## 3 Design of the Measurement System
### Design of the System for Gain and Energy Resolution Measurement  
In order to measure the APD \(G_{APD}\) and \(\delta E/E\), it is necessary to have a photon source, a high voltage generator to properly bias the APD, electronic circuits for the APD output signal conditioning, and a measurement instrument for the data readout [21]. These different parts are here described in detail:  
* _Photon generators_: In order to measure the quantities of interest, the APD under test must be properly excited. In this work, we used the following radioisotopes as X-ray photon sources: \({}^{55}\)Fe, \({}^{109}\)Cd, \({}^{241}\)Am, and \({}^{57}\)Co. Details on the sources are given in Table 1[32]. * _APD biasing system._ As discussed in Section 2, APDs are designed to work with a large inverse polarization voltage. Thus, a high-voltage (HV) source is needed. In this work, the HV is provided by the DT1471 HET CAEN HV generator. It is a four-channel HV power supply with 5.5 kV maximum output, with an accuracy of \(\pm 2\)% of the read value \(\pm 2\) V and 20 mVpp maximum voltage ripple. In order to reduce the voltage instabilities, the supply voltage filter of the CR-150 evaluation board is used. It is a dual-pole RC filter composed of a first 100 M\(\Omega\) with 0.01 \(\mu\)F low-pass RC filter followed by a second filter realized with a 200 M\(\Omega\) resistor and 0.01 \(\mu\)F capacitor. The poles of this filter can be changed by shunting the resistors depending on the dark current of the connected APD. The dark current of the APD under test is nominally 5 nA at 25 \({}^{\circ}\)C, and thus no shunt resistors are added to the filter accordingly with the datasheet of the CR-150 evaluation board [34]. * _APD signal output conditioning system._ For each electron that is possibly generated by the absorption of an incident photon in the APD sensitive region, a charge pulse is \begin{table}
\begin{tabular}{c c c} \hline \hline
**Radioactive Isotope** & **Half-Life (y)** & **Main (Effective) Emitted Line (keV)** \\ \hline \({}^{55}\)Fe & 2.737 & 5.9 \\ \({}^{109}\)Cd & 1.263 & 22.6 \\ \({}^{241}\)Am & 432.6 & 59.5 \\ \({}^{57}\)Co & 0.745 & 123.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of the radioisotopes used as X-ray sources [33]. Figure 1: Block diagram of the designed and tested measurement system. generated by the avalanche phenomenon. The measurement of the quantity of charge contained in the pulse would allow us to directly determine \(G_{APD}\). A possible alternative would consist of transforming the physical quantity that carries the information into another quantity more easily measured: for example, the voltage. In this work, the second possibility is chosen due to the more common, and generally available, laboratory instrumentation measurement. Thus, the charge pulses are passed to the charge preamplifier which is mounted on the CR-150 evaluation board [34]. The CR-110R2.2 is a charge-sensitive preamplifier widely used with several kinds of radiation detectors [35]. The equivalent circuit of the CR-110R2.2 is shown in Figure 2. It is a dual-stage amplifier: the first one is a charge preamplifier followed by a low-gain voltage amplifier. Thus, each charge pulse is integrated by the \(C_{f}\approx 1.4\) pF capacitor. Since the preamplifier output voltage is \(v_{out}=q/C_{f}\), with \(q\) the quantity of charge, then the preamplification conversion factor is \(V_{out}/q=1/C_{f}=0.7\) V/pC. The feedback resistor \(R_{f}\) slowly discharges the capacitor, giving the exponential decay of the output voltage. The characteristic time is \(\tau=R_{f}C_{f}\approx 140\)\(\upmu\)s. Thus, the detector output charge pulses should be limited in duration to a few microseconds, since for longer pulses the exponential decay would distort the shape and carried information. The pulse rise time of this preamplifier is 7 ns and the overall gain of the dual-stage preamplifier is \(\approx\)1.4 V/pC.