Laboratorio di Elettronica
Modulo preamplificatori:
misure di guadagno, linearità, rumore
Preamplificatori
• Preamplificatore (PA) = primo elemento della catena di
amplificazione
• In genere il PA è seguito dallo “shaper” (formatore) detto anche
“shaping amplifier” (SA)
• Il segnale prodotto in un rivelatore di radiazione è quasi
sempre un segnale in corrente i(t)
• Molto spesso l’informazione è data dalla carica
Q = ∫i(t)dt
• I preamplificatori più usati in questo campo sono di corrente
(CA) o di carica (CSA = Charge Sensitive Amplifier)
• Le caratteristiche più rilevanti dei PA sono la sensibilità o
“guadagno”, la linearità e il rumore
Architetture di lettura (1)
ANALOGICA
S&H
PA
DIGITALE (BINARIA)
PA VTH
DISCRIM.
- vantaggi
 informazione completa
 facile da verificare
- svantaggi
 grande volume di dati
 trattamento analogico
- vantaggi
 semplice, veloce
 piccolo volume di dati
- svantaggi
 difficile da verificare
 informazione ridotta
Architetture di lettura (2)
ANALOGICO/DIGITALE
ADC
- vantaggi
 informazione completa
 robusto
- svantaggi
 grande volume di dati
 sistema misto A/D
La scelta del tipo di architettura e della tecnologia (circuito ibrido,
VLSI, …) dipende tra l’altro da:
• Numero di canali da trattare (da 1 a 10 milioni)
• Limiti sulla potenza dissipata
• Limiti sulla velocità di acquisizione dati e quindi sul volume dati
• Richiesta di risoluzione energetica (MIP, raggi X, raggi γ…)
PA a componenti discreti (ibrido)
AmpTek A225
1 channel
minimum power: 10mW
power supply: 4V to 25V
current: 2.3mA
shaping time: 2.4ms
noise < 280 e- rms
size: 2cm x 1cm
energy
timing
PA su circuito integrato (chip VLSI)
PASCAL (Front – end ALICE SDD)
1 cm
CMOS 0.25mm technology
64 channels
32 10-bits ADC
Power: 8mW/ch
Shaping time: 40ns
Noise < 280 e- rms
Size: 1cm x 0.9cm
Catena di amplificazione CSA + SA
da F. Anghinolfi (2005) - parte 1, slides 7-41
in particolare:
• slides 11-25 => principi generali del trattamento del segnale
funzione di trasferimento H(t)
rappr. nel dominio del tempo e della frequenza
• slides 27-31 => esempi di formatori (shaper): RC, CR, (RC)n-CR
• slides 32-38 => ruolo di PA e SA
• slide 35 => shaper CR-(RC)n
Detector Signal Collection
Typical “front-end” elements
Board, wires, ...
+
Rp
Z
-
Particle Detector
Circuit
Particle detector collects charges :
ionization in gas detector, solid-state
detector
a particle crossing the medium generates
ionization + ions avalanche (gas detector)
or electron-hole pairs (solid-state).
Charges are collected on electrode plates
Amplifierbuilding up a voltage or a
(as a capacitor),
current
Function is multiple :
Final objective :
signal amplification (signal multiplication
factor)
amplitude measurement and/or
noise rejection
time measurement
signal “shape”
Detector Signal Collection
Typical “front-end” elements
Board, wires, ...
+
Rp
Z
-
Particle Detector
Circuit
If Z is high, charge is kept on capacitor
nodes and a voltage builds up (until
capacitor is discharged)
If Z is low charge flows as a current
through the impedance in a short time.
In particle physics, low input impedance
circuits are used:
• limited signal pile up
• limited channel-to-channel crosstalk
• low sensitivity to parasitic signals
Detector Signal Collection
Board, wires, ...
+
Rp
Particle Detector
Tiny signals (Ex: 400uV collected
in Si detector on 10pF)
Z
Zo
-
Noisy environment
Collection time fluctuation
Circuit
Particle Detector
Circuit
Large signals, accurate in amplitude
and/or time
Affordable S/N ratio
Signal source and waveform
compatible with subsequent circuits
Detector Signal Collection
High Z
Low Z
+
Rp
-
• Impedance adaptation
• Amplitude resolution
• Time resolution
• Noise cut
Voltage source
Zo
Z
Circuit
Low Z
T
Low Z output voltage source circuit can drive any load
Output signal shape adapted to subsequent stage (ADC)
Signal shaping is used to reduce noise (unwanted fluctuations) vs. signal
Electronic Signal Processing
Time domain :
X(t)
H
Y(t)
Electronic signals, like voltage, or current, or charge can be described in time
domain.
H in the above figure represents an object (circuit) which modifies the (time)
properties of the incoming signal X(t), so that we obtain another signal Y(t). H
can be a filter, transmission line, amplifier, resonator etc ...
If the circuit H has linear properties
like : if X1 ---> Y1 through H
if X2 ---> Y2 through H
then X1+X2 ---> Y1+Y2
The circuit H can be represented by a linear function of time H(t) , such that
the knowledge of X(t) and H(t) is enough to predict Y(t)
Electronic Signal Processing
X(t)
Y(t)
H(t)
In time domain, the relationship between X(t), H(t) and Y(t) is expressed
by the following formula :
Y(t) = H(t)*X(t)
Where

H(t) * X(t)   H(u)X(t - u)du

This is the convolution function, that we can use to completely
describe Y(t) from the knowledge of both X(t) and H(t)
Time domain prediction by using convolution is complicated …
Electronic Signal Processing
What is H(t) ?
d(t)
d(t)
H(t)
H
H(t)
(Dirac function)
H(t) = H(t)* d (t)
If we inject a “Dirac” function to a linear system, the output signal is the
characteristic function H(t)
H(t) is the transfer function in time domain, of the linear circuit H.
Electronic Signal Processing
Frequency domain :
The electronic signal X(t) can be represented in the frequency domain by
x(f), using the following transformation

x(f)   X(t).exp(-j2ft).dt (Fourier Transform)

This is *not* an easy transform, unless we assume that X(t) can be
described as a sum of “exponential” functions, of the form :

X(t)   ck exp( j 2f k t )

The conditions of validity of the above transformations are
precisely defined. We assume here that it applies to the signals
(either periodic or not) that we will consider later on
Electronic Signal Processing
2
X(t)  exp(at)
X(t)
1.5
Example :
1
0.5
For (t >0)
1
2
3
4
5
1

x(f)   exp (-at) . exp (-j2ft) .dt
0.8
0.6
0
0.4
0.2

-6

-4
-2
x(f)  exp(-(a  j2f)t).dt
4
6
x(f)
0
1
x(f) 
a  j2f
2
1
0.5
-6
-4
-2
2
-0.5
-1
The “frequency” domain representation x(f) is using complex
numbers.
Arg(x(f))
4
6
Electronic Signal Processing
Some usual Fourier Transforms :
– d(t)  1
– (t)  1/jw
– e-at  1/(a+ jw)
– tn-1e-at  1/[(n-1)!(a+ jw)n]
– d(t)-a.e-at  jw /(a+ jw)
1
t
The Fourier Transform applies equally well to the signal representation
X(t)
x(f) and to the linear circuit transfer function H(t)
h(f)
y(f)
x(f)
h(f)
Electronic Signal Processing
x(f)
y(f)
h(f)
With the frequency domain representation (signals and circuit transfer
function mapped into frequency domain by the Fourier transform), the
relationship between input, circuit transfer function and output is simple:
y(f) = h(f).x(f)
Example : cascaded systems
x(f)
h1(f)
h2(f)
y(f) = h1(f). h2(f). h3(f). x(f)
h3(f)
y(f)
Electronic Signal Processing
y(f)
x(f)
1
x (f ) 
j2f
X(t )   (t )
h(f)
h(f) 
1
1  j2f
RC low pass filter
y(f) 
1
j2f(1  j2f)
Y(t )  1  exp(t )
1
R
1
0.8
0.6
t
C
0.4
0.2
1
2
3
4
5
Electronic Signal Processing
y(f)
x(f)
h(f)
Frequency representation can be used to predict time response
X(t) ----> x(f) (Fourier transform)
H(t) ----> h(f) (Fourier transform)
h(f) can also be directly formulated from circuit analysis
Apply y(f) = h(f).x(f)
Then y(f) ----> Y(t) (inverse Fourier Transform)
Fourier Transform
h(f) 

 H(t).e

- j2  . f.t
Inverse Fourier Transform
.dt
H(t) 

j2  . f.t
h(f).e
.df


Electronic Signal Processing
y(f)
x(f)
h(f)
X(t)
Y(t)
H(t)
• THERE IS AN EQUIVALENCE BETWEEN TIME AND FREQUENCY
REPRESENTATIONS OF SIGNAL or CIRCUIT
• THIS EQUIVALENCE APPLIES ONLY TO A PARTICULAR CLASS OF
CIRCUITS, NAMED “TIME-INVARIANT” CIRCUITS.
• IN PARTICLE PHYSICS, CIRCUITS OUTSIDE OF THIS CLASS CAN
BE USED : ONLY TIME DOMAIN ANALYSIS IS APPLICABLE IN THIS
CASE
Electronic Signal Processing
y(f) = h(f).x(f)
d(f)
h(f)
h(f)
d(f)
h(f)
f
f
Dirac function frequency representation
In frequency domain, a system (h) is a frequency domain “shaping”
element. In case of h being a filter, it selects a particular frequency domain
range. The input signal is rejected (if it is out of filter band) or amplified (if
in band) or “shaped” if signal frequency components are altered.
x(f)
y(f)
h(f)
x(f)
y(f)
f
f
h(f)
f
Electronic Signal Processing
y(f) = h(f).x(f)
vni(f)
vno(f)
noise
h(f)
f
f
“Unlimited” noise power
Noise power limited by filter
The “noise” is also filtered by the system h
Noise components (as we will see later on) are often “white noise”, i.e.: constant
distribution over all frequencies (as shown above)
So a filter h(f) can be chosen so that :
It filters out the noise “frequency” components which are outside of the frequency band
for the signal
Electronic Signal Processing
x(f)
y(f)
h(f)
x(f)
Noise floor
f0
y(f)
f
f0
f0
f
f
Improved Signal/Noise
Ratio
Example of signal filtering : the above figure shows a « typical » case,
where only noise is filtered out.
In particle physics, the input signal, from detector, is often a very fast
pulse, similar to a “Dirac” pulse. Therefore, its frequency representation
is over a large frequency range.
The filter (shaper) provides a limitation in the signal bandwidth and
therefore the filter output signal shape is different from the input signal
shape.
Electronic Signal Processing
x(f)
y(f)
h(f)
x(f)
Noise floor
y(f)
f
f0
f0
f
Improved Signal/Noise
Ratio
The output signal shape is determined, for each application, by the
following parameters:
• Input signal shape (characteristic of detector)
• Filter (amplifier-shaper) characteristic
The output signal shape, different form the input detector signal, is chosen
for the application requirements:
• Time measurement
• Amplitude measurement
• Pile-up reduction
• Optimized Signal-to-noise ratio
f
Electronic Signal Processing
Filter cuts noise. Signal BW is preserved
f0
f
Filter cuts inside signal BW : modified shape
f0
f
Electronic Signal Processing
SOME EXAMPLES OF CIRCUITS USED AS SIGNAL SHAPERS ...
(Time-invariant circuits like RC, CR networks)
Electronic Signal Processing
R
C
Vin
Vout
Low-pass (RC) filter
Vout 
Xc
Vin
Xc  R
1
1

j 2fC jwC
Example RC=0.5 s=jw
Xc 
Integrator time function
H (t ) 
2
Vout 
1
Vin
1  RCjw
Integrator s-transfer function
1
e  t / RC
RC
h(s) = 1/(1+RCs)
1.5
|h(s)|
1
1
0.5
0.5
1
2
3
4
5
t
Step function response
0.2
0.1
1
0.05
0.8
0.01
0.6
0.05
0.1
0.5
1
Log-Log scale
0.4
0.2
1
2
3
4
5
t
5
10
f
Electronic Signal Processing
C
Vin
Vout
R
High-pass (CR) filter
R
Vout 
Vin
Xc  R
Xc 
1
1

j 2fC jwC
Vout 
RCjw
Vin
1  RCjw
Example RC=0.5 s=jw
Differentiator time function
H(t )  d (t ) 
1
Differentiator s-transfer function
1 t / RC
e
RC
h(s) = RCs/(1+RCs)
Impulse response
0.5
1
2
3
4
|h(s)|
5
1
-0.5
-1
0.5
-1.5
t
-2
Step response
0.2
0.1
1
f
0.05
0.8
0.01
0.05
0.1
0.5
1
0.6
Log-Log scale
0.4
0.2
t
1
2
3
4
5
5
10
Electronic Signal Processing
HighZ
1
R
Vin
C
Low Z
Vout
C
R
Combining one low-pass (RC) and one high-pass (CR) filter :
Vout 
RCjω
( 1 RCjω) 2
Vin
Example RC=0.5 s=jw
CR-RC time function
CR-RC s-transfer function
H(t )  (1  t / RC)et / RC
h(s) = RCs/(1+RCs)2
1
0.8
|h(s)|
Impulse response
0.6
0.4
0.2
0.2
0.15
1
2
3
4
5
t
-0.2
0.1
0.07
0.05
0.03
0.175
Step response
0.15
0.02
0.125
0.015
0.01
0.1
0.05
0.1
0.5
1
0.075
0.05
0.025
2
4
6
8
10
12
14
t
Log-Log scale
5
10
f
Electronic Signal Processing
R
Vin
R
1
C
1
C
C
Vout
R
n-1 times
Combining (n-1) low-pass (RC) and one high-pass (CR) filter :
RCjw
Vout 
Vin
n
(1  RCjw )
Example RC=0.5, n=5 s=jw
CR-RC4 time function
CR-RC4 s-transfer function
H(t )  (4  t / RC).t 3et / RC
0.01
h(s) = RCs/(1+RCs)5
Impulse response
0.0075
|h(s)|
0.005
0.0025
2
4
6
8
10
-0.0025
t
-0.005
0.02
0.01
0.005
0.002
Step response
0.001
0.0005
0.012
0.0002
0.01
0.0001
0.008
0.001
0.006
0.0050.01
0.05 0.1
Log-Log scale
0.004
0.002
2
4
6
8
10
t
0.5
1
f
Electronic Signal Processing
Shaper circuit frequency spectrum
+20db/dec
-80db/dec
Noise Floor
f
h(s) = RCs/(1+RCs)5
The shaper limits the noise bandwidth. The choice
of the shaper function defines the noise power available at the output.
Thus, it defines the signal-to-noise ratio
Preamplifier & Shaper
I
d(t)
Preamplifier
Shaper
O
Q/C.(t)
What are the functions of preamplifier and shaper (in ideal world) :
• Preamplifier : is an ideal integrator : it detects an input charge burst
Q d(t). The output is a voltage step Q/C.(t). Has large signal gain
such that noise of subsequent stage (shaper) is negligible.
• Shaper : a filter with : characteristics fixed to give a predefined
output signal shape, and rejection of noise frequency components
which are outside of the signal frequency range.
Preamplifier & Shaper
I
Preamplifier
Shaper
O
1
1
0.8
0.8
0.6
0.6
t
0.4
0.2
0.4
0.2
1
2
t
3
4
5
-0.2
1
2
3
4
5
5
2
d(t)
0.5
0.2
Q/C.(t)
f
1
0.15
0.05
0.2
0.03
0.1
0.02
0.2
0.5
1
2
5
f
0.1
0.07
10
0.015
0.01
=
0.1
0.5
1
5
10
CR_RC shaper
Ideal Integrator
T.F.
from I to O
0.05
RCs /(1+RCs)2 = RC/(1+RCs)2
x
1/s
0.175
Output signal of preamplifier
+ shaper with one charge at
the input
0.15
0.125
0.1
0.075
0.05
t
0.025
2
4
6
8
10
12
O(t )  t
14
1
RC
e t / RC
Preamplifier & Shaper
I
Preamplifier
Shaper
O
1
0.01
0.8
t
0.0075
0.6
0.005
t
0.4
0.2
0.0025
2
4
6
8
10
-0.0025
1
2
3
4
-0.005
5
0.02
5
2
d(t)
Q/C.(t)
f
1
0.5
0.01
0.005
0.002
f
0.001
0.0005
0.0002
0.2
0.0001
0.1
0.2
0.5
1
2
5
10
0.001
=
0.05 0.1
0.5
1
CR_RC4 shaper
Ideal Integrator
T.F.
from I to O
0.0050.01
x
1/s
RCs /(1+RCs)5 = RC/(1+RCs)5
0.1
Output signal of
preamplifier + shaper with
“ideal” charge at the input
0.08
0.06
0.04
0.02
t
5
10
15
20
25
30
O(t )  t 4
35
1 t / RC
e
4
RC
Preamplifier & Shaper
Schema of a Preamplifier-Shaper circuit
Cf
N Integrators
Diff
Vout
Cd
T0
T0
T0
Semi-Gaussian Shaper
Vout(s) = Q/sCf . [sT0/(1+ sT0)].[A/(1+ sT0)]n
Vout(t) = [QAn nn /Cf n!].[t/Ts]n.e-nt/Ts
Peaking time Ts = nT0 !
Output voltage at peak is given by :
Voutp = QAn nn /Cf n!en
1
0.8
0.6
0.4
0.2
2
Vout shape vs. n order,
renormalized to 1
3
4
5
Vout peak vs. n
6
7
Preamplifier & Shaper
I
d(t)
T.F.
from I to O
Shaper
Preamplifier
Non-Ideal Integrator
CR_RC shaper
Integrator
baseline
restoration
x
1/(1+T1s)
O
RCs /(1+RCs)2
0.03
Non ideal shape, long tail
0.02
0.01
5
10
15
20
Preamplifier & Shaper
I
d(t)
T.F.
from I to O
Preamplifier
Shaper
Non-Ideal Integrator
CR_RC shaper
Integrator
baseline
restoration
x
1/(1+T1s)
O
(1+T1s) /(1+RCs)2
Pole-Zero Cancellation
0.175
Ideal shape, no tail
0.15
0.125
0.1
0.075
0.05
0.025
2
4
6
8
10
12
14
Preamplifier & Shaper
Schema of a Preamplifier-Shaper circuit
with pole-zero cancellation
Rf
Cf
Diff
Rp
N Integrators
Vout
Cp
Cd
T0
T0
Semi-Gaussian Shaper
By adjusting Tp=Rp.Cp and Tf=Rf.Cf such that Tp = Tf, we
obtain the same shape as with a perfect integrator at the input
Vout(s) = Q/(1+sTf)Cf . [(1+sTp)/(1+ sT0)].[A/(1+ sT0)]n
Considerations on Detector Signal Processing
Pile-up :
A fast return to zero time is required to :
• Avoid cumulated baseline shifts (average detector pulse rate should be known)
• Optimize noise as long tails contribute to larger noise level
0.175
0.15
0.125
0.1
0.075
0.05
0.025
2
4
6
8
10
2nd hit
12
14
Considerations on Detector Signal Processing
Pile-up
• The detector pulse is transformed by the front-end circuit to obtain a signal
with a finite return to zero time
0.175
0.15
0.125
CR-RC :
Return to baseline
> 7*Tp
0.1
0.075
0.05
0.025
2
4
6
8
10
12
14
0.1
0.08
CR-RC4 :
Return to
baseline < 3*Tp
0.06
0.04
0.02
5
10
15
20
25
30
35
Considerations on Detector Signal Processing
Pile-up :
A long return to zero time does contribute to excessive noise :
Uncompensated pole zero CR-RC filter
0.03
0.02
0.01
5
10
15
20
Long tail contributes to the increase of electronic noise (and
to baseline shift)
Segnali e guadagni tipici
• Segnali tipici: Q [e] = E/W ≈ 1000 E[keV] / 3.7
(W = 3.7 eV per Si)
1 elettrone = 1.6 10-4 fC
3.7 keV = 1000 el. = 0.16 fC
92 keV = 25000 el. = 4 fC
(1 MIP in 300 µm di Si ≈ 92 keV)
• Guadagno di un CSA + shaper CR-RC:
G = Avs/(e Cf) [V/C]
con Avs = guadagno in tensione dello shaper, e = 2.71828…,
Cf = condensatore di retroazione del CSA
esempio di guadagno alto [RX64]: G ≈ 20 mV/keV ≈ 500 mV/fC
esempio di guadagno tipico [A250CF]: G ≈ 0.18 mV/keV ≈ 4 mV/fC
Segnale in rivelatori a semiconduttore
Radiation ionization energy (W):
determines the number of primary
ionization events
Band gap energy (Eg):
lower value  easier thermal
generation of e-h pairs
(kT = 26 meV for T = 300 K)
Risoluzione energetica intrinseca
DE (FWHM) = 2.35FEW
il fattore di Fano “F”
quantifica la riduzione
nelle fluttuazioni
rispetto alla statistica
di Poisson
Per Si e Ge:
F = 0.10 ― 0.20
(Fano factor)
W (Si) = 3.6 eV
W (Ge) = 2.9 eV
Rivelatori a microstrip
SEGNALE = numero di coppie
elettrone-lacuna:
ne-h = DE/W,
con W=3.62 eV per il silicio
Substrato di tipo n
Capacità per unità di area:
C

d
1/ 2
 eN D 

C  
 2V B 
DIODO in polarizzazione inversa:
• Regione svuotata => ovvero, libera da
portatori di carica: le coppie e-h possono
essere rivelate (e non riassorbite)
• Tensione di polarizz. (VB) => controlla
lo spessore di svuotamento, cioe’ il volume
attivo
• Capacità della giunzione p-n per unità
di area C: 1/C2 cresce linearmente con VB
=> una misura C-V determina la tensione
di completo svuotamento VFD
Rivelatore + Elettronica
collegamento tipico (accoppiamento AC), elementi circuitali rilevanti
Tensione di polarizzazione
Resistenza in serie
Capacita’ di disaccoppiamento
Principali sorgenti di rumore, ENC
Rumore elettronico e sorgenti
di rumore nei circuiti
da F. Anghinolfi (2005) – parte 2, slides 3-37 + 44
in particolare:
• slides 3-7 => introduzione al rumore
• slides 8-15 => rumore termico (resistori, transistor MOS)
• slides 16-17 => rumore granulare (diodi, transistor bipolari)
• slides 18-19 => rumore 1/f (transistor MOS)
• slides 20-37 => rumore nei circuiti (circuiti equiv. per calcolo rumore)
• slides 44 => conclusione (Equivalent Noise Charge)
Noise in Electronic Systems
Signal frequency spectrum
Circuit frequency response
Noise Floor
f
Amplitude, charge or time resolution
What we want :
Signal dynamic
Low noise
Noise in Electronic Systems
EM emission
Power
Crosstalk
System noise
EM emission
Crosstalk
Ground/power noise
All can be (virtually)
avoided by proper design
and shielding
Shielding
Signals
In & Out
Noise in Electronic Systems
Fundamental noise
Physics of electrical
devices
Detector
Front End Board
Unavoidable but the
prediction of noise power at
the output of an electronic
channel is possible
What is expressed is the
ratio of the signal power to
the noise power (SNR)
In detector circuits, noise is
expressed in (rms) numbers
of electrons at the input
(ENC)
Noise in Electronic Systems
Current
conducting
devices
Only fundamental noise is discussed in this lecture
Noise in Electronic Systems
Current conducting devices
(resistors, transistors)
Three main types of noise mechanisms in electronic conducting
devices:
• THERMAL NOISE
Always
• SHOT NOISE
Semiconductor devices
• 1/f NOISE
Specific
Noise in Electronic Systems
THERMAL NOISE
Definition from C.D. Motchenbacher book (“Low Noise Electronic System Design, Wiley Interscience”) :
“Thermal noise is caused by random thermally
excited vibrations of charge carriers in a
conductor”
R
v 2  4kTR.Df
i 2  4kT
1
.Df
R
The noise power is proportional to T(oK)
The noise power is proportional to Df
K = Boltzmann constant (1.383 10-23 V.C/K)
T = Temperature
@ ambient 4kT = 1.66 10 -20 V/C
Noise in Electronic Systems
THERMAL NOISE
Thermal noise is a totally random signal. It has a normal distribution of
amplitude with time.
Noise in Electronic Systems
THERMAL NOISE
R
v 2  4kTR.Df
i 2  4kT
1
.Df
R
The noise power is proportional to the noise bandwidth:
The power in the band 1-2 Hz is equal to that in the band
100000-100001Hz
Thus the thermal noise power spectrum is flat over all frequency range
(“white noise”)
P
0
f
Noise in Electronic Systems
THERMAL NOISE
R
Bandwidth limiter
G=1
v2
tot
 4kTR.BWnoise
Only the electronic circuit frequency spectrum (filter) limits the
thermal noise power available on circuit output
Circuit Bandwidth
P
0
f
Noise in Electronic Systems
THERMAL NOISE
R
v 2  4kTR.Df
The conductor noise power is the same as the power available from the
following circuit :
R
*
Et  4kTR.Df
gnd
<v>
Et is an ideal voltage source
R is a noiseless resistance
Noise in Electronic Systems
THERMAL NOISE
R
*
Et  4kTR.Df
RL=hi
v 2  4kTR.Df
gnd
R
*
Et  4kTR.Df
gnd
i2 
RL=0
4kT
.Df
R
The thermal noise is always
present. It can be expressed as a
voltage fluctuation or a current
fluctuation, depending on the load
impedance.
Noise in Electronic Systems
Some examples :
Thermal noise in resistor in “series” with the signal path :
v 2  4kTR.Df
For R=100 ohms
v 2  1.28nV / Hz
For 10KHz-100MHz bandwidth :
v 2  12.88mVrms
2
Rem : 0-100MHz bandwidth gives : v  12.80 mVrms
For R=1 Mohms
2
For 10KHz-100MHz bandwidth : v  1.28mVrms
As a reference of signal amplitude, consider the mean peak charge deposited on 300um
Silicon detector : 22000 electrons, ie ~4fC. If this charge was deposited instantaneously
on the detector capacitance (10pF), the signal voltage is Q/C= 400mV
Noise in Electronic Systems
Thermal Noise in a MOS Transistor
Ids
Vgs
The MOS transistor behaves like a current generator, controlled by the
gate voltage. The ratio is called the transconductance.
gm 
DI DS
DVGS
The MOS transistor is a conductor and exhibits thermal noise
expressed as :
id2  4kT
2
..gm.Df
3
or
2
vG2  4kT ..gm 1.Df
3
 : excess noise
factor
(between 1 and 2)
Noise in Electronic Systems
Shot Noise
I
2
ishot
 2qIDf
q is the charge of the electron (1.602·10-19 C)
Shot noise is present when carrier transportation occurs across two media,
as a semiconductor junction.
As for thermal noise, the shot noise power <i2> is proportional to the
noise bandwidth.
The shot noise power spectrum is flat over all frequency range
(“white noise”)
P
0
f
Noise in Electronic Systems
Shot Noise in a Bipolar (Junction) Transistor
Ic
gm 
Vbe
DI C
DVbe
The current carriers in bipolar transistor are crossing a semiconductor
barrier  therefore the device exhibits shot noise as :
2
icol
 2qIcDf
The junction transistor behaves like a current generator, controlled by
the base voltage. The ratio (transconductance) is : gm  qIc / kT
2
icol
 4kT
1
gm.Df
2
or
v B2  4kT
1
gm1.Df
2
Noise in Electronic Systems
1/f Noise
Formulation
v
2
f
A
  .Df
f
1/f noise is present in all conduction phenomena. Physical origins are
multiple. It is negligible for conductors, resistors. It is weak in bipolar
junction transistors and strong for MOS transistors.
1/f noise power is increasing as frequency decreases. 1/f noise power is
constant in each frequency decade (i.e. from 0 to 1 Hz, 10 to 100 Hz, 100
MHz to 1Ghz)
Noise in Electronic Systems
1/f noise and thermal noise (MOS Transistor)
1/f noise
Circuit bandwidth
Thermal noise
Depending on circuit bandwidth, 1/f noise may or may not be contributing
Noise in Detector Front-Ends
Circuit
Detector
How much noise is here ?
(detector bias)
As we just seen before :
Each component is a
(multiple) noise
source
Note that (pure) capacitors or
inductors do not produce noise
Noise in Detector Front-Ends
Detector
Circuit
Rp
Ideal
gnd
charge
generator A capacitor
(not a noise
source)
Circuit equivalent
voltage noise
source
Detector
en
Passive & active
components, all
noise sources
noiseless
Rp
in
gnd
Circuit equivalent
current noise
source
Noise in Detector Front-Ends
Detector
en
Noiseless circuit
Av
Rp
in
From practical point of view, en is a
voltage source such that:
en2

2
Vnomeas
Av2
.Df
when input is grounded
gnd
in is a current source such that:
in2

2
Vnomeas
Av2
.
1
R 2p
Df
when the input is on a large
resistance Rp
Noise in Detector Front-Ends
In case of an (ideal) detector, the input is loaded by the detector capacitance C
Detector
Detector signal node (input)
en
Noiseless circuit
i2TOT is the combination of the
circuit current noise and Rp bias
resistance noise :
Av
Cd
i 2p  4kT.
iTOT
1
Rp
2
iTOT
 in2  i 2p
gnd
The equivalent voltage noise at the input is:
2
einput
 en2 
2
iTOT
Cd
2
 jw 
2
(per Hertz)
Noise in Detector Front-Ends
Detector
input
en
Noiseless circuit
Av
Cd
iTOT
The detector signal is a charge Q.
The voltage noise <e2input>
converts to charge noise by using
the relationship
q  Cd .v
gnd
2
qinput
 en2 .C d 2 
2
iTOT
( jw )
2
(per Hertz)
The equivalent noise charge at the input, which has to be compared to
the signal charge, is function of the amplifier equivalent input voltage
noise <en>2 and of the total “parallel” input current noise <iTOT>2
There are dependencies on C and on w  2f
Noise in Detector Front-Ends
Detector
en
Noiseless circuit
Av
Cd
iTOT
2
qinput
 en2 .C d 2 
2
iTOT
 jw 
2
(per Hertz)
gnd
For a fixed charge Q, the voltage built up at the amplifier input is decreased
while C is increased. Therefore the signal power is decreasing while the
amplifier voltage noise power remains constant. The equivalent noise charge
(ENC) is increasing with C.
Noise in Detector Front-Ends
Now we have to consider the TOTAL noise power integrated over the circuit bandwidth
Detector
en
Noiseless circuit, transfer
function Av (w )
Av
Cd
iTOT
gnd
2
ENCtot
Equiv. Noise Charge at input node (per hertz)
2

i
TOT
2
2
.Av (w )2 .dw
 2  en .C d 
Gp 0 

jw 2 


1


Gp is a normalization factor (peak voltage at the output for 1 electron charge at input)
Noise in Detector Front-Ends
Detector
2
ENCtot

1

 
G p2 0

en2
.C d
2
2

iTOT
.Av (w )2 .dw

 jw 2 
en
Noiseless circuit
Av
Cd
iTOT
gnd
In some case (and for our simplification) en and iTOT can be readily estimated
under the following assumptions:
The <en> contribution is coming from
the circuit input transistor
The <iTOT> contribution is only due to
the detector bias resistor Rp
Input node
Active input device
Rp (detector bias)
Noise in Detector Front-Ends
Detector
en2  4kT
2
gm
3
Input signal node
Cd
gm
in2  4kT
Rp
1
Rp
gnd
2
ENCtot



2
1
4kT 
2
2
1


4
kT
.
gm
.
C

.
.
Av
(
w
)
.dw


d
2
2
3
G p 0 
 jw  Rp 
1

Av (voltage gain) of charge integrator followed by a CR-RCn-1 shaper :
Av(w ) 
RC. jw
(1  RC. jw )
t~(n-1)RC
0.15
0.125
0.1
n
0.075
0.05
0.025
2
4
6
8
Step response
10
12
14
Noise in Detector Front-Ends
For a CR-RCn-1 transfer function, the ENC expression is :
Rp : Resistance in parallel at the input
2
4kT 2
4kT
1 C d
ENC  Fs. 2
gm
 Fp. 2 t
t
q 3
q Rp
2
Series (voltage) Parallel (current)
gm : Input transistor transconductance
t : CR-RCn-1 shaping time
C : Capacitance at the input
Series (voltage) thermal noise contribution ENCs is inversely proportional to the square
root of CR-RC peaking time and proportional to the input capacitance.
Parallel (current) thermal noise contribution ENCp is proportional to the
square root of CR-RCn-1 peaking time
Noise in Detector Front-Ends
Fp, Fs factors depend on the CR-RCn-1 shaper order (n-1):
n-1
Fs
1
0.92
2
0.84
3
0.95
4
0.99
5
1.11
6
1.16
7
1.27
n-1
Fp
1
0.92
2
0.63
3
0.51
4
0.45
5
0.40
6
0.36
7
0.34
0.25
0.35
0.3
0.2
CR-RC2
0.25
CR-RC
0.2
0.15
0.15
0.1
0.1
0.05
0.05
1
2
3
4
5
1
2
3
4
5
6
7
0.15
0.2
0.125
0.15
0.1
CR-RC3
0.1
CR-RC6
0.075
0.05
0.05
0.025
2
4
6
8
10
2
4
6
8
10
12
14
Noise in Detector Front-Ends
“Series” noise slope
“Parallel” noise
(no C dependence)
ENC dependence to the detector capacitance
Noise in Detector Front-Ends
The “optimum” shaping
time is depending on
parameters like :
optimum
C (detector)
Gm (input transistor)
R (bias resistor)
Shaping time (ns)
ENC dependence to the shaping time
(C=10 pF, gm=10 mS, R=100 kΩ)
Noise in Detector Front-Ends
C=15pF
C=10pF
C=5pF
Shaping time (ns)
ENC dependence to the shaping time
Example:
Dependence of
optimum shaping
time to the detector
capacitance
Noise in Detector Front-Ends
ENC dependence to the parallel resistance at the input
Noise in Detector Front-Ends
The 1/f noise contribution to ENC is only proportional to input
capacitance. It does not depend on shaping time,
transconductance or parallel resistance. It is usually quite low
(a few 10th of electrons) and has to be considered only when
looking to very low noise detectors and electronics (hence a
very long shaping time to reduce series noise effect)
ENC2f  K.CD 2
Noise in Detector Front-Ends
• Analyze the different sources of noise
• Evaluate Equivalent Noise Charge at the input of front-end circuit
• Obtained a “generic” ENC formulation of the form :
ENC  Fs.
2
4kT
q
2
Rs
C d2
t
Series noise
 Fp.
4kT
2
q Rp
t
Parallel noise
Noise in Detector Front-Ends
• The complete front-end design is usually a trade off between “key”
parameters like:
Noise
Power
Dynamic range
Signal shape
Detector capacitance
Conclusion
ENC  Fs.
2
4kT
q
2
Rs
C d2
t
 Fp.
4kT
2
q Rp
t
• Noise power in electronic circuits is unavoidable (mainly thermal excitation, diode
shot noise, 1/f noise)
• By the proper choice of components and adapted filtering, the front-end
Equivalent Noise Charge (ENC) can be predicted and optimized, considering :
– Equivalent noise power of components in the electronic circuit (gm, Rp …)
– Input network (detector capacitance C in case of particle detectors)
– Electronic circuit time constants (t, shaper time constant)
• A front-end circuit is finalized only after considering the other key parameters
– Power consumption
– Output waveform (shaping time, gain, linearity, dynamic range)
– Impedance adaptation (at input and output)
ENC: dipendenza da Cd, t
ENC2 = in2Fit + Cd2vn2Fv/t + Cd2FvfAf
in2 = current noise spectral density (A2/Hz)
vn2 = voltage noise spectral density (V2/Hz)
t = shaping time; Fi , Fv , Fvf = shaper form factors
ENC per un sistema CSA + shaper
K1+K2 (dovuti a resistenza del canale e del bulk) ↔ vn2Fv
K3 (dovuto a difetti nel canale) ↔ FvfAf
K4, K5 ↔ in2Fi
da notare rispetto allo schema della pagina precedente:
Cstray, Ci, Cf compaiono in aggiunta alla capacità del rivelatore Cd
Rf compare in aggiunta (in parallelo) alla resistenza di “bias” Rb
Modello di rumore per l’A250 (1)
Modello di rumore per l’A250 (2)
A250CF: configurazione di fabbrica
A250CF: caratteristiche (1)
A250CF: caratteristiche (2)
A225 + A206: caratteristiche (1)
A225 + A206: caratteristiche (2)
Esempio di misura del guadagno
600
mV
a = 5.1 ± 1.8
b = 0.064966 ± 0.000486 mV/elettrone
m (mV)
500
400
300
200
2000
3000
4000
5000
6000
7000
8000
9000
Elettroni in ingresso
Ne = Q/e = Ct Vt / e
RX64: Ct = 75 fF (integrata sul chip)
Esempio di misura del rumore
caso dell’architettura digitale (binaria)
x0
= 291.4 ± 0.446
sigma = 11.34 ± 0.51
15
150
100
200
10
50
150
5
100
0
0
240
260
280
300
320
340
Soglia (mV)
Conteggi
Conteggi
200
240
50
260
280
300
320
340
Soglia (mV)
0
240
260
280
300
320
340
Soglia (mV)
1
Obtain Counts vs.
Discriminator Threshold
(threshold scan)
2
Smoothing of Counting
Curve

Error function Fit,
or …
3
Differential Spectrum

Gaussian Fit

extract mean and s
Risultati con il chip RX64
Sn
450
Retta calibrazione con la sorgente
Retta calibrazione con il tubo
m (mV)
400
Ag
Ag
350
Mo
300
Mo
250
Rb
Ge
200
Cu
150
8
10
12
14
16
18
20
22
24
Energia (keV)
6xRX64 + fanout +
detector
GAIN
ENC30
ENC50
62.8 mV/el.
154 el.
179 el.
X-ray tube
63.7 mV/el.
151 el.
182 el.
internal calib.
64.6 mV/el.
141 el.
164 el.
241Am
source