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Table of Contents
spectrometric chain
Introduction
The aim of this laboratory course is to understand how spectrometric chain works. You will look at behavior of components like preamplier or shaper. Next step will be to measure basic parameters of such a chain like gain or noise performence.
Circuit description
Building blocks
The most often used front-end electronics configuration consists of the charge
sensitive preamplifier and 
the filter known as pseudo-gaussian shaper.
The filter is called pseudo-gaussian because the response of such
filter to step function becomes exactly gaussian when its order n approches infinity.
The pseudo-gaussian shaper is frequently used because of its simplicity and because
it allows to obtain close to optimum S/N ratio [ref-gatti,radeka].
Neglecting initially 
the preamplifier output voltage can be written as:
where 
is a Laplace transform of a sensor pulse and assuming
that 
is high enough (reasonable assumption since the gain of
operational amplifiers is of the order 
).
The shaper output signal can be then writrten as:
where 
is the shaper transfer function and again
assuming high enough gain of the operational amplifiers 
.
Writing equations for the first shaper stage:
and for the following shaper stages:
and setting equal the differentiating and the integrating time constant 
one obtains the shaper transfer function:
The transfer function of the whole circuit may be written as:
Usually, a very good assumption for sensor pulse shape is a dirac delta i.e. 
with its integral equal to a total charge 
deposited in the sensor. This assumption reflects the fact that the charge collection time in the sensor is much shorter than the front-end electronics shaping time.
Under this assumption the front-end response in time domain equals:
which has the maximum amplitude at time 
which is equal to:
To pass from the theoretical considerations to a practical circuit the only
thing needed is to add a resistance in parallel to 
. It is needed
for two purposes: first to set the DC level of the preamplifier input and
second, to assure that the feedback capacitance gets discharged
and so it will not saturate the preamplifier output after a number of
subsequent pulses from the sensor. The derived above transfer function becomes
practically unaffected if the value is sufficiently large (
in this work).
Sensor, Preamplifier and Shaper - Noise Analysis
There are two main sources of noise which deteriorate signal to noise ratio
in the proposed spectrometric system: the sensor noise and the front-end electronics
noise. The sensor noise comes mainly from a shot noise caused by the sensor
leakage current 
and its spectral power density is equal to:
The front-end electronics noise is mainly due to the preamplifier and the feedback
resistance . In properly designed system the following shaper stages
give small contribution to the total noise (since the signal is already amplified).
Array
The overall effect of preamplifier noise is described in terms of an equivalent
input noise expressed by series (voltage) and parallel (current) noise sources
with spectral densities respectively 
and 
.
The feedback resistance is characterized by its thermal noise with
spectral density:
The noise diagram of the front-end electronics with the equivalent input noise sources
is shown in fig
.
Knowing that for the circuit with transfer function 
the input noise
spectral power density 
is transformed to the output as:
where 
,
one can derive the preamplifier output spectral power density.
This is easily done using the superposition principle.
Since in practical systems is large and the 
time constant is much larger than the signal duration the contribution from
may be calculated neglecting
the . To calculate the contribution from 
one assumes that 
in the signal bandwith.
With such approximations the preamplifier output noise density equals:
and subsequently the shaper() output noise density equals:
In order to obtain the voltage noise 
at the output the spectral noise density needs to be integrated in the frequecny domain:
For this integration the explicite dependence of 
on frequency is needed. The main contribution are white (constant) voltage and current noise sources 
. In JFET or CMOS technology also the flicker noise voltage component should be taken into account so one can assume the equivalent spectral noise densities as: 
and 
.
For the proposed pseudo-gaussian shaping (fig.~\ref{fig:sch_front})
the above integral can be calculated analytically as:
where 
is the Gamma function.
Sensor, Preamplifier and Shaper - Signal to Noise
Knowing the signal amplitude 
(eq. 
) and the noise value (eq. 
) the 
ratio may be easily obtained. This is rarely done. Instead the noise performance is usually expressed in a slightly different way i.e. by means of an equivalent input noise charge (ENC), in the number of electrons. The ENC is calculated dividing 
by the amplitude obtained for single electron input charge 
and gives in result:
It is seen that in order to minimize the noise one should minimize the sensor leakage curent and use the highest possible value. It is also seen that the voltage white noise decreases with shaping time on the contrary to the current white noise which increases with 
. For a given filter order one can minimize the ENC finding an optimum
shaping time:
In such case the minimum ENC equals:
For practical applications it is often more convinient to study the noise performance
as the function of 
instead of . In this case the formula 
will be espressed as:
Now, instead of optimizing the shaping time the optimization
of the peaking time can be done in order to equalize the
contribution of voltage and current noise. As before the voltage component decreases
with the increase of while the current component increases
with increasing .
The noise dependence on the shaping order is more complicated and in general it is
better seen with numerical simulations. Qualitatively one can only say that the ratio
of voltage to current noise contribution increases with increasing shaping order.
To get feeling about the real noise performance it is instructive to plot the ENC as the function of peaking time, for different shaping orders. This is done below setting all required parameters to the reasonable values corresponding to this work.
As expected the optimum peaking time exists. Reagarding the shaping order it is seen that in general the higher order the better noise performance.
Assuming that the further signal processing stages add negligible contribution
to the overall system noise one can estimate the expected resolution of the spectrometric system in terms of the number of electrones, on the basis of eq. 
.
For example one can assume second order shaping, 
, 
, 
, 
and spectral noise densities of Texas Instruments OPA657 operational amplifier 
, 
.
In such case the total system noise with ENC below 200 electrons may be expected.
Requesting the S/N of 10 for good separation between the signal and noise one can expect
that the signals corresponding to 2000 or more electrons realased in the sensor should be well seen by such system.
As a practical example a copper(
) Xray fluorescence 
line may be considered. Each Xray photon should release in silicon sensor about (8050 eV)/(3.7 eV)=2175 electron-hole pairs which is above the requested minimum
for the given readout electronics specifications. It means that the specified readout
system should be able to detect the 
line
and higher energy photons.
Practical realisation
Schematic diagram of the implemented spectrometric chain is shown below on figure## .
Array
One can see a few differenses betwen figures XX and ZZ.
Laboratory Setup
During this laboratory we use following equipment:
- laboratory module
- digital sampling oscilospcope (TEKTRONIX TDS3034)
- pulse generator (AGILENT 3320)
- True RMS meter (HP3400)
- Power Supply (AGILENT 3631A)
- cables
- PC wich proper software
The connection of the setup is shown in the figure X.
After connecting setup do following steps:
- set supply voltages to +/- 7V
- set chanel 1 input impedance to 50 Ohm
- set rectangular wave on pulse generator
- switch on pulse generators output
- set triger to proper chanel
Measurements
Observing pre-amplifier output
Linearity check of pre amplifier
Observing shaper output
Linearity check of shaper
Noise performance
References
E. Gatti i P.F. Manfredi, Processing the Signals from Solid-State Detectors in Elementary-Particle Physics, Revista Del Nuovo Cimento (1986).
