LHCb note 2005-029
19. May 2005
Abstract: An attempt is made to predict the thermal noise and the shot noise for silicon strip detectors connected to the Beetle preamplifier from basic electronic noise principles. The calibration pulse shapes are used to determine the frequency dependant gain function of the Beetle. The calculated noise values are compared with measurements on the prototype ladders. In addition the signal propagation in the very long ladders is studied using a spice simulation. From this the effect of the thermal noise originating from the ohmic resistors of the detector readout strips is estimated.
1 Physics of noise 2
2 Noise of a charge-sensitive amplifier configuration 3
3 Application to small detectors 6
3.1
Numerical gain function determination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.2
Calculation of white serial and shot noise . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4 Configurations with large detectors 8
4.1
Spice simulation and signal propagation in long ladders . . . . . . . . . . . . . . . . . .
9
4.2
Noise simulation for long ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1

The discrete nature of the electrons as charge carriers gives rise to a permanent fluctuation of currents in electrical circuits. One of the earliest work on the statistics of these fluctutation is described in a paper by Norman Campbell [1] in 1909.
Nowadays we distinguish between shot noise and thermal noise:
The fluctuation of a given current with average value I due to the discrete nature of the charge carrier is called the shot noise, and has been described in detail by W. Schottky [2] in 1918. The variations are well approximated by a Gaussian distributions and are independant of the frequency f (white noise).
The spectral density of the variance is given by d < ∆ I 2 >
= 2 · e · I d f where e denotes the electrical charge of the electron.
(1)
In devices, where moving charges generate a voltage drop there is in addition the so called thermal noise or Johnson noise. The voltage and currents generated by thermal fluctuations of the velocity of the charge carriers are also independant of the frequency. The spectral power density of these currents is given by d < P >
= 4 · k T d f
(2) where k denotes the Boltzmann constant and T the absolute temperature of the device. This effect was described in detail by J.B. Johnson in 1928 [3]. Using the electrical power laws P = RI 2 = V 2 /R , the thermal noise in a resistor can be described either by the variance of a serial noise voltage by the variance of a parallel noise current I
R shown in Figure 1 (a) and (b):
V
R or d < V 2
R d f
>
= 4 k T R or d < I d f
2
R
>
= 4 k T depending which is more suited to the analysis of a given circuit.
1
R
(3)
D
R R
I
R
V inp
G
S
V
R
(a) (b) (c)
Figure 1: Representations of thermal noise effects in a resistor as a serial voltage (a), a parallel current (b) and an input voltage of a field effect transistor (c).
Thermal noise occurs always, if the movement of a charge generates a voltage drop. This is the case in ohmic resistors as described above, but also for instance in the conducting channel of a field effect transistor (FET). In this case the thermal noise generated in the channel corresponds to the variance of an equivalent noise voltage V inp at the gate input, as shown in Figure 1 (c), of d < V 2 inp
> d f
= 4 k T
2
3 g m
2
(4)

C f
V in I p C
V s
V _
_
+
A
1
V
1
G(s)
V out
Figure 2: Noise model for a charge-sensitive amplifier coupled to a detector with a pointlike capacity C .
I p represents the effects of parallel noise sources (in our case dominated by the shot noise of I bias
) , V s the effect of the serial noise caused by the channel noise of the input
FET of the amplifier.
where g m
= ∂I drain
/∂V gate denotes the forward transconductance of the FET [4]. Since g 2 m depends linearly on the channel’s geometrical width divided by the length and on its channel current, low noise input amplifier need to be designed with wide channels and high currents in the input FET.
In complex system often additional noise, whith a 1 /f spectrum is observed, which is called flicker noise. It has different physical origins. For the purpose of this note it can be neglected.
The relevant signal of a radiation detection device is in most cases the total free charge generated by the particle entering the device. The electronics attached to the device is supposed to measure this charge. In addition the device should be able to measure particles with a high rate. In nuclear and particle physics “charge-sensitive amplifiers” are used for this purpose, which produce a short pulse, whose height is proportional to the observed charge in the detector. Such an amplifier consists of an integrator and a pulse shaping stage.
For the purpose of noise analysis the amplifier is modelled [6] as an LTI device (linear and time invariant system) consisting of an ideal differential amplifier with feedback capacity C f and open loop gain A
1 followed by a shaping amplifier with transfer function G ( s ) as shown in Figure 2. For such an LTI
, system the usual signal theory of LTI systems [5] can be applied to calculate the total noise at the output of the amplifier.
The parallel noise current in Figure 2 can come from various sources like the shot noise of the detector bias current, the thermal noise of the bias resistor and a possible feedback resistor connected parallel to C f
. The shot noise evaluates numerically to d < ∆ I 2 > d f
/
[ nA ] 2
[ Hz ]
= 2 · e · I ≈ 0 .
32 I bias
/ [ nA ] (5) and the noise due to any resistor R bias connected parallel to C or C f to d < I
2
R d f
>
/
[ nA ]
2
= 4 k T
[ Hz ]
1
R bias
≈
1 .
6 × 10
− 8
R bias
/ [ M Ω]
(6)
Typical sensor bias resistors are of order 1.5 MΩ per channel, the parallel noise from the bias resistor can thus be safely neglected. Typical feedback resistors needed to discharge C f are even larger.
The noise generated by the input FET of the amplifier is represented by fluctuations of V s
. All subsequent steps of the signal amplification generate of course also noise. However, due to the amplification
3

the signal becomes larger and the noise to signal ratio smaller. Thus, in normal cirumstances, the noise of the amplification chain with the exception of the input FET can be neglected as well.
Now lets calculate the expected size of the noise due to V s amplifier: and I bias as observed at the output of the
Any voltage V in will cause the differential amplifier to regulate its output by charging up equilibrium is reached, which is described by
C f until an
V
1
= A
1
· V
−
V
−
+ V s
= V in
V
1
− V in
V in
C
=
C f
(7) using the basic definitions of the diagram of Figure 2.
From these three equations the two variables V
− and V in can be eliminated, ending up with
V
1
=
C f
C + C f
+
C + C f
A
1
V s which describes, how the serial noise is amplified by the integrator.
(8)
The parallel noise current I p behaves exactly like any signal current injected from the detector. The integrator amplifies this input current spectrum I p
( ω ) injected at V in according to the equations
V
1
= A
1
· V in
I p
= I f
+ I
C
(9) where I f
V in
= ( V
1
+ V in
) · iωC f denotes the current flowing through the feedback capacitor and I
C
· iωC the current flowing through the detector capacity. By eliminating V in we end up with
=
V
1
( s ) = s ( C f
1
+
C + C f
A
1
)
· I p
( s ) (10) using now the usual complex frequency variable s = σ + iω . This equation describes in the frequency domain, how the signal current and parallel noise is amplified in the integrator.
The shaper amplifies the V
1 further by V in general, variances obey the simple rule out
( s ) = G ( s ) · V
1
( s ). Although, G ( s ) is a complex function
< V
2 out
( ω ) > = | G ( iω ) | 2 · < V
1
2
( ω ) > (11)
By inserting equation (4) into (8) and (1) into (10), adding these two contributions in quadrature and using (11) we end up with the variance of the total noise spectral density at the output amplifier:
V out of the d < V d
2 out f
>
=
8 k T
3 g m
· ( C + C f
)
2
+ 2 e I bias
·
1
ω 2
·
C f
1
+
C + C f
A
1
!
2
· | G ( iω ) |
2
(12)
To get the total noise at the output, this equation needs to be integrated over the full frequency range.
It is conveniant to express the integrated noise fluctuations relative to the pulse height generated by a real signal of a given charge (calibration signal). The ratio between the square root of the variance of the noise fluctuations and the output calibration signal height per unit charge is called equivalent noise charge (ENC), it is usually given in units of electrons.
Lets denote with V calib
( t ) the output voltage of our system in the time domain due to a calibration signal current I calib
( t ) = Q · δ ( t ) containing the total charge Q . The response of the integrator in the frequency domain is described for the Laplace transform of this calibration current I ( s ) = L ( I ( t ))
4

by equation (10), which has to be multplied by G ( s ) to get the the output calibration signal in the frequency domain:
1
L ( V calib
( t )) = V calib
( s ) = · Q · G ( s ) (13) s ( C f
+
C + C f
A
1
) where we made use of the fact, that the Laplace transform of δ ( t ) is 1.
We observe, that the right hand side of this equation is very similar to the second expression in (12).
Now let v ( t ) be the normalised response function and V max the calibration signal, such that
V calib
( t ) = V max
· v ( t ) be the measured output pulse height of
(14)
In these terms the definition of the equivalent noise charge becomes now
ENC
2
=
R d f · d < V 2 out
V 2 max
/Q 2
> / d f and from equation (13) we get
1
C f
+
C + C f
A
1
!
2
· | G ( iω ) |
2
= |L ( V max
· v ( t )) |
2
· w 2
Q 2
(15)
(16)
Introducing this into (12) and using (15) we get the final result:
ENC
2
=
Z
∞ d ω
2 π
0
·
8 k T
3 g m
· ( C + C f
)
2
ω
2
+ 2 e I bias
·
|L ( V max
· v ( t )) | 2
V 2 max
(17)
All elements of this equation are known quantities. Therefore we can predict the equivalent noise charge of our system, if we calculate the Laplace transform of a calibration signal. In general this will be done numerically.
There is a special case of a shaper, where G ( s ) is analytically known, and the formula (12) can be integrated analytically. The so called first order RCCR shaper has a transfer function of
G
1
( s ) = s · a
( s + a ) 2
(18) where a = 1 /τ and τ = RC is called the shaping time. The delta response function of the charge amplifier gets the form
V calib
( t ) =
C f
+
Q
C + C f
A
1
· t
τ e
− t/τ
(19)
In this case the Integral of equation (12) can be evaluated analytically with the result:
EN C
2
1 e
2
=
8 · τ
·
8 k T
3 g m
· C
2
+ e
2 · τ
8
· 2 q e
I (20) where e is now Euler’s number and q e the charge of the electon and the definition (15) has been used.
The factor in front of the current evaluates to e 2 q e
/ 4 = 11 .
56 electron 2 / nA ns, which is usually quoted to be 12.
However, this formula is strictly speaking only valid for RCCR filter with only one filter step (“first order”) and with only one time constant ( RC ) int
= ( RC ) diff
. It can be found in many modern textbooks
5

[7] and it is usefull to discuss, how parallel shot noise increases with the shaping time, while serial noise decreases. However, the formula should not be used for calculating exact noise, since practical shapers are normally far away from being a first order RCCR. Of course, already in the early literature it has been shown, that noise is not too sensitive to the details of the pulse shape of a charge-sensitive amplifier. Various pulse shapers differ typically a few 10% in their signal / noise performance. But in present day standards this is considered to be a significant difference.
Figure 3: Measured Beetle delta response function (red, partially covered by the green curve
[8]) for V f s
= 400 mV, T = 52
◦
C. For comparison a 1st (pink) and 2nd (blue) order RCCR filter response is also shown, adjusted to the same peak height and using the same time constant of 15.5 ns. The green curve shows a smoothed Beetle curve, which is also prolonged to larger times, using an exponential decay curve.
In this section we describe the application of the formalism to cases, where the detector can be described by one single capacity only. According to equation (17) we have to calculate the laplace transform of the amplifier response signal first, which we do in the following section. After that we are able to predict noise figures for both thermal and shot noise and we will compare them to measurements with
LHCb silicon tracking prototypes.
In the laboratory Beetle delta response functions have been measured in several places. A well defined step pulse with fast risetime is coupled through a small 1 pF capacity to the input of the Beetle amplifier, which produces a short current spike, which widths is determined by the rise time of the step pulse generator. As long as this rise time is small compared to the Beetle response time (order 10 ns), this pulse can be interpreted as a delta response I calib
= Q · δ ( t ) calibration current, which produces a response at the output according to equation (13).
6

Figure 4: Result of the Laplace transform of the Beetle delta response function (green) compared to 1st (pink) and 2nd order (blue) RCCR filter cases of Figure 3. In order to check the
Laplace transform method, the analytic expression of the RCCR networks are also shown (in the same color). The deviations between analytical and numerical results are mainly due to the too short time period of the response function in Figure 3.
We use here such measurements done at the ASIC Lab in Heidelberg [8] using a version 1.3 chip and Q = 22
0
000 e. The transfer function G ( s ) can be varied in a limited range by programming the parameter V fs in the chip. We use data for V fs
= 400, measured at T=52
◦
C.
Figure 3 shows the measured Beetle delta response function together both with a first and second order
RCCR filter response for comparison. The Beetle curve has been smoothed to get rid of HF pickup noise and also prolonged to larger times by an exponential decay, since unfortunately the measurement was not continued to the end of the undershoot.
A numerical Laplace transform of the calibration pulse was performed. Figure 4 shows the result of the Laplace transform. More precisely it shows the quantity L ( V callib
( t )) · ω , which is according to equation (16) proportional to the transfer function | G ( iω ) | of the shaper.
In order to check this method, the same Laplace transform was also applied to the RCCR shaper model, since in this case it can be compared to the analytically known transfer function. As demonstrated in Figure 4 the numerical calculation agrees with the analytical function very well, except at high frequencies, where the amplitude function is very small.
Now we can use equation (17) to predict the effect of the shot noise and the FET input noise on the output signal of the Beetle.
In order to check the method, the same numerical program is also applied to the RCCR shaper functions of Figure 3. The numerical results are compared to the analytical expression (20) and agree within 3%
7

V fs
0 100 400 1000 mV
ν max
ENC shot noise predicted
14.5
14.1
12.9
10.5
MHz
119 121 134 169 e
2
/nA
ENC serial noise predicted 51.2
50.9
49.0
43.0
e /pF
ENC serial noise measured 52.6
51.9
49.4
45.2
e /pF
Table 1: Predicted equivalent noise charge of the Beetle chip as a function of the shaper parameter V f s
. For definition of the ENC quantities see equations (22) and (21).
ν max the frequency, where | G ( ω ) | reaches its maximaum denotes with the latter.
Results for noise predictions are summarised in Table (1) as a function of the shaper parameter V fs
. The step response curves have been measured at a chip temperature of 52
◦
C.
ν max denotes the frequency, where | G ( iω ) | is at its maximum.
The serial noise is given as the constant b in the following formula:
ENC serial
[electrons] = a + b · C [ pF ] (21) b is compared in the table with the measured noise dependance on the input capacitances [9]. The agreement is quite good. The predictions are somewhat lower than the measurements, but within the systematic error of the method, which is estimated to be of order 5%. The model does not allow to describe the constant a .
The shot noise is given as the constant B in the following formula:
ENC shot
[electrons] = p
B · I [ nA ] (22)
The quantity B can be compared to measurements of the shot noise. An irradiated ladder was used in order to see a larger effect. By variing the temperature of this silicon sensors, the bias current was varied, and the resulting noise measured, from which the shot noise component was extracted (see [10] and Figure 5).
The calibration pulses used so far, have been measured without a external capacity connected to the input of the Beetle. However, the derivation of the formula (17) is independant of the input capacity.
Thus, as a check, the noise calculation was also applied to calibration pulses measured with various noise prediction varies by ∆ b = ± 3% and the shot noise prediction by ∆
√
B = ± 5%. These variations give some indications of the systematical uncertainties of our method.
These calculations are valid for discrete detector capacities. More precisely serial noise is valid for cases, where all serial resistances within the detector (e.g. strip resistance) are small compared to the effective Beetle input noise resistance R eq of about 2 / 3 g m
= 68 Ohm (see equation (4)). Shot noise is valid for detectors with serial resistance which are small compared to the Beetle effective input current impedance of about 600 Ohm (see next section). Shot noise is independant of the detector capacity.
So far we have assumed, that the detector can be looked at electronically as a single capacity device.
However, practical silicon ladders as are used for instance in the LHCb silicon tracking system have rather long strips. In this section we study how the finite ohmic resistance and the inductance of the readout strips has to be taken into account for predicting noise values.
8

300
250
200
150
100
50 irrad1 chip 8 irrad1 chip 9 irrad2 chip 8
0
0 200 400 600 800 1000
V fs
[mV]
Figure 5: Shot noise measured on irradiated LHCb ladders, as a function of V fs
, compared to the prediction according to table 1 (dashed line), plot taken form [10].
Table 2 shows the relevant electrical parameters of the various silicon strip prototype ladders. For the calculation of the capacitances and the inductance the Maxwell program package (2d version) has been used [11]. Capacitance and resistance have been compared to measured values and found to be in good agreement.
Simple estimates show immediately, that the inductances are large enough, such that the ladders should have finite transfer line features. Further more the resistances are well comparable to the noise equivalent input resistance of the Beetle (68 Ohm), thus additional noise from the strip resistance has to be taken into account.
In the following section we use a Spice simulation to study the signal propagation in this long silicon ladders connected to the Beetle chip. After this we use the Spice simulation again to determine the noise spectrum of this long ladders. Finally, weighting the noise spectrum with the Beetle gain function results in noise amplitudes predicted at the output of the Beetle.
The signal lines of the long ladders are simulated by RLC Elements, with a granularity of 10 elements per cm, according to Table 2. We use as an example the “CMS3+Flex” ladder, thus the sensor and the stripline part had to be modeled each with different R , L and C values. In order to check, whether the granularity is fine enough, the results have been compared with a simulation which uses only one element per cm. No significant differences could be observed.
Relevant for the signal shape on the ladder is obviously the effective input impedance of the Beetle preamplifier. Two different methods have bin used, to determine this value in the relevant frequency
9

Ladder
LHCb1 (C)
LHCb2 (C)
LHCb3 (C)
CMS
CMS3+Flex
(Flex part)
(sensors)
R/Ohm C/pF L/nH L/cm
32.9
65.9
98.2
118.5
158
39.5
118.5
19.5
37.
54.5
38.3
57
18.7
38.3
90
180
270
260.
580
320
260
11
22
33
28.2
39
28.2
Table 2: Electrical parameters of some prototype silicon ladders. For the simulations described in this section the “CMS3+Flex” ladder was used, which consists of three CMS OB2 sensors bonded together and connected through a 39 cm long flexible stripline to the Beetle preamplifier
Figure 6: Signal amplitude in µ V at the input of the Beetle model from the 3 CMS plus flex ladder. A current signal, starting at t = 10 ns, with a rise time of 1 ns and a decay time of 10 ns, corresponding to a total charge of about 30’000 electrons, is generated at the far end (blue) of the ladder and at the connection between sensor and flex strip line (pink). Also shown (red) is the signal obtained from a discrete capacitor with the same capacity as the ladder (57 pF).
range. First of all a simulation using the full Beetle description has been done with a simple capacitor connected to the input [11]. The ratio of voltage / current at the Beetle input determines the effective input resistor. As an alternative method, the delta response function of Figure 3 has been folded with an exponentially decaying input signal with decay time τ = RC . The result is compared to the Beetle output signals, measured with discrete capacitance C connected to the Beetle input and the same generator step function coupled through a 1 pF to the input as described in the beginning of section
3.1. A fit for τ of the output signals gives the input resistance R . Both methods give comparable input impedance. For large capacitances the impedance is around 500 Ohms, while it is increasing for small capacitances. In both methods the imaginary part of the impedance is neglected, however. For the following simulations an effective input resistance of 500 Ohm has been assumed, except where noted otherwise.
10

Figure 7: The simulated pulses from the 3 CMS + flex ladder from Figure 6 (same colors) folded with the measured Beetle delta response function from Figure 3, using V fs at a temperature of 24
◦
C
= 400 mV
. Vertical scale is arbitrary. Measured data with the infrared laser setup from the 3 CMS + flex prototype are shown for comparison (green circles, scaled by an amplitude factor to match the puls height).
Figure 6 shows simulated signals as they are expected at the Beetle input. The fast part of the signal seems to propagate during about 5 ns from the far end of the sensors to the amplifier input corresponding to a speed of about 14 cm/ns (46% of speed of light). Due to the finite input resistance of the Beetle the strip charges up. The slow decay of this charge signal is very similar to a signal observed from a discrete capacity and is not influenced anymore by the propagation line features.
As a next step this signal has to be folded with the Beetle delta response function from section 3.1.
Figure 7 shows the resulting Beetle output pulses. They differ only slightly for the three different signal cases. The pulse heights of the extended ladder are predicted to be smaller by about 4% compared to a discrete capacitor with the same capacitance. The maximum of a far end pulse is reached about 3 ns later than that of a pulse generated at the near end of the sensors. The shape of the two pulses look very similar, however the pulse from a discrete capacitor decays slightly faster than those of the ladder.
Bearing in mind the crude method applied here, the simulated pulses agree quite well with the measured data, except at the end of the pulse, where the simulation predicts an earlier zero crossing than what is observed in the data. A much better description of the overshoot is obtained, if one takes into account, that the charge integrator part of the Beetle contains a programmable discharge resistor parallel to the integration capacitance. The resulting slow signal decay is turned into an overshoot by the succeeding shaping amplifier [12].
The same spice configuration as described in the last section is now used to perform a noise simulation.
All strip resistor elements and the input FET equivalent noise resistor of 68 Ohm are taken into account.
11

Figure 8: Noise spectral densities squared in units of 10
− 18
V
2
/Hz for 3 CMS sensor plus flex ladder from the spice noise simulation. Shown is the total spectrum at the amplifier input.
The contribution of the input FET noise of the amplifier is independant of the frequency and amounts to about 1 .
15 × 10
− 18
V
2
/Hz.
The resulting spectrum in Figure 8 shows some interesting features. First of all, the input resistor noise is white as expected. At low frequencies there is no effect from the stripline resistors, since no current can flow through the detector capacitances, which have a high impedance at low frequency.
At 100 MHz, corresponding to the propagation time (back and forth) of the signal discussed in the previous section, a strong noise response can be observed, which corresponds to the lowest eigen frequency of the system. This is followed by higher order modes at higher frequencies.
The accuracy of the spectrum has again be checked by using a 10 times worse granularity of the propagation line simulation. No significant differences could be observed, except for frequencies above
1 GHz. These are, however, irrelevant in view of the Beetle’s frequency response function, which falls off rapidly beyond 70 MHz.
At the frequency of 10 MHz, where the Beetle has its maximum sensitivity the ladder noise spectrum is just in its transition phase. Thus the effective noise amplified by the Beetle might be sensitive to the details of the electrical characteristica. We therefore varied both the inductance per unit length and the effective Beetle input impedance to check effect to the spectrum.
Figure 9 shows the bandwidth of the Beetle chip compared with the total noise at the input for nominal values of the cable and sensor inductance and for a variation of ± 30% of the latter. The uncertainty in the knowledge of the inductance has quite a big effect on the noise spectrum. However, a comparision with the amplitude function shows, that this is not too relevant for the output of the Beetle.
Figure 10 shows the effect of the variation of the effective input resistance. This resistance is supposed to describe the finite capability of the Beetle to absorb the signal currents. Since it is not a real resistance it does not generate noise. Its size, however, influences the magnitude of the noise signal
12

Figure 9: Beetle input noise for different values of the inductance of the stripline and the sensors. Pink is for nominal value, red for +30% and yellow for -30% inductance. For comparison the blue curve shows the amplitude function (arbitrary units on vertical axis) determined from the Laplace transformation of the Beetle delta response (see Figure 4).
Figure 10: Beetle input noise for different values of the effective Beetle input resistor. Pink is for 500 Ohms, red for -20% and yellow for +50% resistance. For comparison the blue curve shows the amplitude function (arbitrary units on vertical axis) determined from the Laplace transformation of the Beetle delta response (see Figure 4).
13

Figure 11: Weighting the input noise squared (red) with the Beetle amplitude function (blue, linear display) results in the predicted output noise spectrum squared (green). Vertical units are arbitrary.
from the ladder’s resistors at the Beetle input. The uncertainty of the size of the input resistance results in a significant systematic error in the estimation of the ladder noise.
Finally multiplying the amplitude function with the input noise spectrum gives the output noise spectrum (see Figure 11). By integrating over the frequency range we get the total noise figures. For example for V f s
= 400 mV, T = 66
◦
C the prediction yields an equivalent noise charge of 3250 e for the 3 CMS + flex ladder. For comparison the prediction for a discrete capacitor of 57 pF amounts to
2790 e.
The “offset noise”, the constant a in equation (21), which has been measured for the Beetle to be 540 e [9], is not modelled by any of these calculations, but must be added “ad hoc” to the above values, resulting in about 3800 e for the long ladder and about 3300 e for the discrete capacitor. The ratio of these two numbers yields a 15% increase of noise due to the ladder’s strip resistances.
The total strip resistance amounts to 158 Ohm and is thus 2.3 times larger than the equivalent input is suppressed by a factor 1 / (0 .
15 / as sometimes being discussed.
2 .
3) ≈ 10 in our special configuration and not only by a factor three
We conclude that the amount of noise observed in such configurations depends strongly on the details of the electrical parameters of the ladders and on the gain function of the amplifier used. It can not be estimated accurately enough by simple rules of thumb.
14

[1] N. Campbell: The study of discontinous phenomena , Proceedings of the Cambridge Philosophical
Society, Vol. XV, [Michaelmas Term], page 117 (1908).
[2] W. Schottky: Uber spontane Stromschwankungen in verschiedenen Elektrizit¨ der Physik, Band 57, page 541 (1918).
, Annalen
[3] J.B. Johnson: Thermal agitation of electricity in conductors , Phys. Rev. July 1928.
[4] K.R.Laker, W.M.C. Sansen, Design of analog integrated circuits and systems , formula 1-65b.
McGrawHill, Singapore, 1994.
[5] See any higher level textbook on electronics, or the scriptum Elektronik f¨ , www.physik.unizh.ch/ ∼ /strauman/elektronik/
[6] Charge amplifier noise is discussed in many older textbooks, see for instance E. Kowalski: Nuclear
Electronics , Springer 1970.
[7] see for instance: H. Spieler: Low-noise electronics , in Review of Particle Physics, Phys. Lett.
B
592 , 1 (2004).
L¨ Beetle chip , Thesis in preparation, Heidelberg 2005.
[9] N. van Bakel et al.: The Beetle Reference Manual, chip version 1.3, 1.4 and 1.5
http://wwwasic.kip.uni-heidelberg.de/lhcb/Publications/BeetleRefMan v1 3.pdf
Heidelberg, November 24, 2004.
[10] M. Needham et al., Laboratory Measurements on Irradiated Prototype Ladders for the LHCb Inner
Tracker , LHCb note 2004-112.
[12] Stefan Koestner, Large Area Silicon Tracking Detectors with fast signal readout for LHC Experiments , PhD. thesis, Vienna 2005.
15