob2 A0
' Z3_ pc2 -JwA3b 3
Fig. 6. Lumped element circuit for Fig. I at low frequencies.
somewhat more tedious than eqns (12) to (16) but still much simpler than the exact solution. The result can be written as follows:
p~=P3-[2-(kb2) 2V'-V2+V3 +jkbz V' + V2+ V3V2
j(kb2) 3VxV3q~-z2j
(18)
where Vv= b~A,. is the volume of element v. From eqns (16) and (18) the transmission loss is calculated as: TL = 10.log P'- 2 P3 Calculations have been made for bl = bz = b and shows the result.
(19)
A1/A o = A3/A o = 10. Figure 7
TRANSMISSION OR INSERTION LOSS OF AN ACOUSTIC FILTER
353
TLdB 60 b 1 =b2=b3=b
50
A1/A 0 = A3/A 0 = I0
/.0 30
20 10 0 0
t -
-
Fig. 7.
2
~3
/
/*
5 kb radians
= complete solution = approximate solution from equation(16) . . . . . . . . . . . . . approximate low-frequency solution from equdion(18) -
-
~
-
-
Calculated transmission loss for the two-chamber acoustic filter of Fig. 1.
As another example consider the insertion loss of an expansion chamber connected to a constant velocity source and terminated by an end pipe as shown in Fig. 8. Without a chamber, i.e. with//1 = Ao, we have: u° U 2 o - c o s k ( b ° + bl + b2)
(20)
whilst the approximate solution with a chamber is: --
U 0
(21)
U2 - A I
- - c o s k b o sin k b 1 sin k b 2 Ao
The insertion loss is: IL = 10. log u2-~° 2
(22)
lU2I
In the low frequency region we use a discrete element representation. Area A 1 Area A0
J
I AreaA0
L.no,,
Fig. 8.
.
u2
Constant velocity source with expansion chamber.
354
STIG S()DERQVIST IL dB
bo/b1:0,3
b 2/
bl=0,/.
AlIA 0 =I0
40
~
30
20
/i
~"
""
10 0 -
10
-20
-0
, 1
3
2
- -
= complete solution
------
= approximate
The approximate
4
5
kb,
radiQns
sotution from equations (20}-[21)
~ow-frequency solution coincides with the complete
solution.
Fig. 9.
Calculated insertion loss for the arrangement of Fig. 8.
In Fig. 9 this approximate solution is compared with the complete solution for bo/b 1 = 0 . 3 ; b2/b I = 0 . 4 and AI/A o = 10.
CONCLUSIONS
The method presented here permits the extraction, by simple means, of the term that dominates the greatest part of the frequency region. This is the term containing the highest power of Acha,nb¢r/Apipe. The higher this figure, the better will be the approximation. However, at the system resonances this term no longer dominates because it contains sine or cosine factors that tend to zero. Thus, the approximation is not valid at the resonances. In practice this is not a very serious limitation. In complicated systems the loss close to resonances is determined by the balance between two or more terms of about equal size. Small changes of dimensions or sound velocity are sufficient to change this balance radically and these figures are usually not known with sufficient precision. It is also in the resonance regions that resistive losses due to pressure drops may influence the result.