2.2 Signal Parameterization
2.2.3 Electronics Parameters
The Majorana collaboration designed an optimized detector electronics readout and
amplification system in an effort to minimize noise while achieving low radioactive back- grounds, which is shown schematically in Figure 2.18. The readout is performed with a resistive feedback circuit, containing an input field effect transistor (FET) and amorphous
Figure 2.18: Schematic of theDemonstratorsignal readout chain. The low mass front end
(LMFE) board is located within the cryostat, very near to the detector point contact, as seen in Figure 1.4. The LMFE contains a resistive feedback element (RC element at top), field effect transistor with drain and source lines, and capacitively coupled pulser input (bottom). The four lines are bundles into one cable, which runs ∼2 m to additional amplification elements outside the shield. Figure from [33].
germanium resistor, called the “low mass front end” (LMFE) board [19]. The LMFE is lo- cated within a few millimeters of the detector, reducing stray input capacitance and therefore series noise. However, given this positioning, the LMFE hardware must be must be minimal and constructed only with very radiopure materials, while the rest of the preamplifier sys- tem is located outside of the cryostat. Approximately 2 m of 0.4 mm diameter cable runs between the LMFE and first stage of the preamplifier, thereby creating a feedback loop 4 m long. A second stage preamplifier, also outside the cryostat, is capacitively coupled to a second stage of amplification. The signal is then digitized with 14 bit precision at 100 MHz with GRETINA digitizer boards [56], which were originally developed for the GRETINA experiment.
Given the extremely long feedback loop, and because the FET response depends sen- sitively on temperature, it is difficult to accurately model the response of the electronics readout with a circuit simulator. A capacitively-coupled pulser line runs to the input of the
FET, but the pulser input is also shaped by 2 meters of cable running into the cryostat. Without a direct measurement of the transfer function, we have developed an empirical parameterization.
The LMFE and inductance from the cable attenuate high frequency components of the signal, and can be modeled as a low-pass filter. A transfer function of order n can be generically modeled as a digital filter in the z domain
H(z) = cnz
n+cn−1zn−1+. . .+c0 dnzn+dn−1zn−1+. . .+d0
(2.6)
To determine which order was appropriate for the Demonstrator electronics, we at-
tempted to fit waveforms with a generic filter of first, second, and third order. Qualitatively, the second order filter was found to match the shape of the observed waveforms significantly better than a first order filter. The third order filter showed no dramatic improvement over the second order. For this reason, we chose to use a second order transfer function.
The fit to the second order preferred values of c0 near to zero, giving us a total digital filter function with four free parameters
Hlow(z) =
az2+bz
z2+ 2cz+d2. (2.7)
In the time domain, this corresponds to a convolution with a decaying oscillatory kernel,
H(t)∝e−tcos(ωt+φ), (2.8)
with the correspondence between the time and z domain is given by:
ω = cos−1c d and φ = tan−1 ac−b a√d2−c2 (2.9)
0 200 400 600 800 1000 Time since charge created [ns]
0.0 0.2 0.4 0.6 0.8 1.0 V oltage [A.U.] No electronics shaping With electronics shaping
Figure 2.19: Simulated effect of the electronics chain on waveform shape. Note that the risetime is increased by ∼100 ns.
The steady-state, or DC, gain for the filter is given by
lim z→1 az2+bz z2+ 2cz+d2 = a+b 1 + 2c+d2 (2.10)
and therefore increasing the value ofa+bonly linearly scales the amplitude without otherwise affecting the shape.
Capacitative couplings, most notably between the first and second preamp stages, cause the waveform to exponentially decay. Empirically, we observe two strong coupling constants with different strength. We model this with three additional parameters describing a linear combination of two exponential decay functions:
Hhi(z) =c z−1 z−exp(−τT 1) + (1−c) z−1 z−exp(−τT 2) . (2.11)
Here, τ1 is set by the RC constant of the coupling between the first and second stage ampli- fiers, usually around 72µs, andτ2 is a∼2µs coupling we believe is intrinsic to the digitizer. The constant cexpresses the fractional mixing between the decay constants.
103 104 105 106 107 108 Frequency (rad/s) −40 −30 −20 −10 0 Amplitude (dB) Low-Pass Hi-Pass
Figure 2.20: Bode diagram of preamp gain using the model described in Section 2.2.3. The total gain (on this logarithmic plot) is the sum of the two curves.
Hhi. The shaping introduced by the electronics chain using this model, based on parameters observed in theDemonstrator, is shown in Figure 2.19. The length of cable in the feedback
loop is responsible for the dramatic increase in rise time [33]. A Bode diagram showing the frequency response is shown in Figure 2.20.
The process of digitizing can also change the shape of the recorded waveform. There is some inherent nonlinearity in the relationship between analog amplitude and digitally con- verted amplitude due to limitations of the analog-to-digital converters on board the digitizer. For the Demonstrator, this nonlinearity of the GRETINA digitizer has been studied and
shown to introduce no more than ≤ 1 ADC unit error for any given sample. Because this is less than the noise amplitude, which is several ADC units, we ignore nonlinearity in our model.