Thermal mechanical noise in small microphones
Dag Wang, Ib-Rune Johansen, Matthieu Lacolle, Thor Bakke and Ole Herman Bjor
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Outline
•
•
•
•
•
Motivation
Johnson noise (electrical)
• Resistance
• Impedance
• Transfer function
Fluctuation dissipation theorem
Thermal noise in the acoustic domain
• Acoustic impedance
• Acoustic transfer function
• Noise as function of size
Interpretation
Z
Johnson noise
Acoustic noise
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Motivation
•
•
•
Microtechnology can be utilized to make
small and compliant structures
Optical readout requires a small area for
readout
Optical read out noise can be minimised
=> Need to get a grip on the thermomechanical noise
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Johnson (Nyquist) noise
Noise:
Electric energy -> Heat
<V2>=4kTRΔf
<V>=0
Electric field
V=RI
Resistivity/dissipation:
V
Heat -> Electric energy
Johnson + Nyquist Phys. Rev. 32 (1928)
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Current and voltage representation
R
<V2>=4kTRΔf
G=1/R
<i2>=4kTGΔf
R
MacDonald Noise and fluctuations, an introduction, Wiley (1962)
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Noise generator
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Filtering of noise by a network
𝑉𝑜𝑢𝑡 = 𝑉𝑛
C
∆𝜔
=
2𝜋
1
𝑗𝜔𝐶
1
𝑅+
𝑗𝜔𝐶
2
R
𝑉𝑜𝑢𝑡 2 = 𝑉𝑛 2
Vn
𝑅𝑒 𝑍 = 𝑅𝑒
Z
1
4𝑘𝑇𝑅
∆𝑓 =
∆𝑓
𝑗𝜔𝑅𝐶 + 1
(𝜔𝑅𝐶)2 +1
1
1
𝑅 + 𝑗𝜔𝐶
=
𝑅
1 + (𝜔𝑅𝐶)2
𝑉𝑜𝑢𝑡 2 = 4𝑘𝑇𝑅𝑒(𝑍)∆𝑓
Similarly
1
𝑖𝑜𝑢𝑡 2 = 4𝑘𝑇𝑅𝑒( )∆𝑓
𝑍
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Charge fluctuation
Impedance
Assume
i (t )  i0e
Charge
Transfer function
𝑞2
jt
and V (t )  V0e
jt
q(t )   i (t )dt   i0e jt dt 
H
defined
Z
V (t ) V0

i (t ) i0
1
i (t )
j
q
i
1
 0 
V
jV0 jZ
1 2
1
1
1
−1
= 2 𝑖 = 2 4𝑘𝑇𝑅𝑒
∆𝑓 = 2 4𝑘𝑇𝑅𝑒 𝑗𝜔𝐻 ∆𝑓 =
4𝑘𝑇𝐼𝑚 𝐻 ∆𝑓
𝜔
𝜔
𝑍
𝜔
𝜔
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Fluctuation dissipation theorem
Callen & Welton 1951
𝑉 2 = 4𝑘𝑇𝑅𝑒(𝑍)∆𝑓
𝑞
Wikipedia
2
−1
=
4𝑘𝑇𝐼𝑚 𝐻 ∆𝑓
𝜔
Response function
Sx( )
Sx( )
Fluctuation density
q
x
V
V
F
P
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Fluctuating quantities
Electrical
Mechanical
Acoustical
Force
𝑉2
𝐹2
𝑃2
Speed
𝑖2
𝑣2
∆𝑉 2
Position
𝑞2
𝑥2
∆𝑉 2
4𝑘𝑇𝑅𝑒(𝑍)∆𝑓
1
4𝑘𝑇𝑅𝑒( )∆𝑓
𝑍
−1
4𝑘𝑇𝐼𝑚 𝐻 ∆𝑓
𝜔
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Model system
•
•
𝑚 = 𝑚𝑓 𝐴
𝐹 = 𝑘𝑥 = 𝑘𝑓 𝐴𝑥
Spring stiffness per unit area, kf
Mass per unit area, mf
 Radius can be changed without
affecting sensitivity or
eignefrequency of the
mechanical part
 Can investigate the effect of the
(size dependent) acoustic
impedance alone
 Use the impedance of a baffled
piston, Zm (Kinsler)
1.5
X1 ( x)
1
R1 ( x)
0.5
0
0
2
4
6
8
10
x
Som vi skal se senere er det den dissipative delen av impedansen, R-leddet, som gir opphavet til st
Den øker først kvadratisk med x og svinger seg inn mot planbølge impedansen, A*ZAir, fra omtrent x
Den reaktive delen av impedansen, X-leddet, øker linært med x, og når en topp for x rett over 2. Den
reaktive delen forsvinner for høye frekvenser.
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Den reakive delen av impedansen kan forstås hvis man ser for seg systemet i sendermodus. Platen
Transfer function and noise density
1
kf A
H( )
1
1
1  j
2
 ZAir    
      

 R1 

j

X

1      
0 kf mf
  m 
 m  0 
H ( 2  f )
0.1
m
N
0.01
1
Im( H( ) )
4
110
100


 ZAir     



 R1 


0 kf mf
m
 1






2
kf A
2
    2  ZAir


Z



      Air  R     
1 


X1 


 
 1
  
  0

   0 kf mf  m   0 kf mf   m   
f
3
110
Sx( )
ZAir


   


 R1 


kf
m
 1





2 kb T

2
kf A
2
    2  ZAir
  ZAir      

 

1


X


 R

  0  0 kf mf 1  m   0 kf mf  1  m   

  

 
 
  
Sx ( 2 f )
nm
Hz
5
110
7
110
9
110
1
4
110
100
f
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Noise equivalent pressure (NEP)
•
Define the pressure reqiured to cause the fluctuation as the noise equivalent pressure
4 k T
Im( H( 2 f ) )
2 f
H( 2  f ) A
NEP ( f )
NEP ( f )
4 kb TZAir   2 Rm2 f  
 R1 

2
 Rm
  cAir  
eten forenkles uttrykket betraktelig:
4
110
6
110
NEP ( 20000Hz x)
Pa
Hz
NEP ( 100Hz x)
Pa
Hz
7
110
110
Pa
110
NEP ( x 1mm)
Pa
9
110
Hz
6
7
110
8
110
9
110
 10
 10
110
110
Hz
8
5
NEP ( x 5mm)
7
110
4
110
110
0.1
1
4
110
100
x
x
Membrane radius (m)
Frequency (Hz)
6
110
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Where does the noise come from?
To find the noise temperature of a receiving system:
Imagine using the system as a transmitter. The noise
temperature is the temperature where the dissipation
takes place.
Håvard Nes, private communication.
Noise temperature
The noise is both intrinsic and transmitted at the
same time.
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Averaging area
8
110
Limiting area
2

2 
Rm

2 
9
NEP (Pa/Sqrt(Hz))
110
 10
110
 11
110
f  100Hz
 12
110
NEP
Small membrane limit
Large membrane approximation
 13
110
7
110
5
110
3
110
0.1
10
3
110
Membrane radius (m)
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Further dissipation mechanisms
P0
P0+pa
y
P0+pb
P0+pm
P0+pi
VB
Vi-ΔVi
MB-ΔMB
Mi-ΔMi
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