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Noise mechanisms in electronic devices:
physical origin and circuit model
Chiara Guazzoni
Politecnico di Milano and INFN Sezione di Milano
e-mail: [email protected]
www: http://home.dei.polimi.it/guazzoni
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Table of contents
Noise definitions, noise analysis and theorems
Noise physical sources
Noise modeling in electronic devices
Equivalent Noise Charge Definition and Calculation
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise analysis – definitions – I
inherent noise
• it refers to random noise
signals due to fundamental
properties of the detector
and/or circuit elements;
• therefore it can be never
eliminated;
• it can be reduced through
proper choice of the
preamplifier/shaper design.
interference noise
• it results from unwanted
interaction between the detection
system and the outside world or
between different parts of the
system itself;
• it may or may not appear as
random signals (power supply noise
on ground wires – 50 or 60 Hz,
electromagnetic interference
between wires, …).
 We will deal with inherent noise only.
 We will assume all noise signals have a mean value of zero.
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise analysis – definitions – II
For those more rigorously inclined, we assume also that random signals
are ergodic therefore their ensemble averages can be approximated by
their time averages.
from S. Cova
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Time-domain noise analysis - I
Noise derives from superposition of a very high number of elementar process,
under good approximation mutually uncorrelated.
Probability distribution is normalised
amplitude
Mean value is zero
Pvn  
vn,rms
time
• noise rms (root mean square) value
vn, rms
1 T

  0 vn2 t dt 
T

• stationary noise
probability density constant with time


vn
exp  
2
2 vn, rm s
 2v n, rms

1





“central limit”,
i.e. Gaussian distribution
where T is a suitable averaging time interval. A longer T
usually gives a more accurate rms measurement.
It indicates the normalized noise power of the signal.
• non stationary noise
probability density varies with time
• full noise description:
 marginal probability pm(vn1,t1) for every instant t1
 joint probability pj(vn1, vn2,t1,t2)=pj(vn1, vn2,t1,t1+t)
stationary noise: pm does not depend
on time
stationary noise pj depends only on
time interval t
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Time-domain noise analysis - II
• autocorrelation function of the noise Rxx
vn t1 vn t 2   vn t1 vn t1  t   Rxx t1, t1  t   Rxx t1, t 2 
 function of the interval t between two instants and for non stationary noise also
function of t1
vn2 t   Rxx t ,0
vn2  Rxx 0
noise variance is the autocorrelation
function value in 0
• signal-to-noise ratio (SNR) (in dB)
 v x2,rms 
 v x ,rms 
 signal power 
SNR  10 log 

10
log

20
log
 2 



 noise power 
 vn,rms 
 vn,rms 
• noise summation
vn1(t) vn2(t)
2
vno
vno(t)=
vn1(t)+vn2(t)
1

T
0 vn1t   vn2 t  dt
T
2
2 T
vn1 t vn 2 t dt
 
0
T


 vn21, rm s  vn22, rm s 
individual m ean
squared values

correlation between
the two signal sources
noise sums “quadratically” (in power)
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Frequency-domain noise analysis I
• noise spectral density:
average normalized noise
power over 1-Hz
bandwidth, measured in
V2/Hz or A2/Hz.
Vn2  f 
frequency
The rms value of a noise signal can be obtained also in the frequency domain:
 2
2
2
vn, rm s  vn  Vn f df
0
2
Vn  f  is the Fourier transform of the autocorrelation function of the timedomain signal vn(t) (Wiener-Khintchine theorem).

 
One-side spectral density: noise is integrated only over positive frequencies.
Bilateral spectral density: noise is integrated over both positive and negative
frequencies.
The bilateral definition results in the spectral density being divided by
two since, for real-valued signals, the spectral density is the same for
positive and negative frequencies.
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Frequency-domain noise analysis II
• white noise
Vn(f)
Vn2
f 
Rnn(t)
2
 Vnw
where Vnw is constant
Vnw
2
2
t   Vnw
Rnn
  t 
f
t
A noise signal is said to be white if its
spectral density is constant over a given
frequency, i.e. if it has a flat spectral
density.
In the time domain white noise shows no
correlation at any finite time interval τ, no
matter how small.
• 1/f (or flicker) noise
Vn(f)
-10dB/dec
1/f noise
corner
Vn2
f
f 
Af
f
where Af is a constant.
The noise power of the 1/f noise is constant in every decade of frequency:
10 f *
f *
Vn2
10 f *
 f df   f *
10 f *
df  A f ln *  A f ln10
f
f
Af
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise analysis – useful theorems
• Carson’s theorem
noise source with bilateral power spectrum N (w)
superposition (in the time
domain) of randomly distributed events with Fourier transform F(w) occurring at
an average rate l
N w   l Fw 
2
• Campbell’s theorem
the r.m.s. value of a noise process resulting from the superposition of pulses of a
fixed shape f (t), randomly occurring in time with an average rate l is:
vn 
2
1/ 2

1/ 2


2
 l  f t dt 
 

• Parseval’s theorem
 2
 h
t  dt 
1 
2
2





H
w
d
w

2
H
w
df

0
2
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise physical sources – Thermal noise
• Thermal noise (also known as Johnson or Nyquist noise - 1928):
 present in all dissipative systems, as a consequence of the fundamental
mechanisms ruling their energy state;
 due to thermal excitation of charge carriers in a conductor;
 from fundamental thermodynamics laws (first and second) and from
Planck's law, can be seen as the black body radiation in a single
propagation mode.
 from thermodynamics, the power spectral density of the thermal noise
is
where k = 1,38 10-23 J/K (Boltzmann’s constant)
h = 6,624 10-34 Js (Planck’s constant)
10
1
S(f)/kT


hf


kT

S ( f )  kT 

 hf  
exp

  1

 kT  

0.1
0.01
3
10 0. 01
01
0.1
1
1
hf/kT
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
10
Noise physical sources – Shot noise
• Shot noise (first studied by Schottky in 1918 in vacuum tubes):
 due to the granularity of charge carriers forming the current flow;
number of emitted (or collected) electrons shows statistical
fluctuation;
 white spectral density and dependent on the DC bias current, easily
derived from statistical considerations (or Campbell’s theorem)
f(t)=qh(t)
mean
mean
square
current
current
t
it   I 
i t 
0

f   pd  qp h d  pq
0
2  2
q
h
0

noise
mean
square
value
2

random sequence of independent pulses
f(t) = q h(t), with h(t) normalized pulse shape
pdt probability that a pulse starts in (t, t+dt)
p=const, independent of other pulses
  pd
2  
q
h
0 0
  pd h  pd 
2  2
pq
h
0
 d  I 2
 

 2
2
2
2
2  2
ni t   i t   it   pq  h  d  qI  h  d
0
0
if we neglect correlation on time scales
shorter than the transit time, h(t)(t)
S n  f   qI H  f   qI
2
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
I n2  K
I
f
Noise physical sources – Flicker noise
• Flicker noise (commonly referred to as 1/f noise or pink noise):
 it is a “fundamental” noise, present in different processes;
 least understood of the noise phenomena;
 usually arises due to traps in the semiconductor, where carriers
constituting the DC current flow are held for some time period and then
released;
 power spectral density:
I
I n2  K
f
with 0.8<<1.3; K process dependent and 1<<2.
Frequency domain
from M. Bertolaccini
Time domain
from M. Bertolaccini
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
___
I n2 
I 2
(1  f 2 )
Noise physical sources – G-R noise
• Generation-Recombination noise
 in semiconductors due to fluctuations in the carriers number due to
thermal generation and recombination, due to trapping/detrapping or
due to direct transition from valence to conduction band
 non-white noise with components of the power spectral density
t
where t is the transition time constant;
(1  w 2t 2 )
 if only one carrier type is involved and only one process dominates (as it
occurs in practical cases) the power spectral density is
___
I 2
2
In 
2 2 2
(1  4 t f )
where  is a constant depending on the
technology and on the physical
properties of the semiconductor
AlGaN/Gan HFET
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise physical sources – Burst noise
• Burst noise (also known as Pop-Corn noise or Random Telegraph Noise)
 in semiconductor is due to fluctuations of carriers number due to
trapping/detrapping of a large number of carriers, not of single
particles.
 the power spectral density is similar to the one of GenrationRecombination noise
I 2
I 
(1  f 2 )
___
2
n
where  e  are constants depending on
the physical characteristics of the
semiconductor and on the fabrication
process
from M. Bertolaccini
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise in Electronic Devices: Resistors I
 Resistors exhibit thermal noise.
 The power spectral density of such voltage fluctuations was originally
derived by Nyquist in 1928, assuming the law of equipartition of energy
states that the energy on average associated with each degree of freedom is
the thermal energy.
R (noiseless)
Sv w   2kTR
R (noisy)
R (noiseless)
2kT
Si w  
R
Only physical resistors (and not
resistors used for modeling)
contribute thermal noise.
1 kW resistor exhibits a root
spectral density of 4nV/Hz
(4pA/Hz) of thermal noise at
room temperature (300 K).
k: Boltzmann’s constant (1.3810-23 J/K)
T: absolute temperature in Kelvin
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise in Electronic Devices: Resistors II
 At frequencies and temperatures where quantum mechanical effects are
significant (hf~kT) each degree of freedom should on average be assigned the
hf
hf kT
energy:
kT
hf


 kT  S v w   2kTR
h  6.63  1034 J  s
  hf  
  hf  
exp

1
exp    1


k  1.38  1023 J / K



  kT  
  kT  
v
noise power spectral density S (V2/Hz)
10
10
10
10
-16
R=1kW
T

T
h
1 
kT
T

T
-18
 300 K
  6 1012 Hz  6THz
 30 K
  6 1011 Hz  600 GHz
 0.3K
  6 10 9 Hz  6GHz
 3 10 3 K   6 10 7 Hz  60 MHz
-20
T=
T=
T=
T=
-22
0
10
300 K
30 K
3K
0.3 K
2
10
4
10
6
8
10
10
frequency (Hz)
10
10
10
12
14
10
at “practical” frequencies
and temperatures resistors
thermal noise is
independent of frequency
 white noise
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise in Electronic Devices: MOSFET
D
source
Source
G
LL
Gate
gate
(Van der Ziel - 1986)
drain
Drain
W
W
Ossido
oxide
S
n
1/f noise
n
substrato p
p-sub
Metal
Oxide
Semiconductor
F
E
T
Thermal noise (the channel can be treated as a
resistor whose increment resistance is a function
of the position coordinate):
ohmic region
2kT
Si 
Rch
(due to random capture and
release of carriers by a large number of
traps with different time constants):
S I D w  
The thermal noise current in the channel is
equal to Johnson noise in a conductance
equal to gm where  =2/3 for long channel
and  = (VGS-VT) for short channel MOSFETs.
w
 SVg w  
K
WLCox w
P-channel MOSFETs feature lower 1/f
noise than N-channel MOSFETs.
D
saturation region
Si  2kT gm
B
G
2kTgm
B
w
S
G
K
WLCox w
2kT
D
gm
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
S
Noise in Electronic Devices: JFET
(Van der Ziel - 1962)
G
D
S
G
t
D
Shot noise
(due to leakage current IG
across the gate-channel junction):
SIG w   qIG
D
S
G
Thermal noise (the channel may be treated as a
resistor whose increment resistance is a function of
the position coordinate):
2kTgm
qIG
2kT
G
A f
D
gm w
S
saturation region
qIG
Si  2kT gm
The thermal noise current in the
channel is equal to Johnson noise
in a conductance equal to gm
where  =2/3.
B
w
S
1/f noise
(due to random capture and
release of carriers by traps in the device):
S I D w 
B
w
However, much lower than in MOSFET
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Noise in El. Dev.: lossy capacitor
C
Im
(lossy)
1
ZC 
wC
  r  ji
o A
1
 jw C

R
d
i
tan 
r
in
Re
Si w 
R
vn
C
(loss-less)
2kT
 2kTC w tan 
R
Power spectral density of the
thermal noise current generator
R2
2 kT
Sv w   Si w 

sin  cos 
2 2 2
wC
1 w R C
 o r A
C
d
d

ZR  R
Y  jw  r  j i 
R
(Van der Ziel - 1975)
1
 i ow A w C tan 

As far as the loss angle () is independent of frequency, the
output voltage noise shows a 1/f spectrum.
At low frequency the loss resistance is merely a measure of the conductivity (s) of
the dielectric
Sv(w) shows a frequency dependence of the form
1
1  w R C 
2 2
o
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
2
Si 
kTC noise
2kT
R
Q(t)
R
voltage noise r.m.s.
s v2



S v w df 
Sv(f) high R
1/2RC
kT
C
noise power spectral density
C
S v w   Si w 
R2
1  w 2 R 2C 2
 2kTR
1
1  w 2 R 2C 2
charge fluctuation:
C=10fF  40 electrons r.m.s.
s q2  Cs v2  kTC
C=100fF  125 electrons r.m.s.
independent of R!!
filter
transfer
function
1/2RC
low R
f
C=1pF
 400 electrons r.m.s.
C=10pF  1250 electrons r.m.s.
In the case of switched capacitors
(reset switch, analog memories, …)…
…kTC noise sets ultimate limit of dynamic range
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
ADC: quantization noise
Quantization intrinsically introduces
an error (an input voltage range is
represented by a single output code):
2
s quant

1
LSB
LSB
0
ERROR  
2
LSB 2
d 
12
Typical input-output characteristics of a
3-bit ADC with the indication of the
quantization error
quantization error is less for higher resolution
ADCs, lowering the quantization noise and leading
to a higher maximum theoretical SNR
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - I
o Identification of the detector and preamplifier noise sources
Radiation
Shaping
Amplifier
Preamplifier
Detector
shaping
amplifier
preamp.
Q   t 
qI L 
2 kT
Rf
Cdet
Ci
2kTC tan w
(capacitor
dielectric losses)
(leakage current
feedback resistor)
2kT 
qIG
gm
(white voltage noise)
 A f (1/f voltage noise)
w
(gate current
shot noise)
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - II
o Equivalent circuit for ENC calculation
noiseless
preamp.
Q   t 
qI L 
Cdet+Cpar
2 kT
 qIG
Rf
(leakage current
feedback resistor
gate current)
shaping
amplifier
Ci
2kTC tan w
(capacitor
dielectric losses)
2 kT 
2 2
2
w
gm CT A f w CT (1/f voltage noise)
(white voltage noise)
Equivalent Noise Charge is the value of charge that injected across the
detector capacitance by a -like pulse produces at the output of the
shaping amplifier a signal whose amplitude equals the output r.m.s. noise,
i.e. is the amount of charge that makes the S/N ratio equal to 1.
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - III
o Time domain representation of signal and noise
 pulse
 pulses
‘ pulses doublets
Current
step
random walk
signal
white parallel
 pulses
Charge
white series
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - IV
o ENC calculation in presence of white parallel noise
noiseless
transamp
b  qI L
shaping
amplifier
ENC 2p
 pulses
Q   t 

  N w  H  jw  df 


 qI L  H  jw  df  qI L  ht  dt
CT
h(t)
0
T


2
Parseval’s
theorem
gated integrator
1
2

 ht  dt  T
2

ENC 2p  qI LT  qN el q
ENC 2p  b A3t
h(t)
1
0t
ENCp2  t
2

RC-CR shaping

 ht 

2
e2
dt  t  1.85 t
4
ENCp2 independent of CT
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - V
o ENC calculation in presence of white series noise
a
2 kT 
gm
shaping
amplifier
noiseless
transamp
2kT 2 2
w CT
gm
Q   t 
CT
CT
noiseless
transamp
shaping
amplifier
‘ pulses
doublets
Parseval’s theorem
1

2kT 2   2
2kT 2   ' 2
2
2
2
ENCs   N w H  jw df 
CT  w H  jw df 
CT  h t  dt
gm
gm



h(t) triangular shaping
RC-CR shaping
h(t)
1

2

2
2
e2 1
'
0 T 2T
 h t  dt 

ENC s2
T
a
0t
CT2 A1
1
t
ENCs2  CT2
'
 h t  dt 

ENCs2 
1.85
4 t
t
1
t
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model

Equivalent Noise Charge - VI
o ENC calculation in presence of 1/f noise and/or dielectric losses
 Af

w
w
noiseless
transamp
c
c  CT2  w   A f wCT2 noiseless
transamp
shaping
amplifier
d  w  2kTC tan w
Q   t 
d  w  2kTC tan w
CT
CT
shaping
amplifier




2
2
2
2
ENC1 / f   N w H  jw df   A f C T  2kTC tan   w H  jw df


h(t)
RC-CR shaping
triangular shaping
1

0
T 2T
ln 2


w
H
j
w
df

4
 0.88



2
ENC12/ f


cCT2

 d A2
h(t)
1
0t

2
 w H  jw  df  1.18

ENC12/ f independent of shaping time
ENC12/ f  CT2
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - VII
o ENC calculation in presence of white and 1/f + dielectric noises
 

h' t  dt



2
ENC 
aCT2
2

series
Introducing x



b  h t  dt



2
parallel
 t t , where t is a typical width of h(t) as the peaking time or the FWHM:
 aCT2  
  h'  x  2 dx
ENC  
t  




2



2
2
cCT  d  w H w df



1 / f  dielectric
series
 aCT2 

A1 



 t





2
2
cCT  d  w H w df



1 / f  dielectric


A2 cCT2  d


bt  h  x  dx








A bt
3

parallel
A1, A2 , A3 are shape factors depending only on the shape of the filter:

2
parallel
1 / f  dielectric
series



A1   w H w  df   h' t  dt A2   w H w  df A3   H w  df   ht  2 dt

2
2

2

2

2

C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - VIII
o ENC vs. shaping time (t)
5mm2 SDD (on-chip JFET)
Cd  0.15 pF
100
CG  0.15 pF
ENC [electrons]
gm  0. 3mS
IL  1.6 pA
A f  1.1  1011 V 2
10
1
0.1
1
10
100
t, shaping time [s]
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
Equivalent Noise Charge - VIII
o ENC vs. shaping time (t)
5mm2 SDD (on-chip JFET)
Cd  0.15 pF
100
CG  0.15 pF
ENC [electrons]
gm  0. 3mS
IL  1.6 pA
A f  1.1  1011 V 2
10
5mm2 pn-diode (NJ14 JFET)
Cd  1.7 pF
C par  2 pF
CG  2 pF
gm  6 mS
1
0.1
1
10
t, shaping time [s]
100
IL  1.6 pA
A f  1  1015 V 2
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model
C. Guazzoni – Advanced School and Workshop on Nuclear Physics Signal Processing - November 21, 2011
Noise mechanisms in electronic devices: physical origin and circuit model