## 3 Results and Discussion
### Gallium Nitride (GaN) Readout Circuit for Position-Sensitive Photomultiplier Tube (PSPMT)  
With the various modules designed using our proposed OPA and verified, the complete GaN-based readout circuit is built as shown in Figure 9. Block X1 is discretized position circuit; X2, X3, X4, X5 are the transimpedance amplifiers; X6 is the adder; X7, and  
Figure 8: (**a**) Integrator circuit. (**b**) Simulation output. Figure 7: (**a**) Adder circuit. (**b**) Simulation output. X8 are the adder-subtractors. The outputs from the adder and adder-subtractor circuits VS1, VS2, and VS3 will be sent to the data processing unit as shown in Figure 1 for computing the position of the radiation event. The output VS1 is also fed to the input of integrator whose output VIE will be fed to the data processing unit for computing the energy of the radiation event. To verify the behavior of the readout circuit, the current obtained from the SiPM due to radiation interaction is mathematically modeled in LTSpice. The PM can be represented by a current source in parallel to a capacitor and the current is given by the Equation (11) [42; 43]:  
\[I_{s}=\frac{Q_{e}\times N_{ph_{\times}}PDE\times G}{\tau_{d}-\tau_{r}}\bigg{(} e^{-\frac{t}{\tau_{r}}}-e^{-\frac{t}{\tau_{d}}}\bigg{)} \tag{13}\]  
where \(Q_{e}\) is the charge of electron, \(G\) is the gain of PMT, PDE is photon detection efficiency of the scintillation crystal, \(N_{ph}\) is the number of the photons detected, \(\tau_{r}\) and \(\tau_{d}\) are the rise and decay time of the scintillator. The value of the parameters in Equation (13) for S11828-3344M SiPM and CsI(TI) crystal is given in Table 1[43]. Figure 10 shows the equivalent circuit of PMT and the corresponding current pulse generated by Equation (13), and in our case, the current pulse is 40 \(\mu\)A. This pulse was applied randomly to different channels of DPC mimicking random radiation events, and the corresponding current signal (I1, I2, I3, I4) obtained from the four channels of the DPC is shown in Figure 11a. These current signals are then converted into voltage signals V1, V2, V3 and V4 by the TIAs shown in Figure 11b. The simulation result shows that the output signal obtained from the TIA has minimal shape distortion and follows Equation (1). Figure 12 shows the outputs of adder X6, and integrator, respectively. Figure 12 shows that the reset signal remains low until the integration is performed, after that it toggles to the high stage and discharges the capacitor so that the integrator returns to initial state and the next signal will be integrated. The outputs VS1 is the sum of the input currents obtained from DPC as given in Equation (5). Suppose Q1, Q2, Q3 and Q4 are the integral of currents I1, I2, I3 and I4; \(Q_{T}\), which is the sum of Q1 to Q4 representing the total charge collected by the PSPMT and proportional to the energy of the prompt gamma signal. Thus, VS1 has the information of the energy of the gamma signal. The output VIE of the integrator X9 is given as:  
\[VIE=a\int VS1 dt+c \tag{14}\]  
where \(a\) and \(c\) are constant. The output of the integrator is then feed to the data processing unit for extracting the energy information. The positioning of the radiation event from the DPC circuit can be determined by the following equations [44; 45]:  
\[X=\frac{(A+B)-(C+D)}{A+B+C+D} \tag{15}\]  
\[Y=\frac{(A+D)-(B+C)}{A+B+C+D} \tag{16}\]  
where \(A\), \(B\), \(C\) and \(D\) represents the peak values of the signals obtained from the four corner channels of the DPC. To verify if the circuit provides the correct position information, the current pulse given by Equation (13) was injected sequentially to each of the input positions of the DPC circuit mimicking the radiation event, and 64 such simulations were independently performed. The detected position information is computed using Equations (15) and (16), where the numerator of Equation (15) is VS2, the denominator of Equation (15) is VS1, the numerator of Equation (16) is VS3, and the denominator of Equation (16) is VS1.