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Harmonic and intermodulation distortion in condenser microphones
Applied Acoustics 25 (1988) 1-9
Harmonic and Intermodulation Distortion in Condenser Microphones
M u h a m m a d Taher Abuelma'atti Department of Electrical Engineering and Computer Science, Faculty of Engineering, University of Bahrain, PO Box 32038, lsa Town, Bahrain (Received 30 September 1987; accepted 18 November 1987)
ABSTRACT A simple formula is presented for the input-output characteristic of the condenser microphone. Also, a closed-form expression is derived for the output voltage of a condenser microphone excited by a multitone acoustic input signal.
INTRODUCTION Condenser microphones are complex electromechanical devices in which mechanical, electromechanical and electrical elements interact to transform sound energy into electrical signals. Obviously, any non-linearity in one or more of the elements will produce undesirable output products. To evaluate the performance of any condenser microphone it is therefore necessary to predict the levels of the output harmonic and intermodulation products resulting from exciting the condenser microphone by a multitone acoustic signal. Assuming a single-tone input, a rigorous theoretical analysis has been given by Hunt, 1 which leads to rather complicated expressions for the amplitudes of the output harmonics. Simpler, but empirical, expressions for the second and third harmonics have been given by Gayford. 2 Djuric 3 used the input-output (i.e. displacement-voltage) characteristic of the microphone to describe a method for obtaining the amplitudes of the output harmonics. However, no closed-form expressions have been given. 1 Applied Acoustics 0003-682X/88/$03-50 © 1988 Elsevier Science Publishers Ltd, England. Printed in Great Britain
2
M. 7". Abuelma'atti
Diaphragm Backplate
pc/////////,/] /, /,
~/
Housing
Wt O
E*e (t) O
Fig. 1. Condensermicrophone As shown by Djuric, the voltage between the terminals of the condenser microphone shown in Fig. 1, can be expressed as: E + e(t) [ Q / ( C o + Cp)]
I -- 6(t) . . . .
1 - ~6(t)
i! !
where E is the polarising voltage which produces the electrostatic field in the gap; e(t) is the variable voltage across the microphone terminals; Q is the electrostatic charge; 6(t) = d(t)/d o, the relative incremental displacement of the diaphragm (where d o is the quiescent distance between the diaphragm and back plate); ~ = Cp/Cp + Co; Cp is the passive capacitance between the stationary parts of the electrodes; and C o is the capacitance for d(t) = 0 and Cp = 0. From eqn (1) it is obvious that, except for the special case of Cp =--0, the relationship between the incremental displacement 6(t), resulting from sound energy, and the produced variable voltage, e(t), is non-linear. Eqn ( 1) can therefore be used to predict the harmonic and intermodulation performance of the condenser microphone. However, in its present torm, eqn (1) will not allow closed-form expressions for the harmonic and intermodulation products output levels. The purpose of this paper is, therefore, to present a simple formula for the normalised displacement-voltage characteristic of the condenser microphone. This formula fits the condenser microphone characteristic and permits closed-form expressions for the output voltage products resulting from multisinusoidal acoustic input.
Harmonicand intermodulationdistortion
3
PROPOSED F O R M U L A The development of this model has proceeded along empirical lines by comparing the truncated Fourier-series model ofeqn (2) with the normalised displacement-voltage characteristic of eqn (1) for each value of a: 5
E+e(t) =(1--e)-5(t)+ [Q/(Co + cp)]
Bnsin -~(5(t)+h)
(2)
The parameters T and B. were obtained by the 12-point method. 4 This procedure yields a family of parameters B., e, h which depend on a. These parameters (see Figs 2-4) were fitted to simple closed-form analytic expressions: B1 = 2.02666~t - 1.7066a 3 (3) Bs = 0.092~ - 0.2560e3
(4)
e = 1.68a- 1.28~t3
(5)
h = 1.0
(6)
and T = 3,
B2 =
Ba = B4 = 0
(7)
Now, using eqns (2)-(7), calculations were made and are shown in Fig. 5, from which it is obvious that the proposed model accurately represents the normalised diplacement-voltage characteristic of the condenser microphone. However, higher accuracies can be obtained, if required, by using the values of the parameters of eqns (3)-(7) as an initial guess set in computer optimisation, using least-squares fitting between the normalised displacement-voltage characteristic and the results obtained using eqn (1).
H A R M O N I C A N D I N T E R M O D U L A T I O N ANALYSIS Consider the case of a condenser microphone excited by a multitone signal of the form: K
5(0 = ~
Ak sin ~Okt
k=l
where K
Z
k=l
Ak < 1
(8)
4
M . T . Abuelma'atti
1.o
0.5
o Fig. 2.
I
0.25
c~
l 0.50
V a r i a t i o n o f the p a r a m e t e r B 1 with ~.
0-02
ID
o.ol
I
0
Fig. 3.
0.25
I
ct
0-50
V a r i a t i o n o f the p a r a m e t e r B~ with :~.
0,8
0.4
0
Fig. 4.
,,t 0-25
Ct
I 0.50
V a r i a t i o n o f thc p a r a m e t e r ~: with ~
Harmonic and intermodulation distortion
5
C(.= I14
o
O. =112
o o 'o
E z
I -1
0
1
6 it)
Fig. 5. Normalised voltage-displacement characteristic of the condenser microphone. , calculated, eqn (I); O, calculated, eqn (2).
Substituting eqn (8) into eqn (2), with
B4 = 0, then
B 2 = B 3 =
K
(Q/(Co + Cp))= l - e -
K
AkSincokt+ ~ Bl sln -~ k=l
Ak sin e)kt
k=l K
x/~
+ --~--B l
cos (nXAkSincokt) ~k=l K
+ ~B5 sin -~-
)
Ak sin CORt k=l K
~X~B, cos ( ~ Z
Ak sin °h,' ) k=l
Using the identities sin (fl sin cot) = 2[Jl(fl) sin cot + J3(fl) sin 3cot + . . . ] cos (fl sin cot) = Jo(fl) + 2[J2(fl) cos 2cot + J4(fl) cos 4cot +-" "]
(9)
6
M.T. Abuelma'atti
eqn (9) reduces to K
E + e(t)
[Q/(Co +
C~)]=
1 --e-- ~'AkSintOkl
k=1 x,
K
+,'I ,.v [.,X,o.(:~;~,.,, ~ll,m2
.....
ttlk
~
-
~
k~
1
K
K
k=l
k-I
+..,E,.(~e)l.,,n(E~,~.,,-) l X~3
+2trll,m2....,mk
Jmk
B1
z..a -~- -
~
k--
K
~3!
I
h
- ~.1 l,.('~;,,)],osf2.......,>] k-1
k
!I()i
1
Noting that
J _.(Z) = (-)" J.(Z) the amplitude of the normalised output-voltage component of frequency K
K
~ mk(')k and order ~ 'mk', k=l
k=l
where m k is a positive or negative integer or zero, will be given by K
h
K
for V k=l
k
k=l
and K
(Vm,.m, .......
),.:~/3 ",
J~m,~\ 3 } k= t K
h
5ZAk
.,FI~.,(T)I k~l
,
for Z link[ e v e r k:
1
Imk]odd I
Harmonic and intermodulation distortion
7
Eqn (11) is correct only if the subscripts on Vare not 0,..., 0, 1, 0,..., 0, as an additional term is required in that case; that is the term Using eqns (1 l) and (12), the amplitudes of any harmonics and intermodulation products of any order can be easily obtained.
Ak.
SPECIAL CASE In this section the special case of a single-tone input signal of the form 6(0 = A sin cot will be considered. Using eqns (11) and (12), the amplitude of the normalised output voltage component at frequency co will be given by V1 = A - [ B I J I ( ~ - - ) - ~ - B 5 J I ( ~ - - ) I
(13)
the amplitude of the normalised output voltage component at frequency 2co will be given by
V2=x//3[B,Ju(~-)-BsJu(~A-)]
(14)
and the amplitude of the normalised output voltage component at frequency 3co will be given by V 3 = BIJ a ~ -
For
+ BsJ 3
(15)
5~A/3<< 1 the Bessel function can be approximated by J.(x) = (x/2)"/n!
and eqns (13)-(15) reduce to gA V1 = A - y [ O
V2 -
1 -,~ 5B5]
x / ~ 2 A 2 [B 1 - 25B5] 72
(16)
(17)
and 7~3A3 V 3 = 1-~-~--[Ol + 125B5]
(18)
Combining eqns (16)-(18), the relative second-harmonic distortion will be given by H D 2 - X//3~2A[Bx -- 25B5] 12{6 -- ~[B 1 + 5B5] }
(19)
8
M . T . Abuelma'atti
-20
_
_
C~ : 112
J
-40
"1"
-60 -
-80 -
J
I 0,02
-100
Fig. 6.
I 0-04 Relative
0,, = 1t4
f
J
I 1 0.06 0.08 displacement
I 0.10
i 0,12
V a r i a t i o n o f the s e c o n d - h a r m o n i c d i s t o r t i o n w i t h d i s p l a c e m e n t .
-50
C)(,= 112
-60 =
-70 "o v
& I
-80
-90
-100
Fig. 7.
0-02
0.04 Relative
0.06 displacement
0.08
0'10
Variation o f the third-harmonic distortion with d i s p l a c e m c m
Harmonic and intermodulation distortion
9
and the relative third-harmonic distortion will be given by rt3A2[Bl + 125B s] HD3 = 216{6 -- rt[B 1 + 5Bs] }
(20)
The relative second-harmonic and third-harmonic distortions expressed by eqns (19) and (20) are plotted in Figs 6 and 7 as a function of the relative incremental displacement of the diaphragm. It is obvious from Figs 6 and 7 that the second harmonic is predominant. However, eqn (19) shows that, for B~ = 25B5, the second harmonic vanishes and only the third and higher harmonics will appear. This may lead to the design of condenser microphones that are free from the second-harmonic distortion. Moreover, it is obvious from Figs 6 and 7 that the harmonic distortions decrease with the decrease of the parameter ct. This means that as the passive capacitance Cp decreases and/or the capacitance C Oincreases, the harmonic distortions will decrease. These results may help in optimising a condenser microphone design with minimised harmonic distortions.
CONCLUSIONS In this paper a simple formula has been proposed for the input-output characteristic of the condenser microphone. The formula fits the condenser microphone characteristic very well. The simiplicity of the proposed formula resulted in a closed-form expression for the output voltage of a condenser microphone excited by a multi-tone acoustic signal. In general, the second-harmonic distortion is predominant. However, proper design of the condenser microphone can reduce the second-harmonic distortion, if not totally eliminating it.
REFERENCES 1. Hunt, F. V., Electroacoustics, Harvard University Press, Princeton, NJ, 1954. 2. Gayford, M. L., Acoustical Techniques and Transducers, Macdonald and Evans, London, 1961. 3. Djuric, S. V., Distortion in microphones. In International Conference on Acoustics, Speech, and Signal Processing. IEEE, 1976, pp. 537-9. 4. Dodes, I. A., Numerical Analysis for Computer Science. Elsevier North Holland, New York, 1978.
Harmonic and Intermodulation Distortion in Condenser Microphones
M u h a m m a d Taher Abuelma'atti Department of Electrical Engineering and Computer Science, Faculty of Engineering, University of Bahrain, PO Box 32038, lsa Town, Bahrain (Received 30 September 1987; accepted 18 November 1987)
ABSTRACT A simple formula is presented for the input-output characteristic of the condenser microphone. Also, a closed-form expression is derived for the output voltage of a condenser microphone excited by a multitone acoustic input signal.
INTRODUCTION Condenser microphones are complex electromechanical devices in which mechanical, electromechanical and electrical elements interact to transform sound energy into electrical signals. Obviously, any non-linearity in one or more of the elements will produce undesirable output products. To evaluate the performance of any condenser microphone it is therefore necessary to predict the levels of the output harmonic and intermodulation products resulting from exciting the condenser microphone by a multitone acoustic signal. Assuming a single-tone input, a rigorous theoretical analysis has been given by Hunt, 1 which leads to rather complicated expressions for the amplitudes of the output harmonics. Simpler, but empirical, expressions for the second and third harmonics have been given by Gayford. 2 Djuric 3 used the input-output (i.e. displacement-voltage) characteristic of the microphone to describe a method for obtaining the amplitudes of the output harmonics. However, no closed-form expressions have been given. 1 Applied Acoustics 0003-682X/88/$03-50 © 1988 Elsevier Science Publishers Ltd, England. Printed in Great Britain
2
M. 7". Abuelma'atti
Diaphragm Backplate
pc/////////,/] /, /,
~/
Housing
Wt O
E*e (t) O
Fig. 1. Condensermicrophone As shown by Djuric, the voltage between the terminals of the condenser microphone shown in Fig. 1, can be expressed as: E + e(t) [ Q / ( C o + Cp)]
I -- 6(t) . . . .
1 - ~6(t)
i! !
where E is the polarising voltage which produces the electrostatic field in the gap; e(t) is the variable voltage across the microphone terminals; Q is the electrostatic charge; 6(t) = d(t)/d o, the relative incremental displacement of the diaphragm (where d o is the quiescent distance between the diaphragm and back plate); ~ = Cp/Cp + Co; Cp is the passive capacitance between the stationary parts of the electrodes; and C o is the capacitance for d(t) = 0 and Cp = 0. From eqn (1) it is obvious that, except for the special case of Cp =--0, the relationship between the incremental displacement 6(t), resulting from sound energy, and the produced variable voltage, e(t), is non-linear. Eqn ( 1) can therefore be used to predict the harmonic and intermodulation performance of the condenser microphone. However, in its present torm, eqn (1) will not allow closed-form expressions for the harmonic and intermodulation products output levels. The purpose of this paper is, therefore, to present a simple formula for the normalised displacement-voltage characteristic of the condenser microphone. This formula fits the condenser microphone characteristic and permits closed-form expressions for the output voltage products resulting from multisinusoidal acoustic input.
Harmonicand intermodulationdistortion
3
PROPOSED F O R M U L A The development of this model has proceeded along empirical lines by comparing the truncated Fourier-series model ofeqn (2) with the normalised displacement-voltage characteristic of eqn (1) for each value of a: 5
E+e(t) =(1--e)-5(t)+ [Q/(Co + cp)]
Bnsin -~(5(t)+h)
(2)
The parameters T and B. were obtained by the 12-point method. 4 This procedure yields a family of parameters B., e, h which depend on a. These parameters (see Figs 2-4) were fitted to simple closed-form analytic expressions: B1 = 2.02666~t - 1.7066a 3 (3) Bs = 0.092~ - 0.2560e3
(4)
e = 1.68a- 1.28~t3
(5)
h = 1.0
(6)
and T = 3,
B2 =
Ba = B4 = 0
(7)
Now, using eqns (2)-(7), calculations were made and are shown in Fig. 5, from which it is obvious that the proposed model accurately represents the normalised diplacement-voltage characteristic of the condenser microphone. However, higher accuracies can be obtained, if required, by using the values of the parameters of eqns (3)-(7) as an initial guess set in computer optimisation, using least-squares fitting between the normalised displacement-voltage characteristic and the results obtained using eqn (1).
H A R M O N I C A N D I N T E R M O D U L A T I O N ANALYSIS Consider the case of a condenser microphone excited by a multitone signal of the form: K
5(0 = ~
Ak sin ~Okt
k=l
where K
Z
k=l
Ak < 1
(8)
4
M . T . Abuelma'atti
1.o
0.5
o Fig. 2.
I
0.25
c~
l 0.50
V a r i a t i o n o f the p a r a m e t e r B 1 with ~.
0-02
ID
o.ol
I
0
Fig. 3.
0.25
I
ct
0-50
V a r i a t i o n o f the p a r a m e t e r B~ with :~.
0,8
0.4
0
Fig. 4.
,,t 0-25
Ct
I 0.50
V a r i a t i o n o f thc p a r a m e t e r ~: with ~
Harmonic and intermodulation distortion
5
C(.= I14
o
O. =112
o o 'o
E z
I -1
0
1
6 it)
Fig. 5. Normalised voltage-displacement characteristic of the condenser microphone. , calculated, eqn (I); O, calculated, eqn (2).
Substituting eqn (8) into eqn (2), with
B4 = 0, then
B 2 = B 3 =
K
(Q/(Co + Cp))= l - e -
K
AkSincokt+ ~ Bl sln -~ k=l
Ak sin e)kt
k=l K
x/~
+ --~--B l
cos (nXAkSincokt) ~k=l K
+ ~B5 sin -~-
)
Ak sin CORt k=l K
~X~B, cos ( ~ Z
Ak sin °h,' ) k=l
Using the identities sin (fl sin cot) = 2[Jl(fl) sin cot + J3(fl) sin 3cot + . . . ] cos (fl sin cot) = Jo(fl) + 2[J2(fl) cos 2cot + J4(fl) cos 4cot +-" "]
(9)
6
M.T. Abuelma'atti
eqn (9) reduces to K
E + e(t)
[Q/(Co +
C~)]=
1 --e-- ~'AkSintOkl
k=1 x,
K
+,'I ,.v [.,X,o.(:~;~,.,, ~ll,m2
.....
ttlk
~
-
~
k~
1
K
K
k=l
k-I
+..,E,.(~e)l.,,n(E~,~.,,-) l X~3
+2trll,m2....,mk
Jmk
B1
z..a -~- -
~
k--
K
~3!
I
h
- ~.1 l,.('~;,,)],osf2.......,>] k-1
k
!I()i
1
Noting that
J _.(Z) = (-)" J.(Z) the amplitude of the normalised output-voltage component of frequency K
K
~ mk(')k and order ~ 'mk', k=l
k=l
where m k is a positive or negative integer or zero, will be given by K
h
K
for V k=l
k
k=l
and K
(Vm,.m, .......
),.:~/3 ",
J~m,~\ 3 } k= t K
h
5ZAk
.,FI~.,(T)I k~l
,
for Z link[ e v e r k:
1
Imk]odd I
Harmonic and intermodulation distortion
7
Eqn (11) is correct only if the subscripts on Vare not 0,..., 0, 1, 0,..., 0, as an additional term is required in that case; that is the term Using eqns (1 l) and (12), the amplitudes of any harmonics and intermodulation products of any order can be easily obtained.
Ak.
SPECIAL CASE In this section the special case of a single-tone input signal of the form 6(0 = A sin cot will be considered. Using eqns (11) and (12), the amplitude of the normalised output voltage component at frequency co will be given by V1 = A - [ B I J I ( ~ - - ) - ~ - B 5 J I ( ~ - - ) I
(13)
the amplitude of the normalised output voltage component at frequency 2co will be given by
V2=x//3[B,Ju(~-)-BsJu(~A-)]
(14)
and the amplitude of the normalised output voltage component at frequency 3co will be given by V 3 = BIJ a ~ -
For
+ BsJ 3
(15)
5~A/3<< 1 the Bessel function can be approximated by J.(x) = (x/2)"/n!
and eqns (13)-(15) reduce to gA V1 = A - y [ O
V2 -
1 -,~ 5B5]
x / ~ 2 A 2 [B 1 - 25B5] 72
(16)
(17)
and 7~3A3 V 3 = 1-~-~--[Ol + 125B5]
(18)
Combining eqns (16)-(18), the relative second-harmonic distortion will be given by H D 2 - X//3~2A[Bx -- 25B5] 12{6 -- ~[B 1 + 5B5] }
(19)
8
M . T . Abuelma'atti
-20
_
_
C~ : 112
J
-40
"1"
-60 -
-80 -
J
I 0,02
-100
Fig. 6.
I 0-04 Relative
0,, = 1t4
f
J
I 1 0.06 0.08 displacement
I 0.10
i 0,12
V a r i a t i o n o f the s e c o n d - h a r m o n i c d i s t o r t i o n w i t h d i s p l a c e m e n t .
-50
C)(,= 112
-60 =
-70 "o v
& I
-80
-90
-100
Fig. 7.
0-02
0.04 Relative
0.06 displacement
0.08
0'10
Variation o f the third-harmonic distortion with d i s p l a c e m c m
Harmonic and intermodulation distortion
9
and the relative third-harmonic distortion will be given by rt3A2[Bl + 125B s] HD3 = 216{6 -- rt[B 1 + 5Bs] }
(20)
The relative second-harmonic and third-harmonic distortions expressed by eqns (19) and (20) are plotted in Figs 6 and 7 as a function of the relative incremental displacement of the diaphragm. It is obvious from Figs 6 and 7 that the second harmonic is predominant. However, eqn (19) shows that, for B~ = 25B5, the second harmonic vanishes and only the third and higher harmonics will appear. This may lead to the design of condenser microphones that are free from the second-harmonic distortion. Moreover, it is obvious from Figs 6 and 7 that the harmonic distortions decrease with the decrease of the parameter ct. This means that as the passive capacitance Cp decreases and/or the capacitance C Oincreases, the harmonic distortions will decrease. These results may help in optimising a condenser microphone design with minimised harmonic distortions.
CONCLUSIONS In this paper a simple formula has been proposed for the input-output characteristic of the condenser microphone. The formula fits the condenser microphone characteristic very well. The simiplicity of the proposed formula resulted in a closed-form expression for the output voltage of a condenser microphone excited by a multi-tone acoustic signal. In general, the second-harmonic distortion is predominant. However, proper design of the condenser microphone can reduce the second-harmonic distortion, if not totally eliminating it.
REFERENCES 1. Hunt, F. V., Electroacoustics, Harvard University Press, Princeton, NJ, 1954. 2. Gayford, M. L., Acoustical Techniques and Transducers, Macdonald and Evans, London, 1961. 3. Djuric, S. V., Distortion in microphones. In International Conference on Acoustics, Speech, and Signal Processing. IEEE, 1976, pp. 537-9. 4. Dodes, I. A., Numerical Analysis for Computer Science. Elsevier North Holland, New York, 1978.